Advances in Molecular Structure Research, Volume 1 (Advances in Molecular Structure Research)

ADVANCES IN MOLECULAR STRUCTURE RESEARCH

Volume

1

9 1995

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ADVANCES IN MOLECULAR STRUCTURE RESEARCH Editors" MAG DOLNA HARG ITTAi Structural Chemistry Research Group Hungarian Academy of Sciences Budapest, Hungary ISTV,~N HARGITTAI

Institute of General and Analytical Chemistry Budapest Technical University Budapest, Hungary VOLUME 1

91995

@ Greenwich, Connecticut

JAI PRESS INC.

London, England

Copyright 91995 by JAi PRESSINC 55 Old Post Road, No. 2 Greenwich, Connecticut 06836 JAi PRESSLTD. The Courtyard 28 High Street Hampton Hill, Middlesex TWl 2 1PD England All rights reserved. No part of this publication may be reproduced, stored on a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, filming, recording, or otherwise, without prior permission in writing from the publisher. ISBN: 1-55938-799-8 Manufactured in the United States of America

CONTENTS

LIST OF CONTRIBUTORS INTRODUCTION TO THE SERIES: AN EDITOR'S FOREWORD

Albert Padwa

PREFACE

Magdolna Hargittai and lstv~n Hargittai

vii

ix xi

MEASURING SYMMETRY IN STRUCTURAL CHEMISTRY

Hagit Zabrodsky and David A vnir

SOME PERSPECTIVES IN MOLECULAR STRUCTURE RESEARCH: AN INTRODUCTION

lstv~n Hargittai and Magdolna Hargittai

ACCURATE MOLECULAR STRUCTURE FROM MICROWAVE ROTATIONAL SPECTROSCOPY

Heinz Dieter Rudolph

GAS-PHASE NMR STUDIES OF CONFORMATIONAL PROCESSES

Nancy S. True and Cristina Suarez

33

63

115

FOURIER TRANSFORM SPECTROSCOPY OF RADICALS

Henry W. Rohrs, Gregory J. Frost, G. Barney Ellison, Erik C. Richard, and Veronica Vaida

THE INTERPLAY BETWEEN X-RAY CRYSTALLOGRAPHY AND AB INITIO CALCULATIONS

Roland Boese, Thomas Haumann, and Peter Stellberg

157

201

vi

CONTENTS

COMPUTATIONAL AND SPECTROSCOPIC STUDIES ON HYDRATED MOLECULES

Alfred H. Lowrey and Robert W. Williams

EXPERIMENTAL ELECTRON DENSITIES OF MOLECULAR CRYSTALS AND CALCULATION OF ELECTROSTATIC PROPERTIES FROM HIGH RESOLUTION X-RAY DIFFRACTION

Claude Lecomte

ORDER IN SPACE: PACKING OF ATOMS AND MOLECULES

Laura E. Depero

INDEX

227

261

303 339

LIST OF CONTRIBUTORS David A vn ir

Department of Organic Chemistry The Hebrew University of Jerusalem Jerusalem, Israel

Roland Boese

Institute of Inorganic Chemistry University-GH of Essen Essen, Germany

Laura E. Depero

Department of Mechanical Engineering University of Brescia Brescia, Italy

G. Barney Ellison

Department of Chemistry University of Colorado Boulder, Colorado

Gregory J. Frost

Department of Chemistry University of Colorado Boulder, Colorado

Istvan Hargittai

Institute of General and Analytical Chemistry Budapest Technical University and Hungarian Academy of Sciences Budapest, Hungary

Magdolna Hargittai

Structural Chemistry Research Group of the Hungarian Academy of Sciences E~tv6s University Budapest, Hungary

Thomas Haumann

Institute of Inorganic Chemistry University-GH of Essen Essen, Germany

Claude Lecomte

Laboratory of Mineralogy Faculty of Sciences University of Nancy Vandoeuvre-les-Nancy, France

vii

viii

LIST OF CONTRIBUTORS

Alfred H. Lowrey

Laboratory for the Structure of Matter Naval Research Laboratory Washington, D.C.

Erik C. Richard

Department of Chemistry University of Colorado Boulder, Colorado

Henry W. Rohrs

Department of Chemistry University of Colorado Boulder, Colorado

Heinz Dieter Rudolph

Department of Chemistry University of UIm UIm, Germany

Peter Stellberg

Institute of Inorganic Chemistry University-GH of Essen Essen, Germany

Cristina Suarez

Department of Chemistry University of California Davis, California

Nancy S. True

Department of Chemistry University of California Davis, California

Veronica Vaida

Department of Chemistry University of Colorado Boulder, Colorado

Robert W. Williams

Department of Biochemistry Uniformed Services University of the Health Sciences Bethesda, Maryland

Hagit Zabrodsky

Department of Computer Science The Hebrew University of Jerusalem Jerusalem, Israel

INTRODUCTION TO THE SERIES" AN EDITOR'S FOREWORD The JAI series in chemistry has come of age over the past several years. Each of the volumes already published contains timely chapters by leading exponents in the field who have placed their own contributions in a perspective that provides insight to their long-term research goals. Each contribution focuses on the individual author' s own work as well as the studies of others that address related problems. The series is intended to provide the reader with in-depth accounts of important principles as well as insight into the nuances and subtleties of a given area of chemistry. The wide coverage of material should be of interest to graduate students, postdoctoral fellows, industrial chemists and those teaching specialized topics to graduate students. We hope that we will continue to provide you with a sense of stimulation and enjoyment of the various sub-disciplines of chemistry. Department of Chemistry Emory University Atlanta, Georgia

Albert Padwa

Consulting Editor

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PREFACE Progress in molecular structure research reflects progress in chemistry in many ways. Much of it is thus blended inseparably with the rest of chemistry. It appears to be prudent, however, to review the frontiers of this field from time to time. This may help the structural chemist to delineate the main thrusts of advances in this area of research. What is even more important though, these efforts may assist the rest of the chemists to learn about new possibilities in structural research. This series will be reporting the progress in structural studies, both methodological and interpretational. We are aiming at making it a "user-oriented" series. Structural chemists of excellence will be critically evaluating a field or direction including their own achievements, and charting expected developments. The present volume is the first in this series which is expected to grow about one volume a year. We would appreciate hearing from those producing structural information and perfecting existing techniques or creating new ones, and from the users of structural information. This would help us gauge the reception of this series and shape future volumes. Magdolna and Istvfin Hargittai Editors

xi

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MEASU Ri NG SYMMETRY IN STRUCTURAL CHEMISTRY**

Hagit Zabrodsky and David Avnir

Io II. III. IV. V.

VI. VII. VIII.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous S y m m e t r y Measure: Definition . . . . . . . . . . . . . . . . . . . . Evaluating the S y m m e t r y Transform . . . . . . . . . . . . . . . . . . . . . . . Proof of the Folding Method . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples o f Measuring S y m m e t r y in 3D: Tetrahedricity and Rotating Ethane A. Tetrahedricity of Phosphates . . . . . . . . . . . . . . . . . . . . . . . . . B. The Rotating Tetrahedra of Ethane . . . . . . . . . . . . . . . . . . . . . . Point Selection for Representation of Contours . . . . . . . . . . . . . . . . . . S y m m e t r y o f Occluded Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . S y m m e t r y of Points with Uncertain Locations . . . . . . . . . . . . . . . . . . A. The Most Probable Symmetric Shape . . . . . . . . . . . . . . . . . . . . B. The Probability Distribution o f S y m m e t r y Values . . . . . . . . . . . . . . Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Note Added in Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

*Dedicated to Professor Jochanan Blum on the occasion of his 60th birthday. *Part 3 of Contimtous Symmetry Measures. 1,2

Advances in Molecular Structure Research Volume 1, pages 1-31. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8

2 2 4 6 8 . 10

10 12 16 18 21 21 24 27 27 27 31 31

2

HAGIT ZABRODSKY and DAVID AVNIR

ABSTRACT Symmetry is treated as a continuous property. A continuous symmetry measure (CSM) of structures is defined to be the minimum mean-squared distance required to move points of the original structure and change it to a symmetrical structure. This general definition of a symmetry measure enables a comparison of the amount of symmetry of different shapes and the amount of different symmetries of a single shape. We describe a general method of obtaining the minimal distance to the desired shape and apply it to any symmetry element or symmetry group, in two or three dimensions. Various examples of the application of the CSM approach to structural chemistry are presented. These include symmetry analysis of distorted tetrahedra, and of rotating ethane, symmetry analysis of contours of equi-property such as molecular orbital representations, reconstruction of incomplete structural data, and symmetry analysis of structures with uncertain point-location, such as encountered in X-ray data analysis.

i. I N T R O D U C T I O N A traditional working tool in structural chemistry has been symmetry analysis. Symmetry point groups and space groups have been used as reference configurations which either do or do not exist in the structure under study. We have argued recently [1,2] that this traditional approach fails to capture the richness of shapes and structures, both static and dynamic, which is found in the molecular and supramolecular domains. Most of these are not symmetric. At most, they are approximately symmetric, either permanently or if the time-resolution of observation is sufficiently narrow. Consider, for instance, the very weak ( E m a x = 200) forbidden x ~ x* transition to the lowest lying singlet in benzene (A 1 ~ B12,,)and compare it with the carbon skeleton of toluene: The D6h symmetry of the benzene hexagon changes to a distinctly different point group, Czu, yet the extinction coefficient increases only to E m a x - 225. Current wisdom of accounting for the discrepancy between the major symmetry change and the small effect in the "allowedness" of the transition is to use such arguments that "the methyl perturbs the rc system only to a small extent", i.e., the day is saved by resorting to "local" symmetry. Another example is the vibrating water molecule. This is a C2~ molecule and its v~ and v 2 vibrational modes preserve this symmetry. But what about v3? This vibrational mode distorts the C2~ symmetry and again, a legitimate question is by how much does the molecule deviate from C2~ after 1% of one cycle, after 10% of it, and so forth. Yet another example is the well known phenomenon of removal of the degeneracy of energy levels of a chemical species whenever contained in an environment of symmetry other than its own (a certain arrangement of ligands or a certain packing in the crystal). The degree of removal of degeneracy is directly linked to the "decrease" in the symmetry of the environment, compared to the isolated chemical species. Traditionally, this problem is treated in terms of

Measuring Symmetry in Structural Chemistry

3

jumps in the symmetry point group. For instance, the splitting of the degenerate p-orbitals increases from a2u + e, in a D4h environment to a 1 + b 1 + b 2 in a C2~ environment [3]. Our next example is the principle of conservation of orbital symmetry which has caused a quantum leap in the understanding of reaction pathways in organic chemistry. It suffices to take one very basic problem to illustrate our point: Consider two ethylenes approaching each other for a [2+2] reaction. The answer to the question of whether that reaction is allowed thermally or photochemically, or whether a suprafacial or antarafacial process will take place, or whether the reaction will take place at all, is very much dependent on the symmetry of alignment of the two reacting molecules or moieties. The extremes are D2h for a parallel approach and C 2 for an orthogonal approach, and it is predicted successfully [4] that the former is needed for a suprafacial photochemical formation of cyclobutane. Most of the time, however, the two ethylenes are not in an ideal D2h arrangement: This may be due to an intramolecular frozen conformation of the two double bonds, to nonsymmetric sterical hindrance caused by substituents on the double bond, and to the dynamical nature of the system (rotations and translations, especially in viscous media). These are but few examples which illustrate the need for a continuous scale of symmetry. A general approach which answers this need was layed out in refs. [1] and [2]. Here we summarize its main features, show how the above mentioned examples are approached, and extend our theory of continuous symmetry measures (CSM) to three new applications: 1. The CSM of (e.g., molecular orbital) contours. 2. The symmetry of occluded shapes, such that appear in microscopy studies of materials. 3. The CSM of points with uncertain locations, such as can be found in X-ray diffraction analyses in molecular structure determinations. Several previous studies were led by the need to relax the current strict language of symmetry. Hargittai and Hargittai emphasize repeatedly in their book [5] the limitations of exact symmetry in the description of many structural problems in chemistry. Murray-Rust et al. [6] and more recently Cammi and Cavalli [7] have suggested the use of symmetry coordinates to describe nuclear configurations of MX 4 molecules that can be regarded as distorted versions of the T d symmetrical reference structure. Mezey and Maurani [8,9] extended the point symmetry concept for quasi-symmetric structures by using fuzzy-set theory (terming it "syntopy" and "symmorphy"), and provided a detailed demonstration of its application for the case of the water molecule. In another recent study [10] Mezey used a resolution based similarity method of polycube filling to measure approximate symmetry of molecular distributions. Other relevant contributions are perturbation analyses in spectroscopy [11], and measures on convex-sets [12]. As will be evident below, our

4

HAGIT ZABRODSKY and DAVID AVNIR

approach to the problem of non-ideal symmetry is quite different, being guided by three principles: 1. Nonsymmetric shapes should not be treated as a perturbation of an ideal reference. Such shapes, as well as perfectly symmetric ones, should appear on a single continuous scale with no built-in hierarchy of subjective ideality. Assessing symmetry should be detached from referencing to a specific shape; yet the shape of the nearest configuration with the desired symmetry, should be obtainable. 3. It should be possible to evaluate the symmetry of a given configuration with respect to any symmetry group, such as the closest one. .

These guidelines are implemented as described in the following section.

Ii. C O N T I N U O U S SYMMETRY MEASURE" DEFINITION We define the continuous symmetry measure (CSM) as a quantifier of the minimum effort required to turn a given shape into a symmetric shape. This effort is measured by the sum of the square distances each point is moved from its location in the original shape to its location in the symmetric shape. Note that no a priori symmetric reference shape is assumed. Denote by ~ the space of all shapes of a given dimension, where each shape P is represented by a sequence of n points {Pi }n-1 i=0" We define a metric d on this space as follows:

d: ~x~----~R

d(P,Q) -d({Pi},

n-1 1 {Qi}) = n Z

IIPi- QilI2

i--o This metric defines a distance function between every two shapes in ~. We define the symmetry transform (ST) as the symmetric shape ifi closest to P in terms of the metric d. The CSM of a shape is now defined as the distance to the closest symmetric shape:

s - d(P,i~) The CSM of a shape P = {Pi}'7-1 i=0 is evaluated by finding the symmetry transform i~ of P and computing: n-1

n Z IIPi- Pill2

S= 1

A

i=0

Measuring Symmetry in Structural Chemistry

5 symmetry

normalize

P~

P~

^

Po

transform

P, a.

-\

1,~ A

Po o

d@

S(C3)= 12.80

~

A Pz

P,

Figure 1. Calculating the CSM of a shape: (a) Original shape {Po, /:'1, P2}. (b) Normalized shape {Po', PI', P2'}, such that maximum distance to the center of mass is one. (c) Applying the symmetry transform to obtain a symmetric shape A A 2). {Po, ~1,P2}. (d) S(C3)= 1/3(IIPo' - ,goll2 + lIP1' - ^,~ + lIP2'- ,g211 CSM values are multiplied by 100 for convenience of handling.

This definition of the CSM implicitly implies invariance to rotation and translation. Normalization of the original shape prior to the transformation additionally allows invariance to scale (Figure 1). We normalize by scaling the shape so that the maximum distance between points on the contour and the centroid is a given constant (in this chapter all examples are given following normalization to 1; however, CSM values are multiplied by 100 for convenience of handling). The normalization presents an upper bound on the mean-squared distance moved by

S ( C 2) = 1.87

S ( C 3) = 1.64

S ( C 6) = 2.53

S ( ~ ) = 0.66

Figure 2. Symmetry transforms of a 2D polygon and corresponding CSM values.

6

HAGIT ZABRODSKY and DAVID AVNIR

points of the shape. Thus the CSM value is limited in range, where CSM = 0 for perfectly symmetric shapes (see Appendix A). The general definition of the CSM enables evaluation of a given shape for different types of symmetries (mirror symmetries, rotational symmetries, and any other symmetry groups--see Section V). Moreover, this generalization allows comparisons between the different symmetry types, and expressions such as "a shape is more mirror symmetric than rotationally symmetric of order two" is valid. An additional feature of the CSM is that we obtain the symmetric shape which is "closest" to the given one, enabling visual evaluation of the CSM. An example of a two-dimensional (2D) polygon and its symmetry transforms and CSM values are shown in Figure 2.

II!. EVALUATING THE SYMMETRY TRANSFORM In this section we describe a geometric algorithm for deriving the ST of a shape represented by a sequence of points {Pi} n-_~. In practice we find the ST of the shape with respect to a given point-symmetry group (see Appendix B for a review of algebraic definitions). For simplicity and clarity of explanation, we describe the method by following some examples. Mathematical proofs and derivations are detailed in Section IV. Following is a geometric algorithm for deriving the symmetry transform of a shape P having n points with respect to rotational symmetry of order n (C,,-symmetry). This method transforms P into a regular n-gon, keeping the centroid in place as follows:

1. Fold the points {Pi} n-1 (Figure3a) by rotating each point Pi counterclockwise about the centroid by 2xi/n radians (Figure 3b).

{Pi}in___~(Figure3c).

2. Let J~0 be the average of the points

^

eo

l

~

, o

P2

",,,

o

"~,

a.

_A

,,/ : ,,,~t

ol

0

'%',..

"'"

b.

j~:,

0

c.

.,,

d.

The Q-symmetry transform of 3 points (a) original 3 points {pi}20 . (b) Fold {pi}20 into {~i}2_o . (c)Average {F'i}2_o 2 0 ~i. (d) Unfold the _ obtaining ~'o = 1/3 Z/= 2 average point obtaining {]bi}i=o. F i g u r e 3.

:

Measuring Symmetry in Structural Chemistry

7 A

~ p,

^ ~ P, ^

,

Po

P' P

a.

~

^

b

Ps

vo Figure 4. Geometric description of the C3-symmetry transform for 6 points. The centroid of the points is marked by @. (a) The original points shown as two sets of 3 points" So = {Po, P2, P4} and $1 = {P1, P3, Ps}. (b) The obtained C3-symmetric configuration.

. Unfold the points, obtaining the C n symmetric points {Ai} n_~ by duplicating ~0 and rotating clockwise about the centroid by 2rci/n radians (Figure 3d). A 2D shape P having qn points is represented as q sets {Sr}q~ of n interlaced points Sr= {Pin+r}n-li-O"The Cn symmetry transform of P is obtained by applying the above algorithm to each set of n points separately, where the folding is performed about the centroid of all the points (Figure 4). The procedure for evaluating the symmetry transform for mirror symmetry is similar: Given a shape represented by m = 2q points and given an initial guess of the symmetry axis, we apply the folding/unfolding method as follows (see Figure 5): P~o ,,v

mirror axis

^po

P~- mirror axis

mirror axis

/

1 /s

t; r

9e .

.

.

.

.

.

.

Vo~0 Figure 5. The mirror-symmetry transform of a single mirror pair for angle O (a) Mirror-fold the pair {Po, ,~ obtaining {~o, ~1}. (b) The transformed-Po denoted Po is A A the average of 75oand ~1. (c) The transformed-P1 denoted P1 is Po reflected about the symmetry axis. Center of mass of the shape is assumed to be at the origin.

8

HAGIT ZABRODSKY and DAVID AVNIR

1. For every pair of points {P0, P1 }: (a) F o l d u b y reflecting across the mirror symmetry axis obtaining {P0, P1}. ^ (b) Average---obtaining a single averaged point P0. (c) Unfold--by reflecting back across the mirror symmetry axis obtaining 2.

Minimize over all possible axes of mirror symmetry.

The minimization performed in step 2 is, in practice, replaced by an analytic solution (derivation and proof can be found in Appendix C). This method extends to any finite point-symmetry group in any dimension, where the folding and unfolding are performed by applying the group elements about the center of symmetry (see derivations in Section IV). The minimization is over all orientations of the symmetry group. In 2D, the minimization is performed analytically; in 3D a minimization process is used. A detailed description of the extension to 3D and to any symmetry group appeared in ref. [2] and two illustrative examples are given in Section V. We briefly mention the case where the number of points m is less than n, i.e., less than the number of elements in the symmetry group G with respect to which we measure symmetry. In this case, m should be a factor of n such that there exists a subgroup H of G with n/m elements. In this case, we duplicate each point n/m times and fold/unfold the points with elements of a left coset of G with respect to H. Following the folding/unfolding method, the relocated points will align on symmetry elements of G (on a reflection plane or on a rotation axis for example). Further details of this case and proof can be found in Ref. 2.

IV. PROOF OF THE FOLDING METHOD As described in Section I, the CSM of a set of points with respect to a given symmetry group G is evaluated by first finding the set of points which is G-symmetric and which is closest to the given set in terms of the average distance squared. We must thus prove that the folding method indeed finds the closest symmetric set of points. The group-theory definitions which are used in this section, are briefly reviewed in Appendix B. Given a finite point-symmetry group G centered at the origin and ordering of its m elements {gl = I . . . . . gin} and given m general points P1 . . . . . Pm' find m points ~1 ~m and find rotation matrix R and translation vector w such that ~1 . . . . . ~m form an ordered orbit under G' (where G' is the symmetry group G rotated by R and translated by w) and bring the following expression to a minimum: . . . . .

m

A

~ IIPi- Pill2 i=1

(1)

Measuring Symmetry in Structural Chemistry

9 A

Since G has a fixed point at the origin and G' has the centroid of orbit Pi asAafixed point (see gemma 1, Appendix B) we have that w is the centroid of orbit Pi: m

1

A

w =-Z Pi m

(2)

i=1

(Note that in the cases where the fixed points of G form an axis or plane, w can be any vector moving the origin to the (rotated) axis or plane passing through the centroid of orbit ~i- Thus also in these cases w can be considered the centroid of orbit ~i). The points ~l J~m form an orbit of G', thus the following must be satisfied, . . . . .

A

A

A

Pi = gi'P1 = RtgiR(P1 - w) + w

(3)

i = 1 ... m

wheregi'isthematr ixrepresentation ofthei thsymmetry element of G' and is equal to the ith symmetry element gi of G rotated by R and translated by w. Using Lagrange multipliers with Eqs. 1-3 we must minimize the following, m

in

m

A

A

A

__

A

Z IIPi- Pi 112+ 2 ~'~(Pi- RtgiR(P1 - w) + w) + c(w - 1 Z Pi) m

i=1

i=1

i=1

where e, ~i for i = 1 . . . m are the Lagrange multipliers. Equating the derivatives to zero we obtain, m A

Y__, ( P i - Pi) = 0

(4)

i=1

and using the last constraint (Eq. 2) we obtain, m

1

(5)

w = - -m ~_ P.t i=1 A

A

i.e., the centroid of P1 . . . . Pm coincides with the centroid of P1 . . . . . Pm (in terms of the symmetry measure defined in Section I, the centroid of a configuration and the centroid of the closest symmetric configuration is the same for any point symmetry group G). Noting that gi" a r e isometries and distance preserving, we have from the derivatives: m Z

i=1

m Pt

g i(Pi

_

A

^

P/) = Y_-,Rtg~R(Pi- Pi) = 0 i=1

Expanding using the constraints we obtain,

10

HAGIT ZABRODSKY and DAVID AVNIR A

m

m

mP1 = Z Rtg:gPi- Z gtg:gW i=1

i=1

or,

m P^ l - W = ~1 ZRtglR(Pi

m

w)

(6)

i=1

The geometric interpretation of Eq. 6 is the folding method as described in Section III, thus proving that the folding method results in the G-symmetric set of points closest to the given set. Given n = qm points (i.e., q sets of m points) {P~ . . . . . P J } i = 1 . . . q we obtain the result given in Eq. 6 for each set of m points separately; i.e., for j = 1 . . . q, ^"

PJ1 - w = -

1

m

m

~

RtglR(P i - w)

(7)

i=1

where w = nl ]~q=l ~]mi=l pji is the centroid of all n points. The geometric interpretation of Eq. 7 is the folding method as described in Section III for m = qn points.

V. EXAMPLES OF MEASURING SYMMETRY IN 3D: TETRAHEDRICITY AND ROTATING ETHANE A. Tetrahedricity of Phosphates We retum now to the general question: given any number of vertices in space, what is its symmetry measure with respect to any symmetry group, subgroup, or class. As explained in the previous section, the generalized approach is to divide the given points into sets and to apply the folding/unfolding method separately on each set, while evaluating the CSM value over all the given points. For example, let us analyze the tetrahedricity of a tetrahedron with a branched connected set of 5 points P1 ... P5 as shown in Figure 6a, which models a tetrahedron with a central atom and apply the CSM folding/unfolding method to evaluate its Ta symmetry. The connectivity constrains the division of points into sets and restricts the center point (P5) to be a one-point set. We thus divide the points into two sets: {P1 ... P4} and {Ps}. The closest symmetric configuration will have point P5 relocated to a position where all 24 of the Ta-symmetry group elements leave it in place. The only such position is at the origin (centroid of the configuration marked as an open circle in Figure 6) where all symmetry planes and axes intersect. Points P1 . . . P4 will be relocated to form a perfect Ta-symmetric configuration of four points, i.e., each point will lie on a C3-rotation axis (see Figure 6b). S(T d) (or any S(G)) is then calculated by considering the full set of ~l . ' . ~5. To illustrate it, we now analyze the distorted phosphate tetrahedron CdzP207 [6] using our method. We first recall that our method evaluates the distance from

Measuring Symmetry in Structural Chemistry

a.

11

b.

Figure 6. A distorted tetrahedron with a central atom, analyzed as a connected configuration of 5 points. (a) The open circle marks the centroid of the configuration; (b) the closest Td-symmetric configuration.

tetrahedricity and not from a specific tetrahedron; and that rather than reporting the deviation in terms of a table of many coordinates (as done in ref. [6]), we provide a single (S(Td)) value. To obtain it, the 3D position coordinates of the four oxygens and phosphorus were taken from ref. [ 13] (also used by Btirgi et al., p. 1790 in their paper) as, P1 - (

0.0

P2 = (

0.0

0.0

1.645)

1.518860 -0.347028)

P3 - ( -1.286385 -0.700083 -0.391603) P4 = ( 1.179085 -0.755461 -0.372341) with an additional center point 0.0. By applying the folding method as described above, the symmetry measure obtained in this example is S(Td) = 0.17 and the closest symmetric shape is a regular tetrahedron with arm length 1.537 ~. (By comparison, a set of 10 symmetry displacement coordinates is used in ref. [6] to report the deviation of this tetrahedron from ideality). In a further example Murray-Rust et al. used the symmetry coordinates to evaluate the threefold axes of 1-methyl- 1-silabicycloheptane (Section V in ref. [6]). They found that the distorted SiC 4 structure (Figure 5 in ref. [6]) is better described with the threefold axis passing through one vertex (point C 1 in their notation) rather than through another (C 2 in their notation). Using the CSM method with respect to C3~-symmetry we easily support their conclusion as follows. Given the coordinates, P~ = C~ = (

0.0

0.0

1.645)

P2 = C2 =(

0.0

0.87461971

-0.48480962)

P3 = C3 = (

0.75128605 -0.39338892 -0.52991926)

P4 = C4 = ( -0.75128605 --0.39338892 --0.52991926)

12

HAGIT ZABRODSKY and DAVID AVNIR

with Si at the origin, the S(C3u) of the configuration was calculated by the method described in Section V and found to be S(C3u) = 0.02 when the threefold axis passes through point C 1, compared to S(C3~) = 1.16 when the threefold axis is constrained to pass through point C 2. Using the folding method we can also measure the C3~-symmetry of the configuration with the constraint that three of the configuration points are equatorial. In this case the S value increases to 5.26, with the threefold axes passing through point C 2.

B. The Rotating Tetrahedra of Ethane 1 Another mechanism which strongly affects molecular symmetry is intramolecular rotation. Consider, for instance, one of the most basic examples; namely, the rotation of the two ethane tetrahedra around the C-C bond (Figure 7a). Current wisdom allows an exceedingly poor description of that process from the symmetry point of view: Ethane is D3dwhen staggered (Figure 7d), D3h when eclipsed (Figure 7b), and D 3 anywhere in between, including the rotamer which is only 1 ~ away from any of the extremes. Doesn't physical intuition dictate that it is more natural to ask about that 1~ rotamer, how much D3h or D3a it contains? Or for that matter, how much D3h and D3a exists at any point in a full 360" cycle? As has already become evident throughout this paper, the CSM method allows one to select any symmetry group and follow its gradual changes along such a full 360 ~ cycle of rotation. We demonstrate it on two perfect tetrahedral structures connected along one of the tetrahedral arms and rotating with respect to each other around the

P4

ao

P4Vl

P1 ~,,

1"6

P2

b. Ps

l~6

~

P2J

c.

as/'

: ~P3

d.

Figure 7. Modeling the C-C rotation in ethane. (a) Only the right hand tetrahedron moves; (b) the cycle starts with the eclipsed D3h rotamer; (c) one of the six chiramers (see text); (d) the D3d staggered rotamer.

Measuring Symmetry in Structural Chemistry

13

connecting arm. We model this by stabilizing one of the tetrahedra and rotating the other, beginning the cycle with the two tetrahedra perfectly aligned (eclipsed) and rotating the second tetrahedron anti-clockwise. For simplicity in evaluating the S value, we considered only the tetrahedral arms not involved in the C-C bond. Figure 8 displays the result where the S value is given as a function of the cycle. (Figure 8a shows a full 360 ~ cycle and Figure 8b shows a detail). The following observations are made: 9 The D3h profile of the rotation and the D3d profile are similar, but shifted from each other by 60 ~ That is, there is as much D3d-ness in the eclipsed structure as there is D3h-ness in the staggered structure. This is intuitive since the distance (rotation or projection) from an e~ position to the 0 ~ position (eclipsed) which determines the S(O3h) value, is equivalent to the distance (rotation or projection) from a 60 ~ + e~ position to the 60 ~ position (staggered) and which determines the S(D3a) value. 9 The maximal S(D3h) value is at the 60 ~ staggered rotamer which is the farthest away from the perfect D3h eclipsed rotamer (0 ~ S(D3h) = 0). Three such maxima are observed in a full cycle, corresponding to the three staggered rotamers. Similarly, the maximal S(D3a) value is for the eclipsed rotamer at 0 ~ and again, there are three maxima in a full cycle corresponding to the three eclipsed rotamers. 9 Figure 8 points to a rotamer (Figure 7c) which is neither eclipsed nor staggered, but in between, at 30 ~ + n.60 ~ We term these special chiral (?) rotamers at 30 ~ + n.60~ (Figure 7c). There are six of these in a full cycle, compared to three eclipsed and three staggered (which are, of course, achiral). These six chiramers, which are at the crossing of the continuous D3h and D3d profile lines, are also the six maxima of the continuous C3u profile line. Note that the C3u line (which is equivalent to the ~y-line) coincides with either the D3h line (00-30 ~ 90~ ~. . . . ) or with the D3a line (30~ ~ 150~ ~. . . . ), whichever gives the lower S value. To understand this, we note that C3u is a subgroup of both D3hand D3dand that in ethane, the nearest C3u object at any point of the full cycle must be either of the two achiral rotamers. Thus, the chiramer is also the most chiral rotamer of ethane. Finally, we wish to make a brief preliminary comment on what seems to us an important application of our approach: Many thermodynamic and kinetic quantities vary cyclically with internal rotations. A commonly presented quantity is the (repulsion) potential. It is then interesting to see, how this property varies with the symmetry rather than with the traditional torsion angle. The results for a model sinusoidal potential (Figure 9a) are shown in Figure 9b. Let us first detail how the potential follows this new process coordinate: The D3h potential line varies smoothly with S, starting at the eclipsed S = 0 value and dropping to zero potential at the staggered S = 22.22 value; then it reverses and climbs back up to the maximum

14

HAGIT ZABRODSKY and DAVID AVNIR . . . . . . . . . . . . . . . . . . I I I

30

I D3d D3h C3v

25 =

I

staggered

staggered

staggered

20

15 10

,::'

'.

9/

\ X

ao

60

1_. '~

,,,,t

,

120

180

240

,,

300

360

Torsion Angle

25

........

i

i. . . . . . . . . .

l . . . . . . . .

1

...... i D3d

staggered ~ ~ 20

,

D3h

..':..

i

C3v ----

...y.///......':""y......

15

N

....."

10

eclipsed b.

0

0

j""

chirarners

/,," ,,~

20

40

'...

t/ ,,,,

%\%

60

80

eclipsed

100 120 Torsion Angle

Figure 8. D3d ( ~ ) , D3h (. . . . ), and C3v( . . . . ) for rotating ethane. (a) A full cycle; (b) a detail of one third of the cycle. potential completing 120 ~ of the cycle. This drop and rise in potential along the symmetry coordinate is repeated continuously, completing a full 360 ~ cycle. The behavior of D3d is a mirror image, starting with the maximal potential at the eclipsed S = 22.22 value and dropping to zero potential at the staggered S = 0 value. The

--.

.

--

-~

--.

--'~.

~:~

0

~

~D

"/'1

0

o

="

~D

.~o

~w

_,~-~. ..~

0 rD

0

rD

~_.o

03

5"o.

~D

rD

~D

to

~'~

o~

o

*

'

I

to

I

.1~

'

I,,

ox

I ......

00

o

1

,,

1

b~

1

4~

Potential (arbitrary units)

i

o~

I

'ii

g:r g~.

oo

o

0

k~

o

o

~

~I

OX

o ~

o

,

t.~

f~ ~

....~.....

"~

~

~

.,,.i.t

~.".~

f"

f

f

"'-. ......

"--~L

~

Symmetry Measure

~ ~

........

~

~

.

~

-.

9

-

~

~

.

~

....

"...

-

..

..

tO

o

~

"~

.....

.~

tO

~

"a

c~

O~

1

~"

o

............ , ........

.............. [=:::. ~'

~

. .:.:_. ..........................

71

'

~-,

o

and Potential (arbitrary units)

1

, ......

o

t~

16

HAGIT ZABRODSKY and DAVID AVNIR

behavior of C3~ is interesting: up to S = 5.95 it follows the D3h line, but then it continues to drop along the D3d line back towards the S = 0 value (the line then climbs back up). Perhaps most notable is that the lines of the two symmetry groups bifurcate at the 30~ What does such symmetry/potential bifurcation mean? In general, it may mean that for symmetry-governed processes, such a crossing point is where the process may select to proceed one way or the other depending on which symmetry is preferred.

Vi.

POINT SELECTION FOR REPRESENTATION OF CONTOURS

As symmetry has been defined on a sequence of points, representing a given shape by points must precede the application of the symmetry transform. Thus, for the case of equi-property contours, such as electronic orbital contours, one represents it as a string of equally spaced points (as dense as one wishes) and then perform the CSM folding-unfolding procedure as usual. As described in Section III, when a multiple of n points are given (where n is the number of elements in the symmetry group), the points must be divided into sets of 17points. In general, this problem is exponential. However when the points are cyclically connected or ordered such as along a contour, the ordering of the points restricts the possible divisions into sets. For example in 2D, points along the contour of a C,,-symmetric shape form orbits which are interlaced as shown in Figure 10a for C3-symmetry. Thus, given a set of m = n q ordered points there is only one possible division of the points into q sets of n points--the q sets must be interlaced (as was shown in Figure 4). In the case of D,,-symmetry the m = 2 n q ordered points, form q orbits which are interlaced and partially inverted as shown in Figure 10b for D4-symmetry. Thus, given a set of m = 2nq ordered points there are m / 2 n = q possible division of the points. In Figure 11 we demonstrate the application of the contour CSM analysis on the lone-pair orbital of a distorted water molecule (perhaps a frozen moment of a vibration, or a water molecule in a matrix of amorphous ice, or a water molecule trapped in a micropore). The ratio of length of the two O-H bonds is 0.9 (instead

bo

Figure 10.

W

W

Dividing m selected points into interlaced sets" (a) Cn-symmetry--one possibility. (b) Dn-symmetry--one of the m/2n possibilities.

Measuring Symmetry in Structural Chemistry

17

/ i

a.

b./ \

, .,4:
,..,,

, ,~ 'lvk'12,\xttll,lll

,

S(o) =0.116 all

bQ

S(o) = 0.213

Figure 11. Two equi-amplitude contours of the wave function of the lone-pair orbital of distorted water molecule are shown. 14'15 The two contours are spaced by 0.05 Bohr-3/2, and the value of the outer one is 0.576 Bohr-3/2. S((~) values are indicated in the figure. The CSM value for the next inner contour (not shown) is 0.248.

of 1.0) and the H - O - H angle is 104 ~ [ 1 4 ] . Two contours are shown, each of which has been represented as a string of about 200 points. The CSM with respect to mirror symmetry was evaluated by the above-mentioned m = n q pairing. It is seen quantitatively (Figure 11) that the distortive effects of the unequal bonds, fades away from the inner to the outer contours.

18

HAGIT ZABRODSKY and DAVID AVNIR

Vii.

SYMMETRY OF OCCLUDED SHAPES

Here we address a problem which is commonly encountered in microscopy studies of particulate materials" Given a collection of similar objects, such that the individual shapes occlude each other, how can the symmetry of these objects be extracted using methods of automated image analysis. The method we apply to solve the problem is to evaluate the symmetry of the occluded shapes, locate the center of symmetry, and reconstruct the symmetric shape most similar to the unoccluded original. As described in Section VI, a shape can be represented by points selected at equal distances along the contour. Another method of representing a shape is by selecting points at regular angular intervals (selection-by-angle) about a point (Figure 12). Angular selection is usually about the centroid of the shape. However, angular selection of points about a point other than the centroid will give a different symmetry measure value (Figure 13). We define the center ofsymmetry of a shape as that point about which angular selection gives the minimum symmetry measure value. When a symmetric shape is not occluded the center of symmetry aligns with the centroid of the shape. However, the center of symmetry of truncated or occluded objects does not align with its centroid but aligns with the (unknown) centroid of the unoccluded shape. Thus the center of symmetry of a shape is robust under truncation and occlusion. To locate the center of symmetry, we use an iterative procedure of gradient descent that converges from the centroid of an occluded shape to the center of symmetry. Denote by center of selection that point about which points are selected using angular selection. We initialize the iterative process by setting the centroid as the center of selection. At each step we compare the symmetry value of points angularly selected about the center of selection and about points in its immediate neighborhood. That point about which angular selection gives minimum symmetry

I..... i , ~

i

i

', ',

\ \

Figure 12. Selection at equal angles. Points are distributed along the contour at regular angular intervals around the centroid.

Measuring Symmetry in Structural Chemistry ~"

\',,, \

:

19 4o,'~

,,,/

"-,"

; _l

Figure 13. The symmetry value obtained by angular selection about the center of mass (marked by +) is greater than the symmetry value obtained by angular selection about the center of symmetry (marked by (9).

value, is set to be the new center of selection. If the center of selection does not change, the neighborhood size is decreased. The process is terminated when neighborhood size reaches a predefined minimum size. The center of selection at the end of the process is taken as the center of symmetry. The closest symmetric shape obtained by angular selection about the center of symmetry (Figure 14c) is visually more similar to the original (Figure 14a) than that obtained by angular selection about the centroid of the occluded shape (Figure 14b). We consider the former a reconstruction of the unoccluded shape. The process of reconstructing the occluded shape can be improved by altering the method of evaluating the symmetry of a set of points. As described in Section III the symmetry of a set of points is evaluated by folding, averaging, and unfolding about the centroid of the points. We alter the method as follows: 1. The folding and unfolding (steps 1 and 3) will be performed about the center of selection rather than about the centroid of the points.

(9+

a

Figure 14. (a) Original occluded shape, its centroid (+) and its center of symmetry ((9). (b,c) The closest C5-symmetric shapes following angular selection about the centroid (b) and about the center of symmetry (c).

20

HAGIT ZABRODSKY and DAVID AVNIR

:',yn ,

9 /',,..

l

..,*

,

-","

i

;

/

," 9

#

9

i

~

i..

9. . . .

,

l

"S'"

',,....... ..y

Q

,

~ +~ i

+",+,,.

i

,

a.

V

,

~

,...... ",,,

b.'~'

"%'.,,.

t

. . . . .,,

.

',

1"-

"', ..... -..

,/

i ....... " " t

d. ',,/

c.','

Figure 15. Improving the averaging of folded points. (a) An occluded shape with

points selected using angular selection about the center of symmetry (marked as @). (b) A single set (orbit) of the selected points of (a) is shown. (c) Folding the points about the centroid of the shape (marked as +), points are clustered sparsely. (d) Folding the points about the center of symmetry of the shape, points are clustered tightly. Eliminating the extremes (the two furthest points) and averaging will result in smaller averaging error and better reconstruction.

2.

Rather than averaging the folded points (step 2), we apply a more robust clustering method: we average over the folded points, drop the points furthest from the average (this is justified by noticing that such points are due to occlusion), and then reaverage (see Figure 15).

The improvement in reconstruction of an occluded shape is shown in Figure 16. This method improves both the shape and the localization of the reconstruction.

8,,

"'. .....

.'

be

",,,.,+. . . . .

/

Co

, ......

/

Figure 16. Reconstruction of an occluded almost symmetric shape. The original shape is shown as a dashed line. The reconstructed shape is shown as a solid line. (a) The closest symmetric shape following angular selection about the centroid. (b) The closest symmetric shape following angular selection about the center of symmetry. (c) The closest symmetric shape following angular selection about the centroid and altered symmetry evaluation (see text).

Measuring Symmetry in Structural Chemistry

21

VIii. SYMMETRY OF POINTS WITH UNCERTAIN LOCATIONS Information obtained from any analytical instrument has a certain degree of uncertainty. In structural chemistry, the uncertainty may be in the location of the atoms, as obtained by, e.g., diffraction methods, due to unknown causes (crystal imperfections, thermal motion, etc.) [16]. We address ourselves now, to this problem, again focusing on symmetry issues. Quite often the data is given as a collection of probability distribution functions of point locations. Given points with such uncertain locations, the following questions are of interest: 9 What is the most probable symmetric shape represented by the data? 9 What is the probability distribution of symmetry measure values for the given data?

A. The Most Probable Symmetric Shape Figure 18a shows a configuration ofpoints whose locations are given by a normal distribution function (marked as rectangles having width and length proportional to the standard deviation). In this section we show a method of evaluating the most probable symmetric shape closest to the data. For simplicity we derive the method with respect to rotational symmetry of order n (Cn-symmetry). The solution for mirror symmetry is similar (see Appendix D). Given n points in 2D whose positions are given as normal probability distributions: Oi ""N(Pi,Ai) i = 0 . . . n - 1, we find the C,,-symmetric configuration of points {pi}~)-i which is most probable. Denote by co the center of mass of ~i: n-1

1

A

O ~ = -rl- Z

Pi .

i=o A

Having that {Pi}~-I are Cn-symmetric, the following must be satisfied, A

A

P i - Ri(Po- ~) + ~

(8)

for i = 0 . . . n - 1 where R i is a matrix representing a rotation of 2xi/n radians. Thus, given the measurements Q0. . . . . Q,,-1 wAeneed to find the most probable J~0and o). We find i~0 and c0 that maximize Prob({ Pi} 7~ ] to,~0} under the symmetry constraints of Eq. 8. Considering the normal distribution we have, n-1

]

A

I'I ki exp(- -~-(Pi i=o

A - Pi)tAsl l(Pi - Pi)

22

where

HAGIT ZABRODSKY and DAVID AVNIR 1

ki = ~- n ]Ai[ 1/2. Having log being a monotonic function, we maximize: n-1 1

^

A

log I I ki exp(- ~- (Pi- Pi)tATl(Pi - Pi)

i=o Thus we need to find those parameters which maximize, n-1

1

^

^

- "2 E (Pi- Pi)tATl(Pi- Pi) i=o under the symmetry constraint of Eq. 8. Substituting Eq. 8, taking the derivative with respect to P0, and equating to zero we obtain: n-1

n-1

n-1

A

E RIAT~IRiP~+ E R~Ai-1(1- Ri)o = E RiAi t -1Pi i=0

i=0

,i

(9)

i=0

B

e

Note that R0 = I where I is the identify matrix. When the derivative with respect to 03 is zero: n-1

n-1

n-1

A

E (I- Ri)t'kiRiPo + ~ (I- Ri)tAT,l(l - Re)03= E (I-- Ri)tATtlPi i=0

'

i=0

b

t ,

,

(10)

i-O

D

'

'

F"

'

Note that when all A i are equal (i.e., all points have the same uncertainty, which is equivalent to the cases in the previous sections where point location is known with no uncertainty), Eqs. 9-10 reduce to Eqs. 5-6 in Section IV. Reformulating Eqs. 9 and 10 in matrix formation we obtain"

U V z Noting that U is symmetric we solve by inversion V = U-1Z and obtain the parameters 03 and P0, and obtain the most probable Cn-symmetric configuration, given the measurements {Qi} n-1 i=0" Similar to the representation in Section III, given m = qn measurements { Qi} m-1 i=0, we consider them as q sets of n interlaced measurements" {aiq+j } n~ for j = 0 . . . q - 1 (see Figure 17). The derivations given above are applied to each set of n measurements separately, in order to obtain the most probable Cn-symmetric set of points {Pi}im_fo 1. Thus the symmetry constraints that must be satisfied are,

Measuring Symmetry in Structural Chemistry

23

P0 s

s"

P1

,,, "

,,, "

i

"s

; o.

i

-:

;'-

i

,

/

/

', p

/

'

5

t

p"

................. .

,,

~"

""

P4

Figure 17. A configuration of 6 measurements represented as two interlaced sets of three measurements. One set is marked by the solid lines and the other set is marked by the dotted lines.

~Siq+j= Ri(Pj - 0)) + O) for j = 0 . . . q - 1 and i = 0 . . . n - 1 where, again, Ri is a matrix representing a rotation of 2xi/n radians and co is the centeroid of all points {J6i}mol. As derived in Eq. 9, we obtain f o r j = 0 . . . q - 1, n-1 Z

n-1 t -1

n-1

A

Rt.A-I /9 RiAiq+YiPj + Z RiAiq+j(It -I Ri)o)= Z i--iq+j, iq+j i=o i=o i_--o

Aj

(11)

kj

Bj

and equating to zero, the derivative with respect to w, we obtain, similar to gq. 10: q-1 n-1

q-1 n-1

(12)

q-1 n-1

A

~_~~ (1- Ri)tAiq+jRiPj+ Z Z (I- Ri)tA~l+j(l-Ri)o)= Z E (i- Ri)tA~l+jPiq+j j=0 i---O

j=0 i=0

j---O i=0

b

k

Reformulating Eqs. 11 and 12 in matrix formation we obtain"

~Ao

Bo A1

B~

f

A

Po A P1

rEo~ E1 ~

A

Aq_l Bq_l CO C 1 " " C q _ 1 D

Pq-1 CO

Eq-1

F k

J

ii

U

V

z

24

HAGIT ZABRODSKY and DAVID AVNIR

/ a.

1 " ' ~ ....

Figure 18. (a) A configuration of 6 measurements given by a normal distribution function (marked as rectangles having width and length proportional to the standard deviation). The most probable symmetric shapes with respect to: (b) C2-symmetry. (c) C3-symmetry. (d) C6-symmetry. (e) Mirror-symmetry.

Noting that U is sffmmetric we solve by inversion V = U-1Z and obtain the parameters co and {Pj}q~, and obtain the most probable C,,-symmetric configuration, {Pj}jmf_ol given the m e a s u r e m e n t s {Oi} m-1 i=0. Several examples are shown in Figure 18 where for a given set of measurements (Figure 18a), the most probable symmetric shapes are shown (Figure 18b--e for C 2, C 3, C 6 and mirror symmetry respectively). Figure 19 shows an example of varying the probability distribution of a measurement on the resulting symmetric shape. Figure 19a shows the most probable C2-symmetric shape for the set of measurements ofFigure 18a. Figures 19b-d show the most probable C2-symmetric shape after varying the distribution of the bottom measurement.

B. The Probability Distribution of Symmetry Values Here the question of interest is not the closest symmetric configuration, but rather the symmetry measure or the probability distribution of the symmetry measure values given the probability distributions of the point locations. Consider the configuration of measurements in 2D given in Figure 20a. Each measurement Qi is a normal probability distribution Qi "" N(Pi,Ai). Without loss of

Figure 19. The most probable C2-symmetric shape for a set of measurements after varying the probability distribution of the bottom measurement. Distributions are normal distributions marked as rectangles having width and length proportional to the standard deviation.

Measuring Symmetry in Structural Chemistry

25

Qo

~

Q2. Qo

\, ii 2~.

:

/

,: ,,

"t % ',

+

i

"

i

Figure 20. (a) A configuration of 6 measurements Qi. (b) Each measurement Qi was rotated 2xi/6 radians about the centroid (marked as '+') obtaining measurement Oi.

generality, we assume the centroid of the configuration is at the origin 9In order to evaluate the Cn-symmetry distribution (in our case n = 6) we rotate each measurement Q.z by 2xi/n radians about the origin obtaining the configuration of measurements Qi as in Figure 20b. Denote by X i the 2D random variable having a normal distribution equal to that of measurement Qi, i.e.,

E(Xi) = RiP i Cov(Xi) = RiAiR I where Ri denotes (as in Section III) the rotation matrix of 2~i/n radians. Denote by Yi the two-dimensional random variable, n-1

1

Yi-- Xi--~ Z Xj j=o

in matrix notation,

Xo

ro =A

i-I

k

Y

or Y = AX where Y and X are of dimension 2n and A is the 2n x 2n matrix: n-1 o -1 A =.1_

o n-1 0

-1 0 9

o

0 -1 0

-1 0 -1

. . . "~ ... 999

n

n-1

HAGIT ZABRODSKY and DAVID AVNIR

26 And we have"

ov(Xo) Cov(X) =

E(X) =

Cov(X._l)

(Xn-1) Cov(Y) = ACov(X)A t

E(Y) = AE(X)

The matrix ACov(X)A t, being symmetric and positive definite, we find the 2n x 2n matrix V diagonalizing Cov(Y), i.e., V A C o v ( X ) A t V t = D,

where D is a diagonal matrix [of rank 2(n - 1 )]. Denote by Z = (Z 0. . . . . 2n-dimensional random variable VAX:

"'"

o~-~176 ,..-

'

~

o ',

,..-

......... 350

-'"'"

!

.. ',

\

oo-"~

\

,...'"

o

"

b."'"|

Zn_l) t the

.,o

"""

l

,

v'"

!

d. "'"-[ ....................

| ..............

\l

..~..

|

--......

'a'

reobaNlity nsxty

b'

'c'

~

300

- -

----- - -

......

250

200

150

\\ // ''~ /," . ......... " \ " -~.:.%

I00

/,..

/

.4

o .........

0.015

..~.~'z ,j ...;';5" "-:'-r ";c O. 02

.

"""

"~\

,

O. 025

h

0.03

,,

..i.. .....

""" "'l --

-- L"

0.035

Z "" "-.""-'r-- -_- . . . . . . . . . . . .

I "" . . . . . . . . . . . . . . . . . . 0.04 0. 045

Symmetry Value

0

Figure 21. (a-d) Some examples of configurations of measurements. (e) Probability distribution of symmetry values (with respect to C6 symmetry) for the configurations (a-d).

Measuring Symmetry in Structural Chemistry

27

E(Z) = VAE(X) Cov(Z) = V A C o v ( X ) A t V t = D Thus the random variables Z i that compose Z are independent and, being linear combinations of X i, they are of normal distribution. The symmetry measure, as defined in Section III, is equivalent, in the current notations, to S = y t y . Having S orthonormal we have: S -- ( V A X ) t V A X - Z t Z

If Z were a random variable of standard normal distribution, we would have s being of a Z2 distribution of order 2(n - 1). In the general case Z i are normally distributed but not standard and Z cannot be standardized globally. We approximate the distribution of s as a normal distribution with: E(S) = E(Z)tE(Z) + traceDtD

Cov(S) = 2trace(DtD)(DtD) + 4E(Z)tDtDE(Z) In Figure 2 le we display distributions of the symmetry measure as obtained for the examples in Figure 21 a-d. Application of the method described here to thermal ellipsoids in X-ray analysis is in progress.

NOTE 1. This section is also Erratum to Section 4.4 in ref. 2. We thank A. Cotton and Y. Pinto for drawing our attention to an error.

ACKNOWLEDGMENTS We thank Prof. S. Peleg for his guidance and Prof. S. Shaik for his continuous interest in this project [14]. D.A. is a member of the F. Haber Research Center for Molecular Dynamics and of the Farkas Center for Light-Energy Conversion.

APPENDICES Appendix A: The Bounds of S Values Following the definition of the continuous symmetry measure (CSM) in Section II, the CSM values are limited to the range 0 . . . 1 (where 1 is the normalization scale). The lower bound of the CSM is obvious from the fact that the average of the square of the distances moved by the object points, is necessarily non-negative. The upper bound of the average is limited to 1 since the object is previously

28

HAGIT ZABRODSKY and DAVID AVNIR

normalized to maximum distance of 1 and by translation of all vertex points to the center of mass, a symmetric shape is obtained. The upper bound on the CSM can be tightened for specific cases. For instance in 2D one can show that the maximum S value for a triangle, with respect to C 3 is 1/3" consider the three vertices of a normalized triangle P1, P2, P3 in 2D (the centroid is at the origin). W.l.g. assume P1 = (0,1) and that P2 has a positive x-coordinate and denote by (x,y) the coordinates of P2" Given the constraint that the centroid is at the origin, one has P3 = ( - x , - 1 - y ) . In fact P2 is limited to a circle sector due to the centroid constraint and the normalization constraint (limiting all Pi's to be in the unit circle). Given these notations, we have that the S value of the triangle with respect to C3-symmetry, is given by:

1

+

y2 + y - "43x + x 2)

Considering the limited range of the P2 coordinates, the maximum value is obtained when P2 = (0,0) or P2 = (0,-1) (which are equivalent cases) and the maximum CSM value is 1/3. The maximum CSM value is actually obtained for extreme cases such as a polygon of m vertices (m = qn) whose contour outlines a regular q-gon (i.e., every q-th vertex of the m-gon coincides with a vertex of a regular q-gon). For details, see Appendix in Part I [1].

Appendix B: Orbits We first review some basic definitions required for our proofs and derivations: 9 The orbit ofx under a group G is the set { g x i g ~ G}. 9 x and y belong to the same orbit if y = gx for some g ~ G. Given a finite group G and given an ordering of its elements" gl, g2. . . . . gm' the orbit under G of a point x in Euclidean space is x 1. . . . . x m such that x i = gix for i = 1 . . . m. If gl = e (the identity element of G) then x i = gixl i = 1 . . . m.

Lemma 1. The centroid o f an orbit o f finite point-symmetry group G is invariant under G.

A point x ~ X is a general point (or is in general position) with respect to G if for all g ~ G, g g: e (where e is the identity in G) we have gx r x . L e m m a 2. I f x is a general point with respect to G then all points in the orbit o f x are general points. Furthermore f o r gl,g2 ~ G gl g: g2 ~ gl x g: g2x"

Measuring Symmetry in Structural Chemistry

29

Thus if x is a general point its orbit contains N(G) different points [N(G) is the number of elements in group G]. Lemrna 3. If the orbit o f x has a point in common with the orbit of y ureter G, then the two orbits are equal. For any x ~ X the group G x = {g ~ G [ g x - x} is called the isotropy subgroup of G at x and it contains all elements of group G that leave x invariant. If x is a general point, its isotropy subgroup contains a single element of G u t h e identity, i.e., G x = {e}. L e m m a 4. If G is finite, the number of different points in the orbit containing x is N(G)/N(GX). Proof is immediate from the 1-1 relationship between points in the orbit of x and the left cosets of Gx. Each left coset of G x consists of all elements of G that map x to a specific point y).

Appendix C" Finding the Optimal Orientation in 2D Following the derivation in Section IV we derive, here, an analytic solution to finding the orientation (rotation matrix R) which minimizes Eq. 1 under the constraints given in Eqs. 2-3. In Part I [1] (Appendix A.2) we gave the derivation for the specific case of the D 1 group having the two elements" {E, (5}. In 2D there are two classes of point symmetry groups: the class C n having rotational symmetry of order n, and the class D n having rotational symmetry of order n and n reflection axes. The problem of finding the minimizing orientation is irrelevant for the C n symmetry groups and R is usually taken as I (the identity matrix). We derive here a solution for the orientation in the case where G is a D n symmetry group. The 2n elements of the Dn-symmetry group (gl . . . . . g2n) are the n elements E, C,1~, cZn. . . . . C,~-1 (gl . . . . . gn respectively), and the n elements obtained by applying a reflection (5 on each of these elements: Cy,6Cln, ~CZn. . . . . cyC~-1 (g,,+l . . . . . g2,, respectively). We denote the orientation of the symmetry group as the angle 0 between the reflection axis and the y axis. Thus: ( c o s O sinO0) R - [-sin 0 cos Without loss of generality we assume the centroid (w) is at the origin. The matrix representation of the rotational elements of D n is then g[ = Rtgi R = gi for i = 1 . . . n. The matrix representation of the operation 6 is given by:

30

HAGIT ZABRODSKY and DAVID AVNIR

/c~ 0 - s i n 00//-10 01/( c~ sin 00/= (-c~ 20 - sin 20 / ~-sin 0 cos [--sin 20 cos 20

RI= Rtgn+lR = ~,sin 0 cos p

and gi = Rf gi-n for i = n + 1 . . . 2n. Thus from Section IV we must minimize the following over 0: 2n

2n A

A

E IlPj- Pill2 = E llg~Pj- PIll 2 j=l

j=l 2n

2n

1 = y_~ IIg,tjPj- -~n E g,t)Pjl12

j=l 2n

i=1 n

2n

= E II E glPi + E Rf glPi- 2ng'~Pjll2 j=l

i=1

(13)

i=n+l

Denoting by Xi,Yi the coordinates of the point g ,tp i i and taking the derivative of Eq. 13 with respect to 0 we obtain: n

tan20=

E E i=1

2n

(xiYj+ xjYi) j=~+l 2n

En E i=1

(14)

(xixj- yiy,) j=n+l

which is an analytic solution for the 2D case of orientation. However in higher dimensions a minimization procedure is used.

Appendix D" The Most Probable Mirror Symmetric Shape In Section VIII we described a method for finding the most probable rotationally symmetric shape given measurements of point location. The solution for mirror symmetry is similar. In this case, given m measurements (where m = 2q), the unknown parameters are {i~ i_ }q_-~,03and 0 where 0 is the angle of the reflection axis. However these parameters are redundant and we reduce the dimensionality of the problem by replacing two-dimensional 03 with the one dimensional x0 representing the x-coordinate at which the reflection axis intersects the x-axis. Additionally we replace Ri, the rotation matrix with: (cos 20 sin 20 R - ~sin 20 -cos 20) the reflection about an axis at an angle 0 to the x-axis. The angle 0 is found analytically (see ref. [1]) thus the dimensionality of the problem is 2q+l (rather

Measuring Symmetry in Structural Chemistry

31

than 2q+2) and elimination of the last row and column of matrix U (see Section VIII.A) allows an inverse solution as in the rotational symmetry case.

NOTE ADDED IN PROOF Part 4 in this series appeared, and is devoted to a detailed analysis of the implications of the CSM approach to chirality: Zabrodsky, H.; Avnir, D. J. Am. Chem. Soc. 1995, 117, 462.

REFERENCES AND NOTES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14.

15. 16.

Zabrodsky, H.; Peleg, S.; Avnir, D. J. Am. Chem. Soc. 1992, 114, 7843. Zabrodsky, H.; Peleg, S.; Avnir, D. J. Am. Chem. Soc. 1993, 115, 8278. Table 6-12 in ref. 5. Fleming, I. Frontier Orbitals and Organic Chemical Reactions; Wiley: Chichester, 1987. Hargittai, I.; Hargittai, M. Symmetry Through the Eyes of a Chemist; VCH: Weinheim, 1986. Murray-Rust, P.; Biirgi, H.B.; Dunitz, J.D.Acta Cryst. 1978, B34, 1787. Cammi, R.; Cavalli, E. Acta Co'st. 1992, B48, 245. Mezey, P.G. In New Theoretical Concepts for Understanding Organic Reactions; Bertr~, J.; Csizmadia, I.G., Eds.; Kluwer: Dordrecht, 1989, p. 55 and p. 77. Maruani, J.; Mezey, P.G. Compt. Rend. hebd. SFtanc. Acad. Sci. Paris, H 1987, 305, 1051 (Erratum: ibid, 1988, 306, 1141); Mezey, P.G.; Maruani, J. Molec. Phys. 1990, 69, 97; idem, Int. J. Quant. Chem. 1993, 45. 177. Mezey. P.G.J. Math. Chem. 1992, 11, 27. E.g. Bunker, P.R. Molecular Symmetry and Spectroscopy; Academic Press: New York, 1979, Chapter 11. Grtinbaum, B. Proc. Pure Math: Am. Math. Soc. 1963, 7, 233. Calvo C. Can. J. Chem. 1969, 47, 3409-3416. The contours were computes as Slater-type orbitals, represented by three Gaussian functions (STO-3G) using GAMESS (General Atomic and Molecular Electronic Structure System) program. 15 We thank Dr. David Danovich and Prof. Sasson Shaik for suggesting this molecule and for performing the calculations. Schmidt, M.W.; Baldridge, K.K.; Boatz, J.A.; Jensen, J.H.; Koseki, S.; Gordon, M.S.; Nguyen, K.A.; Windus, T.L.; Elbert, S.T. Quant. Chem. Program Exchange Bulletin 1990, 10, 52. Stout, G.H.; Jensen, L.H. X-ray Structure Determination, 2nd ed.; Wiley: New York, 1989.

This Page Intentionally Left Blank

SOME PERSPECTIVES IN MOLECULAR STRUCTURE RESEARCH"

AN INTRODUCTION

istv~n Hargittai and Magdolna Hargittai

Abstract

I~ II.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

The Importance of Molecular Geometry

III.

Looking Back

IV.

C o m p a r i s o n of Structures

V.

. . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34 34 35 37

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

A.

Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

B.

Differences

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Chemical Shape

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

VI.

Intramolecular Interactions

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

46

VII.

Intermolecular Interactions

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

VIII. IX.

Crystal Engineering

Supramolecular Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Note

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Advances in Molecular Structure Research Volume 1, pages 33-61. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8 33

34

ISTV,g,N HARGITTAI and MAGDOLNA HARGITTAI

ABSTRACT Over the past years the accuracy of molecular structure determination has increased. This is due not only to improved experimental and computational facilities, and to the combined application of various techniques, but also to a better understanding of the physical meaning of structural information from different sources. Critical assessment of structural information facilitates its application to investigate intramolecular and intermolecular interactions and their consequences on the rest of the molecular structure. Supramolecular chemistry poses new challenges to accurate molecular structure determination at various levels of complexity of chemical systems.

I. I N T R O D U C T I O N Much of chemistry is structural chemistry. Conversely, structural chemistry is also part of the science of structures. The discovery of the stable C60 structure is a case in point. H. W. Kroto [1] describes eloquently how his previous encounters with Buckminster Fuller's work, and, in particular, the geodesic dome of the U.S. Exhibition Hall at the 1967 Montreal Expo, have assisted him to arrive at the highly symmet-

Figure 1. The sphere decorated by regular hexagonal pattern with pentagons inserted here and there. This sphere is found under the paw of a lion-guard in front of one of the palaces in the Forbidden City, Beijing, China (photograph taken by one of the authors in August, 1993).

Molecular Structure Research

35

rical truncated icosahedral structure. This was a lucky synergy. For mathematicians, though, this has been a familiar structure. We quote here from the Introduction of Gasson's Geometry of Spacial Forms [2] which appeared on the eve of Kroto's discovery: ... it is impossible to construct a faceted spherical or part spherical surface (a dome for instance) if one has but hexagonal panels at one's disposal. There are always twelve pentagonal panels in a completely spherical ball and a set of quanta of hexagons.... For another nonchemical illustration, Figure 1 shows a sphere decorated by regular hexagons with pentagons inserted here and there in the pattem. The truncated icosahedron has 20 hexagons in addition to the 12 pentagons, and it is one of the semiregular solids of Archimedes. All carbon substances whose cage molecules contain 12 pentagons and various numbers (except one) hexagons, are called fullerenes, and C60 has the special name, "buckminsterfullerene". What Gasson [2] states about the importance of structure is a truism both in geometry and in structural chemistry: "structural pattern is present in all things . . . . The study of geometrical structure is universally all-important". We augment this with Kepler's succinct statement, "Ubi materia, ibi geometria" [3(a)]. 1

il. THE IMPORTANCE OF MOLECULAR GEOMETRY The hypothesis of the truncated structure for the remarkably stable C60molecule was followed by infrared [4] and N M R spectroscopic [5] evidences for molecular shape and symmetry. However, the ultimate proof of the structure came with the determination of the molecular geometry of buckminsterfullerene (Figure 2). Table 1 has the bond lengths from different techniques [6-9]. Molecular geometry is determined by the relative positions of the atomic nuclei in the molecule. It is most conveniently described by the so-called internal coordinates, i.e., bond lengths, bond angles, and the angles of torsion. Let us use some quotations on the importance of determining the geometry of molecules. The first one is from a book on the story of polywater by Felix Franks

[10]: The central problem in the identification of a new chemical compound is the determination of its molecular structure, the linkage structure of the various atoms in the molecule, the length and strength of the bonds between the atoms, and the general shape of the molecule. This type of information is basic to an understanding of how such a molecule can interact with other molecules, how it takes part in chemical reactions, how the substance crystallizes on freezing, and how the molecules might interact in the liquid state. The added importance of this quotation is that molecular structure was exactly the kind of information that was never obtained (and could not have been) for polywater.

36

ISTVfi,N HARGITTAI and MAGDOLNA HARGITTAI

i v

v

Figure 2. The truncated icosahedral structure of buckminsterfullerene, C60.

Another eloquent, and much quoted, statement is by Roald Hoffmann [11]: There is no more basic enterprise in chemistry than the determination of the geometrical structure of a molecule. Such a determination ..... ends all speculation as to the structure and provides us with the starting point for the understanding of every physical, chemical and biological property of the molecule.

Table 1. Bond Lengths in Buckminsterfullerene Lengths of Shared Edges of Rings

Gas ED a

Neutron Crystallogr. b

X-ray Crystallogr. c

1000K

5K

ll0K

rg

re

re

re

1991

1991

1992

1991

C(5)--C(6),eA

1.458(6)

1.455(12)

1.445(5)

1.45

C(6)-C(6),fA

1.401(10)

1.391(18)

1.399(7)

1.39

Notes:

aGas-phaseelectron diffraction [6], for r8, see Section IV.A. b[7], for ra, see Section IV.A. c[8], for ra, see Section IV.A. d[9], for re, see Section IV.A. eC(5)-C(6) is the bond shared by a pentagon and a hexagon. fC(6)-C(6) is the bond shared by two hexagons.

Ab initio MO Calculs. d

Molecular Structure Research

37

There is then a recent statement by Marlin Harmony [12]: ... surely the most characteristic attribute of a molecule is its three-dimensionalstructure, i.e., the geometricalarrangementof its constituentatoms.Therecan be no doubtthatthe development of chemistry in the 20th century has been paralleled, if not led, by advancement in our quantitative knowledgeof molecularstructure. Two succinct but powerful statements are the following, one by C. A. Coulson [13]: "No one really understands the behaviour of a molecule until he knows its structure that is to say: its size, and shape, and the nature of its bonds." The other is attributed to Linus Pauling: "The most important characteristic of a chemical bond is its length." It is to be stressed, of course, that the geometry of the molecule is only one of the three major aspects of molecular structure. The other two are the intramolecular motion, which comprises the relative displacements of the atomic nuclei with respect to their equilibrium positions, and the electron density distribution. Furthermore, molecular structure is only one of the characteristics structural chemistry is dealing with. Since our introductory chapter deals primarily with molecular geometry, it covers only a relatively small, though fundamentally important segment of structural chemistry. Various aspects of molecular geometry are also discussed in other chapters in this volume along with other characteristics of molecular structure and other branches of structural chemistry. We are aiming at the broadest coverage of structural chemistry over the years in this and future volumes.

iil.

L O O K I N G BACK

Perspectives mean not only the future but the past as well. We find it prudent to review the origins of structural considerations in chemistry. Here, however, we are going to mention only a few important moments without any attempt to be comprehensive. We have already quoted Kepler [3(b)] who first considered packing as he was examining the beautiful snow crystals. The atomistic views date back, of course, to the Greek philosophers of whom Democritos (460-370 B. C. E.) stated: "Nothing exists except atoms and empty space; everything else is opinion" [14]. It is also noteworthy that the representation of the internal structure of snowflakes by Kepler, as densely stacked balls (Figure 3), appeared 200 years before Dalton and 300 years before X-ray crystallography began. Close packing was also invoked in Dalton's papers (Figure 4) as he was illustrating his views on gas absorption [15]. Looking back, some historical facts may appear in a different light today than they did yesterday. As science progresses, the relative importance and the significance of some discoveries may change. Avogadro's law has been a basic tenet for a long time in chemistry: "Equal volumes of all gases at the same temperature and pressure contain the same number of molecules." However, to Buckminster Fuller, Avogadro's law was more than what is stated literally. He saw in it a proof that

~.~ O0

0

0-

0

"0

0

0"

0 "-~ Go

0

f-

9 _o

4~

"n

0"

O_

9

~..K_.~J

MAGYAR

POSTA

Molecular Structure Research

39

chemists considered volumes as material domains and not merely as some abstractions [16]. This was important to Fuller as he advocated a physical kind of geometry and found an especially synergistic science in chemistry. In today's world of ever-increasing importance and penetration of computational chemistry, it is instructive to quote Gay-Lussac [17]: "We are perhaps not far removed from the time when we shall be able to submit the bulk of chemical phenomena to calculation." This statement is not only remarkable for its prescience but to us it is also a caveat. Today we are closer to Gay-Lussac's target by more than a century but it is still far away for many purposes. The most immediate roots of structural chemistry are in the well-known works and discoveries of Pasteur, van't Hoff, and others [18]. The four valences in a tetrahedral arrangement of carbon, however, were first described by Emanuele Patern6 [19] in an obscure journal. Since he did not develop all the consequences of his hypothesis, the credit, justifiably, belongs to van't Hoff (and Le Bel). Yet it is worthwhile to quote Patern6 (after [20]) because there is even direct reference in his work to what would be called conformational isomers today: ... one of the fundamental principles of the theory of the constitution of organic compounds, based on the atomicity of the elements and particularly on the notion of the tetra-atomicity of carbon, is that of the identical chemical function of the four valences of the carbon atom, which is not possible unless there exists only one methyl chloride, one methyl alcohol, etc .... As for the three C2H4Br2 isomers, given that they really exist, they are easily explained without having to assume a difference between the four affinities of the carbon atom . . . . . when the four valences of the atom of this element are assumed to be arranged in the sense of the four angles of a regular tetrahedron. Then the first modification would have the two bromine atoms (or any other monovalent group) connected to the same carbon atom; while in the other modifications each of the two bromine atoms would be bonded to a different carbon atom, with only the difference that in one of the two cases the two bromine atoms would be arranged symmetrically but not in the other.

l,

,

Figure 5. Illustration from Paterno's 1869 paper [ 19].

40

ISTV,~N HARGITTA! and MAGDOLNA HARGITTAI

This is made clearer with the drawings in Figure 5 in which the bromine atoms are represented by a and b. It was not, of course, until more than 70 years later that Odd Hassel published his conformational analyses of cyclohexane and derivatives by gas-phase electron diffraction [21]. Of the 20th century's development of structural chemistry, we mention the discovery of the electron-pair covalent bond by Lewis [22] which remains a fundamental tenet. It is remembered in every line we have drawn to represent a linkage and is present in most models of molecular structure, such as, for example, the valence shell electron pair repulsion (VSEPR) model [23].

IV. COMPARISON OF STRUCTURES Comparison of structures has always been a rich source of information to account for and predict chemical behavior. This is because the chemical variation between molecules often reveals itself in the details of their geometry. Recently, MurrayRust [24] noted the importance of comparative approach. He estimated that Linus Pauling [25] had access to less than 0.01% of the structural information of the early 1990s when he was writing the first edition of The Nature of the Chemical Bond in the late 1930s. Yet his ideas on structure and bonding have stood the test of time. The seemingly obvious question of how similar is one molecule to another and one structure to another is far from trivial to answer. This has also been discussed recently by Murray-Rust [24]. Especially interesting are the multivariate statistical techniques enabling us to look for patterns and clusters in the structural data. Data banks are of great use. They include the Cambridge Structural Database at the University of Cambridge, the Inorganic Crystal Structure Database at the University of Bonn, and the Protein Data Bank at the Brookhaven National Laboratories (see e.g., [24,26]). For gas-phase molecular geometries, three volumes in the Landolt-B0rnstein series constitute the data bank containing information on 2900 molecules through 1990 [27].

A. Representations Today there is an arsenal of experimental physical techniques and computational methods for the determination of molecular geometry [28]. The respective precisions are often better than the various operational effects influencing these parameters, such as the consequences of the specificities of the matter/irradiation interaction and of molecular vibrations. Different physical techniques utilize matter/irradiation events of different nature and may yield different features of the same structure. A schematic but pointed illustration of the problem is depicted in Figure 6 after Grimmer [29]. There are also different averaging procedures over molecular vibrations influencing the determined parameters. The first exposure of this problem was done by Bartell [30]

Molecular Structure Research

41

I Figure 6. Grimmer's view of different features of the same object given by different experiments. Reprinted by permission of Kluwer Academic Publishers.

and it has been expanded and refined over the years with increasing accuracy requirements (see e.g., [28], and in particular, [31]). The so-called operational parameters are the direct output of experimental studies. They do not have well-defined physical meaning. The most important and common ones are the following: Effective internuclear distance, obtained directly from the analysis of electron diffraction intensities. Its conversion into rg distance (see below) is simple with a very good approximation, r s -- ra + 12/ra, where I is the mean vibrational amplitude. In other words, there is no need to use ra in any comparison; it is preferable to use r s. r0 Effective internuclear distance, obtained from the rotational constants; usually refers to the ground vibrational state. Since it depends strongly on the isotopic composition, it may differ from the equilibrium distance by a couple of hundredths of an angstrom. rs Effective internuclear distance determined from the isotopic substitution coordinates of the respective atoms. Since it depends slightly on the isotopic compositions, it may differ from the equilibrium distance by a few thousandths of an angstrom. ra

Internuclear distances with well-defined physical meaning are the following" re

Equilibrium internuclear distance between equilibrium nuclear positions in the minimum position of the potential energy function. No experiment yields d i r e c t l y this parameter. All computed geometries, in principle, correspond to this distance, but only in principle, of course. Basis-set

42

ISTV/i,N HARGITTAI and MAGDOLNA HARGITTAI

Table 2. Factors Influencing Internuclear Distance Parameters

Distance Type

Deformation Motion (e.g., bending, out-of-plane puckering) Effect of Temperature

Effect of Isotope Composition

rg

+

+

+

r~

-

+

+

r~rz

-

-

+

re

~

~

choice, approximations, and all computational conditions may influence the results. rg Distance-average incorporating the effect of all vibrations at temperature T. This is the parameter attainable in a straightforward way from electron diffraction. r~rz Distance between average nuclear positions in the ground vibrational state; r ~ and rz have the same meaning; rz originates from rotational spectra applying vibrational corrections, ra is the distance between average nuclear positions averaged over all vibrational states at temperature T. ra and r ~ are obtained from electron diffraction applying vibrational corrections. The most unambiguous representation of molecular geometry is the r e equilibrium structure. Another excellent representation of bond lengths is rg since it is a real distance averaged over molecular vibrations. Distances ra and r~/rz are less meritorious for characterizing bond lengths as they are projected averages, projected, that is, onto the direction of the lines connecting equilibrium nuclear positions. These representations are, however, the most suitable for characterizing bond angles. (On the other hand, a bond angle calculated from rg distances has no well-defined meaning). A summary of effects of intramolecular motion on the various distance representations is given in Table 2.

B. Differences The rg/r e differences increase with increasing floppiness of the molecule and with increasing experimental temperatures. However, these differences may extend beyond experimental error even for relatively rigid systems studied even at low temperatures. There are various ways to reduce the experimentally determined r g distance to r e distance, at least to a good approximation. A few examples are collected in Table 3 after Kuchitsu [31]. Recently Harmony [12] reviewed the possibilities of correcting spectroscopic information for vibrational effects, and made the following statement: "Finally after more than a half-century of spectro-

43

Molecular Structure Research Table 3. Examples of rg and re Bond Lengthsa'b

r~(A)

re(A)

Z~(r)(A)

C - H in CH4

1.107(1)

1.0870+0.0007

0.020

Rather floppy

B - F in BF3

1.3133(10)

1.3070(1)

0.006

Rather rigid

C - O in C12CO

1.184 + 0.003

1.1766(22)

0.007

Rather rigid

C-C1 in C12CO 1.744 + 0.001

1.7365(12)

0.008

Rather rigid

Notes:

aAfterKuchitsu [31]. bThroughout this paper, the experimental errors are quoted in the following way: Least-squares standard deviations in parentheses as units of the last digit, e.g., 1.107(1)/~, and estimated total errors are quoted as error limits, e.g., 1.184 + 0.003/k.

scopic structural studies, it now appears possible to obtain near-r e bond lengths and angles for organic polyatomic molecules of modest size (6-8 heavy atoms)." Careful considerations of the differences on the physical meaning of the parameters are needed as an increasing amount of experimental and computed structural information are being compared. As we stated: "For truly accurate comparison, experimental bond lengths should be compared with computed ones only following necessary corrections, bringing all information involved in the comparison to a common denominator" [32]. For floppy systems, such as many metal halide molecules, the rg/r e differences may be even much greater than those listed in Table 3. Alkaline earth metal, zinc, and transition metal dihalides, for example, have been extensively investigated by gas-phase electron diffraction (see e.g., [33-36]). The structure determinations have involved a joint electron diffraction/vibrational spectroscopic analysis (cf. [37]). Depending on the model potential used, and among them on the manner in which anharmonic effects are taken into account, even the "re" distances are rather different. This is illustrated by the data of Table 4. The r e distances obtained from experimental data applying various model potentials [32] have the following Table 4. Different Types of Bond Distances for Linear MX2 Triatomic Molecules (in ~)a,b MX2 T(K)

ZnBr2 600

MnCl2 1000

SrBr2 1400

rg

2.204 + 0.005

2.202 + 0.004

2.783 + 0.006

ra h re ch re a re M re

2.185 + 0.008

2.162 +_0.008

2.649 + 0.024

Notes:

2.181 + 0.005

2.153 + 0.005

2.204 + 0.004

2.196 + 0.004

2.196 +_0.005

2.184 + 0.005

2.196 + 0.006

2.186 _+0.005

aReferencesto the original publications: ZnBr2 [35]; MnC12 [36]; SrBr2 [34]. bSee footnote b tO Table 3.

2.771 + 0.006 2.738 +_0.013

44

ISTVfi,N HARGITTAI and MAGDOLNA HARGITTAI

Table 5. Bond Distances for Bent MX2 Triatomic Molecules (in ~)a,b SiCl2

Molecule

T(K)

SiBr2

1470

1470

rg

2.089 + 0.004

2.249 + 0.005

ra h

2.084 + 0.004

2.244 + 0.005

re

2.080 + 0.004

2.239 + 0.005

ch re

2.081 + 0.004

2.239 + 0.004

2.076 + 0.004

2.227 + 0.006

a

re Notes:

aSeefootnote b to Table 3. bFrom

[43,44].

meaning: ~ harmonic approximation with rectilinear coordinates [37], 4 hharmonic approximation with curvilinear coordinates [38], ~ anharmonic approximation [39], and r~eedistance with Morse-type anharmonic stretching correction [40]. The molecules listed in Table 4 have a linear equilibrium structure (except for SrBr2 which is best labeled quasilinear [34,41,42]). Table 5 lists similar data for two

V(em

-~)

_

200000

-

1 0 0 0 0 0

-

ZnC

O

C'rBrz ~---,~

9~ .

~

0

50

I00

Qe

Figure 7a. Comparison of bending potential functions, linear models of ZnCI2 and SrBr2.

Molecular Structure Research

45

silicon dihalides which have a highly bent equilibrium configuration [43,44]. For all these systems it is especially important to define the type of distances that are being compared. Any unambiguous determination of molecular geometry always includes considerations of motion. Its importance is illustrated here by the structure of three symmetric triatomic molecules involving linear, quasilinear, and bent ones [45]. The effective structure observed directly from the electron diffraction experimental data is invariably bent. The decisive difference appears in the shape of the potential energy function describing the bending motion. Examples are shown in Figure 7. The bending potential energy functions of ZnC12 and SrBr 2 are shown in Figure 7(a); Pe = 0~ corresponds to the linear configuration. The minimum of potential

v(cm -1) 60-

40 30000

2O

I

20000 .~--

"I

I

o

1o

!

20

I

~0

Q~

$113r2 9

4

10000 1

0

T

0

i

i

I

i

I

50

I

i00

(?e

Figure 7b. Comparison of bending potential functions, bent models of SrBr2 and

SiBr2.

46

ISTV,g,N HARG ITTAI and MAG DOLNA HARG ITTAI

energy appears to be at Pe = 0~ for both molecules. It is also seen though that the minimum is much more shallow for SrBr 2 than for ZnC12. Figure 7(b) shows the bending potential energy functions of SiBr 2 and, again, SrBr 2. The relatively high barrier at Pe = 0~ for SiBr 2 indicates an unambiguously bent configuration. Further enlarging the scale reveals a small barrier at Pe = 0~ for SrBr 2, so small that it lies below the level of the ground vibrational state; hence the quasilinear designation for such a structure.

V. CHEMICAL SHAPE Although the most unambiguous representation of molecular geometry is the equilibrium structure it may not always be the most useful one. Real molecules in real reactions spend very little time in or near their equilibrium structure, especially if they are characterized by large-amplitude vibrations. Levine [46] calls it the task of dynamical stereochemistry to determine the chemical shape of molecules. He suggests to distinguish the physical and chemical shapes, and finds that the interplay between the two accounts for much of the detail provided by experiments and computational studies. According to Levine [46], the chemical shape describes how molecular reactivity depends on the direction of approach and distance of the other reagent. On the other hand, the physical shape corresponds to a hard space-filling model. The concept of chemical shape is important whenever any kind of interaction between molecules is involved. Thus, Levine's characterization of chemical shape is fully consistent with Legon's description of molecular recognition [47]: At the fundamental level, molecular recognition involves the specific interaction of one part of a molecule with a particular part of another molecule. This interaction will be defined by a relative orientation and by a separation of the two subunits that confer on the system as a whole a lower energy than other conformations. An understanding of the fundamentals of molecular recognition therefore requires a knowledge of the properties of intermolecular interactions and in particular how the energy varies with relative orientation and separation.

VI.

INTRAMOLECULAR INTERACTIONS

Following the structural consequences of ligand substitution in series of substances has been a rewarding approach in investigating correlations between structure and bonding and other properties. The S-C and Se-C bond lengths, for example, are sensitive to the valence state of the participating carbon [48] (Figure 8). The effects are large and the differences may exceed 0.1 ,~. Ligand substitution on carbon causes appreciable, though lesser, bond-length changes in the sulfides and selenides. On the other hand, ligand substitution on carbon may cause changes again up to0.1/k in the S-C bond lengths in sulfones. Thus, for example, S-C changes from 1.763(5) to 1.865(6) ,~ upon CH3/CF 3 substitution, from CH3SO2C1 [49] to CF3SO2C1 [50]. The difference is

Molecular Structure Research r(S-C),

1 9 1 I C=C

C-C I

1.20

r(Se-C). ~,

~k

1.80 -

1.70 - I

47

I C-C

2.00 -

C - C .=

1.90 -

C=C

C=C

I

t 1.30

I 1.40

"C-C

I 1.50 r(CC), A

i 1.20

I, 1.30

I 1.40

I 1.50

r(CC),

Figure 8. S-C and Se-C bond lengths in sulfides and selenides with various carbon valence states (after [48]).

interpreted as a consequence of the electron releasing ability of the methyl group and the electron withdrawing ability of the trifluoromethyl group. There is a much smaller but still significant difference in the C-C bond lengths of adamantane, C10H16 [51], and perfluoroadamantane, C10F16 [52], depicted in Figure 9. The bond lengthening is thought to be a consequence of the electron-withdrawing ability of the fluorine ligand as compared with hydrogen. Very weak interactions may also have appreciable geometrical consequences. An example is the structure of N,N-dimethyl-formamide [53] (Figure 10). There is a

rs CIoHI6

1.542+0.002A

CloFI6

1.560•

A

Figure 9. C-C bond length of adamantane [51] and perfluoroadamantane [52].

48

ISTV/~N HARGITTAI and MAGDOLNA HARGITTAI

O.o.H 2.40(3) A

CH

2 123.5(6)o12018(3)/ 3 1.224(3)A ~C'-~ ~ / .

- - s - - i -'~ 113.9(5)~

1.391(7) A

H

!' 122.3(4)~

357~

1.453(4) ]k \ CI-I 3

H

C--------~ 11 • 4 ~

16 + 5 ~ ~ C

Figure 10. N,N-dimethylformamide structure from gas-phase electron diffraction [53].

difference between the two (O)C-N-C angles. The one syn to the C--O bond is somewhat smaller than the other. The C = O . . . H - C nonbonded distance, 2.40 + 0.03 A, indicates some attractive interaction, although it is far too long to consider it a hydrogen bond. Comparison with formamide [54] itself, which cannot have an O.--H interaction, supports the notion of the presence of some attractive interaction in the N,N-dimethyl derivative. Formamide has a somewhat shorter C--O bond, 1.212(2)/k, and a somewhat greater N-C--O angle, 125.0(4) ~ than N,N-dimethylformamide. The O..-H nonbonded interactions are manifested in yet shorter O...H distances in N,N-dimethylacetamide (2.21 ,~,) and N-methylacetamide (2.33 /k) according to recent high-level ab initio calculations [55,56] (the O--.H distances given here were calculated from the published geometries of refs. [55] and [56], respectively). In Section IV.B, we have stressed the importance of the physical meaning of parameters in a demanding comparison. There are other situations where trends and patterns in the parameters and their changes are sought. This can be done even without corrections for vibrational effects or other conversions. This is especially so when looking for patterns in data collected by the same technique in the same laboratory. Table 6 presents some geometrical parameters of 2-fluorophenol [57], 2,6-difluorophenol [58], and tetrafluorohydroquinone [58]. There are several geometrical features that can be ascribed to the consequences of some weak hydrogenbond formation. They also seem to indicate a trend according to which the hydrogen bonds may strengthen from 2-fluorophenol toward tetrafluorohydroquinone. On

Molecular Structure Research

49

Table 6. Selected Geometrical Parameters of 2-Fluorophenol, 2,6-Difluorophenol, and Tetrafluorohydroquinone a 2-FluorophenoP

2,6-Difluorophenol c

Tetrafluorohydroquinone d

HI3-"F9, ,g,

2.125 + 0.055

2.054 + 0.079

2.015 + 0.069

O7"-'F9, ,h-

2.735 + 0.022

2.715 + 0.067

2.657 + 0.054

Z C - O - H , deg

101.9 + 3.9

96.7 + 4.2

98.2 + 2.4

r(C2-F9), ,~

1.353 + 0.012

1.358 + 0.056

r(C6-F10), ,~

o

1.346 + 0.048

/ C 3 - C 2 - F 9 , deg

120.3 + 4.8

/C5--C6-F10, deg tilt, deg

u

1.343 + 0.013

120.1 + 2.3

122.1 + 1.7

118.5 + 3.8

119.6 + 0 . 9

- 0.7 + 4.0

r(C1-OT), A

1.350 + 0.012

0 (assumed)

1.378 + 0.010

1.362 + 0.036

2.1 + 1.2 1.353 + 0.009

ZOT-H13.-.F9, deg

120.8 + 4.5

127.1 + 5.1

123.8 + 2.9

ZC2-F9...H13, deg R

79.0 + 1.7 0.0280

77.7 + 3.3 0.0281

80.2 + 1.6 0.0269

Notes: aDistancesare rg values. bRef. [57]. CRef. [58]. dRef. [57].

t113

S

07 F10

F9

H12

HI1

the other hand, no geometrical indication pointing to hydrogen bonding was detected in 2,6-difluorobenzenamine and 2-fluorobenzenamine [59]. Relatively strong intramolecular hydrogen bonding was indicated by the geometry of 2-nitroresorcinol [60] and 2-nitrophenol [61] from gas-phase electron diffraction studies. There are considerable bond-length changes in these molecules as compared with nitrobenzene [62] and phenol [63]. These changes are consistent with strong resonance-assisted hydrogen bonding. Such resonance-assisted hydrogen bonding has been described for a number of crystal molecular structures [64]. Schemes 1 and 2 show the resonance forms of 2-nitroresorcinol and 2-nitrophenol that are supposed to contribute strongly to the molecular structure. The experimental evidence is unambiguous for the longer N--O bonds and shorter N-C bond in 2-nitroresorcinol than in nitrobenzene, and for the shorter C-O bonds and longer O-H bonds than in phenol. The experimental geometries are characterized in Table 7. Differences in the geometrical parameters are collected in Table 8. Along with

H'"O~N~O'"H

H ..0 %N//0 .."H

o

?

w

o

H".-0%N/O.. "H m

o

Scheme

1.

O

O%N~O'"H

o'

~'N/O

""H l+

Scheme 2.

Table 7. Geometrical Parameters of Phenol, Nitrobenzene, 2-Nitrophenoi, and 2-Nitroresorcinol from Gas-Phase Electron Diffraction a Parameter

C 1-C2, ,~ C2-C3, ,~ C3-C4, A N=OI4, .A, N--O15, A C-N, ,g, C-O, ,~, O-H, ,~, LC-N--O14, deg / C - N = O I 5 , deg / O - - N - - O , deg LC--O-H, deg LN-C1-C2, deg / O - C 2 - C 1 , deg / C - C 1 - C , deg / C - C 2 - C , deg CO, tilt, deg (N--)O...H(-O), A (N--)O- ..O (-H), ,A ZN'--O...H, deg LO-H-..O, deg Notes:

Phenol b

Nitrobenzene c

1.400(3) 1.399(3) 1.396(3) q --1.381 (4) 0.958(3) ~ --106.4(17) -121.2(12) 121.6(2) 118.8(2) +2(1) -----

1.223(3) 1.486(4) ~ -117.3(1) 125.3(2) ~ 117.4(2) ~ 123.4(3) 117.7(3) -m ~ -m

2-Nitrophenol d

1.411 (12) 1.406(13) 1.388(21) 1.241(9) 1.225(9) 1.464(5) 1.359(9) 0.969(12) 118.2(10) 118.6(10) 123.3(4) 104.4(22) 120.8(7) 123.9(8) 121.4(5) 119.4(8) +3.6(7) 1.72(2) 2.58(1) 104.3(15) 147(3)

2-Nitroresorcinol e

1.426(5) 1.393(4) 1.239(3) 1.449(7) 1.354(4) 1.038(15) 119.3(3) 121.4(5) 116(3) 120.5(4) 122.8(7) 119.1(7) 120.4(5) +2.9(5) 1.76(4) 2.56( 1) 110.5(15) 131(5)

aDistances(rg bond lengths) in ,g,,angles in degrees. bRef. 63. CRef.62. aFrom a concerted analysis incorporating constraints from MP2(FC)/6-31G* ab initio calculations, ref. 61. eRef. 60. 50

Molecular Structure Research

51

the experimental data, the differences referring to quantum chemically calculated parameters are also given [61,65]. Emphasis is on the comparison of differences which are free of a variety of problems that the absolute values of both the experimental and calculated results may suffer from. Of 2-nitroresorcinol and 2-nitrophenol, the former has higher symmetry and is an easier target for gas-phase electron diffraction. On the other hand, the structure of 2-nitrophenol could not have been determined without the introduction of some constraints from quantum chemical calculations. These constraints, however, have always meant differences between parameters rather than actual bond lengths or bond angles. Heretofore, we have mentioned the geometrical consequences of the strong hydrogen-bond formation in 2-nitroresorcinol and 2-nitrophenol. Tables 7 and 8 list the O.-.H (N=O---H-O) and O...O (N=O.-.O--C) nonbonded distances that can be considered direct evidence of the strong hydrogen bonds. Usually such distances, and especially the O...H distance, can be inferred only indirectly in an electron diffraction study because of their small relative weight among interactions in the electron scattering process. Figure 11 shows that, by a rare favorable coincidence, there is a broad valley in the vicinity of the contribution of the O---H distance on the radial distribution of 2-nitroresorcinol. The radial distribution is an expression of the probability density distribution of the internuclear distances obtainable from the electron diffraction data [66]. In this case, a distinct feature, however slight, can be ascribed to the contribution of the O...H distances in the

Table 8. Differences of Selected Parametersa Demonstrating Geometrical

Consequences of intramolecular Hydrogen-Bond Formation 2-Nitrophenol/Phenol

Parameter

Electron Diffraction

Calculation MP2(FC)/6-31G*

2-Nitroresorcinoi/Phenol Electron Diffraction

Calculation MP2(FC)/6-31 G*

O - C , ,~,

- 0.022

- 0.024

- 0.027

- 0.024

( O ) C - C ( N ) , ,~,

+ 0.012

+ 0.014

+ 0.027

+ 0.028

O--C-C(N), deg

+ 2.7

+ 2.8

+ 1.6

+ 1.9

C O flit, deg

+ 1.6

+ 1.3

+ 1

+ 1.2

2-Nitrophenol/Ni trobenzene Parameter

Electron Diffraction

Calculation MP2(FC)/6-31G*

2-Nitroresorcinol/Nitrobenzene Electron Diffraction

Calculation MP2(FC)/6-31G*

N--C, ~,

- 0.022

- 0.015

- 0.037

- 0.032

( O ) C - C ( N ) , ~,

+ 0.011

+ 0.018

+ 0.026

+ 0.033

N--C-C(O), deg

+ 2.5

+ 2.4

+ 3.1

+ 2.6

O-N-C(O), deg

+ 0.9

+ 0.7

+ 2.0

+ 2.0

C N flit, deg

+ 1.5

+ 1.8

Note:

aFor references, see text.

--

m

52

ISTV,g,N HARGITTAI and MAGDOLNA HARGITTAI

f(r)

0

2-Nitrores~176

1/

2

4

'r,A

,o%~0 '~

H/O~O~H O

1.5

O

2

Figure 11. 2-nitroresorcinol: radial distributions [60].

structure with the hydrogen bonds. This is made clear by the comparison with the curve calculated for another form in which there is no such hydrogen bond. Another example of geometrical consequences of intramolecular interactions can be demonstrated by the structure of a few carbon-cage molecules. Whereas the adamantane molecule [51] has only one kind of C-C distance, due to its high symmetry, there is a distribution of C-C distances in, for example, heptacyclotetradecane (Figure 12(a)) [67] and fenestrane (Figure 12(b)) [68] due to intramolecular nonbonded interactions. A final example of intramolecular interactions concerns the geometrical changes upon internal rotation. In a variety of situations, such as, e.g., conformational equilibria in the gas phase or structures of the same molecule in different phases, such effects may be of importance. Thus, for example, the C - C - X bond angles of 1,2-dihaloethanes were calculated to change up to 4 ~ during internal rotation [69]. Ignoring this change may cause an error up to 11 ~ in the determination of the gauche angle of torsion for a mixture of anti and gauche conformers.

Molecular Structure Research

53

rg

c

1-c8

1.586~0.004

A

Cl-C2

1.528:t:0.006 A

c 1-c9

1.553-~0.004 A

c9-c13

1.532~0.004 A

(a)

1

'

rg

C 1--C2

1.558-~0.005 A

C2-C6

1.539+0.008 A

C6--C7

1.542•

.,~


I 16.2~0.5 ~


I06.2~0.2 ~

6

(b)

7

Figure 12. (a) C-C bond lengths of heptacyclotetradecane [67]. (b) C-C bond lengths of fenestrane [68].

VII.

INTERMOLECULAR INTERACTIONS

Until some time ago, crystallographers used to assume that the molecules have the same structure in the crystal as the free molecules in the vapor (see, e.g., [70]). With increasing capabilities of structure determination the gas/crystal structure differences are gaining importance. Their elucidation is rapidly becoming a most important source of information on the intermolecular interactions in the crystal (see, e.g., [71-73]). The origin of gas-phase structures is not only the experimental techniques, such as electron diffraction and high-resolution rotational spectroscopy but, increasingly, high-level ab initio molecular orbital calculations. The result of the latter also refer to the free molecule.

54

ISTV,/~N HARGITTAI and MAGDOLNA HARGITTAI

Again, as with any comparison of structural information, first all operational effects must be eliminated before any difference is to be ascribed to truly structural effects. Thus, for example, further corrections are necessary before making conclusions from a comparison of rg electron diffraction and ra X-ray diffraction bond lengths. The true ra parameter refers to distances between average nuclear positions. As is well known, X-ray diffraction provides distances between the centroids of the electron density distribution rather than internuclear distances. However, for spherically symmetrical electron density distributions, coincidence with the nuclear positions can be assumed. Thus following corrections for the so-called asphericity effects (as well as for rigid-body librations), the X-ray results can be considered as an ra structure. Thus, especially for bond angles, the comparison may provide meaningful information. Considering the energy requirements of structural changes and the energy contents of crystal-field effects, conformational changes, angular changes, and bond length changes may be expected in diminishing order. It has been estimated [72] for a carbon-carbon chain that a typical bond stretching of 0.1 ,~ requires about 15 kJ/mol, a bond-angle deformation of 10 ~ about 5 kJ/mol, and a torsional distorsion of 15 ~ about 1 kJ/mol. However, as was mentioned at the end of the previous section, these changes (e.g., torsion and angle bending) do not occur separately but should be considered as parts of the overall structure relaxation and treated in a concerted way, especially beyond certain accuracy requirements. There are now at least a few well-documented cases of gas/solid structure differences involving the consequences of intermolecular hydrogen bonding and other interactions in the crystal. In addition to gas/solid comparison, comparison of crystallographically independent molecules in the same crystal, analysis of the structure of molecules whose symmetry is lower in the crystal than in the vapor, and comparison of molecular structures in different polymorphic modifications are the principal venues to investigate the influence of intermolecular interactions on molecular structure. These strategies were identified by Kitaigorodskii [74] at a very early stage of accurate crystallographic studies. It was yet another example of the prescience of this great crystallographer, and especially so since previously he had dismissed the possibility of appreciable gas/solid structural changes. Incidentally, a memorial collection of papers with the title, Molecular Crystal Chemistry, was recently published as an homage to him [75]. The understanding of changes in molecular structures is important not only in the investigation of intermolecular interactions in crystals. Another area is the investigation of large and often biologically important systems. Considerations of the chemical shape, molecular recognition, and the energy costs of changes converge. According to Legon [47]: For the purpose of modeling large systems, it is of interest to follow an approach familiar in chemistry, i.e., to consider the larger systems to be composedof groups, each group having its own characteristic properties. Thus, we might enquire into the preferred angular and radial

Molecular Structure Research

55

geometry for the interaction of one groupwith anotherand, given that in a larger unit it might not be possibleto achievethe preferredarrangement,we mightthen ask aboutthe cost in energy of small angularand radial distortionsfor this conformation. As a final note in this section we mention a recent attempt to establish some correlation between gaseous molecular structures and their source crystal structures of highly ionic substances [76]. For some metal halides both monomers and dimers occur in the vapor phase, while for others only the monomer is present in appreciable amounts. Systematic comparisons reveal that dimers are not detected in the vapor if the dimeric molecule cannot be recognized as a unit in the crystal structure. On the other hand, if the presence of dimeric molecules is discernible already in the crystal, it will be observed in the vapor only if its heat of vaporization does not exceed that of the monomer by more than, say, 10 kcal/mol.

VIII. CRYSTAL ENGINEERING Different changes in the molecular structure of analogous compounds observed in gas/crystal comparisons reflect differences in molecular packing. An example is the different impact on the benzene ring deformation by intermolecular interactions in the crystals of p-dicyanobenzene [77] and p-diisocyanobenzene [78] as illustrated in Figure 13. There is a network of dipole-dipole interactions between the antiparallel cyano groups in the crystal of p-dicyanobenzene. The packing of p-diisocyanobenzene molecules, on the other hand, allows charge-transfer interactions. The understanding of crystal packing in terms of intermolecular interactions and using this understanding to design crystals with perceived packing and properties is the subject of crystal engineering [79]. It also follows from the foregoing that the understanding of relatively weak interactions and their geometrical consequences in molecular structure is one of the determining factors in crystal engineering. Intermolecular hydrogen bonding is perhaps the single most important one among the possible weak interactions. It has been called to be the most dominant mechanism of molecular recognition in crystals [80]. Also, intermolecular hydrogen bonds may not be so weak, especially when acting in large numbers. Their cooperative action was nicely illustrated by Jeffrey and Saenger [81] who showed Gulliver, the giant, constrained by a multitude of weak bonds (Figure 14). With all the progress, however, it is still difficult to predict crystal structures. This frustration was powerfully expressed by Maddox [82]: "One of the continuing scandals in the physical sciences is that it remains in general impossible to predict the structure of even the simplest crystalline solids from a knowledge of their chemical composition." There is also considerable progress during the past few years in this respect, mainly due to the utilization of the wealth of information retrievable from the data banks, and especially the Cambridge files [83].

~3.63 ....

-

. . . . .

-..9

(~_~<...

"

"+:.%

\

N~.\

1

\\

_+ +

......

\

,...+~

':i.

+ +-

%

Figure 13. Arrangements of molecules in crystalline p-dicyanobenzene [77] and p-diisocyanobenzene [78] demonstrating a marked difference in packing (see text). 56

Molecular Structure Research

57

Figure 14. After Jeffrey and Saenger's [81] idea to illustrate cooperative action" Gulliver: a giant, constrained by a multitude of weak bonds. Illustration by V. Kubasta in J. Swift, Gulliver Lilliputban, 9Artia, Praha.

The design of drugs and extremely strong and long-lasting materials are two conspicuously important domains of crystal engineering. The determination of structures, the elucidation of structure/activity correlations involving advanced computational techniques allow the design of desired drugs based on stereochemical principles. Yet the elimination of some illnesses may necessitate not only drugs but the solution of social and environmental problems [84]. Concerning the design of new materials, environmental concerns have enhanced the requirements against which their applicability can be measured [85].

58

ISTV,~N HARGITTAI and MAGDOLNA HARGITTA!

IX. SUPRAMOLECULAR STRUCTURES A molecular crystal is a result of molecular self-assembly but it is not the only example of such molecular organizations. The living organism has ranges of self-assembled structures [86]. Needless to say, structural information from the simplest molecules to those with increasing complexity are decisive in designing and understanding such molecular self-assemblies. Whereas two-dimensional and three-dimensional networks have previously been of primary interest to the inorganic chemist (see, e.g., Wells [87]), this has recently become a focal point in organic chemistry as well [88]. Again, there is emphasis on weak interactions of a great variety. According to Lehn [89]: "Beyond molecular chemistry based on the covalent bond lies supramolecular chemistry based on molecular interactions--the associations of two or more chemical entities and the intermolecular bond." There is then a complete analogy between the atoms linked by covalent bonds into molecules and the molecules linked by intermolecular interactions into supermolecules [90]. Lehn has also established a linkage between molecular and supramolecular chemistries through molecular recognition when he stated succinctly [91] that "molecular recognition implies the (molecular) storage and (supramolecular) retrieval of molecular structural information." The investigation of these molecular and supramolecular systems represent true challenges for structural chemistry.

NOTE 1. Latin. In English: "Where there is matter, there is geometry"

REFERENCES 1. Kroto, H.W. Angew. Chem. bzt. Ed. Engl. 1992, 31, 111. 2. Gasson, P.C. Geometry of Spacial Forms; Ellis Horwood: Chichester, 1983, pp. ix-x. 3. (a) Kepler, J., quoted after A.L. Mackay. A Dictionary of Scientific Quotations. Adam Hilger: Bristol, 1991, p. 139. (b) Kepler, J. Strena, seu De Nive Sexangula, 1611. English translation: The Six-Cornered Snowflake; Clarendon Press: Oxford, 1966. 4. Bethune, D.S.; Meijer, G.; Tang, W.C.; Rosen, H.J.; Golden, W.G.; Seki, H.; Brown, C.A.; de Vries, M.S. Chem. Phys. Lett. 1991, 179, 181. 5. Johnson, R.D.; Meijer, G.; Bethune, D.S.J. Am. Chem. Soc. 1990, 112, 8983. 6. Hedberg, K.; Hedberg, L.; Bethune, D.S.; Brown, C.A.; Dorn, H.C., Johnson, R.D.; de Vries, M. Science 1991, 254, 410. 7. David, W.I.E; Ibberson, R.M.; Matthewman, J.C.; Prassides, K.; Dennis, T.J.S.; Hare, J.P.; Kroto, H.W.; Taylor, R.; Walton, D.R.M. Nature 1991, 353, 147. 8. Btirgi, H.-B.; Blanc, E.; Schwarzenbach, D.; Liu, S.; Lu, Y.-J.; Kappes, M.M.; Ibers, J.A.Angew. Chem. Int. Ed. Engl. 1992, 31,640. 9. Hasser, M.; Almlt~f,J.; Scuseria, G.E. University of Minnesota Supercomputer Institute Research Report; UMSI91/142, May 1991. 10. Franks, E Polywater; MIT Press: Cambridge, MA, 1981.

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1I. Hoffmann, R. In Determination of the Geometrical Structure of Free Molecules; Vilkov, L. V.; Mastryukov, V.S.; Sadova, N.I., Eds.; MIR: Moscow, 1983. 12. Harmony, M.D.Acc. Chem. Res. 1992, 25, 321. 13. Coulson, C.A. The Shape and Structure of Molecules; Oxford University Press, 1973. 14. Democritos [of Abdera], quoted after A.L. Mackay. A Dictionary of Scientific Quotations; Adam Hilger: Bristol, 1991, p. 71. 15. Dalton, J. Manchester Memoirs 1805, 6. 16. Fuller, R.B. Synergetics: Explorations in the Geometry of Thinking; Macmillan: New York, 1975. 17. Gay-Lussac, J.L. Memoires de la Societe d'Arouei11888, 2, 207. 18. Ramsay, O.B. Stereochemistry; Heyden, London, 1981. 19. Patemb, E. Giornale di Scienze Naturali ed Economiche; 1869, Vol. VI, pp. 115-122. 20. Natta, G.; Farina, M. Stereochemistry; Longman: London, 1972. 21. Hassel, O. Tidsskr. Kjemi Bergv. Metall. 1943, 3, 32. English translation, 1970, 30, 25. 22. Lewis, G.N.J. Am. Chem. Soc. 1916, 38, 762. 23. Gillespie, R.J.; Hargittai, I. The VSEPR Model of Molecular Geometry; Allyn and Bacon: Boston, 1991. 24. Murray-Rust, E In Computer Modelling of Biomolecular Processes; Goodfellow, J.; Moss, D. S., Eds.; Ellis Horwood: New York, 1992, p. 19. 25. Pauling, L. The Nature of the Chemical Bond; Cornell University Press: 1st ed., 1939. 26. Allen, E H. in Ref. 28, p. 355. 27. Structure Data of Free Polyatomic Molecules; Landolt-B6mstein Numerical Data and Functional Relationships in Science and Technology. New Series Volumes II/7, II/15, and II/21; SpringerVerlag: Berlin, Heidelberg, 1976, 1987, and 1992. 28. Domenicano, A.; Hargittai, I., Eds. Accurate Molecular Structures: Their Determination and Importance; Oxford University Press, 1992. 29. Grimmer, A.-R.; In J. A. Tossell, Ed. Nuclear Magnetic Shieldings and Molecular Structure; Kluwer: Dordrecht, 1993, pp. 191-201. 30. Bartell, L.S.J. Chem. Phys. 1955, 23, 1219. 31. Kuchitsu, K. in Ref. 28, (p. 14). 32. Hargittai, M.; Hargittai, I. Int. J. Quant. Chem. 1992, 44, 1057. 33. Vajda, E.; Hargittai, M.; Hargittai, I.; Tremmel, J.; Brunvoll, J. hTorg. Chem. 1987, 26, 1171. 34. Hargittai, M.; Kolonits, M.; Knausz, D.; Hargittai, I. J. Chem. Phys. 1992, 96, 8980. 35. Hargittai, M.; Tremmel, J.; Hargittai, I. Inorg. Chem. 1986, 25, 3163. 36. Hargittai, M.; Subbotina, N. Yu.; Kolonits, M.; Gershikov, A.G.J. Chem. Phys. 1991, 94, 7278. 37. Spiridonov, V.P.; Gershikov, A.G.; Zasorin, E.Z.; Butayev, B.S. In Diffraction Studies on NonCrystalline Substances; Hargittai, I.; Orville-Thomas, W.J., Eds.; Elsevier: Amsterdam, 1981. 38. Gershikov, A.G. Zh. Strukt. Khim. 1984, 2514], 30. 39. Gershikov, A.G. Khim. Fiz. 1992, 1,587. 40. Bartell, L.S.; J. Chem. Phys. 1979, 70, 4581. 41. Seijo, L.; Barandiaran, Z.; Huzinaga, S. J. Chem. Phys. 1991, 94, 3762. 42. Kaupp, M.; Schleyer, P.v.R.; Stoll, H.; Preuss, H. J. Am. Chem. Soc. 1991, 113, 6012. 43. Hargittai, I.; Schultz, G.; Tremmel, J.; Kagramanov, N. D.; Maltsev, A. K.; Nefedov, O. M. J. Am. Chem. Soc. 1983, 105, 2895. 44. Gershikov, A.G.; Subbotina, N. Yu.; Hargittai, M. J. Mol. Spectrosc. 1990, 143, 293. 45. Hargittai, M.; Hargittai, I. In Structures and Conformations of Non-rigid Molecules; Laane, J.; Dakkouri, M.; van der Veken, B.; Oberhammer, H., Eds.; NATO ASI Series C.: Mathematical and Physical Sciences, Vol. 410, pp. 465-489, Kluwer Academic Publishers: Dordrecht, Boston, London, 1993. 46. Levine, R.D.J. Phys. Chem. 1990, 94, 8872. 47. Legon, A.C. In Molecular Recognition: Chemical and Biological Problems; S. M. Roberts, Ed.; Royal Soc. Chem.: Cambridge, 1992, p. 1.

60 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62.

ISTVAN HARGITTAI and MAGDOLNA HARGITTAI

Hargittai, I. Pure & Appl. Chem. 1989, 61, 651. Hargittai, M.; Hargittai, I.J. Chem. Phys. 1973, 59, 2513. Brunvoll, J.; Hargittai, I.; Kolonits, M. Z. Naturforsch. Part A 1978, 33, 1236. Hargittai, I.; Hedberg, K.J.C.S. Chem. Commun. 1971, 1499. Hargittai, I.; Brunvoll, J.; Sonoda, T. (to be published). Schultz, G.; Hargittai, I. J. Phys. Chem. 1993, 97, 4966. Kitano, M.; Kuchitsu, K. Bull. Chem. Soc. Japan 1974, 47, 67. Duffy, E.M.; Severance, D.L.; Jorgensen, W.L.J. Am. Chem. Soc. 1992, 114, 7535. Guo, H.; Karplus, M. J. Phys. Chem. 1992, 96, 7273. Vajda, E.; Hargittai, I. J. Phys. Chem. 1992, 96, 5843. Vajda, E.; Hargittai, I. J. Phys. Chem. 1993, 97, 70. Csakvari, E.; Hargittai, I. J. Phys. Chem. 1992, 96, 5837. Borisenko, K.B.; Hargittai, I. J. Phys. Chem. 1993, 97, 4080. Borisenko, K.B.; Bock, C.W.; Hargittai, I.J. Phys. Chem. 1994, 98, 1442. Domenicano, A.; Schultz, G.; Hargittai, I.; Colapietro, M.; Portalone, G.; George, P.; Bock, C.W. Struct. Chem. 1990, 1, 107. 63. Portalone, G.; Schultz, G.; Domenicano, A.; Hargittai, I. Chem. Phys. Lett. 1992, 197, 482. 64. See, e.g., Bertolasi, V.; Gilli, P.; Ferretti, V.; Gilli, G.J. Am. Chem. Soc. 1991, 113, 4917. 65. Bock, C.W.; Hargittai, I. J. Comput. Chem. (submitted). 66. See, e.g., Hargittai, I. In Stereochemical Applications of Gas-Phase Electron Diffraction. Part A, The Electron Diffraction Technique; Hargittai, I.; Hargittai, M., Eds.; VCH: New York, 1988, p. 1. 67. Hargittai, I.; Brunvoll, J.; Cyvin, S.J.; Marchand, A.P.J. MoL Struct. 1986, 140, 219. 68. Brunvoll, J.; Guidetti-Grept, R.; Hargittai, I.; Keese, R. Helv. Chim. Acta 1993, 76, 2838. 69. Scha(.fenberg, P.; Hargittai, I. J. MoL Struct. 1984, 112, 65. 70. Kitaigorodskii, A. I. Molecular Crystals and Molecules; Academic Press, New York, 1973. 71. Hargittai, M.; Hargittai, I. Phys. Chem. Minerals 1987, 14, 413. 72. Bernstein, J. in Ref. 28 (p. 469). 73. Domenicano, A.; Hargittai, l.Acta Chim. Hung.--Models in Chemistry 1993, 130, 347. 74. Kitaigorodskii, A.I. InAdvances in Structure Research by Diffraction Methods; Brill, R.; Mason, R., Eds.; Pergamon Press: Oxford, 1970, Vol. 3, p. 173. 75. Hargittai, I.; Kfilmfin, A. (Guest Eds.). A. L Kitaigorodskii Memorial Issue on Molecular Crystal Chemistry. Acta Chim. Hung.--Models in Chemistry; 1993, Vol. 130, Nos. 2 and 3. 76. Hargittai, M.; Jancs6, G. Z. Naturfortsch. 1993, 48a, 1000. 77. Colapietro, M.; Domenicano, A.; Portalone, G.; Schultz, G.; Hargittai, I. J. Mol. Struct. 1984,112, 141. 78. Colapietro, M.; Domenicano, A.; Portalone, G.; Torrini, I.; Hargittai, I.; Schultz, G. J. Mol. Struct. 1984, 125, 19. 79. Desiraju, G.N. Crystal Engineering: The Design of Organic Solids; Elsevier, Amsterdam, 1989. 80. Bernstein, J.; Shimoni, L. CollectedAbstracts; XVI Congress and General Assembly, International Union of Crystallography; Beijing, China, 1993. Paper MS-06.01.01 (p. 164). 81. Jeffrey, G.A.; Saenger, W. Hydrogen Bonding in Biological Structures; Springer-Verlag: Berlin, 1991. 82. Maddox, J. Nature 1988, 335, 201. 83. Brock, C.P. Collected Abstracts; XVI Congress and General Assembly, International Union of Crystallography; Beijing, China, 1993. Paper MS-06.01 (p. 4). 84. Johnson, L.N. Collected Abstracts; XVI Congress and General Assembly, International Union of Crystallography; Beijing, China, 1993. Paper ML-05.01 (p. 4). 85. Dixon, D.A. Abstracts; Second Conference on Current Trends in Computational Chemistry; Vicksburg, MS, 1993. 86. Mathias, J.P.; Seto, C.T.; Zerkowski, J.A.; Whitesides, G.M. In Molecular Recognition: Chemical and Biochemical Problems H; Roberts, S.M., Ed.; Royal Soc. Chem: Cambridge, 1992, p. 35.

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87. Wells, A.E Structural Inorganic Chemistry; 5th ed.; Clarendon Press: Oxford, 1984. 88. Roberts, S.M. In Molecular Recognition: Chemical and Biochemical Problems H; Roberts, S. M., Ed.; Royal Soc. Chem.: Cambridge, 1992 (Preface). 89. Lehn, J.-M. Science 1985, 227, 849. 90. Lehn, J.-M. J. Inc. Phenom. 1988, 6, 351. 91. Lehn, J.-M. Science 1993, 260, 1762.

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ACCU RATE MOLECU LAR STRUCTU RE FROM MICROWAVE ROTATIONAL SPECTROSCOPY*

Heinz Dieter Rudolph

Abstract

Io II.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction

64

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

A. B.

68 72

Inertial M o m e n t T e n s o r . . . .9 . . . . . . . . . . . . . . . . . . . . . . . . General Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . .

III.

S u b s t i t u t i o n rs-Structure; rs-Fit Structure . . . . . . . . . . . . . . . . . . . . .

78

IV.

Effective (r0) Structure; t o - D e r i v e d Structure . . . . . . . . . . . . . . . . . . .

92

V. VI.

O t h e r Structures

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C o n c l u d i n g R e m a r k s and O u t l o o k

104

. . . . . . . . . . . . . . . . . . . . . . .

110

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

tDedicated to Professor Helmut Dreizler on occasion of his 65th birthday

Advances in Molecular Structure Research Volume 1, pages 63-114. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8 63

64

HEINZ DIETER RUDOLPH

ABSTRACT For later reference, the basic facts and equations are presented, (1) for the inertial moment tensor of a molecule modeled as a rigid set of mass points, and (2) for the general linear least-squares procedure. These tools are extensively used in the recent versions of the two classical methods of determining molecular structure from rotational spectra. The two methods are: (1) the m-method which basically determines the "effective" or r0-structure in internal coordinates by a least-squares fit, while its more recent variants yield the p-Kr (or rzx/-) and the r/E-structure, and (2) the rs-method which basically determines the "substitution" or rs-structure in Cartesian coordinates from the inertial moments of the parent molecule and all singly substituted isotopomers (by the repeated application of Kraitchman's equations), while the more recent variant, the rs-fit method, includes also all available multiply-substituted isotopomers in a least-squares fitting process. These methods and their variants have the least restrictive requirements, regarding the set of isotopomers whose inertial moments must be known, and are also more easily applicable when the structures of larger molecules are to be determined. The effect of the generally unknown rovibrational contributions to the inertial moments is discussed and it is shown that the variants presented make allowance for the constant, isotopomer-independent, part of these contributions. The equations for the true derivatives of the inertial moments with respect to the internal or Cartesian coordinates of the atoms, as used in the particular fitting processes, are given. The role of the covariance matrix of the input data (the inertial moments) is discussed, likewise that of the output when structures expressed in internal and Cartesian coordinates are to be compared, inclusive of errors, in a consistent fashion. Short accounts are also given of the "mass dependence" or rm-method, where the rovib contribution is explicitly expanded in terms of the atomic mass change upon isotopic substitution and where first-order terms are correctly included and of the closely related "complementary" or re-method, where also the second-order terms are approximately compensated for by the selection of complementary sets of isotopomers. The rm- and re-methods are in practice applicable only to small and simple molecules. Mention is also made of the r~,, -method, a simplified and hence more widely applicable derivate of the rm-method, for which a recent series of detailed papers is available.

!. I N T R O D U C T I O N The determination of accurate molecular structure from molecular rotational resonance (MRR) spectra has always been a great challenge to this branch of spectroscopy [1]. There are three basic facts which make this task feasible: (1) the free rotation of a rigid body is described in classical as well as in quantum mechanics by only three parameters, the principal inertial moments of the body, Ig, g = x, y, z; (2) the molecule can, within limits, be modeled as a rigid set of atomic mass points; (3) the relative geometry of this set of mass points is, to a high degree of approximation, independent of the isotopic substitution of one or more of the atoms. Statement (1) indicates that no more than three relevant parameters of molecular

Microwave Rotational Spectroscopy

65

mass geometry can be obtained from the evaluation of one MRR spectrum. Statement (3) suggests to aggregate experimental information by measuring the spectra of an adequate number of molecular isotopomers which all have the same relative geometry but different inertial moments due to the (known) isotopic mass differences and their pulling effect on the principal inertial axis system. Unfortunately, the connecting statement (2) is the weakest of them. The molecule is a rotating a n d vibrating system. The vibration is much faster than the rotation, and it is permissible to average the relevant parameters of molecular mass geometry over a rotational cycle. The molecule is thus far "effectively rigid". However, the effective inertial moments so obtained depend on the vibrational state v of the molecule, I gv , and each vibrational state has its own rotational spectrum. Usually rotational transitions ("vibrational satellites") for the lower, and hence more populated vibrational states, can be observed in the spectrum closely adjacent to the transitions in the vibrational ground state. Due to the zero-point vibrations, even the inertial moments of the vibrational ground state, 0 I g , are different from those of the hypothetical, vibrationless, and truly rigid "equilibrium configuration", Ig. It is possible only for very simple molecules to obtain the inertial moments I ~ for the vibrational ground state a n d for the vibrationally excited states I gV of every normal vibration and to extrapolate from them to the equilibrium moments I ge . For the larger molecules, the differences e eg - Ig0 - - Ig, g - x, y, z, must be accepted as unknown rovibrational (rovib) contrie butions to the equilibrium moments I g. The readily obtainable experimental 0 e ground state moments Ig = Is + eg hence only approximate the desired but elusive e e 0 m o m e n t s I g . If the u n k n o w n I g are replaced by the k n o w n I g , a systematic error is the unavoidable consequence. It is a particularly distressing fact that the unknown eg are expected to be larger than the experimental errors of the I g0 by orders of magnitude. The E g depend on a large number of force constants, quadratic as well as cubic. It has to date been practically impossible to obtain the eg for larger molecules from the information gained from vibrational and rotational spectroscopy alone. However, progress has recently been made in the ab initio calculation of rovib contributions to the inertial moments for small molecules [2,3], which appears to be a promising way of combining computer power and practical spectroscopy towards the goal of greatly improved accuracy of molecular structure determinations. Meanwhile, efforts must continue to at least estimate the resulting errors of the molecular structures obtained by the available methods or to develop variants which are less error prone due to a (partial) compensation of the rovib contributions or to devise methods to include them in some way in the model used. The experimental inertial moments may also be affected by mechanisms other than the rovib interaction discussed above; e.g., by contributions from large-amplitude internal motions or by the coupling of the molecular angular momentum and spin momenta caused by the presence of internal magnetic or electric fields. We shall assume in this review that these generally much smaller effects have been

66

HEINZ DIETER RUDOLPH

properly corrected or are small enough to be neglected. It will also be assumed that the centrifugal distortion in the MRR-spectra has been adequately accounted for when the inertial moments were evaluated from the spectra. The two major routes followed in the attempt to determine consistent molecular structures from the experimental inertial moments of the MRR-spectra began with the r0-method and the seemingly very different substitution or rs-method [4,5]. Both methods require only the ground state inertial moments of a sufficiently large number of isotopomers if a complete molecular structure without any assumptions is to be determined. This is, nonetheless, a demanding requirement for all but the simplest molecules. The price paid for getting along without any additional knowledge of the vibrational parameters of the molecules is the fact that the "r0-structure" as well as the "rs-Structure" and all their variants are only operationally defined by the process of their calculation, quite in contrast to the equilibrium configuration or "re-Structure" which is the location of the global minimum of the molecular potential energy function and has a clear physical meaning. In the original version of the r0-method, ground state inertial moments calculated for all isotopomers in terms of internal coordinates are least-squares fitted to the experimental moments I g0. The internal coordinates represent a reference system which is identical for all isotopomers and the resulting "r0-structure" is obtained as the final set of internal coordinates determined by the criterion of optimum fit. All atomic positions must either be included in the list of those to be determined, or estimated values must be supplied and then kept fixed in the fit. The result depends on these assumed values. Schwendeman has suggested a useful ro-derived variant [6], the "p-Kr method", where the isotopic differences between the calculated inertial moments of the isotopomers and the parent species are fitted to the respective experimental differences in the attempt to compensate (the isotopomerindependent) part of the rovib contribution. The same result is achieved explicitly by the rl~-method, a r0-derived variant which is presented later in this chapter, where the calculated inertial moments plus three isotopomer-independent rovib contributions Eg are fitted to the experimental ground state m o m e n t s I 0g . The original r~-method is based on equations presented by Kraitchman [4]. He had noticed that the numerically dominant part of the inertial moment tensor for an isotopomer in which (exactly) one atom has been substituted, can be replaced by the known inertial moments of the parent molecule, and that the remainder of the tensor then depends only on the Cartesian coordinates of the substituted atom. Equating the roots of the secular equation for this "hybrid" tensor to the experimental inertial moments of the isotopomer, Kraitchman obtained his famous equations which, quite in contrast to any r0-type method, always aim at the determination of the Cartesian coordinates of just this one substituted atom. The rest of the molecule is irrelevant, it does not influence the result and need not be known. Costain has checked, for several small molecules, the invariance of rs-determined bond lengths when employing different pairs of parent molecule and isotopomer [1]. He found it superior to the comparable r0-data and recommended to prefer r~-structures for

Microwave Rotational Spectroscopy

67

better consistency. The original rs-method has been later developed by Typke [7] into a least-squares procedure including all available isotopomers of a molecule, also multiply substituted species, while retaining all of Kraitchman's basic principles. Widely used treatises and monographs on the theory and the applications of MRR-spectroscopy are available [8-11], also comprehensive reviews on all aspects of obtaining molecular structure from MRR-spectra [6,12,13], including rovib interactions [14] and the reliability of the results [15-17]. Molecular structural and other data obtained from MRR spectroscopy have been compiled over the past decades [18,19]. The most recent compilation is MOGADOC (short for "Molecular Gas Phase Documentation"), a computerized database and retrieval system that is updated periodically and today contains more than 20,000 references, which were critically selected and evaluated by means of keywords. Included is work done by MRR-spectroscopy, electron diffraction of gases, and molecular radio astronomy; the documentation refers to more than 6000 compounds. As an additional feature, MOGADOC contains explicit numerical data on the structure of approximately 2000 compounds. A detailed description can be found in ref. [20]. This chapter is not intended as a general review. The intention is rather to present the most recent versions or variants of both, the r s- and the r0-method, on a small common background of theory but in sufficient depth and detail to enable the reader to follow, e.g., the flow of programs coded for the purpose. The recent versions of these two methods are the least restrictive with respect to size and composition of the set of isotopomers required and are applicable also to the largest molecules that can be treated by molecular rotational spectroscopy (perhaps ten "heavy atoms" plus hydrogen). We shall not go into the problems of the "average-structure" (also called r z- or r~0 -structure), the average molecular configuration for the vibrational ground state [21-23]. For the evaluation, the harmonic part of the molecular potential energy function must be known, i.e., a great deal of information from vibrational spectroscopy (or ab initio calculations) must be available. Also, it is not independent of isotopic substitutions, although corrections can be introduced. Valuable features of the r_-structure are the clear physical meaning and the readiness with which it can be compared with the counterpart supplied by the other important and physically very different method of determining the structure of free molecules, the investigation of electron diffraction patterns of gases. Likewise, we shall not pursue here in detail the fundamental work of Watson [24,25] who introduced the "mass dependence" or r,,-structure, where for the first time an explicit compensation of part of the rovib contributions was attempted, based on the first-order expansion of the eg in terms of atomic masses. On the basis of Watson's work, Nakata and colleagues [26] have developed the "complementary" or rc-method where, by a judicious selection of complementary sets of isotopomers, the effects of also the second-order terms of the expansion can be almost completely removed, so that the rc-Structure is expected to be very nearly equal to the re-Structure. In practice, the r,,,- and rc-methods are limited to very small molecules.

68

HEINZ DIETER RUDOLPH

Also based on Watson's earlier rm-method, Harmony and colleagues have collected arguments for the application of particular scaling factors pg to the ground state inertial moments I s0 in order to let them approach the desired equilibrium moments. The structure so obtained is called the "r m~ -structure" and is not limited to small molecules. A recent series of comprehensive papers is available [27-31]. A short account of the rm-, rc-, and rO~ m -methods is given in the last section of this chapter. The following contains a minimum of theory but conveniently referenced. The equations and statements come from two fields: (1) the description of the inertial properties for the rotation of a molecule, modeled as a rigid set of mass points, and (2) the application of the general linear least-squares procedure to fit observations that can be expressed as known functions of variables which are to be determined by the fitting process.

I!.

THEORETICAL BACKGROUND A. Inertial Moment Tensor

In mechanics, the inertial properties of a rotating rigid body are fully described by its inertial moment tensor I. We can simplify the subsequent equations if we employ in place of I the closely related planar moment tensor P, apparently first used by Kraitchman [4]. At any stage of the calculations, however, an equivalent equation could be given which involves I instead of P. The principal planar m o m e n t s Pg (g = x, y, z) are the three eigenvalues of the planar moment tensor P and the principal inertial moments Ig the eigenvalues of the inertial moment tensor I. Pg, Ig, and the rotational constants Bg =f/Ig are equivalent inertial parameters of the problem investigated (fconversion factor). Let s be any of a number of rigid mass point sets, s = 1, N s, which all have the same (relative) geometry but differ in the mass mu(s) of one or of some of the mass points oc = 1, N a. The points are described by their masses m~(s) and their positions r~ with respect to an arbitrary Cartesian reference coordinate system. This ensemble of mass point sets is the model for a number of isotopomers of a molecule, often called the substitution data set (SDS). This SDS is required for the determination of the molecular structure (the relative geometry) by any of the current methods. The total mass M(s) and the position of the center of mass rcm are then given by: Na

M(s) - E mc~(s)

(1)

o~

Na

rcm(S) = M(s) 1 E ma(s) ra

(2)

Microwave Rotational Spectroscopy

69

Let further the positions of the mass points be r~ ( - r'a(s)), when expressed with respect to a reference axis system which is parallel to the first system but with the origin shifted into the center of mass of the particular sets: p

r~= r -

rcm, all a , rcm=0

(3)

The three conditions for this particular translation, the three "first moment relations", are, from Eq. 2: Na

Z ms(s ) g~= ' O, g' = x ' , y ' , z '

(4)

The rotational motion of the rigid set of mass points about any axis through its center of mass in the absence of exterior forces is known as the "free rotation" of the rigid body. The planar moment tensor for this motion, with the position vectors ra referred to an arbitrary basis system, can be compactly written as a dyadic (T denotes transposition) [8,32], Na

P(s) = ~ m.(s)r=rS - M(s) rcmrcm,'r

(5)

(X

explicitly:

1 P(S) = E m~(s)r=rT-- M(s) Na

=Z

ma(s)ra

Na

/N

mla(s)rla

Na

I ~" Z ma(s)mp(s) r~r~ ma(s)r=rV~- M(s----~

a

a

(6)

~

If use is made of the shifted reference system with the origin at the center of mass, the second term in Eqs. 5 and 6 vanishes: Na

P(s) = Z rn~(s)'I'i-'-'T

(7)

O(

This is a symmetric, in general nondiagonal, tensor with explicit elements: 2

Z,n=(s)xdy

y=o(s)y;= is===

p

p

y_mo(s)zo=I

(=)

70

HEINZ

DIETER RUDOLPH

While the elements of the tensor P(s), Eqs. 5 and 6, are numerically invariant with respect to a parallel translation of the reference axis system, they do depend on the orientation (the directions) of the reference axes. If the mass point system is rigidly rotated by an orthogonal transformation T (with T v = T-i), r~'= TTr~ = TTrr

TTr~m (all r

(9)

the planar moment tensor is transformed correspondingly, P"(s)= TrP(s)T. The most important orthogonal transformation is the principal axis transformation of the moment tensor, here denoted as T(s), which transforms the coordinate system r' a of the mass point set s into its principal axis system (PAS) _Is] IFO~ ,

rt4_ TV(s)rr = TV(s)r,~_ TV(s)r~m (all r

(10)

and hence diagonalizes the tensor P(s), Na

~, 9 /~-.[s]-.[s] ptSl(s) = Tr(s)P(s)X(s)- ]~ ,,,~w.~ .~ r

( 11 )

or, showing the eigenvalues explicitly: [4 ~m~(s")x,~

ptsl(s)-/ 0

0

~

0

~_ma(s)yt~ 1-

0

0

Zma(s)z[~]-

(12)

The principal axis transformation requires that the off-diagonal elements (the deviation moments or inertial products) vanish in the PAS, the resulting three conditions are known as the "second moment relations": ~__m~(s)x[~ly[~] - ~2_}natsjya ,, [sl_[s]_ ~ .~ 2 m a t s, ,) x ~[s]_[s] ~ = 0

(13)

Since the tensor P(s), Eq. 5, is numerically invariant with respect to a parallel translation ofthe general reference system ra, the diagonalizing transformation T(s) is likewise independent of such a translation. The superscript [4 is used here to indicate that a quantity is expressed with reference to the PAS of the mass point set s. PE4(s) is hence diagonal by definition. However, P[Sl(s'), the planar moment tensor of a different set s' of mass points, expressed with respect to the PAS of the mass point set s, is, in general, nondiagonal. The need for this distinction will later occur frequently when s enumerates the different isotopomers. The eigenvalues, the diagonal elements of Eq. 12, are Pg[4(s) ( g - x, y, z), where the single subscript implies that the quantity is an eigenvalue, not a general tensor element, and where the superscript must be dropped because an eigenvalue does not depend on the reference system used.

Microwave Rotational Spectroscopy

71

If the set of mass points possesses any overall symmetry, a symmetry plane of the set will always be also a principal plane of the moment tensor, a symmetry axis always a principal axis, and the number of independent mass point coordinates is less than 3N,~- 6. Group theory shows that the number of independent internal coordinates (bond lengths, bond angles, and dihedral angles) required for the description of the mass point set equals the number of totally symmetric normal (displacement) coordinates [33]. The number of independent Cartesian PAS coordinates equals the number of independent internal coordinates plus the number of non-trivial first and second moment equations (those that are not satisfied alone by symmetry). If the set includes a group of mass points which are symmetrically equivalent with respect to any three- or higher-fold local symmetry axis, it can be shown that the moment tensor is insensitive to the torsional position of this group of points about the local symmetry axis fixed in the otherwise rigid set of mass points. The inertial moment tensor I and the planar moment tensor P are related by, I - Tr(P).l - P

(14a)

P = 89

(14b)

- I

[where 1 is the unit matrix and the scalar Tr(P) is the trace of the tensor P]. From Eq. 14 it is obvious that I and P are diagonalized by the same orthogonal transformation. This relation holds, no matter which reference system has been used, i.e. for Eqs. 6, 7, 11, or 12. If Eq. 12 is used, one obtains from Eq. 14 the relations between the eigenvalues of the two tensors, here shown as transformations:

ll~l~l(s)

I

0 1 1 IPxt~l(s) (15a)

1 tPzl,l(s)

I

-'-' tC,s,(sj 1

(15b)

1

The quantity,

Ag[sl = -2P gtsj = 21gf4 -11xl.~l + lrt~l + I:.l.~lI

(15c)

has been called the (pseudo-) inertial defect. For a planar molecule, Ag[slis the true inertial defect for axis g perpendicular to the molecular plane. It vanishes when the equilibrium moments Ige are used in Eq. 15c and is a small quantity for the ground state m o m e n t s I 0g. For any molecule, the pseudo-inertial defect Agt~l does not

72

HEINZ DIETER RUDOLPH

change, under the same conditions, for isotopic substitution on the principal inertial plane to which g is perpendicular. Following convention, the principal axes are usually denoted as a, b, c in place of x, y, z, in such a way that I a < I b < I c. This ordering corresponds to Pa > Pb > Pc" For the prolate symmetric top, Pb = Pc, and for the oblate symmetric top, Pa = Pb"

B. General Least Squares We shall adhere, as far as possible, to the notation of the excellent tutorial by Albritton, Schmeltekopf, and Zare [34]. Suppose that n observable quantities Yi exist which depend on m independent variables Bj:

Yi- Yi(",Bj,") , i - 1,n, j - 1,m, m < n It is convenient to let the n • 1 vector Y be composed of the observables m • vector B of the variables Bj: Y = Y(B)

(16)

Yi and the (16a)

Suppose that the variables Bj are to be determined by a least-squares fit of the relations, Eq. 16, to the measured values Yi,exp (vector Yexp)" Assume that the measurements Yexp are unbiased (E(Yexp) = Ytrue where E( ) represents the mean or expectation value) and that the measurement errors and their correlations are described by the positive-definite n x n variance-covariance matrix Oy which can be written as the dyadic: exp

OYexp=E((Yexp-E(Yexp))(Yexp-E(Yexp))T I

(17)

Assume that an adequate approximation for the desired vector B is known, B (~ near enough to the final result for an approximate linearization of the original Eqs. 16, 16a. The vector Y calculated for this B (~ y(O) = y(B(0)), is a constant, nonrandom vector (y(0) = E(y(0))). The linearization of the Eqs. 16, 16a leads to:

m (~yi I

(Bj - BJ0,) i = l ,n

(18)

Abbreviating Yi - YI~ by Yi (vector y : Y - y(0)) and Bj- BJ~ by 13j (vector [I = B B(~ and collecting the partial derivatives in the Jacobian n x m matrix X, with elements

-

liOYiI

i= 1,n j - l,m

Xij:~-~j~

(19)

. . . Bj (~ . . ).

the system of "model equations" of the problem is: y = Xp

(18a)

Microwave Rotational Spectroscopy

73

We continue to call y the observations, and [~ the variables. The Jacobian X is a rectangular, in general "high", matrix (n > m). For further treatment it has to have maximum rank (= m), which requires that the [3j be independent variables. The columns of X, the "fit vectors", span the m-dimensional "fitspace", a subspace of the n-dimensional space of the observations and their errors. The Jacobian X is a constant (nonrandom) matrix which depends on the functional type but not on the measured value of each of the observations. For the practical application, we wish to substitute in y - Y - y(0) the vector Y by the vector Yexp which gives Yexp= Yexp- y(0). Since y(0) is a nonrandom vector, the covariance matrix of the observations is Oyoxp = OV~xp. The observations Yexpare also unbiased (E(Yexp) = Ytrue) since the Yexp were assumed unbiased. Due to n > m, the system of Eqs. (18, 18a) would, in general, be compatible only for the true, but unknown, observables and variables (Ytrue = Xl]true). To make it compatible when the Yexp are used, the system of Eqs. 18, 18a must be supplemented by the yet unknown (random) vector of residuals ~; (= t:true), which gives Yexp- Xll + ~:. (This is explicitly: Yexp = Xlitrue + I;true = Ytrue+ s Since Yexpis unbiased and Ytrue is a nonrandom vector, we have E(t:) = E(Yexp) - E(Ytrue) = 0, and also OE = OYexp.)The residuals t: should not be confused with the rovib contributions to the inertial moments, eg, introduced in the preceding chapter. The covariance matrix of the observations, Oy~xp = Oe, is an integral part of any least-squares problem and the subsequent result depends on Oy~p, but only to within a scale factor, ~2, the variance of the fit. Dropping the index and writing y instead of Yexp, the "error equations" in the usual notation are: y = Xp + t;,

Oy = Or; = oZM

(20)

The general least-squares treatment requires that the generalized sum of squares of the residuals, the variance 02 , be minimized. This is, by the geometry of error space, tantamount to the requirement that the residual vector be orthogonal with respect to fit space, and this is guaranteed when the scalar products of all fit vectors (the rows of X T) with the residual vector ~ vanish, XTM-l~ = 0, where M -1 is the metric of error space. The successful least-squares treatment [34] yields the following minimum-variance linear unbiased estimators (^) for the variables, their covariance matrix, the variance of the fit, the residuals, and their covariance matrix" = (XTM-1X)-IXTM-ly A

A

(21a)

O~ = O'2(XTM-1X)-1

(2 lb)

~2 = (y _ x~)T M-l(y _ X~)/(n - m)

(21c)

s^

(21d)

74

HEINZ DIETER RUDOLPH A

A

O~ = (y2(M- x(xTM-IX)-Ix T)

(21e)

The vector ~ is the vector of corrections which must be applied to the first approximation B (~ of the desired variables to give an improved approximation to be used in the next iteration step, B (~ + ~ ~ B (~ The iterations are to be continued until convergence is obtained. Note that O~ is then not only the covariance matrix of increments t] of the last iteration step but also of the final variables B, ~ = ~ , by arguments similar to those which let Orexp= OYexp. I f M is a diagonal matrix (experimental errors, but no correlations), M -1 is called the weight matrix, and the solution (Eqs. 21) is that for the weighted, uncorrelated least-squares problem. If M is introduced in Eqs. 21 as the unity matrix, Eqs. 21 solve the equally (unity-) weighted, uncorrelated problem. In this special case, the covariance matrices^(Eqs. 2 lb and 2 le) depend on the measurements only through the common factor ~2 (Eq. 2 lc). The solutions (Eqs. 21 a, b), are correct also for the limiting case where m = n. X is then a square matrix, X -1 exists, which simplifies the equations greatly. However, a program coded for the general least-squares procedure ^ can still be used, if provision is made for assigning unity to the then irrelevant ~2. After convergence has been achieved, Eq. 21 a gives the solutions and Eq. 2 lb their variances-covariances, correctly propagated from M. (All elements of ~ and ~)~ vanish.) If the system of error equations (Eq. 20) is "ill-conditioned", i.e., the matrix XTM-1X of the "normal equations", XTM-IXII = XTM-ly (cf. Eq. 2 la) is near-singular and has near-zero eigenvalues, care must be taken to choose a stable algorithm for the least-squares solution. The numerically stable inversion of the matrix XTM-1X is then difficult. The system is ill-conditioned when the fit vectors, the columns of X, are "not sufficiently independent" in error space (with metric M -l), that means, one or more linear combinations of them almost vanish and the corresponding linear combinations of variables 13j are indeterminable with the particular selection of observations that yield this Jacobian X. It may then become necessary to assume fixed values for one or more of the ~j involved in these combinations and eliminate them from the list of variables to be determined by the fit. Useful algorithms for arriving at the solution (Eq. 2 la) that are more stable than the direct matrix inversion are available [35-37]. In problem cases the "singular value decomposition" [38] can be applied which affords a method [39,40] that yields a numerically stable solution also in the cases where the matrix of the normal equations XTM-1X is near-singular. But it is particularly useful in such cases where this matrix is truly singular, when X has less than maximum rank e.g., because more parameters 13jare to be determined than the number of observations Yi warrants (m > n). Instead of eliminating so many of the parameters 13j from the fit by keeping them fixed till the offending matrix is no longer singular, an unambiguous solution of the underdetermined system is enforced by the additional requirement that the vector of solutions, 11, shall have minimum length. For this purpose the method selects linear combinations out of the number of original variable parameters and

Microwave Rotational Spectroscopy

75

keeps them fixed at their initial values. The singularity threshold can be set at the user's judgement, the elements of II can be weighted prior to the application of the minimum length criterion, and, due to the automatic choice of the linear combinations kept fixed, all initial parameter values must represent the best possible estimates. All this requires a certain amount of a priori insight into a problem to be solved by this method. If the original model is sufficiently perfect, the linearization of the problem adequate, the measurements unbiased (no systematic error), and the covariance matrix of the observations, O., a true representation of the experimental errors and A ,Y their correlations, then o 2 (Eq. 21c) should be near^ unity [34]. If Or is indeed an honest assessment of the experimental errors, but 0 2 is nonetheless (much) larger than unity, model deficiencies are the most frequent source of this discrepancy. Relevant variables probably exist that have not been included in the model, and the experimental precision is hence better than can be utilized by the available model. Model errors have then been treated as if they were experimental random errors, and the results must be interpreted with great caution. In this often unavoidable case, it would clearly be meaningless to make a difference between a measurement with a small experimental error (below the useful limit of precision) and another measurement with an even smaller error (see ref. [41]). A deliberate modification of the variance-covariance matrix Or towards larger and more equal variances might then be indicated, which results in a more equally weighted and less correlated matrix. In most practical cases the original relations (Eq. 16) are nonlinear and the linear least-squares treatment must be iterated to obtain convergence. The elements of the Jacobian X must be recalculated with each new iteration step. Although the least-squares procedure is said to be rather tolerant with respect to the precision of the Jacobian X, true derivatives should be used if ever possible, because finite difference schemes will most often require detailed considerations with respect to the allowed step width. Even then the results may show a tendency to oscillate long before a convergence limit due to the algorithms used orthe number of digits carried is reached. With true derivatives, however, this limit is attainable. The least-squares solution ~ (Eq. 21c) is independent of any non-singular, linear, constant (e-independent) transformation of the observations, y' = Cy. The transformed error equations are: y' = Cy = CXp + Ce --- X' p +e'

(22)

% , = O C y - COyC T = 6eCMC T = oZM ' 1 Since C- 1 exists, we have M pl- = ( C"1" ) - 1M - C - . 1 Ifthe primed quantities are inserted In^Eqs. 2^1, it is easily seen that ~, ~ , and o 2 remain unchanged, while e = Ce and O~, = CO~C T are correctly transformed. The relations between the observations, Y' = Y'(Y), need not necessarily be linear, provided they can be sufficiently well .

.

.

.

ix

.

.

Ap

A

76

HEINZ DIETER RUDOLPH

linearized (at y,(0) = y,(y(0))) for the linear relation y' = Cy to hold between the increments y' and y. It may happen that a set of variables B~-B~(..,Bk,..), j - 1,m [i.e., B ' = B'(B)], exists, different from, but equivalent to, the set of variables Bj that have been determined by fitting a particular set of observations. We assume that the relation between the two sets of variables can be sufficiently well linearized at B '~~ = B'(B ~~ to admit the linear relation between the increments, p' = Cp. The matrix of the partial derivatives of the observations with respect to the new variables B' is then X' = XC -1. The error equations (Eq. 20), A

y = X' ~' + ~; = XC-1CI~ + ~; = X[I + ~;, O~ = Oy = 62M

(23)

have the solutions (cf. Eq. 21), l]'= (X'TM-1X')-Ix'TM-ly- c ( x T M - 1 X ) - I x T M - l y - CO A {~,A2 A

A (y2(X'TM-IXp) A

A -1 = ( y 2 c ( X T M - 1 X ) - I c T

--

A C~C T,

(24a)

(24b)

while (~, ~;, and O~ remain unchanged. From this it can be seen that fitting an available set of observations y to determine a new (equivalent) set of variables I]' = CI], is an operation that could be replaced by properly transforming the old set of variables ~ and their covafiance matrix. In a practical case where convergence requires several iterations, the new values would be calculated from the old final values, B#-B#(..,B/,..), while ~ , - ~)~,, follows from Eq. 24b. As an example, consider the r0-type fit to determine the independent internal coordinates of a molecule. These variables need not all be true bonding coordinates. Therefore, different, but equivalent sets are often possible. It is obviously not necessary to set up a new least-squares procedure, if the molecular structure, complete with errors and correlations, is desired with reference to a new system of independent internal coordinates. Often one or more additional internal coordinates B#, not contained in the original complete basis of independent variables (..,Bk,..) but depending on them, or differences or sums of such internal coordinates, are wanted. The correct treatment of error propagation then requires the application of Eq. 24b where, in this case, C is a rectangular matrix, the number of rows depending on the number of additional coordinates desired. In the limiting case of one additional variable (e.g., the difference between two bond lengths), C has a single row. Simplified "error propagation formulae" may lead astray if they do not take proper account of the covariances. In situations, where the observables Yi d e p e n d on p a r a m e t e r s , R(x) "-9 ' j = m + 1,m + mtx), that have been kept fixed, while others, Bj, j - 1,m, were determined by the least-squares fit,

Microwave Rotational Spectroscopy

77

y = y(B;B(~)),

(25)

one often needs to estimate the additional errors caused by the fact that the fixed parameters themselves may be known only within (estimated) error limits. Let C denote the n x m matrix relating the least-squares solution 1~ and the observations y in Eq. 21a, C = (xTM-1X)-IxTM -1,

AB = CAY

(26)

where AB has been used instead of 1~ and AY instead of y in Eq. 21 a to avoid the impression that the following discussion is actually part of the least-squares procedure. Let, analogous to the Jacobian X, the matrix X (x) be composed of the derivatives of the observations Yi with respect to the parameters BJX):

OY i ~})

=

t~gBj:r))(..~..~3)..~ \ }

~, 1,n,j 1,m :m+l,m+m(X)

(27)

9~} R(x) ..), are given by the experiment and do The observations, Yi = AYi = Yi - Yi(" ,-9..=-9, not change when the effect of changes of the "fixed" parameters on the result of the fit is being investigated, hence: AY = X AB + X(X)AB(x) = 0

(28)

Note that CX = 1 from Eq. 26. After left multiplication of Eq. 28 by the non-vanishing matrix C it is seen that the change AB of the parameters determined by the fit, caused alone by a potential change AB (x) of parameters B (x) kept fixed in the fit, is, AB = - C X (x) AB (x)

(29)

when least-squares conditions are to be maintained. The additional covariance matrix of the result B, alone due to finite errors of the fixed parameters, represented by the covariance matrix OB(X),is hence: OB = CX(X)OB,x)X(x)rcT

(30)

In practical cases, it will probably be difficult to estimate the covariances within the OB~X)and even more difficult to estimate any correlations between the parameters and B (x). If the latter are neglected, the covariance matrix of the parameters determined by the fit, which includes also the errors due to the "fixed" parameters, will then be the sum of Eqs. 21 b and 30: O B -- O~ + CX(X)OB(x)X(x)r C r

(31 )

78

HEINZ DIETERRUDOLPH Iil.

S U B S T I T U T I O N rs-STRUCTURE; rs-FIT STRUCTURE

The basic equations of the rs-method will be presented later within the framework of the more general "rs-fit" problem. A rigid mass point model, which is strictly true only for the equilibrium configuration, is assumed. The application of Kraitchman's equations (see below) to localize an atomic position requires: (1) the principal planar moments (or equivalent inertial parameters) of the parent or reference molecule with known total mass, and (2) the principal planar moments of the isotopomer in which this one atom has been isotopically substituted (with known mass difference). The equations give the squared Cartesian coordinates of the substituted atom in the PAS of the parent. After extracting the root, the correct relative sign of a coordinate usually follows from inspection or from other considerations. The number, identity, and positions of nonsubstituted atoms do not enter the problem at all. To determine a complete molecular structure, each (non-equivalent) atomic position must have been substituted separately at least once, the MRR spectra of the respective isotopomers must all have been evaluated, and as many separate applications of Kraitchman's equations must be carried out. If the planar moments P ge for the equilibrium configuration were available, Kraitchman's equation would give the true re-Structure. When the ground state m o m e n t s P g0 are used in Kraitchman's equations, the structure obtained is called the rs-Structure. The rs-Structure is expected to approximate the elusive re-Structure better than the r0-structure does. For the bond lengths r of linear molecules, rs--- ( r e + r0)/2 has been demonstrated, with the ordering r e < r~ < r o when bond stretching is dominant and r 0 < r s < r e when bending is more important [5]. For several small molecules with a larger than minimum number of isotopomers available for the determination of the structure, the r~-structure has been shown to be less dependent on the particular selection of isotopomers than the r0-structure [1]. Although the Pg0 are contaminated by rovibrational contributions, part of these contributions is compensated by the rs-method. From Kraitchman's equations, the square of a coordinate is found to be roughly proportional to the isotopic differences of corresponding moments, P ~ ( s ) - P~ where s is the current isotopomer and 1 the parent, while the moments themselves play a lesser role. The equations are hence dominated by the isotopic differences of the moments, and the unknown rovibrational contributions to the experimental moments cancel to the extent to which they are equal for parent and isotopomer. It is essentially for this reason, that the substitution or r~-method has to date been the preferred method for accurate molecular structure determination from MRR spectra. However, there is also a drawback. Small coordinates (< 15 pm) cannot be reliably determined because a small coordinate with its even smaller square is the result of a very small inertial moment difference which, due the experimental errors of the moments and the less than perfect compensation of the rovib contributions, may no longer be significant. The r~-method does not use the first or second moment equations (which would hold strictly only for the equilibrium configuration). The

Microwave Rotational Spectroscopy

79

nontrivial moment equations can hence serve as an approximate check of the quality achieved. If the nontrivial relations are assumed to be valid, they can occasionally be used to calculate a single (or at most a few) accidentally small and hence indeterminate, atomic coordinate(s) from the rs-coordinates of the remaining atoms, provided the latter are sufficient in number and large enough for accurate determination. It means, of course, that the coordinate(s) so obtained must then make up all the inherent differences between the respective moment equations written in r sand in re-COordinates. While Kraitchman's original equations require the moments of the parent and a singly substituted isotopomer, Chutjian [42] has extended the application to the case of an isotopomer which is multiply substituted in symmetrically equivalent positions. Nygaard [43] has given simplified disubstitution formulae for different types of symmetry. Arguments have been advanced for preferring Chutjian's equations also for the single substitution on a plane or axis of symmetry [44]. One of the first computer programs written to obtain the Cartesian atomic coordinates referred to the PAS of the parent by means of a least-squares fit to the inertial or planar moments of a number of isotopomers (also multiply substituted) appears to have been the program STRFIT coded by Schwendeman [6]. It is a versatile r0-type program incorporating many useful features, it is not a "rs-fit" program in the sense in which this term is used in this paper. Typke has introduced the rs-fit method [7] where Kraitchman's basic principles are retained. A system of equations is set up for all available isotopomers of a parent (not necessarily singly substituted) and is solved by least-squares methods for the Cartesian coordinates (referred to the PAS of the parent) of all atomic positions that have been substituted on at least one of the isotopomers The positions of unsubstituted atoms need not be known and cannot be determined. The method is presented here with two recent improvements: true derivatives are used for the Jacobian matrix X, and the problem of the observations and their covariances, which is rather elaborate, is fully worked out. The equations are always given for the general asymmetric rotor, noting that simplifications occur in more symmetric situations, e.g. for linear molecules, which could nonetheless be treated within the framework presented. Let s = 2, N s be a set of isotopomers of a parent molecule s = 1, and let o~ = 1, N a, enumerate the atoms in the molecule. (Eventually, only substituted atoms will be relevant.) In the present notation, the sites of the atoms are referred to the PAS of the parent s = 1, and are hence defined by the position vectors r~ J (in the rigid mass point model). Let the mass change upon substitution of atom ~ be Ama(s) for isotopomer s, we then have: m~(s) = m~(1) + Ama(s ) for O; = 1,N a and s = 2,N s

(32)

In a practical case, many of the Ama(s) will vanish (for an atom that has never been substituted, all Area(s) vanish). The planar moment tensor of the parent, with reference to its own PAS (cf. Eq. 11),

80

HEINZ DIETER RUDOLPH

(33)

P[1](1) : E m~(1) r[1]r[llr

is diagonal by definition, the eigenvalues are Pgt~(1), g = x, y, z, and the first and second moment conditions are satisfied due to the principal axis system used. However, the corresponding planar moment tensors of the set of isotopomers, when expressed with respect to the PAS of the parent, will be neither diagonal nor will the respective centers of mass coincide with the origin of the basis system used. Therefore, the general Eq. 6 must be applied. For isotopomer s,

/N /T

ma(s) r[llr[l] r ....

pO](s ) = ~

-~

-~

!

ma(s)r~]

M(s)

ml~(s)r~ll

(34)

and with Eq. (2), remembering that ~ m~(1)r~ 1-0: N

o~

a

pill(s) = ~ ma(1) r~lr[2 Iv

+ E Ama(s) r[1]r[l]r

1 ~ ~' Am,~(s)Am~(s) "E~'f~;~ M(~) 9 05)

a

a

f5

The first term of Eq. 35 is the planar moment tensor of the parent, ptl](1) (Eq. 33). Kraitchman's basic idea was to introduce into this first (diagonal) tensor term of Eq. 35 the three experimental principal planar moments of the parent Pgtq,exp(1) as obtained from the MRR spectrum and treat them as independent experimental information. This is the essential distinguishing feature between any G-type and any r0-type method and all differences between the two types of treatment may be [1] traced back to this fact. The first term of Eq. 35 is now written as Pexp(1). It is then convenient to replace the notation for the planar tensor Pt~](s) (Eq. 35) by ll[l](s) to distinguish this new function (Eq. 36) from Eq. 35. Note that Eq. 36, in contrast to Eq. 35, depends explicitly on the positions of only those atoms that have actually been substituted in the isotopomer s: Na

n[1](s) = P~lx]p(1) + ~_~Am~(s)r~]r [1Iv-

N (t

1 E M(s---)

Na

E

(36) Am~(s)Amf~(s)r~]r~1Iv

Let T(s) be the orthogonal transformation that diagonalizes l-I[1](s) with eigenvalues l-Igisl(s).Kraitchman equates these eigenvalues to the experimental principal planar moments Pgtsl,exp(S) of the isotopomer s:

MicrowaveRotationalSpectroscopy

81

PgH(s) ----)17gH(S)= (TT(s)I-l[1](s)T(s)~t~lgtS]

(37)

If only one atom ct has been substituted in isotopomer s = 2 (Kraitchman's original proposition), the last two terms of Eq. 36 reduce to kt(2)r~ ]ra[1iv, where lu(2) = M(1)Ama(2)/(M (1) + Am~(2)) is the "reduced mass of substitution". Equation 37 is then explicitly,

21

o

o

Py(2) 0 = 0 Pz(2)

IPx(1) + [Lt(2)x~ ]2 TT(2) 9 /

~1,(2)xtl]y[1] ct ct

[1]z~[1] l.t(2)x~

..[l].y,~tl] ~t(2)..,~

..[1]..tl] ~t(2)..,~ ~,~

Py(1) + 0(2)y~ ]

~(2)y~]z%,

(38) 9T(2)

Pz(1) + g(2)Z~ kt.(2~v[l]'[1] ,.,~ .~

[where Pgt~J,exp(S ) has been abbreviated to Ps(S)]. Equation 38 is the basis of Kraitchman's equations [4]. When forming the secular equation for the tensor Hill(2) on the right-hand side of Eq. 38, it is immediately seen that the coefficients of the polynomial will contain only the squares proper of the coordinates [1]-.. y~]-, z~]-, but no mixed squares. The left-hand side of Eq. 38 displays the roots XO~ of the secular equation. Comparing coefficients, Kraitchman obtained his equations (for the general asymmetric rotor):

x[l12_ l tPx(2)-Px(1))(Py(2)-Px(1))(Pz(2)-Px(1))l(x,y,z cycl.) (39) a t.t(2) (Py(1) - ex(l))(P:(1) - Px(1)) usually the P_(s) differ much more for different g than for different s, the

Since right-hand side is indeed approximated by x~ ]2--- ~ (P.(2)- Px(1)), i.e., by an isotopic planar moment difference, as was mention~'bel~ore. Kraitchman did not actually work out the transformation T(2) of Eq. 38, although this matrix as well as the shift rcm, Eq. 2, can be given in closed form with no more knowledge than required for Eq. 39 [45]. For symmetrically equivalent multiple substitution [42,43] and for substitution on a principal plane or axis [44], the matrices 1-I[1](2) and T(2) are block-diagonal with ensuing simplifications for Eqs. 39. The sole advantage of Kraitchman's equations over the G-fit method to be discussed presently is the ease with which the equations can be applied to just a pair of molecular species, s = 1, 2, by a hand calculation as compared to the more elaborate least-squares procedure requiring a coded program. Equations 36 and 37 are the basis for Typke's r~-fit method [7]. By Eq. 36, the eigenvalues Hgt~l(s)of Eq. 37 depend on the positions r- [a1 ] of only those atoms that have actually been substituted on the isotopomer s. In accordance with Kraitchman's ideas, the eigenvalues Hgt~(s) are equated to the experimental planar mo-

82

HEINZ DIETER RUDOLPH

ments of the isotopomer s (see Eq. 40). It is supposed that the atomic coordinates [I] (~ are known to a first approximation, r(~ . The tensor Hill(s) in Eq. 36, calculated with these coordinates, is denoted as IItll(s) (~ The linearization of the three eigenvalues HuH(S ), g = x,y,z, of Eq. 37 then yields for the isotopomer s, Pg,,) e,p(S)~ HgM(S) = HgH(s)(O) + 2 |

Ah~]

(40)

h,cz

where the eigenvalues HgL,)(s)(~ of the t e n s o r n[1](s) (0) are obtained after diagonalization by means of T(s)(~

Hg[q(s)(O) = (,i'(o)r(s)l-l[1](s)(O)T(s)(O)~t.,lg[,1

(41)

The h~], h = x,y~., a = 1, N a in Eq. 40 are the components of the position vectors r~ l (PAS of parent). It is convenient for later use to define a long-vector r Ill (3N., components) which is composed of the individual vectors rtaII, r [11" [1] T [1] T [1] T [1] (r 1 ..... r a ..... r N ) .LetAr be the corresponding long vector of the coordinate increments. By the Hellmann-Feynman theorem [46], the derivative of an eigenvalue Hgl,l(s) of a hermitian (here, real symmetric) matrix II[l](s) with respect to a parameter h~ l, is given by the diagonal element of the matrix product, a

Ohm]

011[1](S)'T(s)

-

3h~ ]

/,

(42)

[Slg[Sl

where the transformation T must be chosen to diagonalize the matrix II[l](s), II[Sl(s) = TT(s)IItll(s)T(s). Differentiating Eq. 36, we have,

0h~]

= Area(s) ~ 13

a~

M(s)

~."h "[3 +

, (z - l 'Na

(43)

where 8c~13is Kronecker's delta and @1 is the unit vector in h-direction in the PAS of the parent, @ IT = (Sh#IU, 8h.ym, 8h,:i,0. Noting that the general tensor element is,

+

f"

=Sj,if[~ +ff~Shf, '

f,f,=x[1],y[1],Z[1]

(44)

a closed formula, Eq. 45, can be given for the derivative, Eq. 42. The superscript (0) has been attached where required by Eq. 40,

Ohm]

-2Ama(s ) T,/l,gH(S)Z~

~-

Microwave Rotational Spectroscopy

83

with g,h = x,y,z; s = 2, Ns; o~ = 1, N a

(45)

The expression is easily coded, since T (~ Eq. 41 and the r Ill~ are known. It simplifies for substitution on a principal plane or axis and for symmetrically equivalent multiple substitution, because several of the elements of the matrix T will then vanish. It is clear from Eq. 45 that the derivatives are nonvanishing only for those atoms cz that have actually been substitued in the particular isotopomer s. Therefore, the Jacobian matrix X generated from these derivatives is, in general, a sparse matrix. If the SDS consists of singly substituted isotopomers only, the rs-fit method is, in principle, insensitive to the sign of the initial approximation r~14. for the coordinates, just as Kraitchman's equations are. The final coordinates r~ ] simply accept the sign of the initial input quantities. The reversal of the sign of one or more of the input coordinates has no implications on the result or the quality of the fit. If, in addition to the single substitutions, groups of two or more atoms are "bound together" by multiple substitution in any one isotopomer of the substitution set, the statement is still true as long as the sign (of a particular coordinate component, x, y, z) is reversed for all atoms of the respective group. The fit is otherwise unaffected. If the signs of the coordinates in the group are reversed in a disconcerted fashion, the rs-fit may nonetheless converge, though to a local minimum and with a much larger standard deviation. Again, the atomic positions outside the group remain unaffected. However, all coordinates of the atoms in the group, not only those of a particular component, are then incorrect. Therefore, if the sign relation of a coordinate component, x, y, or z, of two atoms in a molecule is in doubt, caution is in order. Convergence is no guarantee. Multiple substitution helps to assess the correct relation. For a least-squares solution of the system of Eqs. 40 for all s = 2, N s, we have to identify the components of the vector of observations y, the components of the vector of variables [I and the elements of the Jacobian matrix X as shown below (Eqs. 46-48). A left arrow has been used instead of a sign of equation to indicate that, in general, the dimensions of [I, X, and y are preliminaryand must be reduced before least-squares processing can take place: some of the [~hma- Ah~ ] may not be independent because symmetrically equivalent atoms have been substituted. Other coordinates may be kept fixed intentionally (e.g., at zero when an atom is known to lie on a principal plane or axis). The respective component(s) Ah~ l must then be eliminated from the vector of variables. Also, one or more of the observations y"gM,s may have to be dropped in order to comply with the recommendations given for the Chutjian-type treatment of substitutions on a principal plane or axis [44], Yi ~-"

Ygt~"s = eg'sl,exp(S) - I-Igl4(S)(O)' g = x,y,z, s = 2, N~

[~j ~ [~h[1]or

=

Ah.~ 1,

h - x,y,z, ~ = 1, N a

(46) (47)

84

HEINZ DIETER RUDOLPH

With the long-vector Ar [11 introduced earlier, we have ~ = Artll. While in Eq. 20 the covariance matrix | of the observations, y = Yexp- y(0), is simply that of Yexp because y(0) is a constant nonrandom vector, the I]gvJ(s)(~ of Eq. 46 must be treated as random quantities because they go back, via Eqs. 41,37, and 36, to the three experimental moments of the parent, Pg~,exp(l ). The construction of the covariance matrix of the observations (Eq. 46), is hence more elaborate. The required matrix O r can be obtained from the available information by the following arguments. For the sake of a concise notation, let, for each isotopomer s = 1, N s, the three planar moments Pg~,Lexp(S) be the components of a 3 • 1 vector p(s):

pT(s) = (Pxlsl,exp(S), py[sl,exp(S),pzl4,exp(S))

(49)

The 3 • 3 covariance matrices Or ) for each isotopomer are known from the evaluation of the spectra, they will usually have been calculated via the covariance matrices of the respective inertial moments by the application of the transformation given in Eq. (15b). (Note that Op(s) will be non-diagonal even if the inertial moments or rotational constants have been assumed uncorrelated.) Let a long vector p (no argument) be composed of the p(s), omitting the parent p(1 ), pT = (pT(2). . . . . pT(s) . . . . . pT(Ns)) and let the corresponding notation for the quantities

(50)

Hgf~(s) (~ s = 2, N s, of Eq. 46

be: nr(s) = ( Hxt~(s)(~ ,H?~,(s)(~

(~

(49a)

~T = (rtT(2) . . . . . ~T(s ) . . . . . ~T(Ns) )

(50a)

The vector of observations, with the components of Eq. 46, is then y = p - n. Provided the rotational parameters (rotational constants or moments) of the isotopomer s, as evaluated from the spectrum, are not correlated with those of any other isotopomer s', in particular not with those of the parent (which is a general supposition in MRR-spectroscopy), the required covariance matrix O r is: Oy - Op_,~= Op + O,~

(51)

This can be seen from the definition (Eq. 17) because mixed p, rt-covariance terms will then vanish. Op is a block-diagonal matrix with blocks Op~s).Note that Oy does not consist of Op alone; this would only be true if the vector n were (erroneously) taken to be a constant (nonrandom) approximation (comparable to y0; see Eq. 18).

Microwave Rotational Spectroscopy

85

In contrast, the vector n is, by Eqs. 41, 37, and 36, a function of the random quantity p(1). If the transformation between the increments of p(l) and those of n is denoted as U, the required covariance matrix O,~ is found by transforming O~1 ) by the application of U, n = n(p(1)), An = UAp(1), O n = UOr

T

(52)

U is a "high" matrix with row numbering g = x,y,z, s = 2, N s, and with only three columns for x [1], y[l], z[l]. The general element of U is the derivative of any component of the vector n with respect to any of the three components of the vector p(1) and is again obtained by the Hellmann-Feynman theorem, cf. Eq. 42: (0)

--/~..cxp.()//g)(0)ISl(S)(,.I.(0)T(s). t ~~le]x(p~!l)/.T(0)(s)/)glslg(53) lsl The right-hand derivative and the resulting Ug[~,],hmare very simple. From Eq. 36:

~l'I[1](S) = diag(Sxh, 6~.h, ~-h)' ,. C)Ph[1],exp(1)

Ug[s],h[~J /J0) LI h[llg[sl),~2 =

(54)

Note that U and, therefore, O n = UOr T (Eq. 52) is a dense matrix. But O n is clearly structured by the different magnitude of its elements due to the special form of U discussed below. By Eq. 51, Oy is also a dense matrix. We have found Or to be positive-definite in all practical applications. With few exceptions, the diagonalizing orthogonal matrix T is dominated by the diagonal elements and approximates a 3 • 3 unit matrix because the angle required to rotate the PAS of the parent into that of the isotopomer is usually small: Ug~'Lh[~j= ( [~Jg[']) --" ~hg" The matrix U is then similar to a stack of unit matrices. This w o u l d be strictly true if, instead of the quantities of Eq. 46, Pg[sl,exp(s) --/-/gl~l(S) (0), the characteristic quantities of a r0-type "pseudo-Kraitchman" fit, P#,)exp(S) - Pgm exp (1), were fitted (see next section and also [47]). To avoid confusion let us recapitulate that the vector of observations, y = p - n (Eq. 46) has the elements Pg[s],exp(S) -//g[4(S) (0) (or Pgl4,exp(S ) - Pg[sl,calc(S)(O)if we revert, for the moment, from the/-/- back to the P-notation). In seeming contrast, the covariance matrix Or of the vector y very nearly equals that of a vector with elements Pgtq,exp(S ) -Pg[,l,exp(1). This is due to the "hybrid" character of the tensor II[l](s) (Eq. 36) which is composed of experimental data and nonrandom quantities. Experience has shown that the true covariance matrix,

Oy -- Op

+UOp(1)U r

(55)

is sufficiently different from Op to make also a significant difference in the result of the subsequent least-squares fit.

86

HEINZ DIETER RUDOLPH

The usual nearness ofT to a unit matrix has another important consequence which was mentioned before. From Eq. (36) it is seen that the tensor II[l](s) contains on the diagonal the experimental ground state planar moments of the parent, part of which are the rovib contributions. If the transformation matrix T (Eq. 37) is near enough to the unit matrix, the eigenvalues of Eq. 36 will then contain these rovib contributions with little modification. These eigenvalues are equated to the experimental ground state planar moments of the isotopomer s (Eq. 37), which also include their rovib contributions. That means that the rovib contributions will cancel to the extent to which they are equal for parent and isotopomer. This is the essential advantage which is claimed for the G-method. However, in exceptional situations (e.g., substitution in nearly symmetric rotors), where the gross rotation by T can be large and T then no longer resembles a unit matrix, the rovib compensation may even be reversed and turn into a magnification of the errors

[14]. The dimensions of ~ - A1~11and 'X must be reduced when symmetrically equivalent, and hence dependent, atomic positions have been substituted. A way which is convenient for the demonstration and discussion (though not necessarily for coding a program) can best be illustrated by a simple example. Assume that two positions, ct' and ~", exist which are symmetrically equivalent with respect to a base atom t~; one such position, y ' , with respect to atom y, and none for atom ~.. All positions, including the dependent ones, could be generated from the independent ones, ct, y, ~,, by the transformation matrix D, here shown as a table (Eq. 56).

Y

y'

y

k

1

0

0

0

1

0

0

0

1

C(c~')

0

0

C(~")

0

0

0

C(~,')

0

Likewise the coordinate increments [I, including the dependent ones, are obtained from the independent variables II of the last-squares procedure by Eq. 56a: Ar~1] - ~ - Dp

(56a)

The entries in the table are 3 • 3 matrices (many of them zero or unit matrices). For example, a threefold-symmetric equivalence of the positions ct, t~', and t~" with respect to the x[ll-axis, and a reflection symmetry of the positions y and y' with respect to the x [11, ztll-plane would be taken care of by the orthogonal matrices C(t~'), C(t~"), and C(y') in D of Eq. 57,

Microwave Rotational Spectroscopy

87

r

1 C(a') =

0

0

- - ~

2

~

2

, C(a")

2

,gO

0

0 =

D ~

,/ 0 !/

(57)

0

1

w

,5-

C(7') =

.....

2

2

2

2

2

,

0

1 ~

-~

.J

The increment of a coordinate that is kept fixed, e.g. fixed at zero for the substitution on a principal plane or axis, must not be a component of the vector p of variables. The required elimination is most easily implemented by letting the matrix D of Eq. 56a be followed by a matrix E, which consists of a unit matrix where only the columns (not the rows) corresponding to any components of 13to be dropped have been eliminated: Artl] = ~ - DEll

(56b)

Both, D and E, are "high" matrices, the number of rows exceeding the number of columns. If, in the above example, the base atom a of the threefold symmetric group is assumed to lie on the xtl], ytll_plane ' za[1]..is fixed at zero and Az~ 1 vanishes. D would be a 18 • 9 matrix, E a 9 • 8 matrix, 1~a 18 • 1 vector, and p a 8 • 1 vector. (Schemes for taking account of symmetries and eliminations other than using matrices D and E are, of course, possible.) A closer inspection of the derivatives given by Eq. 45 shows that the elements of the entire row of the Jacobian matrix X, which corresponds to the principal axis gtS] of an isotopomer s, will vanish if this isotopomer has been substituted (singly or multiply) only on that principal plane to which gtS] is perpendicular. This fact justifies the elimination of the observation Pg,4,exp(S ) - Hgl4(s) (0) which corresponds to this row from the least-squares fit. It is related with the symmetry arguments provided in [44] for excluding the corresponding experimental planar moment, Pg~,~,exp(S), from a problem involving a zero-valued coordinate. In these cases the respective Kraitchman equation would often yield a negative square instead of a perfect zero due to the imperfections of the experimental planar moments. As reported before [7], the rs-fit method then fails to converge and tends to "oscillate" between small positive and negative values for the zero-valued coordinate. Let us hence assume that any offending observation, Pgt4,exp(S ) Hg[sl(S)(0), and the corresponding empty (zero-filled) row of X have been properly eliminated, that means the observations y have been reduced to y for further use. This is particularly important for linear molecules where there is only one observation per isotopomer, where the sum over the axis directions reduces to (say) y only, and the matrix T degenerates to the scalar 1. Also, this case can be handled correctly by means of the above matrix E and the proper elimination of components from y and rows from X. It may happen, that an inertial moment of one of the isotopomers could not be evaluated from the MRR-spectrum and must be left out of the fit. However, in the -

-

88

HEINZ DIETER RUDOLPH

above discussion the least-squares system has been written for the planar, not the inertial moments. Since the result of the least-squares fit is independent of any nonsingular transformation of the observations, provided the covariance matrix of the observations is correspondingly transformed and the Jacobian modified (see Eq. 22), one can transform the three components ofy and the three rows of X, which belong to this particular isotopomer s, by the application of Eq. 15a from planar moment components into inertial moment components without implications for the results ~ and ~li (~ and ~ must be retransformed). After this is done, the unknown inertial moment component of y and the corresponding row of X can be eliminated (which will then modify the result). Of course, this transformation from planar to inertial moments could be carried out for all isotopomers if desired. The model equations of the least-squares treatment (cf. Eq. 19), can now be expressed as: y = Xp = XDEI~ = X[I

with

XDE = X

(58)

With the Jacobian X prepared as indicated and the vector of variables I~ limited to the independent coordinates, X has maximum rank and the problem can be solved by the iterated least-squares treatment. After each iteration step, I~ should be expanded to obtain I1 by means of Eq. 56b for the correction of the independent and the dependent coordinates. Due to the presence of E in Eq. 58, 13has the required zero component wherever a coordinate has to be kept fixed and must not be changed. The corrected coordinates are required to recalculate ~ and X (the quantities HgH(S)(~ and (3Hg~s~(s)/3h~])(~ for the next iteration step. In contrast to most applications of the least-squares procedure, the covariance matrix of the (effective) observations, O~ (Eq. 55) must also be recalculated because O~ depends on U which changes (though probably very little) with each step (Eq. 53). After the iterations have converged, the matrix product DE (Eq. 56b) can be put to further good use. Noke that the product is a constant, nonrandom matrix. The final covariance matrix O~ of the variables ~ can hence be expanded to obtain ~ : A

A

O~p= DEO~ETDr

(59)

As mentioned above, this is the covariance matrix not only of the coordinate increments of the last iteration but also of the final coordinates as determined by the fit, ~r~U= ~Artll = ~ . By means of ~rtl;, the errors and correlations of the independent and dependen~ coordi2aates can be assessed. Because of the presence of E in Eq. 59, the matrix OArtll- Or[ll will contain an empty (zero-filled) row and column where a coordinate has been kept fixed. It is easy to account for what has been called "Costain's errors" [1,5] by adding to each variance o2(h~ 1) on the diagonal of ~)Ar[ll an additional variance term (Ah~l) 2. Costain has estimated the uncertainty of a rs-COordinate with absolute magnitude lhl (in pm) as Ah = +15/Ihl pm (when the reduced mass of substitution is assumed as 1). Errors so introduced would be correctly propagated to any further processing of ~zxr~l~. There is still another, perhaps more fundamental way, to

Microwave Rotational Spectroscopy

89

include Costain's error by manipulating the covafiance matrix Oy of the observations instead of that of the results, ~6rf~. Costain s error estimate can be derived from the experimental findings [6] that the isotopic difference of the pseudo-inertial defect (Eq. 15c) which should vanish (if equilibrium moments were used) for a substitution on, say, the xy-plane, scatters by approximately 8[P_(s)- Pz(1) ] = 2 8 [ Az(s) - Az(1) [ - _+0.003 u/k2. This scattering range, if accepted as an estimate for the inherent planar moment uncertainty for any axis g, can be easily introduced into the rs-fit method. From the discussion above, we recall that the covariance matrix of the observations, Or, is very nearly that of the vector of the isotopic planar moment differences, P~, 0 Pg.exp(1). If the square of the above estimate, [8(Pg(s) - Pg(1))]2, is added to every diagonal element of O r, and the covariance matrix | is not merely used for weighting the observations, but is truly propagated fixing ~2 at unity), the covariance matrix of the resulting coordinates, = ~zxrI~- ~ , will then include the effect of Costain's estimate. (As a side effect, the additional variances added on the diagonal of Or will serve as "adjustment variances", as discussed in the next section, because they will often be larger than the original experimental variances.) There is another important reason why the expanded form of the covariance matrix, ~rr~l = ~i~, is useful. Working through an rs-fit as described here, with a sufficient number of isotopomers, will yield a complete molecular structure with consistent error estimates and correlations. This is different from the results obtained after separately solving a number of distinct Kraitchman equations, but quite similar to the output after finding the r 0- (or a r0-derived) structure based on the same input information. Therefore, it would be highly desirable to compare the rs-fit and the r0-structures, each obtained by a compact but different operational procedure. That means, the information gained by the rs-fit, the Cartesian position vector r [11, composed of the vectors tI~1, should also be available in the form of internal coordinates (variousbond distances, bond angles, and dihedral angles), complete with errors and correlations for an easy comparison with the r0-data. Internal coordinates and their errors and correlations are also indispensable for the truly chemical assessment of the structure obtained. For the correct error propagation, the expanded covariance matrix, ~,j~J = 6a,.t,1 = ~ , is required. Any number of desired internal coordinates St (t = 1, N,) can be calculated from the final Cartesian position vectors ..[tl , a , most conveniently by known vector formulae. For the transformation of the covariance matrix of the Cartesian coordinates into that of the internal coordinates, the derivatives of any particular internal coordinate are required with respect to all Cartesian coordinates that actually participate in the motion of this internal coordinate:

6rl(bY

~h~] = Bt,ha This problem has already been solved by the early infrared spectroscopists who calculated the transformation between the atomic Cartesian displacement coordi-

90

HEINZ DIETER RUDOLPH

nates and the internal displacement coordinates [48-50]. The array of derivatives has become known as the B-matrix (not to be confused with B as used earlier (Eq. 16a etc.). Each row of B for a particular internal coordinate S t is composed of Wilson's s/a-vectors for the participating atomic positions ~. The increments of the Cartesian coordinates (vector Ar [1] = I~) are transformed by the matrix B into the increments of the internal coordinates (vector AS), and the covariance matrix ~}rt~l= ~}Arm = ~ of the Cartesian coordinates into that of the internal coordinates, AS = B Ar [11,

A

O~ = BOrt~JB "r

(61)

As mentioned above, the expanded covariance matrix ~ r m of the results contains a zero-filled row and column when a Cartesian coordinate h~ 1has been kept fixed. A judiciously chosen variance o2(h~ l) can be entered on the respective diagonal of ~ r m, prior to the transformation (Eq. 61 ) if the fixed coordinate is afflicted with an (estimated) error and the propagation of this error to any of the derived internal coordinates is to be studied. For this reason it may even be practical to carry along in the expanded vector of variables p an atom that has never been substituted (and will hence drop out of the fit by the application of E), but whose position can be estimated and is required for the calculation of certain bond lengths and angles involving atoms that were substituted. While the basic assumption of identical geometry for parent species and isotopomers is generally a good one, deuterated X-H bonds make an exception because the relative mass change is so large and the X-D bond so much shorter that noticeable errors would occur in any of the presented methods if the bond shortening upon deuteration remained uncorrected. The required corrections have become known as "Laurie's corrections" [6, 22]. Suppose that the X-D bond is shorter than the X-H bond by 8r. The uncorrected rs-method would then let the X-H bond appear too short by --- 28r [51]. After the rs-coordinates r~ l for atom X and preliminary rs-coordinates for the apparent location r~ l of H, have been obtained from the uncorrected planar moments, a hypothetical position H' of the hydrogen atom after an intentional corrective X-H' bond elongation by 25r (usually chosen near 2 • 0.3 pm) is assumed as,

r[1] = r[~] + 2~ir o[l] H' "XH"

~[11

"XH'

=

/

'~H'

(62)

where ",~[1] is the unit vector in the X-H' bond direction. The coordinates, first of XH' [1] are inserted into the right-hand side of Eq. 38, and the -H'[11, then of position rH,, respective tensors are diagonalized. The differences of the respective left-hand sides (eigenvalues) will then give the three required correction terms to be applied to the experimental planar moments of the deuterated isotopomer. The final rs-fit should now give the correct H position. The corrections can be computed by a small separate program and could be immediately converted into corrections for the inertial moments by Eq. 15a, or for the rotational constants. The method described requires only the experimental planar moments of the parent, the X-substituted, and

Microwave Rotational Spectroscopy

91

the deuterated isotopomer. A more elaborate treatment within the framework of the rs-fit method, which can take account of changes of all bonds lengths in the molecule, has been given by Typke [51]. Simplified approximation formulae for Laurie's corrections have been reported also by Berry and Harmony [30]. As is to be expected, the results of the rs-fit method are identical with those of Kraitchman's original equations when a problem with one isotopomer and one substituted atom is treated; the rs-fit method simply offers a mathematically equivalent solution of the same secular problem. This is also true when ChutjianNygaard type symmetrically equivalent multiple substitutions are present. The results of a "substitution sequence" (e.g., AB --) A*B ~ A'B*), as suggested earlier [45], also turn out to be identical with those of the corresponding rs-fit problem. The merits of the rs-fit method over the repeated application of Kraitchman's equations are: (1) the better least-squares balancing of the errors, which can be expected when a large number of (redundant) isotopomers is included in the investigation, and; (2) the greater flexibility regarding nonequivalent multiple substitutions. Note, however, that the method cannot locate a pair of nonequivalent atomic positions that appear either both unsubstituted or both substituted in the same way in all isotopomers present. The rs-fit method has retained several of the essential features of Kraitchman's original substitution method: the (partial) compensation of the unknown rovib contributions of the rotational parameters, but also the lower limit for a coordinate (approx. 12 to 15 pm [5]) for the determination with still significant accuracy, and the fact that for complete structures the nontrivial first and second moment conditions are not necessarily satisfied. Using the rs-fit method with a substantial body of information (i.e., with a significantly larger than minimum number of singly and multiply substituted isotopomers) generally alleviates the limitations. It may improve the accuracy of small non-zero coordinates, and also tends to average out gross deviations from the moment conditions, as has been suggested by the (still limited) experience with the rs-fit method and its recent developments presented here ([52, 53] and unpublished results). For the problems investigated, the convergence properties of the iterative least squares treatment were good when not impaired by the presence of too small a coordinate. The condition number (the ratio of the largest to the smallest eigenvalue of the symmetric, positive-definite matrix of the normal equations) was better by orders of magnitude than for r0-derived fits with the same input (see below). Excepting small coordinates, the accuracy of the coordinates determined was comparable with that of the (r0-derived), rl~-fit. However, the correlation between the coordinates was much less, probably a consequence of the sparse occupation of the Jacobian X (although this property may be considerably modified by the multiplication with the inner factor M -1 and the subsequent inversion; see Eq. 2 lb). In the cases studied, the values of the atomic coordinates determined by the competing methods, rs-fit and r1~,differed by perhaps <1r to 5cy.

92

HEINZ DIETER RUDOLPH

The first and second moment conditions can be very easily introduced into the

G-fit method as least-squares constraints [7,54] if the number of isotopomers is sufficient for a complete rs-structure. The effect on the coordinates is not expected to be particularly unbalanced unless the moment conditions are required for the sole purpose of locating atoms that could not be substituted (e.g., fluorine or phosphorus) or that have a near-zero coordinate. While all coordinates may change, the small coordinates will, of course, change more. In the cases tested, the coordinate values of the rs-fit with constraints and those of the corresponding ri~-fit (not of the r0-fit), including errors and correlations, differed by only a small fraction of the respective errors, i.e., much less than reported above. This was true under the provision that all atoms could be substituted and that the planar moments that were excluded from the rs-fit because of substitution on a principal plane or axis, were also omitted from the rt~-fit. With these modifications, the basic physical considerations and the input data are the same in both cases, and the results should be identical in the limit where the number of observations equals that of the variables. Of course, the practical identity of the two structures from different origin does not necessarily permit a well-founded statement regarding the nearness of this structure to the physically meaningful equilibrium structure. In this respect, one should assess the Costain errors for small coordinates; and for the larger ones (if the Costain errors turn out to be unreasonably low) continue to rely on the estimates for the generally agreext upon accuracy of a well-done substitution structure, perhaps between 0.1 and 1 pm for Cartesian coordinates or bond lengths and between 0.01 and 0.1 ~ for bond angles. A computer program coded along the lines presented above is well suited to replace repeated separate calculations using Kraitchman's equations (Eq. 39). If desired, the program can do just what the latter equations do, but it could do a good deal more if the situation permits it. Take a simple case where, e.g., the parent molecule and two isotopomers with different single substitutions are available. Two separate Kraitchman calculations would have to be carried out to find the two atomic positions involved. However, only a program including all three molecular species would show the correlations between the coordinates of the two positions which are due to the same parent molecule for both substitutions. The correlation will usually influence the error of the distance between the two positions. The sole exception precluding the application of a program like the one described here, would be special problems when performing r m- or r.~ -calculations where explicit use is made also of negative squares obtained by Kraitchman's equations for zeroor near-zero valued coordinates.

IV. EFFECTIVE (to) STRUCTURE; ro-DERIVED STRUCTURE Molecular structures obtained from least-squares fitting of experimental rotational parameters (or isotopic differences of such) to the corresponding quantities calculated from an approximate, but iteratively improved structure, have been called

Microwave Rotational Spectroscopy

93

"effective" or r0-structures. Almost exclusively, the molecular structure is described by using independent internal coordinates. In contrast to the rs-method, it is a constituent feature of all r0-derived methods that the description of the structure must include all atomic positions. The "parent part" of the planar moment tensor of an isotopomer is not replaced by the experimental moments of the parent (cf. Eqs. 35 and 36) and, as a consequence, all atomic positions are explicitly present in the tensor of Eq. 6 and must hence either be determined by the fit or must be supplied and kept fixed. Since the early investigations of Costain, the rs-structure has been preferred over the r0-structure because of its better consistency and because small partial structures can be determined from a very limited SDS, neglecting the rest of the molecule. However, the limitation of Kraitchman's equations to a substitution set of singly substituted (or symmetrically disubstituted) isotopomers made it desirable to have a method which could also include isotopomers that do not satisfy these restrictive requirements, and which could also handle the problem of small coordinates in a more balanced fashion than the rs-method. The rs-fit method, described in the preceding chapter, offers greater flexibility than Kraitchman's equations, but can, in principle, not locate unsubstituted atoms. Calculating an effective or r0-structure introduced, for the first time, least-squares fitting into the determination of molecular structure from MRR spectra. Unpublished computer programs for this purpose must have been in use by various groups for a long time. A more widely used program with many options, STRFTQ, has been coded by Schwendeman [6]. It uses internal coordinates and fits inertial or planar moments, but also isotopic differences of moments. When the differences Pg,exp(S) Pg,exp(1) are fitted, the structure is called a "pseudo-Kraitchman" or p-Kr structure, because these differences are the quantities that also dominate Kraitchman's equations. Therefore, the p-Kr structure should approach the rs-Structure. By the criterion adhered to in the present paper (see preceding section), the p-Kr structure is a r0-type structure. N6sberger et al. have described their r0-derived program GEOM [39] which used the possibilities of the singular value decomposition algorithm for the first time. The applications appear to have been based mostly on a p-Kr type fitting of isotopic differences. The authors point out the predictive capabilities which permit a stepwise progress leading from a situation with a small number of isotopomers measured and a correspondingly larger number of (good) guesses of structural details, via the prediction of new rotational constants and the search for the spectra of the respective isotopomers to a complete SDS and a final reliable structure. In a recent paper [55], the multitude of possible r0-fitting schemes have been ordered under systematic aspects. Any of the three major types of rotational parameters, principal inertial and planar moments, and rotational constants, or isotopic differences of these quantities between differently chosen members of the available substitution set, could be r0-fitted. The basic experimental information evaluated from the MRR-spectrum of any molecular species will usually consist of -

-

94

HEINZ DIETER RUDOLPH

the three ground state rotational constants or inertial moments and their 3 • 3 covariance matrix which depends on the number and selection of the transitions measured and the precision of the spectrometer. In the least-squares process, the covariance matrices act as (reciprocal) generalized weight matrices of the respective observations. In all practical cases tested, the r0-fit to this set of raw data for the available number of isotopomers,Awith the usual experimental accuracy assumed, yielded a standard deviation 6 of the fit far in excess of unity, a clear indication that systematic errors were presentNno doubt due to rovib contributions that were not explicitly included in the model. In this situation it would be meaningless to differentiate between a precise measurement and a still more precise measurement when both are better than the useful limit of precision. The possibly very different experimental weights of isotopomers that have been measured under different experimental conditions (e.g., the parent and a less abundant isotopomer) should then be equalized by deliberately adding on the diagonal of the individual covariance matrices for all isotopomers the same "adjustment variances" (possibly different values for g = x,y,z) so as to reduce the standard deviation of the subsequent r0-fit to approximately unity (or any other reasonably chosen small number; see also [41]). This will enhance greatly the numerical dominance of the diagonal in the individual covariance matrices and often practically suppress the correlations. The experimental rotational parameters and their (adjusted) covariance matrices should then be considered as the input information for any type of further data processing. The arguments following Eq. 22 will take care of the fact that the structural results of subsequent r0-fits are identical, no matter whether (equivalent) sets of rotational constants Bg(s), inertial moments Ig(S), or planar m o m e n t s Pg(S) are fitted, provided the covariance matrices of the respective input quantities have been correctly transformed. The relations for the resulting structures are, in symbolic notation: r 0 -r~ = r e = rs; that means, there is only one unique r0-structure. In contrast, corresponding sets of rotational constants, inertial moments, and planar moments, each used in a unity-weighted r0-fit, would all give different molecular structures because the three calculations then do not use equivalent input information (e.g., a unit covariance matrix for rotational constants is not equivalent to a unit covariance matrix for the planar moments). Note also that a diagonal covariance matrix for the lg necessarily leads to a nondiagonal matrix for the Pg and vice versa, the transformation matrix C of Eq. 22 being given by Eqs. 15. While the relations between the inertial and planar moments are strictly linear and constant, the relation between the increments of the rotational c o n s t a n t s Bg and the moments, say lg, is a truncated series expression and only approximately linear, Z~lg = (-f/B~)&Bg. Also, the transformation coefficientsf/B~ are not strictly constant (nonrandom), although usually afflicted with only a very small relative error. The approximations are, in general, good enough to satisfy the requirements for Eq. 22, and for the above statement r I = r e = r 8 to be true for all practical purposes.

Microwave Rotational Spectroscopy

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The simple matrix for the linear transformation of, e.g., the planar moments a SDS into the isotopic differences As'lpg = Pg(S) - Pg(l) is non-square and hence singular because there are less differences than moments. The transformation of the problem is formally described by Eq. 22, but since the transformation matrix C is now singular, the arguments following Eq. 22 are not applicable. Therefore, symbolically: rap 4: rp, where rap is the/p-Kr structure, which is thereby shown to be different, in principle, from the %-structure. The non-singular matrices of Eqs. 15 apply for the linear transformation not only between the Pg and Ig but also between the APg and A/g. Therefore, symbolically, rap = ra/. However, due to the reciprocal relation between the rotational c o n s t a n t s Bg and the moments Ig, there is no linear and constant transformation between the ABg and the A/g; therefore, we have for the three r0-derived structures using isotopic differences: rM ~: rae = rap The isotopic differences need not be taken as differences with respect to the parent, Pg(S) - Pg(1). Any arrangement, Pg(S") - Pg(S'), which can be transformed by a non-singular matrix C into the former set will result in the same rAp-structure. Occasionally, %-type fits have been reported where the isotopic differences Pg(S) - Pg(1 ) for the isotopomers s = 2, Ns, g = x,y,z, plus the three moments of the parent, Pg(1), were fitted, in the hope of compensating for part of the rovib contamination of the moments. However, the simple linear transformation between this combination of observations and the complete set of individual moments Is(l) .... Pg(S),. 9 Pg(N), g= x,y,z, is constant and non-singular, and the result of this type of fit must be the unmodified r0-structure (not the rap-structure ), if the covariance matrices of the observations have been handled properly. Demaison and Nemes [56] have reported a linear regression investigation of the rovib contributions l~g-/~g- lg evaluated from the spectra of a sizable number of diatomic and very small polyatomic molecules, from which the two-parameter relation,

Pg(S) o f

log c - c 1 log/0 + c2

(62)

with different sets of constants for the diatomic and the polyatomic molecules could be established. The relation was not meant to be directly applicable to large molecules. The report prompted the attempt to introduce constant (i.e., isotopomerindependent) eg into the framework of the %-structure as three additional variables to be determined by the fit to the experimental inertial moments [55]. A constant would be the first term in any expansion of Eg as a function of whatever parameter. The true relations, for s = 1, N s,

l~

- leg(s) + eg(S)

(63a)

P~

- Pe~(s) + rig(S)

(63b)

or

for the planar moments, were thus modeled as

96

HEINZ DIETER RUDOLPH

10,exp(S ) ._ Ig,calc(S) pe -t- Eg

(64a)

pOg,exp(S)= Fg,calc(S pe ) -k-TIg

(64b)

or

respectively. The calculated structure, called the "r/z-structure", is expected to reproduce a set of "pseudo-equilibrium" inertial moments ~e(s) instead of ground state moments. As noted before (Eq. 22), proper transformation of the observations and their covariance matrix by Eq. 15b is assumed when Eq. 64b is to replace Eq. 64a. The Eqs. 15 also apply to the transformation between the rovib contributions rig to the planar moments and those to the inertial moments, ~8. Either type of contribution can be transformed after the fit, as shown by the arguments following Eqs. 24. The usefulness of rovib contributions eg which are independent of the particular isotopomer s of a molecule could be questioned. The expansion of the true e~g(S)for the ground vibrational state is (to first order) [8],

3N-6 1

~g(S)--~ E ~ i)(s)di

(65)

i where i numbers the normal modes and d i is the degeneracy of the ith mode. From the experience with smaller molecules, where the terms e~i)(s) could be obtained from vibrational spectroscopy or by ab initio computation, they are expected to differ in sign, with e(gi)>0 where bond stretching dominates and e~)< 0 where bending is more important. The terms e~i)(s) do depend on the particular isotopomer s, and if they accidentally sum up to almost zero, the outcome may indeed be an erratic change of eg(S) from isotopomer to isotopomer. However, in the cases studied to date (most of them were investigations of larger molecules, [52,53] and unpublished work; also [57]) an improvement was achieved by the r0-derived rlE-fit over the unmodified r0-fit. This was shown by the lower standard deviation of the fit and smaller errors of the structural parameters, even in situations where truly significant values for the I~g could not be obtained. The l~g could be determined with significant accuracy ( =10 %) only when a large SDS was available. When all eg were positive, the bond lengths were shorter than the r 0 bond lengths and hence presumably nearer to the re bond lengths. Since the angles were hardly affected, the molecule appeared to be downscaled in size with only little distortion. As noted above, the constants eg could be considered as the first constant terms of an expansion for each of the isotopomer-dependent contributions eg(S). Unfortunately, the expansion cannot be extended to terms linear (or nearly linear) in the inertial moments, in the manner of F~g(S)= F~gJr Cg(lg(S)- Ig(1)), with ~g and Cg - (3eg(S)/~Olg)~ 1 as parameters to be determined by the fit. No matter whether

Microwave Rotational Spectroscopy

97

pe the Ig in the corrective term are chosen as/~g,expor Ig,calc, Eq. 63a can then be shown 0 pe to degenerate to l g,exp(S) ~-Cglg,calc(S ) plus terms independent of s, where the constant Cg is a function of Cg. Since the inertial moments are homogeneous functions of the (Cartesian) coordinates, any change of Cg in the fitting process could be counteracted by a corresponding common relative change of the coordinates, i.e., Cg is not a sufficiently independent variable and the fit diverges. However, if the coefficients Cg, which act as a sort of scaling factor for the inertial moments, were already known from other sources and need not be determined by 0 the fit, the model Ig,exp(S) -- c~ecalc(S) + Eg could possibly be a useful extension of the rte-method (see also the r e -'method, next section). For the linear molecule OCSe, LeGuennec et al. [57] have successfully expanded e(s) in terms of the isotopic mass changes upon substitution of Am~(s), e(s) = e + E~ c a Am~(s), where c a - (i)e(s)/t)ma)s=_l (the index g on 1~and ca, for the linear molecule g = b, has been dropped). There are four rovib parameters in this example: ~, c o, co and Cse. The three derivatives c a are related by the constraint equation (Euler's theorem), Z a ma ~e(s)/i)ma=e(s)/2 [24], because e(s) is a homogeneous function of degree 1/2, so that only two independent derivatives remain. The expansion of e(s) in terms of the isotopic mass changes appears to be a most reasonable choice. However, for the general polyatomic molecule (index g - x,y,z), the number of rovibrational parameters would then be prohibitively large: three constants Eg plus 3N a - 3 constants Cga =- (t)s (note: 3 constraint equations). At any rate, a model for the change of the eg(S) from isotopomer to isotopomer can be potentially successful only when: (1) any new variable introduced is truly independent of those already present and, of course, when; (2) the number of variables is not increased unreasonably. Although the r~E-fit and the p-Kr rap (=rat) fit are not equivalent (the former determines three more variables), it could be shown [55] that the molecular structures determined by the rz~-fit and the ra/-fit are strictly identical, including the covariance matrix. This is due to the specific form of the Jacobian matrix X of the coupled least-squares problem r~, which permits a decomposition by a non-singular transformation into a smaller least-squares problem r~ plus a subsequent direct calculation of the constant rovib contributions eg. The rzx/-part of the problem alone determines the molecular structure which must then be used (including the covariance matrix of the structural parameters) for the calculation of the contributions eg. When rotational constants of new isotopomers are to be predicted from the structure determined, the rl~-method performs much better than the ra/-method due to the presence of the additional rovib parameters eg. As discussed in the preceding section, the r~-method (Kraitchman's equations or the r~-fit) is practically insensitive to a (small) change of a planar moment component, as represented by the rovib contributions, if the change is common to both the parent and the isotopomer s, and if the rotation by the diagonalizing matrix T(s) (Eq. 41) is small. Therefore, it usually does not matter whether experimental ground 0 state planar moments Pg,exp(S) or, after the rig have been obtained from a preceding

98

HEINZ DIETER RUDOLPH

rlE-fit, rovibcorrected moments d,exp(S) - qg [= epe(s), Eq. 64b] are used as input data for the rs-method. The situation is different when the rotation by T(s) is large for the substitution of a particular atom. Then, using again the rs-method, the 0 different type of input data, Pg,exp(S) and pPge(s), respectively, may localize this atom in apparently different rs-pOsitions. A good agreement of the earlier rlCdetermined position with the rs-position obtained from the Ppge(s) data, despite the large rotation by T, can then be taken as an indication that the Ppge(s) are obviously good approximations to the equilibrium moments Pg(S) (which would yield identical structures by whatever method chosen irrespective of any rotation by T). In the following, we present the essential equations of the rl~-fit. They are simply those of the r0-method, with three more variables added. For the sake of a unified development, all relations are expressed in planar moments. The model for the structural part is the rigid set of mass points; only when the calculated moments are equated to the experimental ground state moments, are they supplemented by three constant rovib contributions. Let the set of masses be described, as before, by the atomic masses of the parent species, ma(1), and the mass changes upon substitution, Ama(s), for the atoms ot = 1, N a and the isotopomers s = 2, N s (see Eq. 32). The structure is defined by N s (not to be confused with N s) internal coordinates S t (composed into a vector S): bond lengths, bond angles, and dihedral (or torsional) angles for each atom. The (minimum) number of internal coordinates (i.e., components of S) is 3N a - 6. The ~oms are numbered, and a list showing to which atom each atom is attached, is a constituent part of the description. It defines the connectivity in the molecule. The list is arranged as a "tree" or branched chain, one atom must be the initial atom, and the chain must not have loops. The connectivity so defined need not, and sometimes (annular structures) cannot, be a true image of chemical bonding in the molecule. The list should be organized in a way which facilitates the description of symmetrical equivalence of atoms (both, overall and local symmetries) by preserving the molecular symmetry as far as possible. The introduction of (massless) "dummy atoms" is often helpful when the deviation from local symmetry of a whole group of atoms (e.g., the tilt of a methyl group) is to be described (then N s is larger than minimum). The vector S of internal coordinates is the same for all isotopomers, with the sole exception that Laurie corrections for the X-H bond shortening upon deuteration can be arranged for with particular ease by simply reducing the respective bond length coordinate for the D-isotopomer. For the calculation of the principal moments, the structural description of a particular isotopomer must be translated into Cartesian coordinates. This can be done by an algorithm which has been given an efficient and widely applicable form by Thompson [58]. Each atom is considered to be the origin of a local Cartesian base system whose axis directions are fixed by a prescription which references the two preceding atoms in the chain (and a few conventions if necessary). The positional description by the local base systems is then retransformed, in a stepwise fashion along the chain (by a translation plus a rotation), into that of the local base

Microwave Rotational Spectroscopy

99

system of the initial atom, which eventually serves as the reference system for the whole set of mass points. This Cartesian description can be expressed as r~ = r~(S) (all (~) or

r = r(S)

(66)

T ., r[1]T~ when the notation rT - (r T. . . . r~,. N~,, is used. It is identical for all isotopomers (excepting Laurie corrected bonds). With the known masses m~(l) and Eq. 6 for the planar moment tensor referred to an arbitrary base system, the tensor P(1 ) is calculated and diagonalized by means of matrix T(1 ), Eq. 11. With T(1) known and rc,,,(1) from Eq. 2, the base system of Eq. is transformed into the principal axis system of the parent, r~ ] (Eq. 10),

66

rill= TV(1)r~ - TT(1)rcm(1) (all t~)

(10a)

also denoted as r [1IT- (r~1IT.... 10~"[I]T . r~. ]T) as. in the. preceding . section This system will be used in this paper as referefice system for calculating the planar moment tensors of all isotopomers, P[q(s). The eigenvalues are obtained after [1] ), Eq. 2, will be required diagonalization by means of T(s). Also, the vectors rcm(S later. However, prior to the least-squares procedure, only an initial approximation for the structure is available, S ~~ Therefore. only approximations for the Cartesian description r ~~ = r(S(~ for the vectors r~~ for the PAS of the parent r IlIa~, and for the eigenvalues of the isotopomers, can be calculated. The expansion of the eigenvalues is, when S r176is assumed to approximate the true structure S well enough for truncating the expansion after the linear term,

pgfs~(s)

Pgt~(s)~~

Pgt~J(s)

Us ~(OPgt~'(s) Pg[4(s)= Pg,.,,(s)(~+ ~ ASt

(67)

PgN(S)(0)

Laurie corrections would be applied by calculating the initial v a l u e s for shorter X-D bonds. From the preparation of the P(s) and from Eq. 66 it follows that the are indirect functions of the internal coordinates S, via the Cartesian coordinates r~] 9

Pg~sl(s)

Pg's'(s)=Pgt"(s)(~+ ~ ~'~(h~ bh~]

~ bS,

As in Eq. 42, the derivative of the principal planar m o m e n t s the PAS coordinates of the parent, h Ill is given by

AS'

Pgf4(s)with respect to

aP?~(s) /Tv al~l](s) 0h~]

=

(s).

(68)

Oh[al]

The derivative of the tensor p[ll(s) (Eq. 5) with reference to the PAS r~ ] is,

(69)

100

HEINZ DIETER RUDOLPH

k"h ~"s -- r[1] ,,,,T

()pill(s) =

- - %,,,[l]s m())

e~l] T)

(70)

fcm (s)))

(71)

with elementf, f ' = x[ll,y[1],z[1],

(~P[1](s) {)h~l Ij , : m s ( s )

(~)fh(fs

'-

L,,,(' S ))+ 8f'h(fa

The closed formula for the desired derivative after the required superscripts (o) have been attached, is:

(bpp,(s)]C~ ~h~ ]

~,;',~,1.(0)

- 2 ms(s ) ~~

lJ,lgIS,

g,h = x,y,z

s = 1, N s

(s)(f[lal(o)_ f[l~O,(s) ~, cm

c t - 1, N a

) (72)

It is convenient to arrange these derivatives as a 3Ns x 3N a matrix F (~ The derivatives of the Cartesian PAS coordinates h~ ] of the parent with respect to the internal coordinates S t, required for Eq. 68, are well known from molecular vibrational theory as the derivatives of the respective displacement coordinates. They are usually arranged as the elements of a 3N a x N s dimensional matrix, which has been called A [59] and gives the linear relation between the increments of the internal and Cartesian coordinates: At= A AS

(73)

Several authors have shown how to calculate the matrix A [58,60-63]. A is the pseudo-inverse of the matrix B of Eq. 60: B. A = 1, where 1 is the N s x N s unit matrix. Polo [64] appears to have been the first to express the matrix A directly by constructing, for each atom c~ and each internal coordinate t, the 3 x 1 vector:

po,0-f

(74)

Each column t of A is then composed of N a vectors pO, just as each row t of B is made up of N a Wilson-type vectors sts. The pseudo-inverse relation of B and A is thus given by:

N a

Z Sto~pOt -- ~)tt,

(75)

s The superscript ofPolo's p~ indicates that the Eckart conditions are satisfied [64], which is required for the PAS coordinates. For the calculation of the vectors

Microwave Rotational Spectroscopy

101

p0, the above mentioned tree of connectivity is needed with the local base reference systems attached to each atom. The matrix A (~ is superscripted to show the dependence on S (~ The inner sum of Eq. 68 is the element of the 3N s x N s matrix product: X ' = l~~176 (76)

P glSls.h[1]o~ We pause to state that the treatment which led us from Eq. 67 to the matrix X' of Eq. 76 is not unique. Instead of Eq. 68, the indirect dependence of the planar m o m e n t s Pg[4(s) on the internal coordinates St, could have been expressed by:

"g,,l(s) = "g,,l(s)(0) + ~ ~h,,

c3h~l J t -~t ) ASt

(68a)

The first factor in each term of the sum is simpler than in Eq. 68 and would be given by Eq. 70 with the superscript Ill replaced by [4 and noting that rcm(S)[4= 0. However, in contrast to Eq. 76, the second factor would then have to be calculated separately for each s. The system of Eq. 77, obtained from Eq. 67 by any treatment chosen, is the starting point for a r0-type fit (here a re-fit),

Ns

Pgtsl(s) =

Pgt4(s) (0) + Z X'g ['lst' ASt

'

gs - = x,y,z I,N s

(77)

t

when the right-hand sides are equated to the experimental ground state principal 0 planar moments, Pg,exp(S), and the appropriate covariance matrix is used. For the rl~-fit, Eq. 77 must first be supplemented by terms for the (constant) rovib contributions rig:

Ns

Pg[4(s) + q g - Pg[4(s)(~ + rl~~ + Z Xtg Isls,t ASt + A]lg

(78)

t It is then convenient to append to the matrix X' a stack of N s 3 • 3 unit matrices, thus forming the composite Jacobian matrix X of dimension 3N s • (N s + 3). Likewise, the N s • 1 vector_ AS and the 3 • 1 vector All are combined to the composite vector of variables II of dimension (N s + 3) • 1. The component index of II and the column index of X is 7"= 1, N s + 3.

102

HEINZ DIETER RUDOLPH

ri

1

":" AS

'X= X'

~=

(79)

A11

k

\

J

In contrast to the rs-method, the part X' of the Jacobian is, in general, a dense matrix with no apparent structure. The following equations show the identification with the quantities of the least-squares procedure (the vl~~ will most often be chosen as zero) 9

Yi +-- Yg"l,s = Pg'~",exp(s) - Pg'"(s) (0)- l"l(g0)' [~j <-- [3r - AS, (for F<_Ns),

g = x,y,z s= l , N s g- l , N s + 3

= Arlr_Ns (for F> Ns) (~PgH(S)) (0) Xij ~ Xg%'r= ~ 3S, (for F< Ns),

(8o)

(81)

g - x,y,z s- l,N s F- 1,Ns+ 3

-

-

'~g,F_Ns

(for F> Ns)

(82)

Prior to the application of the least-squares procedure, the number of variables 13r(and the number of columns of X) may have to be reduced because symmetrically equivalent positions are present or because certain variables must be kept fixed. Also, one or more observations ~gl.q(and corresponding rows of X) may have to be excluded from the fit in order to comply with the rules for planar substitution or because they could not be measured. This can again be done with the aid of matrices D and E. Other than in the preceding section, the matrix D must now relate the increments of symmetrically equivalent internal (or bond) coordinates with the increment of the respective base internal coordinate. Let us return to the example, Eq. 56, and let the matrix D, as depicted there, be a portion of the whole matrix D. The row and column headings of each 3 x 3 block of D for a particular atom now specify the increments of bond length Ar, bond angle Aq0, and dihedral (or torsional) angle AO used for the attachment of this atom to the chain, instead of the increments of Cartesian coordinates of the atom. It is assumed that the description by internal coordinates has been chosen for the best possible preservation of molecular symmetry. The three symmetrically equivalent atoms ~, ~', and c~" will then be attached to an interatomic bond along the x-axis (or, by convention, to an equivalent

Microwave Rotational Spectroscopy

103

axis stub if the immediately preceding atom is the initial atom of the branched chain). The increments of the bond coordinates for the three symmetrically equivalent atoms ix, ix', and or" must be i d e n t i c a l Ara=Ara,=Ara,,, Aq~ = Aq)~, = Aq~., AO~ = AO~, = A O ~ , - and the 3 x 3 matrices C(~') and C(ct") will now be unit matrices. Let the symmetrically equivalent atoms y and y' be attached to an interatomic bond lying on the x,z-plane, whence: Arc=Art,, Aq)~,= A~,,, but AO~,=-AO.t,. The respective 3 x 3 block of the table of Eq. 56 is now C(y ) = diag(1, 1, - 1 ) . The matrix E must take care of the fact that the inertial tensor is, in principle, insensitive to the torsional position of a C3-symmetric group (here atoms c~, ix', and ~"). Therefore, the irrelevant dihedral angles Oa, Oa,, and Oa., are kept fixed at their initial values and the variable AOa (with the symmetrically equivalent partners) is eliminated from the fit by leaving out the respective column of the matrix E. In practice, the matrix product DE is readily generated from the input to the actual problem. The input must specify which bond coordinates of which atoms are free to be determined by the fit and, in the case of symmetrically equivalent bond coordinates which must be varied simultaneously, if the increments are to be taken in uniform or opposite directions. The matrix DE then serves as a bidirectional list relating the complete set of all internal coordinates required for the description of the molecule with the smaller set of the truly independent internal coordinates of the fitting problem. The model equations for the iterated least-squares treatment are given, reduced to these independent variables, by, y = XII = XDEp = Xp

with

XDE= X

(58)

where it has been assumed that also the observations y have been reduced to those,,,., appropriate for the situation. After each iteration step, I] should be expanded to II (Eq. 56) for the correction of the independent and the dependent internal coordinates, S (~ + ~ ---->S (~ which results in an improved approximation for the next step which starts again with Eq. 66. After convergence, the description of the molecular structure is available in internal coordinates ~, obtained from S (~ after adding repeated corrections ~, and in Cartesian coordinates rtl], from Eq. 10a of the last iteration cycle. The expanded covariance matrix of the variables (Eq. 59) is that of the internal coordinates, ~ = ~ . The covariance matrix of the Cartesian coordinates is obtained by the transformation of ~ with matrix A (Eq. 76) for the parent species from the last iteration cycle: A

A

OrB1-AO~A r

(83)

A

Experience has shown the covariance matrix Or[,J to be conspicuously different when the same problem was treated by either the rs-method (withoutenforcing the first and second moment conditions) or any of the r0-derived methods. For the former method the errors of the coordinates were much less correlated. This, as well as the better condition number of the normal equation system, is no doubt a

104

HEINZ DIETER RUDOLPH

consequence of the different structure of the Jacobian matrices X. The Jacobian of the rs-method (without the moment conditions) is a sparse matrix and gets denser only to such a degree as more and more multiply substituted isotopomers are included in the SDS. The results of the r s- and the r/~-methods agree well regarding the effect of applying Laurie's corrections when the evaluation can be based on a sufficiently large SDS. In several cases tested, the inspection of the structure in internal coordinates showed that the effect was almost completely limited to the corrected X - H bond length. Compared with the result obtained without correction, this bond was elongated by --2 8r when the X-D bond length in the input for this isotopomer was described as shorter by 5r than the X-H bond length of the parent This is the effect required to compensate for the error incurred when the bond length change remains unattended [51]. However, when the correction was applied within the original r0-method, bond length changes were distributed over several bonds, although the X - H bond was affected most. In contrast to the rs-method, where unsubstituted atomic positions are indeterminable, r0-derived methods can locate also unsubstituted atoms, although with less accuracy than for substituted positions. In practical cases, this capacity is limited to one or two unsubstituted atoms and yields useful results only when the SDS is otherwise large enough. Coordinates that are (for whatever reason) not included in a r0-type fit must be assigned estimated values, which are kept fixed in the fit but affect the result. In these cases, one should not do so without also setting error limits to these coordinates in order to find out, by Eq. 30, which additional errors of the positions are due to the error afflicted "fixed" positions. If symmetrically equivalent atomic positions are among those kept fixed, the relations established between the independent and the expanded set of variables (Eq. 56b) now denoted as B (x) and ~(x), respectively, are applicable in the form" AB (x) = D(X)E(S)AB(X),

X(X)D(X)E (x) = X (x)

(84)

The expanded covariance matrix of the "fixed" parameters (cf. Eq. 59) is, Oh(x) _ D(X)E(X)OB(.,.)E(X)tD(x)t

(85)

and the additional effect on the variables determined by the least-squares fit, (cf. Eq. 30) is given by OB = CX(X)O~(x)X(X) C r _ CX(X)OB(x)X(x)Tcr

(86)

V. OTHER STRUCTURES In this section we shall first give a short account of the "mass dependence" or rm-Structure introduced by Watson [24], and also of the "complementary" or

Microwave Rotational Spectroscopy

105

re-Structure developed by Nakata et al. [26] on the basis of Watson's method. The central point in both methods is the expansion of the rovib contributions Eg(S) of the single-substituted isotopomer s with respect to the isotopic mass change Ama(s) = m~(s) - m~(1) of the substituted atom o; (g = x,y,z), OEg

1 C)2Eg

Amo +

A,n +...

(87

where the derivatives are to be evaluated with the set of masses m~ of the parent. Watson's original rm-method takes into consideration Z~g up to the first order in the expansion; the rc-method additionally arranges the required isotopomers into complementary sets in the attempt to also compensate the effect of the second order terms of the expansion. Both methods require a very large number of isotopomers for all but the smallest molecules. They are important for theoretical reasons and they are practical for the highly precise determination of the structure of very small molecules (where the results can often be compared with the available re-Structure), but they are hardly applicable to the larger polyatomic molecules. They have, therefore, not been treated at length in the preceding sections. The basic principles of the rm-method can be exemplified most clearly by the case of the diatomic molecule, c~ = 1, 2, where, by Kraitchman's equation in the simple form for linear molecules, the equilibrium and the substitution coordinates with respect to the PAS of the parent, x~ and ~ , respectively, are given by,

a

2

A/~, (xS)2 A/~,,

- s

(88)

where A indicates the isotopic difference between the inertial moment of each of the singly {x-substituted isotopomers, c~ = 1, 2, and that of the parent. From i0 = Ive + ev, we have, .

=

+--B--~a-

5 + ~9m----~+ 5 5

~)m2" + "'"

(89)

since the expansion (Eq. 87) can be reformulated with the reduced mass Ba = MAmJ(M + Ama): Agy_ ~8,, _

g~

.

Ama ~2(M 8v) +

+...

c)m~ 2 M

(90)

~m2

Eq. 89 can be summed up over the two single-substituted isotopomers, ot = 1,2, to give: 1,2

2

1,2

/'e'~ 2

1,2

Zm.CxS.)-Zm.~-~)+Zmo~

~ey+

1

Omc~ 2---M

1,2

Z m . Am.

~)2(M Ev) (91) " +...

am~

106

HEINZ DIETER RUDOLPH

The left-hand side of Eq. 91 is computed from the substitution coordinates and is hence defined as the "substitution inertial moment" ~, of the parent. The first term of the right-hand side is the equilibrium moment ly of the parent:

1,2

aE3,

1,2

1

a2(Mgy)

I~,=l~e,+Ema - ~ m a + - ~ E m a A m a ~ am+ 2 ot

...

(91a)

o~

The substitution moment I~,s differs from the equilibrium moment lye by first order terms of the expansion. Since Eg is a homogeneous function of degree one-half of the atomic masses [24], the second term on the right-hand side of Eq. 91a is, by Euler's theorem:

1,2

~1~,, 1

E ms 0m~

1 (i 0 _ le )

2 ~Y=2

-

(92)

-

Equation 91 a can hence be rewritten as,

1,2

r

lym - 21y- l~ lye +-~1 E ma Ama ~

+ ...

(93)

o~

where the left-hand side is defined as the "mass dependence inertial moment" of the parent which differs from the equilibrium moment only by second order terms. If the bond length r m is then calculated from this mass-dependence moment/.'~ = 2I~~,- ~, using, (','= ~

mlm 2

r2m

(94)

/711 + m 2 (which is the r0-type procedure applied to ly), r m should be closer to the equilibrium value re than the substitution bond length r s = ~ + ~2" Watson [24,25], who tested his method with a number of linear and bent XY 2 molecules, could show that the rm-Structure was indeed a good approximation to the re-Structure, except for molecules containing hydrogen. The relative mass change upon deuteration is so large that it does not suffice to take into account only first order terms of the expansion with respect to the atomic masses. If the planar moment components, Pg (g = x,y,z), and their rovib contributions, TIg = ~1 (ex + ev. + 8,,) - eg, are used, the corresponding equations required for the rm-method can be found also for the general polyatomic molecule (asymmetric top). The equations are approximate and hold under the provision that the PAS rotation upon isotopic substitution is small [24],

2

2

Aqg iaa

2

~qg

Ama ~2(Mqg)

~

2M

~)m2

9

(95)

Microwave Rotational Spectroscopy

107

where g~ and g~ (g= x,y,z) are the equilibrium and substitution coordinates, respectively, of atom ot with respect to the PAS of the parent. Summing over all single-substituted isotopomers yields, N

N

N

a(gsl2~(gel2C~

Ema

~l, lg q..

N -a

1

(96)

~ 2 ( M F_,g)

a =~.~ ma a + E m a ama -~--~EmaAma

~

~9m2

-

-

~

-

,,.

which is, after defining the left-hand side as the "substitution planar moment" of the parent, ps. g" N

N

brig

1

ang

1

"

a2(M rig)

+2 PSg=peg+~-~ma~mu+2-M~-~makma ~ am

...

(96a)

Ot

With Euler's theorem, N

o

1

o

ma am a = ~- rig = ~-(P g -

,,~)

(97)

one eventually obtains the three "mass dependence planar moments" of the parent: N

1 c)2(M 'lqg) Pg =- 2pss - P g = P g + -'M Z ma kma .......am 2 + . . . m

0

e

"

(98)

As before, the substitution moments differ from the equilibrium moments by terms of first order in the expansion, and the mass-dependence moments only by terms of second order. Only in the rare case of very small or highly symmetric molecules will the small number (< 3) of planar moment components Pgm (1) of the parent alone suffice to compute the molecular rm-Structure. To make available for this purpose the three mass-dependence; planar moment components ~'(s) = 2~g(S) - P~ of only one additional isotopomer s, all isotopomers which are single-substituted with respect to this isotopomer s must have been measured and evaluated in order to find Pg(S). For all but very small molecules the expense involved is prohibitive. Nakata and colleagues [26] have noticed that the term for any atom cz in the sum of Eq. 93 changes sign when the role of the substituted and unsubstituted (parent) species is exchanged with respect to this atom" A m a = m ~ - m a - - A m a = (ma - ma). While Watson's rm-method for a diatomic requires the three molecular species XY (parent), X'Y, and XY* to evaluate ?v~(XY) for the parent by Eq. 93, Nakata and colleagues also suggested the evaluation of the "complementary set" of isotopomers, X ' Y * (parent), XY*, and X'Y, in order to also find l'vm (X * Y * ). Assuming mffM and ,92(M Sy)/Om2 to be sufficiently invariant with respect to XY -

108

HEINZ DIETER RUDOLPH

and X'Y*, the second order terms of Eq. 93 are then approximately equal in magnitude and opposite in sign for I~m (XY) and lym (X'Y*). The bond lengths computed from Eq. 94 are, therefore, expected as rm(XY) = r e + Ar and rm(X*Y*) = r e - A r , respectively, from which the "complementary" bond length rc(XY) is obtained by the average:

'(

)

rc(XY) = ~ rm(XY) + rm(X*Y*) = re(XY )

(99)

The method is an improvement over the rm-method since it approximately compensates also the second order terms of the expansion of Eq. 98. It has been shown to work well even for molecules containing hydrogen. For the r c bond length of a diatomic molecule, which presents the simplest possible case, the inertial moment of one additional isotopomer, X'Y*, would have to be known over those required for the original rm-method: XY, X'Y, and XY*. Nakata, Kutchitsu, and Mills [65, 66] have later given details and shown that the bond length rm(X*Y*) need not be calculated from the spectroscopically derived inertial moment Ism (X'Y*) of this additional isotopomer, but can indeed be calculated from the inertial moments of the parent XY and the singly substituted isotopomers, X*Y and XY*. The method is restricted to a limited class of very small molecules (essentially diatomics and bent XY 2 molecules) and is based on the additivity rule [67] for vibrationally averaged, i.e. rz-type, bond length changes 8r z upon isotopic substitution of either one or both of the bonded atoms: r_(X*Y) - rC( X Y ) + 8rz(X*Y) ,.

rz(XY*) - rz(XY ) + 8rz(XY* ) rz(X*Y*) = rz(XY) + 8rz(X*Y ) + 8rz(XY*)

(100)

Therefore, it is the vibrationally averaged inertial moment I~(X*Y*) for the diatomic which is calculated from I~.(XY), 9 Iv(X = *Y), and I~(XY*) by the authors. Splitting the rovib contribution into a harmonic and anharmonic part, the relation between I ~ I e, and I= is given by [8, 22]: Ig0 - - Ige + 8g(harm) + 8g(anh) - Fg + 8g(harm)

( 101 )

The moments I:g = Ig0 - - E g (harm) can be calculated from the ground state moments if the harmonic force field is known, although the present application is not very sensitive to uncertainty in the force-field [65]. The moments l:g are geometrically more consistent than the I g0 (e.g., the inertial defect, Eq. 15c, expressed by the l:g practically vanishes for a planar molecule). For the present purpose, the s the substitution m o m e n t s I gs, and the mass dependence substitution coordinates g~, m o m e n t s I gm must be adequately redefined in accordance with the use of the moments IZg instead o f I g0, but the important result, Eq . 99, remains unchanged .

Microwave Rotational Spectroscopy

109

There are very few cases where the moments Ig' of one molecular species (at most three moments) suffice to derive all structural parameters; this is, e.g., possible for the diatomic where one moment is obtainable and required for one bond length, and for the bent (planar) XY 2 molecule where two independent moments are obtainable and required for one bond length plus one angle. In these few cases, the relevant rm-type parameter(s), obtained for both, the parent, XY., and the "complementary parent", X'Y*.*, can be averaged according to Eq. 99 to obtain the r c values. If the moments I gm of more than one species are required to determine the molecular structure, this averaging is no longer feasible. The rc-Structure must then be obtained by direct calculation (or by a least-squares fit, if possible) using a substitution data set composed in a completely balanced manner of complementary sets of isotopomers. The linear XYZ molecule, e.g., is a problem which must be treated in this way. This rc-Structure is still expected to approximate the re-Structure better than the rm-Structure does. The number of multiple-substituted isotopomers, which are required to form complementary sets for a rc-type structure determination, increases enormously with the number of constituent atoms of the molecule. Therefore, Nakata and Kutchitsu [68] have developed a method to estimate the (ground state) inertial moments of all multiple-substituted isotopomers only from the moments of the parent and the single-substituted species. In effect, the method is also based on the additivity rule for increments of vibrationally averaged bond lengths upon isotopic substitution. A precise knowledge of the molecular structure is not required. Where the estimates could be checked, they were good enough to replace the spectroscopically obtained values when calculating the rc-structure. The results were a little less than optimum where a rotation of the PAS upon isotopic substitution would have to be taken into account. Since the extension offered by Nakata and Kutchitsu's estimates of the inertial moments of multiple-substituted isotopomers is still limited to a few classes of small and simple molecules, the application of the rc-method remains restricted, but where it can be applied it appears to yield the best possible approximation to the re-Structure. The r~, -method, which is applicable to larger polyatomic molecules, was not discussed in the preceding sections because a recent series of detailed papers is available [27-31]. The method requires the ground state rotational constants of the parent and all single-substituted isotopomers. Therefore, it is less restrictive than the r m- and re-methods, but somewhat more so than the rs-fit and r0-derived methods which permit the inclusion of multiple-substituted isotopomers. In contrast to the r m- and r~-method, the substitution inertial moment is required only for the parent, I sg (1). For this purpose the single-substituted isotopomers are indeed sufficient. From Igs (1) the set of three numbers,

Pg --

igO(1)'

g = x,y,z

(102)

110

HEINZ DIETER RUDOLPH

is obtained which are then used for the corrective scaling of the ground state moments of all isotopomers by a factor of 29g - - 1 ,

IPm,g(S) - ( 2 9 g - 1 ) ~ ( s ) - ( 2

Ig(1)10g (l) _ 1)ig0(S)

(103)

before a final r0-type fit to the IP,,g(S) is carried out which yields the ~,-structure. The scaling of the moments is such that the substitution moments of the isotopomers are approximated by lSg(s) = 9gI~ and that the relations (Eq. 93) for the mass-dependence moments Ig'(S) are approximately satisfied by IP,,g(S) for s ~ 1, and exactly for s = 1. This scaling variant alleviates some of the difficulties characteristic for the determination of the moments Ig(S). Laurie's corrections must be applied for deuterated isotopomers [30]. The r~, -method requires the same input information as the evaluation of the rs-Structure by Kraitchman's equations and is essentially the attempt to implement rovib corrections which were previously computed by the scaling process and do depend, in a restricted sense, on the particular isotopomer without, however, increasing the number of variables:

lgn,g(S) -(2pg

- 1)l~(s)=

l~

2(1 -

pg)l~(s) = l~(s) - Egm,g(S)

(104)

In effect, the rOO m -method introduces rovib contributions which are linear in the inertial moments, eOm,g(S) = 2(1 - 9g)l~ Promising results of the r~m -method have been obtained for several small molecules where the re-Structure was available for comparison. A corrective scaling of all moments could be easily introduced into the r0-procedure as described in the preceding section; it would require a careful analysis of the propagation of the input errors to the values of the p g, g = x,y,z, which are unique for all isotopomers, and from there to the final structural results. The recent paper by LeGuennec et al. [57] mentioned earlier is noteworthy also for the fact that it reports for the small molecule OCSe the structures obtained by all presently known methods and compares them with the known re-Structure.

VI. CONCLUDING REMARKS AND OUTLOOK From what has been shown in the preceding sections (cf. Eqs. 61 and 73, 83), it is possible to present the molecular structure resulting from both the rs-fit method and any of the r0-derived methods in a convenient and easily comparable form, as a structural description in both Cartesian and internal coordinates, and with consistent errors and correlations (for small and larger molecules). A detailed comparison would require a sufficiently large SDS to determine a complete molecular structure, but the requirements are still the least restrictive of all methods presented. The input data must include the covariance matrix of the rotational constants or moments. This matrix may have to be adequately modeled to avoid grossly different weighting of isotopomers which is usually not warranted. The definition of the input data set

Microwave Rotational Spectroscopy

111

also requires a decision which moments, if any, should be excluded to comply with the rules for planar substitution. Costain's error criterion can be introduced at different levels and must be correctly propagated. Attention must further be paid to the additional errors of the solution caused by the uncertainties of any parameters kept fixed in the fit. When the input data sets are identical in the above sense, the results of the rs-fit method and of the rl~-method (or p-Kr-method) tend to converge within the error limits. Experience with sufficiently comprehensive investigations is limited, and the relevant work has been mostly done on larger molecules where the corresponding equilibrium structures for a critical comparison are not available. The present rs-fit and r0-derived methods for the determination of a molecular structure from MRR ground state spectra, accurate as this structure may then appear from a chemist's viewpoint, cannot fully benefit from the extremely high precision with which ground state moments can now be measured. This is because the input data to either method are still contaminated by at least part of the elusive rovib contribution. Information on its dependence on isotopic substitution is very scarce and has only recently become the subject of ab initio calculations which are still limited to very small molecules. As they progress, useful suggestions could possibly be hoped for on how to also treat the rovib contributions for larger molecules with a little more detail. In the earlier conventional MRR-spectroscopy, the application of the double resonance technique has been absolutely crucial for the successful aggregation of a sufficiently large SDS, because this technique allowed the measurement of the weak spectra of isotopomers in natural abundance despite the presence of strong transitions of the parent species. The modern Fourier-transform MRR~ [69,70] also makes use of double resonance techniques by adequately chosen pulse sequences of pump and signal frequency [71,72]. In the beginning, Fourier-transform MRR-spectroscopy with its greatly enhanced resolution has been used preferably for the previously very difficult investigations where complex multiplets due to interactions between the molecular rotation and various properties of the constituent atoms had to be resolved and analyzed. Recent developments of the instrumentation have simplified the construction and facilitated and speeded up the scanning operation. The admittance of sample molecules into the spectrometer as a molecular beam (supersonic jet) with a very low rotational temperature has increased the sensitivity in a way which lets this technique appear ideally suited for the measurement of ground state spectra, and also, in combination with double resonance operation, for the measurement of spectra of isotopomers in natural abundance. The low temperature favors the rotational spectrum in the vibrational ground state over the rotational spectra in vibrationally excited states---even low-J over high-J ground state transitions--and greatly increases the spectral intensity due to its inverse relation with respect to temperature.

112

HEINZ DIETER RUDOLPH

It is to be hoped that this development will promote the interest to successfully apply these modern and efficient MRR-Fourier-transform techniques to the very fundamental task of determining accurate molecular structures.

ACKNOWLEDGMENTS The author wishes to thank Dr. J. Demaison and Dr. V. Typke for a critical reading of the manuscript and helpful hints. I also gratefully acknowledge the support by the Fonds der Chemischen Industrie.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

19. 20. 21. 22. 23. 24. 25.

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26. Nakata, M.; Sugie, M.; Takeo, H.; Matsumura, C.; Fukuyama, T.; Kutchitsu, K. J. Mol. Spectrosc. 1981, 86, 241. 27. Harmony, M.D.; Taylor, W.H.J. MoL Spectrosc. 1986, 118, 163. 28. Harmony, M.D.; Berry, R.J.; Taylor, W.H.J. Mol. Spectrosc. 1988, 127, 324. 29. Berry, R.J.; Harmony, M.D.J. Mol. Spectrosc. 1988, 128, 176. 30. Berry, R.J.; Harmony, M.D. Struct. Chem. 1989, 1, 49. 31. Harmony, M.D. Acc. Chem. Res. 1992, 25, 321. 32. Goldstein, H. Classical Mechanics; Addison-Wesley: Reading, Mass., 1980. 33. Watson, J.K.G.J. MoL Spectrosc. 1972, 41, 229. 34. Albritton, D.L.; Schmeltekopf, A.L.; Zare, R.N. In Molecular Spectroscopy: Modern Research; Rao, K.N., Ed.; Academic Press: New York, 1976, Vol. II, p.1. 35. Wilkinson, J.H. In Linear Algebra; Wilkinson, J.H.; Reinsch, C., Eds.; Springer: Berlin, 1971, p. 1. 36. Businger, P.; Golub, G.H. In Linear Algebra; Wilkinson, J.H.; Reinsch, C., Eds.; Springer: Berlin, 1971, p. 111. 37. Bauer, EL. In Linear Algebra; Wilkinson, J.H.; Reinsch, C., Eds.; Springer: Berlin, 1971, p. 119. 38. Golub, G. H.; Reinsch, C. In Linear Algebra; Wilkinson, J. H.; Reinsch, C., Eds.; Springer: Berlin, 1971, p. 134. 39. N6sberger, P.; Bauder, A.; Gtinthard, Hs. H. Chem. Phys. 1973, 1, 418. 40. Brodersen, S. J. MoL Spectrosc. 1990, 142, 122. 41. van Eijck, B. P. J. Mol. Spectrosc. 1982, 91,348. 42. Chutjian, A. J. Mol. Spectrosc. 1964, 14, 361. 43. Nygaard, L. J. MoL Spectrosc. 1976, 62, 292. 44. Rudolph, H. D. J. Mol. Spectrosc. 1981, 89, 460. 45. Rudolph, H. D. J. MoL Spectrosc. 1981, 89, 430. 46. Landau, L. D.; Lifshitz, E. M. Quantum Mechanics; Pergamon: London, 1959; eq. (72.2). 47. Hirose, C. BulL Chem. Soc. Jpn. 1974, 47, 976. 48. Eliashevich, M. Compt. Rend. Acad. Sci. U.R.S.S. 1940, 28, 604. 49. Wilson Jr., E.B.J. Chem. Phys. 1941, 9, 76. 50. Wilson Jr., E.B.; Decius, J.C.; Cross, P.C. Molecular Vibrations; McGraw-Hill: New York, 1955, Chapter 4. 51. Typke, V. J. MoL Spectrosc. 1978, 72, 293. 52. Meyer, M.; Grabow, J.U.; Dreizler, H.; Rudolph, H.D.J. Mol. Spectrosc. 1992, 151,217. 53. Epple, K.J.; Rudolph, H.D.J. Mol. Spectrosc. 1992, 152, 355. 54. Hamilton, W.C. Statistics hi Physical Science; The Ronald Press: New York, 1964, Chapter 4. 55. Rudolph, H.D. Struct. Chem. 1991, 2, 581. 56. Demaison, J.; Nemes, L. J. Moi. Struct. 1979, 55, 295. 57. LeGuennec, M.; Wlodarczak, G.; Demaison, J.; BUrger, H.; Litz, M.; Willner, H. J. Mol. Spectrosc. 1993, 157, 419. 58. Thompson, H.B.J. Chem. Phys. 1967, 47, 3407. 59. Woodward, L.A. hltroduction to the Theory of Molecular Vibrations and Vibrational Spectroscopy; Clarendon: Oxford, 1972. 60. Jacob, E.J.; Thompson, H.B.J. MoL Struct. 1972, 12, 298. 61. Thompson, H.B.J. Chem. Phys. 1970, 53, 3034. 62. Reichman, S.; Ginn, S.G.W.J. MoL Struct. 1969, 3, 418. 63. Pulay, P.; Sawodny, W. J. Mol. Spectrosc. 1968, 26, 150. 64. Polo, S. R. J. Chem. Phys. 1956, 24, 1133. 65. Nakata, M.; Kutchitsu, K.; Mills, I. M. J. Chem. Phys. 1984, 88, 344. 66. Nakata, M.; Kutchitsu, K. J. Chem. Soc. Jpn. 1986, 1446. 67. Kutchitsu, K.; Oyanagi, K. Faraday Discuss. Chem. Soc. 1977, 62, 20. 68. Nakata, M.; Kutchitsu, K. Bull Fac. Gen. Ed. Univ. Tokyo A&T 1990, 27, 153.

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69. Dreizler, H. Mol. Phys. 1986, 59, 1. 70. Bauder, A. In Vibrational Spectra and Structure: A Series of Advances; Durig, J. R., Ed.; Elsevier: Amsterdam, 1993, Vol. 20, p. 158. 71. Stahl, W.; Fliege, E.; Dreizler, H. Z. Nat,rforsch. A. 1984, 39, 858. 72. Vogelsanger, B.; Bauder, A. J. Chem. Phys. 1990, 92, 4101.

GAS-PHASE NMR STU DIES OF CONFORMATIONAL PROCESSES

Nancy S. True and Cristina Suarez

I. II.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gas-Phase N M R Studies of Conformational,

116 116

Tautomeric, and General Isomer Equilibria . . . . . . . . . . . . . . . . . . . A. Conformer Equilibria and Thermodynamic Barriers Using Direct Measurements . . . . . . . . . . . . . . . . . . . . . . . .

117

III.

IV.

Conformer Equilibria and Activation Barriers via Analysis of Long-Range Coupling Constants ..................... Gas-Phase N M R Studies of Conformational Kinetics . . . . . . . . . . . . . A. Temperature-Dependent Gas-Phase N M R . . . . . . . . . . . . . . . . . B. Pressure-Dependent Gas-Phase D N M R . . . . . . . . . . . . . . . . . . C. Bimolecular Rate Constants and Collision Efficiencies . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

118

B.

Advances in Molecular Structure Research Volume 1, pages 115-155. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8

115

122 124

126 135 145 151 152

116

NANCY S. TRUE and CRISTINA SUAREZ

ABSTRACT Gas-phase NMR spectroscopy can, in many cases, provide thermodynamic and kinetic data for chemical exchange processes which are comparable to those obtained in liquid-phase studies. Gas-phase studies of chemical exchange processes including internal rotation about partial double bonds, keto-enol tautomerism, ring inversion in six-membered rings, Berry pseudorotation in SF4, and the degenerate Cope rearrangement are reviewed. Part II of this review describes gas-phase conformer and tautomer equilibrium studies. Part III describes gas-phase kinetic studies of chemical exchange processes at high pressures (unimolecular limit), in the falloff region, and at low pressures (bimolecular limit).

I. I N T R O D U C T I O N Much of our present understanding of conformational equilibria and conformational interconversion processes in liquids and solids is the result of NMR studies. [1] NMR spectroscopy has several advantages in conformational studies. In many cases, spectral assignments can be made unambiguously. Multinuclear studies, selective decoupling, relaxation measurements, coherence transfer studies, and measurement of long-range coupling constants can aid in making spectral assignments. [2] NMR spectral intensities are directly related to the relative populations of the absorbing species. Magnetic dipole transition moment matrix elements depend on the identity of the nucleus which is being studied and are not significantly affected by its local environment. Also, for many molecular processes with activation energies in the 20-85 kJ mo1-1 range, rate constants can be obtained under equilibrium conditions. Most NMR studies of condensed phases have been performed at ambient pressure and have reported isobaric activation parameters. Recent studies by Jonas et al. [3] have employed pressure as an experimental variable and have reported isobaric as well as isoviscosity activation parameters which provide information about dynamic solvent effects. Recent work has shown that extension of NMR spectroscopy to studies of gas-phase systems can result in a better understanding of the intramolecular dynamics of many conformational exchange processes as well as the direction and magnitude of environmental perturbations. NMR spectra of dilute gases can be obtained with signal to noise ratios and resolution which are comparable to those obtained in the best liquid-phase studies. In general, 1H and ~3C spectral of most molecules can be obtained with line widths of 1 Hz or less in the gas phase. Spectra can be obtained for 1H, and enriched 13C present at 0.25 torr or more partial pressure in the presence of a few torr of a bath gas. It is difficult to obtain spectra of samples where the total pressure is below 1 or 2 torr, since the transverse relaxation time, t 2, is generally very short at these pressures and results in significant line broadening. 13C spectral acquisition is actually facilitated in the gas phase since the longitudinal relaxation time, t l, for this nucleus is much shorter in the gas phase

NMR Studies of Conformational Processes

117

than in condensed phases. For example, at 500 torr, the 13C t 1 of benzene is 0.07 s, compared to ca. 30 s in the pure liquid [4]. These short tlS make it possible to collect transients 100 to 1000 times faster than is possible in condensed phases. Proton tlS are somewhat shorter in the gas phase than in liquid solutions and exhibit a characteristic nonlinear pressure-dependence. [4-7] Gas phase NMR studies have provided thermodynamic and kinetic parameters for many low-energy molecular processes. In addition, several gas-phase NMR studies have reported measurement of rate constants as a function of pressure. Pressure-dependent rate constants provide information about the extent and rate of intramolecular vibrational redistribution in the reacting molecules. Also rate constants in the low pressure bimolecular region can be used to determine collisional efficiencies for the chemical exchange process. To date, more than 50 studies in this area have been reported addressing topics including internal rotation in alkyl nitrites, amides, and nitrosamines; keto-enol equilibria; ring inversion in several six-membered rings, and Berry pseudorotation in inorganic and organic molecules. The following review discusses these studies. This review is divided into two major parts. The first part describes gas-phase NMR studies of conformational equilibria by direct and indirect methods. The second part describes studies of gas-phase kinetics of chemical exchange processes including studies at the high pressure limit, pressure-dependent studies, and studies in the bimolecular region. Other reviews focusing on gas-phase NMR spectroscopy include: Jameson's review on gas phase NMR studies of relaxation, and chemical shifts, as well as gas-phase conformational studies [6]; Armstrong's review of gas-phase NMR relaxation studies [7]; and other general gas-phase NMR spectroscopy reviews by Harris and Rao, [8] and Govil [9].

II. GAS-PHASE NMR STUDIES OF CONFORMATIONAL, TAUTOMERIC, AND GENERAL ISOMER EQUILIBRIA The development of NMR methods for the study and analysis of gas-phase systems has taken two main approaches. In one, the experimental applications have been concerned mostly with measurement of gas-phase chemical shifts, coupling constants, and relaxation times of small molecules with high vapor pressures, and with theoretical studies of pressure and/or temperature dependence of relaxation and nuclear shielding. C. J. and A. K. Jameson have contributed extensively to this area with comprehensive investigations of absolute shielding values for diverse small molecules containing different magnetically active nuclei [10,11]. The essential isolation of small molecules in the gas phase provides an excellent experimental counterpart to theoretical models used in the development of shielding and relaxation theories. Important work in this field includes" Govil's calculation of vicinal coupling constants for ethane, methylamine, and methanol using extended HUckel theory [12]; Schindler's ab initio calculations of chemical shifts in small gaseous molecules [13,14]; theoretical studies of relaxation and line-shapes at low pressures

11 8

NANCY S. TRUE and CRISTINA SUAREZ

by Sanctuary [15]; D. Kouri's study of relaxation in He and o-H 2 [16]; as well as Armstrong and McCourt's work on the pressure dependence of the line shape H 2 and its isotopes [17]. Comprehensive reviews of the field by C. J. Jameson contain many other examples [6,18].

A. Conformer Equilibria and Thermodynamic Barriers Using Direct Measurements A second approach has also developed in the last decade in which gas-phase NMR is applied not only to the study of individual nuclear magnetic properties, but also to the analysis of the dynamic processes that some of these molecules undergo. In the absence of solvent-solute interactions, gas-phase NMR is the method of choice in investigations studying conformational, tautomeric, and general isomeric equilibria because the isomer populations can be identified and measured correctly without perturbing the isomeric equilibrium. The NMR advantages of unambiguous spectral assignment, and spectral intensities being directly related to the relative populations of the absorbing species, make the study of many isomeric processes with moderate activation energies in the range of 20 to 85 kJ mo1-1 routinely accessible. Furthermore, variable temperature gas-phase NMR experiments have allowed the complete thermodynamic and kinetic characterization of such equilibria. Early examples of the NMR determination of gas-phase equilibrium constants are given by Harris for the Z-E isomerization in acetaldoxime and keto-enol tautomerism of acetylacetone [19] where signals for each independent isomer and tautomer respectively yielded information on their population ratios. More recently, Folkendt and co-workers have done extensive work on the keto-enol isomerization of a series of [~-diketones including acetylacetone (R 1 = R 2 = CH3), methyl acetoacetate (R 1 = CH 3, R 2 = OCH3), and ethyl acetoacetate (R 1 = CH2CH 3, R 2 = OCH3). [20] The general structure of the keto and enol tautomers being:

o

o

II

II

RI/C.cH2....I C. R2 -"

O''"H~o _

II

II

RfC.cH2...~C. R2

The high energy barrier to tautomer interconversion permits observation of distinct gas-phase NMR spectra of both forms at temperatures well above ambient thus allowing the quantitative gas-phase thermodynamic data to be obtained for the first time for these relatively nonlabile molecules. For each molecule, the vinyl resonance was integrated to determine the population of the enol form, and the methylene resonance was integrated to determine the keto form. The enol tautomer showed more stability for all three 13-diketones. A mean value for the equilibrium constant, Keq= enol/keto, was obtained at 19 different temperatures (372.8 K-444.9 K). The resulting temperature dependence of Keq was subsequently analyzed via a linear regression fit to the van't Hoff equation to obtain the thermodynamic parameters:

NMR Studies of Conformational Processes

119

ln(Keq)=_[---~-III]+[A~R]andAGO=AHO_TASO

(1)

Table 1 summarizes these parameters characterizing the keto-enol equilibria, where A refers to the difference between the enol and keto forms. The enol forms are significantly more stable, consistent with the inclusion of an intramolecular hydrogen bond in the structures and concurrent resonance stabilization. The low frequency torsional vibration of the keto forms can account for their significantly greater relative entropy. In solutions the keto-enol energy difference is much smaller than that observed in the gas phase and the observed phase dependence can be accounted for by electrostatic factors. Another example of the application of gas-phase NMR techniques to conformational equilibria is the investigation of conformational effects on the relative maxima in the N-O bond internal rotation potential function in primary nitrites of one to five carbon atoms Alkyl nitrites (ONOR) exist as mixtures of and conformers which differ in the orientation of the alkyl group with respect to the nitrosyl group.

[21,22].

anti

N

syn

.......

0

/

N

o//

syn

R

O anti

The magnitude of the N--O internal rotation barrier (ca. 40-48 kJ/mol) and the magnetic inequivalence of the t~-lH's and t~-13C's caused by anisotropy of the shielding tensor of the nitrosyl group make these systems especially amenable to NMR investigation. However, determination of equilibrium constants via integration of slow exchange 1H NMR resonances, as done for the keto-enol analysis, is complicated by the slightly lower barriers to intemal rotation of the larger alkyl nitrites, which necessitate low temperatures (below ca. 200 K) to obtain slow exchange spectra. Moreover, gas-phase slow exchange population measurements of alkyl nitrites are even more difficult to obtain due to the limitations of sample volatility. An indirect method, based on the temperature dependence of chemical

Table1. Gas-PhaseThermodynamic Parameters(EnoI-Keto)from Integrated 1H NMR Spectra

Compound

Acetylacetone Ethylacetoacetate Methylacetoacetate

AG~

mo1-1

-9.20 -0.33 -0.33

zS]I~

mol -I

-19.50 - 13.26 -12.72

z~k~~ m o l "-1 g -1

-34.56 --43.43 -41.55

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NANCY S. TRUE and CRISTINA SUAREZ

shifts is used to obtain the equilibrium constants. The temperature dependencies of 1H chemical shifts in the gas phase, which are due to changes in collision frequencies and changes in vibrational partition functions [23,24], are linear and small (< 1 • 10-4 ppm/K upfield relative to gaseous tetramethylsilane) in comparison with effects due to changing equilibrium constants. This is not the case in nitrite solutions due to the strong temperature dependence of the slow exchange chemical shifts which is similar in magnitude to the temperature-dependent changes in equilibrium constants. Consequently, the determination of gas-phase equilibrium constants is done from rigorously first-order, fast-exchange NMR spectra assuming that the observed methylene proton shifts in the alkyl nitrite spectra at fast exchange are weighted averages of the slow exchange syn and anti chemical shifts, Vobsd =

(2)

PsynVsyn + PantiVanti

where: Psyn + Panti = 1

(3)

geq = anti / syn = (Vobsd- Vsyn) / (Vanti - Vobsd)

(4)

The result is:

The gas-phase thermodynamic (van't Hoff) parameters M/~ AS~ and AG~ for the syn-anti conformer interconversion of these gaseous alkyl nitrites are presented in Table 2. For MeONO this data compared well with those obtained by direct integration of slow exchange spectra. [21,22] The agreement of these gas-phase NMR thermodynamic parameters with microwave and theoretical data reinforce the validity of this technique applied to the syn-anti equilibria. Comparison of these gas-phase data with solution values yields information on the effects solvents have on the conformer equilibria, which in the particular case of these alkyl nitrites reflect a balance of steric and dielectric effects.

Table 2. Gas-PhaseThermodynamic Parameters (syn-antl) from Fast-Exchange cz Proton Chemical Shifts

Compound

AG~

tool-s

Methyl nitrite Ethyl nitrite n-Propyl nitrite Isopropyl nitrite n-Butyl nitrite lsobutyl nitrite Neopentyl nitrite

1.49 (0.10) -1.29 (0.10) -1.64 (0.10) -2.99 (0.21) -1.51 (0.21) -2.26 (0.21) -2.80 (0.21)

AH~

mo1-1

3.48 (0.08) 0.72 (0.08) 0.46 (0.08) -4.60 (0.08) 0.86 (0.17) -0.24 (0.17) -1.33 (0.17)

zk~~ mo1-1 K-1

6.69 (0.42) 6.74 (0.63) 7.03 (0.63) -5.40 (0.63) 7.03 (1.67) 6.78 (1.67) 4.85 (1.67)

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121

One final example of conformer equilibria studied in the gas-phase by NMR techniques is the ever popular internal rotation about the C-N partial double bond in substituted asymmetric amides. Isbrandt and co-workers as early as 1973 were able to measure the barriers to internal rotation in solution in a series of asymmetrically N,N-disubstituted amides. [25] Presently, our group has undertaken the first attempt to characterize in the gas-phase this rotational equilibria in a pair of asymmetrically N,N-disubstituted amides (XCONCH3(CH2CH3); X = CF 3 and H). Analysis of the conformer population has been the focus of much theoretical work aimed at elucidating the role of the higher energy cis peptide bond in stable protein conformations [26,27]. Values of Pcis and Ptrans in N,N-ethylmethyl trifluoroacetamide (EMTFA) and formamide (EMF) (where cis and trans assignments were made in the formamide for the preferred isomer to have the bulkier substituent trans to the carbonyl oxygen, while in the other amides, the bulkier group is cis to the carbonyl) are obtained directly from signal integrations at each temperature. Preliminary results show that these values show no consistent variation (5% or less) throughout the temperature ranges analyzed. Gas-phase signalto-noise ratios and the large errors associated with integration techniques preclude an in-depth analysis of the enthalpy and entropy energies associated with the conformational process; therefore, listed in Table 3 are only the average values of Keq = Pcis/Ptrans and AG~ = - RTlnKeq for the equilibrium. Substituents should decrease AG~ in the same order that they lead to decreases in the rotational barrier. We observed such a trend in our liquid data. On the other hand, our gas-phase AG~ s show an opposing trend. This might be due to the absence of solute-solvent and solute-solute interactions. There is an extensive amount of work in the literature, both theoretical and experimental, dealing with the structure of amide solutions and their influence on barrier heights [28-30]; however, no such experimental work has been available for gaseous amides because of low vapor pressure conditions (N,N-ethylmethylacetamide has less than 0.4 torr at room temperature) and, in some cases, the absence of conformer separation. An increase in AG~ (619 J mo1-1) going from gas to liquid in EMF shows how important hydrogen-bonding perturbations are on the internal rotation potential function. Clearly, a systematic analysis of these perturbations and other solvent interactions would require further gas-phase conformational studies which at the moment are restricted to these few values.

Table 3. Gas- and Liquid-Phase Free Energies at 300 K for Internal Rotation in EMTFA and EMF

AG~300 J mo1-1(gas) AG~300 J mol-l (1% DMSO)

EMF

EMTFA

326 (6) 946 (19)

833 (16) 427 (8)

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B. Conformer Equilibria and Activation Barriers via Analysis of Long-Range Coupling Constants As described previously, dynamic NMR analysis (DNMR) has become the method of choice in the evaluation of conformational, tautomeric and isomer equilibria in systems with potential energy barriers ranging from 20 to 85 kJ mol -l. In the specific case of rotational isomerism, DNMR analysis provides direct information on the relative energy of rotamers, the potential barrier to internal rotation and information on chemical shifts (8) and coupling constants (J). Systems such as the amides and nitrites previously described have potential barriers large enough to yield NMR spectra representative of different stable rotamers, displaying various degrees of exchange. On the other hand, systems such as substituted ethanes and other substituted alkanes, have barriers to rotation sufficiently low (12-41 kJ mol -l) to allow rapid hindered internal rotation in the NMR scale; consequently, the observed spectra are weighed averages over the various rotamers involved. [31] Determination of physical parameters related to internal rotation about the C-C bond, such as relative energies of the rotational isomers and the potential barrier to internal rotation, from these rotationally averaged spectra is difficult [32]. The most important indirect method to obtain these parameters is based on the analysis of the temperature dependence of long-range coupling constants ("J" method) [33]. Other examples of indirect approaches used are the previously described temperature dependence of chemical shifts in the fast-exchange region used in the calculation of anti-syn equilibrium constants in nitrites and the pressure dependence of chemical shifts used to determine association constants for dimerization of acetic and trifluoracetic acid [34], and hydrogen bonding in methanol [35]. The J method has been extensively applied to the measurement of conformer equilibria and barrier heights in solutions. Gas-phase high resolution NMR spectroscopy can also yield well-resolved spectra suitable for complex spin system analysis. In the gas phase, spin-lattice relaxation occurs primarily through spin-rotation interactions. For large molecules at moderate pressures, tls are long and contribute negligibly to the observed line widths. Under these conditions, coupling constants can be measured in the gas phase with the same accuracy as in solutions. Inomata and Abe [36] have just recently obtained gas-phase values of 3JHH for 1,2-dimethoxyethane at different temperatures ranging from 5.80 to 5.85 Hz with a probable error of less than 0.05 Hz. Even though the temperature dependence of coupling constants in these compounds is small (<1 Hz per 100 range), a standard deviation in their temperature analysis of no greater than 0.05 Hz can also be achieved without difficulty. Furthermore, the innate absence of solvent effects in the gas phase provides the base reference to model completely the solvent dependency of such energies and barriers. The theory underlying the dependence of vicinal coupling parameters on internal rotation has been widely studied. Most approaches have followed the best fit method pioneered by Gutowsky et al. [37,38] which assumes a rotational isomeric

NMR Studies of Conformational Processes

123

state (RIS) model relating the experimentally determined coupling constants, <J>, to the relative populations, fi, and the corresponding parameters of the individual rotamers by <J> = Zf.Jj where Zf/= 1. This approach assumes that the temperature dependence of <J> results only from changes in the equilibrium proportions of the rotamers and searches for a set of parameters characteristic of each stable conformer to find the best fit to the experimental data. The conformational energy difference for the conformers i and j, AEi_j, can be related to the conformers' population fractions by assuming simple Boltzmann factors in the corresponding energies as f / / f j - exp (-AEi_j/RT). Classical examples of this type of analysis in the gas phase are provided by Hirano and Miyajima in the study on 1,2-difluoroethane and 1,2-disubstituted propanes [39-42]. More recently, Inomata and Abe have published a rigorous study of the conformation of 1,2-dimethoxyethane in relation to the gauche oxygen effect [36] associated with the internal rotation around the OC-CO bond. [43] The observed gas-phase IH-IH and 13C-lH NMR vicinal coupling constants were compared with those previously determined in nonpolar solvents. From these reported values and the conformational energies determined using the RIS, they reached the conclusion that the gauche oxygen effect is not caused by surrounding solvent molecules. The authors note that the value of the conformational energy E~ for the constituent bond OC-CO estimated to be-1.7 kJ mo1-1 is lower than results of recent MO calculations [44] by 4.2 kJ mo1-1. They also indicate that there is only a qualitative agreement between the fraction of the conformers estimated in their work and electron diffraction measurements. It is important to note that other authors have found discrepancies within results yielded by this "static" method. For instance, Govil and Berstein, [45] Heatley and Allen [46], and more recently Pachler and Wessels [47] have mentioned that this method yields inconsistent results and can only give accurate results with considerably difficulty. Another major approach developed by Lin [48] expresses the observed vicinal coupling constants as weighed averages over a cycle of rotation (i.e., ~ = 0 to = 27t). The potential energy as a function of the internal rotational coordinate ~, V(~), is taken as the weighting function. The observed vicinal coupling constant, <J>, is then given by, [49], J J(~))exp (-V(~) / R T ) d , <J>=

o

(5)

2x ; exp (-V(~)/RT)d 0

where J(~) is a modified Karplus equation. This rotational averaging approach has the advantage of sampling the entire population distribution as opposed to the RIS model which assumes that only stable

124

NANCY S. TRUE and CRISTINA SUAREZ

conformers are important. Lin's theory has been successfully applied in liquidphase studies to substituted ethanes of the type CHzX-CH2Y [50] and CH2XCHXY [51]. In the gas phase, Pachler and Wessel's study of the rotational isomerism of 1-bromo-2-chloroethane [47] further reaffirms these discrepancies of the "static" versus the "dynamic" approach. Analysis of the average vicinal coupling constants, measured in different solvents, assuming a "dynamic" model yielded values for the vicinal coupling constants of the individual rotamers in excellent agreement with those obtained from their gas-phase data. These values are at 305 K: trans rotamer Jr= 13.6 Hz; J'r= 4.9 Hz; and gauche rotamer 1/2 (Jcl + Jc2) = 0.9 Hz, 1/2 (J'cl + J'c2) = 7.8 Hz. Js are temperature dependent with Jr changing about four times as much as the other coupling constants. In this context, Pachler and Wessels explore the dependency of these vicinal coupling constants with temperature and solvent dielectric constant (e). This is based on a proposed modified Karplus equation, J(~) = a + B cos ~ + C cos ,2~ + D sin~ + E sin 2~

(6)

where the constants A...E are, to a first approximation, linearly related to the electronegativities of the substituents. The foundation for this work, and the latest developments of this approach are due to the extensive work done by Abraham and co-workers [52,53]. In their latest work [54], proton coupling constants are reported for 1,1,2-trichloroethane in 32 protic and aprotic solvents, as well as in the gas phase. The conformational equilibrium is thus analyzed in terms of long-range coupling constants using multiple regression analysis through the AbrahamKamlet-Taft equation, J / H z = c + d~5+ s ~ + a(z I + b[31 + h(~i~)1

(7)

where ~i (polarizability correction term), ~ (solvent dipolarity); c~l (solvent hydrogen-bond acidity); ~l (solvent hydrogen-bond basicity) and 82//(Hildebrand cohesive energy density). In this case, gas-phase data provides a decisive data point in the analysis of the variables, because of the absence of solvent effects and constrictions in the system. As more gas-phase systems are studied, more work will be expected to appear on the application of this J method to the temperature and solvent dependence of internal rotation barrier parameters and their parametrization, allowing for the extrapolation of theoretically calculated values. I!1.

GAS-PHASE NMR STUDIES OF CONFORMATIONAL KINETICS

Since its early beginnings, probably the best known application of DNMR spectroscopy has been to the evaluation ofkinetic barrier heights. The methods by which rate constants for the conformational processes have been obtained from NMR spectra have changed over time. In the vast majority of DNMR studies, the desired

125

NMR Studies of Conformational Processes

385.9 K .

~__~ .

.

.

.

.

.

338.1

329.1 K

320.1 K

i

5.41

,,

i

5.21

i

5.01

i

4.81

i

4.61

ppm Figure 1. Temperature dependence of gas-phase 500 MHz 1H NMR spectra of

N, N-ethyl methyl trifluoroacetamide. Top and bottom traces representcalculated and experimental spectra at the indicated temperature.

kinetic information has been obtained by analysis of exchange-broadened spectra. The NMR spectral lineshape is determined by a set of five parameters: (1) conformer chemical shifts in the absence of exchange, 6i; (2) spin-spin coupling constants, Ju; (3) relative populations of each conformer, Pi; (4) transverse relaxation times in the absence of exchange, t2i; and (5) the exchange rate constant, k. An iterative nonlinear least-squares regression analysis is performed using a computer program (we use DNMR5 [55]) to obtain the best fit of a calculated spectrum to the experimental spectrum, the so-called bandshape analysis technique. An excellent description of this and other methods can be found in the work of SandstrOm [56]. Figure 1 illustrates an example of the DNMR5-calculated (top) and experimental (bottom) traces for exchange-broadened spectra of N,N-ethylmethyl trifluoroacetamide gas at different temperatures.

1 26

NANCY S. TRUE and CRISTINA SUAREZ

A. Temperature-Dependent Gas-Phase NMR The evaluation of kinetic barrier heights (21-105 kJ mo1-1) from the temperature dependence of rates has been one of the most important contributions of DNMR to conformational processes. However, only a handful of these studies have addressed gas-phase processes, mainly due to the need for instrumentation improvements just recently achieved as described above. It has become customary to discuss exchange processes in terms of the Arrhenius equation and transition state theory (TST) of reaction rates [57] which is summarized by the Eyring equation. The Arrhenius equation in the following form is used to obtain the activation energy (Eact) and frequency factor (A) from the slope and intercept, In k -

~'

+ In A

(8)

where k is the rate constant, R is the gas constant, and T is the absolute temperature. This equation is valid at all pressures [57]. The kinetic activation parameters are determined from the Eyring equation in a similar fashion, In k -

-

+ In --h-- + --R--

w h e r e ~cis the transmission coefficient, kb is the Boltzmann constant, h is Planck's constant, and z~* and AS* are the enthalpy and entropy of activation respectively. Since this derivation involves transition state theory, it is valid only at the high pressure limit. If the precision of some experimental measurements is poor, only a AG~* determined from Eq. 10 at the coalescence temperature is generally reported: kbT AG* = - RTIn ~ hk

(1 O)

Comparison of the barrier heights in the gas phase with those found in the liquid phase can elucidate the role of solvent internal pressure on conformational interconversion when dielectrical interactions are minimal. The magnitude of these interactions can be estimated by considering an activation volume, AV*, for the process, defined as,

L

l

where 8 P i is the difference in internal pressures between the solution and gas phases (Pi for gases = 0). As mentioned above, only a handful of these studies have addressed gas-phase processes. Presently, the majority of the published work in the area investigates such conformational processes as internal rotation in several substituted N,N-dial-

NMR Studies of Conformational Processes

127

kylamides, alkyl nitrites, and N,N-dimethylnitrosamine, ring inversion in six-membered tings, pseudorotation in SF4, and the Cope rearrangement in bullvalene. At the high-pressure limit, all of these conformational processes exhibit first-order kinetics and the temperature dependence of the rate constants yields information, as previously described, about the kinetic parameters characterizing the process" AG*, AH*, and AS*.

Gas-Phase NMR Studies of Amides Variable-temperature NMR spectroscopy is a well-established technique in the study of conformational exchange kinetics in liquid amides. [1,58] Using DNMR spectroscopy, it is possible to measure the temperature dependence of interconversion rates in order to yield a complete set of kinetic parameters characterizing the internal rotation about the carbon-nitrogen bond. In the last several years, the development of gas-phase NMR methods has provided a valuable tool for the elucidation of the magnitude of intrinsic and environmental contributions associated with this conformational exchange. In order to characterize these contributions, a series of molecules with systematic variations in substituent size and polarity must be analyzed both in the liquid and gas phase. The determination of gas-phase amide rotational barriers encounters several difficulties. In addition to low vapor pressure at exchange temperatures (usually below 1 torr at 298 K), gas-phase limiting chemical shift differences between conformers (Act) are usually smaller than in the liquid because solvents and other intermolecular effects, not present in the gas phase, contribute to increased anisotropy of the proton environments. One dramatic case is N, N-dimethylformamide, which has an 83 Hz proton-limiting chemical shift separation in CC14 (10 vol. %) at 500 MHz, but shows magnetic equivalence of its proton dimethylamino resonances in the gas phase [59]. Also, smaller limiting shift differences result in lower slow-exchange and coalescence temperatures. This, combined with low vapor pressures, can make gas-phase 1H NMR study unattainable. Furthermore, the gas-phase rotational barriers of all the amides studied so far are lower than in the liquid. This increases the rate constant at a specific temperature, further lowering the temperature range which must be studied. Work on amides has included Feigel's barriers to hindered internal rotation in a series of N,N-dialkylamides (R'CONR 2 : R = CH 3, R' = F, C1, CH 3, CF3; R = i-Pr, R' = CH3; R = C2H 5, R' = H) and in N,N-dimethylthioacetamide (CH3CSN(CH3)2) [60]. A line shape analysis of their gas-phase 1H NMR spectra yielded AG*s which are at least 4-9.6 kJ mol -~ smaller than the values reported in solution; the differences could not be correlated with the structure and the barrier heights of the amides. Ross also studied the gas-phase NMR temperature dependence of a series of monoalkyltrifluoroacetamides in search of line-broadening effects indicative of chemical exchange [61]. LeMaster et al. [59,62] also reports that the barrier to rotation in N, N-diisopropylacetamide is lower in the gas phase than in solution, and furthermore in formamides it decreases with increasing size of the alkyl

128

NANCY S. TRUE and CRISTINA SUAREZ

substituents on the amide nitrogen. Gas-phase 13C NMR spectra of dimethylformamide have also been obtained and studied [63]. In our last work [64], a series of N-substituted trifluoroacetamides was chosen to study the effect of N-substituent size on the amide rotational barrier. Substituents ranged from methyl to relatively large isobutyl groups. Also, we are working on the analysis of asymmetrically substituted amides which is of special interest because they can display conformer population differences as discussed in a previous section. They have been the focus of much theoretical work aimed at elucidating the role of the higher energy cis-peptide bond in stable protein conformations [65,26,27]. Medium effects on conformer relative Gibbs energies reflect the stabilizing or destabilizing role of solvent interactions upon the minima in the peptide bond torsional potential function [66,67]. For nonpolar molecules and polar molecules in nonpolar environments, steric effects due to liquid packing forces can be quite large, approaching those of electrostatic interactions, and contribute considerably to the phase dependence of the conformational Gibbs energy. Amide-amide self-association, solvent interaction, solvent polarity, and internal pressure have proven to affect the rotation rate and may mask the intrinsic influence of substituents. Gas-phase values are very useful particularly as test cases and parametrization data in the development of molecular mechanics and semiempirical theoretical calculations designed to model

Table 4. Summary of Gas- and Liquid-Phase Free Activation Energies (298 K) for

Several Substituted Amides (R1CONR2R3)

Rl

R2

CF3 CF3 CF3 CF3 CF3 H H H H CH3 CH3 CH3 N3 F Cl Br CH2F CHF2

methyl ethyl ethyl i-propyl /-butyl methyl ethyl ethyl i-propyl methyl ethyl i-propyl CH3 CH3 CH3 CH3 CH3 CH3

R3

methyl ethyl methyl i-propyl /-butyl methyl methyl ethyl i-propyl methyl ethyl i-propyl CH3 CH3 CH3 CH3 CH3 CH3

AG ~ gas (kJ mo1-1) 69.0 (0.4) 67.4 (0.4) 68.6 (0.4) 66.1 (0.4) 68.6 (0.4) 81.2 (0.4) 82.8 (0.4) 80.3 (0.4) 79.5 (0.4) 65.7 (0.4) -59.8 (0.4) 69.0 (0.4) 71.5 (0.8) 64.4 (0.8) 59.0 (0.8) 63.6 (0.4) 72.8 (0.4)

AG ~ liq (kJ mo1-1) 75.7 74.5 (0.1) 75.1 (0.2) 68.2 (0.1) 72.4 (0.1) 86.6 85.3 (0.3) 85.3 86.6 76.1 74.0 65.3 74.1 75.7 (2.5) 69.0 (2.1) 65.7 (1.7) 71.1 78.7

Solvent

Ref

CC14 CCl4 CCi4 CCl4 CCI4 C2C14 DMSO Neat o-DCB Neat o-DCB o-DCB CCI4 CCl4 CC14 Neat ClPh CIPh

68 64 69 64 64 63 69 61 61 70 62 71 71 71 7l 71 71

NMR Studies of Conformational Processes

129

0.5 0.0 V

-0.5

V

-1.0

T

[]

V

-1.5 E~

-2.0

o

,,.4

O

V

o

vn

[] []

V

-2.5

o

D Y

[] 9

v

o

V

un

O

V

o []

9

-3.0

ff

[]

v

O O

-3.5

9

[]

-4.0 -4.5

.......

2.8

2.9

l

t

3.0

3.1

.....

l

3.2

'"I ..........

3.3

~

3.4

looo/T

figure 2. Eyring plot of gas- and liquid-phase exchange rate constants for disubstituted trifluoroacetamides (TFAA). o-N,N-diethyl TFAA gas; T-N,N-diisopropyi TFAA gas; m-N,N-diisobutyl TFAA gas; o-N,N-diethyl TFAA liquid; ~z-N,N-diisopropyl TFAA liquid; []-N,N-diisobutyl TFAA liquid.

this C-N torsional potential function. Table 4 summarizes the AG*s obtained for all the amides described in this section, and Figure 2 illustrates the difference between liquid and gas Eyring plots for three of the trifluoroacetamides (diethyl, diisopropyl, and diisobutyl).

Gas-Phase NMR Studies of Aikyl Nitrites Table 5 contains the gas-phase activation parameters for syne-~anti conformational exchange available for six primary alkyl nitrites. Two trends are apparent in the gas-phase activation parameters listed in this table. First, the barrier to syne--~anti

1 30

NANCY S. TRUE and CRISTINA SUAREZ

Table5.

Activation Parameters for syn(-~anti Conformational Exchange in Primary Alkyl Nitrites AG~298

AH ~t

AS~t

Compound

kJ mol -l

kJ tool -1

J g -1 mol --1

Ref

Methyl nitrite Ethyl nitrite n-Propyl nitrite n-Butyl nitrite Isobutyl nitrite Neopentyl nitrite

50.6 46.0 47.3 47.7 47.3 46.9

48.9 (1.7) 45.6 (1.2) 45.2 (1.7) 44.8 (1.2) 44.8 (1.7) m

-3.8 (1.7) -2.5 (5.8) -7.9 (7.9) -8.8 (5.0) -7.5 (6.3) m

72,73 74 74 75 75 74

(0.4) (0.4) (0.8) (0.8) (0.8) (0.4)

conformer inversion in methyl nitrite is higher by 2.9 to 4.6 kJ mo1-1 than that in the other primary alkyl nitrites studied. Second, gas-phase activation Gibbs energies, enthalpies, and entropies in the other five nitrites are remarkably similar within their relative errors. The Gibbs activation energies for the conformer inversion seem to parallel the relative stabilities of the syn conformer. Destabilization of the syn conformer would be expected to result from steric effects on the ground state. Substitution of the methyl substituent on the nitrosyl oxygen by larger alkyl groups would tend to bend the nitrosyl N-O bond out of the plane, decreasing its double bond character and thus the rotational barrier around this bond. It is the bulkiness of the substituents directly attached to the carbons bonded to the nitrosyl oxygen, not the overall size of the substituent, that affects the magnitude of the barriers. A similar trend was seen for the N,N-disubstituted trifluoroacetamides described in the previous section. Comparison of available solution data with the gas-phase values obtained show that the destabilization of the larger freely rotating transition state by solvent packing forces must be important in these substituted nitrites; therefore, the unexpectedly small phase dependence may be attributed to dielectric effects which are opposite in direction but similar (or slightly lower) in magnitude to these steric forces. For methyl nitrite AG~298is lower in solution than in the gas phase (AAG~298 = 0.8 kJ tool -~) while for the other primary alkyl nitrites is slightly higher (AAG?298 0.0 to 2.5 kJ mol-1).

Gas-Phase NMR Study of N,N-Dimethylnitrosamine Harris and Spragg were the first group to report a gas-phase DNMR study determining the barrier to internal rotation in N,N-dimethylnitrosamine [76]. The CH 3

/, CH 3

O

NMR Studies of Conformational Processes

1 31

free energy of activation, AGf, for the internal rotation was obtained from the coalescence temperature (431 K) spectrum as 88.3 kJ mo1-1. Two mechanisms are possible for chemical exchange in N,N-dimethylnitrosamine. Molecular orbital calculations demonstrate that rotation about the N-N bond has a lower energy barrier than inversion at the nitroso nitrogen. CNDO/2 calculations [77] yielded barriers of 76.1 and 306.7 kJ mo1-1 for rotation and inversion and MNDO calculations [78] with complete geometry optimization of both the ground and transition states yielded barriers of 54.4 and 334.7 kJ mo1-1 for rotation and inversion, respectively. Chauvel and co-workers reanalyzed this nitrosamine system in order to apply the complete bandshape method to the DNMR spectra obtaining first-order chemical exchange rate constants which are ca. 25 times faster than those observed in neat liquids at corresponding temperatures [79]. The associated kinetic parameters [Eact(Oo): 85.8 (4.6) kJ mol-I; AH~:: 82.4 (4.1) kJ mol-1; AG*: 88.3 (1.7) kJ mo1-1] are consistent with a process proceeding via a freely rotating transition state.

Gas-Phase NMR Studies of Ring Inversion in Six-Membered Rings The structures and results of a systematic study of the gas-phase conformational dynamics of six-membered ring compounds in which substitution and heterocyclicity have been studied in the gas phase are presented in Table 6 and Table 7, respectively. The values for the barriers to ring inversion in six-membered rings from Table 7 are in the case of non- and mono-substituted rings slightly higher in the gas phase than in the liquid. As substitution increases, the trend reverses. The di- and trisubstituted nitrogen heterocycles have higher gas-phase barriers to inversion than their solution-phase counterparts. Secondly, rings containing the N-CH 3 group have slightly higher barriers to inversion than cyclohexane or tetrahydropyran. Phase differences for all the ring inversion processes studied are small and can be the result of many different factors.

Table 6. Summary of Six-membered

Rings Studied

Name

Cyclohexane Cyclohexylfluoride Tetrahydropyran N-Methylmorpholine N-Ethylmorpholine N-Methylpiperidine N-Methylpiperazine N,N-Dimethylpiperazine 1,3,5-Trimethylhexahydro-1,3,5-triazine

Code

CHX CHF THP MM EM MP MPZ DMPZ TRZ

132

NANCY S. TRUE and CRISTINA SUAREZ

Table 7. Summary of Kinetic Parameters for Chair to Twist-Boat Ring Inversion in Six-Membered Rings

Compound

AG~298 kJ tool-l

AH~ kJ mo1-1

CHX CHF THP MM EM MP MPZ DMPZ TRZ

43.5 (0.8) 44.8 (0.4) 43.9 (0.8) 49.8 (0.4) 47.7 (0.3) 50.2 (0.4) 51.5 (1.2) 53.6 (0.8) 51.9 (0.8)

50.6 (2.1) 48.5 (5.4) 48.5 (7.9) 49.8 (0.4) 47.1 (2.3) 50.6 (0.8) 51.9 (1.7) 61.1 (1.3) 55.2 (2.09)

AS~ J K -1 tool -1 23.8 (2.1) 13.8 (5.4) 25.1 (33.5) 0.0 (2.1) -1.9 (4.9) 1 (4) 2.1 (1.7) 25.5 (4.6) 11.72 (7.5)

Ref

80 81 82 83 84 85 86 87 88

Gas-Phase NMR Studies of Structural Exchange in Sulfur Tetrafluoride and Bullvalene There is an extensive body of work concerned with the nature of the fluorinefluorine exchange mechanism in sulfur tetrafluoride (SF4) [89]. Studies of very pure liquid samples demonstrate that the temperature dependent spectral effects observed are attributable to an intramolecular process having the permutational character of a Berry pseudorotation [90]. Catalytic amounts of HF have been observed to increase the magnitude of the rate constants characterizing this process; therefore, the values of the gas-phase rate constants provide a way to elucidate the magnitude of the solvent effects and clarify the kinetic model of the process. Spring has studied the temperature dependence of the axial ~ equatorial fluorine exchange rates in SF4 and found them to be consistent with an activation energy for axial-equatorial exchange of 50.6 (29) kJ mo1-1 [91]. In the gas phase this activation energy is higher than that found in solution measurements which is compatible with a process occurring via a transition state smaller than the equilibrium configuration. The degenerate Cope rearrangement of bullvalene (tricyclo(3.3.2.0)deca-2,7,9triene) has also been studied extensively in condensed media and has just recently been reported in the gas phase. The dynamic mechanism can involve any of the three cyclopropane bonds that are present in the molecule, the chemical identity of bullvalene remaining unchanged during this process. Figure 3 shows one of these three possible rearrangement pathways. The direction and magnitude of solvent effects on the Cope rearrangement can be determined by comparing once again rate constants for gas phase samples with those obtained for bullvalene solutions. Previous studies have shown that increasing the pressure applied to a solution of bullvalene in CS 2 by 500 MPa increases the rearrangement rate constant by 10% at 19.8 ~ [92]. The small negative activation volume, AV*, of -0.5 cm 3 mo1-1, consistent with these results, indicates that significant charge separation does not

NMR Studies of Conformationai Processes

'Z4

133

3

4

"1

Figure 3. One of the three possible pathways in the degenerateCope rearrangement of bullvalene. occur in the transition state in solution. In the gas phase, rate constants characterizing the Cope rearrangement in bullvalene are found to be 15% lower than in the liquid [93]. This agrees with a prediction based on the pressure-dependent solution-phase results, which indicates a predominantly intramolecular process which is facilitated by solvent interactions. Gas-phase kinetic parameters for the rearrangement are: AG*298 = 54.8 (0.8) kJ mol-1; M-fl = 55.2 (1.2) kJ mol-1; and AS* = 1.7 (5) J mo1-1 K-1.

Gas-Phase NMR Ring Inversion Barrier of Cyclohexene Lastly, the use of gas-phase NMR measurements can help elucidate the magnitude of barriers even in situations where experimental conditions are not optimal for a complete analysis of the temperature dependence of the rate constants. Recently, the barrier to ring inversion to cyclohexene was reported to be 43 kJ mo1-1 using a two-dimensional potential energy surface for ring twisting and ring bending consistent with assigned gas-phase vibrational frequencies [94]. For a solution of cyclohexene in bromotrifluoromethane, M-fl for ring inversion, determined from analyses of exchange broadened 1H NMR lineshapes, is 22 kJ mo1-1 [95], which is 21 kJ mo1-1 lower than the value determined from the vibrational analysis. If the magnitude of the ring inversion barrier were as high as the vibrational analysis suggest, gas-phase NMR analysis of the exchange-broadening due to the process would be detectable at 298 K. NMR spectra were acquired at two different temperatures (195 K and 298 K) in order to compare our values with those obtained from the solution and the vibrational studies. At 195 K, the rate constant associated with the ring inversion is consistent with a Gibbs energy of activation AG~195 of 27 kJ mo1-1. Assuming that solvent effects on the limiting axial and equatorial chemical shifts of cyclohexene are comparable to those observed for other six-membered rings, and AS* for the ring-inversion process is small, the barrier to ring inversion is 30 kJ mol -~ or less, in agreement with the solution study, but in disagreement with the vibrational analysis.

1 34

NANCY S. TRUE and CRISTINA SUAREZ

The above examples show that comparison of gas and solution phase rate constants reveals the direction and magnitude of solvent effects on conformational processes. At present it is not possible to attribute the observed solvent effects to specific causes for several reasons. Qualitative trends can be determined, but the lack of detailed studies in the liquid phase limits a quantitative analysis. First of all, liquid-phase studies generally do not obtain data which allows static and dynamic solvent effects to be separated [96,97]. Static solvent effects produce changes in activation barriers. Dynamic solvent effects induce barrier recrossing and can lead to modification of rate constants without changing the barrier height. Dynamic solvent effects are temperature and viscosity dependent. In some cases they can cause a breakdown in transition state theory [96]. Typically in solutions, conformer rate constants are measured as a function of temperature at constant pressure. In these isobaric experiments, rate constants change as a result of both changes in temperature and temperature-dependent changes in dynamic solvent effects. Jonas [98] has recently shown that rate constants obtained as a function of temperature at constant viscosity allow static solvent effects to be elucidated. These measurements are obtained by adjusting the pressure applied to the liquid sample. For cyclohexane, NMR studies of liquid samples at atmospheric pressure reported activation energies of 42.7 +_0.8 kJ mo1-1 in CS 2 solution. Jonas et al. [98] have shown that measurement at constant sample viscosity produce a significantly higher activation energy of 47.19 + 0.04 kJ mo1-1 in CS 2, very close to the gas-phase value of 52.47 + 2.0 kJ mo1-1 obtained at 2500 torr. Activation energies of ring inversion in other rings have not been measured at constant sample viscosity. Another factor which might complicate the analysis of phase-dependent differences in rate constants of conformational processes is a possible phase-dependent difference in internal vibrational redistribution (IVR) rate constants. Pressure-dependent studies of most conformational processes in the gas phase can be adequately modeled using Rice-Ramsperger-Kassel-Marcus's (RRKM) theory which assumes that IVR in critically energized molecules is ergodic on a time scale which is shorter than the reaction time. Recent molecular dynamic studies have suggested that in liquid solutions the cyclohexane ring inversion process may be non-RRKM. Since the density of states of molecules at the threshold energies required for conformational processes are low, efficient coupling mechanisms are required for RRKM theory to provide a valid description of these processes. Vibrational redistribution mechanisms which depend on rotational angular momentum coupling can play an important role in IVR processes in the gas phase, but are probably not effective in liquid solutions where rotational motion is largely quenched. If a conformational process does not follow statistical kinetic theories in one or both phases, this factor, as well as steric and dielectric effects, can influence the phase dependence of the rate constants for the conformational process. At present, it is not possible to determine all the factors responsible for the relatively small phase-dependent differences observed in ring inversion rate constants.

NMR Studies of Conformational Processes

135

B. Pressure-Dependent Gas-Phase DNMR One of the most interesting applications of gas-phase NMR spectroscopy is the study of the unimolecular kinetics of conformational and other low energy molecular processes. In the gas phase, the rate constants of many chemical exchange processes are pressure dependent at pressures high enough (above a few torr) to allow their measurement from the analysis of the exchange-broadened NMR lineshapes. This dependence occurs as a falloff of the first-order rate constant with decreasing pressure and an eventual change to second-order kinetics at very low pressures. For a thermally initiated unimolecular process, the observed pressure dependence is due to competition between bimolecular deactivation, A+M ~

k

k.

A* + M

(12)

and reaction of energized molecules (A*), (13)

A*~E-~p

where ka, ka, and k(E) are the activating, deactivating, and internal energy dependent reaction rate constants, respectively; M is any collider, i.e., another A molecule, a bath gas molecule, a reference gas molecule, or the NMR tube wall, and P is the final product. Solution of this Lindemann mechanism yields the macroscopic unimolecular reaction rate constant, kun~, E

-ku~i-

d[A] [

1 [A] dt - -

k(E)ka / k d

1+ k(E) / kd[M] dE

(14)

E*=0

which reduces in the high pressure limit t o kun i = k(E)ka[k d and in the low pressure limit to kun i = k a [M]. Several statistical kinetic theories, most importantly RRKM, have approached the dependence with some degree of success. The major feature that distinguishes RRKM theory from the others is the assumption that energy is redistributed rapidly and randomly among the various degrees of freedom of an energized molecule before that molecule reacts or is deenergized. All the internal energy within a reacting molecule is available for the reaction process. For this condition to be fulfilled, IVR within the reacting molecule must be ergodic and all the IVR rate constants, kivR, must exceed k(E), the energy dependent rate constant. According to RRKM theory: k(E) = (l/h) [aG*(E- Eo)Ip(E)] where o~ is the reaction path degeneracy; G*(E- E 0) is the sum of the states in the transition state evaluated at energy E; p(E) is the density of states in the reactant; and E 0 is the threshold energy. From the Fermi Golden Rule, klvR from a state "s" to a manifold of states "/" is kiv~ = (l/h) (Z,p,(E) I~s,I 2) where 9t is the state density in the manifold and the I Vst/

136

NANCY S. TRUE and CRISTINA SUAREZ

are the coupling matrix elements [96,99,100]. Since k(E) is inversely proportional to p(E), at high-state densities the relationship klvR >> k(E) is usually valid and thermal gas-phase reactions such as isomerization and decomposition reactions have been successfully modeled over limited pressure ranges using statistical theories such as RRKM theory [100]. At lower state densities, kivR for at least some of the possible vibrational redistribution pathways within the molecule may be smaller than k(E) as calculated from statistical theories and the fundamental assumption common to all statistical kinetic theories may not be valid. At the energies required for conformational conversions and other exchange processes which are amenable to study by NMR spectroscopy, the reacting molecules have state densities which are much lower than those of molecules undergoing isomerization and decomposition reactions which are generally found to obey RRKM kinetics. Whether these systems can be modeled with RRKM theory is a question of current interest. Table 8 lists molecules for which pressure-dependent gas-phase chemical exchange rate constants have been obtained. Reported gas-phase NMR studies have compared experimental pressure-dependent rate constants obtained from lineshape analyses with those calculated using RRKM theory which assumes stochastic IVR. This method is sensitive to significant departures from RRKM theory but cannot distinguish smaller departures due

Table 8. Threshold Energies, State Densities and Approximate k(E) Values of

Molecules Undergoing Low Energy Processes E0

Molecule Methyl nitrite (syn) Ethyl nitrite (syn) n-Propyl nitrite (syn) n-Butyl nitrite (syn) Cyclohexane Cyclohexyl fluoride Tetrahydropyran N-Methylmorpholine N,N-Dimethylpiperazine 1,3,5-Trimethylhexahydro-l,3,5-triazine N-Methyl piperidine N-Methyl piperazine Sulfur tetrafluoride Bullvalene

p(E0) a

(kJ mo1-1) (states/cm -1) 53.1 (0.8) 34 48.1 (1.7) 405 48.9 (0.8) 1.7 x 105 48.1 (0.8) 6.9 x 105 52.3 1500 48.1 3100 50.2 700 51.0 9800 61.5 1.3 x 106 55.2 5.9 x 106 51.5 2.4 x 104 53.1 1.8 x 104 53.1 60 54.8 (0.8) 16550

<

k(E) >b (s-l)

2.6 x 1.2 x 1.6 x 2.8 x 2.0 x 3.7 x 3.5 x 9.6 x 3.0 x 3.0 x 9.3 x 8.2 x 3.0 x 5.7 x

109 109 108 107 109 109 109 107 107 105 107 107 108 106

Ref 101,102 75 75 75 80 81 82 83 87 88 85 86 91 93

Notes: aCalculatedfrom modified direct count procedures. bEstimated from experimental values of Pla, the pressure where the pseudounimolecular rate constant is 1/2 its high pressure limiting value, assuming hard-sphere or modified hard-sphere collision diameters.

NMR Studies of Conformational Processes

137

to uncertainties in transition-state structural and vibrational characteristics which are necessary for the RRKM calculations and which must be estimated. It is usually assumed that RRKM kinetic theory provides the best description of thermal unimolecular reactions with activation energies above ca. 210 kJ mo1-1 [103]. In small molecules with lower activation energies, where state densities are low to moderate (<106 ] cm-l), the vibrational redistribution process is not well understood. The modeling of the pseudounimolecular rate constant's falloff with decreasing pressure can provide useful information in regard to the redistribution. After the high-pressure limit has been established, the unimolecular rate constants (kuni) obtained are plotted versus pressure to generate the experimental falloff curve. The RRKM specific rate constants [103] are calculated from:

~_~9(g~)

(15)

hN(E;) The unimolecular rate constants are calculated from,

oo Ep(E~)e_ET/RTdE, e l l k(ET/ kuni- hQiQ2 1+

(16)

E*=E0 where Q~ is the partition function for extemal rotation o-l (IAIBIc)U2 where o is the rotational symmetry number, h is Planck's constant, Q2 is the partition function for the internal active modes (vibrations), E* is the energy of the molecule, R is the gas constant, T is the absolute temperature, and o~ is the collision frequency. The superscript zl: denotes the transition state, while nonsuperscripted variables pertain to the ground state molecule. The computer program RRKM written by Hase and Bunker [104] calculates, by the Rabinovitch-Setser direct count option, the sums and densities that are needed for the program FALLOFF [105]. A direct analysis of the fit between the computer generated falloff curve and the experimental dependence will determine the applicability of the statistical IVR and the RRKM modeling.

Alkyl Nitrites The first conformational process for which pressure-dependent rate constants were obtained using gas-phase NMR spectroscopy was the syn-anti exchange process of methyl nitrite (MeONO). Figure 4 shows gas-phase 1H NMR spectra of pure MeONO as a function of pressure at 262 K. At low pressures the rate of the exchange process is slow on the NMR timescale and the spectrum is resolved into resonances of the syn and anti conformers. As the pressure increases, the exchange rate increases and causes the syn and anti resonances to ultimately collapse into a single line. Our first study reported pressure-dependent rate constants obtained for pure MeONO at pressures between 4.6 and 364 torr, and for MeONO/CO 2 mixtures

96.6

......

A

175.3

31.2

56.7

21.3

" " "18w0() ' "16104) " 14100 ' 12100 " "IC~00" "8(~0"" "6(~0"' "4~0

Hz

Figure 4. Pressure dependent 1H NMR spectra of pure methyl nitrite gas at 262.2 K. Labels indicate pressure values in torr. The anti resonance is downfield.

138

NMR Studies of Conformational Processes

139

containing 15.3 torr of MeONO and C O 2 pressures between 12.9 and 701 torr [72]. In this early study, pressure-dependent rate constants for the syn-anti conformational exchange process were calculated using RRKM theory. The curvature and location of the calculated falloff curves are sensitive to the threshold energy, collisional cross section, and the state density in the reactant and transition state. Satisfactory agreement was obtained between calculated and experimental pressure dependent rate constants using a collision diameter of 3.0 ,~, and a threshold energy of 52.3 kJ mo1-1 which are reasonable for this process. State densities were calculated from experimental vibrational frequencies and the transition state model used vibrational frequencies which were the averages of those of the syn and anti conformers. This study concluded that the syn-anti conformational exchange process of MeONO is adequately modeled using RRKM kinetics, and therefore IVR is ergodic in critically energized MeONO. Our first study of MeONO used standard RRKM theory, which does not take into account the reversible nature of the con formational process to calculate rate constants. Modifications to the standard RRKM theory for reversible reactions were reported by Bauer et al. [106,107]. An additional study by Bauer et al. reported discrepancies between RRKM predictions and rate constants in the low-pressure bimolecular region for MeONO and concluded that certain regions of vibrational phase space are poorly coupled to the reaction coordinate [108]. This study compared theoretical calculations to experimental rate constants which were later shown to be significantly in error. The results reported by Bauer prompted a detailed reinvestigation of the bimolecular kinetics of the syn-anti exchange process of MeONO at pressures between 5 and 70 torr. This study reported a bimolecular activation energy of 46.9 (1.3) kJ mo1-1 and demonstrated that RRKM and reversible RRKM calculations can be used to model the process in this pressure region [109]. This study also demonstrated that the bimolecular rate constant, kbi = kuni/[MeONO], appears to increase as the pressure is decreased below 10 torr. This may indicate that IVR is not completely ergodic in this system and that additional degrees of freedom, poorly coupled to the reaction coordinate, may participate in the process when the time between collisions is sufficiently long, as suggested originally by Bauer. The quality of the data obtained in this low pressure region was poor and large uncertainties made it impossible to draw definitive conclusions. Recently, some additional low-pressure rate constants for the syn-anti process in MeONO have been obtained [110]. Figure 5 shows pressure-dependent rate constants (k/koo) for syn~--~anti exchange in pure MeONO at 262.2 K. The solid line corresponds to RRKM calculated values using E 0 = 49.8 kJ moF 1 and ~ = 5.4 ]k. Pressure-dependent rate constants for the syn-anti conformational process in larger alkyl nitrites provide a further test of the ability of RRKM theory to successfully model the kinetics of the internal rotation process in these molecules. Solution of the Lindemann mechanism shows that at the pressure where the rate constant is one-half of its limiting high-pressure value, P1/2, the frequency of deactivating collisions is comparable to , the average rate that critically

140

NANCY S. TRUE and CRISTINA SUAREZ 0.0

8

-0.5

-

-1.0

-

-1.5

.-4

-20 J-I

-2.5

-3.0

-

-3.5 2

I

I

3

4

I

I

5

6

7

Ln(P)

Figure 5. Pressure dependent rate constants (k/koo) of methyl nitrite gas. Solid line represents RRKM calculated rate constants as described in the text.

energized molecules become products. RRKM theory predicts to be inversely proportional to the total state density of the reacting molecule, p(E). In a homologous series of molecules which undergo the same reaction or process, an increase in the available state density should cause a corresponding decrease in and P1/2, if other factors such as the sum of states in the transition state remain relatively constant. Pressure-dependent gas phase NMR studies of syn-anti exchange in alkyl nitrite/CO 2 mixtures reported P1/2 values of 312, 166, 26, and <3 torr for MeONO, ethyl nitrite, n-propyl nitrite, and n-butyl nitrite, respectively [75]. For all the nitrites studied, experimental rate constants were modeled using RRKM theory. This study concluded that the agreement between the rate constants calculated using RRKM

NMR Studies of Conformational Processes

141

theory with reasonable values of vibrational frequencies, collision diameters, collision efficiencies, and threshold energies and the experimentally obtained pressure dependent rate constants strongly supports the conclusion that IVR in this series of alkyl nitrites is rapid compared to and statistical. This indicates that the increased number of vibrational states due to each additional methylene group may participate in the IVR process and the energy acquired in these states is available for the conformational exchange process in all the nitrites studied.

Cyclohexane Pressure-dependent studies of cyclohexane (CHX) ring inversion employed SF 6 as a bath gas [80]. The samples studied contained 1 torr of cyclohexane and SF 6 pressures ranging from 5.9 to 4000 torr. The high pressure (unimolecular) limit occurred at pressures above 2500 torr and bimolecular kinetics were obtained at pressures below 200 torr. CHX ring inversion has a threshold energy of--41.8 kJ mo1-1 and a vibrational state density of-- 2500 states/cm -1 at the threshold energy. The rate constants obtained from analysis of 1H exchange-broadened lineshapes were compared to those calculated using RRKM theory using the transition state model proposed by Pickett and Straus [111]. The experimental falloff curve was adequately reproduced by the calculation indicating that IVR in critically energized cyclohexane is ergodic or nearly so over the pressure range studied. Recently, several studies have concluded that ring inversion in cyclohexane does not follow RRKM kinetics in solutions. Jonas has recently shown that cyclohexane ring inversion constants obtained as a function of temperature and pressure indicate that coupling of cyclohexane molecules to the solvent bath is weak and causes a breakdown of RRKM theory [98]. Molecular dynamics studies of cyclohexane ring inversion conclude that the solvent causes a qualitative breakdown of transition state theory and the RRKM picture of unimolecular kinetics for this process in solution [112,113]. In the gas phase, Coriolis coupling, as well as the recently described vibrationally induced Rotational axis switching mechanism [114], facilitate the IVR process. These are both rotationally mediated IVR mechanisms. In solutions, rotational motion is essentially quenched and this may also impede IVR in solvated cyclohexane, causing the reported non-RRKM behavior. Pressure-dependent studies of ring inversion of the six-membered rings previously described in the temperature-dependent section (see Table 6 for their structures and code names) have also been reported. Table 8 lists threshold energies and state densities for these molecules. In all cases, the pressure-dependent ring inversion rate constants could be adequately modeled using RRKM theory.

Aziridine Nitrogen inversion in aziridine and 2-methylaziridine was studied using 1H NMR by Bauer et al. [ 115]. This study observed no dependence of the nitrogen inversion rate constant of aziridine on pressure down to a pressure of 5 torr, in sharp contrast to RRKM calculations which indicated that this system should be in the bimolecular

142

NANCY S. TRUE and CRISTINA SUAREZ

regime at pressures below 100 torr. A less extensive set of measurements was made with 2-methyl aziridine, which did show a small but clear fall off in the rate constant with pressure. These results were interpreted in terms of a regional phase space model, previously proposed for methylnitrite.

Bullvalene The degenerate Cope rearrangement of bullvalene (tricyclo(3,3,2,0)deca-2,7,9triene) exchanges protons in vinylic and allylic environments. It was possible to obtain pressure-dependent exchange-broadened lineshapes of bullvalene at 355.8 K at its vapor pressure, ca. 1 torr, as functions of the pressure of SF 6 which was used as a bath gas [93]. Measurements were made at SF 6 pressures between 7 and 2580 torr. Low vapor pressure and large linewidths precluded spectral acquisition below 330 K and it was not possible to observe slow exchange spectra in the gas phase. This resulted in additional uncertainties in calculated rate constants since limiting chemical shifts could not be determined in the gas phase and were estimated from liquid studies. Unimolecular rate constants were observed at pressures above ca. 250 torr. P1/2 is estimated to occur below 5 torr and it was not possible to observe rate constants in the bimolecular region for this molecule. To determine if the observed pressure dependence of these rate constants indicated statistical behavior, pressure-dependent rate constants were calculated using RRKM theory. Transition states with either biradicaloid or aromatic character have been proposed for the Cope rearrangement of bullvalene, and their structures, and relative energies have been determined from AMPAC calculations [116]. These calculations indicated that the aromatic transition state is considerably more stable. Rotational constants and vibrational frequencies of the ground state and both transition states were calculated using the computer program AMPAC using the AM 1 Hamiltonian. The RRKM calculations used threshold energies of 56 and 54.4 kJ mo1-1 for the biradicaloid and aromatic transition states, respectively, which are compatible with the experimentally determined gas phase activation energy. In Figure 6 the top solid line indicates the RRKM calculated curve for the biradicaloid transition state model and the lower curve represents the RRKM calculated falloff curve for the more flexible aromatic transition state model. Both curves were calculated with a collision diameter of 3.6 /k which is somewhat larger than expected, based on previous studies and considering the hard sphere cross sections ofbullvalene and SF 6. Additional calculations were performed which demonstrated that an unreasonably large collision diameter of 6.4 A is required to force an agreement between the RRKM calculation using the aromatic transition state model and the observed pressure-dependent rate constants. The biradicaloid transition state provided better agreement with the experimental data. It is unlikely that this model is the correct transition state for the process, however, since it has a much higher energy than the aromatic transition state model, according to the reported AMPAC calculations. Several factors can account for these results. First the AMPAC calculated frequencies of both transition state

NMR Studies of Conformational Processes 0.1

143

-0.2

8 .-t

=

-0.5

-0.8

-1.1

/ ....... I

I

i

2

3

4

.....

.....

5

1

1

. . . . . . I.

6

7

8

9

Ln (P)

Figure 6. Logarithmic plot values of kunJkooversus pressure for the Cope rearrangement of bullvalene (torr at the experimental temperature of 356 K). Experimental values are signified by solid circles. Pressures are the total sample pressure at 356 K. Errors in kuni/k~ a r e reported to 2o. The solid (upper) line represents the values calculated from RRKM theory using the biradicaloid transition state model. The lower line represents calculated rate constants using the aromatic transition-state model. The collision diameter was 3.6 ,~, in both cases.

models may be too low. Higher vibrational frequencies in the transition state shift the calculated falloff curves to lower pressures. The aromatic transition state model can fit the experimental pressure dependent rate constant data if the four lowest frequencies are assumed to be ca. 50 cm -l higher than their AMPAC calculated values. Secondly, the assumptions of RRKM theory may not be valid in this case. The data are not inconsistent with the more flexible AMPAC calculated transition state if a few of the vibrations are remoded from the state density count. In this

144

NANCY S. TRUE and CRISTINA SUAREZ

molecule some IVR pathways are forbidden based on symmetry arguments. Since only slight modifications in transition state models or calculational methods are required to achieve an acceptable agreement between the observed rate constants and RRKM calculations using the aromatic transition state model, this study concluded that IVR in critically energized bullvalene molecules is statistical or nearly so.

Sulfur Tetrafluoride The Berry pseudorotation process of SF4 is the first example of a low energy molecular process whose kinetics could not be modeled using RRKM kinetic theory. The Berry pseudorotation process exchanges the axial and equatorial fluorine atoms and has an activation energy of 50.6 kJ mo1-1 in the gas phase. Rate constants were obtained from analysis of exchange-broadened 19F NMR spectra between 298 and 356 K at pressures ranging from 55 to 2004 torr. Extensive sets of pressure dependent rate constants were obtained at 308, 328, and 348 K. P1/2, the pressure where the pseudounimolecular rate constant is 1/2 its limiting high pressure value, was approximately 170 torr. For this small 5-atom molecule, state densities at the threshold energy required for the pseudorotation process to occur are -60 states/cm-1 and based on results obtained for other systems, one would expect significantly larger P1/2 values for this molecule. Pressure dependent rate constants were calculated for SF4 using RRKM theory. Using reasonable values of these parameters~i.e., a threshold energy of 50.2 kJ mol -~, consistent with the experimentally determined bimolecular activation energy; a collision diameter of 2.2/k consistent with the experimental spin-rotation cross section; reactant state densities calculated from vibrational data; and reasonable transition state state densities~produced an RRKM calculated falloff curve with P1/2 of 2250 torr. Additional calculations varied the threshold energy, collision diameter, and transition state state density to determine the sensitivity of the RRKM calculation to these parameters. Clearly unreasonable values of these parameters were required to force an agreement between the experimentally determined and RRKM calculated pressure-dependent rate constant data. It was suggested that IVR for critically energized SF 4 molecules is slower than and may be the limiting factor in determining the overall reaction rate constant. This may be due to low-state density at the internal energy required for the reaction, coupled with symmetry restrictions which may prevent IVR between some vibrational modes. For most of the molecules discussed above, the experimentally determined pressure-dependent gas-phase rate constants can be modeled adequately with RRKM theory. SF4 and possibly aziridine are the exceptions. Due to uncertainties in the model parameters used in RRKM calculations, and the sensitivity of the calculations to these parameters, only qualitative conclusions can be drawn from the observed agreement. Since major departures are not observed, it can be

NMR Studies of Conformational Processes

145

concluded that IVR in the reacting molecules is statistical or nearly so. Regional phase-space theories, using different model parameters, can also provide satisfactory models for many of these processes if only small subsets of vibrational states are considered to be weakly coupled to the reaction coordinate. The suggestion of an increase in bimolecular rate constants with decreasing pressure observed in some data supports these models. Better data in the low-pressure region is necessary in order to draw any definite conclusions. Further tests of kinetic theories for conformational processes require extensive bimolecular and high pressure data. In the high-pressure region, Kramer's theory predicts a slow rise in the observed rate constant with pressure, followed by a decline at pressures above the turnover point [96]. Non-RRKM effects can change the dependence of the rate constants on pressure in this region [117]. With the exception of isomerization of trans-stilbene in an electronically excited state [118], rate constants of conformational processes have not been measured in the gas phase at pressures above a few atmospheres. This is clearly an area for future work.

C. Bimolecular Rate Constants and Collision Efficiencies The main interest in unimolecular reactions in the low-pressure region is concentrated on the change in order, shape, and position of the falloff Curve, and, in favorable cases, with the bimolecular rate constant. In order to study these characteristics, it is imperative to reduce the scatter of data due to temperature gradients and poor signal-to-noise ratios. As seen in Eq. (9) temperature has an exponential effect on the rate constant, whereas pressure dependence is, at most, linear. Temperature gradients across the active volume should be <0.2 K with measurements accurate to 0.1 K. Moreover, transverse relaxation times, tzS, decrease with pressure, t2 line broadening becomes more significant than that due to exchange, introducing large errors in the rate constant. Therefore, ca. 5 to 20 torr is an experimental lower limit for the determination of these rate constants. The addition of an inert bath gas to increase sample pressure can be accounted for if the collision efficiency of the bath gas is taken into consideration. Rate constants in the low-pressure bimolecular region are determined by the rate of activating collisions and are dependent on the nature of the collider. Relative energy-transfer efficiencies can be obtained from rate constants measured as a function of bath gas pressure. Bath gas relative collisional efficiencies have been measured for the methyl nitrite (MeONO) conformer conversion [119] and for ring inversion in cyclohexane (CHX) [120] using gas-phase NMR spectroscopy. They are the first conformational processes to be extensively studied in the bimolecular region. The bimolecular kinetics of the methylisocyanide (MIC; CH3NC) isomerization, which has a much higher activation energy, was studied earlier using conventional kinetic techniques [121].

146

NANCY S. TRUE and CRISTINA SUAREZ

Rabinovitch and co-workers found that the Lindemann mechanism is adequate for modeling the pressure dependence of bimolecular region unimolecular rate constants for extracting collision efficiencies for the methyl isocyanide isomerization [122]. For the conformer conversion of molecule A at constant temperature, it can be written as, k.~ A + M _2!_. A* + M k~ k3

A, --->A where M is any collider, i.e., another A molecule, a bath gas molecule, a reference gas molecule, or the NMR tube wall. Then, in the bimolecular region the unimolecular rate constants can be expressed as, kNMR= kuni - ~ [ a ] + k~l[M] + ko

(17)

where [A] and [M] refer to the densities of the molecule and bath gas. In the absence of a bath gas (or in the presence of a small fixed pressure of a bath gas) and at constant temperature, where only the density of A is varied, Eq. 17 can be written as,

kuni =/r [a] + ko

(18)

where k0 may incorporate the additional term k~l[M] 0 for the small, constant amount of bath gas. This expression is valid as long asthe total effective pressure remains in the bimolecular region. Analysis of pressure-dependent unimolecular rate constants for conformer conversion in samples of pure A at pressures below PI/2 (unimolecular-bimolecular transition point) using Eq. 18 via linear regression yields a slope ofk~l. If the pressure of A is held as low as possible and fixed whereas the pressure of the bath gas is varied, Eq. 17 becomes, kun i = kml[M] + k0

(19)

where k0 incorporates the additional term ~[A] 0 for the small, constant pressure of A. Analysis of pressure dependent unimolecular rate constants for A/bath gas mixtures in the bimolecular region using Eq. (19) via linear regression yields a slope of k~l. The collisional energy transfer efficiency of the bath gas relative to A in the activation of A in the conformer conversion process on a pressure per pressure basis is then written as 13p= k~l / k~l. Corrections for the relative velocities of the colliding partners and their cross sections produces the quantity, ~C "- [~p X (~AM/~AA) 0"5 X (SAA/SAM)2

(20)

NMR Studies of Conformational Processes a

147

6.7

5.0

i

800

i

600

i

i

400

Frequency

200 Hz

Figure 7. Pressuredependent gas-phase 1H NMR spectra, acquired at 300 MHz and 249,1 K, of exchanged-broadened axial and equatorial CHX protons. The labels indicate the total sample pressure in torr at 298 K. Samples contained (a) 0.5 torr of gaseous TMS as a reference and increasing amounts of CHX gas and (b) 0.7 torr of TMS, 3.0 torr of CHX, and increasing amounts of xenon as a bath gas. The x axis is in units of Hz referenced to gaseous TMS (0 Hz). (continued) which is the collisional energy transfer efficiency on a collision per collision basis. Since the quantities SAA and SAMwere not known for CHX and MeONO, collision efficiencies [3~ = ]3p x (laaM/PAA) ~ were calculated for comparison purposes. In their investigation of MIC isomerization, Rabinovitch and co-workers found that when looking for factors influencing trends in collision efficiency, [3~twas a more reliable parameter to study than [3c. Typical experimental spectra and rate constants for CHX are shown in Figure 7a and 7b. Figure 7a shows the effect of increasing pressure of CHX gas on the NMR line shape. The spectra were acquired with samples of 0.5 torr of TMS reference and between 1.1 and 7.6 torr (298 K ) o f C H X . Figure 7b shows the dependence of the NMR line shape on increasing pressure of xenon gas with a constant pressure of CHX and TMS (3.0/0.7 torr). This concentration of CHX was chosen so the dilution factors would be as large as possible, while keeping signal-to-noise ratios adequate for analysis. Figures 8a and 8b show

148

NANCY S. TRUE and CRISTINA SUAREZ

b

.6

_j

I

800

..

I

600

I

400

Frequency

Figure 7.

I

200 Hz

(continued)

representative plots of kuni versus pressure for the pure CHX and CHX-Xenon mixtures, respectively. For all the mixtures studied, the rate constants obtained exhibit a linear pressure dependence. Table 9 lists 13~tvalues for the 16 bath gases reported in the MeONO study and for the 18 bath gases reported in the CHX study. 130 values of these gases measured relative to MIC for the methyl isocyanide isomerization are also shown for comparison purposes. For MIC and MeONO similar trends in bath gas relative efficiencies are observed. For these processes, strong correlations between bath gas collision efficiency and polarizability and bath gas collision efficiency and boiling point were observed. This result indicates that the attractive portion of the intermolecular potential is important in determining collision efficiency. It was found in the case of MeONO conformer conversion and isomerization of MIC and ethyl isocyanide that there is a high correlation (R > 0.9) between these properties and 13~tfor all bath gases studied. For nonpolar CHX, [3~ also correlates positively with polarizability for the 18 bath gases studied when they are treated collectively. Separate trends and correlations are observed for noble gases, diatomics, and polyatomic molecules. There is no correlation between 13~ and polarizability for the five noble gases

NMR Studies of Conformational Processes

149

cyclohexane/no

bath gas

200

I

150 0 (i) o9 ~

r

100

50 2

4

6

8

10

Pressure (Torr ot 298 K)

Figure 8. Plots of kuni (S-1) versus total sample pressure (298 K) for (a) pure CHX and (b) for CHX-xenon mixtures which contained 3.0 torr of CHX, 0.7 torr of TMS, and varying pressures of Xe. The linear nature of the plot shows that the effective pressures are in the bimolecular regime. (continued)

studied, but 13~tcorrelates positively (R = 0.99) with polarizability for the four homonuclear diatomic molecules studied when they are treated as a group. If the linear triatomic molecule CO 2 is added to the set of diatomics, the correlation coefficient is not significantly decreased. The observed collision efficiencies of the larger polyatomic molecules are strongly correlated with polarizability, showing a greater dependence than the linear polyatomics. 13~tas a function of boiling point displays similar trends. This suggests that for monoatomic gases colliding with CHX, the attractive portion of the intermolecular potential is not the dominating factor in collisional energy transfer. As the molecular complexity of the bath gas increases, however, it appears to play a more important role. General relative trends are similar in all three systems; [3~tincreases with molecular complexity, with most

150

NANCY S. TRUE and CRISTINA SUAREZ xenon ......

I

180

w,-

I

o (19 co 1 5 0 V ~ c"

120

I

I

50

100

Pressure

............ ! ...........

150

i

200

(Tort at 298 K)

Figure 8. (continued) CHX-bath gas collision efficiencies being approximately 20-30% of those determined for MeONO and MIC. Overall, the values of I]~tfor this series of bath gases colliding with CHX are much smaller than those obtained in the MeONO and MIC systems. This may be due in part to the absence of a dipole moment in CHX. For the He/MeONO system, the temperature dependence of [3rt for He was determined [119]. An 18% decrease in [3rtover a 31 K increase in temperature was observed. This trend is similar to that observed previously for MIC [122] and can be explained by considering the size of the vibrational partition function of MeONO which increases by 19% over the 31 K temperature interval studied. Previous studies of collisional energy transfer in MeONO have shown that the vibrational degrees of freedom of the colliders is important. Thus, an inert gas such as helium, which has only translational degrees of freedom, would be expected to show a decrease in collision efficiency relative to MeONO on itself with increasing temperature.

NMR Studies of Conformational Processes

151

Table 9. Bath Gas Collision Efficiencies, 131a,for Cyclohexane (CHX), Methyl Isocyanide (MIC), and Methyl Nitrite (MEONO) Bath Gas

CHX He Ne

Ar Kr Xe 2H2 H2 N2 02 CO CO2 NH3 SF6 CBrF3 CBrH3 CCIF3 CCIH3 CCIFCF2 CH2CF2 Acetone-F6

CHX

1.00 (04) 0.0185 (13) 0.0182 (13) 0.0299 (19) 0.0272 (20) 0.0191 (09) 0.0256 (19) 0.0287 (42) 0.0461 (151) 0.0520 (111 ) 0.0745 (44) 0.114 (06) 0.155 (19) 0.176 (15)

0.0722 (09) 0.0976 (25) 0.135 (04) 0.134 (05) 0.147 (06) 0.100 (05) 0.087 (7) 0.19(01) 0.47 0.33 0.58 0.52

(5) (01) (01) (03)

MEONO

0.086 (5) 0.24 (3) 0.37 (5)

0.13 (1) 0.41 (6) 0.44 (7) 0.23 0.55 (4) 0.47 (3) 0.79 (7)

0.60 (05) 0.180 (25) 0.51 (01) 0.277 (17) 0.53 (02) 0.286 (15) 0.95 0.35 0.56 0.64

Acetone

CH4 C2H6 C3H8 C4F8 n-C4H10 n-CsH12

MIC

(05) (4) (4) (5)

0.33 0.52 0.64

0.485 (35) 0.74 (5) 0.85 (6)

0.89 0.99

IV. C O N C L U S I O N Gas-phase NMR spectroscopy has been used to obtain equilibrium constants and rate constants for many low-energy molecular processes. These data have been used to address questions regarding the relative stability of conformers and tautomers in the gas phase, the kinetics of exchange processes in the gas phase, and the direction and magnitude of solvent effects on these equilibria and processes. Most of the studies have appeared in the last 10 years. Continued progress in NMR instrumentation and techniques as well as considerable recent developments in kinetic theory ensure that the next 10 years will see many novel applications of gas-phase NMR spectroscopy.

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REFERENCES A N D NOTES 1. Jackman, L.M.; Cotton, EA. Eds. Dynamic Nuclear Magnetic Resonance Spectroscopy; Academic: New York, 1975. 2. See for example: Ernst, R.R.; Bodenhausen, G.; Wokaun, A. Principles of Nuclear Magnetic Resonance in One and Two Dimensions; Oxford University Press: New York, 1987. 3. See for example: Campbell, D.M.; Mackowiak, M.; Jonas, J. J. Chem. Phys. 1992, 96, 2717. 4. Weiss-Lopez, B.E.; Winegar, E.D.; True, N.S.J. Phys. Chem. 1988, 92, 4052. 5. Folkendt, M.M.; Weiss-Lopez, B.E.; True, N.S.J. Phys. Chem. 1988, 92, 4859. 6. Jameson, C.J. Chem. Rev. 1991, 91, 1375. 7. Armstrong, R.L. Magn. Reson. Rev. 1987, 12, 91. 8. Harris, R.K.; Rao, R.C. Org. Magn. Reson. 1983, 21,580. 9. Govil, G. AppL Spectrosc. Rev. 1973, 7, 47. 10. Jameson, A.K.; Jameson, C.J. Chem. Phys. Lett. 1987, 134, 461. 11. Jameson, C.J.; Jameson, A.K.J. Magn. Reson. 1985, 62, 209, and references therein. 12. Govil, G. hldian J. Chem. 1971, 9, 824. 13. Fleischer, U.; Schindler, M. Chem. Phys. 1988, 120, 103. 14. Schindler, M. J. Am. Chem. Soc. 1987, 109, 5950, and references therein. 15. Sanctuary, B.C.; Selwyn, L. J. Chem. Phys. 1978, 69, 2157, and references therein. 16. Kouri, D.J.J. Chem. Phys. 1978, 69(11), 4999. 17. McCourt, ER.; Tenti, G. Chem. Phys. 1978, 32, 23, and references therein. 18. Jameson, C.J. Nucl. Magn. Resort. 1985, 14, 1. 19. Harris, R.K.; Rao, R.C. Org. Magn. Reson. 1983, 21(9), 580. 20. Folkendt, M.M.; Weiss-Lopez, B.E.; Chauvel, J.P. Jr.; True, N.S.J. Phys. Chem. 1985, 89(15), 3347. 21. Chauvel, J.P; True, N. S. J. Phys. Chem. 1983, 87, 1622. 22. Conboy, C.B.; Chauvel, J.P.; Moreno, P.O.; True, N.S.; Ott, C.M.J. Phys. Chem. 1986, 90, 4353. 23. Jameson, C.J.; Jameson, A.K.; Cohen, S.M.; Parker, H.; Oppusunggu, D.; Burrell, P.M.; Wille, S. J. Chem. Phys. 1981, 74, 1608. 24. Jameson, C.J.; Jameson, A.K.; Oppusunggu, D. J. Chem. Phys. 1984, 81, 2571. 25. Isbrandt, L.; Tung, W.C-T.; Rogers, M.T.J. Magn. Reson. 1973, 9, 461. 26. Sapse, A.M.; Mallah-Levy, L.; Daniels, S.B.; Erickson, B.W.J. Am. Chem. Soc. 1987, 109(12), 3526. 27. Mirkin, N.G.; Krimm, S. J. Am. Chem. Soc. 1991, 113(26), 9742. 28. Perrirn, C.L.; Dwyer, T.J.; Rebek Jr. J.; Duff, R.J.J. Am. Chem. Soc. 1990, 112, 3122. 29. Jorgensen, W.L.; Gao, J. J. Am. Chem. Soc. 1988, 110, 4212. 30. Spencer, J.N. et. al. J. Chem. Phys. 1981, 85, 1236. 31. Gutowsky, H.S.; Belford, G.G.; McMahon, P.E.J. Chem. Phys. 1962, 36, 3353. 32. Abraham, R.J.; Bretschneider, E. Internal Rotation in Molecules; Orville-Thomas, W. J. (Ed); John Wiley and Sons: London, 1974, Ch. 13, and refs. cited therein. 33. Jackman, L.M.; Cotton, EA., Eds. Dynamic Nuclear Magnetic Resonance Spectroscopy; Academic: New York, 1975. 34. Lumbroso-Bader, N.; Coupry, C.; Baron, D.; Clague, D.H.; Govil, G. J. Magn. Reson. 1975,17(3), 386. 35. Clague, D.H.; Govil, G.; Bernstein, H.J. Can. J. Chem. 1969, 47(4), 625. 36. Inomata, K.; Abe, A. J. Phys. Chem. 1992, 96, 7934. 37. Gutowsky, H.S.; Belford, G.G.; McMahon, P.E.J. Chem. Phys. 1962, 36(12), 3353. 38. Vold, R.L.; Gutowsky, H.S.J. Chem. Phys. 1967, 47(7), 2495. 39. Miyajima, T.; Hirano, T.; Sato, H. J. Molec. Struc. 1984, 125, 97. 40. Hirano, T.; Miyajima, T. J. Molec. Struc. 1985, 126, 141.

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83. LeMaster, C.B.; LeMaster, C.L.; Tafazzoli, M.; Suarez, C.; True, N.S.J. Phys. Chem. 1990, 94, 3461. 84. Tafazzoli, M.; Gerrard, S.; True, N.S.; LeMaster, C.B.J. Mol. Struct. 1994, 317, 131. 85. Tafazzoli, M.; Suarez, C.; True, N.S.; LeMaster, C.B.; LeMaster, C.L.J. Mol. Struct. 1994, 317, 137. 86. Tafazzoli, M.; Suarez, C.; True, N.S.; LeMaster, C.B.; LeMaster, C.L.J. Phys. Chem. 1992, 96, 10201. 87. LeMaster, C.B.; LeMaster, C.L.; Suarez, C.; Tafazzoli, M.; True, N.S.J. Phys. Chem. 1989, 93, 3993. 88. LeMaster, C.B.; LeMaster, C.L.; Tafazzoli, M.; Suarez, C.; True, N.S.J. Phys. Chem. 1988, 92, 5933. 89. Jesson, J.P.; Muetterties, E.L. In Dynamic Nuclear Magnetic Resonance Spectroscopy; Jackman, L.M.; Cotton, R.A. Eds.; 1974, 253. 90. Klemperer, W.G.; Krieger, J.K.; McCreary, M.D.; Muetterties, E.L.; Traficante, D.D.; Whitesides, G.M.J. Am. Chem. Soc. 1975, 97, 7023. 91. Spring, C.A.; True, N.S.J. Am. Chem. Soc. 1983, 105, 7231. 92. Schrtider, G. Chem. Bei: 1964, 97, 3140. 93. Moreno, P.O.; Suarez, C.; Tafazzoli, M.; True, N.S.; LeMaster, C.B.J. Phys. Chem. 1992, 96, 10206.

94. 95. 96. 97. 98. 99. 100. 101.

Rivera-Gaines, V.E.; Leibowitz, S.J.; Laane, J. J. Am. Chem. Soc. 1991, 113, 9735. Anet, EA.L.; Haq, M.Z.J. Am. Chem. Soc. 1965, 87, 3147. Hanggi, P.; Talkner, P.; Borkovec, M. Rev. Modern Physics 1990, 62, 251. Hynes, J.T. Ann. Rev. Phys. Chem. 1985, 36, 573. Campbell, D.M.; Mackowiak, M.; Jonas, J. J. Chem. Phys. 1992, 96, 2717. Forst, W. Theory of Unimolecular Reactions; Academic Press: New York, 1973. Robinson, P.A.; Holbrook, K.D. Unimolecular Reactions; Wiley-Interscience: New York, 1972. Chauvel, J.P. Jr.; Friedman, B.R.; Van, H.; Winegar, E.D.; True, N.S.J. Chem. Phys. 1985, 82(9), 3996. 102. Chauvel, J.P. Jr.; Conboy, C.B.; Chew, W.M.; Matson, G.B.; Spring, C.A.; Ross, B.D.; True, N.S. J. Chem. Phys. 1984, 79(6), 1469. 103. Hase, W. Dynamics of Molecular Collisions. Part B. Modern Theoretical Chemistry; Miller, M. H. Ed.; Plenum: New York, 1976, p. 121. 104. Hase, W.L.; Bunker, D.L. Program No 234, Quantum Chemistry Program Exchange, Indiana University, Bloomington, IN 47405. 105. (a) Hase, W.L. (personal communication); (b) LeMaster, C.B.; Ph.D. Dissertation, University of California, Davis, 1988. 106. Bauer, S.H.; Lazaar, K.I.J. Chem. Phys. 1983, 79, 2808. 107. Bauer, S. H. hltl. J. Chem. Kinetics, 1985, 17, 367. 108. Lazaar, K. I.; Bauer, S. H. J. Phys. Chem. 1984, 88, 3052. 109. Chauvel, J.P. Jr.; Friedman, B.R; True, N.S.J. Chem. Phys. 1986, 84, 6218. 110. LeMaster, C.B.; True, N.S. (in preparation). 111. Pickett, H.M.; Strauss, H.L.J. Am. Chem. Soc. 1970, 92, 7281. 112. Wilson, M.A.; Chandler, D. Chem. Phys. 1990, 149, 11. 113. Kuharski, R.A.; Chandler, D.; Montgomery, J.A. Jr.; Rabii, E; Singer, S. J. J. Chem. Phys. 1988, 92, 3261. 114. Gregory, H.L.; Ezra, S.; Philips, L.A.J. Chem. Phys. 1992, 97, 5956. 115. Borchardt, D.B.; Bauer, S.H.J. Chem. Phys. 1986, 85, 4980. 116. Dewar, M.J.S.; Jie, C. Tetrahedron 1988, 44, 1351. 117. Borkovec, M.; Straub, J.E.; Berne, B.J.J. Chem. Phys. 1986, 85, 146. 118. Lee, M.; Holton, G.R.; Hochstrasser, R.M. Chem. Phys. Lett. 1985, 118, 359.

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119. Chauvei, J.P. Jr.; Friedman, B.R; True, N.S.; Winegar, E.D. Chem. Phys. Lett. 1985, 122, 175. 120. Moreno, EO.; True, N.S.J. Phys. Chem. 1991, 95, 57. 121. Chan, S.C.; Rabinovitch, B.S.; Bryant, J.T.; Spicer, L.D.; Fujimoto, T.; Lin, Y. N.; Pavlou, S. E J. Phys. Chem. 1970, 74, 3160. 122. Chan, S.C.; Bryant, J.T.; Rabinovitch, B.S.J. Phys. Chem. 1970, 74, 2055.

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FOURIER TRANSFORM SPECTROSCOPY OF RADICALS

Henry W. Rohrs, Gregory J. Frost, G. Barney Ellison, Erik C. Richard, and Veronica Vaida

Abstract

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Io Introduction II. llI. IV. V. VI. VII. VIII. IX.

Xo XI.

158 158

The M i c h e l s o n Interferometer and Fourier Transforms . . . . . . . . . . . . . 159 Practical Considerations and Necessary Corrections . . . . . . . . . . . . . . 163 A d v a n t a g e s and Disadvantages of Fourier Transform Spectroscopy . . . . . . 168 Special Considerations in the Infrared . . . . . . . . . . . . . . . . . . . . . 170 Special Considerations in the Visible and Ultraviolet . . . . . . . . . . . . . . 171 A d v a n t a g e s and S o m e Details of Jet S p e c t r o s c o p y . . . . . . . . . . . . . . . 173 M a k i n g Radicals for S p e c t r o s c o p y . . . . . . . . . . . . . . . . . . . . . . . 176 F o u r i e r Transform Absorption S p e c t r o s c o p y for the Study of Photoreactive Radicals . . . . . . . . . . . . . . . . . . . . . . . . . 179 Fourier T r a n s f o r m Infrared Spectroscopy of Jet-Cooled Radicals . . . . . . . 180 Visible and Ultraviolet Fourier Transform S p e c t r o s c o p y o f Reactive Species . 186 A.

Spectroscopy of OC10

B.

Spectroscopy of CN

. . . . . . . . . . . . . . . . . . . . . . . . . . .

187

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

193

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 195

Advances in Molecular Structure Research Volume 1, pages 157-199. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8

157

158

ROHRS, FROST,ELLISON, RICHARD, and VAIDA

ABSTRACT Fourier transform methods have come into their own as a means of studying the optical spectra of gas-phase radicals. Both infrared (FTIR) and ultraviolet/visible spectroscopy (FrUVNIS) are now used to scrutinize these reactive molecules. We discuss the underlying principles of Fourier transform spectroscopy (F-I'S) with particular emphasis on the advantages and drawbacks of FTIR and FTUV/VIS measurements. Extensive tables are presented of metastable molecules that have been studied by Fourier transform methods.

!. I N T R O D U C T I O N The spectroscopic study of radicals is a vexing business. The most common approach is to isolate these unstable species in a matrix at low temperatures. But if one is interested in atmospheric radicals (such as OC10 or CIOO) or combustion species (CH 3 for example), one has to deal with gas-phase radicals. Direct observation of gas-phase radicals is quite difficult; they are not stable, commercially available samples. Since radicals are reactive intermediates, they are destroyed by each other at nearly diffusion controlled rates [ 1]. As a result, one typically has only small concentrations of radicals (roughly 1012 cm -3) to work with. This chapter reviews Fourier transform (FT) methods [2] to monitor the infrared (IR) and ultraviolet/visible (UV/VIS) spectra of gas-phase radicals. Why are FT methods attractive compared to laser based approaches? Lasers are wonderful devices to probe the optical properties of radicals, but, in order to be useful, the spectra of these radicals have to be studied at a resolution of approximately 0.01 cm -1. Consequently a laser is best used when one knows the spectrum of the target radical quite well. When sweeping the frequency of a laser [say for laser induced fluorescence (LIF), optogalvanic spectroscopy, multiphoton ionization, etc.], about the fastest one can scan is 1 cm-l/min; with signal averaging this can easily become 1 cm-1/15 min. Realizing that modern ab initio theoretical methods can find polyatomic vibrational frequencies only to about +5%, it is a difficult problem to search for unknown lines. For example, to search for the C-H stretches of the benzyl radical (C6HsCH 2) requires laser scans of [5%- (3000 cm-1) 95 min/cm -1 ] minutes, or roughly 12 hours. Fourier transform spectroscopy enables one to record spectra of mtorr samples of reactive species at a resolution of 0.005 cm -1 or better. It is possible with this technique to collect these FTS spectra over 500 cm -1 windows in the UV/VIS or 200 cm -1 windows in the IR in about 3 hours. Consequently, we believe that FTS has real advantages to study the spectroscopy of gas-phase, reactive, polyatomic molecules. We will mainly discuss studies of radicals to give an overview of this area of research. We briefly describe FT spectroscopy in the infrared and ultraviolet/visible and discuss a few of the experiments carried out in our laboratories. We compare

FTS of Radicals

159

FTIR and FTUV/VIS to other available means of studying transient species. Finally, we provide tables listing some of the species which have been studied by these means. There are several reviews of high resolution spectroscopy of transient species [3]. The reader interested in studies of unstable species by methods other than Fourier transform spectroscopy should consult these articles and the references in them.

Ii. THE MICHELSON INTERFEROMETER A N D FOURIER TRANSFORMS Most Fourier transform spectrometers are designed with Michelson interferometers [4]. This device consists of a beamsplitter, a moving mirror, a fixed mirror, a source, and a detector (Figure 1). A discussion of how it works usually begins by considering the theoretical signal at the detector as a function of the position of the moving mirror when the source is a perfectly collimated beam of monochromatic light of frequency, v, and the beamsplitter is ideal (its reflectance and transmittance are both Moving Mirror

i I I I I ! I ! I I I I

2\\

,,J

Fixed Mirror

..,,

I I I

BF

"

2rt

~ _ ~ % _ ~_- r2 + 1 2 Beamsplitter

poou

i

Collimating Mirror

r

l)etector

Figure 1. Schematic of a Michelson interferometer. The dashed lines show the paths of light which return to the source and the solid lines show the rays which propagate to the detector. The signal at the detector is the result of two light waves which have each been reflected and transmitted once by the beamsplitter. BF and BM are the respective distances of the fixed mirror and the moving mirror from the beamsplitter. Note that 5 = 2 ( B M - BF).

160

ROHRS, FROST,ELLISON,RICHARD, and VAIDA

50%). We start with the moving mirror and the fixed mirror equidistant from the beamsplitter. The light wave from the source can be expressed by, E(z) = E o exp[-2rci~z]

(1)

where (y = 1/L = v / c is wavenumber and z is the distance the beam has propagated. When this wave hits the beamsplitter it is divided in two" Ef(z) = t E o exp[-2rtRyz],

(2)

Em(Z) = r E o exp[-2rti6z]

(3)

and

Ef(z) and Em(Z) are, respectively, the waves travelling toward the fixed and the

moving mirrors and t and r are the amplitude transmission and reflection coefficients. These should not be confused with the transmittance and the reflectance. When theses waves return to the beamsplitter they will be divided again. Some of the light will propagate back toward the source and the rest to the detector. The waves which are detector bound are now, Ef(z) = r t E o exp[-2r~i6(z

+ zf)]

(4)

and Em(Z) = t r E o exp[-2rti6(z + Zm)]

(5)

where zf equals 2BF and Zm equals 2BM. Adding these waves together yields: ER(Z) = t r E o {exp[-2rcio(z + zf)] + exp[-2rti{~(z + Zm)] }

(6)

We set 8 = z m - zf and, making our independent variable 8, square this equation to get: 1(5) = 2trEo[1 + cos 2rcc~8]

(7)

We now have an equation for the intensity of the signal as a function of 8, the path difference, or retardation. We now include the dependence of E on ~ and incorporate most of the constants into a single term, I(~), to arrive at: I(~5) - 1I(~)[1 + cos(2rc~io)]

(8)

If we employ Eq. 8 and start with 5 equal to zero, l(~i) is a maximum. If the moving mirror is displaced by )d4, where )~ is the wavelength of the light, 8 = L/2, there is complete destructive interference at the beamsplitter and the signal at the detector is zero. When 5 = ~ (or n)~) we again get constructive interference at the beamsplitter and a maximum signal at the detector. Scanning the moving mirror at a constant velocity produces a time-dependent signal at the detector (Figure 2). The DC component in Eq. 8 is usually discarded to give, 1(8) - ll(G)cos(2rtSG)

(9)

o

--

0~

::r"

0

u,,

"~:r

0

0

9

---t

~

--~

0

0

~n

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0

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3

::7-o

o

o

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~

_~.

~

g-

Signal

e'~t3

~t !

! ~5=0

I

. . . . . . . . . . . . . . .1. . . . . . .

Retardation,

Figure 3. Signal at the detector from a polychromatic source. This interferogram shows the form of the signal about a DC offset. Note the sharp "centerburst" at 5 = 0, the only point where all of the frequencies interfere constructively. This feature is the reason gain ranging is necessary in FT absorption spectroscopy.

162

163

FTS of Radicals

which is called the interferogram. If the nonideality of the experiment is now taken into account the expression becomes, 1(8) = R(t~)cos(2n:fcy)

(10)

where R(6) includes the factor of 1/2 and instrumental effects like the true beamsplitter characteristics and the detector response that modify the signal. 1(8) and R(~y) are a cosine Fourier transform (FT) pair. The same arguments as above apply to polychromatic sources. However, the interferogram will now be a sum of all the interferograms for the light at the various frequencies and it will be sharply peaked around 8 = 0 since this is the only position where all of the light interferes constructively (Figure 3). For a continuum source the FT pair is, oo

(11)

1(8) - ; R(6) cos(2rl:86)d~, --oo

and: oo

R(~) = ; 1(8) cos(2rt86)d8

(12)

-.-oo

Since the integrand in Eq. (12) is even, oo

R(6) = 2~ 1(8) cos(2rc86)d8

(13)

0

Thus, we need only take a one-sided interferogram. From Eq. 13 we can also see why spectroscopy associated with the Michelson interferometer is called Fourier transform spectroscopy. Movement of the mirror which corresponds to a change in retardation provides a signal which is a function of distance. This signal is then decoded by a Fourier transform to give a spectrum which is a function in the reciprocal space. This is why wavenumbers (cm-1) are such a convenient unit to use with this type of spectroscopy.

!il.

PRACTICAL CONSIDERATIONS AND NECESSARY CORRECTIONS

Equation (11) tells us that in theory we could measure the entire spectrum at an infinite resolution. This would require one to attain a retardation of infinity and to store an infinite number of points for the Fourier transform. Since this is obviously impractical, the retardation is truncated and the signal is gathered at certain discrete points. These necessary procedures have effects on the spectrum.

164

ROHRS, FROST,ELLISON, RICHARD, and VAIDA

In practice one can only scan the moving mirror a finite distance which we will call L. Thus, Eq. 6 becomes, oo

e03) = 2;I(8)A(8) cos(27tS~)d8

(14)

0

where A(8) is defined as follows: A(8) = 1 when -2L < 8 < 2L A(8) = 0 when 8 > 12LI

(15)

A(8) is called the boxcar function. This limit on the retardation leads to a limit on the resolution of 1/2L, so if L = 100 cm, the highest resolution attainable is 0.005 cm -1. By the convolution theorem, the product of two functions in one space is the same as the convolution of the Fourier transforms of the two functions in the reciprocal space. The effect of multiplying by this boxcar function is to convolve each point in the reciprocal wavenumber space with a sinc function [sinc(x) = sin(x)/x; Figure 4]. An undesirable feature of the sinc function as a lineshape is the large amplitude oscillation (the first minimum is -22% of the maximum). This ringing can make it difficult to get information about nearby peaks and leads to anomalous values for intensities. This ringing can be removed by the process known as apodization. Apodization is done by simply choosing a different form for the function A(8). Unfortunately, the sinc function has the narrowest full width at half maximum (FWHM) and choosing any other function leads to broader lineshapes; in effect, a loss of resolution. It is thus up to the spectroscopist to strike a balance between how much ringing can be tolerated and how narrow the peaks should be. Many such functions are available for these purposes. Norton and Beer have analyzed over 1000 functions, some of which fall near an empirical boundary [5]. The most useful functions fall on this boundary as they maximize resolution for a given secondary maximum or minimize the side lobe amplitude for a given linewidth. It is worthwhile to note that many popular apodization functions (such as the Happ-Genzel, or Hamming function; Figure 4) are not on this empirical boundary and their use can result in the loss of spectral information. As mentioned earlier, it is also impractical to collect data at infinitesimally spaced values of 8. Data in Fourier transform spectroscopy is usually collected at discrete steps determined by a coincident interferometer with a mode-locked helium-neon (He-Ne) laser at )~ = l/(y = 632.8 nm. The fringes of this laser are detected with a diode and sampling at every zero crossing of this line will provide data for retardation values spaced at 316.4 nm. In addition, one must insure that the mirror is moved slowly enough that the detector can relax between each sample so the data for each point is independent.

1.0--

1.0-

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Figure 4. Apodization functions and their Fourier transforms. The top left function is the boxcar function and its FT is the sinc function. Note the large amplitude of the secondary minimum and the narrow full width at half maximum, Ao. The bottom pair of figures show the Hamming function and its FT. The secondary oscillations are smaller but the width has grown.

165

166

ROHRS, FROST,ELLISON, RICHARD, and VAIDA

Another important detail is the number of data points taken. This is determined by the Nyquist criterion which states that the minimum sampling to unambiguously determine a spectrum is twice the frequency of the highest frequency component, or two times per wavelength of this wave. Sampling at a lower rate than this results in a phenomenon called aliasing, or foldover (Figure 5). If the interferogram were sampled twice per wavelength of the dashed wave in Figure 5, the higher frequency solid wave would be folded into the spectrum. For example, in the IR the highest frequency component will have an energy of about 4000 cm -1 (2500 nm). Sampling twice per wavelength requires that the maximum distance between samples is 1250 nm. Thus we could completely determine IR spectra by sampling once per laser wavelength. A quick calculation will show that the shortest wavelength we can uniquely specify when sampling twice per He-Ne wavelength is 632.8 nm, which corresponds to 15,803 cm -1. Obviously, spectroscopy in the UV/VIS will require that another method is employed for sampling. Discrete sampling will have little effect on the Fourier transform other than changing the integral to a series. The process can also be thought of as multiplication of the interferogram by a shah, or comb, function. Discrete sampling is a necessity for digital computer processing. In fact, if the total interferogram consists of exactly 2" points, where n is an integer, Cooley-Tukey type fast Fourier transform algorithms increase the speed of the numerical transform [6]. Since the files that are being transformed may consist of 500,000 points or more, this is very important. A problem that arises when the data is taken discretely is the determination of zero-path displacement (ZPD). How can one be sure that the first sample is taken when the mirrors of the interferometer are equidistant? In practice, the data sampling is triggered by a white light source which follows the interferometric path. At ZPD this polychromatic source has a very large peak but it dies off quickly as a function of retardation (see Figure 3). Still, due to electronic and optical factors, this centerburst is not necessarily at ~ = 0. To correct this, one takes an interferogram that is double'sided about ZPD. An algorithm then computes phase factors which correct the interferogram to make it symmetric [7]. If the phase correction is done properly it is possible to get very accurate line positions from Fourier transform spectroscopy [8]. Another problem that arises in absorption spectroscopy is the huge dynamic range of the experiment. At ZPD all of the light constructively interferes leading to a large signal at the detector. The amplification and detection must be set such that this signal does not saturate the electronic system. However, as soon as we move far away from ZPD the signal gets very weak in comparison to this centerburst. This region of the interferogram at large retardation contains the high resolution information. In order to use this data, most FT spectrometer systems employ gain ranging. The He-Ne fringes are counted, and at a certain point the gain is boosted. After the signal has been collected, this is accounted for when the interferogram is processed.

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168

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

The last important consideration we shall mention is collimation. There are specific relations which determine how large the aperture (see Figure 1) can be for a given resolution. In general, as the aperture gets smaller the resolution gets better since off axis rays will have to travel a greater path difference to be out of phase with the collimated radiation. The maximum beam half angle, o~, that can be passed through the interferometer without degrading the fringe contrast is related to the throughput of the interferometer. The throughput may be thought of as a function of the solid angle subtended by the source. It is also proportional to the resolving power, R = (Ymax/A(y, where Ae~ is the resolution and (Ymax is the maximum wavenumber in the spectrum. When these two expressions are combined an equation can be derived for the resolution as a function of aperture size, (Ymaxd2 A~ = ~

(16)

8F 2

where F is the focal length of the interferometer optics and d is the diameter of the aperture.

IV. ADVANTAGES AND DISADVANTAGES OF FOURIER TRANSFORM SPECTROSCOPY There are three well known advantages one gains by doing Fourier transform spectroscopy that are usually discussed in terms of their relationship to grating spectroscopy. They are called the Fellgett advantage, the Jacquinot advantage, and Connes accuracy. Regardless of the spectroscopic method used, the signal level generated by each spectral element, de~, is dependent on the product of the spectral intensity in dey, B(e~) (with units of W/cm-1), and the observation time [9]. In FT spectroscopy, the whole spectrum is collected at once, so that each spectral element is sampled for the whole observation time, t. Then the signal from each spectral element is proportional to: Svr <,<:B(6) t d6

(17)

In contrast, the dispersive technique samples each spectral element for only a fraction, 1/N, of the total scan time, where N is the number of spectral elements. Thus the signal for dispersive spectroscopy is proportional to: t

SDis oc n(o) ~ do

(18)

Next we must consider the different types of noise which can be present in the spectrum [2,9]. Detector noise is generally due to random fluctuations in the detector, such as thermal noise, and is therefore independent of the signal level. This type of noise is proportional to the square root of the amount of time a given

FTS of Radicals

169

spectral element is observed. This source of noise is rather small in a modern experiment. Another type of noise is photon or shot noise, due to fluctuations in the amount of photons reaching the detector. The noise is proportional to the square root of the number of photons. A third kind of noise, called source or modulation noise, can result in experiments with refractive index fluctuations such as long-path atmospheric or astronomical measurements, or when there is a design flaw in a laboratory instrument. Modulation noise is directly proportional to the signal level. Detectors which are photon noise-limited act as photon "buckets", and collect nearly all the photons emitted from the source. In the dispersive instrument, the signal-to-noise ratio (SNR) is then easy to calculate. The signal from d(~ is given above, while the noise is proportional to the square root of the signal, so: t SNRDis '~ 4B((~) ~ d(~

(19)

The SNR from d(~ measured on an Fourier transform instrument is more complicated. The signal is given in Eq. 17. The noise is given by the square root of the signal, but is uniformly distributed throughout the whole spectrum, thus making the noise in d(~ dependent on all of the other spectral elements. We approximate the shot noise due to the whole spectrum as, Noise = ~/BNd(~ t

(20)

with B being the average intensity [10] in the whole spectrum. Then: B((~)d(~ t SNRzT or ~/~ Nd(~ t

(21)

The multiplex gain in the shot-noise limited case is then: Gvr / D i s

oc

4 B ( (~1 B

(22)

In the photon-noise-limited case, this distributive multiplex gain [11] is a function of the line intensities and the overall spectral structure [12]. In absorption, there is a continuous incident intensity, causing B to be large. For weak spectral lines, B(o) and B are approximately equal, so the multiplex gain is essentially 1. For strong absorption lines, the "gain" may actually be less than one. In an emission experiment, the background is very low, and therefore so is B. For strongly emitted lines, the multiplex gain can be quite significant, but for weak emission lines the gain will be small. In addition, strong emission lines will increase B and cause weaker lines to be obscured. Fellgett's, or multiplex, advantage is the result of the detector recording the signal from all of the frequency channels during the course of the entire experiment. This may be contrasted with a grating experiment where the grating and the slits only allow the signal from one channel at a time to be recorded. Of course, this means

170

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

that noise is recorded throughout the course of the experiment as well. It turns out that the important consideration here is the source of the noise in the system. Fellgett's advantage is only available when the experiment is detector noise limited. This is almost never the case as detectors, especially in the IR, have improved such that they are limited by the Johnson noise of a resistor in some amplification scheme. Jacquinot's, or etendue, or throughput advantage is not dependent on the wavelength of light one is interested in. It is the result of the Michelson interferometer processing an axisymmetric beam. In a grating instrument the slits drastically reduce the throughput of the machine. Slits are not necessary in a Fourier transform device. The entire beam can be imaged onto the detector. The gain in signal due to this advantage is usually about a factor of 200. Connes' accuracy is the result of the ability to carefully measure an accurate interferogram. If the data acquisition and phase correction is done properly it is possible to measure transitions in a species to very high accuracy. It is common to see line positions accurate to 0.0005 cm -1 or better. Accurate intensity measurements are possible as well [13]. As mentioned in the introduction, a major advantage that Fourier transform spectroscopy has over laser spectroscopy is that it is straightforward to record the entire spectrum of a species at once. Diode lasers in the infrared are not continuously tunable and have mode gaps which can only be filled by switching diodes. Many ultraviolet lasers are not continuously tunable either. Tunable difference frequency methods and diode lasers involve much longer scan times than are necessary with a Fourier transform device. For example, the Bomem DA3.002 can scan a bandwidth of 100 or more wavenumbers in the mid-IR at a resolution of 0.005 cm -1 in less than 3 minutes. A diode laser which scans in 20 MHz steps may require more than a day to scan the same spectral region. The main disadvantages of Fourier transform methods are usually discussed with reference to laser spectroscopy. A laser may be as much as 106 times more sensitive mostly due to its high photon flux. Thus, weak absorbers may be impossible to see with an FT device but it is often easy to study these species with a laser. Tb,e lines from a laser are also much narrower and so they allow one to get resolution of at least a factor of 10 better than that available with Fourier transform methods. The limit of resolution for Fourier transform techniques with commercially available spectrometers is approximately 0.001 cm -1 in the IR [14].

V. SPECIAL CONSIDERATIONS IN THE INFRARED Performing an FTIR experiment requires some specialized equipment. First, a broadband radiation source is needed. Typically a SiC globar is used that reaches temperatures of 1200-1500 K. When studying species that are weak absorbers or are in low concentration a brighter source is needed and high temperature sources (>2500 K) for such experiments have been developed [ 15]. These devices are nearly

FTS of Radicals

1 71

blackbodies. A specialized beamsplitter is needed and typically C a F 2 is used from 1200 cm -1 to 4000 cm, l, KBr is used down to about 400 cm-1, and Mylar, or pellicle, beamsplitters extend to the far IR. For detectors, one uses liquid nitrogen cooled InSb down to 1850 cm -1, liquid nitrogen cooled HgCdTe, liquid helium cooled CuGe, or liquid helium cooled SiB down to about 400 crn-1, and bolometers for lower energy measurements. As mentioned above, modern detectors are usually not the noise-determining part of the experiment. It should be possible to set up any Fourier transform experiment in the IR such that it is limited by the background photon flux or photon noise. When using the Bomem DA3.002 in the IR at the highest energy (4000 cm -1) at a resolution of 0.004 cm-1, with F = 325 mm, Eq. 16 indicates that the largest the aperture can be is about 0.6 mm or a lack of collimation will degrade the resolution. This is easily accomplished. A problem that arises when taking IR spectra in the laboratory is the presence of atmospheric absorbers, notably H20 and CO 2. To remove these species from the experiment one must either evacuate the entire interferometer or purge it with a suitable gas. When evacuating, very low pressures (10-4 Torr) will be required for high resolution work since the path length for absorption is quite long. Unless the system is specifically designed for evacuation, such an approach is likely to be difficult. The advantage of purging is that it will tend to pressure broaden atmospheric transitions making them readily discernible if they appear in the spectrum. The disadvantage of purging is that it is very difficult to remove all of the unwanted molecules unless one resorts to a continuous (24 hours/day, 7 days/week) purge.

VI. SPECIAL CONSIDERATIONS IN THE VISIBLE AND ULTRAVIOLET The principle components of a FT spectrometer that are different in the UV/VIS as compared to the IR include light sources, beamsplitters, and detectors. For operation in the visible region (up to 25,000 cm-1), a quartz halogen lamp is typically used. The advantages of this lamp are its very continuous, flat spectrum and its stable output. At the shorter visible wavelengths and in the near-UV, we use a high-pressure, 75 W xenon arc lamp. Xe lamps are an order of magnitude more intense than quartz halogen sources at about 30,000 cm -1, but arc lamps are also much noisier. An alternative source for the UV is a deuterium arc lamp, which is more stable than Xe but is one to two orders of magnitude less intense below 40,000 cm -~. A quartz beamsplitter with the appropriate coatings can be used throughout the visible and the near- and mid-UV, up to at least 45,000 cm -1. A silicon photodiode detector is used for the visible and near-UV to about 30,000 cm -1, and a SbCs cathode photomultiplier tube (PMT) covers higher energy regions. Both of these detectors have high quantum efficiencies, so that the detection becomes short-noise limited. The various types of noise will be discussed below. The high gain characteristics of the PMT make it the optimum detector for low light level

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FTS of Radicals

1 73

conditions, which include weak emission from radicals produced in a discharge source. For visible and near-UV absorption spectroscopy, the lower gain Si photodiode and high brightness Xe lamp are well matched. It should be possible to conduct a UV/VIS experiment such that it is photon-noise limited. An important consideration upon going to higher energy is the effect of imperfect collimation of the source. The lack of perfect collimation, or finite aperture size, has a much larger effect as the frequency of the light increases. For example, the smallest aperture diameter we normally use in the UV/VIS is 0.6 mm. The focal length of the collimating mirror is 325 mm. Then the highest resolution that can be achieved at an energy of 30,000 cm-1 is 0.026 cm-1. Another important difference in FT operation between the IR and UV/VIS is the sampling rate of the interferogram. As was mentioned earlier, the proper sampling rate in the IR is insured by sampling twice per He-Ne cycle. The shortest wavelength that can be uniquely specified with this rate is 632.8 nm. In order to operate in the UV/VIS, the sampling rate must be increased. Bomem DA3 spectrometers use successive interpolation [16] to decrease the size of the sampling interval beyond just every zero crossing. Because the He-Ne interferogram has very precise zero crossings as a function of mirror displacement, the interval between two adjacent zero crossings can be subdivided into four equally spaced subintervals. Figure 6 will help to explain the technique used. After the white light ZPD triggers, the first mirror displacement region between the first and second zero crossings (Region 1) is divided into four equally spaced intervals. These interpolated subintervals are then used as data sampling intervals between the second and third zero crossings (Region 2). After the data is sampled in Region 2, this region is subdivided into four intervals. These subintervals are used in Region 3, and the process repeats. This method is highly precise because the interpolations are done with the known zero crossings rather than an external, uncorrelated triggering device. With successive interpolation, there are a total of eight digital sampling points for every complete cycle of the He-Ne waveform. The Nyquist condition then allows the maximum optical frequency that can be transformed without spectral aliasing to increase to about 63,200 cm -1. This frequency corresponds to a minimum optical wavelength of 158 nm, in theory allowing operation into the vacuum UV region of the spectrum.

VII. ADVANTAGES AND SOME DETAILS OF JET SPECTROSCOPY A problem with high resolution spectroscopy is the analysis of the enormous amounts of very complicated data FT spectrometers can produce. If the spectrum is taken at or above room temperature many rotational states of the radical will be populated and a huge number of transitions will be seen (Figure 7). One means of simplifying the spectrum is to cool the species that is being studied. A very useful way of accomplishing this is to seed the target radical in a supersonic expansion of

174

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

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Room temperature absorption spectrum of CF3. The top shows a broad region while the bottom shows a small portion of the spectrum demonstrating how congested room temperature spectra can be.

a neutral carrier gas or even by itself. One can attain very low temperatures in a jet, even below 1 K For example, if one observes CO at room temperature and sees about 60 transitions, at 20 K with the same spectroscopic setup, only about 15 transitions will be observed (Figure 8). When the species being studied is not well characterized such cooling can make it much easier to assign the spectrum. A supersonic expansion is achieved by expanding gas through a nozzle from a high pressure volume into a low-pressure volume [17]. The pressure drop is maintained by having the gas expand through a small orifice and having adequate

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175

176

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

pumping (e.g., Roots pump or diffusion pump) on the low-pressure side. The gas expands out of the high-pressure region into a collisionless envelope in the low-pressure region. During this expansion process the molecule is collisionally cooled, especially its rotational degrees of freedom. The nozzle geometry can have an effect on jet experiments [18] and the nozzle configurations used by the authors will be discussed below.

VIii. MAKING RADICALS FOR SPECTROSCOPY There are several ways to generate radicals for use in spectroscopic studies. We discuss four of them: controlled chemical reaction, discharge sources, pyrolysis, and photolysis. Using a controlled chemical reaction to make the transient species has been used most effectively by flow tube chemists. The application of this technique to high resolution IR spectroscopy by experimenters at NOAA in Boulder has proven to be very fruitful (Figure 9). In this technique a precursor is metered into a flow tube fitted with White cell optics [19]. At various places along this flow tube reactants are bled in. For example, fluorine atoms generated in a microwave discharge may be added to react with ammonia to generate the NH 2 radical. It is possible to conduct multiple reactions in this way to generate the target species, and much of the chemistry involved in these gas phase reactions is well documented. A necessary complication of these experiments is that the other reaction products can absorb or emit in the sample. Discharge sources have long been used to generate unstable species. Bernath and co-workers have used hollow cathode discharges and other discharge sources to generate unstable species (e.g., see Table 1, ref. 30). This technique is well suited to emission studies as the transient species one generates are often very hot. Engelking's group made use of a corona discharge supersonic expansion to generate radicals [20], and Richard et al. have successfully coupled this source to a Fourier transform ultraviolet experiment [21]. In the past pyrolysis has been avoided because the temperatures involved usually preclude emission and the technique usually made a large number of products. It was not a selective technique. However, the pioneering work of Chen et al. has demonstrated that it is possible to employ flash pyrolysis to generate unstable species in high enough concentrations to make spectroscopy viable [22]. This technique involves heating small ceramic tubes such as alumina, zirconia, or silicon carbide to a very high temperature (>1000 K). When this tube is used as a supersonic expansion source, the residence time in the hot nozzle can be adjusted such that a precursor seeded in an inert cartier gas has sufficient time to decompose but not enough time to undergo significant recombination before it passes into the relatively collisionless environment of the supersonic expansion. Typical residence times in the nozzle are on the order of 100 las while the half-life for radical recombinations is on the order of 1 ms.

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1 78

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

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SiC nozzle for pyrolysis. The idea behind this radical source is the short residence time of the radical precursor in the heated portion of the nozzle. The hot length is varied so that the parent molecule undergoes e,lough collisions to dissociate before it expands from the throat of the nozzle. However, the precursor is not in the heated portion long enough for significant recombination to occur.

A drawing of a silicon carbide nozzle is shown in Figure 10. An advantage of using ZrO 2 tubes is that this material has a negative coefficient of resistance. The hotter it gets the more current it draws. If the tube is placed in series with a ballast resistor having a positive coefficient of resistance (a light bulb with a tungsten filament) the nozzle will be self-regulating. The advantage of zirconia is that it can achieve a higher temperature (about 2300 K) and can handle oxidizing species. SiC has this same resistive property as ZrO 2 and it is mechanically stronger, but it only can be heated to about 1800 K. Photolysis is a selective way to make radicals from a suitable precursor. It is very clean and particularly useful in emission experiments. The energy required to photolyze a precursor is typically about 6 electron volts. This means that photolysis sources will be pulsed lasers or resonance lamps. Lasers such as XeC1 (~ = 308 nm) or ArF (~ = 193 nm) with pulse widths of about 10 ns and repetition rates of around 100 Hz will thus make transients species in short bursts. It is a difficult task to couple such a system to a Fourier transform spectrometer but it has been done in a dynamics study [23]. Resonance lamps, while providing the appropriate energy, usually do not provide the flux necessary to generate the high concentrations of unstable species needed for a successful spectroscopy experiment [24].

FTS of Radicals

1 79

IX. FOURIER TRANSFORM ABSORPTION SPECTROSCOPY FOR THE STUDY OF PHOTOREACTIVE RADICALS With the appropriate changes of sources, optics and detectors, the FT spectrometer can be used to obtain absorption spectra in the IR, visible, and UV (20 gm to 200 nm). Our current apparatus allows for a resolution of 0.004 cm -1 in the IR and 0.03 cm -1 in the UV/VIS. Thus, it is possible to investigate and characterize the ground and excited states of a molecule. Supersonic expansions allow the preparation of reactive species in a collision-free environment. Inhomogeneous effects including Doppler broadening, rotational congestion, and cluster-induced spectral perturbations can be minimized with the proper choice of the expansion conditions. Therefore, the combination of FT absorption spectroscopy with a supersonic expansion is an excellent method for the study of photoreactive molecules in a very well-controlled environment. In particular, FTIR is a powerful diagnostic probe for characterizing the expansion, such as the rotational temperature [25] and degree of cluster formation [26]. With FT spectroscopy in the UV/VIS, excited electronic states can be studied, and the dynamics of photoreactions on these excited states can be derived from the homogenous linewidths. These abilities become invaluable for the study of photoreactive excited states where the lifetimes are generally too short to obtain this information accurately using other spectroscopic methods. There are many advantages to using direct absorption spectroscopy to study photoreactive systems. Although the technique suffers from an inherent lack of sensitivity, the information that can be obtained for cases where emission and multiphoton ionization (MPI) techniques are inefficient can be invaluable. This generally applies to photoreactive systems where short excited state lifetimes preclude the use of such indirect techniques [2 7]. A good example of such a situation is the near-UV electronic transition of OC10, in which the excited state dissociates rapidly [28a, b,c]. Emission is observed from the low-energy, near-origin levels, but as the photochemistry becomes more efficient at increasing energy, emission yields and up-pumping rates make emission and MPI signals negligible. Direct absorption spectroscopy proves to be an excellent method for investigating the excited-state structure and reaction dynamics in the higher energy part of the transition. Previous direct absorption free jet experiments in our laboratory and others have had to rely on phase sensitive detection [3c]. The light levels detected are generally low as a result of the spectral dispersion and slit discrimination. For this reason, the operational resolution achievable has been limited (typically >10 cm-1). This constraint therefore limits the types of systems which can be studied to those involving excited-state lifetimes short enough to give homogeneous linewidths larger than the instrumental resolution. However, this situation is not the case for interferometric detection where an increase in resolution does not produce a corresponding decrease in optical throughput. Using FT spectroscopy, strong absorbing systems can be studied under very dilute expansion conditions which

180

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

minimize inhomogeneous effects. Additionally, much weaker absorbers can be studied under high resolution.

Xo FOURIER TRANSFORM INFRARED SPECTROSCOPY OF

JET-COOLED RADICALS

Table 1 lists many of the short-lived species detected in the gas phase with Fourier transform infrared spectroscopy. Two prominent groups are those headed by Bernath, now at the University of Waterloo, and by Howard at the National Oceanic and Atmospheric Administration (NOAA). The former group has used IR emission to study unstable diatomics produced in discharge sources or furnaces. The molecules studied in this group tend to be of astrophysical interest. The research team at NOAA mainly studies short-lived molecules of atmospheric significance. They employ a long flow tube fitted with White cell optics and coupled to a Bomem DA3.002 spectrometer. They usually make the transient they are interested in by performing a carefully controlled series of chemical reactions. A recent experiment may provide a general technique for observing jet-cooled radicals [29]. A schematic of the apparatus is shown in Figure 11. The experiment is centered around a Bomem DA3.002 high resolution Fourier transform spectrometer. For the IR experiments an internal SiC globar source is used with a CaF2 beamsplitter and a liquid nitrogen cooled InSb detector. With the exception of the vacuum chamber the entire apparatus is purged with nitrogen. The light from the interferometer is focused into a vacuum chamber downstream from the throat of a nozzle in a supersonically expanded molecular beam. The vacuum chamber is pumped by a Kinney Roots pump (800 L/s @ 0.1 torr). A suitable precursor is seeded into 1-2 atmospheres of an inert gas before it passes through a SiC nozzle heated to > 1000K and expands into a vacuum of about 0.10 torr. The idea is to decompose this precursor into the species of interest while it is in the heated portion of the nozzle and then have the short-lived molecule expand into the vacuum before it undergoes significant recombination. A question might arise as to whether or not a hot nozzle will give rise to cold molecules. The top spectrum in Figure 12 shows the high resolution (0.005 cm -1) spectrum of jet-cooled NO. This spectrum was recorded with a sample mixture of 10% NO in He passed through a room temperature 1 mm ID SiC capillary nozzle at ambient temperature. Data for only one interferogram was recorded. The stagnation, or backing pressure was about 700 torr and the background pressure in the vacuum chamber was 40 m torr. There are two spin-orbit states for NO, 21-I1/'2, and 2I-I3/2. The 21-13/2state is 121.1 cm -1 above the ground state 21-I1/2 and it has been frozen out of the cold part of the spectrum. The higher temperature rotational distributions in the spectrum derive from NO outside the jet expansion (but still in the vacuum chamber) at about 270 K. The cold part of the spectrum is at 9 K. The bottom spectrum in Figure 12 shows a spectrum from the same experimental

Table 1. Species Studied by FTIR Transient Species AIC1 AlF AIH BaC1 BaH Bill Bill BiF BiCI BiBr BiI BH arO Can C2 C2 CC1 CH CH CN CN CP CP CS Cs2 Cs2 Cs2 Cs2 CIO CIO CrO CoO

Cull FO FO FeO Gall GaF He2 InF IF Li2 Li2 Li2

Transition

Spect. Method Synth. Method

xl~ + X1Z+ X1Z+ C2FI1/2 - A'2A3/2 X2Z+ X 0+ A l-X0 + X2 l-X10 + X2 l-X10 + X2 l-X10 + X2 l-X10 + X1Z+ X2FI3/2 X2Z+ A 1Flu - X1Z~ B 1Ag-X1Z~,B'E~-A 1Flu X2H3/2, 2H1/2 A 2A-X2H, B 2Z--A Ill X 2FI A 2FI-X 2Z+ A- X A 2Hi-X2Z+ A 2Hi-X2Z+ X 1Z+ 1Z+__lH E lv-~ In g E D 1Eu+,C,..,u-~g 1Flu-(1) 3]~ D 1Eu+-(2)1E~ X 2H3/2, 2FI1/2 X 2FI3/2,2FI1/2 A 5E-X 5FI A 4I-li-X 4Ai X 1~21-13/2, 2FI1/2 2FI3/,2, 21-I1/2 X 5Ai X 1E+ X X X 1Z1 1Ag-B 1Flu c lFlu-1 1Fig, 2 1Z+-I 2 1Z+-2 1Z~ lHg

181

FTE FTE FTE FTLIF FTE FTE FTLIF FTCIF FTCIF FTCIF FTCIF FTE FTA, WC FTE FTE FTE FTA, WC FTE FTE FTLIF FTE FTE FTE FTA, WC

TF HCD TF TF TF TF MD,TF FR FR FR FR MD FR TF MD MD FR CO MD MD CDNE MD MD FR

FTLIF FTLIF FTLIF FTLIF FTA, WC FTA, WC FTE FTE FTE FTE FTA, WC FTE FTE FTE FTE FTE FTE FTLIF FTLIF FTLIF

TF TF TF TF FR FR MD HCD CLR FR MD TF TF TF FI TF TF TF

Ref 1

2 3 4 5 6 7 8

44 44 44 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4O 41 (continued)

Table 1. (continued) Transient Species Li2 NiO NH NH NS NaLi Na2 Na2 OH OH PH PtH PtO Rb2 ScN SiC SiC Sill Sill SiS SH SO Tell TeF TeCI TeBr Tel XeH BO2 C2H C3 C3 CF2 CF2 FO2 HCO HNC HNO HNO HNSi HO2 NH2

Transition + 1Eg+ 2 1Eu-2

Spect. Method Synth. Method

A 3FIi-X3]~X 3EX 3y7 X 2H 3 1Z+ ( C ) - 2 lz+ (A) many (1) 1Ag-B(I ) 1Hu X21-I X 21-I X 3Zf~'= 1.5- Xf~- 1.5 A' 31-I- X 3EA 1E+ - X 1Zg+ A IE+ - X 1Z+ d 1Z+ - b 1H A 3Z- - X 3I-I X21-I X2FI X 1E+ X2FI X 3Z-, a 1A X2 21-I1/2- X1 2I-I3/2 X2 2I-I1/2-X1 21-I3/2 X2 21-I1/2- X1 2I-I3/2 X2 2I-I1/2- XI 2I-I3/2 X2 21-I1/2 - XI 21-I3/2 C 21-I-B 2E+,D 2Z+-C 2I-I X 2Fig; Vl, v3 v3 b 31-Ig- ~3Flg Vl, v3 Vl vl, v2, v3 v3 Vl, v2, v3 Vl v l, v2 v! v2, v3 Vl, v2, v3

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FTLIF FTE FTE, WC FTE, WC FTA, WC FTCIF FTLIF FTLIF FTE FTE, FTA, WC FTE FTE FTE FTLIF FTE FTE FTE FTE, WC FTE, NYC FTE FTE, NYC FTA, WC FTE FTE FTE FTE FTE FTE FTA, WC FTE FTE FTE FTA, WC FTA, WC FTA, WC FFA, WC FTA, WC FTE FTE FTE, WC FFA, WC FTA, FTE, WC

TF HCD RFD RFD FR TF

F1 CLR, FR MD HCD TF HCD HCD HCD RFD RFD TF RFD FR MD, FR MD, FR MD, FR MD, FR MD, FR HCD FR RFD SE MD FR FR FR FR FR RFD MD, FR RFD FR FR

Ref 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 100 100 100 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79

(continued)

183

FTS of Radicals Table 1. (continued) Transient Species

Transition

Spect. Method Synth. Method

Ref

NH2

v2

FTA, W C

FR

N3

v3

FTA, W C

FR

81

Vl, v2, v3

FTA

FR

82

BH3

v3

FI'A, W C

ED

83

NO3

many

FTA, W C

MD, FR

84

CH2NH

v5, v6

FTA, W C

P

85

v3

FTA, W C

MD

86

CH3CP

Vl, v2, v3, v5, v6, v7

FTA

P

87

CH2CS

Vl, v2, v3, v4, v7, v8

FI'A, W C

P

88

v3

FTE

SE

89

FCP

C3H2

C5

80

Notes: Species are arranged first by the number of atoms they contain and then alphabetically. The first column lists the short-lived molecule of interest, the second lists the transition, the third lists the spectroscopic method, the fourth lists the synthetic method used to produce the species, and the last column lists the reference. Abbreviations: CD = Cossart discharge. CDNE = Corona discharge nozzle expansion. CLR ED

= Chemiluminescence reactor. = Electric discharge.

F1

= Flame.

FR = Flow reactor. FTA = Fourier transform absorption. FFCIF = Fourier transform detection of collision induced fluorescence. FTE

= Fourier transform emission.

FI'LIF = Fourier transform detection of laser induced fluorescence. HCD

= Hollow cathode discharge.

MD P RFD

- Microwave discharge. = Pyrolysis. = Radiofrequency discharge.

SE TF WC

= Stellar emission. = Tube furnace. = White cell.

arrangement except that the SiC nozzle was heated to 1520 K. The spectrum is very similar but the temperatures are slightly different. The higher temperature distribution is about 310 K and the cold part is at 11 K. This demonstrates the viability of hot nozzles for cold expansions. Figure 13 shows two more spectra of NO. These were obtained by pyrolyzing ethyl nitrite seeded into a beam of He: a CH3CHzO-NO --~ CH3CH20 + NO

(23)

184

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

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The temperature of the nozzle was approximately 1400 K. The rotational temperature of the 21-I1/2state in the cooled portion of both of these spectra is about 60 K, a number obtained by analysis of the R branch of the spectra. It is apparent that there are several temperature distributions in these experiments. This is mostly due to heating of the background gas by convection. The top spectrum was collected at a resolution of 0.1 cm-1 and the bottom spectrum at 0.01 cm-l, both in less than 1 hour. Our ability to detect this jet cooled absorption spectrum of an unstable species made by the pyrolysis of a suitable precursor is very promising. It demonstrates that it is possible to get enough decomposition to observe reactive intermediates. It should be possible to make a variety of precursors that will thermally decompose to produce interesting radicals. If these radicals have cross sections comparable to NO we should be able to observe their jet-cooled spectra.

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Figure 12. Spectra of NO in jet. The top spectrum shows 10% NO coexpanded in helium through a 1 mm ID SiC nozzle at ambient temperature. P = P branch, Q= Q branch, R = R branch and H20 signifies peaks due to water. The bottom shows NO passed through the same nozzle at high temperature. See text for more detail.

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i 1.06

1.05

-

I

I

1840

1860

I

I

1880 1900 Wavenumber, c

I

I

1920

1940

Figure 13. Spectrum of NO in jet. This shows the spectrum of NO obtained by

pyrolyzing CH3CH2ONO in a 1.0 mm ID SiC nozzle at about 1400 K. The top spectrum was taken at a resolution of 0.1 cm -1 and the bottom spectrum at 0.01 cm - 1

XI. VISIBLE AND ULTRAVIOLET FOURIER TRANSFORM SPECTROSCOPY OF REACTIVE SPECIES Recently there have been many studies of reactive molecules making use of FT spectroscopy in the UV/VIS (see Table 2). FT emission spectra have successfully characterized the electronic states of many radicals and other reactive species Produced in a variety of sources. Other workers have used FT emission and FT

FTS of Radicals

187

UV/VIS detection coupled with laser-induced fluorescence in similar studies. Table 2 highlights some of the work using these techniques.

A. Spectroscopy of OCIO The FT direct absorption spectra [28] of OC10 provide an example of the capabilities of FT spectroscopy in the visible and ultraviolet regions for the study of short-lived species. In Figure 14, part of the near-UV absorption spectrum is Table 2. SpeciesStudied by UV/VIS FT Methods Transient Species BC BH CC1 CCN CCN CH3N CHaN CN CN CN CN CS Cu2 CuD Li2 MgH NCO NH NH OC10 TiO ZnD

Transition

Spect. Method

Synth. Method

Ref

B4Z- - X4Z A1H-X1Z + A2A - X2H ,~2A- X 2I-I ,a,2A- X 2H , ~ 3 E - X 3A2 ~ , 3 E - X 3A2 B2E + - X2Z + B2E + - X2Z + A2Hi - X2Z + A2Hi, B2E + - X2E + d3Ai- a3Flr B 1 Zu+ - X1Z+ AlE + X1Z ~ 13Ag - b3Flu A2H X2Z + ,~2E+- X 2I-Ii A3FI - X3Z c 1FI - a 1A ,~2A2- X 2B1 C3A, B3H - X3A A2FI - X2Z +

FTE FTE FTE FTLIF FTE FTE FFE FTE FTE FTE FTE FTE FTE FFE FTCIF FTE FTE, FTLIF FTE FTE FTA FTLIF FTE

HCD HCD CONE MD CDNE CDNE CONE CDNE, CD CONE CONE CDNE ED HCD HCD TF HCD RFD HCD HCD FR HCD HCD

90 91 92 93 94 95 96 97 21 98 54 99 100 101 102 103 104 105 106 28a,b,c 107 108

Notes." The first column lists the short-lived molecule of interest, the second lists the transition, the third lists the spectroscopic method, the fourth lists the synthetic method used to produce the species, and the last column lists the reference. Abbreviations: CD = Cossart discharge. CONE = Corona discharge nozzle expansion. ED = Electric discharge. FR = Flow reactor. FrA = Fourier transform absorption. FTCIF = Fourier transform detection of collision induced fluorescence. FrE = Fourier transform emission. FTLIF = Fourier transform detection of laser induced fluorescence. HCD = Hollow cathode discharge. MD = Microwave discharge. RFD = Radiofrequency discharge. TF = Tube furnace.

188

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

Room Temperature

Jet=cooled

.

.

.

i

.

2413(1(I

.

.

~

.

.

.

.

1

,

25000 26~ 27000 Energy / cm-1

,

,

i

,

,

.

280(10

Figure 14. Comparison of direct absorption spectra ofthe ~, <-- ~ transition of OCIO. The top spectrum was measured at room temperature and the bottom spectrum was measured under jet-cooled conditions. shown under both room temperature and jet-cooled conditions. The expansion conditions used in the jet-cooled spectrum are 3% OC10 in helium with a total pressure of 1.5 atm. These conditions were chosen because they achieved the greatest rotational cooling, resulting in narrower vibronic bandwidths. Also, no vibrational hot bands were detected under these conditions. That direct absorption spectra with high signal-to-noise can be obtained with such dilute expansions demonstrates the superior sensitivity of the FTUV technique as compared to a conventional dispersive method. Based on the OC10 A +--X absorption cross section of about 3000 L mo1-1 cm -1, we calculated [28a] that the FTUV method ha s 15 times better sensitivity and three orders of magnitude higher resolution than our best dispersive absorption, supersonic jet technique [3c],. To date, systems with absorptivities as low as 8 -- 200 L mo1-1 cm -~ have been studied with FT spectroscopy in a free jet expansion with reasonable signal-to-noise at spectral resolutions as low as 0.1 cm -1.

189

FTS of Radicals

1~)

3

4

v 2 lo3 0 2 1~)2~ 3 102~30~ -

_

I

23oo0

1

5

3

6 4

4

7 5

5

8 6

6

7 7

8

_

I

240o0

,

1

250oo

~

I

260oo

I.

I

270oo

Energy / cm-1

Figure 15. Expanded portion of the jet-cooled spectrum of OCIO with vibrational assignments.

Three excited states of chlorine dioxide, 2A2, 2B2, and 2A 1, are located at about 2.5 eV in the near UV. Electronic dipole selection rules predict that the transition into the 2B 2 state is dipole forbidden, while the other two transitions are allowed. Experimentally, the observed transition is the parallel polarized .A (2A2) ~ ]~(2B1). The perpendicularly polarized (2A1) <----]~ (2B l) transition should be weaker and has not been observed experimentally. The most intense progression in the near UV spectrum, shown in Figure 15. is caused by excitation of the symmetric stretch (Vl). Combinations involving v 1 and the bend (v2) and even quanta of the asymmetric stretch (v 3) are also seen. The splitting in each vibronic band is due to the O35C10 and O37C10 isotopomers. There has been some question about the assignment of the progressions involving the asymmetric stretch [30]. The measurements performed with FT spectroscopy in our laboratory allowed a reinterpretation of these progressions. The conditions of the expansion and the high resolution of the FT technique allowed an examination of the rotational structure of several bands. Rotational band contours could be modeled with an asymmetric rotor program, and this served as a method for determining both the rotational temperatures achieved in the expansion and structural parameters of the various electronic states. As shown in Figures 16 and 17, under quite dilute conditions the rotational temperature derived from a Boltzmann fit to the (5,0,0) band was 30 K, while for neat expansions of OC10, the temperature determined from this fit yielded a rotational temperature of 80 K. High resolution spectra, such as those in Figure 18, were used to revise the excited state

190

ROHRS, FROST, ELLISON, RICHARD, and VAIDA

structure [28b, c]. The geometries of the 2B 1 electronic ground state and the 2A2 excited state are somewhat different. In the 2B~ state the angle is 117.6 ~ and closes to 107 ~ in the 2A2 state, while the bond length increases from 1.47 to 1.63 A. Note that because of the rotational cooling afforded by the jet, the individual isotopic bands could be evaluated separately. Due to the jet cooling of the spectrum, the rotational congestion of the vibronic bands was greatly reduced from that in the room temperature spectrum. The rotational cooling caused the two isotopic components of each vibronic band to become distinct (see Figure 15), and sharp bandheads for each isotopomer resulted (see Figures 16 and 17). Thus one benefit of the rotational cooling was accurate relative vibronic band positions. In addition, in FTS the entire spectrum is measured at once, which insures correct relative intensities. As a result, relative vibronic band areas, and therefore relative band intensities, can be accurately measured. Correct Calculation

Expedment

Energy / cm-t

Figure 16. A 0.5 cm -1 resolution spectrum of the (5,0,0) band. The top spectrum is calculated from an asymmetric rotor program with a Boltzmann rotational temperature of 30 K. The bottom spectrum is from an experiment with expansion conditions of 3% OCIO in 1.5 atm of helium.

FTS of Radicals

191 Calculation

Experiment

24380

24420 24~ Energy / cm-1

24500

Figure 17'. A 0.5 cm -1 resolution spectrum of the (5,0,0) band. The top spectrum is calculated with a Boltzmann rotational temperature of 80 K. The bottom spectrum is from an experiment with expansion conditions of 200 torr of neat OCIO.

relative vibronic band positions and intensities are necessary for the calculation of electronic potential surfaces. The vibronic bands give information about the structure of the excited electronic state relative to that of the ground state via a Franck--Condon analysis. Due to the activity in all of the vibrational modes, and the fact that the most intense vibronic bands involve many quanta of the symmetric stretch, it can be deduced that the bond lengths and bond angle in the 2A2 excited state are significantly different than those of the 2B 1 ground state. Along with a consideration of the electronic promotion occurring in the ,~(2A2) ~ X(2B1) transition, the vibronic analysis indicates that the bond length is larger and the bond angle is smaller in the excited state relative to the ground state. These observations of the vibronic bands confirm the results of the structural determination based on a rotational analysis. The spectra can further be interpreted to describe the excited-state dynamics [28b, d] The advantage of our cold molecular jet experiment is that inhomogeneous effects are minimized. In absorption, all lines are lifetime broadened because of the efficient photochemistry of OC10. The magnitude of this broadening does not depend on the rotational quantum numbers, but is strongly dependent on the

192

ROHRS, FROST, ELLISON, RICHARD, and VAIDA Experiment

Calculation

24470

24480

24490

Energy / cm- 1

Figure 18. The top spectrum shows part of the (5,0,0) band for the O35CIO isotopomer, measured at high resolution (0.05 cm -1) under expansion conditions of 10% OCIO in 1 atm of helium. The bottom spectrum is from a 100 K Boltzmann rotational temperature calculation. The experimental rotational linewidth is consistent with a 0.28 cm -1 width calculated Lorentzian lineshape.

vibrational quantum numbers. The lifetime of the A (2A2)excited state is estimated from the spectral linewidths to range from 20 picoseconds to several hundred femtoseconds. The rotational linewidths for the corresponding combination bands of v 1 with the bending mode v 2 and the asymmetric stretch v 3 are always much larger than those of v 1 alone. Based on this result, both v 2 and v 3 are thought to serve as promoting modes for the UV photochemistry of OC10. The bending mode in the A(2A2) state may be especially important [31] in bringing the oxygen atoms closer together, possibly resulting in the formation of C1OO and ultimately the production of CI and O 2. Indeed, the spectroscopic results were confirmed by photofragment studies [32] and ab initio calculations [31,33] showing that two photoreactions occur when OC10 is excited in the near-UV:

FTS of Radicals

193

(24)

OCIO + hv --->CIO + 0 ~ CI + 0 2

B. Spectroscopy of CN An example of an alternative use of FT technology in the UV/VIS is our work on the X2E- ~ B2E§ emission spectrum of jet-cooled CN [21]. These experiments were made possible by the development of the corona-excited supersonic expansion source by Engelking [20]. The Engelking source creates radicals in a continuous discharge, followed by immediate cooling in the expansion. Ahigh number density of rotationally and translationally cold radicals in excited electronic and vibrational states is produced. As a result, excited vibronic states of reactive species can be studied with a minimum of rotational congestion. This source works extremely well in combination with a FT spectrometer: the continuous, high-density production of rotationally cold radicals is matched with the high resolution and sensitivity of the spectrometer. This setup avoids the problems with timing and electronic noise that often occur in pulsed radical experiments. The entire spectrum is measured at one time, eliminating the need for precise control over radical production conditions such as flow rates. The X ~ B emission spectrum of CN measured at 0.25 cm -1 resolution is shown in Figure 19. The CN radical in its B state was produced by coexpanding 100 torr of acetonitrile (CH3CN) with 1 atm of helium in the corona discharge source. The spectrum includes both the 0-0 and 1-1 transitions. An analysis of the rotational distributions in both the v' = 0 and 1 levels revealed a Boltzmann temperature of

m g~

1!

..........

l

t 25755

:

l 25785

25815 Energy

1 25845

Z ~

1 25875

~__

/ cm- i

Figure 19. Jet-cooled emission spectrum of the X <-- B transition of CN measured at 0.25 cm -1 resolution. The expansion conditions were 1O0 Torr of acetonitrile seeded in 1 atm of helium. Both the 0-0 and 1-1 transitions are shown. The arrows indicate the small perturbations due to A state rotational levels (see text).

194

ROHRS, FROST,ELLISON, RICHARD, and VAIDA 15

I0

5

45

it

25740

50

I

25754

I

P(K)

55

59

i

25768 25782 Energy ! cm-I

25796

25810

Figure 20. Emission spectrum of the P-branch of the CN X ~ B transition at 0.25 cm -1 resolution. The expansion conditions were 100 tort of acetonitrile with ] atm argon buffer gas. A rotational assignment is shown with the spin-splittings indicated in the higher levels.

75 K, though for higher rotational levels the populations were higher than predicted by a Boltzmann model. The population ratio of the v' = 1 level to the v' = 0 represented a vibrational "temperature" of about 2200 K, which, though nonBoltzmann, is expected from the corona discharge source. Thus two vibrational levels of CN(B) could be studied relatively free from rotational congestion. The same electronic transition was also studied by expanding 100 torr of CH3CN in 1 atm of argon, as shown in Figure 20. Less effective rotational cooling in Ar compared to He is apparent in the enhancement of the P-branch bandhead. Due to population of higher rotational levels, spin-rotation doublets became resolved. These doublets were observed for rotational levels as high as K = 66 in the excited electronic state. From the line positions and accurate ground state constants [34], new rotational constants for the B state of CN were derived. In addition, B state rotational level perturbations by rotational levels in t h e 21-Ii electronic state [35] were investigated. Extra lines (shown by arrows in Figure 18) in the B state emission spectrum result from transitions beginning in the perturbed A state rotational level. Drastically different fluorescence lifetimes of the A and B electronic states give rise to a "feeding" process from the A to the B state through the perturbed rotational levels. Higher pressures of rare gas decrease the intensity of perturbed lines because of collisions within the lifetime of the B state fluorescence following the A ~ B feeding [36]. The perturbation has only a small effect in Figure 18 because of collisions between He and CN before the expansion. This

FTS of Radicals

195

result a g r e e s with the o b s e r v a t i o n o f c o o l e d C N radicals and c o n f i r m s that the c o o l i n g p r o c e s s for the m o s t part f o l l o w s the f o r m a t i o n o f the r a d i c a l s in the discharge.

ACKNOWLEDGMENTS The authors thank Dr. J. B. Burkholder, Dr. C. J. Howard, and Dr. E E Bemath for providing lists of their work. We also thank Dr. J. W. Brault for useful discussions and for providing us with some of his excellent notes. We thank Sarah Gallagher for donating her computer and her computer expertise. GBE and H W R are supported by a grant from the National Science Foundation (CHE-9215164) and the Petroleum Research Foundation (PRF 22664AC4-C). GJE ECR, and VV would like to thank the National Science foundation for their support (CHE-9013037) and the Petroleum Research Foundation (PRF25732-AC6-C).

REFERENCES AND NOTES 1. A major loss channel for organic radicals will be the dimerization or disproportionation of the radicals, R, with each other; R + R --->products. The rate of disappearance of R will be second order in R; --d[R]/dt = 2kn[R] 2. This yields an expression for the half life, 1;, of an initial radical sample, [Ro], of 1: = {2kll[Ro ] }-1. Thus the harder one works to increase the density [Ro]) the faster the radicals destroy themselves. Suppose that one ~nerated large samples (say 1 mtorr) of A. 10 1 a radical such as CH~. If we use the gas kinetic rate for k , 10- cm" sec- , we find that haft of 13 -3 "~ . the 3 x 10 " cm CH 3 radmals are destroyed in 1: = 0.3 msec. 2. For readers interested in greater detail, Fourier transform techniques are treated in the following references: (a) Marshall, A.G.; Verdun, ER. Fourier Transforms in NMR, Optical, and Mass Spectrometry; Elsevier: Amsterdam, 1986; (b) Griffiths, ER., DeHaseth, J.A. Fourier Transform Infrared Spectrometry; Wiley-Interscience: New York, 1986; (c) Chamberlain, J. The Principles oflnterferometric Spectroscopy; Wiley-Interscience: Chichester, 1979; (d) Bell, R. J. Introductory Fourier Transform Spectrometry; Academic Press: New York, 1972. 3. (a) Hirota, E.Annu. Rev. Phys Chemistry; Strauss, H.L. Babcock, G.T.; Leone, S.R., Eds.; Annual Reviews: Palo Alto, 1991, Vol. 42, pp. 1-22; (b) Bernath, EE Annu. Rev. Phys. Chem. 1990, 41, 91-122; (c) Vaida, V. Acc. Chem. Res. 1986, 19, 114-120; (d) Engelking, EC. Chem. Rev. 1991, 91, 399-414; (e) Robiette, A.G.; Duncan, J.L. Annu. Rev. Phys. Chem. 1983, 34, 245-273; (f) Jacox, M.E.J. Phys. Chem. Ref." Data 1984, 13, 945-1068; (g) Jacox, M.E.J. Phys. Chem. Ref. Data 1988,17, 269-511; (h) Northrup, EJ.; Sears, T.J.Annu. Rev. Phys. Chem. 1992, 43, 127-152. 4. (a) Michelson, A.A. Phil. Mag. 1891, 31,256; (b) Michelson, A.A. Phil. Mag. 1892, 34, 280. 5. Norton, R.H.; Beer, R../. Opt. Soc. Am. 1976, 66, 259-264. 6. Cooley, J.W.; Tukey, J.W. Math Comput. 1965, 19, 297. 7. (a) Mertz, L. Transformations in Optics; Wiley: New York, 1965; (b) Mertz, L. Infrared Phys. 1967, 7, 17; (c) Forman, M.L.; Steele, W.H.; Vanasse, G.A.J. Opt. Soc. Am. 1966, 56, 59. 8. Brault, J.W. Mikrochim. Acta 1987, 3, 215-227. 9. Nordstrom, R.J. In Fourier, Hadamard, and Hilberr Transforms in Chemistry; Marshall, A. G., Ed.; Plenum Press: New York, 1982. 10. The average spectral intensity B is given by:

196

ROHRS, FROST, ELLISON, RICHARD, and VAIDA fmax B(~)do -

B= -

Cmin ~maxd~J amin

11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

24.

25. 26. 27. 28.

29. 30.

31. 32.

33.

where ~min and ~max are the minimum and maximum spectral wavenumbers, respectively. The average spectral intensity has units of W/cm-1. Hirschfeld, T. Appl. Spectrosc. 1976, 30, 68. Luc, R; Gerstenkorn, S.Appl. Opt. 1978, 17, 1327. Johns, J.W.C. Mikrochim. Acta 1987, 3, 171-188. Private communication with Dr. Lenian Shen of Bruker Instruments Inc. (a) Spanbauer, R.; Fraley, RE.; Rao, K.N.AppL Opt. 1963, 2, 1340-1341; (b) Burkholder, J. B.; Hammer, RD.; Howard, C.J.J. Phys. Chem. 1987, 91, 2136-2144. (a) Baudais, EL.; Buijs, H. Bomem Technote 226; Bomem, Inc., Vanier, Quebec, Canada, 1986; (b) Bomem Training Manual; Bomem, Inc., Vanier, Quebec, Canada, 1987. Miller, D.R. In Atomic and Molecular Beam Methods; Scoles, G., Ed.; Oxford University Press: Oxford, 1988; Vol. 1, 14-53. Murphy, H.R.; Miller, D.R.J. Phys. Chem. 1984, 88, 4474-4478. White, J.U.J. Opt. Soc. Am. 1942, 32, 285. (a) Engelking, RC. Rev. Sci. Instrum. 1986, 57, 2274; (b) Droege, A. T.; Engelking, R C. Chem. Phys. Lett. 1983, 96, 316-318. Richard, E.C.; Donaldson, D.J.; Vaida, V. Chem. Phys. Lett. 1989, 157, 295-299. (a) Kohn, D.W.; Clauberg, H.; Chen, P. Rev. Sci. lnstrum. 1992, 63, 4003-4005; (b) Blush, J. A.; Park, J.; Chen, P. J. Am. Chem. Soc. 1989, 111, 8951-8953; (c) Minsek, D. W.; Chen, P. J. Phys. Chem. 1990, 94, 8399-8401. (a) Murphy, R.E.; Sakai, H. In Aspen International Conference on Fourier Spectroscopy; Aspen, Colorado, 1970, p. 307; (b) Sloan, J.J.; Aker, P.M. In Time Resolved Vibrational Spectroscopy: Springer Proceedings in Physics, Laberau, A.; Stockburger, M., Eds.; Springer Verlag: Berlin, 1985; Vol. 4, p. 6; (c) Leone, S.R.Acc. Chem. Res. 1989, 22, 139-144. (a) Gordon, R.; Ribbert, R.E.; Ausloos, P. Rare Gas Resonance Lamps; NBS Technical Note 496, 1969, U. S. Government Printing Office; (b) Kanofsky, J. R.; Lucas, D.; Gutman, D. Symp. [hit.] Combust. 1974, 14, 285. Amrein, A.; Quack, M.; Schmitt, U. J. Phys. Chem. 1988, 92, 5455. Gough, T. J. Chem. Phys. 1987, 86, 6012. (a) Vaida, V. Nato ASI Ser. 1987, 200, 253; (b) Vaida, V.; McCarthy, M.; Rosmus, P.; Werner, H.; Botschwina. P.J.J. Chem. Phys. 1987, 86, 6669. (a) Richard, E.C.; Wickham-Jones, C.T.; Vaida, V. J. Phys. Chem. 1989, 93, 6346; (b) Richard, E. C.; Vaida, V. J. Chem. Phys. 1990, 94, 163; (c) Richard, E.C.; Vaida, V. J. Chem. Phys. 1989, 94, 153; (d) Vaida, V.; Richard, E.C.; Cooper, L.A.; Flesch, R.; Ruhl, E. Ber. Bunsenges Phys. Chem. 1992, 96, 391. Rohrs, H.W.; Wickham-Jones, C.T.; Berry, D.; Ellison, G.B. Chem. Phys. Lett. (to be published). (a) Coon, J.B.; Cesani, EA.; Loyd, C.M. Discuss. Faraday Soc. 1963, 35, 118; (b) Richardson, A.W.; Redding, R.W.; Brand, J.C.D.J. Mol. Spectrosc. 1969, 29, 93; (c) Coon, J.B.; Ortiz, E. J. MoL Spectrosc. 1957, 1, 81. Gole, J.L.J. Phys. Chem. 1980, 84, 1333. (a) Ruhl, E.; Jefferson, A.; Vaida, V. J. Phys. Chem. 1990, 94, 2990; (b) Bishenden, E.; Hancock, J.; Donaldson, D.J.J. Phys. Chem. 1991, 95, 2113; (c) Bishenden, E.; Donaldson, D.J.J. Phys. Chem. (in press); (d) Davis, H.E; Lee, Y. T. J. Phys. Chem. 1992, 96, 5681. (a) Peterson, K.A.; Werner, H. J. Chem. Phys. 1992, 96, 8948; (b) Jafri, J. A.; Lengsfield III, B. H.; Bauschlicher Jr., C.W.; Phillips, D.H.J. Chem. Phys. 1985, 83, 1693.

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34. Cemy, D.; Bacis, R.; Guelachvili, G.; Roux, E J. Mol. Spectrosc. 1978, 73, 154. 35. (a) Osaki, Y.; Nagata, T.; Suzuki, K.; Kondow, T.; Kuchitsu, K. Chem. Phys. 1983, 80, 73; (b) Long, S.R.; Reilly, J.P.J. Phys. Chem. 1982, 86, 56. 36. Osaki, Y." Kondow, T." Kuchitsu, K. Chem. Phys. 1983, 77, 223. 37. Hedderich, H.G.; Dulick, M.; Bernath, P.E J. Chem. Phys. submitted. 38. Hedderich, H. G.; Bernath, P. F. J. Mol. Spectrosc. 1992, 153, 73-80. 39. White, J.B.; Dulick, M.; Bernath, P.F.J. Chem. Phys. submitted. 40. Amiot, C." Verges, J. Chem. Phys. Lett. 1991, 185, 310-312. 41. Walker, K.A.; Hedderich, H.G.; Bernath, P.E MoL Phys. 1993, 78, 577-589. 42. Hedderich, H.G.; Bemath, P.E J. Mol. Spectrosc. 1993, 158, 170--176. 43. Fink, E.H.; Setzer, K.D.; Ramsay, D.A.; Vervloet, M.; Brown, J.M.J. Mol. Spectrosc. 19911, 142, 108-116. 44. Fink, E.H.; Setzer, K.D.; Ramsay, D.A.; Vervloet, M. Chem. Phys. Lett. 1991, 179, 95-102. 45. Pianalto, ES.; O'Brien, L.C.; Keller, P.C.; Bernath P.F.J. MoL Spectrosc. 1988, 129, 348-353. 46. Orlando, J.J.; Burkholder, J.B.; Bopegedera, A.M.R.P.; Howard, C.J.J. MoL Spectrosc. 1991, 145, 278-289. 47. Frum, C.I.; Pickett, H.M.J. MoL Spectrosc. 1993, 159, 329-336. 48. Douay, M.; Nietmann, R.; Bemath, P.E J. MoL Spectrosc. 1988, 131,250-260. 49. Douay, M.; Nietmann, R., Bernath, P.E J. MoL Spectrosc. 1988, 131,261-271. 50. Burkholder, J.B.; Sinha, A.; Hammer, P.D.; Howard, C.J.J. MoL Spectrosc. 1988, 127, 61-69. 51. Bemath, P.E; Brazier, C.R.; Olsen, T.; Halley, R.; Fernando, W.T.M.L.; Woods, C.; Hardwick, J.L. J. Mol. Spectrosc. 1991, 147, 16-26. 52. Bemath, P.E J. Chem. Phys. 1987, 86, 4838-4842. 53. Amiot, C.; Verges, J. Chem. Phys. Lett. 1983, 95, 189-194. 54. Rehfuss, B.D.; Suh, M.; Miller, T.A.; Bondybey, V.A.J. MoL Spectrosc. 1992, 151,437-458. 55. Ram, R.S.; Bernath P.E J. MoL Spectrosc. 1987, 122, 282-292. 56. Ram, R.S.; Tam, S.; Bemath, P.E J. Mol. Spectrosc. 1992, 152, 89-100. 57. Burkholder, J.B.; Lovejoy, E.R.; Hammer, P.D.; Howard, C.J.J. Mol. Spectrosc. 1987, 124, 450-457. 58. Amiot, C.; Crepin, C.; Verges, J. Chem. Phys. Lett. 1983, 98, 608-610. 59. Amiot, C.; Crepin, C.; Verges, J. J. MoL Spectrosc. 1984, 107, 28-47. 60. Amiot, C.; Verges, J. Chem. Phys. Lett. 1985, 116, 273-278. 61. Amiot, C.; Crepin, C.; Verges, J. Chem. Phys. Lett. 1984, 106, 162-165. 62. Burkholder, J.B.; Hammer, P.D.; Howard, C.J.J. Geophys. Res. 1989, 94, 2225-2234. 63. Burkholder, J.B.; Hammer, P.D.; Howard, C.J.; Maki, A.G.; Thompson, G.; Chackerian Jr., C. J. MoL Spectrosc. 1987, 124, 139-161. 64. Cheung, A.S.C.; Zymicki, W.; Merer, A.J.J. MoL Spectrosc. 1984, 104, 315-336. 65. Ram, R.S.; Jarman, C.J.; Bernath P.E J. Mol. Spectrosc. 1993, in press. 66. Ram, R.S.; Bernath, P.E; Brault, J.W.J. Mol. Spectrosc. 1985, 113, 269-274. 67. Hammer, P.D.; Sinha, A.; Burkholder, J.B.; Howard, C.J.J. MoL Spectrosc. 1988, 129, 99-118. 68. Burkholder, J.B.; Hammer, P.D.; Howard, C.J.; McKellar, A.R.W.J. MoL Spectrosc. 1986, 118, 471-480. 69. Taylor, A.W.; Cheung, A.S.C.; Merer, A.J.J. MoL Spectrosc. 1985, 113, 487-494. 70. Campbell, J.M.; Dulick, M.; Klapstein, D.; White, J.B.; Bemath, P.E J. Chem Phys. (submitted). 71. Uehara, H.; Honai, K.; Nakagawa, K.; Suguro, H. Chem. Phys. Lett. 1991, 178, 553-557. 72. Rogers, S.A.; Brazier, C.R.; Bernath, P.E Mol. Phys. 1988, 63, 901-908. 73. Uehara, H.; Horiai, K.; Mitani, T.; Suguro, H. Chem. Phys. Lett. 1989, 162, 137-139. 74. Frum, C.I.; Engleman Jr., R.; Bernath, P.E Chem. Phys. Lett. 199t), 167, 356-361. 75. Linton, C.; Martin, E; Crozet, P.; Ross, A.J.; Bacis, R. J. Mol. Spectrosc. 1993, 158, 445-454. 76. Linton, C.; Martin, F.; Bacis, R.; Verges, J. J. MoL Spectrosc. 1990, 142, 340-375. 77. Linton, C.; Martin, E; Bacis, R.; Verges, J. J. MoL Spectrosc. 1989, 137, 235-241.

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78. Ram, R.S.; Bernath, P.F.J. Mol. Spectrosc. 1992 155, 315-325. 79. Chackerian Jr., C.; Guelachvili, G.; Lopez-Pineiro, A.; Tipping, R.H.J. Chem. Phys. 1989, 90, 641 --649. 80. Boujaadar, D.; Brion, J.; Chollet, P.; Guelachvili, G.; Vervloet, M. J. Mol. Spectrosc. 1986, 119, 352-366. 81. Sinha, A.; Burkholder, J.B.; Hammer, P.D.; Howard, C.J.J. Mol. 82. Fellows, C.E.; Verges, J.; Amiot, C. J. Chem. Phys. 1990, 93, 6281-6290. 83. Verges, J.; Effantin, C.; d'Incan, J.; Topouzhhanian, A.; Barrow, R.F. Chem. Phys. Lett. 1983, 94, 1-3. 84. Barrow, R.F.; Amiot, C.; Verges, J.; d'Incan, J.; Effantin, C.; Bernard, A. Chem. Phys. Lett. 1991, 183, 94-97. 85. Hardwick J.L.; Whipple, G.C.J. MoL Spectrosc. 1991, 147, 267-273. 86. Nelson Jr., D.D.; Schiffman, A.; Nesbitt, D.J.; Orlando, J.J.; Burkholder, J.B.J. Chem. Phys. 199{t, 93, 7003-7019. 87. Ram, R.S.; Bernath, P.F.J. Mol. Spectrosc. 1987, 122, 275-281. 88. McCarthy, M.C.; Field, R.W.; Engleman Jr., R.; Bernath, P.F.J. Mol. Spectrosc. 1993, 158, 208-236. 89. Frum, C.I.; Engleman, R.; Bernath, P.E J. MoL Spectrosc. 1991, 150, 566--575. 90. Amiot, C.; Crozet, P.; Verges, J. Chem. Phys. Lett. 1985, 121,390-394. 91. Ram, R.S.; Bernath, P.F.J. Chem. Phys. 1992, 96, 6344-6347. 92. Bernath, P.F.; Rogers, S.A.; O'Brien, L.C.; Brazier, C.R.; McLean, A.D. Phys. Rev. Lett. 1988, 60, 197-199. 93. Brazier, C.R.; O'Brien, L.C.; Bernath, P.F.J. Chem. Phys. 1989, 91, 7384-7386. 94. Knights, J.C.; Schmitt, J.P.M.; Perrin, J.; Guelachvili G. J. Chem. Phys. 1982, 76, 3414-3421. 95. Betrencourt, M.; Boudjaadar, D; Chollet, P.; Guelachvili, G.; Morillon-Chapey, M.J. Chem. Phys. 1986, 84, 4121--4126. 96. Frum, C.I.; Engleman Jr., R; Bernath, P.F.J. Chem. Phys. 1990, 93, 5457-5461. 97. Benidar, A.; Farrenq, R.; Guelachvili, G.; Chackerian Jr., C. J. Mol. Spectrosc. 1991,147, 383-391. 98. Burkholder, J.B.; Lovejoy, E.R.; Hammer, P.D.; Howard, C.J.J. Mol. Spectrosc. 1987, 124, 379-392. 99. Fink, E.H.; Setzer, K.D.; Ramsay, D.A.; Vervloet, M. J. Mol. Spectrosc. 1990, 138, 19-28. 100. Fink, E.H.; Setzer, K.D.; Ramsay, D.A.; Vervloet, M. Chem. Phys. Lett. 1991, 177, 265-268. 101. Douay, M.; Rogers, S.A.; Bernath, P.F. Mol. Phys. 1988, 64, 425-436. 102. Maki, A.G.; Burldzolder, J.B.; Sinha, A.; Howard, C.J.J. Mol. Spectrosc. 1988, 130, 238-248. 103. Vervloet, M.; Herman, M. Chem. Phys. Lett. 1988, 144, 48-50. 104. Hinkle, K.H.; Keady, J.J.; Bernath, P.F. Science 1988, 241, 1319-1322. 105. Sasada. H.; Amano, T.; Jannan, C.; Bernath, P.F.J. Chem. Phys. 1991, 94, 2401-2407. 106. Burkholder, J.B.; Howard, C.J.; Hamilton, P.A.J. MoL Spectrosc. 1988, 127, 362-369. 107. Wormhoudt, J.; McCurdy, K.E.; Burkholder, J.B. Chem. Phys. Lett. 1989, 158, 480--485. 108. McKellar, A.R.W.; Burkholder, J.B.; Sinha, A.; Howard, C.J.J. Mol. Spectrosc. 1987, 125, 288-308. 109. McKellar, A.R.W.; Burkholder, J.B.; Orlando, J.J.; Howard, C.J.J. Mol. Spectrosc. 1988, 130, 445-453. 110. Burkholder, J.B.; Sinha, A.; Hammer, P.D.; Howard, C.J.J. Mol. Spectrosc. 1988, 126, 72-77. 111. Petersen, J.C.; Vervloet, M. Chem. Phys. Lett. 1987, 141,499-502. 112. Johns, J.W.C.; McKellar, A.R.W.; Weinberger, E. Can. J. Phys. 1983, 61, 1106--1119. 113. Elhanine, M.; Farrenq, R.; Guelachvili, G. J. Chem. Phys. 1991, 94, 2529-2531. 114. Burkholder, J.B.; Hammer, P.D.; Howard, C.J.; Towle, J.P.; Brown, J.M.J. Mol. Spectrosc. 1992, 151,493-512. 115. McKellar, A.R.W.; Vervloet, M.; Burkholder, J.B.; Howard, C.J.J. Mol. Spectrosc. 1990, 142, 319-335.

FTS of Radicals 116. 117. l l8. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144.

199

Burkholder, J.B.; Howard, C.J.; McKellar, A.R.W.J. Mol. Spectrosc. 1988, 127, 415-424. Brazier, C.R.; Bernath, EE; Burkholder, J.B.; Howard, C.J.J. Chem. Phys. 1988, 89, 1762-1767. Ohno, K.; Matsuura, H.; Murata, H. J. Mol. Spectrosc. 1983, 100, 403-415. Kawaguchi, Kentarou J. Chem. Phys. 1992, 96, 3411-3415. Kawaguchi, K.; Ishiwata, T.; Tanaka, l.; Hirota, E. Chem. Phys. Lett. 1991, 180, 436-440. Duxbury, G.; LeLerre, M.L.J. MoL Spectrosc. 1982, 92, 326-348. Hirahara, Y.; Masuda, A.; Kawaguchi, K. J. Chem. Phys. 1991, 95, 3975-3979. Kroto, H.W.; McNaughton, D. J. Mol. Spectrosc. 1985, 114, 473-482. Kroto, H.W.; McNaughton, D. J. Mol. Spectrosc. 1985, 114, 473-482. Bernath, EE; Hinkle, K.H.; Keady, J.J. Science 1989, 244, 562-564. Fernando, W.T.M.L.; O'Brien, L.C.; Bernath, P.E J. Chem. Phys. 1990, 93, 8482-8487. Femando, W.T.M.L.; Bernath, P.E J. MoL Spectrosc. 1991, 145, 392-402. O'Brien, L.C.; Brazier, C.R.; Bernath, P.F.J. Moi. Spectrosc. 1987, 124, 489-493. Brazier, C.R.; O'Brien, L.C.; Bernath, P.E J. Chem. Phys. 1987, 86, 3078-3081. Oliphant, N.; Lee, A.; Bernath, P.E; Brazier, C.R.J. Chem. Phys. 1990, 2244-2247. Carrick, P.G.; Brazier, C.R.: Bernath, P.E J. Am. Chem. Soc. 1987, 109, 5100-5102. Brazier, C.R.; Carrick, P.G.; Bernath, P.F.J. Chem. Phys. 1992, 96, 919-926. Prasad, C.V.V.; Bernath, P.F.; Frum, C.I.; Engleman Jr., R.J. MoL Spectrosc. 1992,151,459-473. Prasad, C.V.V.; Bernath, P.E J. Mol. Spectrosc. 1992, 156, 327-340. Choe, J.; Rho, Y.; Lee, S.; LeFloch, A.C.; Kukolich, S.G.J. MoL Spectrosc. 1991, 149, 185-213. Ram, R.S.; Jarman. C.N.; Bernath, P.E J. Mol. Spectrosc. 1992, 156, 468-486. Femando, W.T.M.L.; O'Brien, L.C.; Bernath P.F.J. MoL Spectrosc. 1990, 139, 461-464. Linton, C.; Bacis, R.; Crozet, P.; Martin, E; Ross, A.J.; Verges, J. J. Mol. Spectrosc. 1992, 151, 159. Bernath, P.E, Black, J.H.; Brault, J.~. Astrophysical J. 1985, 298, 375-381. Hemmerling, B.; Vervloet, M. Mol. Phys. 1993, 78, 1423-1447. Brazier, C.R.; Ram, R.S.; Bernath, P.E J. Mol. Spectrosc. 1986, 120, 381-402. Ram, R.S.; Bernath, P.F.J. Opt. Soc. Am. B 1986, 3, 1170-1174. Gustavsson, T.; Amiot, C.; Verges, J. J. Moi. Spectrosc. 1991, 145, 5665. O'Brien, L.C.; Fernando, W.T.M.L.; Bernath, P.E J. Moi. Spectrosc. 1990, 139, 424-431.

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THE INTERPLAY BETWEEN X-RAY CRYSTALLOGRAPHY AND AB INITIO CALCULATIONS

Roland Boese, Thomas Haumann, and Peter Steliberg

I. II. III. IV. V. VI.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ab Initio Studies as a Tool for Rationalizing Unexpected Experimental Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expansion of the Theoretical Approach to Reproduce Experimental Data from X-ray Studies . . . . . . . . . . . . . . . . . . . . . Modification of the Experimental-Based Model Prompted by Ab Initio Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Verification of Experimental Results by Ab Initio Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References and Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 1, pages 201-226. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8

201

202 202 206 210 212 215 223 224 224

202

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG

ABSTRACT X-ray crystallography and molecular orbital (MO) calculations are today the most important methods for the exact determination of the geometries of small molecules. Although both methods are based on different models and consequently must provide different results, there seems to be some competition instead of acceptance that both methods are complementary in many respects and may fertilize each other. With selected examples from our own experience, we describe in this review how fruitful the interplay between the two methods can be. One of the advantages of ab initio calculations is the fact that it is possible to calculate nonexisting molecular conformations that are not accessible for the experiment. The method to change molecular geometries, starting from the experimentally determined ones to nonexisting conformations, may help to rationalize unexpected geometries. Sometimes an extreme deviation may be found between the structure as determined in the crystal lattice and by ab initio calculations. This calls for more appropriate theoretical models to consider the environment of a molecule which may have a significant influence on its geometry, in solution as well as in the solid state. Another topic where ab initio calculations may help to select a more probable model for crystal structure determinations are the problems connected with the phenomenon of disorder. Bond distances from X-ray structure determinations must differ from those attained by ab initio calculations. This problem can be overcome by comparing differences of distances rather than distances themselves. Even small influences of conjugation and strain can be detected in such a way which is demonstrated in a series of conjugated cyclopropyl- and spirocyclopropylderivatives taking spiropentane as a reference system.

I.

INTRODUCTION

The development of modern equipment for X-ray diffractometry on the experimental side and the extended computational capacities of modern computer systems on the theoretical side allows the achievement of high accuracy structural information. For small molecules, even for those which are gaseous or liquid at e,mbient conditions, where in situ crystallization techniques directly on the diffractometer [1] can be applied, precise structural parameters are accessible. Especially these small molecules are simultaneously of adequate size for high level [2] ab initio calculations. Considering these two structure-determination methods, it seems as if there is a competition and, when reading between the lines in some publications, the reader gets the impression that the experts of two competing methods argue which one is the more appropriate. Among the structural chemists in these two disciplines, two apparently contradicting opinions exist. To an increasing extent, theoreticians believe that'it is unnecessary to perform expensive experiments for molecular structure determinations, whereas others point out that it is much more convenient, precise, and reliable to calculate the geometries and properties of the molecules of interest with the

X-ray Structures Versus Ab Initio Geometries

203

appropriate tools of quantum chemical calculations. Up to some years ago theoreticians needed the experimental structure determinations in order to prove the reliability of the method. Today, more confident in their methods and strategies, they start to check the experimental results and sometimes even claim that these must be wrong in cases when they do not match the calculated geometries [3]. But still, and to a decreasing extent, the experimentalists claim that only they can provide the "truth" because they suspect that theoreticians might be able to manipulate the results using various methods or basis sets until they fit the experimental results. This chapter is written to reconciliate the diverging opinions and to address the possibilities which allow a fruitful interplay, giving justification for both methods. The treatise herein is restricted to single crystal X-ray structure determinations and high level ab initio calculations because both claim to provide reliable information on the structures of molecules. We intend to demonstrate that both methods are biased by handicaps in providing essential information to chemistry and that the use of bidirectional thinking is sometimes much more appropriate for the problems instead of one-way approaches. Whenever standpoints tend to diverge it often starts with semantic problems and these are to be found mostly among preparative chemists. This is so because the majority of them are pure experimentalists and consequently they believe exclusively in the truth of experiments. Therefore they do not accept that theoretical studies on molecular structures may be called "structure determination" because calculations do not provide an "ultimate" picture of a molecule. On the other hand, many experimentalists rely almost blindly on the results of X-ray structure determinations, not being aware of the fact that the nice ORTEP plots which they are keen on are derived from a model with all the restrictions a model can have. The plots are very often taken as a real display of a molecule and not that of a model. And of course, a model may be wrong, which will be demonstrated in Section IV. However, by single crystal structure determinations, the model is based on a vast number of experimental data called "observed intensities." No other experimental structure determination method is based on so many observed data, which results in the high accuracy of the method. But it should be considered that it is only a statistical accuracy, neglecting many systematic errors. Even when experimental standard deviations of bond distances are given at 0.001 ,~,, they are derived from an X-ray beam with a wavelength of approximately 1 /k, and generally the resolution cannot be higher than that of the applied wavelength. Consequently, the low standard deviations achieved are a result of the immense overdetermination of parameters during the least-squares refinement cycles. X-rays are scattered by electrons only and this produces problems when hydrogen atom positions are determined. Although the exact position of the hydrogen atoms is important for many molecules, the fact that these can be determined with low accuracy is a main drawback of X-ray structure determinations.

204

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG

Most chemists are well aware that molecules in the solid state may be biased by the surrounding, but they should note that the molecules are always influenced more or less by the neighboring molecules. Nevertheless, the so-called packing effects are only reported if a significant distortion is found in the expected geometry. It is widely assumed that packing effects mostly influence torsion angles of free rotating groups, and then, to a lower extent, the angles between atoms are biased and almost no influence is found on the bond distances. This is in accord with the decreasing force constants of the respective parameters. However, it will be demonstrated in Section III that even bond distances may be influenced significantly if the potential energy surface of a bond is flat and highly influenced by the crystal lattice. The expected local symmetry, as assumed from theoretical calculations, is generally not established in the crystal lattice. Instead it is usual that the crystal packing distorts the local symmetry of the molecule. Thus, if the molecule has no local symmetry within the 36 criterionmi.e., geometric parameters obey the assumed local symmetry within the threefold of the given standard deviationmthe free molecule should be considered adopting the local symmetry. On the other hand, there exist many examples in which the crystal symmetry requires a local symmetry which should not exist for the free molecule. This very often holds for molecules which reveal disorder in the crystal lattice, another problem which is outlined in detail below in Section IV. The common interest of both theoreticians and experimentalists is generally to find the geometry attributed to the global energy minimum of a molecule. However, the experimentalists can never claim to provide information about a free molecule because of the surrounding, which always causes a local minimum for the molecule under consideration. Therefore they need the help of the theoreticians who can prove, by careful analysis of the force constants of considered conformers, which geometry is related to the global minimum in respect to a given surrounding. As a result it seems that the experimentalists only need the assistance of theoreticians. However, theoreticians mostly obtain the ideas about what kind of molecules should be calculated, which molecule is associated with a specific problem, or what kind of reaction should be investigated from the experimentalist. The experimental results may help because they are expected to find a molecule at least close to the minimum energy of a free molecule, hence the comparison of both results must be carried out critically. The computational methods can go one step further: they can calculate geometries which are not observed in the crystalline state. Molecules which represent the ground state, transition states, or even excited states can be calculated and this will help to find the answer to many questions which arise with the chemical behavior or structural investigations. In any case both methods are expected to provide comparable results for a specific conformation in respect to intramolecular distances. When comparing, one has to be aware of the deficiencies of each method applied. The difference of interatomic distances between theoretical and experimental structure investigations has been treated in numerous publications and is therefore

X-ray Structures Versus Ab Initio Geometries

205

not repeated herein [4]. It is clear by the methodological differences that the bond distances are not expected to match exactly, independently from the precision of the experimental structure determination and also independently from the level of calculation. If the results match perfectly, this must be considered much more as a coincidence than the consequence of precision of both methods. Ab initio calculated bond distances at a HF/6-31 G* level are generally found to be shorter than those from X-ray diffraction because the former incorporates deficiencies in respect to the Hartree-Fock procedure, and the latter determines the structure in a vibrational excited state. The (more accurate) data from low-temperature diffraction experiments tend to provide even longer bond distances than from room temperature experiments (unless these data are corrected for libration). The shorter bond distances at room temperature are attributed to a systematical error of the model, generally applied and not an effect of a real change of the geometry of molecules. Consequently the more precise low-temperature structure determination provides even longer bond distances than the computational structures calculated by using the Hartree-Fock method. In spite of all these problems it is suggested to compare trends of interatomic distances applying the same method on a series of molecules. For this, structural parameters of a carefully chosen molecule must be taken as a reference and only the changes of these parameters within the series should be considered. Applying this strategy, a comparison of differences provides essential information how well both methods describe the effects controlling the molecular structure. This is demonstrated below in Section V. Very often we find an unexpected geometry for a molecule by experiment, and even careful inspection of intermolecular distances does not explain the findings. In this case, the only possibility remains to consult computational methods. Provided they reproduce the geometry as outlined above, further options exist, e.g., to change the conformation and rationalize the formerly unexpected geometry (see Section II). What can be done if discrepancies exist between both methods which are not explained by intermolecular contacts? Another theoretical approach which considers the surrounding of a molecule may help to understand experimental results. By doing this, it seems that theoreticians manipulate their method until it fits the experiment, but it is justified if the unexpected geometry is reproduced and it can be readily explained why the modification is necessary in a particular case (see Section Ill). Also, the experimental model applied may be modified if the results of both methods diverge significantly. Especially for those cases in which no further help exists to decide which experimental model is the most reliable one, theoretically based structures may provide a worthy help for such a decision. Of course, this does not happen very often, but if disorder or pseudosymmetry is detected in a crystal lattice, the determination of the correct model is difficult.

206

ROLAND BOESE, THOMAS HAUMANN, and PETER STELLBERG

A series of examples from various fields of structural chemistry is given below to outline the previous statements.

Ii. AB INITiO STUDIES AS A TOOL FOR RATIONALIZING UNEXPECTED EXPERIMENTAL GEOMETRIES With the structure determination of triethylboroxin (Figure 1) we focused on the geometry of the six-membered ring. After detecting a solid-state phase transition we were attracted by the mechanism of the change of the assembly of the molecules in the crystal lattice [5], totally disregarding the geometry of the attached ethyl groups. Reviewing the structural data we found that the B-C-C angles were surprisingly large (119.4~ This intrigued us to reexamine other molecules containing a B-C-C fragment and we found that the widened angle was common to all structures--however to a varying extent, [e.g., in B-triethylborazine [6], 116.1 (1)o]. In most of these cases the expansion of the B-C-C angle could not be explained as the result of strong repulsive intermolecular [7] or intramolecular [8] interactions. In triethylboroxin, for example, the shortest intermolecular H-..H distance is 2.39/~, [5]. Therefore we suspected that hyperconjugational effects might be responsible for the widening of the B-C-C angle. In order to compare the B--C-C angles in

9

~_Bla

01 B1

C1 #zO 02

Clb ~

01o

Figure 1. Molecular structure of triethylboroxin, presentation with thermal probability plots of 50%. Mean values of bond distances (A) and angles (degrees). Standard deviations in brackets: BO 1.387(1)/1.380(1), BC 1.564(1), CC 1.423(1), OBO 118.3(1), BOB 121.7(1), BCC 116.8(1).

207

X-ray Structures Versus Ab Initio Geometries

C2 Cl C4 C5

(

C2 Cl C

'

'~

C1'

Figure 2. Molecular structures of Et3B and Na+Et4B-, presentation with thermal

probability plots of 50%. Mean values for selected bond lengths (A) and angles (degrees). Standard deviations in brackets: EtgB: BC 1.573(1), CC 1.529(1), BCC 118.9(2); b)Na+Et~ :BC 1.646(4), CC 1.540(6)/1.519(7), BCC 116.4(2). molecules with and without potential hyperconjugation, we carried out the X-ray structure determination of Et3B [9] with a vacant p-orbital at the boron atom, and with sodium tetraethylborate (Na§ B-) [9,10], which has no vacant p-orbital and therefore no hyperconjugation is expected (Figure 2). In Et3B we found a B-C-C angle in the same range as in triethylboroxin, but surprisingly in tetraethylborate the angle ( 116.4 ~ was also significantly widened compared to the tetrahedral angle (109.47~ The question arises why the angle in tetraethylborate is that large and in triethylborane even larger?

208

ROLAND BOESE, THOMAS HAUMANN, and PETER STELLBERG

Ab initio geometry optimizations with MP2/6-31G* [10] level set provided the answer. The strength of computational methods is demonstrated because not only the experimentally determined molecules in their observed conformations could be calculated. Also, various conformations which did not exist in the crystal structure as well as other main group elements of the first and second periods instead of boron could be included in these calculations. The experimental structure of Et3B, which is only slightly distorted from the expected C3h-symmetry reveals B-C bond distances of 1.573(1) A (mean value) and B - C - C bond angles of 118.9(2) ~ The minimum nonbonded H-..H distances are 2.50 A (intramolecular) and 2.44 /k (intermolecular) and are not expected to have a major influence on the molecular structure. The EtaB- ion has (crystallographic) C2 symmetry and has greater B-C bond distances [1.646(4)/k, mean value] as typical for tetracoordinated boroncarbon compounds. The B-C-C bond angles [ 116.4(2) ~ are smaller than in Et3B, but in the same range as in related compounds [11]. The shortest intermolecular (2.41/k) and intramolecular (2.35 .A,) H---H distances, likewise, should not affect the molecular geometry significantly. Consequently, the question arises for the main effect on the geometry if there is no strong steric repulsion nor a reasonable possibility for hyperconjugation. In the large set of ethyl compounds which were computed in order to find an answer, a good correlation between the X - C - C bond angle and the electronegativity of X [12] (according to Allred Rochow, where element and group electronegativities are very similar) resulted (Figure 3).

117

[] Li -.

116

Na

118

115

[] MP2/6-31G** ~ BHgecl. [] ""'-[] A1H2ecl. MgH ~ .....

~lPH2

9 Exp. angle

stag []

Bell " ' " " - - - .

~ SH gauche

9 NH2stag.

114 oSiH 3 ~ BH

113 112 |

d

111

o A1H2 perp.

....- . m "'---. CH3 o". . . . .

9

[]pH 2

110

9 Ci

-'--...... ~ NH 2

[] SH anti

109

[] OH gauche

Ci

9 "--..

F ""---

[]

108 6 OH anti

107 106

BH2Perp.

105

.

0,80 1,00 1,20 1,40 1,60 1,80 2,00 2,20 2,40 2,60 2,80 3,00 3,20 3,40 3,60 3,80 4,00 EN

Figure 3. Electronegativity EN (according to AIIred-Rochow) and the X-C-C angles

(MP2/6-31G*: black dots; experimental data" triangles) of ethyl-main-group elements (ecl. = eclipsed, stag. = staggered, perp. = perpendicular). For more details see text and ref. [22].

209

X-ray Structures Versus Ab Initio Geometries

Table 1. Bond Angles txa Determined by X-ray Crystallography for Compounds

Containing the Fragments Et-X

X

nb

(x -I- o[ ~

~min[ ~

~max[ o]

Be

2

115.35 + 0.95

114.67

116.02

Mg

7

120.73 + 6.98

111.65

129.58

B

88

115.50 + 3.38

100.08

128.07

A1

51

116.82 + 4.78

105.59

133.74

C

3431

113.50 + 4.53

74.88

159.66

Si

24

116.69 + 3.88

112.87

132.03

N

3105

118.60 + 3.41

75.42

145.88

P

1410

115.64 + 3.89

95.85

137.47

O

1751

110.01 + 5.44

83.96

164.32

S

245

112.69 + 3.99

88.83

139.53

Notes:

a x - c - c , for X = main group element.

A simultaneously performed search in the Cambridge Structural Data Base revealed that expanded X-C-C bond angles are more or less common (see Table 1) [13-15]. Generally, the compounds with more electronegative substituents, X, have smaller X-C-C bond angles and the more electropositive substituents induce larger angles. As a result, the expanded X-C-C angles (e.g., the computed X-ray structure angles in Et4B- ) are primarily the consequence of the o-inductive effect of the boron atom in the absence of a vacant p-orbital. Deviations from this regular behavior are the consequence of hyperconjugation effects, which are conformation-dependent. If, for example, in EtBH 2 the empty boron p-orbital is in the B-C-C plane with a perpendicular conformation, the possible hyperconjugation with the C-C bond results in a reduction of the B-C-C bond angle to 105.3 ~. In the eclipsed conformation, however, where the empty boron p-orbital is perpendicular to the B--C-C plane, hyperconjugation with the ct-C-H bonds occurs and the B-C-C angle is influenced indirectly. The antibonding interactions between the hydrogen atoms are reduced by withdrawing the electrons from the orbital with r~-symmetry at the CH 2 group, and the orbital with H...H character has more influence. Both result in the decrease of the H - C - H angle and as a consequence the B-C-C angle is expanding (Thorpe Ingold hypothesis) [16]. At the beginning of this study there was no understanding of the widened B-C-C angles. By consulting the theoreticians, the results of the computational work led to the understanding of the background which is responsible for the observed geometries. This example demonstrates how effective the combination of experimental and theoretical work can be in order to tackle problems which could not be solved with only one discipline alone.

210

ROLAND BOESE, THOMAS HAUMANN, and PETERSTELLBERG

III. EXPANSION OF THE THEORETICAL APPROACH TO REPRODUCE EXPERIMENTAL DATA FROM X-RAY STUDIES Ammoniaborane, first described in 1926 [17], is the isoel~tronic and isosteric counterpart to ethane, which crystallizes in the monoclinic space group P21/n [18]. An early X-ray structure determination of ammoniaborane assigned the tetragonal space group 14mm (No. 107) with two independent molecules in the asymmetric unit and yielded B-N distances of 1.56(5) [19] and 1.60(20) ,~ [20], but in both molecules the B and N atoms were found to be disordered. A reinvestigation revealed the orthorhombic space group Pmn21 (No. 31) with a B-N distance of 1.565(7) ,~ and no disorder of the hydrogen atoms or B and N atoms was detected [6] (Figure 4). In the common refinement procedure only the scattering power of neutral atoms with a spherical electron density distribution are taken into account, thus the results can be interpreted as follows. The displacement parameters of boron and nitrogen reproduce the polarity of the B-N bond established by a greater axis of the ellipsoids at boron and a smaller axis at the nitrogen atom in the bond direction. The crystal packing of ammoniaborane is quite different from that of ethane, caused by the molecular dipole which should also be responsible for the higher calculated density of the B-N compound (ethane: 0.719 gcm -3 [18]; ammoniaborane: 1.142 gcm-3 [20]). The arrangement of the ammoniaborane molecules in the crystal lattice produces a polar environment for the molecules, where the local dipole of the molecules is antiparallel to the environmental dipole (Figure 5). This induces the decrease of the B-N distance in comparison with the results of the microwave spectroscopic study in the gas phase (r 0 = 1.672/k; r s = 1.657/k) [21]. Ab initio calculations at various levels [22] are in good agreement with the MW value for the B-N distance. The same holds for crystal structures and ab initio calculations of amineborane derivatives, and in a recent review a reference value of 1.58 ,~ for the B-N bond is recommended [23]. If the B-N distance from the

B

N

9

Figure 4. Molecular structure of H3BNH3, presentation with thermal probability plots of 50%, BN bond distance: 1.567(7) ,~.

X-ray Structures Versus Ab Initio Geometries

211

•

i

@ Figure 5. Crystal packing of H3BNH3 in Pmn21 viewed along the y axis.

X-ray structure determination is used as a constant in the quantum chemical calculation (and all other parameters fully optimized), the energy is only 1.5 kcal/mol higher than that of the free optimized molecule as a consequence of the flat potential energy surface for the B-N bond. For a large number of borane complexes, the calculated (IGLO [24]) chemical shifts are in satisfactory agreement to the experimental chemical shifts, [25] without any solvent dependency. The only molecule which does not fit this correlation is the parent ammoniaborane. Remarkable at this point is that the calculated chemical shifts for the B-N distance gained from the X-ray structure determination are in good agreement with the experimental chemical shifts. There are two experiments: (1) the X-ray structure determination, and (2) the NMR investigations in which the results do not match the calculated properties of the fully optimized molecule. Assuming that in both cases the molecules are strongly influenced by the polarity of the environment, the surrounding field should be included in the quantum chemical calculations in order to match the experiments. The geometry, and as a result the calculated chemical shifts, depend on the environment; this was suggested and investigated by calc:ulations including an Onsager solvent reaction field [26]. Simulations for hexane and water as solvents were carried out for ammoniaborane and resulted in a reduction of the B-N bond distance (1.62/k for hexane and more dramatic 1.57 /k for water). The IGLO calculated chemical shifts especially with the geometry in aqueous solution gave a much better agreement with the experimental data. The influence on the chemical shifts is considered to be indirect, because the solvent molecules or the polar environment is expected to be too remote to influence the nuclear shielding directly. Instead, solvation can change the geometry and electronic structure which results in a change of chemical shifts.

212

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG

To our knowledge, the discrepancy between atomic distances found in solid state and in the gas phase for ammoniaborane is one of the most extreme examples to be found in the literature. This demonstrates that especially for systems with very flat potential energy surfaces of polar bonds, the interpretation of precise X-ray data has to be carried out very carefully and all possible perturbations of the environment must be taken into account. The comparison with the structures of free molecules is not appropriate. On the other hand, the theoretical model for the correct interpretation of such structures in polar mediums must be expanded; for example, with the calculation of an Onsager solvent reaction field.

IV. MODIFICATION OF THE EXPERIMENTAL-BASED MODEL PROMPTED BY AB INITIO CALCULATIONS The structures of cyclopropenes are well understood and theoretically investigated [27]. Even most of the homolog heterocyclopropenes with several main group elements have been synthesized and structurally characterized. Some general tendencies in these structures can be summarized to have a strong correlation to the electronegativity of a fragment Z which is introduced in a Y3 ring as a substituent for one Y (Y: CR 2, SiR 2, GeR 2, SnR2; Z: CR 2, SiR 2, NR, O, S, Se, Te) [28]. In case the electronegativity of Z is higher than that of Y, the Y-Y bond distance will appear shortened in comparison with that in an Y3 ring. Simultaneously the angle between substituent and double bond increases. For the fundamental description of the bonding situation in these three-membered heterocyclic compounds, several models can be applied. Without discussing details, the main features are based on a description of olefin complexes by Dewar, Chatt, and Duncanson, with the bonding situation described by a r~-complex of the Y = Y bond and the fragment Z (Figure 6) [29]. The shortened Y-Y bond is a result of the weakened back-donation from an occupied p(Z)-orbital to the x*-orbital of the Y--Y group. Alternatively, the structure of three-membered rings can be understood because of a high contribution of the o'-bridged-rc-orbital out of the binding molecular orbitals [30]. This model

Q ....

Figure 6. Schematic drawing of the Dewar-Chatt-Duncanson model (left, middle) and the c-bridged-x-MO (right).

X-ray Structures Versus Ab Initio Geometries

213

was translated by Cremer and Kraka [31] into the electron density distribution within the plane of the ring. Qualitatively there is good agreement with the x-complex model, and the same deformation of the o-electron density can be obtained. The resulting electron density is quite similar to that of a re-bond and is delocalized in the plane of the three-membered ring (surface delocalization). Generally the maximum electron density cannot be found on a line connecting the two nuclei. Depending on the substituent Z, it is shifted inward or outward of the ring. Cyclopropene is an example for outward-shifted maxima of electron density and a protonated oxirane represents the case of inward-shifted maxima. Therefore a shorter Y-Y distance is not only the result of a stronger r~-bond, it is also a consequence of the bent bonds. With these models it is quite easy to develop simple rules for the prediction of the geometries of heterocyclopropenes and the effects of substituents [12a,30,31b]. The first oxadiborirane synthesized was bis(trisyl)oxadiborirane, and it was structurally characterized by an X-ray crystal structure determination [32] which confirms the three-membered ring structure with an almost linear C - B - B ' - C ' moiety. The oxygen atom was found to be disordered at the crystallographic inversion center and therefore refined with half occupation factors on each side of the B-B' bond (model A, Space Group: C2/c, No. 15). This disorder is connected with the centrosymmetry in the crystal lattice, which is also strictly valid for the C(SiMe3) 3 substituents. In spite of the disorder of the oxygen atom in the crystal, the bond length of the B-B' bond could be determined with satisfactory precision [ 1.601(7) /k]. The tendency for linearization of the R-B" B'-R' chain is reflected by deviations of the C-B-B' and B-B'-C' angles from 180 ~ by +2.3(3) ~ The B-O bond distances of 1.545(5) and 1.510(6)/k are remarkably long and there exists a large discrepancy to the calculated distances for the parent compound (R = H) of 1.409/k (HF/3-21G basis set [30]) or 1.403/k (MP2/6-31 G* [33]). A different refinement model (model B) with an additional disorder had been applied, where disorder was also assumed for the boron atoms and a B-O distance of 1.38/k resulted (Figure 7). Because the geometry of substituted compounds is often very different from the parent compound, and there was no improvement in the R-values, model A (the conventional refinement) was chosen for publication [31]. A new series of quantum chemical calculations [34] (see Table 1), in which substituted derivatives [R = Me, C(SiH3)3] on HF/DZP level and the parent compound on higher levels of theory including electron correlation [35] has been performed for clarification of this problem. As shown in Table 1 there is no substantial change in the geometric features. The calculations point out that generally the B-O bond distance should not be larger than 1.39 ,~. Moreover, from the energetical point of view there is no longer any support for model A: a partial optimized structure of the parent compound with a fixed BBO ring, based on the X-ray data and optimized hydrogen positions, exhibits to be 12.4

214

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG

C9~.~ C

B1"",

,,'~B"

C2 "~"~ C3

Figure 7. Disordered molecular structure of bis(trisyl)oxadiborirane in the solid state (model B), presentation with thermal probability plots of 50%. Bond lengths and angles see text and Table 4. kcal/mol less stable than the fully optimized geometry (CISD/DZP niveau [36]) with C2v-symmetry. An additional argument is that the average IGLO calculated liB NMR chemical shift [37] based on the experimental model, which gives 5 = 81.1, is not close to the experimentally observed region of ~i = 65.7. However, the fully optimized structure shows very good agreement: 8 = 67.0 (II'//MP2/6-31G*) and 66.3 (II'//CISD/DZP) [38]. Because of the described deficiencies of model A, we selected model B as the more appropriate one with oxygen and boron atoms refined in splitted positions. The final molecular geometry (Figure 7) can be understood as an overlay of two independent molecules, disordered at the crystallographic inversion center. The B-O distances in model B, 1.347(7) and 1.365(7)/k, respectively, are now in much better agreement with the ab initio data (1.378 * , Table 2). We suppose that the enlarged B-O distances in model A are a result of the averaging of the atomic positions of the disordered molecules. In contrast to model A, these B-O distances are now in the normal region for trigonal coordinated boron (1.28 to 1.43/k, mean value 1.365 ~,) [39] and are very close to the B-O distances in Me2B-O-BMe 2, 1.359(4)/k [40], which can be taken as a acyclic reference molecule. The B - B - C bond angles in model B [172.2(5) and 171.9(5) ~ are in better agreement with the ab initio values (172.8 ~ Table 2) than that of model A. Finally the calculated energy for R = H with the ring dimensions of model B is only 1.5 kcal/mol higher than that of the fully optimized structure.

X-ray Structures Versus Ab Initio Geometries

215

Table 2. Structural Data for Oxadiboriranes a R

Basis Set

H

HF/3-21Gb MP2/6-31G *c CISD/DZP H F/DZP HF/DZP Exp. Model Ac

CH3 C( Sill 3)3 C(SiMe3)3

Exp. Model B d

B-O

B-B

B-R

1.409 1.403 1.391 1.378 1.378 1.545(5) 1.510(6) 1.365(7) 1.347(7)

1.594 1.562 1.575 1.584 1.587 1.601(7)

1.169 1.183 1.175 1.561 1.549 1.544(4)

1.599(9)

1.510(7) 1.607(7)

B-B-R

173.4 175.2 174.6 173.8 172.8 177.7(3) 182.3(3) 172.2(5) 171.9(5)

Notes: aBonddistances in/~,, bond angles in degrees. bRef. [30]. CRef. [32]. ~Ref. [34].

In summary the refinement of model B gives a much better agreement between theory and experiment and should therefore represent a more reliable molecular geometry than that of model A. For model A we achieved an R value of 0.0424 with 163 parameters refined based on 1799 intensities; for model B these values are R = 0.0420, 172 parameters. Based on the ab initio data in Table 2, we suggest 1.39 /~ as a reference value for the B-O bond distance in oxadiboriranes. If crystallographic criteria (e.g., R values) give no help for the decision between the two alternative models A and B, we have demonstrated with the theoretical results that model B gives the more realistic geometric parameters for the structure of bis(trisyl)oxadiborirane. It is shown that, if the experimental results do not agree with that of high level ab initio calculations, the experimentalists should take other possible interpretations into account, especially if they lead to a much better agreement between experiment and theory.

V. VERIFICATION OF EXPERIMENTAL RESULTS BY AB INITIO CALCULATIONS In spiro compounds the cyclopropane ring exhibits significant geometry distortions caused by conjugation, additional strain, and electron donor/acceptor effects [41,42]. Among these compounds, we were interested in small and medium sized hydrocarbons which are accessible to high-level ab initio calculations and simultaneously to experimental structure determinations, provided that good crystals for diffraction data are available. In order to achieve the highest precision, we determined the structures at low temperatures (110-125 K) and included data from

216

ROLAND BOESE, THOMAS HAUMANN, and PETERSTELLBERG 1.5z9(2)

t.499(1)

2

3

t.53v(0

1

Figure 8. X-ray bond lengths of spiropentane (1), [3]rotane (2), and vinylcyclopropane (3) in A.

high-angle reflection intensities (2Omax >_ 80 ~ attaining standard deviations for bond lengths less than 0.001-2 A and for bond angles less than 0.1 ~ In spiropentane (1) (Figure 8) the reported [43] bond lengths of 1.537(1)/~, for distal [44] and 1.484(1) /k for vicinal [44] bonds reflect the consequence of spiroannelation of two cyclopropane rings which causes additional strain and as a consequence rehybridization with an increased s-character at the spiroatom [45], subsequently attributed as the "spiropentane effect." The same effect is apparent in the X-ray structure of[3] rotane (2) at 120 K where the distal bond length is 1.529(2) ~, and the vicinal bond is 1.478(1)/k [46] (mean values). In the central ring of this molecule with D3h-symmetry the bond length of 1.480(2) g, is close to the "usual" C(sp2)--C(sp 2) single bond in olefinic hydrocarbons [47]. Another interesting perturbation at the three-membered ring is introduced by an attached n-system as illustrated by the structure of vinylcyclopropane (3) [48]. The bond length distortions of the cyclopropane unit in 3 show the opposite behavior, with the vicinal bond tending to be lengthened to 1.514(1) A, whereas the distal bond does not differ from the distance reported for the X-ray structure of cyclopropane with 1.499(1)/k [48,49]. This distortion is obviously caused by conjugation of the cyclopropane ring with the C = C double bond. This leads to a maximum interaction (orbital overlap) [50] in the bisected conformation which is apparent in the crystal structure of 3. A strong influence on the ring geometry occurs at the alkylidenecyclopropanes 4 and 5 (Figure 9). With an introduction of a double bond exocyclic to the three-membered ring, a rehybridization occurs on the central carbon atoms. Similar to spiropentane (1), the vicinal bond lengths in methylenecyclopropane (4) [ 1.460(1 ) A] and bicyclopropylidene (5) [ 1.469(1) ~,] are reduced, whereas the distal bonds [4:1.526(1) A, 5:1.544(1) A] are lengthened [51,52] relative to cyclopropane. Compared to a standard C(sp2)--C(sp 2) double bond length of 1.335/k [47] the observed double bonds are significantly shorter [4:1.316(1)/k, 5:1.314(1 )

X-ray Structures Versus Ab Initio Geometries 1.526(2)

217 1.544(1) 1469(1)

4

5

Figure 9. X-ray bond lengths of methylenecyclopropane (4) and bicyclopropylidene

(5) in ,g,.

/k]. The above mentioned different effects are combined in compounds 6, 7, and 8 [53] (Figure 10). Cyclopropylidenespiropentane (6) contains a central three-membered ring with an exocyclic double bond which is able to conjugate through the spiro center. In ?-cyclopropylidenedispiro[2.0.2.1]heptane (7) and 1-cyclopropylidenedispiro[2.0.2.1]-heptane (8) the additional spiro-connected three-membered ring introduces more strain and changes of hybridization. Compounds 5, 6, 7, and 8 are liquids at room temperature and a low temperature in situ crystallization was performed on the diffractometer using a miniature zone melting procedure with focused infrared light or CO 2 laser beam [1]. All bond lengths given are corrected for libration [54]. The crystal structure of 6 [55] (Figure 11) presents the expected bond length distortion in the central ring (C2-C5-C6) caused by the exocyclic double bond. The reduced distances of 1.446(1) ~ (C2-C6) and 1.495(1) ~ (C2-C5) as well as the lengthening of bond C5-C6 [1.517(1) ,&] compared to the unsubstituted spiropentane (1) demonstrates the strong rehybridizational effect at C2 as observed in molecules 4 and 5. A weaker but highly significant influence is introduced by the double bond conjugation at the vicinal bonds of the terminal spiropentane ring. The lengthening of the mean distance of C6-C7 and C6-C8 [ 1.493(1 ) A] of 0.009

6

7

8

Figure 10. Spirocyclopropyl substituted derivatives of 1,1'-bicyclopropylidene (title compounds).

218

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG

c7

(~c2

Figure 11. Molecular structure of cyclopropylidenespiropentane (6); presentation with thermal probability plots of 50%. ,~ in 6 is close to the lengthening of 0.015 ,~, as observed in vinylcyclopropane (3). The double bond length of 1.315(1)/~ and the distances in the terminal cyclopropylidene ring (C1-C3-C4) do not reveal significant deviations in comparison with the structure of 5. The crystal structure of 7 [56] (Figure 12) has a crystallographic mirror plane passing through the double bond C 1-C2 orthogonal to C3-C3a. The observed mean

Figure 12. Molecular structure of 7-cyclopropylidenedispiro[2.0.2.1]-heptane (7); presentation with thermal probability plots of 50%.

X-ray Structures Versus Ab Initio Geometries

219

c2

Figure 13. Molecular structure of 1-cyclopropylidenedispiro[2.0.2.1]-heptane (8); presentation with thermal probability plots of 50%.

distances for C2v-symmetry are 1.464(1) A for C2-C3 where the distal and vicinal bond distortions of two adjacent spiropentane units cancel out to a slightly shortened vicinal bond compared to 5. The central bond C3-C3a [ 1.485(1) ,4,] is subject of the shortening through the double vicinal "spiropentane effect" and the lengthening caused by the additional strain from the opposite exocyclic double bond. For the vicinal bonds [1.488(1)/~] of the terminal spiropentane rings the lengthening of 0.004/k due to conjugation is less significant than in 6 but still observed. In the crystal structure of 8 [57] (Figure 13) the main distortions are taking place in the whole central spiropentane unit. The bicyclopropylidene unit C 1 to C6 do not reveal significant deviations from the geometry of 6. Therefore the ring C6-C7-C8 is disturbed by the terminal spiro ring attached to C8 which can also be considered as a central ring of a dispiro[2.0.2.1]heptane unit. Considering the unsubstituted dispiro[2.0.2.1]heptane [58] with C2,,-symmetry, this fragment in 8 reveals strong deviations from the ideal symmetry with a difference of 0.015/k for bond lengths of 1.516(1)/~ (C6-C7) and 1.501(1) .& (C7-C8). This is obviously not caused by additional strain at spiro atom C6 which is demonstrated by the angle 60.7 ~ (C2-C6-C5) compared to 62.2 ~ (C9-C8-C10) because this would lead to the lengthening of bond C7-C8 rather than bond C6-C7. The influence of conjugation to the double bond is more favorable in 6 and 8 because of a stronger interaction induced by a shorter distance of the overlapping Walsh- and n-orbitals [in 6 1.446(1) ,~, in 8 1.443(1)/k compared to 1.464(1)/k in 7]. The experimentally determined geometries of compounds 6, 7, and 8 do not reveal any obvious discrepancies. Some weak packing effects resulting from intermolecular H-.-H and H---C contacts could be detected by deviations of torsion angles but they seem to be neglectable.

220

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG Table 3. Bond Lengths (,g,)and Angles (Degrees) for the Cyclopropyl idenespiropentane Unita Structure (6) in Cs

Structure (7) in C2v

Strltcfttre (8) in C1

(A)

X-ray

6-31G*

X-ray

6-31G*

X-ray

6-31G*

C=C

1.315

1.297

1.311

1.298

1.313

1.298

a

1.495

1.482

1.464

1.456

1.497

1.482

b

1.446

1.439

1.464

1.456

1.443

1.440

c

1.517

1.501

1.485

1.480

1.517

1.502

d e

1.493 1.493

1.480 1.480

1.488 1.488

1.480 1.480

1.469 1.516

1.459 1.498

f

1.530

1.512

1.525

1.513

1.501

1.488

g

1.472

1.463

1.468

1.463

1.470

1.463

h

1.473

1.464

1.468

1.463

1.468

1.463

i

1.546

1.527

1.542

1.527

1.540

1.527

60.6

60.5

59.6

59.5

60.7

60.5 60.4

[3

61.6

61.5

61.6

61.5

60.3

y

62.0

61.8

60.9

61.8

62.1

61.8

~5

63.3

62.9

63.3

62.9

63.4

62.9

Note: aX-raybond distances are corrected for libration and merged to ideal molecular symmetry,esd's for angles are less than 0.1 ~ for bond lengths less than 0.001 A.

The question arises if the small but significant deviations of distances--mostly in the three-membered rings--are reproduced by ab initio calculations. For these it is necessary to use split valence basis sets including polarization functions to approximate the bent bond model by Walsh [59]. A commonly employed basis set is specified as 6-31 G* [60], where the geometries of several three-membered ring compounds are predicted with generally good accuracy as reported in the literature [61]. The structures of molecules 6 and 7 have been optimized with C s- and C2v-symmetry, respectively, and the results for the common cyclopropylidenespiropentane fragment are given in Table 3 according to the numbering scheme given in Figure 14. For bond angles o~, 13, and y, the calculated values are in very good agreement with the experimental data; the differences of 0.1-0.2 ~ are in the range of the experimental standard deviations. The terminal bicyclopropylidene angles 8 are predicted to be 0.4-0.5 ~ smaller than experimentally observed, but this is correlated with the most underestimated (0.013-19 ,~) ring bond distance i. Our interest is mainly focused on the bond lengths of the spiropentane fragment (bonds a-f) where the different distortions of the chosen molecules take place at high significance as described above. To avoid comparing apples with bananas, we only compare experimental bond length differences A[,~,] of this fragment with the theoretical values of A[,~]. The reference system for this analysis is the unsubsti-

221

X-ray Structures Versus Ab Initio Geometries

f ~ e ,

,,"^' ',,

#

",,,, ,,,,,' d b

a

i Figure 14. Labeling scheme for bonds and angles at the common cyclopropylidenespiropentane unit in 6, 7, and 8.

tuted spiropentane (1). In the calculated differences (see Eqs. 1-4) we consider bonds a and f as distal and b, c, d, and e as vicinal. The results are given in Table 4. A[/~] =

A[,~] =

dx.ray 1.537 [,~] (1)

dx.ray 1.484 [,~] (1)

a,f [,~] (6), (7), (8)

(1)

b,c,d,e [,~] (6), (7), (8)

(2)

d i s t a l - dx_ray

v i c i n a l - dx.ray

A[/~] = d6_31G, 1.517 [/k] (1) distal- d6_31G, a,f [/k] (6), (7), (8) A[/~] = d6_31G.

(3)

14.75 [/~] (1) vicinal - d6_31G* b,c,d,e [/~] (6), (7), (8)

(4)

Table 4. Differences A of Bond Lengths [,~] of the Spiropentane Unit in 6, 7, and 8

to the Bond Lengths of the Unsubstituted Spiropentane (1)a Structure 6 in Cs

Structure 7 in C2v

Structure 8 in Cl

A [A]

X-ray

6-31G*

X-ray

6-31G*

X-ray

6-31G*

a

-0.042

-0.035

-0.073

-0.061

-0.040

-0.035

b

-0.038

-0.036

-0.020

-0.019

-0.041

-0.035

c

0.033

0.026

0.001

0.005

0.033

0.027

d e

0.009 0.009

0.005 0.005

0.004 0.004

0.005 0.005

-0.015 0.032

-0.016 0.023

f

-0.007

-0.005

-0.012

-0.004

-0.036

-0.029

Note: aa and f are distal and b, c, d and e are vicinal bonds (calculated values from Table 1).

222

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG 0.04

f

--

e

0.03 0.02 0.01

\

~I

c

-0.01

-

*"" ~" -..~..

f

b

a

d

-0.02 -0.03

"-

-0.04

X-ray(libr.)

.... x.... 6-31G*

-0.05

Figure 15. Bond lengths differences for 6 compared to spiropentane. Derived from Table 4, theoretical calculated values of A[/~] reveal generally a good agreement with the experimental values calculated from the X-ray structures. For graphical presentation the A-values of each molecule in one diagram (Figures 15 to 17) were combined; the bond lengths on the x-axis are arranged from the strongest lengthening +Amax to the shortening-Ama x. As shown in the above diagrams, the theoretical calculations on molecules 6, 7, and 8 reproduce the above described distortions on the spiropentane fragment remarkably well. Generally, systematic smaller values for bond length differences are observed for the ab initio results (with the dotted curves closer to the x-axes in the diagrams). The mean deviation between experimental and theoretical differences in bond distances (AA value) is _+0.004 A for all 18 bonds. 0.01 . . . . . . . . . . . . . . . . . .

[h]

)K.

f

b

a

0 o0.01

d

e

-0.02 -0.03 -0.04 -0.05 ~ -0.07

:

X-ray (libr.)

.... x.... 6-31G*

-0.08

Figure 16. Bond lengths differences for 7 compared to spiropentane.

223

X-ray Structures Versus Ab Initio Geometries 0.04 0.03

- -

0.02

...... ~',,,~,

0.01

zx[A]

0

t

-0.01 -0.02

-0.05

,\

e

9

o

~~.

t f

....

t

,

b

I a

~

"0"03[ -0.04

i

c

x'ray (libr') '

L

. . . . :~. . . . ~ 3 1 G * [

. . . . . . . . . . . x. . . . . . . . . . . . x

I

Figure 17. Bond lengths differences for 8 compared to spiropentane. We demonstrated that by comparison of differences between only experimental data and only calculated data the values match extremely well. A direct comparison of theoretical and experimental data the average of bond lengths differences would lead to AA = _+0.012 ]k caused by the expected systematically underestimated bond lengths of ab initio calculations using HF/6-31 G* method/basis set combination. For the title compounds, which incorporate a complex interaction of mutual influencing substituents, it can be shown that the HF/6-31 G* basis set detects even small changes in geometry, each in the right direction.

VI.

CONCLUSIONS

Molecular orbital calculations have left the stage of simply reproducing experimental structure determinations a long time ago. Today it is possible to predict and to understand molecular structures by means of theoretical approaches. However, a detailed knowledge of the properties of the applied model and basis set is essential. Together with the knowledge of the background of experimental as well as the theoretical structure determinations, both methods can provide a fruitful interplay. If the results are comparable under the restrictions of methodological deficiencies, theoretical structure determinations can help to rationalize experimental geometries which remain obscured or otherwise purely descriptive. One of the main advantages of theoretical approaches is that structures can be calculated which are not accessible to the experimentalists. However, high quality ab initio calculations including large basis sets and electron correlation are restricted concerning the number of electrons and therefore the size of molecules. With increasing computing capacities there is a dramatic increase in the size of molecules that can be calculated even at correlated high ab initio levels.

224

ROLAND BOESE,THOMAS HAUMANN, and PETERSTELLBERG

In case of large discrepancies between experimental and theoretical results, the experimental-based model or an extension of a theoretical model must be applied in order to conciliate both methods. The challenge for the theoretician is still the experiment, either to establish the experimental data or to make predictions which are again a challenge for the experimentalists. Thus, there is an intrinsic demand in chemistry to support the interplay between experimentalists and theoreticians.

ACKNOWLEDGMENTS We gratefully acknowledge the fruitful collaboration with P.v.R. Schleyer and the help of D. Bl~iser and A. H. Maulitz, as well as the support from the Fonds der Chemischen Industrie and the Deutschen Forschungsgemeinschaft.

REFERENCES A N D NOTES 1. Boese, R.; Nussbaumer, M. hi situ crystallization techniques. In Organic Crystal Chemistry, Jones, D.W., Ed.; Oxford University Press: Oxford, England, 1993 (in press). 2. We apply the expression "high level" ab initio calculations for basis sets 6-31G* or higher, being aware of the fact that of the course of increasing computing capacities this basis set might be consequently considered as medium level. 3. Biihl, M.; Schleyer, P. v. R. J. Am. Chem. Soc. 1992, 114, 47%-491. 4. Hargittai, M.; Hargittai I. Int. J. Quantum Chem. 1992, 44, 1057-1067. 5. Boese, R.; Polk, M.; Bl~iser, D. Angew. Chem. 1987, 99, 239-241; Angew. Chem. Int. Ed. Engl. 1987, 26, 245-247. 6. Boese, R.; Niederpriim, N.; Bl~iser, D. Molectdes in Natural Science and Medicine; Maksie, Z.B., Eckert-Maksic, M. Eds.; Ellis Horwood: New York, 1991, Chapter 5, p. 103. 7. K6ster, R.; Seidel, G.; Boese, R.; Wrackmeyer, B. Chem. Ber. 1988, 121,597-615. 8. K6ster, R.; Seidel, G.; Boese, R. Chem. Ber. 1990, 123, 1013-1028. 9. Boese, R.; Bl~iser, D.; Niederpr~im, N.; Ntisse, M.; Brett, W.A.; Schleyer, P.v.R.; Biihl, M.; Hommes, N.J.R.v.E.Angew. Chem. 1992, 104, 356; Angew. Chem. hit. Ed. Engl. 1992, 31,314. 10. Hehre, W.J.; Radom, L.; Schleyer, P.v.R.; Pople, J.A. Ab initio Molecular Orbital Themy; Wiley: New York, 1986. 11. (a) K6ster, R.; Seidel, G.; Boese, R. Chem. Be~: 1990,123, 2109; (b) Yalpani, M.; Boese, R.; K6ster, R. ibid 1990,123, 713; (c) K6ster, R., Seidel, G.; MUller, G.; Boese, R.; Wrackmeyer, B. ibid 1988, 121, 1381; (d) Yalpani, M.; Boese, R.; K6ster, R. ibid 1990, 123, 707; see also Ref. 1. 12. (a) Clark, T.; Spitznagel, G.W.; Klose, R.; Schleyer, P. v. R. J. Am. Chem. Soc. 1984, 106, 4412; (b) Schleyer, P.v.R. Pure Appl. Chem. 1987, 59, 1647. 13. The data were obtained from the Cambridge Structural Database (CSD), Version from 8.5.1991 with 90296 entries, using the Cambridge Structural Database System (CSDS) Version 4.40 [14]. Only crystal structures containing the element X-Et (X = second or third-period element) with R-values between 0.001 and 0.08 were considered. The valencies of the specified elements were not taken into account. Despite the relatively high standard deviations, the t-test [15] shows that most of the comparisons among the angles involving the various elements are significantly different (except for Be and Si). 14. Allen, EH.; Kennard, O.; Taylor, R. Acc. Chem. Res. 1983, 16, 146. 15. Kaiser, R.; Gottschalk, G. Elementare Tests zur Beurteihmg von Mefldaten; B. I. Wissenschaftsverlag: Mannheim, 1972, p. 25.

X-ray Structures Versus Ab Initio Geometries 16. 17. 18. 19. 20. 21. 22. 23. 24.

25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

35. 36.

37.

38. 39. 40. 41.

225

Schleyer, P.v.R.J. Am. Chem. Soc. 1961, 83, 1368. Stock, A.; Pohland E. Ber. dtsch. Chem. Ges. 1926, 59, 2215. Van Nees, G.J.H.; Vos, A. Acta Co,stallogr. 1978, B34, 1947. Hughes, E.W.J. Am. Chem. Soc. 1956, 78, 502. Lippert, E.L.; Libscomp, W.N.J. Am. Chem. Soc. 1956, 78, 503. (a) Suenram, R.D.; Thorne, L.R. Chem Phys. Lett. 1981, 78, 157; (b) Thorne, L.R.; Suenram, R. D.; Lovas, EJ. J. Chem. Phys. 1983, 78, 167. Biihl, M." Steinke, T.; Schleyer, Ev.R." Boese, R. Angew. Chem. 1991, 103, 1179; Angew. Chem. hzt. Ed. EngL 1991, 30, 1160. Haaland, A.Angew. Chem. 1989, 101, 1017; Angew. Chem. hit. Ed. EngL 1989, 28, 992. (a) Kutzelnigg, W. lsr. J. Chem. 1980, 19, 193; (b) Schindler, M.; Kutzelnigg, W. J. Chem. Phys. 1982, 76, 1919; for a review see (c) Kutzelnigg, W.; Fleischer, U.; Schindler, M. NMR, Basic Principles and Progress; Springer: Berlin, 1990, Vol. 23, p. 165; for IGLO applications to boron compounds see (d) Schleyer, Ev.R.; Biihl, M.; Fleischer, U.; Koch W. lnorg. Chem. 1990, 29, 886; (e) Biihl, M.; Schleyer, Ev.R. Angew. Chem. 1990, 102, 962; Angew. Chem. hit. Ed. Engl. 1990, 29, 886; (f) Biihl, M.; Schleyer, Ev.R. Electron Deficient Boron and Carbon Clusters; Olah, G. A.; Wade, K.; Williams, R.E., Eds.; Wiley: New York, 1991, p. ll3. (a) Ntith, H.; Wrackmeyer, B. NMR, Principles and Progress; Springer: Berlin, 1978; Vol. 14; (b) Wrackmeyer, B. Ann. Pep. NMR Spectrosc. 1988, 20, 61. Wong, M.W.; Wiberg, K.B.; Frisch, M.J.J. Chem. Phys. 1992, 95, 8991. For various reviews see: The Chemistry of the Cyclopropyl Group; Rappoport Z., Ed.; Wiley & Sons: Chichester, 1987. Griitzmacher, H.Ange~,: Chem. 1992, 104, 1358;Angew. Chem. Int. Ed. Engl. 1992, 31, 1329 and cited literature. See also: Greenberg, A.; Liebman, J.E Strained Organic Molecules; Academic Press: New York, 1978, p. 280; and cited literature. Liang, C.; Allen, L.C.J. Am. Chem. Soc. 1991, 113, 1878-1884. (a) Cremer, D.; Kraka, E.J. Am. Chem. Soc. 1985,107, 3800; (b) Cremer, D.; Kraka, E. ibid. 1985, 107, 3811. Paetzold, E; G6ret-Baumgarten, L.; Boese, R. Angew. Chemie. 1992, 104, 1071; Angew. Chem. hit. Ed. Engl. 1992, 31, 1040. Schleyer, Ev.R.; Biihl, M. Universit~it Erlangen-Niirnberg, (private communication). Biihl, M.; Schaefer III, H.E; Schleyer, Ev.R.; Boese, R. Angew. Chem. 1993, 105, 1265; Angew. Chem. hzt. Ed. Engl. 1993, 32, 1154; Hampel, E In Landolt-BOrnstein, New Series, Vol. 22b; Schleyer, Ev.R., Ed.; Springer-Verlag: Heidelberg, 1994, p. 16. The geometries were fully optimized with C2v-symmetry for R - H, and for R - CH 3 and C(SiH3) 3 with C2-symmetry. For details on the applied basis sets see references 5 and 6 cited in [34]. Optimization with the Turbomole-Program: (a) H~iser, M.; Ahirichs, R. J. Comput. Chem. 1989, 10, 104; (b) Ahlrichs, R.; B~ir, M.; H~er, M.; Horn, H.; K61mel, C. Chem. Phys. Lett. 1989, 162, 165. NMR chemical shifts were determined with the IGLO-method (Individual Gauge for Localized Orbitals): Kutzelnigg, W. Isr. J. Chem. 1980, 19, 193; Schindler, M.; Kutzelnigg, W. J. Chem. Phys. 1982, 76, 1919) with application of the following contracted basis sets: Huzinaga, S. Approximate Atomic Wave Functions; University of Alberta, Edmonton, 1971: II' (9sSpld/Ss4dl d) for B (ctd = 0.5) and O (o~d = 1.0), (3s,2s) for H. See also Kutzelnigg, W.; Schindler, M.; Fleischer, U. NMR, Basic Principles and Progress; Springer Verlag: Berlin, Heidelberg, 1990, p. 165. Similar Values were obtained with P. Pulay's GIAO-Program under usage of a TZP-basis set: Sulzbach, H.; Schleyer, Ev.R. (unpublished results). Wells, A.E Structural hlorganic Chemistry, 5th ed.; Clarendon Press: Oxford, 1984. Gundersen, G.; Vahrenkamp, H. J. Mol. Struct. 1976, 33, 97. Allen EH.Acta Co,st. 1980, B36, 81-96.

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42. lmgartinger, H.; Gries, S.; Klaus, Ph.; Gleiter R. Chem. Ber. 1992 125, 2503-2512. 43. Boese, R.; Blaeser, D.; Gomann, K.; Brinker, U.H.J. Am. Chem. Soc. 1989, 111, 1501-1503. 44. We use the terms "distal" and "vicinal" to distinguish between the C-C bonds of a cyclopropane ring, which can be opposited (distal) or attached (vicinal) to the spiro- or vinylsubstituted atom. 45. Gleiter, R.; Krennrich, G.; Brinker, U.H.J. Org. Chem. 1986, 51, 2899-2901. 46. Boese, R.; Miebach, Th.; de Meijere, A. J. Am. Chem. Soc. 1991, 113, 1743-1748. 47. Rademacher, P. Strukturen OrganischerMolekiile; Klessinger, M., Ed.; Verlag Chemie: Weinheim, 1987, p. 56. 48. Nijveldt, D.; Vos, A.Acta Cryst. 1988, B44, 281-296. 49. For two attached unsaturated groups as in spiro[2.4]heptadiene and 1,1-diethynylcyclopropane, a stronger interaction causes significant shortening of the distal bond. Boese, R.; Haumann, Th.; Kozhushkov, S.I.; de Meijere, A. (unpublished results). 50. Hoffmann, R. Tetrahedlon Lett. 1970, 33, 2907. 51. (a) Boese, R. Structural studies of strained molecules, lnAdvances in Strain in Organic Chemistry; Halton, B., Ed.; JAI Press: Greenwich, CT, 1992, Vol. 2, pp. 191-254; (b) Boese, R.; BlUer D.; Haumann, Th. Abstracts of the Fuzhou Symposium oll Molecular Structure (China) 1993, AL5. 52. The single crystal structure determination of (5) was biased by and a solid state phase transition (DSC: -40.2~ AH = 0.16 kJ/mol heating curve). All previous attempts to maintain a single crystal by cooling through the phase transition failed. Therefore we cooled the sample to 190 K in a capillary and applied a miniature zone melting procedure with a focused CO 2 laser beam [1]. This enabled us to bypass the high temperature phase and the study of both polymorphous structures. Structure data for C6H 8 (5) at LT-phase. Ce I1 dimensions a = 4.024( 1), b = 12.498(4), c = 4.964(3) ,~, oc =90, 13=95.30(3) ~ V= 248.6(2) ,~3, Pcal = 1.070 g/cm 3. SG: P21/n, Z = 2, T= 140 K, 1248 unique, 1111 observed intensities, (F 0 > 4~(F)), 2Omax = 80 ~ (MoKct), R = 0.041, R w = 0.044, T(crystal growth) = 190 K. Further details of the crystal structure investigations are available on request from the Fachinformationszentrum Karlsruhe, Gesellschaft ftir Wissenschaftlich-technische Information mbH, D-76344 Eggenstein-Leopoldshafen, on quoting the depository number CSD-400403 for (5) HT-Phase, CSD-400401 (5) LT-Phase, the authors' names, and the full citation. 53. De Meijere, A.; Kozhushkov, S.I.; Spaeth, Th.; Zefirov, N.S.J. Org. Chem. 58 (1993) 502-505. We gratefully acknowledge A. de M. and S.I.K. for providing the samples of (5), (6), (7), and (8). 54. Schomaker, W.; Trueblood, K.N. Acta Cryst. 1968, B24, 63. 55. Structure data for CsH10 (6). Cell dimensions a = 9.101(2), b = 5.142(1), c = 13.394(3) * , ct = 90,13=92.46(2) ~ V =632.4(2)/~3, Pcal = 1.115 g/cm 3. SG: P21/c, Z = 4, T= 115 K, 4867 unique, 3347 observed intensities, (F0 > 4~(F)), 2Omax = 90 ~ (MOKoc), R = 0.04 1, Rw = 0.048, T(crystal growth) = 216 K. Further details are available on quoting the depository [52] number CSD-400402. 56. Structure data for C10HI2 (7). Cell dimensions a = 5.093(1), b = 11.567(2), c = 6.738(2) ,~, ~ = 90,13 = 93.88(2)o, V = 369.0(1) ~3, Pcal = 1.109 g/cm 3. SG: P2 l/m, Z = 2, T = 120 K, 3329 unique, 2683 observed intensities, (F 0 > 4t~(F)), 2Omax = 90 ~ (MoK~), R = 0.044, Rw = 0.054, T(crystal growth) = 253 K. Further details are available on quoting the depository [52] number CSD-400405. 57. Structure data for C10H12 (8). Cell dimensions a = 5.721(1), b = 8.073(2), c = 8.797(2) ,~, ct = 79.51(2), 13 = 75.41(2), y = 87.44(2) ~ V = 386.6(1) ,~3, Pcal = 1.136 g/cm 3. SG: PT, Z = 2, T = 110 K, 5562 unique, 4462 observed intensities, (F 0 > 46(F)), 2Omax = 85 ~ (MoKt~), R = 0.042, R w = 0.046, T(crystal growth) = 264 K. Further details are available on quoting the depository [52] number CSD-400404. 58. X-ray structure determination of dispiro[2.0.2.1 ]heptane: Boese, R.; Haumann, Th.; Kozhushkov, S.I.; de Meijere, A. (publication in preparation). 59. Walsh A.D. Trans. Faraday Soc. 1949, 45, 179. 60. Hehre, W.J.; Ditchfield, R.; Pople, J.A.J. Chem. Phys. 1972, 56, 2257. 61. Wiberg, K.B. In The Chemistry of the Cyclopropyl Group; Rappoport, Z., Ed.; Wiley & Sons: Chichester, 1987, p. 1-24.

COMPUTATIONAL AND SPECTROSCOPIC STUDIES ON HYDRATED MOLECULES

Alfred H. Lowrey and Robert W. Williams

Io II. III.

IV.

V. VI.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characterization of Solvation . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Background . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Molecular Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Monte Carlo Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . C. Free Energy Perturbation Methods . . . . . . . . . . . . . . . . . . . . . D. Semiempirical Molecular Orbital Calculations . . . . . . . . . . . . . . E. Ab hlitio L C A O Molecular Orbital Studies . . . . . . . . . . . . . . . . E Self-Consistent Reaction Field Theory Using Ab Initio Methods . . . . . Spectroscopic Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Approaches Based on Band Narrowing . . . . . . . . . . . . . . . . . . B. Approaches Based on Pattern Recognition . . . . . . . . . . . . . . . . . Scaled Quantum Mechanical Force-Field Method . . . . . . . . . . . . . . . Effects of Hydration of Scale Factors for Ab Initio Force Constants . . . . . . A. Simple Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 1, pages 227-260. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8

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B. Supermolecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Isotopic Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. N-Methylacetamide and Glycine . . . . . . . . . . . . . . . . . . . . . . E. Larger Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

248 249 252 255 256 256

ABSTRACT The vibrational spectra of molecules dissolved in water are different in significant ways from the spectra of these molecules in the gas phase. The study of water solution spectra is particularly important for molecules of biological significance because their structure and properties are often determined by the presence or absence of water. Computational techniques have been developed that relate computationally determined structure and associated properties such as force constants to experimental information such as vibrational frequencies. Experimental vibrational studies have been used to elucidate information about such problems as the secondary structure of proteins in water solution. A brief review of the computational and experimental techniques is presented. Our work, which builds on the essential combination of theoretical and experimental information, is then reviewed to outline our ideas about using computational studies to investigate the complicated problems of amino acids and proteins in water solution. Finally some suggestions are presented to show how computational techniques can enhance the use of experimental techniques, such as isotopic substitution for the study of complicated molecules.

!. I N T R O D U C T I O N This chapter is primarily devoted to a review of our research involving vibrational spectroscopy of molecules dissolved in water. Because these spectra exhibit features distinct from the related gas phase spectra [1], there is unique information which is particularly important in relation to biochemistry and the molecules of living organisms. We have been particularly concerned with amino acids and small molecules with functional groups resembling amino acids, such as acetic acid and methyl amine [2-7]. Our long range interest lies in the problems of secondary structure in proteins [8], and our approach to these problems is to build up from smaller molecules a base of information about the scaled quantum mechanical force constants and subsequently the transition dipole coupling interactions for short peptides in water. The focus of our research has been on combining the techniques of experimental Raman and FTIR spectroscopy (reviewed by Braiman and Rothschild [9] and Surewicz and Mantsch [10]) with the information obtained from computational chemistry using scaled quantum mechanical force-field (SQM) methodology [11] in analogy to its applications to gas-phase molecules. This review will discuss some of the ideas concerning problems associated with hydrated

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molecules and cover in some detail our research which yields important information for understanding these problems.

II. CHARACTERIZATION OF SOLVATION The language used to describe solvation evokes much of the history of chemistry. Solvation is often equated with "hydration". Solution energies and forces due to liquid structure are often invoked simultaneously with the term solvation. Long ago, Derjaguin called solvation, "the structural component of disjoining pressure" [12,13]. So many phenomena are dictated by hydration forces that it is difficult to create a useful description in general terms. A useful discussion of this issue is given by Ninham [14]. His article provides a visualization of the problem in terms of forces between molecules acting at distances greater than molecular dimensions (i.e., external to a molecular surface) and less than those at which matter can be treated as a continuum. In this view, a key question is that of the structure of a liquid near an interface and the consequent forces on that structure. It is clear that solvation or hydration forces, considered at small distances, depend crucially on the surfaceinduced structure of the liquid at the interface. At local dimensions, solvation must perturb proximal liquid structure (densities, hydrogen bonding, dipolar orientation, vibrational frequencies, etc.) such that interactions must be propagated in a stress field passing from molecule to molecule. These forces must arise from the components of intermolecular interactions, such as electrostatics, polarization, exchange forces (important at quantum dimensions in the vicinity of the van der Waals surface), and dispersion forces. They can be attractive or repulsive. They can be classified as enthalpic or (particularly for biological systems) entropic in origin. However, it is clear that whatever the nature of the particular interactions under consideration the very existence of phenomenon associated with solvation or hydration necessitates the existence of distinct structural features associated with the collection of molecules involved [14]. Rigorous formulations of the problems associated with solvation necessitate approximations. From the computational point of view, we are forced to consider interactions between a solute and a large number of solvent molecules which requires approximate models [15]. The microscopic representation of solvent constitutes a discrete model consisting of the solute surrounded by individual solvent molecules, generally only those in close proximity. The continuous model considers all the molecules surrounding the solvent but not in a discrete representation. The solvent is represented by a polarizable dielectric continuous medium characterized by macroscopic properties. These approximations, and the use of potentials, which must be estimated with empirical or approximate computational techniques, allows for calculations of the interaction energy [15]. The challenge of understanding the effects resulting from the structure of the solvation complex continues to motivate large fields of chemistry and physics [16]. The molecular view of these phenomenon is important, particularly for under-

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standing solvation thermodynamics [17]. Spectroscopic results are well known to depend on the fundamental nature of the interactions between the solute and the molecules of the surrounding medium [18]. New techniques are elucidating the intricate details of the molecular interactions involved in creating these effects [19,20]. This article will discuss our approach of using the SQM method to relate spectroscopically observed solvation effects to ab initio calculations on small molecules and small molecule clusters with water. Our research is primarily concerned with solvent effects as simulated by scale factors for ab initio force constants and discrete representations of solvation in our use of supermolecule calculations. The resulting force-field and calculated potential energy distributions will be discussed in context of interpretations of spectral data useful for understanding problems of secondary structure in proteins [1]. i11.

COMPUTATIONAL

BACKGROUND

Developments in experimental and computational science have shed light on phenomena in bioenvironments and condensed phases that pose significant challenges for theoretical models of solvation [21]. Tapia [22] raises the important distinction between solvation theory and solvent effects theory. Solvation theory is concerned with direct evaluation of solvation free energies; this is extensively covered by recent reviews [16,17]. Solvent-effect theory concerns changes induced by the medium onto electronic structure and molecular properties of the solute. Solvent-effect theory is concerned with molecular properties of the solvated molecule relative to the properties in vacuo; as such it focuses on chemical features suitable for studying systems at the microscopic level [23]. Extensive reviews of different computational methods are given in a book by Warshel [24].

A. Molecular Mechanics Computational simulations of molecular structure almost always begin with a mathematical model for the potential energy surface which determines the nuclear positions of the atoms. In the molecular mechanics method, this potential energy surface is generally considered to be a function of forces acting between atoms in a pairwise manner [25,26]. This formulation chooses to ignore specific electronic structure effects characteristic of molecular orbital techniques, and instead focus on the determination of an empirical force-field and a reference geometry for atom-atom interaction. This simplification of the molecular potential energy surface has made calculations on biological molecules with thousands of atoms possible. A typical potential energy function minimally includes terms to describe bond-stretch and bond-angle deformation, hindered rotation about single and partial double bonds, and nonbonded interactions. Bond-stretch and bond-angle deformation forces are usually represented with simple harmonic potential functions although often a cubic term will be included for congested molecules.

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Torsional potentials are generally expressed as a truncated Fourier series. Nonbonded interactions are most often represented as Lennard-Jones (van der Waals) and Coulomb (electrostatic) potentials. Based on the origins of these ideas in the science of vibrational spectroscopy, various schools often include different types of cross terms in the potential energy function that describe interactions between geometrical parameters, such as stretch and bends [27]. The description of these pairwise potential functions include two basic terms: a geometrical parameter, such as a bond length, which is assigned an average standard value considered to be characteristic for the two atoms involved, and a potential energy function and associated force parameter, which describes the functional dependence of geometrical deformation away from this reference value. The reference geometry is defined as the state in which all bond lengths, valance angles, and dihedral angles are equal to chosen reference values and where all nonbonded interactions are turned off [28]. It is important to recognize that the reference geometries and the force parameters are empirically determined; they represent average values which produce a best fit for a variety of chemical information. In general, an initial guess is made for the parameters by use of data from model systems. This guess is refined by trial and error and via nonlinear least-squares techniques. For biological molecules, this procedure is most feasible because of the small number of distinct atoms involved (i.e., H,C,N,O,S,P) and limited types of bonding associated with proteins, nucleic acid bases, lipids, and carbohydrates [29,30]. A variety of information is used for this parameterization. Experimental sources include viscosity data, scattering data, crystal structures, vibrational spectroscopy, gas-phase structural data, and chemical intuition [31]. High-level quantum-chemical calculations have become increasingly important as substitutes for experimental data [27]. Experience has shown that useful information can be obtained by employing the presently available functional forms and parameters sets. However, it must always be remembered that whatever the precision of the motional or thermodynamic properties are obtained from these simulations, the ultimate accuracy depends on the potential function used. For example, if different types of molecules are of interest, such as might be synthesized to obtain a more potent drug, inhibitor, or antibiotic analogue, new terms and associated parameters may have to be introduced into the potential energy function [31]. Because of the ease with which molecular mechanics calculations may be obtained, there was early recognition that inclusion of solvation effects, particularly for biological molecules associated with water, was essential to describe experimentally observed structures and phenomena [32]. The solvent, usually an aqueous phase, has a fundamental influence on the structure, thermodynamics, and dynamics of proteins at both a global and local level [31]. Inclusion of solvent effects in a simulation of bovine pancreatic trypsin inhibitor produced a time-averaged structure much more like that observed in high-resolution X-ray studies with smaller atomic amplitudes of vibration and a fewer number of incorrect hydrogen bonds [33]. High-resolution proton NMR studies of protein hydration in aqueous

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ALFRED H. LOWREY and ROBERT W. WILLIAMS

solution show that there are two qualitatively different types of hydration sites: a well-defined small number of water molecules in the interior of the protein are in identical locations as in the crystal structure with residence times of 10-2 to 10- 8 s while water molecules on the surface were characterized by residence times in the subnanosecond range [34]. Solvent effects in molecular mechanics have been treated in the continuum approach by means of the solvent's dielectric constant

[35]. The continuum approach based on the Onsager reaction field has also been formulated [36]. In the discrete approximation, inclusion of water in molecular mechanics and molecular dynamics simulations is generally accomplished by adding a large number of additional particles or molecules with carefully optimized potential function to the set of nuclear centers for the molecule being studied [34]. Systematic inclusion of water is an art more than a science. This is because of the difficulties in establishing potential functions. No realistic potentials can be described accurately by pair potentials, particular in a system of polar molecules. In practice, pair-additive potentials are effective potentials in the sense that they include average effects of the many particle interactions. Thus they are of limited validity, having being tested only for certain ranges of density, composition, and temperature [37]. The site-site potentials developed for liquid water are known to be difficult; they exhibit a molecular dipole moment of 20-30% larger than the experimental dipole moment of water in the gas phase and yield a calculated second virial coefficient almost twice that experimentally observed [38,39]. Thus it is imperative that the results of these simulations be interpreted in terms of the experimental data they are intended to reproduce. Careful comparisons with experimental data are essential for understanding the simulated structures and prop~ erties that these rapid techniques produce.

B. Monte Carlo Techniques Coupled with the development of molecular potential energy functions, Monte Carlo techniques have been developed for calculating molecular structure and other properties based on the relative ease of finding configurations with decreasing energy [40]. Monte Carlo molecular dynamics simulations of aqueous solutions of small molecules have recently been reported [41-43]. Using optimized potentials for liquid simulation and the free-energy techniques discussed below, calculated free energies of hydration for substituted benzenes showed an average difference 0.5 kcal/mol compared with experimental observation [44]. This may be compared with experimental determination of rotational barriers by means of gas-phase electron diffraction. The values derived directly from experimental observation are reliable only to 1.0-1.5 kcal/mole and require other information, such as that derived from computational techniques for more accurate determination [45].

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C. Free Energy Perturbation Methods The formulations of statistical mechanics provide a recipe for differences in free energy between two states [40]; the development of perturbation methods allows computational simulation of free-energy changes in molecular state going from, for example, gas phase to a solvated environment [46,47]. Both Monte Carlo and molecular dynamics techniques can be used to implement this technique. There are two primary methods: (1) the perturbation window approach which simulates the changes in the perturbation in small discrete steps [48], and (2), the thermodynamic integration approach which integrates the potential energy differential as a function of the perturbation over ensemble averages [31]. The umbrella sampling technique is an alternative approach to the perturbation methods for evaluating free energies of solvation [49,50]. The development of techniques for precise computation of free energies in solution has greatly enhanced the opportunities for meaningful comparison of experiment and computation [51,52]. It is important to recognize that this technique is generally applicable solvation chemistry; Rao and Singh have published on problems on hydrophobic hydration [53] looking at systems such as solvation of methanol and dimethyl sulfoxide [54], and solvation of hydrazine and carbon tetrachloride [55].

D. Semiempirical Molecular Orbital Calculations Semiempirical techniques are the next level of approximation for computational simulation of molecules. Compared to molecular mechanics, this approach is slow. The formulations of the self-consistent field equations for the molecular orbitals are not rigorous, particularly the various approaches for neglect of integrals for calculation of the elements of the Fock matrix. The emphasis has been on versatility. For the larger molecular systems involved in solvation, the semiempirical implementation of molecular orbital techniques has been used with great success [56,57]. Recent reviews of the semiempirical methods are given by Stewart [58] and by Rivail [59]. Implementation of MOPAC One popular implementation is the computational package known as MOPAC [60]. MOPAC is a general-purpose, semiempirical molecular orbital program for the study of chemical reactions involving molecules, ions, and linear polymers. It implements the semiempirical HamiltoniansmMNDO, AM1, MINDO/3, and MNDO-PM3mand combines qualitative calculations of vibrational spectra, thermodynamic quantities, isotopic substitution effects, and force-constants in a fully integrated program. Elements parameterized at the MNDO level include H, Li, Be, B, C, N, O, F, A1, Si, E S, C1, Ge, Br, Sn, Hg, Pb, and I. Within the electronic,part of the calculation, molecular and localized orbitals including excited states up to sextets, chemical bond indices, charges, electric moments, and polarizabilities are

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calculated. Both intrinsic and dynamic reaction coordinates can be calculated. A transition-state location routine and two transition-state optimizing routines are available for studying chemical reactions [60]. The focus on versatility and utility has governed the development of semiempirical methods. Computational chemistry produces only models, albeit with wide variations in sophistication, depending on the problem to be solved. There is no advantage in rigorously solving the Schr6dinger equation for a large system if the orbitals used to describe the atoms had to be abbreviated in order to make the calculations tractable. The computational foundation of semiempirical methods is the same Roothaan [61] formulation used in the ab initio self-consistent field calculations in the LCAO approximation [62]. This provides the visualization of the necessary calculation of one- and two-electron integrals over 1,2,3, or 4 atomic centers. The original formulation of these integrals in terms of Slater type atomic orbitals had an important historical consequence: the atomic Coulomb or exchange integrals appear as a combination of a limited number of elementary integrals. In particular, the atomic multiplet structure which corresponds to a large number of electronic transitions can be rationalized by means of these quantities which can be treated as empirical constants" the Slater-Condon parameters [59]. In this formulation for calculating molecular orbitals, the basic semiempirical strategy consists of: (1) using the smallest possible set of atomic orbitals, (2) neglecting a large number of integrals in the electron-electron interaction energy, and (3) setting most of the remaining integrals in a parametric form and adjusting these values on atomic or molecular experimental data [59]. The detailed implementation of these methods are presented in excellent reviews [63]. It is important to realize, however, thatdespite the elegant formalism of molecular orbital theory, the usefulness of any particular calculation is still an indeterminate function of the empirical data and methodology incorporated in the parameterization. For example MINDO/3 produced bond lengths accurate to within 2% for compounds of carbon, hydrogen, nitrogen, and oxygen. However, was supplanted 2 years later by MNDO which was extended to include an additional 16 elements but was unable to correctly reproduce the effects of the hydrogen bond [58]. The development of the Austin Model 1 Hamiltonian increased the number of parameters for each atom from 7 to up to 16. This model reproduced the water dimer hydrogen bond with an energy of 5.5 kcal/mol [64, 65]. In the development of these model Hamiltonians for semiempirical molecular orbital calculations, more and more parameters were changed from relying on atomic spectra to molecular data such as structure and heats of formation. The next development in these methods was the MNDO-PM3 parameters derived from self-consistent optimization of all parameters using a large set of reference molecular data [66]. This method had the advantage of predicting the correct linear geometry for the hydrogen bond in the water dimer. This method uses 18 adjustable parameters per element with the exception of hydrogen which uses 11. In the view of one author:

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PM3 may well be the best semiempiricalmethod available for the time being. Neverthelessthe table of optimized parameters exhibits some intriguing features for those who are familiarwith some degree of regularity along a row of the periodic table. Besides, the {developer} found two different stable sets of p,'u'ametersfor iodine so that the PM3 parameters look more like ad hoc quantities than semiempiricallyevaluated atomic physical data [59]. It is important to recognize that chemical intuition plays a crucial role in determining the parameters used in these calculations and that any result which does not correspond to educated intuition is probably indicative of fundamental problems for the particular molecule in question.

Inclusion of Solvation Effects Using Semiempirical Methods The great utility of semiempirical methods derives from the ability to calculate quantities related to the electronic structure and the ease of including large numbers of atoms. The simplicity of the elements of the Fock matrix and the acceptability of simplifying approximations has allowed specific modeling of solvent effects derived from continuum theories of solvation [5 7, 67, 68]. A brief discussion of some of these basic ideas will demonstrate the usefulness of computational methods in understanding this problem; in particular the partitioning of interaction energies in terms of quantities related to molecular orbital calculations. An excellent review of computational approximations in dealing with solvation is given by Claverie et al. [15]. One computational analysis based on the continuum model for the solvent begins with the ideas of Onsager [69] and Kirkwood [70], commonly called the reaction field model. The dipole moment of the molecule polarizes the surrounding solvent which then reacts back to repolarize the solute. The result is an image dipole in the solvent that is proportional to the solute dipole itself. The solvation energy of the molecule is then described in terms of this dielectric interaction and the radius of the cavity which is occupied by the solute molecule [71]. Karelson et al. [56] describe this interaction as a function of the bulk dielectric constant, include it as a perturbation on the semiempirical Hamiltonian (in this case AM 1), and calculate the geometry and molecular orbitals for the solute by including this as an additional term in the Fock matrix. Revail et al. [68] use the ability of semiempirical techniques to estimate molecular surfaces and use a more intricate estimation of this reaction field to calculate an isoelectric potential for the molecule in the solvent. This calculation is then approximated by a closest fit ellipsoidal surface which is used as a basis for a multipole expansion of the reaction field. These electric multipole terms are than included in the Fock matrix elements to reproduce the effects of solvation on the molecule. Tomasi and colleagues represent the dielectric interaction as a discrete set of point charges located on a cavity surface whose net field is included in the Hamiltonian for the calculation of electronic structure [72, 73]. Cramer and Truhlar [74] use spherical cavities, with radius determined from an empirical fit to aqueous free energies of solvation, to define the polarization free

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ALFRED H. LOWREY and ROBERT W. WILLIAMS

energy using a procedure introduced by Still [75]. Contributions from this polarization effect are than explicitly included in diagonal elements of the Fock matrix. Using these techniques, Karelson et al. [56] find that inclusion of solvent reaction field effects in quantum-chemical theory is obligatory for accurate modeling of relative tautomer energies in solution; Rivail et al. [68] demonstrate the importance of the shape of the cavity and polarization of the solute in using this type of semiempirical approximation; Cramer and Truhlar [74] show the importance of solvent-induced reorganization, primarily of the electronic structure but also the nuclear geometry, on individual free energies of solvation. There is also recognition that this effect may be small enough such that use of gas-phase geometries is a very useful and time-saving approximation [76]. It is important to recognize that these techniques rely on the formalism of the semiempirical Hamiltonian primarily as a means of including experimental solvation energies, dielectric constants, aqueous PK a values, and other solvent-related quantities in parameterization schemes to define quantities primarily related to electronic structure. A useful comparison between methods is given by Alkorta et al. [77].

E. Ab Initio LCAO Molecular Orbital Studies Our own research is developed using molecular orbital calculations based on ab initio techniques; these will be the primary focus of this review. Other excellent reviews of this technology with particular relation to experimental structure determination are given by Boggs [78,79] and is covered in standard textbooks [80]. The development of gradient techniques has been essential for the optimization and convergence problems in ab initio calculations [81] and has been carried over to semiempirical calculations as well [60].

Direct Calculation of Interaction Energies Using Supermolecule Formulation A well-studied method of calculating intermolecular interactions has been the formulation of the problem as a supermolecule which includes the nuclear centers from two or more covalently bonded systems in the formulation of the equations for calculating the molecular orbitals [82]. Despite recent progress in the theory of intermolecular interactions [83], reliable interaction energies can be obtained only for the smallest atomic systems [84]. Attempts to compute by ab initio methods the interaction energy of two systems that are only weakly bound, such as hydrogenbonded water molecules or van der Waals complexes, have proven exceedingly difficult primarily because of the smallness of the quantity being calculated [85]. It is of interest that one of the great difficulties inherent in this problem is a computationally induced effect known as basis set superposition error (BSSE) [82, 86]. Much effort has gone into formulating the appropriate expression for the interaction energies in hopes of minimizing errors from incalculable quantities [87]. A major use of ab initio techniques has been to calculate partial atomic charges

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used in molecular mechanics force-fields to simulate solvation interactions such as hydrogen bonding [88].

Other Electrostatic Representations Much effort has been devoted to other electrostatic representations of molecular interactions [89] using ab initio calculations based on the understanding that this component is the largest portion of the interaction energy [90]. A major application of the detailed analysis of intermolecular interactions provided by ab initio formulations has been an approximate expansion in terms of analytical functions that allow practical calculation for many different intermolecular distances [91]. Alarge scale simulation of potential functions for solvated amino acids has been derived from supermolecule calculations based on one interacting water molecule [92].

F. Self-Consistent Reaction Field Theory Using Ab Initio Methods The advent of supercomputers has stimulated the continuing development of extensive computational algorithms for modeling increasingly complex systems [93]. Progressive elaborations of ab initio molecular orbital theory [94] have resulted in the series of GAUSSIAN programs [95]. Based on the early developments of gradient techniques for Mr theory [96], computational modeling of solvent effects based on the Onsager formalism in the spherical cavity approximation has been incorporated in calculations of molecular properties [97100]. The effects of electron correlation are included at the MP2 level and with the quadratic configuration interaction with the singles and doubles (QCISD) approach. Molecular effects, such as conformational equilibrium of 1,2-dichloroethane and furfural in solution [97], vibrational frequencies of formaldehyde [98], solvent effects on sulfamic acid [99], and tautomeric equilibria of formamide and 2-pyridone [100], were modeled at various levels of theory. The recognition that these calculations were computationally accessible was as an equally important result as the good comparisons between the models and experiment.

IV. SPECTROSCOPIC B A C K G R O U N D Because our research is focused on problems relevant to secondary structure of proteins in solution, this section will briefly review the recent developments in spectroscopic techniques applied to this problem. These techniques are considered low-resolution methods which provide global insight into the overall secondary structure of proteins without being able to establish the precise three-dimensional location of individual structural elements [101] Vibrational spectroscopy has played a pioneering role in studying the conformations of peptides, polypeptides, and proteins [102]. The advent of stable and powerful lasers has led to the development of Fourier transform methods which allows the use of powerful computational techniques for the analysis of spectral data [10,103,104]. Laser

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technology also makes possible the use of Raman spectroscopy [105]. Methods for circular dichroism estimation of globular protein structure are summarized by Yang [106]. Detailed reviews are given by Bandekar [107], Surewicz [101], and Williams [108]; only a brief discussion of essential ideas will be presented here. The early work of Miyazawa [109] described the normal modes of vibration for a polypeptide backbone in terms of the normal modes of N-methyl acetamide (NMA). This established the basis for understanding these complex spectra in terms of normal coordinate analysis (NCA) [110]. A detailed review of the development of this methodology is given by Krimm [111]. The foundation for the use of NCA resides in the useful approximation that the atomic displacements in many of the vibrational modes of a large molecule are concentrated in the motions of atoms in small chemical groups, and that these localized modes are transferrable to other molecules. This concept of transferability is the basic principle for the use of spectroscopic techniques for studying problems associated with peptide structure

[111]. In early studies, efforts were dedicated to identification of such characteristic frequencies and determination of their relation to the structure of the molecule [102]. Current efforts are focused on using amide vibrational modes which arise from the peptide linkages [107]. In analogy with NMA, there are seven amide modes from the vibrations of the peptide linkage. The amide I-amide VII modes range from higher energy stretch modes to lower energy torsional modes in this order: 9 The amide I is primarily CO stretch with some contributions from CN stretch and CCN deformation. 9 The amide II mode is an out-of-phase combination of largely NH in-plane bends and CN stretch. Smaller contributions come from CO in-plane-bend, CC stretch, and NC stretch. 9 The amide III is the combination of NH in-plane-bend, CN stretch, and CH bend, with small contributions from CC stretch and CO in-plane-bend. 9 The amide IV mode is mainly CO in-plane-bend plus CC stretch with a small contribution from CNC deformation. 9 The amide V mode is largely an NH out-of-plane bend with some CN torsion. 9 The amide VI mode is mainly CO out-of-plane bend in terms of the potential energy distribution from various internal coordinates. However, the N and H atoms are also displaced and influence the relative intensity of this mode. 9 The amide VII is a mixture of NH out-of-plane bend and CN torsion. It is related to the barrier of rotation about the CN bond. The amide I mode is most widely used in studies of protein secondary structure [10, 108]. This mode gives rise to infrared band(s) in the region between 1600 cm -1 to 1750 cm -1 and is predominantly due to the CO stretching mode. The major factor responsible for conformational sensitivity of the amide I bond is coupling between

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transition dipoles [107]. The frequencies and intensities of these modes are related to secondary structure [112]. The dipole-dipole interaction is dependant on orientation and distance between interacting dipoles [113] and produces characteristic shifts in frequencies [111]. The concept of transferability and a detailed understanding of these amide modes provides the basis for quantitative estimation of secondary structure for unknown proteins and polypeptides. The quantitative methods currently used to analyze vibrational spectra of proteins can be classified into two categories: (1) methods based on decomposition of band contours into underlying components characterized by distinct frequencies, and (2) methods based on principles of pattern recognition.

A. Approaches Based on Band Narrowing A widely used approach to extract information on protein secondary structure from infrared spectra is linked to computational techniques of Fourier deconvolution. These methods decrease the widths of infrared bands, allowing for increased separation and thus better identification of overlapping component bands present under the composite wide contour in the measured spectra [103]. Increased separation can also be achieved by calculating the nth derivative of the absorption spectrum, either in the frequency domain or though mathematical manipulations in the Fourier domain [114]. An example is the method of Susi [115] which uses second derivative FT-IR spectra recorded in D20 for comparison with similar spectra derived from proteins with known structure. These methods have not yielded quantitative results that are more accurate than those obtained with methods that do not use deconvolution.

B. Approaches Based on Pattern Recognition There are a number of methods based on using the spectra of proteins with secondary structures known from X-ray data [101]. An example is the method of Williams [8,116] which analyzes the amide I band of the Raman spectrum for a protein of unknown structure in terms of linear combinations of amide I bands for proteins with known X-ray structure. Significant correlations were observed between the Raman and X-ray diffraction estimates of helix, ~-strand, turn, and undefined. Correlations were also observed between or-helix and disordered helix, and between parallel and antiparallel ~-sheets. Both approaches are empirical. They depend on comparing unknown spectra with spectra which represent presumably known structures. They give relatively accurate percentages of helix, 13-sheet, reverse turn, and unfolded structure, but quantitate only the average secondary structure content [108]. The relative success of these spectroscopic methods gives confidence that more detailed information about specific vibrational characteristics of peptides and proteins will provide valuable and useful contributions to the study of these problems. The developments

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in computational science described earlier have provided powerful tools for providing complementary information relating to such details of vibrational structure. Such computational techniques were successful in analyzing uniform, residue dependent, shifts in amide III frequencies for diamino acid peptides [1,117]. This prompted our detailed investigations based on the SQM methodology [11,78].

V. SCALED QUANTUM MECHANICAL FORCE-FIELD

METHOD

The vibrational potential energy of a molecule can be expanded as a function of internal displacement coordinates, qi, in the following way: 1

V - Wref -4- Z i

1

giqi + -2 Z Z F(i qiqj + -6 Z Z Z FiJkqiqflk +"" i

j

i

j

(1)

k

This is the basic equation used to derive normal coordinate analysis [110] as well as to define the vibrational quantities to be calculated using molecular orbital theory [79,94]. The coefficients, gi, are the forces acting on the nuclei, which are zero at equilibrium geometry. This leaves the quadratic terms Fij as the first term in the change of potential energy with instantaneous vibrational displacement. The quadratic terms Fij, are conveniently ordered as a matrix which is known as the force field or force matrix. These terms correspond to the derivatives of the potential energy V:

FiJ

~OqiOqj)e

When i = j, these coefficients are the diagonal force constants; when i g: j, these terms correspond to coupling constants. The coefficients Fijk are corresponding cubic anharmonic terms. It was the recognition that these terms could be directly calculated from ab initio wave functions [118] that led to the development of gradient techniques [81]; these have played an integral part in the development of structure optimization techniques as well as in the calculation of vibrational properties [119]. Once an approximate wave function is obtained, the expectation values of the force-field can be obtained as well as the corresponding anharmonic constants. Methods have been implemented for calculating these quantities for a variety of types of wavefunctions: Hartree-Fock, C.I., multiconfiguration SCF, etc. [95]. In ab initio methods, these quantities are often calculated for Cartesian displacement coordinates; transformation into internal or symmetry vibrational coordinates is accomplished through techniques implemented in the SQM method [11]. For any coordinate system, the desired normal coordinates are obtained by diagonalization of the appropriate potential energy matrix [110]. Ab initio calculations usually provide force constants for cartesian displacements which can then

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be transformed to provide the desired normal coordinates [120]. Automated transformation of the Cartesian force-field into a symmetry coordinate force-field was developed by the group of Peter Pulay and is part of the implementation of the SQM methodology [11]. Symmetry coordinates are useful in simplifying the understanding of the vibrational spectrum as long as the treatment is restricted to the harmonic oscillator approximation [79]. If harmonic force constants are computed at a modest quantum chemical level (e.g., with the use of a 6-31G** basis set and complete neglect of electron correlation), it is nearly universally found that computed force constants are too large by an amount averaging 15-20% [121]. Errors in calculated frequencies arise both from uncertain selection of equilibrium geometries and from inherent inaccuracies of differentiation techniques [94]. Differences between analytic and numerical differentiation techniques have been show to be of the magnitude of 2-18 cm -1 for ethylene calculated with the HF/6-31G* level of theory [94]; this is troublesome, particularly for larger molecules, but is not in principle a flaw in the techniques. Far more serious are the complex questions that surround the problem of determining the appropriate equilibrium geometry and the reliability of the harmonic oscillator approximation. It is well known, if not widely recognized, that ab initio calculations, even at the highest levels of theory and complexity, do not yield an equilibrium geometry for a molecular structure [78,122]. This means that a direct comparison between calculated and experimental bond lengths is not straightforward [123]. This results from the effects of thermal vibration; a more subtle problem is the transformation of molecular parameters determined from different experimental techniques into common reference values [124]. Further difficulties in comparisons between experimental and calculated vibrational properties arise from anharmonicity in the molecular vibrations [121]. The method of "combined techniques" is a carefully reasoned approach for overcoming these problems in integrating experimental and computational data for gas-phase molecular structure determination [125]. Rationalization of these discrepancies between calculated and observed molecular properties has been achieved by recognizing that the accuracy problems tend to be systematic for the computational methods [126]. For the same levels of theory, the errors in molecular parameters tend to be similar [79] for molecules if a variety of types and size. Building on the ideas of similarities between similar chemical functional groups in different molecules, the SQM procedure develops empirical scaling factors to correct the overestimated harmonic force constants from the computations [11]. These scale factors form a diagonal matrix C which modified the calculated force matrix Fcalc according to the relationship [127]: F = C 1/2FcalcCI/2

(3)

This relationship scales the diagonal force constants by the full value of C i where i is the matrix index corresponding to Fii, and the off-diagonal coupling constants

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ALFRED H. LOWREY and ROBERT W. WILLIAMS

are scaled by (CiCj) 1/2. Vibrational coordinates that are similar in nature, i.e. similar vibrations in similar chemical functional groups are generally not scaled independently. The reduced set of scale factors is adjusted by a least-squares procedure to give the best fit of frequencies predicted by F to the observed fundamentals. The practical utility of deriving an SQM force-field must lie in the degree to which scale factors can be transferred from one molecule to another. The purpose is to understand molecules of unknown structure and spectra by comparison with related molecules for which the spectra are well known. Since the scale factors represent the errors in the quantum mechanical calculation of the force constants, it would be expected that they would be more reliably transferred than would be the force constants themselves. It is also obvious that if the errors are very small (as the scale factors get closer to the value of 1), the accuracy of the method improves. Accordingly it is advantageous to perform the calculations at the highest practical quantum mechanical level. Tests have shown that useful results can be obtained, even from semiempirical calculations, but the scale factors are far from unity and the accuracy of reproduction of spectra is considerably inferior to that which can be obtained from an ab initio molecular orbital calculation with a double-zeta basis set or better [79]. It has also been observed that scale factors provide systematic compensation for inaccuracies arising from using only modest basis sets [122,126], allowing calculations on much larger molecules.

Vi. EFFECTS OF HYDRATION OF SCALE FACTORS FOR AB INITIO FORCE CONSTANTS The purpose of our research has been to understand the vibrational spectra of amino acids and related molecules dissolved in water. There are significant changes in vibrational spectra when molecules go into solution. Cheam and Krimm [128,129] have observed significant changes in the force constants related to intramolecular hydrogen bonds in glycine and alanine dipeptides. Wong et al. [97] report shifts in carbonyl stretch frequencies of the order of 10 cm -1 for formaldehyde, acetaldehyde, acetone, acrolein, acetyl chloride, methyl acetate, and formamide in going from gas-phase to cyclohexane solution, and almost double those shifts in going from gas-phase to acetonitrile solution. Some vibrational modes are not as strongly affected by solvation, and some are affected in different ways. For formaldehyde, Wong et al. [97] calculate acetonitrile solution shifts for six different modes; at the 6-31 ++G(d,p) basis set level, these vary from-28.4 to +30.1 cm -1 with some modes showing shifts of only 2 to 4 cm -1. Our interest evolved from the idea of whether the SQM method could correctly model the vibrational properties of chemical functional groups in aqueous solution and produce a set of transferrable parameters for use in simulating the spectra of proteins [5]. The initial interest in this question occurred in an investigation of the Raman spectra of Ala-X peptides. We observed that the amide III frequencies of the neutral Ala-X peptides shift to a lower frequency as the side chain amino acid X becomes

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larger [1,117]. Optimized structures were calculated with the Gaussian program using the 4-31 G basis set and with MOPAC using the MNDO Hamiltonian. The 4-31G basis set was chosen because it is a recognized standard [94] and is computationally efficient for calculations on large molecules in the spirit of the SQM methodology [126]. Frequency and normal mode calculations were made using scaled ab initio force constants, normal mode methods based on empirical force fields, and using MOPAC. We observed the following: 1. Frequencies calculated using scaled ab initio force constants correlate well with experimental results. 2. Structures of the Ala-X peptides optimized by ab initio methods show clear trends toward lower values of the dihedral angle ~ as the X side chain becomes larger while structures optimized using semiempirical and empirical force fields do not show trends. 3. Computational changes in the dihedral angle ~)of Ala-Ala produce a change in the amide III frequency consistent with experimental results. 4. The experimental frequency shifts cannot be attributed directly to the effects of changing residue mass. For these calculations, the force constants were scaled by a single scale factor of 81.06% to give agreement between the calculated ]]-sheet Ala-Ala amide I frequency and the experimentally observed frequency at 1680 cm -1. The use of a single scaling factor is standard practice for the Pople school of ab initio calculations [94]. While the average accuracy is improved with this approach, some predicted force constants deviate by quite different amounts from values which correspond to experiment. It has been shown that a scale factor as small as 45% is necessary for the N-H wagging motion in pyrrole [130]. The scale factor for a similar motion in imidazole is 49% [131] and in maleimide is 51% [132]. Thus, this large shift in scale factors exists for similar functional groups. However, even though the computational error is fairly large, this error itself does not vary by large amounts between similar molecules [121]. This variation in the values for scale factors is an important feature of the SQM procedure. For benzene, the scale factors obtained were in the range of 80% but they varied between 73.9 and 91.1% for different types of motions [133]. We have shown, as discussed below, that scale factors for motions within a chemical functional group differ by significant amount and that the distribution of the values for these scale factors changes in going from a gas-phase to a hydrated environment [2].

A. Simple Molecules Because our goal is to understand spectra of peptides in solution based on fundamental principles, we chose to investigate the primary functional groups related to peptide backbone structure. We calculated scale factors for ab initio force

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ALFRED H. LOWREY and ROBERT W. WILLIAMS

Figure 1. ORTEP [ 143] drawing of optimized structures for formic acid/water supermolecules at various stages of hydration.

constants for the molecules of formic acid, acetic acid, acetone [2,4], and methyl amine [3] from spectra measured in both acidic and basic solution. For the purposes of this article, the calculations on formic acid will be used to illustrate our results. Figure 1 shows the structure diagrams for ab initio supermolecule calculations of formic acid at various stages of hydration. Table 1 illustrates the calculated effects of hydration on force constants and scale factors for formic acid. Least-squares optimized scale factors are given for the force constants calculated using the 4-131G and 6-31G+ basis sets and fit to the vapor-phase and aqueous solutionphase spectra [2]. Significant changes are observed for those vibrations associated with the carbonyl groups, while the vibrations associated with the CH group are small. These changes have different signs for different symmetry vibrational coordinates but their magnitudes and sign are consistent across basis-set calculations. Similar calculated changes in carbonyl frequencies have been observed by Wong et al. [98] using their Onsager model reaction field techniques. For formaldehyde, their calculated shifts in frequencies are similar across a wide variety of extended basis sets at the 6-31 G level, but are much smaller at the 3-21G and STO-3G level. The shifts in carbonyl frequency are an important test for the computational methods since this is the most sensitive probe for the effects of solvation [134,135].

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245

Table 1. Effect of Hydration on Force Constants and Scale Factors for Formic Acid Scale Factors (%)b

Change in Force Constant a Symmetry c Coordinate

Force d Constant

1 H20

2 H20

3 H20

Vapor

Aqueous

Change

SCF/4-31G CO s

15.064

-3.09

-6.46

-5.05

86.59

76.38

Co H s

7.365

1.92

15.97

10.59

83.46

93.50

12.03

OH s

8.746

-.05

-18.44

-20.58

82.36

58.80

-28.61

CH s

6.212

1.93

1.09

1.64

76.52

77.00

.62

OH b

.803

.58

53.10

321.17

75.64

90.00

18.98

CO sd

1.337

2.29

60.89

67.75

91.80

125.00

36.16

CH ib

.749

-1.31

.63

-.52

80.40

83.00

3.23

CH ob

.522

2.93

6.56

8.71

77.50

77.50

0.00

CO H t

.084

4.59

155.49

39.05

86.43

163.00

88.59

14.390

-2.83

-5.76

91.45

82.00

-10.33

CO H s

7.172

2.52

14.84

86.00

98.50

11.05

OH s

8.820

-.10

-14.85

81.70

58.50

-28.40

CH s

6.279

1.13

.45

76.10

76.00

-0.13

OH b

.774

1.16

47.12

78.00

98.50

26.28

CO sd

1.307

2.58

53.35

92.50

119.00

28.65

CH ib

.728

-.71

1.46

83.00

85.00

2.41

CH ob

.517

3.12

7.18

78.50

74.00

-5.73

CO H t

.081

5.53

146.96

89.90

172.10

91.43

-11.79

SCF/6-31G+ CO s

e

Notes: aResults under this heading are from supermolecule calculations. Change: {[ab initio force constant for molecule with hydration/ab initio force constant molecule with no hydration (from column 1)] - 1.0 } x 100. bScale factors and changes shown under this heading are calculated from experimental measurements of the vapor and aqueous phase spectra using force constants only from the non-hydrated molecule. Results from the supermolecule calculations were not used here. CAbbreviations and symmetry coordinates are defined in ref. [2]. dForce constant: the unscaled diagonal symmetry coordinate force constant from ab initio calculations on the non-hydrated molecule. eCalculations with three water molecules in the 6-31G+ basis were not done.

The frequencies calculated using the scale factors in Table 1 along with the associated potential energy distributions are shown in Table 2 for formic acid vapor phase and Table 3 for aqueous phase. The scale factors are used for several isotopic molecules and average the differing effects of anharmonicity between hydrogen and deuterium. The differences are significantly less than 1% in the region of the spectrum around the carbonyl and corresponding amide frequencies in peptides. The differences are larger in the lower energy modes where anharmonicity and broad band widths make experimental assignment more difficult. The differences are also larger at the high-energy bands where the isotope effects of deuterium make large differences [124]. The use of single sets of scale factors to fit spectra for

Table 2. Frequencies and Potential Energy Distributions for Formic Acid Vapor a Frequency (cm -l ) Observed HCOOD

Calculated mean error

508 556 972 1037 1178 1375 1773 2634 2944 HCOOH

505 550 965 1035 1177 1380 1776 2611 2946 mean

625 642 1033 1105 1223 1387 1776 2943 3570 DCOOD

491 558 873 945 1040 1171 1737 2232 2632 DCOOH

620 628 875 971 1141 1203 1739 2221 3570

-3 -6 -8 -3 0 5 3 -23 2

CO D t( 98 ) CO sd(58), OD b(34), CO D s(7) OD b(60), CO sd(19), CO D s(16) CH ob(98) CO D s(74), CO sd(22) CH ib(95) CO s(90) OD s(100) CH s(99)

2 -1 4 5 17 -4 4 2 18

CO sd(75),OC b(13), c o H s(10) CO H t(96) CH ob(96) CO n s(55), OH_ b(38) OH b(43), CO H s(32) CO sd(24) CH ib(93), CO s(6) CO s(88) CH s(99) OH s(100)

error = 10 cm -1

490 547 868 946 1018 1168 1740 2203 2612 mean

Potential Energy Distribution (%)b

error = 7 cm -1

627 641 1037 1111 1240 1383 1780 2945 3588 mean

Error

= 5 cm -1

-1 -11 -5 1 -22 -3 3 -29 -20

CO D t(100) CO sd(59), OD b(33), CO D s(7) CD ob(100) OD b(41), CD ib(24), CO D s(20), CO sd(14) CD ib(64), OD b(23), CO sd(8) c o D s(68), CO sd(17), CD ib(8) CO s(84), CD s(8) CD s(90), CO s(8) OD s(100)

1 4 -6 13 0 19 4 -17 15

CO sd(76), OH b(13), CO H s(9) CO H t(99 ) CD ob(99) CD ib(79), c o n s(16) c o H s(46), OH b(27), CD ib(l 8), CO s(8) H OH b(54), CO s(24), CO sd(21) CO s(83), CD s(8) CD s(89), CO s(8) OH s(100)

error = 9 cm -1

622 633 869 984 1140 1222 1742 2204 3584

total mean error = 8 cm -1

Notes: aObservedIR frequency assignments for vapor phase formic acid are from R. L. Redington, J. Mol. Spectrosc. 1977, 65, 171. Calculated frequencies for all four molecules were obtained using a single force matrix scaled with factors listed in Table 1. bAbbreviations for symmetry coordinate definitions are listed in ref. [2].

246

Studies on Hydrated Molecules

247

Table 3. Frequencies and Potential Energy Distributions for Aqueous Formic Acid and Formatea Frequency (cm-l) Observed HCOOD

630 693 1011 1053 1240 1396 1695 2955 HCOOH

700 850 c 1063 1213 1380 1400 1709 2947 HCOO-

error = 19 e m

Error -1 -9 -9 35 -4 39 1 1

621 684 1046 1049 1279 1397 1696 2209 2955

0

Potential Energy Distribution (%)b ........

CO sd(52), OD b(38), CO D s(9) CO D t(92), CI-I ob(8) OD b(51), C O ~ s(26), CO sd(13), CO s(9) CH ob(92), CO ~ t(8) CO D s(60), CO sd(30), OD b(7) CH ib(89), CO s(9) CO s(81), CH ib(8) OD s(99) CH s(99)

m e a n e r r o r = 15 c m -1

14 4 7 -35 -18 -1 -2 5

714 854 1070 1178 1362 1399 1707 3034 3034 mean error = 0 cm

765 1065 1350 1380 1600 2947

Notes:

Calculawd mean

765 1065 1350 1380 1600 2947

CO sd(69), OH b(15), c o H s(14) CO H t(78), CH 9_b(22) CH ob(78), CO t/t(22) CO n s(54), OH b(34), CO s(8) OH b(41), c o n s(29), CO sd(27) CH ib(86), CO s(12) CO s(76), CH ib(9), OH b(7) OH s(96) OH s(97)

-1 0 0 0 0 0 0

CO CH CO CO CO CH

ss(14), CO sd(86) ob(100) ss(86), CO sd(15) as(26), CH b(74) as(74), CH b(26) s(101)

aCalculated frequencies for HCOOH and HCOOD were obtained using a single force matrix scaled with factors listed in Table 1. Calculated frequencies for HCOO- were obtained using a different force matrix scaled with factors reported in ref. [2]. bAbbreviations for symmetry coordinate definitions are listed in ref. [2]. CThis frequency, not observed in the Raman spectrum, was assigned as described in ref. [2].

molecules containing deuterium and other isotopic substitution is an important feature of the SQM methodology (see discussion below). However, it is important to exercise caution in comparison with other force-fields for similar molecules as was shown in the case of N-methylacetamide [5]. The differences in calculated force constants is the result of using different basis sets. It is noteworthy that the potential energy distributions are not greatly different in the tables for vapor and aqueous phase. This suggests that alteration of the force

248

ALFRED H. LOWREY and ROBERT W. WILLIAMS

constants is the primary significant change in the normal mode molecular vibration model for the process of going from gas-phase to aqueous solution. For the more complicated molecule of N-methylacetamide, mixing of low-energy modes shows significant variation, but the composition of higher energy modes in the amide I-amide III regions does not show large changes [5]. This demonstrates that the SQM procedure is capable of compensating for the additional systematic errors in comparing force constants calculated using ab initio techniques to those that reproduce experimental frequencies of molecules in aqueous solution.

B. Supermolecules With the success of these calculations for isolated molecules, we began a systematic series of supermolecule calculations. As discussed previously, these are ab initio molecular orbital calculations over a cluster of nuclear centers representing two or more molecules. Self-consistent field calculations include all the electrostatic, penetration, exchange, and induction portions of the intermolecular interaction energy, but do not treat the dispersion effects which can be treated by the post Hartree-Fock techniques for electron correlation [91]. The major problems of basis set superposition errors (BSSE) [82] are primarily associated with the calculation of the energy. Figure 2 shows Interaction energies for formic acid and a water molecule at the SCF and MP2 levels calculated with and without BSSE corrections. Frequency calculations at the minimum energy distance for each curve show that the correction for BSSE did not alter the calculation of vibrational frequencies [4]. Ab initio techniques have been extensively used for calculations on hydrogen bonding [136,137]. Table 4 gives the optimized structure parameters for the formic acid molecules calculated using the 4-31 G basis set [2]. The changes in the carbonoxygen bond lengths show significant changes, in different direction, with increasing numbers of associated waters in agreement with the observed changes in scale factors for force constants associated with carbon-oxygen vibrations, while the C-H bond length is relatively unaffected. Using self-consistent reaction field theory, Wong et al. [98] calculated a lengthening of the carbonyl bond of 0.005 in formaldehyde in going from gas-phase to acetonitrile solution; about half the lengthening observed in our supermolecule calculations for formic acid. According to Bader's theory of atoms in molecules [138,139], this leads to an increase in charge on the oxygen and corresponds to a decrease in the force constant for the C = O stretch. This is reflected by the decrease in the scale factor for this mode shown in Table 1 in going from vapor to aqueous solution. Table 1 also shows the changes in force constants calculated in the supermolecule formulation for differing number of waters. These calculated changes reflect the trends empirically observed in the changes in scale factors. This is further confirmation of the utility of the SQM procedure as a model for hydration. These numbers reflect the differing numbers of waters, but are not sufficiently unique or precisely determined to provide reliable

Studies on Hydrated Molecules

249

-3.0

-3.5

-4.0

-4.5 -

-5.0

-

KCAL MOLE -5.5

~ -6.5

~

-

/

9

_

1.9

2.0

2.1

2.2

2.3

2.4

CO .... HOH D I S T A N C E , Jk

Figure 2. Interaction energy at the SCF and MP2 levels (o) and the corresponding counterpoise corrected levels (o) for a single water molecule approaching the carbonyl oxygen atom of formic acid.

measures of the degree of hydration. As a brief observation on technique, we found it crucial in optimizing the supermolecule clusters to relax all geometrical parameters to find a stationary state. This was essential to avoid the presence of negative frequencies in the vibrational calculations which significantly distort the scaling procedure.

C. Isotopic Substitution A great advantage of the SQM method is the ease of incorporating information obtained by isotopic substitution. For the small molecule studies, deuterium substitution provided experimental frequencies for several different molecular isotopes

Table 4. Internal Coordinatesa for Formic Acid at Various States of Hydration R

Atoms

No. H20

1 H20

2 H20

3 H20

Bond stretches: 1 2

2-1 3-2

1.0724 1.2003

1.0706 1.2074

1.0717 1.2209

1.0713 1.2182

3

4-2

1.3416

1.3370

1.3158

1.3263

4 Angle bends:

5-4

0.9560

0.9560

0.9756

0.9781

5

3-2-1

124.94

124.26

122.40

123.71

6

4-2-3

124.58

123.74

124.41

123.48

7 Torsions:

5-4-2

114.87

114.95

114.93

114.60

8

4-2-3-1

180.00

180.00

180.00

179.98

9

5-4-2-1

180.00

180.00

180.00

-179.84

Note," aOptimizedvalues taken from the Z-matrices, bond lengths in A, angles in degrees. Internal coordinates for water molecules did not show trends and are not shown.

_I I I i I i I I I I I I

' l""

!"I'1""

I'"'1'"" Ala-Ala pH 7 13C= 0

.0

:>.,

1624

12C= 0

1677

r-o__ f---

E IT

LI I I

iillili

,1,,,,1,,,, I,,,,!,,,,i,,,,-

1000 '1:100 1200 1300 1400 '1500 _1600 :1700 1800

Frequency shift (cm-1)

Figure 3. Spectra of ala-ala and difference spectrum for isotopically substituted species. 250

251

Studies on Hydrated Molecules

which were then fit with a single set of scale factors. This reduces the correlations between independent scale factors. Because this least-squares method does not lead to unique solutions, we constrained the scale factors to yield a PED that was in agreement with our assignments of spectral bands based on calculated relative intensities [3]. This allowed prediction of coupled rock/wag bands for methyl amine for isotopic species that included ~3C substitution. Information about the conformation of a specific amino acid in a polypeptide can be obtained through synthesis of the polypeptide with an amino acid incorporating 13C at the position of the main chain carbonyl for that residue [1'08]. By measuring spectra for a number of isotopomers of a peptide, each one having a different 13C, the amide I frequency 9

12

rY Figure 4. ORTEP drawing of fully optimized trans and cis N-methylacetamide at various states of hydration. Only those conformers with no calculated negative frequencies are shown.

252

ALFRED H. LOWREY and ROBERT W. WILLIAMS

for each amide group can be obtained. Using this isotopic substitution approach, assignments ofl3-sheet and o~-helical structures can be made to many specific amino acid residues based solely on empirical rules for the amide I frequencies of these structures. Figure 3 (ala-ala spectra) show the experimentally measured amide I spectra of 12C and 13C isotopomers of ala-ala. A difference spectrum is also shown. The replacement of the amide group carbonyl carbon with 13C identifies the amide I frequency for this group at 1677 cm -1, in good agreement with the calculated frequency. Similar results are found for the ala-ala-ala peptide where substitution of the number 2 amide carbonyl with 13C identifies the amide I contribution from that group at 1651 crn-1 (R.W. Williams and A.H. Lowrey, unpublished results). While empirical rules would fail to yield a correct conformational interpretation of the amide I spectrum, normal mode calculations using the SQM method clearly eliminate several possibilities and put forward a preferred structure for this peptide in water [108]. This approach can be extended to much larger peptides having stable secondary structures. We have collected spectra of several isotopomers of the 23-residue peptide magainin E We observe in the difference spectra that the amide I bands corresponding to specific amino acids are much narrower than they are in the short peptides and clearly identify the amide I frequencies of these groups.

D. N-Methylacetamide and Glycine Figure 4 shows fully optimized trans and cis N-methylacetamide (NMA) in various states of hydration. Sixteen unique 4-31 G optimization and frequency

Figure 5. ORTEP drawing of fully optimized structures for glycinate with 1,2 and 4 waters of hydration. Only those conformers with no calculated negative frequencies are shown.

Studies on Hydrated Molecules

253

Figure 6. ORTEPdrawing of fully optimized structures for cis- and trans-protonated glycine supermolecule with four waters of hydration (stereo view). calculations were performed on four conformational isomers of NMA in different states of hydration. At least two minimum energy conformational isomers were found for each state of hydration. However, only one conformational isomer for each state of hydration (including isolated state) was found to yield no imaginary calculated frequencies; those are shown here [5]. Mirkin and Krimm were able to obtain force fields for all of the four stable conformers of isolated t r a n s - N M A using the 4-31G* basis set [140]. A single set of scale factors for isolated NMA yields relatively correct predictions of the shifts in vibrational frequencies between the trans- and cis-conformers both in vapor phase and in water. This supports the use of the SQM methodology to predict vibrational spectra of a peptide in a variety of

254

ALFRED H. LOWREY and ROBERT W. WILLIAMS

conformational states. The supermolecule calculations simulated the experimentally measured effects of hydration on the spectra in a manner similar to the smaller molecules. Figure 5 shows completely optimized structures for glycinate ion with 1,2, and 4 waters of hydration [7]. These structures yield no calculated imaginary frequencies; we were unable to find an optimized structure for isolated glycinate ion that did not yield calculated imaginary frequencies. Figure 6 shows optimized structures for protonated glycine in acidic solution [6]. Glycine is the next larger molecule between acetic acid and alanine. The importance of these calculations lies in the concept of transferability of scale factors calculated for smaller molecules. The assignment of vibrational modes to bands in the vibrational spectra of molecules in water becomes difficult for molecules much larger than acetic acid. This is due to the weak intensities shown by some bending vibrations in both Raman and IR spectra, to the presence of intense bands that overlap with weak vibrations and the presence of extra bands in the vibrational spectra due to overtones and Fermi resonances. Because of this, it is useful to build a vibrational analysis of large molecules using information obtained from smaller molecules. This is the main purpose of the SQM methodology for molecules in the gas phase [ 11]. Our goal is to use this technique to develop a vibrational force field for large peptides in water. Calculations on glycine allow us to develop scale factors for vibrational modes characteristic of the backbone for peptides separate from the complicating factors

Figure 7.

ORTEPdrawing of optimized hydrated ala-ala peptide.

Studies on Hydrated Molecules

255

arising from the side chain groups. Scale factors for the stretching modes of acetate are within 2% of the related scale factors for glycinate. Because of the calculated imaginary frequencies for isolated glycinate, scale factors could not be obtained by the usual approach but were obtained by extrapolating from the supermolecule calculations for various levels of hydration. Good agreements between calculated and experimental frequencies were obtained for this set of scale factors and scale factors transferred from calculations on methylamine and acetate. Scaling appears to compensate for the systematic effects of hydration on force constants making it possible to obtain reliable frequency predictions for amino acids in water [7].

E. Larger Molecules The success of our calculations and the availability of more extensive computational resources has led. to extending this research to larger molecules. Figure 7 shows optimized supermolecule structures for ala-ala peptide hydrated with water (R.W. Williams, A.H. Lowrey, unpublished results). A scaled force-field has been used to calculate the frequencies for this molecule as a function of discrete values for the torsional angle ~. The amide I frequencies do not change in this harmonic

Figure 8. ORTEPdrawing of fully optimized 6-ala helix.

256

ALFRED H. LOWREY and ROBERTW. WILLIAMS

approximation. This suggests that the transition dipole coupling (TDC) [111], which arises from electrostatic intramolecular interactions not included in the harmonic force field transferred from small molecules, is the major contributor to the experimentally observed shifts. This result is also suggested by anomalously high amide I absorption frequency for purple membrane protein, which the authors hypothesize is due to TDC between peptide bond oscillators in different m-helices [141]. Geometric optimization suggests that a 6-ala residue helix is more stable in a 310 helical conformation rather than the m-helix form (Figure 8). This agrees with the observation that the 310 form is primarily found in short segments [136]. This also agrees with recent experimental results using ESR and FTIR techniques [142]. VII.

CONCLUSION

The continuing developments in spectroscopic and computational techniques are providing more detailed understanding of the spectral features associated with peptides in water solution. We have shown that ab initio techniques can usefully be applied to study these traditionally large molecules. The scaled quantum mechanical force field methodology, combined with isotopic substitution techniques, yield intricate detail about spectral features resulting from local conformation of peptide residues. Hydration of the peptides produces significant changes in their vibrational characteristics; there is excellent correlation between the experimental observations and computational simulations. Building on the foundations of molecular mechanics, the quantitative analysis of amide bands, and the applications of SQM to gas-phase molecules, transferability of scale factors is a systematic approach to unravel the contributions of local structure to the complex vibrational bands observed in experiments. Computational techniques provide a reliable basis for normal mode analysis of large molecules; careful use of potential energy distributions provides diagnostic and predictive evaluations of complex spectra. Methodical computational and spectroscopic studies of amino acids and peptides provide the promise of significant development in molecular mechanics forcefields, and a systematic understanding of local structure features of proteins in aqueous solution.

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79. Boggs, J.E. Quantum Mechanical Determination of Static and Dynamic Structure. In Accurate Molecular Structures; Hargittai, I.; Domenicano, A., Eds. Oxford University Press: New York, 1991, pp. 322-335. 80. Ostland, N.S.; Szabo, A. Modern Quantum Chemistry; New York: Macmillan, 1982. 81. Pulay, P. Analytical Derivative Methods in Quantum Chemistry, inAdvances in Chemical Physics: Ab initio Methods in Quantum Chemistry Part II, Lawley, K.P., Ed.; Wiley-Interscience: Chichester, 1987. 82. Boys, S.E; Bernardi, R. MoL Phys. 1970, 19, 553-566. 83. Kaplan, I.G. Theory of Molecular Interactions; Amsterdam: North-Holland, 1986. 84. Sokalski, W.A.; Roszak, S.; Pecul, K. Chem. Phys. Lett. 1988, 153(2,3), 153-159. 85. Feller, D.; Davidson, E.R. Basis Sets for Ab lnitio Molecul,'u" Orbital Calculations and Intermolecular Interactions, in Reviews in Computational Chemisoy; Lipkowitz, K.B.; Boyd, D.B., Eds.; VCH: New York, 1990, pp. 1-43. 86. Chalasinski, G.; Gutowski, M. Chem. Rev. 1988, 88, 943-962. 87. Morokuma, K.; Kitaura, K. Energy Decomposition Analysis of Molecular Interactions in Chemical Applications of Atomic and Molecldar Electrostatic Potentials; Politzer, P., Ed.; Plenum: New York, 1981, pp. 215-242. 88. Weiner, S.J., et al. J. Am. Chem. Soc. 1984, 106(3), 765-784. 89. Sokalski, W.A., et al. Int. J. Qttantum Chem. Quantttm Biol. Symp. 1987, 14, 111-126. 90. Almlof, J. Geometrical Derivatives of Energy Surfaces and Molecular Properties, in Chemical Application of Energy Derivatives: Frequency Shifts as a Proof of Molecular Structure in Weak Complexes; Jorgensen, P.; Simons, J., Eds.; D. Reidel: Dordrecht, 1986, pp. 289-302. 91. Avoird, A.v.d., hltermolecular Forces and the Properties of Molecular Solids. In Theoretical Models of Chemical Bonding: Theoretical Treatment of Large Molecules and Their Interactions; Maksic, Z.B., Ed.; Springer-Verlag: Heidelberg, 1991, pp. 391-433. 92. Clementi, E.; Cavallone, F.; Scordamaglia, R. J. Am. Chem. Soc. 1977, 99(17), 5531-5545. 93. Jensen, K.E; Truhlar, D.G. ACS Symposium Series: Supercompnter Research in ChemistD' and Chemical Engineering. ACS Symposium Series; Comstock, M.J., Ed.; Washington: American Chemical Society, Vol. 353, 1987, 436 + vii. 94. Hehre, W.J., et al.Ab hfftio Molecular Orbital Theory; New York: Wiley Interscience, 1986. 95. Frisch, M.J., et al. GAUSSIAN 92; Pittsburgh: Gaussian Inc., 1992. 96. Pople, J.A., et al. Int. J. Quanttlm Chem., Symp. 1979, 13, 225-241. 97. Wong, M.W.; Frisch, M.J.; Wiberg, K.B.J. Am. Chem. Soc. 1991, 113(13), 4776-4782. 98. Wong, M.W.; Wiberg, K.B.; Frisch, M.J.J. Chem. Phys. 1991, 95(12), 8991-8998. 99. Wong, M.W.; Wiberg, K.B.; Frisch, M.J.J. Am. Chem. Soc. 1992, 114(2), 523-529. 100. Wong, M.W.; Wiberg, K.B.; Frisch, M.J.J. Am. Chem. Soc. 1992, 114(5), 1645-1652. 101. Surewicz, W.K.; Mantsch, H.H.; Chapman, D. Biochemistry 1993, 32(2), 389-394. 102. Southerland, G.B.B.M. Adv. Protein Chem. 1952, 7, 291-318. 103. Kauppinen, J.K., et al. Appl. Spectrosc. 1981, 35, 271-276. 104. Maddams, W.E; Southon, M.J. Spectrochim. Acta 1982, 38A, 459-466. 105. Tobin, M.C. Raman Spectroscopy. In Methods in Enzymology; Hirs, C.H.W.; Timasheff, S.N., Eds. Academic Press: New York, 1972, pp. 473-497. 106. Yang, J.T.; Wu, C.S.C.; Martinez, H.M. Enzyme Structure 1986, 130, 208-269. 107. Bandekar, J. Biochim. Biophys. Acta 1992, 1120, 123-143. 108. Williams, R.W. Experimental Determination of Membrane Protein Secondary Structure Using Vibrational and CD Spectroscopies. In Membrane Protehl Structure, White, S., Ed.; Oxford University: New York, in press. 109. Miyazawa, T. Poly-a-Amino Acids. In Poly-a-Amino Acids, Fasman, G.D., Ed.; Marcel-Dekker: New York, 1967, pp. 69-103. 110. Wilson, E.B.; Decius, J.C.; Cross, P.C. Molecular-Vibrations; New York: McGraw-Hill, 1955. 111. Krimm, S.; Bandekar, J. Adv Protein Chem. 1986, 38, 181-364.

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ALFRED H. LOWREY and ROBERT W. WILLIAMS

112. Pearson, W.; Serbi, G. Vibrational hltensities in hlfrared and Raman Spectroscopy; Amsterdam: Elsevier, 1982. 113. Krimm, S.; Cheam, T.C.J. Chem. Phys. 1985, 82, 1631-1641. 114. Cameron, D.G.; Moffatt, D.J. Appl. Spectrosc. 1987, 41,539-544. 115. Susi, H.; Byler, D.M. Appl. Spec. 1988, 42, 819-826. 116. Williams, R.W.J. Mol. Biol. 1983, 166, 581-603. 117. Weaver, J.; Williams, R.W. Biopolymers 1990, 30, 593-598. 118. Pulay, P. Theory. Mol. Phys. 1969, 17, 197-204. 119. Pulay, P. Direct Use of the Gradient for Investigating Molecular Energy Surfaces, in Applications of Electronic Structure Theory; H.EI. Schaefer, Ed. ; Plenum Press: New York, 1977, pp. 153-185. 120. Gwinn, W.D.J. Chem. Phys. 1971, 55(2), 477-481. 121. Boggs, J.E. Nuclear Vibrations and Force Constants. In Theoretical Models of Chemical Bonding: Molecular Spectroscopy, Electronic Structure and h~tramolecular Interactions; Maksic, Z.B., Ed.; Springer-Verlag: Berlin, 1991, pp. 1-24. 122. Schaefer, L., et al. Molecular Orbital Constrained Electron Diffraction (MOCED): The Concerted Use of Electron Diffraction and Quantum Chemical Calculations, in StereochemicalApplications of Gas-phase Electron Diffraction; Part A, The Electron Diffraction Technique; Hargittai, I.; Hargittai, M., Eds.; VCH: New York, 1988, pp. 301-319. 123. Hargittai, M.; Hargittai, I. hit. J. Quantum Chem. 1992, 1057-1067. 124. Kuchitsu, K. The Potential Energy Surface and the Meaning of Internuclear Distances inAccurate Molecular Structures; Hargittai, I.; Domenicano, A., Eds.; Oxford University Press: New York, 1991, pp. 14-46. 125. Geise, H.J.; Pyckhout, W. Self-Consistent Molecular Models, in Stereochemical Applications of Gas-phase Electron Diffraction; Part A, The Electron Diffraction Technique; Hargittai, I.; Hargittai, M., Eds; VCH: New York, 1988, pp. 321-346. 126. Pulay, P., et al. J. Am. Chem. Soc. 1979, 101, 2550-2560. 127. Pulay, P., et al. J. Am. Chem. Soc. 1983, 105, 7037-7047. 128. Cheam, T.S.; Krimm, S. J. Mol. Struct. (Theochem) 1989, 188, 15-43. 129. Cheam, T.S.; Krimm, S.J. Mol. Struct. 1989, 193, 1-34. 130. Xie, Y., Fan, K.; Boggs, J.E. Mol. Phys. 1986, 58, 401-411. 131. Fan, K.; Xie, Y.; Boggs, ,I.E. Mol. Struct. (Theochem) 1986, 136, 339-350. 132. Csaszar, P., et al. J. Moi. Struct. (Theochem) 1986, 136, 323-337. 133. Pulay, P.; Fogarasi, G.; Boggs, J.E.J. Chem. Phys. 1981, 74, 3999-4014. 134. Taft, R.W., et al. J. Am. Chem. Soc. 1988, 110(6), 1797-1800. 135. Laurence, C.; Berthelot, M.; Morris, D. Spectrochim. Acta 1983, 39A(8), 699-701. 136. Jeffrey, G.A.; Saenger, W. Hydrogen Bonding in Biological Structures; Berlin: Springer-Verlag, 1991,569+XIV. 137. Scheiner, S. Ab Initio Studies of Hydrogen Bonding, in Theoretical Models of Chemical Bonding: Theoretical Treatment of Large Molecules and Their hlteractions; Maksic, Z.B., Ed.; SpringerVerlag: Heidelberg, 1991, pp. 171-227. 138. Bader, R.E Ace. Chem. Res. 1985, 18, 9-15. 139. Bader, R.E Atoms in Molecules: A Quantum Theory; New York: Oxford University Press, 1990. 140. Mirkin, N.G.; Krimm, S. J. Mol. Struct. 1991, 242, 143-160. 141. Hunt, J.E, et al. Biophys. J. 1993, 64, A293. 142. Fiori, W.R.; Maritinez, G.V.; Millhauser, G. Biophys. J. 1993, 64, A378. 143. Johnson, C.K. ORTEP-H: A Fortran Thermal Ellipsoid Plot Program for Crystal Structure Illustrations; Oak Ridge National Laboratory, 1976.

EXPERIMENTAL ELECTRON DENSITIES OF MOLECU LAR CRYSTALS AN D CALCULATION OF ELECTROSTATIC PROPERTIES FROM HIGH RESOLUTION X-RAY DIFFRACTION

Claude Lecomte

Io II.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Study of Multipole Analysis of the Electron Density . . . . . . . . . A. Deconvolution between Thermal Motion Parameters and Deformation Density Parametrization of the Pseudoatom Model . . . . . B. Accuracy of Experimental Deformation Densities: Comparison with Theory . . . . . . . . . . . . . . . . . . . . . . . . . . C. Size of Molecules Tractable by Experimental High Resolution X-ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Transferability of Multipole Parameters . . . . . . . . . . . . . . . . . . E. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 1, pages 261-302. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8

261

262 262 267 267 270 277 279 281

262

CLAUDE LECOMTE

III. Some Applications of Electron Density Studies in Molecular Compounds . . . 282 A. d Orbitals: Occupancies from Multipole Population Parameters . . . . . . 282 B. Electrostatic Potential Calculation from X-ray Diffraction Data . . . . . . 284 C. Topological Analysis of the Electron Density . . . . . . . . . . . . . . . . 294 IV. Conclusion and Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

ABSTRACT Methods of recovering electron density and electrostatic properties from a high resolution X-ray diffraction experiment are discussed. Application to organic and coordination compounds, mainly to peptides and porphyrins, are given together with comparison with ab initio SCF calculations. Special emphasis is given to the experimental electrostatic potential and to its fit by point charges for application to modeling.

i.

INTRODUCTION

Experimental charge density analysis by crystallographic methods requires accurate low temperature X-ray diffraction measurements on single crystals in order that thermal vibrational smearing of the scattering electron density distribution is small. The experiments, techniques, and processing for accurate data are given in specific references [1-5] and will not be discussed here. These data give a set of accurate moduli of the Fourier components of the thermally smeared electron density in the unit cell Pav (r) which are called structure factors:

F(H) = I Pav (r) e2Xiltrd3r

Pay(r) = I Pstatic ( r - u) P(u)d3u

(1) (2)

where P(u) and Pstatic(r) are the probability distribution function and the static electron density of the atom, respectively. This continuous static density which may be compared to the theoretical density is divided into pseudoatomic charge densities:

N

a

Pstatic(r) = E Pj (r-

Rj)

(3)

j=l where N a is the number of atoms in the asymmetric unit or of the molecule. In the conventional least-squares refinements 9j(r- Rj) is the electron density of the free neutral atomj which has a spherically averaged shape; when this free atom

Electrostatic Properties of Crystals from X-rays

263

density is summed over all the atoms of the molecule (Eq. 3) it is called the promolecule density [ppr~ However, due to chemical bonding and to the molecule-molecule interactions, the electron density is not spherical and the deformation density: ~p(r) = p~

-- ppr~

(4)

can be mapped from accurate low temperature X-ray diffraction data. It reveals the asphericity of the valence electron density due to chemical bonding. In some cases, as shown by Dunitz and Seiler [6] for O-O and C-F covalent bonds, accumulation of bonding density in the internuclear region on 8p(r) maps is not characteristic of the strength of a covalent bond. Deformation density maps can be computed for centrosymmetric crystals after a high order (HO) refinement of the atomic coordinates and thermal parameters which relies on the assumption that the HO data (sin 0/;~ > 0.8 ~-1 or 0.9 ~-1) are mainly core electron scattering and therefore insensitive to chemical bonding (frozen core approximation) [7,8] as shown in Figure 1 which gives the scattering factor of the core and valence (3d,4s) electrons of a free iron atom. However, these density maps which give a dynamic deformation density do not readily lead to numbers describing charges and electrostatic properties; on the other 26 --~f (d)

,),

4-1

"~

o

! 0.2

! o.,

! 0.6

! o.~

, io

,.

!2

,.~

, '6 .

, .'s

Figure 1. Total (a), core (b), 3d (c), and 4s (d) form factors of the iron atom; (~) as a function of sin0/,~.

264

CLAUDE LECOMTE

hand, the spherical atom approximation is not precise enough to estimate the phases of the structure factors of acentric crystals and does not permit the mapping of 89. Alternative and much more elegant methods are those using aspherical pseudoatoms least squares refinements. These refinements permit access to the positional and thermal variables of the atoms as well as to the electron density parameters. Several pseudoatoms models of similar quality exist (9-12) and are compared in reference [13]. In general, these models describe the continuous electron density of the unit cell as a sum over pseudoatom densities centered at the nuclear sites: N a

(5) j:l where pj and 5pj are either the core density and the perturbated refinable nonspherical valence pseudoatom density [9,11] or the free atom total density and the deviations from this density [10,12]. In any case, the P and 8P functions are centered at the nuclei and are the product of a radial function (usually Slater type R n (r') = N r'"e- ~r', sometimes Laguerre function [9]) with a set of orientation dependent functions An(O,cp) defined on local axis centered on the atoms (Figure 2):

~9j (r') = 59j ( r -

Rj)= ~

C,,R. (r') A,,(O',cp')

(6)

where the C n coefficients are obtained from least-squares refinement against the X-ray structure factors. k

lVl

~d f

i

Figure 2. Coordinate system on the atoms.

Electrostatic Properties of Crystals from X-rays

265

The models used in this review are due to Coppens and his co-workers [ 11]. First, the 1s formalism [ lla] permits an estimation of the net charge of the atom and allows for the expansion contraction of the perturbated valence density; for each atom, the density is described as, pa~at(ff) = Pcore( at r , ) +

(7)

at (1s Pv 1s Pval

where patcore and 19atalare the spherically averaged core and valence electron density of the free atom, calculated from the best available wave functions. Pv is the valence shell population and 1s is the expansion or contraction coefficient of the perturbed density. If 1s is larger than one, the observed valence density of the atom at distance / corresponds to the valence density of the free atom at a larger distance, which means that the real observed density is contracted compared to the free-valence distribution. Coppens [8] justified the existence of the 1s parameter by the variation in electron-electron repulsion with electron population and showed that there is a linear correlation between 1s and the net charge obtained from Pv as predicted by Slater rules. Model studies of the 1s refinement procedure were recently made by Brown and Spackman [14]. These authors found this procedure remarkly successful in modeling radial valence density. To take in account the nonspherical shape of the valence electron distribution, the 1smodel has been improved by the addition ofmultipole parameters [11 b]. Then, the pseudoatomic density is written (Molly program), /max

pat(r,)_ Pcore( at r , ) + Pv ~3 pv(1s , ) + ~ 1s RI(1s

i--0

l

Phn Yhn(0 '(P, ) m=-I

I Ylm ] df~ = 2 if I ~: 0 and 1 when l = 0

'

(8a)

(8b)

where the Ytmare the multipolar spherical harmonic angular functions in real form, the R t = Ntr'"ex p - (1s are Slater type radial functions in which N l is a normalization factor. The Plm are the multipole coefficients which are refined in the least-squares process. The normalization of Ytm implies that a Plm value of 1 transfers one electron from the negaive lobe of the Yt,n function to the positive lobe. The parameters are chosen to be consistent with atom or molecule optimized orbital exponents ~ [~ = 2~ since p(r) o~ W2]. The n exponents of the Slater function are chosen with n > I for proper columbic behavior satisfying Poisson's equation as r goes to zero. Values of n for the multipoles were at first suggested by Hansen and Coppens [lib] based on the product of Slater orbitals t~(n'l') W(n"l")which have preexponential radial dependence r n'-l and r ''-I by analogy with hydrogenic orbitals; it leads to Table 1. These values have been found suitable for first-row atoms. For second-row atoms, this suggests a value of n = 4 (n' = n" = 3) for ! = 0 to 4. However, as shown

266

CLAUDE LECOMTE Table 1. Parameters of the Slater Type Radial Functions of the Valence Electron Density Orbital

2s 2s 2p 2p 3d

Product

2s 2p 2p 3d 3d

Density Multipole

l

n

monopole dipole quadrupole octapole hexadecapole

0 1 2 3 4

2 2 2 3 4

later, the optimal value of n must be found for second row atoms by inspection of the residual density maps, APres(r) = V -1 ~

I([Fo 1-IF m ! )ei%,] exp(-2rtiH-r)

(9)

H

where the m suffix designates the multipole atom model in the structure factor calculation; (CPmis the phase of the structure factor calculated with the multipole model. In the multipole model, the refinable parameters are Pv, Pzm' ~c, and ~:'. The limit /max = 4 is usually used for the description of second row atoms and first row transition metal because of the d orbitals (l = 2 for the wave functions) whereas lmax is taken equal to 3 for C, O, and N atoms and 1 for hydrogen. The local axis on each atom is defined by the program's user (see Figure 2); this flexibility is very interesting for big molecules possessing non-crystallographic local symmetry and/or containing chemically equivalent atoms. These symmetry and chemical constraints permit to reduce the number of the ~c, Pv, and Plm electron density parameters in the least-square process (see applications in ref. 13). For example, all atoms of a benzene ring may be constrained to have the same density parameters and a local symmetry mm2 can be applied to each atom. Modeling the electron density by spherical harmonics functions is equivalent to modify the form factor of the atom by adding [13,15]: Af=

it Pzm •

•

Ytm (U,V)

(10)

which is linear in Pzm" Once the multipole analysis of the X-ray data is done, it provides an analytical description of the electron density that can be used to calculate electrostatic properties (static model density, topology of the density, dipole moments, electrostatic potential, net charges, d orbital populations, etc.). It also allows the calculation of accurate structure factors phases which enables the calculation of experimental dynamic deformation density maps [16]:

Electrostatic Properties of Crystals from X-rays APexp ( r ) - W-1 ~

[g-~ l Fo l ei% - I F~lei*s ] e -2rtiH'r

267 (11)

all H up to nsin 0/~,max

where the subscripts m and s designate the multipole and spherical models of the electron density. When in Eq. 11 is replaced by the structure factor calculated from the multipole model the map obtained is called dynamic model deformation density: it filtered out the experimental noise. For dynamic deformation maps, ~)m differs from ~s when the crystal is acentric. Neglecting this phase difference can underestimate the deformation density of a covalent bond by 0.2 e ,~-3 which represents something like one-third to one-half of the deformation density [16]. However, before using extensively the results of the experimental density analysis in other areas of chemistry and physics, one has to answer the following questions"

IFol

IFml,

1. Does the multipole analysis of the structure factors permit a real deconvolution between thermal motions parameters and deformation density parameters? 2. How extensively parametrized a pseudoatom multipole model is necessary for reproducing crystallographic informations of the electron density in molecules containing first, second-row elements, and first-row transition metals? 3. What is the accuracy of the electron density results? How do they compare with sophisticated ab initio theoretical calculations? 4. Is there any possibility to transfer these experimental atomic parameters obtained from one molecule in a crystal to similar atoms in other molecules? The first objective of this review (Section II) is to give a first answer to all these questions; it will be shown that all the answers are positive. Then, it allows the second part of this review to be devoted to the applications of charge density research like electron density topology, electrostatic properties, study of hydrogen bonds, and metal-ligand interactions.

!!. CRITICAL STUDY OF MULTIPOLE ANALYSIS OF THE ELECTRON DENSITY A. Deconvolution between Thermal Motion Parameters and Deformation Density Parametrization of the Pseudoatom Model Moss and Blessing [17] have carried out an extended ab initio calculation essentially equivalent to 6-31G** (p polarization functions for hydrogen and d polarization for first- and second-row atoms) on the phosphoric acid H3PO 4. The

268

CLAUDE LECOMTE

molecular geometry from the neutron study by Cole [18] was adopted. Figure 3 shows the theoretical static deformation density in the O--P-O(H) plane. As expected, bonding densities and oxygen lone pairs are observed in the P--O and P-OH bonds; it may be surprising to see that the peak height in both P-O(H) and P--O bonds are the same (0.5 e,~-3). However, the P--O bonding density is much more extended than that of the P--O(H) bond leading to a larger integrated charge density. From this theoretical calculation, a set of static structure factors corresponding to the superposition of H3PO 4 theoretical static molecules at their crystal structure positions was computed to (sin 0/~)max = 1.5 ~-1 (dmin = 0.333/~). These simulation structure factors do not correspond to experimental data because they define a procrystal in which the intermolecular interactions POH...O-P responsible for the crystal packing are not taken into account.

H2 H4 a) .

,

.,.

",

, ( r'~C~-,,

~.,~~'~~ __. /

;

,

(((g.~}#,'/..,

_.-~;z

." . - ~

i,~,f

.-.. \

P " ~ . ' . . _~:..-.'Jtll/A11Z,1._eVr~\

"~ ', " ~ ~ - . , , k , \ ~ ~ ~ , !

b)

Figure 3. ORTEP view (a) and theoretical static deformation density of H3PO4 in the O--P-O(H) plane (b). Contours interval 0.1 e ~-3; (_) positive contours, (...) negative contours; zero contour omitted.

Electrostatic Properties of Crystals from X-rays

269

Nevertheless, it allowed Moss and co-workers [19] to calculate three sets of thermally averaged, dynamic structure factors at simulated 75, 150, and 300 K temperatures. These four sets of data, including the static structure factors, were used first to investigate radial modeling of the phosphorus using the Molly program [llb] (Eq. 8) and then to understand to what extent multipole modeling was able to reproduce the theoretical density (Figure 3). A series of fitting trials using different sets of phosphorus radial functions, R,,(r) = Nfl~e -~r, with the goal of zeroing the residual density (Eq. 9) led to n = 6,6,7,8 for l = 1,2,3 and 4 compared to 4,4,4,4 in Table 1. This result shows that it is fundamental to adjust the radial functions of atoms to small molecule theoretical calculations when no information concerning the radial function is available. Having secured a set ofn values for phosphorus, the pseudoatom model was fitted to the four simulated data sets to test the effectiveness of the pseudoatoms model's formal deconvolution of multipolar valence density features from thermal vibrations smearing. Results are illustrated in Figure 4 as maps of the model static deformation densities:

""--~": ~-'"''-~,:,.,,. :~,:~'". . ,",'

~ii; .'"~-.-".'...

, , ,, ,

.." ..-..:-':.-

~.. ,'z

i. L,:-.,~.'..' ..:::,,~

i

..:.,.,~ ~:..i~..,~

,,

,......::~" ....~: ........ ,

,

\\

i

\~

.'" .--::-::.'--7

,

\\

I

\

II

a) U i j = 0

b) 7 5 K

- q-re...:.

/

, ,- 2. 3,,6 ~ \

i ~" -- -

_

- "~'~.

--

..,'

I"

--

"...::....:

,

": :

o--. ";": 2:U

----

t"

...... .

,

.... : .

. . . .

t" ..,' :-:"] ?:

,"

'

/

"

,"

,

"~ i I ; o ~ ' ,

~

"I

,,___.,

\

N:, ~..~

, . . : , , # ~

,

....~-: 9

", .- -~...2'

,

I .. 11 I,,'" ! / i," ,"s'

..." ...:--..::.

"

,

..:::,

"...'t

i -, --......,k",...-~',..~::.-::....' ,........ .,...,,-,~ ',,

ii

"-

,,,, "'"

.

i

, i/' I

:

.....

,

,,

\

I t

\ \

I

\ \

/ I

C) 150 K

d) 300 K

Figure 4. Model static deformation densities of H3PO4 in the O--P-O(H) plane from simulated structure factors with uiJ= 0 (a), at 75K (b), at 150K (c), at 300K (d). Contours as in Figure 3.

270

CLAUDE LECOMTE

U 5P star(r) = Z Pj (rj=l

Rj) - p~ (r- Rj)

(12)

where pj andp~ are, respectively, the pseudoatom modeled density and the spherically average free atom density for atom j at rest. Maps from the simulated dynamic data and from the static simulated data are hardly distinguishable from one another or from the theoretical density mapped directly from the extended basis wave function (see Figure 3). Furthermore, the statistical agreements factors are excellent [R(F)] = 3.1, 3.0, 3.2, and 3.1%o, respectively, for the static and 75 K, 150 K, and room temperature simulated data. This shows that the pseudoatom model effectively recovers the theoretical electron density that comes from the dynamic structure factors. A full report, including a refinement against real room temperature structure factors of H3PO 4, is given in references 19 and 20.

B. Accuracy of Experimental Deformation Densities: Comparison with Theory Rees [21] has calculated the effect on the deformation electron density maps of experimental random errors in centrosymmetric crystals"

Ap=po-p~ o2(Ap) = 02(90) + o2(pc) - 2cov (Po,Pc)

(13) (14)

where O'2(po) #

]~O'2(Fo)+ ~ ~

and

oa(p~) # 2: ( ~v~,,) 2 ~2(vm) where vm designates the refined positional and thermal parameters and K the adjusted scale factor: cov(Po,pc ) =-2Po/K ~ m

~)(~vlO(v,n)o(K) Y(vm,K)

(15)

t-m;

If the correlation 7(vm,K)are small, which is often the case, the variance on Ap is the sum of a term constant in the unit cell (~ Z62(Fobs))depending on the quality of the data and of two terms which are large'when Pobs or Pcalc is big (i.e., close to the atomic positions); therefore, the error function peaks at atomic positions and is specially large for heavy atoms. It is important to see that the maps are reliable in the regions where bonding densities appear (middle of the bonds) and in the

Electrostatic Properties of Crystals from X-rays

271

C(101 C(9)

C(8)

C(7)

C(5) ~CC4)

C~ NU)

C(I)

CC3)

C(2]

O~

N(2)

C12) 0C1)

Figure 5. ORTEPview of N-acetyl-o~,[3 dehydrophenyl alanine methylamide (AcPhe). intermolecular region. One should, however, understand that in these calculations, the effect of errors (e.g., extinction and diffractometer positioning) and of the errors on the phases of structure factors in non-centrosymmetric crystals are not included. Another possibility to estimate the accuracy is to use an external reference-like extended ab initio calculation: Lecomte and co-workers [22] have collected low~ 1 temperature, high-resolution (sin O/X < 1.35 A), accurate structure factors on a pseudopeptide molecule, N-acetyl-c~,]3 dehydrophenylalanine methylamide (AcPhe) (Figure 5), monoclinic Cc, 7032 unique MoKo~ data with I > 36(I). A multipole analysis of the electron density was performed JR(F) = 2.19%, [Rw(F) = 1.85%; g.o.f = 0.85] and the resulting static maps were compared to ab initio SCF calculations. These calculations were carried out by B6nard's group using the molecular geometry and the IBM version of the Asterix program [23]. Two different basis sets were used: (1) basis set I was medium size, and (2) split-valence basis set II taken from Huzinaga [24] (9s and 5p gaussian-type orbital contracted into [3s, lp] for the C,N,O atoms). In basis set II, more flexibility has been allowed to the description of the valence shells by adapting a triple-~ contraction completed with one p type (for hydrogen) or one d type (for C,N,O) polarization function. Then two ab initio calculations were carried out: (1) all atoms were described with basis set I, and (2) basis set II was used for all the atoms but those belonging to the methyl and phenyl substituents due to computer limitations. Figures 6 and 7 give respectively the experimental and theoretical static deformation density in the planes of the C3--C 4 double bond and of one of the two peptide links, calculated with the two basis sets, as well as their difference.

Il I

|s

~%

l

, .... ;-.-.-:;'j I

J

s

i ! I

I I

\

: It|; ill ,~

Sj J

oJ

%~,,_ ,,. ,," ~'"

it/,li

"il

s

! i I ~

/ s

i

/

4.oo.

,

,

; ,,

.-.

,.

s\

s

~,I

,

a)

~

.

,,,,,,,~ ~

.

.

.

..........................

.,

~

.,,

.,

J"

i. C)

3 cll

Figure 6. Static deformation densities in the plane defined bythe C3--C4 double bond in AcPhe" Experiment (a), basis set I (b), basis set !1 (c), (c) - (b) = (d). (continued) 272

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

2.00

3.00

4.00

S.O0

6.00

5.00

6.00

b)

1L,I~.

-2.00.

-4.00

-3.00

-2.00

-1.00

0.00

d)

1.00

4.00

Figure 6. (continued) Contours 0.05 e ~-3, solid lines for zero and positive contours; dotted lines negative contours; zero contour omitted on experimental map (reproduced from [22]). 273

1A

:

:

i/ s

I

... .~

s"

9

l ,''~

"

~ #l "l I

I ~

I I

.

~

.

*.,

t

.

.

j)

~" ,r

#%i

-lit%~,778/7/

.

%

%

I

I

i

m m %

.

!

%

........

9

l

I

,. '," q ~ - , , / i N)/illt.' 7 ~ ~ \ \ \ ,, " ; " -'t,'lil~l~iE~-'/t-J))/kkk~k\ ,

.

,,

".

J~ l f i ~

"

-

;v

tt((((((~5~/~-"

", \ \ \ ~ ~ d #

-

"

li~ ', " . " 7 - . . ' ---

.

,

,

.

,'

,i

-

If/lllll

tt ~llll

I

I

;

~ ", I tr

t--

_

-~,

a) .00

X

9 "

~.oo

i

o~

"",

-tOO

'

,"

'

--~.T-.

"",2" :"':,

",

-.

-,

! i

.

.

.

',,

.

.

.

)

"':]'">

t ",

!

-2.00

,

o.=-=.-.-.

-4.

02

~

/ -4.00

-3.00

-~

-tO()

0.00

tO()

c)

"

Figure 7. Static deformation densities in one peptide plane of AcPhe Experiment (a), basis set I (b), basis set II (c), ( c ) - (b) = (d). (continued) 274

/~~,~

,.:,,

/

i X

-.

i -4.00

vw

I i -3.00

i

w wl

--i

-2.00

iw

w w l i l l V l -1.00

iw 0,00

jl

i 1.00

w i w i

i

2.00

i w Vnl 3.00

v w

fl

4.00

d) Figure Z

(continued) Contours as in Figure 6 (reproduced from [22]). 275

276

CLAUDE LECOMTE

Table 2. Comparison of Bond Peak Heights (in e ,~-3) and Bond Distances (,~) in N-Acetyl-oc,13-dehydrophenylalanine methylamide [22] Distance T3pe

a

b

Uncorr

Cor~

Pexper.

Pstatic

Ptheor.

Ptheor.

C1-C2 C3-C11 C4-C5

1.5034 (5) 1.5027 (7) 1.4717 (7)

1.505 1.504 1.474

0.40 0.55 0.55

0.51 0.57 0.57

0.58 c 0.65 0.53 c

0.43 0.44 0.41

C5-C6 C5-C7 C8-C9 C9-C10 C7-C10 C6-C8

1.4044 1.4010 1.3953 1.3927 1.3943 1.3917

(6) (6) (7) (7) (8) (9)

1.407 1.403 1.397 1.395 1.397 1.394

0.57 0.57 0.58 0.60 0.58 0.59

0.66 0.66 0.72 0.72 0.73 0.74

C-C f

C3--C4

1.3473 (7)

1.348

0.73

0.82

0.74

0.56

C-N g

Cll-N2 C2-N1

1.3356 (4) 1.3470 (4)

1.339 1.351

0.50 0.50

0.63 0.64

0.65 0.64

0.38 0.37

C-N h

C3-N1 C12-N2

1.4118 (4) 1.4484 (8)

1.413 1.450

0.37 0.35

0.47 0.39

0.50 0.41 c

0.27 0.24

C-O

C2-O1 Cll-O2

1.2383 (3) 1.2387 (7)

1.242 1.241

0.56 0.61

0.69 0.73

0.65 0.65

0.41 0.41

C-H

Phenyl C4-H4

1.07 1.07

0.38 0.40

0.56 0.52

0.73

0.61 0.54

N2-H14 NI-H3

1.03 1.03

0.50 0.45

0.61 0.64

0.69 0.70

0.46 0.46

C-C d

C-C e

N-H

Bond

0.49 0.49 0.49 0.49 0.49 0.49

Notes: aCalculations carried out with basis set II (triple-~ plus polarization), except for the methyl and phenyl substituents. bCalculations carded out with basis set I (split-valence). COne atom of the considered bond is described with basis set I. dSingle C-C bonds. ephenyl ring C-C bonds. fDouble C-C bond. gC-N bonds with double bond character. hC-N single bonds. iCorrected for whole-molecule and internal torsional librations (see ref. [22]).

As shown in ref. [22], the effect of basis set extension is clearly visible: the height of the peaks increased by 0.18 e/~-3 for the C 3 - - C 4 double bond (Figure 6d), by 0.25 e/~-3 for the C-N bonds, and by 0.24 e ,~-3 for C--O bonds. For the C--O region, extended basis sets reduce the depopulation region close to the oxygen atom, increase the peak height, displace it by 0.16 ,~ towards the oxygen atom, and reduce the lone pairs accumulation by 0.25 e ,~-3. Basis set I provides a fair qualitative agreement with experiment (Figures 6a, 7a) concerning the positions of the maxima and minima of the deformation density, but

Electrostatic Properties of Crystals from X-rays

277

the peaks are systematically underestimated with the bigger discrepancy for the C-N and C--O peaks. The quantitative improvement is impressive with basis set II, which shows an almost quantitative agreement. Discrepancies remain important however in the lone pairs regions (0.6 e ,~-3 versus 1 e/~-3). This discrepancy can be attributed to the finite resolution of the experimental map and also to the fact that theory calculates the electron density of an isolated molecule at rest, whereas experiment shows a pseudo-isolated, pseudo-static molecule removed from the crystal lattice in which crystal field effects--intermolecular hydrogen bonds--are not absent. Most of these small discrepancies are also found in the formamide study by Stevens, Rys, and Coppens [25a]. Similar conclusions were made by Eisenstein in her experimental and theoretical study of cytosine and adenine [25b]. Table 2 compares the peak heights obtained from experiment and theory. All bond peak heights obtained from the experimental static model and from theoretical distributions (basis set II) are equal within less than 0.1 e/~-3 except for one C-H bond where the difference reaches 0.21 e ,-3; good agreement is obtained for the N-H bond peaks. The 0.1 ~*-3 discrepancy is an external measure of experimental error compared to the approximation:

O(Pobs) # V-1 K-' ~ [o 2 IF o I]'/2 = 0.029e ~-3,

(16)

It

or to

~

-- W-1 Z

[OlEoIg-I -

I Fm I)=1

0.032e/~-3 [22]

(17)

H

Then, the discrepancy between theory and experiment appears to be within three estimated os in the bonding region. In conclusion, almost quantitative agreement is obtained between experimental static deformation maps and extended triple-~-plus polarization maps.

C. Size of Molecules Tractable by Experimental High Resolution X-ray Diffraction The difficulty of experimental electron density increases only moderately with the size of the molecule compared to the fourth-power dependence of theoretical calculations. This is a significant advantage for experimental studies, especially when one has to study biological molecules like drugs or polypeptides. Stevens and Klein have experimentally studied chemical carcinogens and opiate molecules with a good precision [26] and the calculation of the experimental electrostatic potential will certainly help to understand their reactivity. Interesting calculations of experimental electrostatic potential of puromycin (C22H31N70~+, 2C1-.5H20 ) are underway in Craven's group [27]. The experimental electron density of leu-Enkephalin, 3H20 (C28NsOvH37-3H20; tyrl-gly2-gly3-phe-leu) in its folded conformation (Fig-

o

o

o

-r-

f,d

r162

e"

eo~

e-

q) 9

>

r~

0

-

:iil1, ~

i!i

~'-,

~l

..........

.,,

"'-'

.,_,

"',

""

m

N

&

~,~

o~

!

~. "~ o~

~

d~

~.~

m0

cE

' ~

..]

t

ti

e"

E.~-

~.1

C

.~:~ ~f - .

co

Electrostatic Properties of Crystals from X-rays

279

ure 8) has been published recently by Pichon-Pesme, Lecomte, B6nard, and Wiest [28] [40,000 reflections measured on a CAD4 diffractometer at 100(5) K; R(F) = 3.79%, Rw(F) = 2.29%, G.O.F. = 0.73] and the maps obtained are very accurate. Figures 9a,b show, for example, the static maps of the tyrosine group and of the phenylalanine residue: the bonding density in the C--C bonds agrees quantitatively with that obtained on smaller molecules (see Table 2). Theoretical studies of such big molecules are at the present time far beyond the computational possibilities, and experimental determination of electron density is the only tool to get accurate electrostatic parameters. However, it may be possible to recover the experimental electron density by performing ab initio fragment calculations [28].

D. Transferability of Multipole Parameters Because electron density is a local property, electron density studies of the peptide-like molecules show that the nonspherical part of the deformation density (i.e., the Plm parameters of Eq. 8) remain essentially the same for a given atom in the same environment (the peptide residue, a phenyl ring, a methyl group...) [29]. The same observation was made for porphyrin ligands [30] and by Brock, Dunitz, and Hirshfeld [31] for naphthalene and anthracene type molecules. All these observations suggest that the multipole parameters are highly transferrable from one atom to a chemically similar atom in different molecules and crystals. A key question is" is it possible to determine for each chemical type of a given atom a small set of pseudoatom multipole parameters, and can such parameters be used to calculate electrostatic properties of new molecules? To answer this question [29], two accurate but low resolution X-ray data sets (sin 0/~ma x = 0 . 6 5 ,~t-1) were

t~trtzolr

me O1

J

/

pro C~E1

0

c

N1

\

\

C A I ~

/

II

Cl ~

02

N2 ~

CA2 ~

I

/

CG1

/

N3 ~

\/

CG2 CD22

CD21

I

pyT

II

C2 ~

I

CE22

CE2~ CZ2

b)

a)

Figure 10. ORTEP view of Pyr-Phe-Pro-tMe (PPP).

CA3~

CG3

C3 m

\/

N4 ~

N6

CA4

280

CLAUDE LECOMTE Table 3. Statistical Indices of the Refinements of PPP no

nv

R(F)% Rw(F)% G.O.E

Spherical atomform factors (conventionalrefinement) Aspherical pseudoatomform factors

3104 3104

270 270

4.69 3.71

3.85 2.90

2.07 1.56

110 K data Spherical atom form factors Aspherical pseudoatomform factors

2286 2286

270 270

3.44 2.33

3.77 2.44

2.22 1.44

Room temperature

selected, one measured at room temperature and one measured at 125 K for peptide-like molecule Pyr-Phe-Pro-tMe (PPP) [29] (an ORTEP illustration is given Figure 10). These data were measured by Pangborn, Smith, and Howell (Medical Foundation, Buffalo, N.Y.). Two or three most significant Plm parameters of each atom type were then chosen from previous work on peptide molecules [16,22,29] and used as fixed parameters in least-squares refinements of PPP. Only the fractional coordinates and the thermal parameters of all atoms were adjusted; the statistical indices of the refinement decreased dramatically as shown on Table 3, compared to those obtained from a conventional refinement using spherical atoms form factors. It confirms the possibility of transferring multipole parameters. Having new xyz U ij parameters, new structure factors phases (~m) were calculated and the new 1A 4-. I

'

,

;' , .,.,

"

IA

(.k'&"-.%~"-

'

,'; _

'

.... ," -., ///~\ ,, ,, ,

i i//...xl!

l \ \\~Jllll ~._ ; ~, , ~ ,, -

a

-

-

-

'l

_ ~,

-._

[

.-

I

i

. . . . . . ,

Lz ,~ ~,

" "

,'--"

"--"

"I

,' .. ~, '...:--

x

h

Figure 11. Low temperature (110 K), low resolution (0.65 ,~-1, sinO/)~) experimental deformation density in the planes of a peptide residue (a) and of a phenyl ring (b) for Pyr-Phe-Pro-tMe using P/m transferability. Contours as in Figure 6.

Electrostatic Properties of Crystals from X-rays i

_ i "x x ,~

1A

281

, ,i

z i i" i -I % l

1A

i

t i l

ii

\1 \ I--i i

,,

i I

,

,

,""

i

I

i ~. ~'

l

, 9

,

i

,,,, x

, ,.

1

,

,',-,, l

i i

', ii

',l,, " -'I !

I

z

.,."

,,

j

_

I

t ' ',-,'.',

tttt.W)/U

t

d .-1

~I

l

I

i

i ( - ,."

,,,

J

|

r

I

b

Figure 12. Same as Figure 11 but with room temperature data.

atomic parameters were used to map the experimental electron deformation density for both room temperature and 110 K data (Eq. 11). The summation was made on all the 0.65 A-1 resolution observed structure factors. The experimental deformation density obtained are shown on Figures 11 and 12 which represent the deformation density of a peptide residue and of a benzene ring temperature at 110 K and at room temperature, respectively. The low-temperature maps agree almost quantitatively with those of very highresolution studies; for example, the C-C and C-H bonding densities are 0.50 + 0.05 e/~-3 and 0.35 + 0.05 e ,~-3, respectively. This shows that this crude modeling permitted to deconvolute almost all the deformation density from thermal motion. The low resolution of the experiment is only detected at the oxygen lone pair region as expected. Room temperature maps are also of good quality and show the effect of thermal motion on the dynamic bonding density: on the phenyl ring (Figure 12b), the bonding density decreases with the distance of the bonds from the center of the molecule, whereas thermal motion increases.

E. Conclusion Pseudoatom multipole modeling reproduces accurately the deformation density within less than 0.05 e/~-3 in the bonding and intermolecular regions; calculations on theoretical thermally smeared structure factors show that deconvolution between density and thermal parameters is effective. The accuracy obtained from a highresolution accurate diffraction experiment compares with extended triple-~ + polarization ab initio SCF calculations. These experiments are tractable for mole-

282

CLAUDE LECOMTE

cules up to 100-150 atoms and the results are transferrable from an atom to chemically similar atoms in different molecules and crystals. These conclusions allow the calculation of the properties of molecules from experimental high resolution X-ray data and their transferability to bigger molecules.

III.

SOME APPLICATIONS OF ELECTRON DENSITY STUDIES IN MOLECULAR C O M P O U N D S

A. d Orbitals: Occupancies from Multipole Population Parameters At the present time, only first-row transition metals may be accurately studied because the energy of the widely used Mo K~ incoming beam (~ = 0.71 A, v = 4.2 x 1018 Hz) is close to the ionization energy of the core electrons when the atomic number Z is bigger than 35 [32]. For first-row transition metals, the populated or depopulated d orbitals of the metal are easily observed by a visual inspectation of the deformation density maps" it is clearly an advantage over theoretical calculations for determining the leading contributing configuration to the metal electronic ground state because different configurations, even very closely spaced in energy, may correspond to very different spatial distributions. As an example, the nature of the ground state of iron II tetraphenylporphyrin bistetrahydrofuran [TPP Fe(THF) 2] was not well established by theoretical or spectroscopic methods; the THF axial ligands are weakly bound and the complex is high spin. Theoretical calculations of the analogous bis-aquo complex by Rohmer [33] showed the 5B2gand ionic states of the iron atom were of comparable energies and therefore may be leading terms of the ground-state configuration. Experimental deformation density maps [34] of TPPFe (THF) 2 showed dz2 orbital depopulation and dxz, dyz density accumulation as well as lack of population in the xy directions (Figure 13). This is in agreement with 5E2g state (Figure 13d) because 5B2g would be compatible with electron populations in the dxy orbitals. Figure 13 shows also excess deformation density in the Fe-N bonds which can be interpreted as donation from the porphyrin ligands and as a contraction of the dx2y2 orbitals due to the negative charges of the nitrogen atoms of the porphyrin. This anisotropic contraction of the d orbitals due to the ligand field was observed also for germanium porphyrins both by X-X [30] and by ab initio SCF calculations [30a]. As described by Stevens [35] and Holladay, Leung, and Coppens [36], d orbital occupancies of the metal atom can be derived from the multipole parameters assuming that the overlap density and the asphericity of any 4p orbital density are small. For first row transition metals, the d orbital expansion and the overlap density between d orbitals and the ligands are small; then, the asphericity of the electron density of the metal atom is mainly due to the d orbital occupancies. From Eq. 8, we can write:

5Ezg

Electrostatic Properties of Crystals from X-rays ,

'~ i

, .~

=-1 c Jt \ _ ~ ~ ' - '

t t..,.. ,,:,, iii # I ~

t',:" s s ,

OI "~

", '~\'~.~]llll ", "..":." , ' , ~ ~

. I#

~~/".

~~~'7///(~

283

-~X

.,

i" ",~ - ~

k:51l !)}11111'

_

", ",

I

"'

Z

--'~-'7"---" .~ ~"

~

_~

', t ,',,]

...../

#,,.

o,

I ~k\\~

9

't,: ,'

7.;

!

. "

I

~Y

I

!'."

F'~

,

b)

a)

5

- -.

I "-~,.

-'Y

.::-,"

i

I

_

r

d)

.___.

Figure

13. Dynamic deformation density on a pyrrole plane (a), in the (x,y) four nitrogen plane (b) and in the (y,z) N-Fe-O (THF) plane (c) in TPPFe(THF)2; corresponding theoretical density in the (x,y) plane for the 5E2g state of PFe(H20)2 [33] d). Contours as in Figure 6 (reproduced from [34]).

4 EPiid~ + E E didjPiJ: E K,3RI(K,r)E PlmYlm(O,~p) i

i ~j

i

(18)

m=0

where Pig are the population of the d i orbitals. Since products of spherical harmonics functions can be expressed as linear combinations of spherical harmonics, it leads to a set of linear equations which can be solved to get the Pii and Pii parameters. This calculation was performed for all the porphyrins studied by Coppens and co-workers [37]. The results for TPPFe (THF) 2 are given in Table 4 compared to the populations of the iron quintet states.

284

CLAUDE LECOMTE

Table 4. Experimental iron Atom d-Orbital Populations of TPPFe(THF)2 Compared

to the Iron II High-Spin State [34]

Term symbol dx2-y 2 dz2 dvz,dxz dxy

5B2g

5Alg

5Big

5E2g

Exp

Spherical

1

1

2

1

1.42

1.2

1

2

1

1

1.04

1.2

2 2

2 1

2 1

3 1

2.52 0.93

2.4 1.2

In the multipole refinement of TPPFe(THF) 2, a D4h local symmetry was imposed on the iron atom which explains that only four d i population parameters were derived; inspection of Table 4 leads to the same conclusion derived qualitatively from the examination of the deformation maps; i.e., t h e 5E2g state is the main contributor to the ground state of the complex. This interesting calculation of d electron population calculations was also performed on other coordination compounds like metal carbonyls [38] and metal carbynes [39].

B. Electrostatic Potential Calculation from X-ray Diffraction Data Electron density mapping permits a direct comparison with theory but does not provide much information about chemical reactivity or intermolecular interactions. This information can be obtained by inspection of the electrostatic potential generated by molecules in their outer part. Stewart was among the first crystallographers to realize that the Fourier components of the electron density can be used to evaluate a variety of electrostatic properties [40] of molecules in the crystal--for example, the electronic part of the electrostatic potential at a point r inside the crystal: p(r') dar, Ve(r) = ~ I r__r, ]

(19)

1

can be calculated by expanding [ r-r'l in reciprocal space, one gets [40,41]" 1 Ve(r) -- ~--g Z F(H) H 2 exp-2irt(I-I-r)

(20)

H

The electrostatic potential V e is the inverse Fourier transform of H -2 F(H). However, there is a singularity for H = 0 [42]. In order to avoid this problem, one calculates the deformation electrostatic potential at r: AV(r) - ~

1

1

Z ~__ (I F m I e i% H

I Fsl

e i*S) exp (-2rl;i Br)

(21)

Electrostatic Properties of Crystals from X-rays

285

where I f m I, *m, I f l, *s are the moduli and phases of the static structure factors (Uq = 0) calculated respectively from the multipole model and from the promolecule. Examples of these calculations are given in references 40, 41, and 42. Another method is to calculate the molecular electronic electrostatic potential by replacing p(r') in Eq. 19 by its multipole formulation (Eq. 8). The quantity obtained represents the electrostatic potential of a molecule removed from the crystal lattice. First calculations have been performed by the Pittsburgh group (Stewart, Craven, He, and co-workers) [43]; electrostatic potential calculations were also derived from the Hansen Coppens [lib] electron density model [41,44]. The atomic total electrostatic potential including nuclear contribution may be calculated as: V(F)-- Vcore(r) 4" Wval(F) 4" AW(r)

(22)

oc

with (Figure 14)

Vc~

Z - I Pcore (r') d 3 r' = [ r - R[ [r-R-r'[

(23)

O

oc

Vval(r ) _ _ ; 0val (rp) d3 r' O

[r-R-r'l

(24)

and AV(r) is the deformation potential due to the asphericity of the electronic cloud (for explicit formulation, see [41]). This aspherical term, calculated from dipolar, quadripolar, octopolar, and hexadecapolar terms of the electron density, vanishes very quickly as soon as the distance between the molecule and the observation point

d 3 ~r

[~

o Origin

Figure 14. Calculation of electrostatic potential.

286

CLAUDE LECOMTE

V (c &-l)

~.00

'

o.s'o

'

3.06

'

t.s~

'

z.o~

'

z.s~

'

3.0~

'

3.s'o

'

4.0~

'

4s'o

'

s.~

r(,~)

Figure 15. Noneffect of the ~ parameter on the electrostatic potential of a negatively charged oxygen atom, Pv = 6.44 ~, K = 1 (continuous curve) and 0.977 (crosses) (reproduced from [41]).

increases; one can use the results of a K:refinement [lla] to estimate with a good precision the electrostatic potential. Furthermore, the effect of the expansion contraction ~ parameter is very small on the electrostatic potential [41,43b] as shown on Figure 15 which represents the radial electrostatic potential of a negatively charged oxygen atom (-0.44 e) with ~z= 1 and 0.977. Then, the electrostatic potential reflects mainly the net charges of the atoms. Craven and co-workers used Stewart's model to calculate the electrostatic potential and dipole moments of several small organic molecules including phosphorylethanolamine [45], the 1:1 complex of thiourea with parabanic acid [46], methyluracil [47], and cytosine monohydrate [48]. In cytosine monohydrate, Weber and Craven [48] showed that the electronegativity of the hydrogen-bonding acceptor sites can be ranked with the water oxygen atom, cytosine N(3), and carbonyl 0(2) in decreasing order. Energies of interaction derived from this analysis and from electron gas theory were reported and discussed by Spackman, Weber, and Craven [49]. For the hydrogen-bonded dimer the experimental electrostatic energy of interaction was found to be-96 (27) kJ mo1-1. This promising calculation as well as the pioneering work of Moss and Feil [50] on pyrazine opens a way to estimate interaction energy derived from an X-ray diffraction experiment. The electrostatic potential of urea and imidazole was calculated by Stewart [32] from X-ray data of Craven [51]. For the pseudo isolated urea molecule (removed

Electrostatic Properties of Crystals from X-rays

287

L

'", a)

~// /

,

b)

Figure 16. Electrostatic potential generated by a N-acetyl (z,[~dehydrophenylalanine methylamide molecule removed from the crystal lattice, in the peptide plane (a) and perpendicular to this plane (b). Contours _+0.1 e A -1-, positive contours: solid line, zero contour short dashed (a), negative contours dashed (reproduced from [52a].

from the crystal lattice), double minimum potential (-0.35 e A-l) were found in the molecular plane near the oxygen atom; this oxygen atom accepts four intermolecular hydrogen bonds to form a four-molecule cluster in the crystal lattice. When this cluster is formed, the electrostatic potential becomes positive everywhere. This is a general result for any sufficiently large cluster because the positive charges (i.e., the nuclei) are localized contrary to the negative electron cloud. The electrostatic potential of peptide-like molecules is being studied by Ghermani, Lecomte, and co-workers [41,52]. Figure 16 gives the electrostatic potential generated by a N-acetyl-o~,13-dehydrophenylalanine methylamide (AcPhe) [22] molecule removed from its crystal lattice in its peptide plane (a) and in a plane perpendicular to the peptide plane passing through C = O (b). In this peptide molecule, the oxygen atom is surrounded in the outer region by a wide and deep negative region (-0.40 to-0.30 e/~-1; 1 e A-l = 332 kcal mo1-1 = 1390 kJ mo1-1) which would favor the approach of electrophilic agents as well as hydrogen bond formation. Another interesting feature of V(r) is that, contrary to the deformation density, the electrostatic potential generated by the oxygen lone pairs does not have a "rabbit ear" shape, but rather has an almost spherical skull shape. This means that lone pairs directionality is in part lost when electrostatic interactions occur and could explain why hydrogen bonds are not always directed along lone pairs (Figure 17b). When calculating the electrostatic potential of two AcPhe molecules interacting via an hydrogen bond as in the crystal (N..-O = 2.869/~), the deep negative region divides in two parts on each side of the H...O hydrogen bond (Figure 17a). Similar

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18.

ORTEP view of tbuCOprohisNHMe. 288

6

Electrostatic Properties of Crystals from X-rays

289

but slightly more positive potentials are found in N-acetyl 1-triptophane methylamide [16] where one oxygen atom of a peptide residue accepts two equal hydrogen bonds (N..-O = 2.885/k) (Figure 17b). Contrary to the experimental electron density, the electrostatic potential is conformation dependent. Figure 18 shows the ORTEP view of t-butylCOprolinehistidine-methylamide (tbuCOprohisNHMe) [53] which exhibit a folded conformation due to an intramolecular hydrogen bond (13-turn) between 01 and N3H (N3-.-O l = 2.935 ,~); as a consequence, hydrogen bond occurs between the histidine N 4 and the N 2 hydrogen (Na...N 2 = 3.205 ,~,). The effect of the [3turn on the electron density has been discussed in refs. 28 and 64. The electrostatic potential calculated for one molecule removed from the crystal lattice in the histidine plane (Figure 19a) shows a very small minimum of potential (-0.18 e/,~-1) around the nitrogen atom which becomes positive when the calculation is made for a cluster of two hydrogen-bonded molecules [53] (Na.-.N 5 = 2.856 /k) (Figure 19b). This very shallow negative potential around the nitrogen atom of the histidine residue is a result of the folded conformation of the molecule. As the multipolar electron density parameters are transferable [29], we have calculated the electrostatic potential in the histidine plane for a single molecule in an extended conformation using the electron density parameters of the folded molecule [54]. Figure 20a shows that in the new conformation ofthe molecule, the N 4 atom ofthe histidine residue is now not interacting with a peptide nitrogen as observed in the crystal, and the resulting electrostatic potential around N 4 is more negative (--0.27 e ,~-1) and resembles that of imidazole [32]. Other examples of electrostatic potentials in peptides and other molecular compounds can be found in refs. 53, 32, and 55. A new and very promising application of the calculation of electrostatic potential from experimental electron density is its modeling by point charges and dipole moments [43b,53,54]. When the potential calculated from a ~: refinement [lla] is fitted by point charges at the atomic sites, the resulting charges are not dependent of the molecular conformation [56] and the fit is excellent outside the van der Waals envelope of the molecule. Figure 21 shows the potential calculated in the peptide plane from the ~ refinement of AcPhe (Eqs. 24,25) and its fitted potential. When the potential is calculated from Eq. 22 (i.e., includes aspherical terms of electron density) the potential is reasonably well reproduced at the van der Waals surface by point charges, as shown in Figure 22 which gives the comparison between the total potential in a peptide plane of tbuCOprohisNHme and the point charges fitted potential. The rms deviation is = 0.03 e/A, and it could be important to include dipolar terms on hydrogen atoms [43b,53]. At the present time, it then seems possible to build a data bank of experimental atomic charges and dipole moments which could be used to parametrize the force fields in the molecular modeling codes.

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294

CLAUDE LECOMTE

C. Topological Analysis of the Electron Density Another possibility to assign charges to atoms is to find schemes for partioning the total charge distribution: the ~: refinement [11] is one of these schemes and leads to charges close to those calculated from a fit to the electrostatic potential [56]; the Stockholder partitioning as proposed by Hirshfeld is another scheme [58]. Bader discovered a method [57] based on the topology of the total electron density, which leads to an atomic classification of the properties of matter: "the form of the total electron distribution in a molecule is the physical manifestation of the forces acting within the system". The characteristics of the total electron density topology may be analyzed by a search of the critical points (minima, maxima, or saddle points) located at given points r for which the gradient of the density is zero. V p(r) = 0

(25)

Whether a function is a minimum or a maximum at an extremum is determined by the sign of its second derivative or curvature at this point; in three-dimensional space for a given set of coordinates axis, the curvature is determined by the hessian matrix which elements are:

3 2 p(r)

(26)

H(ij) - 3 xj ~X i This real matrix may be diagonalized to give the principal axis of curvature and the trace of the hessian matrix; i.e., the laplacian of the density, is an invariant.

329

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(27)

V2p = V-Vp = ~ X2 + ~y2 4-~)Z2 Then, the critical points are characterized by two numbers, co and ry, where co is the number of nonzero eigenvalues of H at the critical point (rank of the critical point) and r (signature) is the algebraic sum of the signs of the eigenvalues. Generally for molecules, the critical points are all of rank 3; then, four possible critical points may exist: 9 three positive eigenvalues ((3, +3) critical point): the electron density at that point is a local minimum; this type of point is found, for example, in the center of a cage; 9 three negative eigenvalues ((3,-3) critical point)--the electron density at that point is a local maximummand they are usually found at the positions of the nuclei; 9 two positive eigenvalues ((3, + 1) critical point). The density is a minimum in the plane containing the two positive curvatures and maximum in the perpendicular direction--this type of critical point is found at the center of a ring

Electrostatic Properties of Crystals from X-rays

295

formed by n atoms covalently bonded (a hydrogen bond may replace a covalent bond); 9 two negatives curvatures ((3,-1) critical point)rathe density is maximum in the plane containing the two negative curvatures and minimum in the perpendicular plane; these points are found in covalent bonds, when associated with a high value of p. The laplacian of the electron density is also related to the total energy by the virial theorem. The sign of the laplacian determines which of the kinetic energy or potential energy values is in excess in the total energy: in regions of space where the laplacian is negative and electronic charge is concentrated, potential energy dominates due to covalent bonds and lone pairs; in the region where the laplacian is positive, total energy is dominated by the kinetic energy (hydrogen bonds, ionic bonds...) and there is a local electron density depletion. (For further information, see reference 57.) Then, analyzing the electron density topology requires the calculation ofV 9 and of the hessian matrix. After diagonalization one can find the critical points; in a covalent bond characterized by a (3, -1) critical point, the positive curvature ~3 is associated with the direction joining the two atoms covalently bonded, and the )~2, )h curvatures characterize the ellipticity of the bond by:

E -"

~1 -- ~2 ~2

(28)

For example, e would increase with the x character of the bonds. This review only focuses on experimental results. The experimental topological analysis is made for molecules removed from the crystal lattice in the same way than for the electrostatic potential calculations (see above). The crystal field effects are therefore not absent. Today, very few topological analyses of X-X experimental densities have been performed. Due to the finite resolution of the experiments they require a combination of experimental results for the valence electron distribution--more diffuse in real space, i.e., more contracted in reciprocal spacemwith theoretical core electron density usually calculated from good quality atomic wave functions. As an example, De Titta and N. Li [59] collected high resolution, very high quality X-ray data at 100 K on two forms of glycouryl Cmcm and Pnma and on biotin and chainless biotin (Figure 23). Glycouryl is a bicyclic, cis-fused ring compound, each ring of which resembles chemically the ureido ring of biotin. Souhassou [60] has performed a topological analysis of the multipolar electron density resulting from a Hansen-Coppens refinement against De Titta data. Figure 23 shows the negative Laplacian maps of the electron density (-V 2p) in the ureido plane of the four glycouryl and biotin molecules, and Table 5 gives the properties of the electron density at the critical points. It is very interesting to see

296

CLAUDE LECOMTE HI

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that a good transferability between the experimental properties of the fragments exists, confirming Bader's work on theoretical densities of peptide bonds. The same type of calculations have been performed using experimental X-ray structure factors on: crystalline phosphoric acid, N-acetyl-c~,[3-dehydrophenylalamine methylamide, and N-acetyl- 1-tryptophan methylamide by Souhassou [60]; on urea, 9-methyladenosine, and imidazole by Stewart [32]; and on 1-alanine [61] and annulene derivatives [62] by Destro and co-workers. The latter authors collected their X-ray data at 16 K [63]. Stewart [32] showed that the positions of the (3, -1) critical points from the promolecule are very close to those of the multipole electron density, but that large differences appear in comparing the density, the Laplacian maps, and the ellipticities at the critical points. Destro et al. [61] showed that the results obtained may be slightly dependent on the refinement model. The analysis of the gradient vector field of the charge density displays the trajectories traced out by V 9 (gradient path). Because P is a local maximum at nuclear position ((3, -3) critical point), all the gradient paths at a proximity of a

Table 5. Characteristics of Electron Density at the Critical Points of the Ureido

Ring in Glycouryl and Biotin [60]

Atom1

C2 C2' C2 C2 C2' C2' C2' C2' C2 C2 C2 C3 C4 C3 C4 C3a C6a C6a C3 C3 C3a C6a

Atom2

dl (,3,)

d2(A)

AeA-5

02 02' 02 02 N3' NI' N3' NI' N3 N1 N1 N3' NI' N3' NI' N3 N1 N1 C4 C4 C6a C6a

0.545 0.540 0.525 0.523 0.581 0.600 0.586 0.602 0.578 0.591 0.584 0.660 0.671 0.641 0.653 0.624 0.628 0.634 0.781 0.776 0.783 0.783

0.699 0.711 0.723 0.723 0.771 0.759 0.765 0.750 0.774 0.760 0.770 0.792 0.787 0.815 0.803 0.821 0.821 0.811 0.788 0.786 0.786 0.783

-24.0687 -23.1579 -25.8573 -26.1926 -20.0554 -18.4142 -19.5166 -18.1998 -20.0305 -19.5934 -19.3876 -11.6718 -10.7932 -9.3812 -11.4275 -11.8925 -12.1237 -12.4460 -12.5621 -11.5225 -12.1513 -10.8470

2,8335 2.8191 2.8394 2.8346 2.3277 2.2751 2.3485 2.3319 2.3028 2.2956 2.2744 1.8478 1.8382 1.8014 1.8875 1.8480 1.8554 1.8665 1.6252 1.6360 1.6018 1.5916

0.0739 0.1739 0.1021 0.1126 0.2341 0.2802 0.2330 0.2370 0.2191 0.2198 0.2300 0.0286 0.0238 0.0674 0.0607 0.0439 0.0507 0.0272 0.0055 0.0069 0.0306 0.1147

C. biotin* Biotin GIy~B Gly_A C. biotin* C. biotin* Biotin Biotin GIy~B GIy~B Gly_A C. biotin* C. biotin* Biotin Biotin Gly_B Gly_B Gly_A C. biotin* Biotin Gly_B Gly_A

0.652 0.653 1.083 0.780 0.671 0.615 0.617 0.617

1.239 1.206 1.634 1.324 1.217 1.212 1.233 1.233

2.4116 2.5109 0.7550 1.5368 2.2234 2.3303 2.6322 2.5293

0.1882 0.1853 0.0674 0.1079 0.1754 0.1953 0.1751 0.1852

0.1147 0.1399 0.2765 0.0602 0.0380 0.0175 0.0109 0.1284

C. biotin* C. biotin* Biotin Biotin Biotin Gly_B Gly_B GIy~A

1.187 1.218 1.187 1.223 1.186 1.214

1.191 1.249 1.189 1.244 1.186 1.254

6.5646

0.3562

Biotin

6.6912

0.3268

GIy--B

6.6761

0.3347

Gly--A

Rho -e ~-3

~

Molecule

Hydrogen bonds H3' HI' *H8B HI' H3' H3 H1 H1

02' 02' S O10B O10B 02 02 02

Ring critical point N3' & C2' N3 & C2 N1 & C2

NI' C3 N1 C3a N1 C6a

Note: *C. biotin: chaineless biotin.

297

298

CLAUDE LECOMTE

....

l i

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Figure 24. Electron density gradient paths in the ureido plane in Pnma glycouryl (a) and Cmcm glycoun/I (b). given nucleus terminate at the nucleus which behaves as an attractor; the shape of the atom is then defined by the basin of the attractor as shown on Figure 24 which gives the trajectories of the gradient vectors of the experimental charge density in the ureido plane of the two forms of glycouryl [60]. Around the C 2 atom, one can see that all the gradient paths which terminate at C 2 define the atomic basin and that the corresponding atomic surface is not crossed by any trajectory of Vp. This surface is referred as zero flux surface which normal vector n satisfies the boundary condition: Vg(r).n(r ) = 0 for all points of the surface

(29)

Then, if one knows how to integrate the charge density into the atomic basin, one can get charges and multipolar moments of the distribution. We are only at the beginning of these promising topological studies of the experimental densities and it requires further experimental work as well as more comparison between experiment and theory.

IV. CONCLUSION AND FUTURE Electron density distributions in molecular compounds by X-ray diffraction methods can be obtained in organic molecules or first transition metal coordination compounds as big as 150 non-hydrogen atoms with an excellent accuracy. The size of these molecules exceeds the possibilities of ab initio computational chemistry due to the dependence of the SCF algorithm on the fourth power of the contracted gaussian basis set and to the need for large polarized basis sets to reach near quantitative agreement with X-X experiments. A theoretical promising approach developed by B6nard's group [28,64] is the calculation on molecular fragments because 9(r) is a local property.

Electrostatic Properties of Crystals from X-rays

299

The multipolar model succeeds very well in deconvoluting thermal motion from nonsphericity of the electron density and gives an analytical representation of the charge density. This analytical representation enables us to calculate electrostatic properties of the molecules in or removed from the crystal lattice, like dipole moments, electric field, electrostatic potential, and field gradient. These quantities, usually obtained from approximate theoretical methods, are fundamental in understanding intermolecular interactions, or molecular reactivity fits of experimental potentials by point charges at the nuclei by dipoles at the hydrogen atom sites, should provide the chemists and physicists with a data bank of experimental electrostatic parameters which will be used in modeling. Building a data bank requires a lot of electron density measurements which may be performed on synchrotron machines; it is then necessary to improve the accuracy of these data (e.g., stability of the beam, monitoring, image plates). It will be realized in the very near future. Another promising application of charge density studies is the topological analysis of 9(r), possibly also of V(r). Very interesting and very new results also come from Coppens et al. [65] who were able to determine the crystal structure at 138 K of sodium nitroprusside [Fe(CN)sNO] 2- ion in an electronic extremely long-lived excited state produced by long term Ar + laser illumination (X = 485 nm) of a single crystal, with K / / a and polarization//c. A metastable population of about one-third excited molecules was obtained. Their analysis shows a 0.060(9) ,~ lengthening of the Fe-N bond and C - F e - N and C - F e - C angles variation. Furthermore, they were also able to analyze their data with a multipolar model and to determine the deformation density of the excited state of the molecule [66], knowing the ground state from another X-ray experiment. These types of experiments which, performed in the future using synchrotron radiation, will certainly enable us to get informations on second-order properties of matter like nonlinear susceptibilities.

ACKNOWLEDGMENTS I would like to thank Professors E Coppens and R.H. Blessing for numerous helpful and stimulating discussions. I am very grateful to Mrs. Anne Gulia and to Mr. Christian Bourdon for technical help in the preparation of this chapter. Much of the work described here was realized with my co-workers Drs. N.E. Ghermani, V. Pichon-Pesme, M. Souhassou, N. Bouhmaida and E. Espinosa who I thank very much. And I thank the University Henri Poincar6-Nancy I and the S.P.M. CNRS Department for support.

REFERENCES 1. Lehman, M.S. In Electron and Magnetization Densities in Molecules and Crystals; Becker, P.J., Ed.; NATO Advanced Studies hTstitute; Plenum: New York, 1980, B48, pp. 287-322 and pp. 355-372.

300

CLAUDE LECOMTE

2. (a) Seiler, P. Static and Dynamic Implications of Precise Structural Information; In Accurate Molecular Structures: Their Determination and Importance; Domenicano, A.; Hargittai, I., Eds.; Oxford University Press: Oxford, 1992, pp. 170-198. 3. Lectures Notes, Tutorial on Accurate Single Crystal Diffractometry; Blessing, R.H., Ed.; Am. Cryst. Assoc. Meeting, New Orleans, LA, Dayton, Ohio, Polycrystal Book Service, 1990. 4. (a) Larsen, EK. In The Application of Charge Density Research to Chemistry and Drug Design; Jeffrey, G.A.; Piniella, J.E, Eds.; NATO Advanced Studies Institute; Plenum: New York, 1991, B250, pp. 187-208. (b) Blessing, R.H.; Lecomte, C. In The Application of Charge Density Research to Chemistry and Drug Design; Jeffrey, G.A.; Piniella, J.E, Eds.; NATO Advanced Studies Institute, Plenum: New York, 1991, B250, pp. 155-185. 5. Blessing, R.H. Cryst. Rev. 1986, 1, pp. 3-58. 6. Dunitz, J.D.; Seiler, P. J. Am. Chem. Soc. 1983, 105, 7056-7058. 7. Coppens, P.; Lehmann, M.S.Acta Cryst. 1976, B32, 1777-1785. 8. Coppens, P. In Electron Distributions and the Chemical Bond; Coppens, P.; Hall, M.B., Eds.; Plenum: New York, 1988, pp. 61-92. 9. (a) Stewart, R.E J. Chem. Phys. 1969, 51, 4569-4577. (b) Stewart, R.E J. Chem. Phys. 1973, 58, 1668-1676; (c)Stewart, R.E Acta Cryst. 1976, A32, 565-574. 10. (a) Hirshfeld, EL. Acta Cryst. 1971, B27, 769-781. (b) Hirshfeld, EL. Isr. J. Chem. 1977, 16, 198-201. 11. (a) Coppens, P.; Guru Row, T.N.; Leung, P.; Stevens, E.D.; Becker, P.; Yang, Y.W. Acta Cryst. 1979, A35, 63-72. (b) Hansen, N.K.; Coppens, P. Acta Co'st. 1978, A34, 909-921. 12. (a) Craven, B.M.; Weber, H.P.; He, X. Tech. Report TR 87-2; Department of Crystallography, University of Pittsburgh, 1987. (b) Epstein, J.; Ruble, J.R.; Craven, B.M. Acta Cryst. 1982, B38, 140-149. 13. Lecomte, C. In The Application of Charge Density Research to Chemistry and Drug Design; Jeffrey, G.A.; Piniella, J.E, Eds.; NATO Advance Studies Institute, 1991, B250, pp. 121-153. 14. Brown, A.S.; Spackman, M.A. Acta Cryst. 1991, A47, 21-29. 15. Craven, B.M.; Stewart, R.E In Studies of Electron Distributions in Molecules and Crystals; Blessing, R.H., Ed.; Trans. Am. Cryst. Ass., 1990, Vol. 2b, pp. 41-54. 16. Souhassou, M.; Lecomte, C.; Blessing, R.H.; Aubry, A.; Rohmer, M.M.; Wiest, R.; B6nard, M. Acta Cryst. 1991, B47, 253-266. 17. Moss, G.R.; Blessing, R.H. Acta Cryst. 1984, A40, C-157 XIII IUCR Congress, Hamburg, Germany. 18. Cole, EE., Ph. D Thesis, 1966, University of Washington, Pullman Washington, Ann Harbor, Michigan: University microfilms international. 19. Moss, G.R.; Souhassou, M.; Espinosa, E.; Lecomte, C.; Blessing, R.H. Acta Cryst. B 1995 (in press). 20. Souhassou, M.; Espinosa, E.; Blessing, R.H.; Lecomte, C. Acta Cryst. B 1995 (in press). 21. (a) Rees, B.Acta Cryst. 1976,A32, 483-488. (b) Rees, B.Acta Cryst. 1978, A34, 254-256. 22. Souhassou, M.; Lecomte, C.; Ghermani, N.E.; Rohmer, M.M.; Wiest, R.; B6nard, M.; Blessing, R.H.J. Am. Chem. Soc. 1992, 114, 2371-2382. 23. (a) Ernenwein, R.; Rohmer, M.M.; B6nard, M. Comput. Phys. Comm. 1990, 58, 305. (b) Rohmer, M.M.; Ernenwein, R.; Ulmschneider, M.; Wiest, R.; B6nard, M. h~t. J. Quantum. Chem. 1991, 40, 723-744. 24. Huzinaga, S., Technical Report, University of Alberta, Edmonton, 1971. 25. (a) Stevens, E.D.; Rys, J.; Coppens, P. J. Am. Chem. Soc. 1977, 99, 265-272. (b) Eisenstein, M. Acta Cryst. 1988, B44, 412-426. 26. Stevens, E.D.; Klein, C.L. In The Application of Charge Density Research to Chemistry and Drug Design; Jeffrey, G.H.; Piniella, J.E, Eds.; NATO Advanced Studies Institute, 1991, Vol. B250, pp. 319-336.

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27. Kampermann, S.E; Arora, S.K.; Ruble, J.R.; Craven, B.M. Amer. Cryst. Ass. Meeting, Albuquerque, NM, PG06, 1993, p. 106. 28. Pichon-Pesme, V.; Lecomte, C.; Wiest, R.; B6nard, M. J. Am. Chem. Soc. 1992, 114, 2713-2715. 29. Pichon-Pesme, V.; Lecomte, C.; Lachekar, H. J. Phys. Chem. 1995, 6242-6250. 30. (a) Benabicha, E; Habbou, A.; Lecomte, C.; Rohmer, M.M. 1995 (in preparation). (b) Benabicha, E Th~se de 3~me Cycle; Universit6 de Nancy I, France, 1986. 31. Brock, C.E; Dunitz, J.D.; Hirshfeld, EL.Acta Cryst. 1991, B47, 789-797. 32. Stewart, R.E In The Application of Charge Density Research to Chemistry and Drug Design; Jeffrey, G.; Piniella, J.E, Eds.; NATO Advanced Studies Institute, 1991, Vol. B250, pp. 63-102. 33. Rohmer, M.M. Chem. Phys. Lett. 1985, 116, 44-49. 34. Lecomte, C.; Blessing, R.H.; Coppens, P.; Tabard, A. J. Am. Chem. Soc. 1986, 108, 6942-6950. 35. Stevens, E.D. In Electron and Magnetization Densities in Molecules and Crystals; NATO Advanced Studies hlstitute; Plenum: New York, 1990, Vol. B48, pp. 823-826. 36. Holladay, A.; Leung, EC.; Coppens, P. Acta Cryst. 1983, A39, 377-387. 37. Coppens, P. In Studies of Electron Distributions in Molecules and Crystals; Blessing, R.H., Ed.; Trans. Amer. Cryst. Ass., 1990, Vol. 26, pp. 91-105 and references therein. 38. Leung, EC.; Coppens, P. Acta Cryst. 1983, B39, 535-542. 39. Spasojevic de Bir6; Dao, N.Q.; Becker, P.; B6nard, M.; Strich, A.; Thieffry, C.; Hansen, N.K.; Lecomte, C. In The Application of Charge Density Research to Chemistry and Drug Design; Jeffrey, G.; Piniella, J.E, Eds.; NATO Advanced Studies Institute; 1991, Vol. B250, pp. 385-399. 40. (a) Stewart, R.E Chem. Phys. Lett. 1979, 65, 335-338. (b) Stewart, R.E God. Jugosl. Cent. Kristallogr. 1982, 17, 1. 41. Ghermani, N.; Lecomte, C.; Bouhmaida, N. Z. Naturforsch. 1993, 48a, 91-98. 42. Spackman, M.A.; Weber, H.P.; J. Phys. Chem. 1988, 92, 794-796. 43. (a) He, X.M., Ph.D. Thesis, 1984, University of Pittsburgh. (b) Stewart, R.E; Craven, B.M. Biophys. J. 1993, 000. 44. Su, Z.; Coppens, P. Acta Cryst. 1992, A48, 188-197. 45. Swaninathan, S.; Craven, B.M. Acta Cryst. 1984, B40, 511-518. 46. Weber, H.P.; Craven, B.M. Acta Cryst. 1987, B43, 202-209. 47. Kloosten, W.T.; Swaminathan, S.; Naumi, R.; Craven, B.M. Acta Cryst. 1992, B48, 217-227. 48. Weber, H.P.; Craven, B.M. Acta Cryst. 1990, B46, 532-538. 49. Spackman, M.A.; Weber, H.P.; Craven, B.M.J. Am. Chem. Soc. 1988, 110, 775-782. 50. Moss, G.; Feil, D.Acta Cryst. 1981, A37, 414-421. 51. (a) Swaminathan, S.; Craven, B.M.; Spackman, M.A.; Stewart, R.E Acta Cryst. 1984, B40, 398. (b) Epstein, J.; Ruble, J.R.; Craven, B.M. Acta Cryst. 1982, B38, 140. 52. (a) Lecomte, C.; Souhassou, M.; Ghermani, N.; Pichon-Pesme, V.; Bouhmaida, N. In Studies of Electron Distributions in Molecules and Crystals; Blessing, R.H., Ed.; Trans. Amer. Cryst. Ass., 1990, Vol. 26, pp. 91-103. (b) Lecomte, C.; Ghermani, N.; Pichon-Pesme, V.; Souhassou, M. J. Mol. Struct. (Theochem) 1992, 255, 241-260. 53. Bouhmaida, N.; Th~se de l'Universit6 de Nancy I, 1993, France. 54. Bouhmaida, N.; Ghermani, N.E.; Lecomte, C. Amer. Cryst. Ass. Meeting; Albuquerque, NM PG 02, 1993, p. 105. 55. Studies of Electron Distributions in Molecules and Crystals; Blessing, R., Ed.; Trans. Amer. Cryst. Ass., 1990, Vol. 26, pp. 23-79. 56. Ghermani, N.E.; Bouhmaida, N.; Lecomte, C. Acta Cryst. 1993, A49, 781-789. 57. (a) Bader, R.EW. Atoms in Molecules. A Quantum Theory; Oxford University Press, Oxford, 1990. (b) Bader, R.EW.; Laidig, K.E. 1990, (in ref [55], pp. 1-21). (c) Bader, R.EW.; Essen, H. J. Chem. Phys. 1984, 80, 1943. 58. Hirshfeld, EL. Theor. Chim. Acta 1977, 44, 129. 59. (a) De Titta, G.D.; Li, N. 1993 (personal communication). (b) Li, N.; De Titta, G.D.; Blessing, R.H.; Moss, G. 40th A.C.A. Meeting, New Orleans, Abst. PD05, 1990, p. 79.

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60. (a) Souhassou, M., 1993 (personal communication). (b) Souhassou, M.; Blessing, R.H. Sagamore Conf. Proc., Konstanz, Germany, 1991. 61. (a) Destro, R.; Bianchi, R.; Gatti, C.; Merati, F. Chem. Phys. Lett. 1991,186, 47-52. (b) Gatti, C.; Bianchi, R.; Destro, R.; Merati, F. J. MoL Struc. (Theochem) 1992, 255, 409-433. 62. Bianchi, R.; Destro, R.; Merati, F. (in Ref. [4], p. 340). 63. Destro, R.; Marsh, R.E.; Bianchi, R. J. Phys. Chem. 1988, 92, 966-974. 64. Wiest, R.; Pichon-Pesme, V.; B6nard, M.; Lecomte, C. J. Phys. Chem. 1993 (submitted). 65. Pressprich, M.R.; White, M.A.; Coppens, P. J. Am. Chem. Soc., 1993, 115, 6444-6445. 66. Coppens, P. 1993 (private communication).

ORDER IN SPACE: PACKING OF ATOMS AND MOLECULES

Laura E. Depero

I. II.

III. IV.

V.

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Geometrical Model of Packing . . . . . . . . . . . . . . . . . . . . . . . A. The Principle of Close Packing . . . . . . . . . . . . . . . . . . . . . . . B. The Packing Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . The A t o m - A t o m Potential Method . . . . . . . . . . . . . . . . . . . . . . . Factors Influencing the Packing . . . . . . . . . . . . . . . . . . . . . . . . . A. Molecular Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Hydrogen Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Packing and Lone Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . Packing and Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Symmetry of the Structural Unit and Crystal Symmetry . . . . . . . . . B. Space Group Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Dipole Moments and Symmetry . . . . . . . . . . . . . . . . . . . . . . D. Packing Symmetry of Inorganic Compounds . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Advances in Molecular Structure Research Volume 1, pages 303-337. Copyright 9 1995 by JAI Press Inc. All rights of reproduction in any form reserved. ISBN: 1-55938-799-8 303

304 304 306 306 309 311 316 316 319 322 324 326 326 328 331 333 334 334

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ABSTRACT In this chapter the geometrical model of packing, with the underlying "principle of close packing" and concept of packing coefficient is first reviewed and discussed, while the necessary energy calculation schemes are provided by the atom-atom potential method with its predictive capabilities. In fact, any empirical potential will predict which structures are impossible, but no potential can consistently predict which structure is the most stable. Four factors that influence packing are taken into consideration: molecular shape, hydrogen bonding, lone pairs, and entropy, all contributing to the minimization of the free energy. In particular, the role of entropy, which is not easily recognized in crystals, can be critical in some packing conditions as in the formation of a liquid crystalline phase, when an increasing concentration of rigid rods is packed into a given volume of space. Finally, as an important consequence of molecular packing, symmetry will be discussed by using Venn diagrams to illustrate the distribution of frequencies of the assigned space groups.

i. I N T R O D U C T I O N In crystal chemistry it is important to derive packing models for interpreting experimental data and for postulating new possible structures [1-4]. The prediction of new materials and their properties, useful for particular applications, can lead to the planning of new syntheses. Moreover, the differences between the model and the experimentally determined structure can show the limitations of the theory used to build the model, while the interpretation of these differences gives a better understanding of the chemical and physical properties of the material studied. The equilibrium configuration for an ensemble of atoms or molecules corresponds to the minimum of the free energy: F=U-TS

where U is the potential energy of interaction, T is the absolute temperature, and S is the entropy. For molecular crystals it is natural to use a molecule as the structural unity, while the energy required for breaking the intermolecular bonds should be used for the lattice energy. Thus, the crystal structure is principally determined by the energy of interaction U among molecules. This is summarized by the principle of maximum occupation of the crystal cell volume by atoms or molecules, i.e., by consideration of optimum packing. The entropy contribution to the free energy is important at relatively high temperature only when very weak interactions exist between molecules, as in the case of liquid crystals (see Section IV.C) The simplest approach for studying the packing is geometrical (Figure 1). In the corresponding model the molecular energy is considered to be in a deep minimum and therefore no changes in the molecular conformation are considered. The shape of a single molecule is given by the geometrical characteristics of the relevant covalent bonds and by the resultant of the sum of rigid van der Waals spheres, while

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Figure 1. Flow-chart of different models and calculation schemes of molecular packing. The various approximations are indicated. the interaction between adjacent molecules does not depend on the atom types. This very qualitative approach can rationalize the structure of a large number of molecular crystals [5]. An approximation to the packing energy can be given by the sum of terms due to all the interactions among atoms belonging to adjacent molecules. This approach is the so-called atom-atom potential method [6] where pairwise potentials are empirically determined and the molecular conformation is kept fixed. This method enables is to estimate the packing energy and to justify crystal properties, such as the enthalpy of formation. The calculations become much more complicated if the assumption of a rigid molecule is dropped. In global force-fields, intra- and intermolecular interactions are included, using a superposition of simple potentials. The basic idea is the same

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as that implied by the atom-atom potential method, but, in this case, all the atoms are considered in the calculations. In the next step toward a better approximation, the SchrOdinger equation has to be applied, taking into account all the atoms in the structure. If one describes the atomic inner electrons by empirical potentials, the calculations can be simplified. Otherwise, real ab initio calculations have to be performed, where all the electrons are considered in the Schr6dinger equation. In principle, ab initio calculations could always be applied, thus giving the correct model for any structure. Even if this were feasible (and many practical problems would have to be overcome first), the real mechanism of packing might be lost, hidden inside the complexity of the calculations. Indeed, it is very important to understand the factors that control the structure, and this can be done only by introducing appropriate approximations, as it is done, for example, in the geometrical or in the atom-atom potential method. In the following two sections (II and III) the geometrical model and the atomatom potential method will be discussed with some of their applications. In the discussion (Section IV) of the factors influencing the packing, molecular shape, hydrogen bonds, lone pairs, and entropy will be considered. As an important consequence of molecular packing, symmetry (Section V) will be discussed by using Venn diagrams to illustrate the distribution of frequencies of the assigned space groups. Very different fields will be shortly reviewed in this chapter and being exhaustive was not in this author's intentions. What follows should be understood only as an attempt to find fundamental similarities between organic and inorganic materials. Indeed, the possibility exists of trying a unitary description of all material structures and the present chapter should also be considered as an effort in this direction.

il. THE GEOMETRICAL M O D E L OF PACKING A. The Principle of Close Packing Kitaigorodsky is a pioneer in the modeling of crystal structures [5, 7, 8]. He has been dealing mainly with organic compounds, but the principles developed by him can be applied more generally. The active interest in the structures and properties of organic crystals is based on the possibility of correlating the arrangement of molecules in a crystal with its properties. Indeed, studies on the forces governing the crystal structure (and hence the crystal properties) can be utilized in other fields of chemical research, like protein folding, drug design, or site-receptor interactions. The first model proposed by Kitaigorodsky was simply geometrical: if molecules are kept together by bonds that are not directional, a crystal can be represented as a close packing of molecules. Molecules pack in such a way that they have the maximum number of lowest energy interactions. If they are assumed to be impenetrable objects, all the intermolecular contacts will be equal to or greater than the sum of the atomic radii (see

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307

Figure 2). In this sense, the mutual orientation of molecules is conditioned by the short distances between atoms of adjacent molecules. Since the sums of the mean intermolecular radii differ somewhat from the actual distances, in the geometrical model some atoms overlap while others do not touch each other. It is always possible to generate an ideal packing (where no molecule is suspended in empty space and none overlaps with others) by translational and rotational shifts of the molecules [5]. From this ideal packing, derived from strictly geometrical considerations, a structural model can be obtained and employed in the calculations in order to find the best fitting of the experimental data. In this approach the mutual arrangement of the molecules in a crystal is always such that the "projections" of one molecule fit into the "hollows" of adjacent molecules. Real structures are among the most closely packed of all those conceivable. Close packing can exist if the molecular coordination number is sufficiently high (usually 12). The real number depends on the specific shape of the molecule. By an array of hexagonal close packed spheres stretching along an arbitrary direction, i.e., transforming them into triaxial ellipsoids, it is possible to obtain a model of packing of arbitrarily shaped bodies in its first approximation. Clearly, in this case too, "layers" of molecules, similar to those derived from the packing of spheres, can be identified; the coordination number of each molecule will be 12, with a distribution of neighbors identical to that existing in the packing of spherical molecules. The geometrical model is fundamental to our understanding of the packing of molecular crystals, as it is the close packing model in metals [9,10] and in ionic structures [11]. Recently, Frank [12] reviewed the definition of this concept for nonrigid spheres in a metal. He shows that the best "occupancy" of space is given, in this case, by body centered cubic (bcc) structures relative to the cubic close packed (ccp) structures. In fact, at equal density, in bcc structures the nuclei are closer to each other, thus better filling the available space. This can be the reason why many metals attain and retain the bcc structure at very high pressure. Here emphasis is given to distances, neglecting the differences in the coordination number (CN) which is a factor that must be taken into consideration when dealing with stability. It is feasible that this observation could be extended to organic molecular structures, when atoms are not considered uncompressible. It is possible that a structure that cannot be described in the "classical" close-packing model is, in fact, close packed in this new approach. Both coordination number and distances have been considered by Pearson [13] in the study of the cohesive energies of simple AB ionic and covalent solids. For a given set of atoms the interatomic distances for CN = 6, typical of ionic compounds, are larger by about 5% than those found forCN = 4, typical ofcovalent compounds. If only small repulsions among ligands are active, it is advantageous to decrease the distance, to the detriment of the coordination number. In the case of large repulsions, the increase in the distance is partially compensated by the increase of the coordination number. This fact can be understood by considering that the

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principal contribution to the free energy of a crystal is made by the potential energy U, i.e., the packing energy. The greater the number of atoms approaching the equilibrium interatomic distances the greater is U. This can be expressed as the geometric principle of the maximum filling of space. An interesting case arises when comparing the crystal structure of CO 2 (a molecular crystal based on the packing of linear O--C--O molecules) with that of SiO 2 (a tridimensional covalent "crystal molecule"). The well-established generalized valence model, described by Brown [14], is not useful here because in this theory it is necessary to know the CN of the atom in order to make any prediction about its coordination geometry. The small C radius allows the formation of short and strong C-O bonds, based on both ~ and rt interactions between orbitals of proper symmetry. The result is the CO 2 molecule. The more expanded valence orbitals of the Si atom generate comparatively long and weak Si-O bonds, a result balanced by the formation of four such bonds. Molecules with carbon atoms bonded to three oxygens are well known [15], but no compound has ever been found with a carbon atom bonded to four oxygen atoms, even if, for carbon, the formation of four bonds is common. In the author's opinion, this fact can be understood as a steric hindrance effect about the C atom, which would generate long and very weak C-O bonds. In SiC each C atom is surrounded tetrahedrally by four Si atoms and this compound is known for its strong bonds. Isolated Si atoms are larger than isolated O atoms, which might suggest even more crowding around C and even weaker bonds. However, Si bonded to C must, in fact, have a smaller radius than O bonded to C because of the relative electronegativities involved. An important application of the geometrical model can be found in the study of homologous series, as for example the naphthalene-anthracene-naphthacene series [5] and p-substituted benzoic acids [16]. In each series one of the cross sections of the unit cell remains approximately constant and one of the parameters increases by a value equal to the "elongation" of the molecule. In general, analogous processes can be applied to polymers when only small chemical differences in the monomers exist. Indeed, on this basis a method of structural analysis was proposed [17,18] which can be applied when differences are systematically introduced into the monomers, thus "designing" a family of polymers structurally related to each other. Similar principles were also developed for the structural analysis of layered crystalline silver thiolate compounds of the type AgSR (R = alkyl or aryl group)

[19]. Packing criteria may be taken into consideration in other cases, as, for example, in that of intercalated compounds. Here the geometry of the repeat mesh of the inorganic layers can be considered constant with the aim of developing reasonable models for their structures [20-22]. The geometrical method, in spite of its inherent low accuracy derived from the drastic approximations introduced, is successful because the equilibrium distance between two atoms is principally fixed by the repulsion term, constituted by a very steep potential "wall". However, the value of the interaction energy EA_B mainly

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EA.B t

k:i ~! /!

R

/

/

..... .....

geometrical approach

..._...__ a t o m - a t o m E3

I i i

potentials

u_

Figure 2. Plot of the interaction energy EA-B VS. the distance R between two atoms A and B in the geometrical model (dashed lines) and in the atom-atom potential method. The stable structure, corresponding to the minimum energy, is given by the sum of all possible interactions.

depends on the attractive terms (Figure 2). Among all the close packings, the stable structure is that corresponding to the minimum energy, as given in the first approximation by the sum of all possible atom-atom interactions.

B. The Packing Coefficient The packing coefficient, k, is defined as the ratio of the sum of the volumes occupied by the molecules in the cell to the volume of the cell. The close-packing principle in the geometrical model is represented by the maximum value of the packing coefficient. In structures built according to the method of the close packing of identical hard spheres, k is approximately 0.74. If we assign a volume Vi to multiatomic structural groups or molecules, then the maximum filling principle will be expressed by the maximum value of the packing coefficient [8]. In general, for calculating the packing coefficient, it is necessary to know the volume of the volume of the molecule, and, therefore, the bond lengths, valence angles and intermolecular distances. Different algorithms, based on knowledge of the coordinates of the constituent atoms and of the atomic radii [23-26], have been suggested to perform this calculation. The simplest and quickest computational method was first suggested by Kitaigorodsky [5]. This method, commonly known as the "cap and spheres method", considers each atom in a molecule as a sphere with a cap cut off for each atom to which it is bound. This method has severe limitations in sterically hindered molecules, where overlapping of the van der Waals surface occurs between atoms in the complex ion which are not bound to each other, and when two caps of the atom intersect. Depending on geometry, the spheres defined by the atomic radii may overlap, making volume calculations cumbersome.

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A simple approach, proposed by Gavezzotti [26], consists of sampling on envelope space, containing the molecule with a large number of probe points, and counting the number of points inside at least one of the spheres. Packing coefficients for the majority of organic crystals are between 0.65 and 0.77. Desiraju stated [27] that if an organic molecule is designed with a shape so awkward that no packing is possible with a coefficient above 0.6, a lowering of temperature results in vitrification rather than crystallization. In open cage-like structures, the low values of the packing coefficients are avoided by the inclusion of the solvent or other guest species. Experimental studies of benzene, naphthalene, and anthracene have shown that, when these substances are in the solid state, their packing coefficient is larger than 0.68. During transformation from a solid phase to a liquid one, k drops to 0.58. A further increase of the temperature up to the boiling point causes a decrease of k down to 0.5. At k < 0.5 these substances become gaseous. Thus, soundly, compressibility was shown to be inversely related to the packing coefficient [5,7]. The calculation of the volume of molecules is fundamental not only for the study of packing, but also in the computer simulation of the dynamics and fundamental physical properties of macromolecules such as proteins and nucleic acids [23]. Likewise, the determination of the void space within any microporous material is of interest in sorption applications [28]. An interesting correlation was found between the crystal density and the packing coefficient was found by Kuzmina [29]. In this study, the structures of 159 molecular compounds, consisting of atoms C, N, O, H, F, C1, and S, and with crystal densities larger than 1.70, were chosen from the Cambridge Database. The mean value of k decreases from 0.672 for crystals with 170g/cm 3 < 9 < 1.80g/cm3 to 0.666 in the region of density 1.90g/cm 3 < p < 2.05g/cm 3. This correlation can be rationalized if one considers that the high density is due to strong (and thus short) bonds, as, for example, covalent and H-bonds that can slightly reduce the packing efficiency. In this connection, the results obtained by Roman et al. [30], using the Cambridge Structural Database as a source of data on organic molecules, are also relevant. Indeed, from their analysis, it is clear that the inter- and intramolecular interactions such as H-bonds and rc-x interactions produce a lower average volume per atom and thus a high density. In protein crystals, due to the large size of the molecule, the empty space can have cross sections of 10-15 ,~ or greater. The empty space between the protein molecules is occupied by mother liquor. This property of protein crystals, shared by nucleic acids and viruses, is otherwise unique among the crystal structures. In fact, the values of the packing coefficient of protein crystals range from 0.7 to 0.2, but the solvent molecules occupy the empty space so that the total packing coefficient is close to 1 [31]. Nevertheless, a detailed theoretical study has been carried out to examine the models of DNA-DNA molecular interactions on the basis of hard-sphere contact criteria. The hard-sphere computations are insufficient for qualitative interpretation of the packing of DNA helices in the solid state, but

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these results are a useful starting point for energy-based studies as well as relevant to the analysis of long range interactions in DNA [32]. It was stated that the packing properties of inorganic compounds depend not only on shape and size, since the intermolecular forces take on a complex form, even for simple molecules. Nevertheless the molecular dimensions of an ion provide a starting point for discussing their solid state properties. For this reason, the calculation of the size and shape was performed for some inorganic molecules and ions [33]. The calculated packing coefficients for molecular salts containing spherical cations and anions lie between 0.64 and 0.76 with an average value of 0.69. These values are only slightly smaller than the packing coefficients commonly observed for organic molecules. An interesting aspect is that salts with discoidal cations and anions generally have higher packing coefficients (0.70-0.76) than those involving spherical cations and anions (0.64-0.71). This is a direct consequence of the fact that the packing coefficients for cylinders are higher than those for spheres. The relative constancy of the packing coefficients for inorganic compounds with spherical cations and t anions has several important implications. A comparison of the calculated volume of the cell with that derived from diffraction experiments should give a confirmation of the content of the unit cell and should indicate the possible presence of solvent of crystallization. In complexes of rare earths, where the covalent bond is not significant, the packing of the ligands around the central atom is an important factor for complex stability [34-36]. As for the molecule in molecular crystals, the complex is not stable below a certain degree of occupancy around the central atom. In this case it is not possible to consider a packing coefficient, but analogous parameters can be based on the calculation of the occupied area around the central atom. These parameters are very useful to evaluate the stability of complexes of rare earths, which are interesting for being precursors of catalysts [37].

Ill.

THE A T O M - A T O M POTENTIAL M E T H O D

The principle of close packing in molecular crystals naturally suggests the idea of describing the energy of molecular interactions as the sum of the interactions of the component atoms. A model has been used widely to predict crystal structures with a greater degree of accuracy and to estimate quantitatively the thermodynamic properties of a crystalline compound: the atom-atom potential method [6,27]. In the atom-atom potential model, molecules are built with slightly compressible atomic spheres connected with spring-like bonds. In molecular mechanics, very similar procedures are applied for calculating the stable arrangement of nonbonded atoms in a molecule [38]. Since the "unit" in molecular crystals is a whole single molecule, it appears appropriate to consider molecule-molecule interactions. This is done in the geometrical model, with the assumption of totally incompressible molecules (see Figure 2). But, since the forces between molecules cannot be considered central, it

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is not appropriate to perform these calculations quantitatively. Instead, the interaction between molecules can be evaluated as a sum of atom-atom potentials, where forces between atoms are supposed to depend only on the atomic species. The packing potential energy is U = - N a E, where N a is the Avogadro number and E is defined as the energy required to take any molecule out of the crystal to infinity, without rearrangement of the crystal matrix around the hole left behind. The packing energy (PE) per mole is the energy released when N a molecules from an infinite distance are brought together to form a perfectly ordered crystal: PE = 1/2 U, where the factor 1/2 avoids counting each interaction twice. A good potential should locate the minimum of the potential curve in the experimentally observed position. In this approach, the effects of the crystalline field on the shape of the molecule and on the intermolecular vibrations are disregarded. If there is no molecular association in the gas phase, PE can be compared with the experimentally determined sublimation energy (enthalpy). The major hypothesis in the atom-atom potential method is that PE may be obtained by summing in a pairwise fashion the energies for all atoms i and j belonging to adjacent molecules: PE = 1/2 ~ ~ E(Ri,kj ) i k) where E(Ri,o) is the potential of atom i belonging to the reference molecule and atom k in the j-th surrounding molecule, and Ri,kj is the distance between the two atoms (Figure 3). In order to have a finite summation in this formula, usually a maximum interaction distance of 7-10/~ is considered. The analytical expression of the atom-atom potential must have these characteristics: (1) a minimum at an intermediate value of Ri,kj, (2) high positive values for small distances, and (3) zero value when atoms are far away.

reference molecule i

O

a t o m k in the reference molecule i

@

atoms of the l molecule

.th

neighbouring

Figure 3. Packing scheme for the definition of the crystal potential in the atom-atom potential method (see text).

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The simplest interaction potential for a pair of atoms i and j separated by rij is the Buckingham potential:

E =-Ar~ 6 + Bexp(-Crij )

(1)

where A, B, and C are constants. The first term is attractive and is an approximation to the dispersion energy in that instantaneous multipoles are limited to dipoles. The second term approximates the exchange repulsion energy. The behavior of this potential at an interatomic distance lower than the equilibrium distance is physically unjustified and is due only to the choice of the mathematical expression. The constants A, B, and C depend only on the specific atoms involved in the interaction and not on their valence state, molecular environment, or other chemical characteristics. These are the so-called universal potentials. The parametrization may be performed by fitting the potentials to observed properties such as crystal structures and heats of sublimation. A theoretical approach to calculating these constants requires solution of the Schr6dinger equation. In the case of multielectron systems, this calculation leads to energy differing noticeably from the experimental data. This is the reason why a purely empirical approach to the choice of the constants of the potential curves is usually preferred, and it leads in many cases to very satisfactory results. It was found that the summation of the potentials had to be extended to large distances before acceptable agreement with experimental sublimation energies was obtained [39]. This simple model is sufficient to reproduce properties such as crystal structure, vibrational frequencies, dispersion curves, and elastic constants [40-43]. These calculations were applied also to other organized media such as monolayer gaseous films on graphite [44, 45], liquid crystals, and Langmuir Blodgett films. Other analytical expressions for the potential have been proposed [27,46]. In particular, for taking into account molecules with hetero atoms and permanent dipoles that cannot be handled well with formula (1), the proposed potential is:

E =-Ar~ 6 + Bexp(-Cr,~/) + D r~1 where A, B, C, D, are empirical parameters. Pertsin and Kitaigorodkii discussed the use of the r -1 term in ref. [6]. Since the crystal properties are governed by the intermolecular potential as a whole but not by its individual constituents, and since the atom-atom representation is not a rigorous model of the true intermolecular potential, these potential parameters, when inferred from the crystal properties, ultimately absorb all constituents of the true intermolecular potential, whatever the analytical form adopted for the atom-atom potential functions [6, 47]. Thus, there is no ground for attributing physical meaning to the individual terms constituting the expression for the intermolecular potential. In fact, the apparent improvement of the results, which can be obtained in some cases by taking more elaborate functions into consideration, is a consequence of the introduction of one or more adjustable parameters into the potential model. For example, in the case of ethylene

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it has been found that the sum of empirical atom-atom coulombic terms has nothing in common with the ab initio electrostatic energy [48]. A more general expression was proposed:

E =-ArTj" + Bexp(-Crij ) + D rij--/17 where also the parameters m and n are fitted to the experimental data. No physical meaning can be attributed to these coefficients. While the atom-atom potential model allows a quantitative evaluation of the packing energy, the geometrical model better describes the tendency for minimization displayed by the potential energy of molecular interaction. Neither model considers the entropy contributions to free energy. For this reason, the crystal structures at 0 K should be in better agreement with the model than the crystal structures at 300 K where experiments are usually made. Moreover, differences in both energy and entropy being small, many polymorphs can exist even at the same temperature and pressure. This makes difficult any structure prediction based on this kind of modeling. It is possible [27,49,50] to improve the predictive capability of the atom-atom potential method by introducing anisotropic potentials--in particular when the structure is based on intermolecular contacts to halogen atoms. In this way one takes empirically into account the orbital interactions in nonbonding molecular contacts

[51]. The atom-atom potential method has been applied recently to organometallic compounds [52,53]. The crystals of neutral first-row transition metal carbonyls fulfill the two basic requirements of the definition of a molecular crystal" the absence of net ionic charge and the presence in the structure of discrete molecular entities (which can be recognized purely on the basis of the intermolecular separations). It was found that neutral metal carbonyls constitute true molecular crystals and pack essentially in accord with the close-packing principle. Moreover, an exact knowledge of the molecular environment can be obtained from the partitioning of the PE between the molecules neighboring the one of reference. This allows a direct study of the geometric features of the CO...CO intermolecular interactions. These qualitative results are encouraging in view of future applications to more complex systems. The structure of amorphous metals, ionic solids, and molecular organic solids (and also some liquids), which are held together predominantly by nondirectional forces, can be described in terms of the canonical "dense random packing of hard spheres" model or "dense random packing of soft spheres", depending on the analytical form of the adopted potential [54]. Also the growth process leading to quasicrystals was studied using interatomic potentials [55,56]. Recently the spherepacking problem in quasicrystals was discussed from the higher dimensional viewpoint [57].

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Catlow and his co-workers [58-60] have investigated the effect of adding a specific bond-bending term to a simple pair potential: E(0) = KB (0 -

00) 2,

where 0 - 00 is the deviation from 00, a standard bond angle, and K B is the bond-bending force constant. This improved pair-potential formula was applied to the modeling of SiO 2 (in this case 00 is the tetrahedral angle). This is an important case for two reasons: first, because of the importance of the material itself, and second because it is possible to extend the model to framework-structured silicates, like feldspars and zeolites. The inclusion of the E(0) terms has a dramatic effect on the calculated properties of SiO 2, thus leading to a very good agreement with the observed crystal properties [58]. More generally, the potential energy of a model can be expressed as a sum of bonded (valence) interactions and nonbonded interactions that depends only on the distance between the atoms according to the expression: Eto t = Eva I -i- Enonbonde d

The valence interactions consist of bond stretching, bond angle bending, and dihedral angle torsion, active in nearly all force-fields for covalent systems. The nonbonded interactions consist of van der Waals, electrostatic and hydrogen bond terms, and the form of each expression depends on the particular force-field [61,62]. Any potential will predict which structures are impossible, but presumably no potential considered here will be accurate enough to consistently predict which structure is the most stable. Anyway, it is possible in some cases to construct new crystalline phases, and the empirical potential energy calculations indicate the likelihood of the existence of these theoretically predicted phases [63-67]. Gavezzotti proposed [68] to build pairs of molecules related by screw axis, glide-plane, or inversion operators as an intermediate step in building a crystal structure. The stability of such pairs has to be judged by their intermolecular potential energy. Optimized structures are always more densely packed than the observed ones. Gavezzotti's work shows that it is possible to construct, for a given molecule, a large number of crystal structures whose PE's differ by less than 10%. The method proposed by Gavezzotti gives, in some cases, a correct prediction of crystal structures. The prediction can be helped when some structural evidence, such as spectroscopic information, is available. In addition the molecular packing in the crystals of benzene, anthracene, and naphthalene were recently analyzed in terms of molecular dimer interactions while usingab initio wave functions [69]. Recently a new packing procedure for the ab initio prediction of possible molecular crystal structures was proposed, based on interatomic potentials and simulated annealing methods [70-73]. Starting from randomly generated arrangements of atoms or molecules, this algorithm progressively enforces the building rules until a minimum energy configuration is obtained. This method ignores

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symmetry constraints and considers only the shape and the size of the simulation box. When there is more than one possibility for the packing, it can be very useful to simulate the diffraction pattern and to compare it with the experimental data. Today this can be easily achieved by commercially available packages [74, 75]. The packing-density and packing-energy methods were also applied in mapping the most favorable reaction paths in organic molecules [76] and in modeling two-dimensional structures [77, 78].

IV. FACTORS INFLUENCING THE PACKING A. Molecular Shape

Single Molecules The shape of a molecule is such an important parameter that the requirements of the minimum packing energy can cause the selection of one of the possible conformations of the molecule or can deform the molecule itself in order to achieve the best packing. Bond lengths and angles cannot be deformed, but rotations about single bonds are possible for obtaining the conformation most favorable for the packing. Indeed, the need to create the maximum number of short contacts between molecules causes a compression of the space between the molecules, and so, other conditions being equal, a conformation is selected for which the packing coefficient is the highest. Moreover, according to Gavezzotti and Desiraju [79], the more symmetrical the shape of the molecule the higher the packing energy and the packing coefficient. The problem is to define in rigorous terms what it is meant by "more symmetrical shape." In order to characterize the shape, it is necessary to find some dominant relation among angles, i.e., the internal symmetry of the figure. Shapes can be compared with several geometrical known standards. In ref. [80], shape descriptors were defined and their applications to the packing problems were found to be interesting. In structural chemistry, the concept of shape is often considered in connection with size. A simple method to calculate the surface was proposed by Gavezzotti [81], who considers the volume-to-surface ratio as a parameter to describe the molecular shape. Mayer [24] proposed two parameters for describing the packing. One is the ratio of the surface area of the sphere with the same volume of the molecule (the equivalent sphere) to the surface area of the molecule. This ratio can be interpreted as a measure of the molecular "globularity". The other parametermthe ratio of the surface area to the volume--affects the steric hindrance that a molecule offers. All information on molecular structure, such as the volume and the shape of a molecule, is contained in the electronic wavefunctions [82,83]. In ref. [84], the volume of an atom in a molecule is defined as a property of the charge density, with

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a particular envelope which contains over 96% of the total electronic charge and lies within the usual range of the van der Waals contact distances. When calculating the surface area in this way, the envelope is smoother than that determined by the hard-sphere model, while the calculations are more complicated. An advantage of this method is that the atoms in the molecules are not rigidly spherical. This is important because it was shown that departures from the spherical shape may amount to about 15% of the van der Waals radius [49], and the use of the anisotropic van der Waals radii in the atom-atom potential method can be justified. An important shape effect appears when one considers the PPE as a function of the number of carbon atoms. Because of the possible isomers and different molecular shapes, the correlation between the PPE and the number of carbon atoms in the molecule is lost for the higher values of this number. In this connection the observation was made [85,86] that the shape factor has to be taken into consideration in the crystal packing of hydrocarbons. Indeed, only 19% of the structures in the database used have an odd number of carbon atoms. However one may also explain this observation in terms of the relative instability of odd AHs (alternant conjugated hydrocarbons with an odd number of carbon atoms) with respect to even AHs of comparable molecular weight and structure [87]. Indeed, all neutral odd AHs are radicals with one unpaired electron in a nonbonding molecular orbital, and this fact obviously leads to high reactivity with eventual formation of even AHs. As a consequence most of the AHs (probably numerous) in the database considered have an even number of carbon atoms, which may help explaining the observed low percentage of odd-term hydrocarbons. This is an example of the difficulty and ambiguity in interpreting results inferred by statistical analysis. The study of the crystal structures of benzene and bis(benzene) chromium reported in ref. [88] is very interesting. The space group of benzene (Pbca) is a non-isomorphic subgroup of that of the metal complex (Pa3) and an evident structural correlation can be established. This result can be attributed to the disk-like shape of the arene fragment and to its spatial requirements. Braga and Grepioni [88] conclude that these chemically different molecules, in the solid state, pack in similar manners, and this suggests that the crystal-building process (at least for small organic and organometallic molecules in the absence of strong directional intermolecular interactions) is essentially a molecular self-assembling process based on the shape of the molecules or component fragments. In the case of large molecules with apolar helices, such as peptides [89], shape selection--that is, bulges fitting into groovesmappears to be the dominating factor in determining the packing. As a consequence of the close-packing principle, chemically different molecules with similar shape and volume should have identical crystal structures [27]. Substances forming mixed crystals must be isomorphous (i.e., not only must they have an identical space group and the same number of molecules, but also a similar molecular packing), otherwise there will be a discontinuity in their solubility curve [5]. For example, benzoic acid and isopropylbenzene, in spite of the similarity in

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the shape and size of their molecules, do not form mixed crystals since their packing is, in fact, different due to the formation of benzoic acid dimers. However, Sacconi, in his early work [90] on some nickel and zinc chelates of N-methylsalicylideneaniline, showed that compounds with different crystal structures may still form mixed crystals since conformational changes in the molecular geometry of the minor component are induced by the dominating major component. The energy required for such conformational changes is presumably small by the formation of the mixed crystal and is, in any case, very small. Moreover, proteins and the molecules of their "mother liquor" also cocrystallize together in spite of the differences in shape and size. In this case the formation of the crystal is not due to substitution but to inclusion phenomena.

Chain Molecules In most chain-molecule aggregates an ordered equilibrium state is not achieved, and the degree of ordering depends on the type and time of condensation. Crystallization from solution often requires very slow cooling, and even small density differences within a solution may cause convection currents that prevent the necessary intermolecular bonds from forming. This is why attempts were made to carry out some difficult crystallizations in experiments on board the space shuttles. For molecules of an arbitrary cross section, two types of close packed layers are possible: one with an oblique cell, and the other with a rectangular cell. The analysis starts with the consideration of the close-packing of two infinite molecular chains. The analytical procedure is described in Kitaigorodsky and Vainshtein [5, 8]. Natta and Corradini [91] have stated the basic principles of the geometric organization of stereoregular polymers. 1. The axis of this macromolecule (i.e., the helix) is parallel to a crystallographic axis and all the monomer units occupy geometrically equivalent positions in relation to this axis. 2. The conformation of the polymer chain in a crystal is approximately the same as that corresponding to one of the minima of the potential energy of the isolated chain, with only small deviations possible. 3. The chains of the macromolecules are parallel and are separated from each other by distances characteristic of low molecular weight compounds, meaning that the principle of close packing applies to polymers. These general rules, that are verified for many compounds, have a very interesting exception in the structure of the y-isotactic polypropylene [92]. This structure comprises layers which are two chains wide (similar to the m-phase) but with the chain-axis directions in adjacent bilayers at an angle of 80 ~ to one another. The packing energy calculations for this structure indicate very nearly identical values for the c~ and y forms of polypropylene [93]. It is interesting that similar packing at large interaxial angles between isochiral chain fragments is also often found in

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globular proteins [94]. Similar packing of helices has been described for peptides

[89]. O'Keeffe and Andersson [95] studied the packing of rods in inorganic crystals. Among these packings, one is very similar to that found by Brtickner for ypolypropylene. This result shows that the principles discussed above are too limiting because they introduce arbitrary restraints, particularly those related to the hypothesis of infinite-chain packing. Geometrical criteria are, obviously, unifying factors that can be used to describe all possible packings.

B. Hydrogen Bonds The crystal structures of many compounds are dominated by the effect of H-bonds, as, for example, in the case of the tridimensional structure of ice, the layer structure of B(OH) 3, and the infinite zig-zag chains in crystalline HF. Many crystal structures are determined with the specific intention of studying the H-bond arrangement. Unfortunately, the H atom is a weak X-ray scatterer and it is not possible to locate it precisely by X-ray diffraction. One way to avoid this problem is the use of neutron diffraction [96]. Hydrogen bonding also leads NH4F to crystallize with a structure different from that of other ammonium (and alkali) halides: NH4C1, NH4Br, and NH4I each have a low-temperature CsCl-type structure and a high-temperature NaCl-type structure, but NH4F adopts the wurtzite (ZnS) structure in which each NH~ group is surrounded tetrahedrally by 4 F- to which it is bonded by 4 N-H..-F bonds (Figure 4). This structure is very similar to that of ordinary ice [97]. An ab initio molecular-orbital calculation for the NH3-HCI dimer was performed and the existence in it of a hydrogen bond was predicted [98-100]. This result was confirmed experimentally when it was found that three N-H bonds are shorter than the fourth one [101]. NH4C1 crystallizes in the CsCl-type structure, where NH~ has tetrahedral symmetry, because of the stabilization effect of the Madelung energy

o

nn

a~ (a)

(b)

Figure 4. Structures of ammonium halides: (a) CsCl-type of structure shown by NH4CI, NH4Br, and NH41; (b) wurtzite (ZnS) structure, shown by NH4F, and induced by the formation of N-H...F hydrogen bonds.

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term. Evidently, in the case of NH4E the Madelung term is not sufficient to break the H-bonds [102]. Hydrogen bonding also vitally influences the conformation and detailed structure of the polypeptide chains of protein molecules and the complementary intertwined polynucleotide chains which form the double helix in nucleic acids. The formation of each H-bond involves a significant lowering of energy and it is, therefore, obvious that many structures will display a tendency to form all possible H-bonds. If any such bond fails to form between two molecules, this can be due only to geometrical restrictions. Experimental evidence indicates that, as a rule, the H-bonds affect, but only slightly, the molecular packing density. For example, in the crystal of trimesic acid, 1,3,5-benzene-tricarboxylic acid, there is a very open arrangement of H-bonds. Such a network is very unsatisfactory from the point of view of the packing. This is the reason why trimesic acid forms inclusion compounds with water, and bromide and iodine ions. Another way to satisfy the requirements of hydrogen bonding and space filling is that performed in the structure of the ~-polymorph. Here six pleated networks interpenetrate and the molecules are hydrogen bonded to each other only in the same sheet. Ice is formed by a three-dimensional network of H-bonds and, for this reason, it is a very open structure. Liquid water has a packing coefficient higher than that of icesince its density is higher. Small energies are sufficient to break H-bonds and to create a closer structure. This shows the tendency to form the maximum number of interactions, as summarized by the close-packing principle. Desiraju suggested [27] that, by analogy with biomolecules, the molecule itself is the primary structure and the network constitutes the secondary structure. The whole crystal is constituted by the packing of the secondary motif that defines the tertiary structure [103]. In the assumption of a known primary structure, the difficulties in the prediction of the crystal structures based on intermolecular H-bonds are due to the possibility of building several reasonable secondary structures and, further, of generating many tertiary structures with very small differences in energy. The actual structure results from an interplay of hydrogen bonding and van der Waals forces. Often it is just the latter interactions that discriminate among various secondary and tertiary patterns. For example, adipamide dispenses with a favorable H-bond arrangement in order to achieve a better van der Waals packing. The energy differences between observed and nonobserved packing modes can be 1-2 kcal/mol, while all structures have reasonable H-bond geometries. Energy partitioning revealed that adiapanide achieved a better interlayer packing, scarifying H-bonds [104]. For conformationally flexible molecules, however, it may be easier to satisfy simultaneously the geometrical requirements of hydrogen bonding and those of the van der Waals interactions. For the description of the classes of H-bond networks see ref. [27]. The nature of the H-bond is still under discussion. However, it seems possible to conclude that electrostatic effects predominate, particularly for weak bonds, but that covalency effects increase in importance as the strength of the bond increases.

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It is also possible to apportion the energy obtained from the ab initio SCF-MO calculations in this way. However, it was recently shown [105,106] that even the C-H..-O interactions, though weak, are not van der Waals in their nature and resemble O-H--.O and N-H-.-O hydrogen bonds in their geometrical properties. The long-range, electrostatic character of the C-H.--O interactions plays an important role in the crystal engineering of some structures [105]. Desiraju performed extensive statistical studies on structures containing this type of interaction [27]. Indeed, geometrical properties of weak intermolecular forces are conveniently studied by using crystallographic data bases. This is because a statistical study of several crystal structures can partly eliminate the distorting effects of other strong interactions which may differ from structure to structure. Different crystal structures may have different total energies if one takes into consideration not only optimum packing requirements, but also possible special interactions like those deriving from dipoles or from hydrogen bonding. In this sense, even weak interactions like C-H...O may be relevant, as mentioned above. The significance of these C-H-..O interactions in a particular structure increases with their number. Indeed, recently, there has been evidence for C-H--.O and C-H...N intermolecular hydrogen bonding [107-110]. H-bonds are easily bent and several hydrogen interactions are found to be attractive [111]. In this sense, the formation of a single bond is not important for the packing, but the maximum number of interactions has to be generated, as stated by the principle of close packing. For a given molecule, different packings can be performed with closely similar values of the packing coefficient. The best packing will be that in which there is a maximum number of bonds and strong interactions. The larger the molecule the greater the number of terms in the summation formula for the energy. The maximization of the resultant total energy is the only factor to be taken into consideration for the stability. However, the importance of H-bonds becomes determining for the minimum energy of conformation in intramolecular interactions, as is the case of polypeptides. When possible, due to the small values (4-8 kJmo1-1) of the single-bond rotation energy, the formation of intermolecular H-bonds (25-35 kJmo1-1) is highly probable. In view of the fact that a H-bond can have very different energies, it is possible to break an intramolecular H-bond and change the conformation to achieve a stronger intermolecular H-bond. For instance, the extended, intermolecularly hydrogen bonded conformation of N-malonyl glycine derivatives in the solid state differs from the intramolecularly hydrogen bonded form favored in a dilute solution of methyline chloride [112]. A new class of compounds in which hydrogen bonding is important is that of the molecular intercalates. These are a special type of inclusion compound, formed by the insertion of molecules into empty sites between layers of a layered structure. All intercalation reactions are characterized by an expansion of the crystal lattice along the c direction perpendicular to the layers to an extent that may be correlated

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with the sizes of the guest molecules and the stoichiometry of the final product. Organic molecules containing at least one functional group are used mostly as guest molecules. This functional group mediates the anchoring of the molecules to the layers. Typical representatives of guest molecules are amines, alcohols, and heterocyclic compounds [20]. Only few intercalates have been prepared as single crystals for X-ray diffraction studies, and for this reason modeling is fundamental in this class of compounds. In general, the guest molecules are arranged between the layers with a minimum increase of the basal spacing. The anchoring of the guest molecule can be realized by the formation of a coordination bond, with the donor being the guest itself. Otherwise the guest species can be anchored by H-bonds or by association to the host structure, leading to the stabilization of the product by ionic bonds between the guest cations and the negatively charged layers. A further possibility for the anchoring of the guest molecules is by proton transfer from the host lattice to the guest molecules. Since the intralayer bonds are very strong, it can be assumed that there are no changes inside the layers. Thus, in many cases one can make reasonable hypotheses on the structure of the molecular intercalates [21,22], leading to useful structural models.

C. Entropy Ordered Phases As stated in Section I, the equilibrium configuration of an ensemble of atoms or molecules corresponds to a minimum of the free energy:

F=U-TS In the previous discussion of the atom-atom potential method (Section III) the contribution of the entropy to the free energy was completely disregarded. However, the role of entropy can be critical in some kinds of packing, as in the formation of a liquid crystalline phase when an increasing concentration ofrigid rods is packed into a given volume of space. This is the basis of Flory's early work [113] on the phase diagrams of lyotropic liquid crystals; he initially predicted quite realistic phase diagrams on the basis of entropic considerations alone, though later refinements also included enthalpic contributions. Phase separation of a nematic liquid from an isotropic liquid in solutions of simple rod-like molecules is promoted by the gain in translational entropy, which significantly offsets the loss of orientational entropy. The literature review in the paper of ref. [114] shows how relationships between entropy and packing have been recognized. A new continuous-placement Monte Carlo (CMC) approach was developed that measures the entropy of rod-like particle configurations having preset global orientation distributions [114-116]. Another context in which packing is driven by entropy is the immiscibility of hydrocarbons and perfluorocarbons of comparable length (e.g., decane and per-

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fluorodecane). One might predict mixing on the basis that the net positive charge on the hydrogen in decane should attract the net negative charges on the fluorines in perfluorodecane. However, the melts are immiscible, and this is a consequence of entropy. Consider a single hydrocarbon molecule surrounded by several equivalent perfluorocarbon molecules in a melt. The number of conformational states that can be occupied by the normally flexible hydrocarbon is severely constrained by the presence of the surrounding rigid perfluorocarbon molecules. Therefore, such mixing is not favored. Similarly, consider a single perfluorocarbon molecule surrounded by several equivalent hydrocarbon molecules in a melt. Again, the presence of a rigid perfluorocarbon molecule constrains the number of conformational states accessible to flexible hydrocarbons, and mixing does not in fact occur.

Disordered Phases In the case of disorder in molecular orientation, diffraction techniques detect the "average molecule" by the superposition of all the molecules located in the same crystallographic position of the average unit cell. One of the most elementary and common examples of this kind of disorder is the formation of centrosymmetric crystals by molecules without a symmetry centerwfor instance, azulene and p-nitrochlorobenzene [117]. In such crystals, as shown in Figure 5, the average molecule is actually obtained by centrosymmetric superposition of two molecules with half-weight atoms. In these cases, disorder is generated by two different orientations. In the case of chain molecules, structures of this type exist, in which disorder is caused by rotations about the long molecular axis or by several different spatial orientations. In each case the entropy increases, thereby lowering the free energy. When all the rotations are possible in the solid state the symmetry increases to hexagonal. This form corresponds to the close packing of spheres or cylinders and the molecule is in a rotational crystalline state, characterized by rigorous order in the arrangement of the center (axes) of the molecules and by disordered azimuthal rotations [118]. If the chain molecules are azimuthally chaotic (they rotate freely around their axes), their average cross sections are circular and, for this reason, they choose hexagonal packing. The ease of rotation of molecules in the crystal depends merely on the molecular shape, as in molecules of an almost spherical shape like methane and ethane derivatives with small substituents, or molecules of a shape close to that of a cylinder (e.g., paraffin-like molecules). The rotational crystalline phase has conceptually the same meaning as that of the smectic liquid crystal [8]. Molecules of substances forming liquid crystals have, as a rule, an anisometrically elongated shape and low symmetry. The main structural feature of the liquid-crystalline state is a parallel array of molecules with very light contacts between them. For example, the structure of poly p-hydroxybenzoic acid (PHBA) at high temperature was recently found to be in a disordered state with two molecules at each lattice point and with an occupancy of 0.5 [119]. At higher temperatures the

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Figure 5. Formation of centrosymmetric crystals by noncentric molecules and relations among ordered state, rigid disordered state, and rotational crystalline state.

PHBA goes through an orthorhombic-to-hexagonal phase transformation (a rotational crystalline phase or a smectic liquid-crystalline phase). Other structures [120-122] were recently described as belonging to the rotational crystalline state. For transitions in the liquid-crystalline state, very simple idealizations of the shape of the molecule are useful for studying changes in the packing [123,124]. It was recently shown that a variety of minerals also display lower symmetries than would be expected from their X-ray diffraction patterns [125]. A static disorder model has also been proposed [126] to justify the high symmetry X-ray diffraction patterns of ZrO 2 nanopowder. In particular, the cubic phase was interpreted as a disordered monoclinic phase, i.e., the phase is stable at room temperature.

D. Packing and Lone Pairs A very interesting model was developed by Gillespie for studying the influence of lone pairs (LPs) on molecular geometry [127-129]. Using this model, it is possible to foresee the coordination geometry of an atom or of an ion. An alternative approach to analyze and systematize the structures of compounds with stereochemically active lone pairs of electrons was developed [130]. In this approach, the space taken by a LP and its cation in a crystal was used to locate the centroid of its negative charge. The LP is often clearly localized in a structuremfor example, in the yellow phase of PbO. When PbO is oxidized, O atoms enter the crystal and take up the positions of the lone pairs, thus obtaining the c~-PbO2 structure [130].

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What is less understood is the extent to which the presence of a LP is responsible for orientation effects in crystals and, more generally, for determining the type, comPactness, and symmetry of crystal packing. The notion that a stereochemically active LP should have spatial extension implies that significant electron density is associated, locally, with the LP, and that this electron density extends radially and in a particular direction from the parent atom over an appreciable portion of space in the crystal. This effectively excludes other atoms from occupation of this space and creates a void in the crystal structure. In crystals the interatomic distances are determined by coulombic and packing factors; the result is that generally LPs will be removed from a potentially interacting atom or ion, more than in the case of the liquid or gas phases. All these effects are considered by Knop et al. [131] for YEL3~ ions (E is the lone pair, Y is a second row element, e_ 1 are simple ligands, and el). However, because of (a) the actual shape and size of the trigonal-pyramidal YEL3~ unit and (b) the relative magnitude of the crystal factors, the presumed orientation of the LP may not be in the most favorable direction as is possible in the isolated pair. Both circumstances would lessen the importance of the LP as a structure-determining factor in crystals. In this respect, the discussion of the LP is reminiscent of the discussion of hydrogen bonding as a structure-determining factor in crystals, especially in ionic crystals. As for the importance of the LP in packing, it is probably better to consider small molecules in molecular structures. Indeed, when considering the LP in ionic crystals, it may seem that the LP contribution to packing is less than that of the coulombic charge of the ions. In this sense, the influence of the LP in packing can be shown by considering the structures of PF 3, PF30, and BC13, shown in Figure 6. [132]. In the case of BC13 we have the packing of the triangular coordination of B. In the projection in Figure 6, the anions are distributed in parallel layers. However, comparing the crystal structures ofPF 3 and PF30 (where the oxygen atom is substitutional for the LP), the two different molecular geometries appear to deform the layer structures in the same way, as if the LP had an influence on the packing comparable to that of the oxygen atom. Thus, this result is in apparent contrast with those mentioned above, indicating that the LPs are not an important factor in packing. However, the distortion induced by a trigonal pyramid and that induced by a tetrahedron are probably similar, which would explain this result. It was reported [53] that the LP influences the molecular and electronic distortions observed in [Cr(C6H6)(CO)3 ] and related carbine and carbene complexes. The mutual orientation of the first-neighboring CO groups in these species reflects more the balance between the molecular shape and the tendency of the molecules to lie as close as possible (close-packing principle), rather than an electronic requirement of the CO-CO intermolecular interactions. The contour maps of the Laplacian distribution of the electronic charge density for HO-H ..... OH 2 shows that the shape of H20 molecules has not been modified significantly by the presence of the LPs of O [84]. This is further proof of the

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go :) (a)

(b)

(c)

Figure 6. The crystal structures of (a) BCI3; (b) PF3; and (c) PF30, where the anions are distributed in parallel layers.

noninfluence of the LPs on the shape of the molecules and, consequently, on packing. However, the influence that the LPs have on charge distributions and molecular shape, and so indirectly on packing, is important. Indeed, it was found that the electric potential for pyrimidine, pyrazine, and s-tetrazine showed a significantly better fit when the LP sites were included [133].

V. PACKING A N D SYMMETRY A. Symmetryof the Structural Unit and Crystal Symmetry Symmetry features of both inorganic and organic molecules have been related to the symmetry of the corresponding crystals [8]. Molecules or, more generally, multiatomic finite structural groupings have a definite point symmetry and the field of the potential interaction energy around them conforms to the symmetry of the grouping itself. As a consequence, the space symmetry of the crystal structure arising from a given structural unit is related to its point symmetry. The symmetry of the Wyckoff position concides with the symmetry of the structural unit, or is a subgroup of it, as can be predicted by the Curie principle of interaction of symmetries [134]. Indeed, in many structures the symmetry of the position occupied by the molecule (or structural unit) is lower than that of the isolated unit. For instance, the benzene molecules with high symmetry (6/mmm) are packed in an orthorhombic structure, and the symmetry of the position of the molecule center is 1_. The SiO 4 group keeps its high symmetry in crystobalite, while in many silicates it occupies positions of lower-symmetry. Since the preservation of a molecular center of symmetry in a crystal does not generate a lower packing density, this symmetry element is always present in the crystal of a centrosymmetric molecule. The preservation of other symmetries in a crystal is usually associated with a lower packing density. In all cases, when a molecule preserves high symmetry in a crystal at the expense of a certain decrease in the packing density, this decrease is not large.

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If this sacrifice in packing density is large, the symmetry is partially or completely lost [5]. The determining principle in the formation of a structure is the principle of energy minimization which, for nondirectional forces, is expressed geometrically as the maximum filling principle. If the minimization of energy is attained when the structural units occupy low-symmetry positions, their proper symmetry does not completely coincide with the symmetry of the crystallographic positions. The position of a structural unit in a crystal may have, statistically, a point group symmetry higher than that of the isolated unit. This can be achieved either by averaging over all the unit cells of the crystal or by reorientation of the molecule. For instance, asymmetric molecules form a centrosymmetric crystal, in which they statistically occupy a position of symmetry 1. This general rule would seem to have some exceptions, as in the case of the N-(p-chlorobenzylidene)-p-chloroaniline (BACL), in which the crystal imposes on the molecule a "conformational symmetry" which the isolated molecule does not possess [ 135]. The space group symmetry regulates the mutual arrangement of the structural units, not only by means of operations of inversion, reflection, and rotation, but also by translations and by symmetry operations with a translational component. Crystals with the lower symmetries, typical of many organic compounds, are built by the stacking of layers of three-dimensional objects; the postulated requirement is that these layers must be closely packed. The close stacking can be obtained

Figure 7. As a consequence of their particular shape many objects (umbrella) pack together, generating space groups that are "impossible" according to Kitaigorodsky's categorization.

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either through a juxtaposition of regularly displaced layers for which the repeat vector forms an arbitrary angle with the layer plane, or by inversion centers, glide planes, and screw axes. For molecules without symmetry elements, close packing is attainable in the space groups P1, P2 l, P21/c, Pbca, Pna2 l, and P212121. For centrosymmetric molecules, the number of groups in which close packing can be achieved is even smaller: they are P1, P21/c, C2/c, and Pbca. In a recent paper, Wilson [136] discusses the Kitaigorodsky categorization of space groups, in which the space groups are divided into four categories: (1) closely-packed (PI, P21, P21/c, C2/c), (2) "limiting close-packed" (C2/m), (3) permissible (P1, C2, Cc, Cm, P21/m), and (4) impossible (P23, Pm, P2/m). This categorization has generally proved very successful, with only a few exceptions. Indeed, as a consequence of a particular molecular shape, some "impossible" space groups become possible; in some space groups, requiting the molecules to capitalize on point-group symmetry, the molecules are in general positions (Figure 7). If the bonding forces are directional, as in a covalent crystal, the energy minimum is attained upon the saturation of the directional covalent bonds. In these structures, atoms have relatively few neighbors, as, for example, in diamond and NiAs. The symmetry of these structures is defined by the coordination symmetry of the constituent atoms. It is interesting to note that the volume per atom in structures with different types of bonds differs only slightly, on the average, since the covalent bonds are usually slightly shorter than the ionic or metallic ones.

B. SpaceGroup Statistics In the cases of close packing, the coordination number of 6 in a layer can be achieved with any mutual orientation of the molecules. The space group P21/c, in which both centrosymmetric and non-centrosymmetric molecules can be packed, is the most frequent. Moreover, according to Kitaigorodsky, from the point of view of close-packing requirements, the remarkable predominance of the space group P21/c in organic crystal chemistry can be explained by the fact that in this group close-packed layers can be built on all three coordinate planes of the cell [5]. Wilson [137,138] proposed that, to a first approximation, the number of structures in each space group of a specified crystal class is given by:

Nsg = Acc exp [-Bcc[2]sg - Ccc[m]sg ] where [2]sg is the number of twofold axes and [m]sg is the number of reflection planes in the cell, Bcc and Ccc are parameters characteristic of the crystal class in question, and Acc is a normalization factor, proportional to the total number of structures in the crystal class. The population of the Cambridge Database contains many biases that have to be considered when studying the statistical distribution of the space groups. In a recent study on the space group distribution, Wilson [137] has considered the question as to whether crystallographers exercise an inherent preference in the choice of the

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Figure 8. in this Venn diagram and in those of the following Figures the ellipses contain the representative points of all the structures with at least all of the symmetry elements characteristic of the indicated space groups. As an example, the ellipse indicated by P-1 contains all the structures with at least one inversion center, even if, commonly only the structures in the dashed area are described in P-1. In this sense the diagram represents the distribution of symmetry elements among the more frequent space groups.

space group of a crystal. Allen et al. [139] comment that there must have been some interesting or unusual feature in a crystal to warrant its structure analysis. However, the rapid increase in the number of structures being solved has resulted in the smoothing out of some of these irregularities. A graphical way to visualize the distribution of the structures can be performed using Venn diagrams, as shown in Figure 8. In the diagram, the set indicated by

,9

/

Figure 9. Venn diagram representing the distribution of symmetry elements among the space groups more frequent among organic structures and showing the groupsupergroup relationships (see text).

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PT contains all the structures with at least one inversion center, and therefore can be described in that space group, even if, commonly, only the structures in the dashed area are described in PT. In Figure 9 the Venn diagrams of the symmetry groups which are most frequent among organic structures are shown [140], taking into account the group-supergroup relationships [141-143]. As an example, the area indicated by Pnma, which is a supergroup of P21/c, Pna2 l, and P212121 , is the intersection of the areas of these three groups. In Figure 10a the diagrams obtained using 51,611 organic compounds [144] are shown, each point representing 1% of the structures. We can compare these diagrams with those obtained for the distribution of compounds with more than one formula unit in the asymmetric unit (Figure 10b), with those obtained for the structures in which the asymmetric unit is constituted by half molecules (Figure lOc), and with those obtained for the special case of proteins (Figure lOd). Since the structures containing more than one asymmetric unit in the unit cell constitute only 8.3% of the total number of the considered structures, the diagrams in Figure 10b can be considered identical to those of the structures with one molecule in the asymmetric unit. In Figure 10a there is a higher percentage of P21/c as compared with Figure 10b. In particular the structures with both an inversion center and a 21 screw axis with one molecule in the asymmetric unit are 43%, with more than one molecule 30%, with half a molecule 29%, and for the proteins less than 1%.

: ;. ") ,,, ..... ] ] -

,

z j

/

o

(a) \

\ (....~-~)~176176

(c)

.

(b) ,

(d)

Figure 10. Each point represents 1% of the total number of the structures. (a) Venn diagram obtained using 51,611 organic compounds. (b) Distribution of structures with more than one formula unit in the asymmetric unit. (c) Distribution of structures in which the asymmetric unit is constituted by half molecules. (d) Distribution of structures of proteins.

Order in Space

3 31

It is worth noting that the domain containing the center of symmetry is empty for proteins (Figure 10d), as it is the case for synthetic polymer crystals where the presence of an inversion center is rare. The helicoidal conformation of proteins (as well as that of synthetic polymers) implies a high frequency of groups with 21 screw axes [31]. It is important to note, however, that the statistics for proteins have been done only on about two hundred structures.

C. Dipole Moments and Symmetry Molecular dipole moments were supposed to be an important factor leading to centrosymmetry in organic crystals [145]. The significance of dipole-dipole interactions in crystals has been considered quantitatively by several groups since the pioneering studies of Kitaigorodsky. It was shown that electrostatic interactions are of fundamental importance in determining the behavior of interacting molecules in the gas phase, in weakly bound complexes, and in condensed phases [49]. The total value of the electrostatic energy cannot be used as a criterion for determining the role of electrostatic interactions in the adoption of the crystal structure. In acenaphtaquinone, for instance, electrostatic forces determine the crystal structure even if their contribution to the total energy is small [27]. This is due to the fact that long-range electrostatic forces at large distances direct the molecules and orient them in their path with the final packing determined also by van der Waals forces. In general, asymmetric shapes achieve the best packing in the crystal by introducing a center of symmetry (Figure 11 a). The necessity, in many cases, of a screw axis may arise from the presence in the molecule of a "tail" and a "head", caused by a displacement between the centers of negative- and positive-charge distribution. Thus, in order to have the maximum number of "tail-to-head" interactions, it is necessary to introduce a screw axis besides the center of symmetry (Figure 11 b). The large number of crystals having both a screw axis and a center of symmetry may mean that even small displacements in the molecular charge can induce the two-symmetry operations. The importance of the atomic charge in packing is shown by the observed non-coplanar arrangements of benzene molecules that cannot be predicted with a simple van der Waals interaction model; an atomic charge of at least 0.09 electrons on hydrogen is necessary to predict the observed crystal structure [146]. The fact is that a one-symmetry operation is not enough for molecules with head-tail interaction can also be derived from the observations of Whitesell et al. [145], who were looking for a relationship between the presence of a center of symmetry and the generation of a dipole moment. They did not find any correlation by looking at the P1 and P21 structures. The average dipole moment for structures in these two space groups is very small, even smaller than the average dipole in P1. This fact may be a confirmation of the necessity of the presence of two elements of symmetry for molecules with a dipole moment. In this sense a statistical study of the dipole moment in the P2~/c group may be more interesting.

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Figure 11. Asymmetric shapes achieving the best packing in a crystal by introducing" (a) a center of symmetry and (b) screw axis, the latter induced by the need of having the maximum number of "tail to head" interactions (see text).

When the tail and the head of the molecule do not lie on the direction of the dipole vector, different symmetry operations are introduced in the structure. In a recent study, Gavezzotti statistically considered molecules containing either a carbonyl or a nitrile group [64]. He concluded that, on the basis of electrostatic packing energy calculations, the dipole-dipole interactions contributed negligibly to the total crystal stabilization. In spite of this fact he found that these molecules have, with respect to hydrocarbons, and, in general, to molecular crystals, significantly different distributions among the most frequent space groups (see the Venn diagrams in Figure 10 and in Figure 12). In these molecules the dipole is nearly perpendicular to the molecular axis" this fact may give rise to screw axes. It is interesting to note that the distribution of this class of compounds lies in between the distribution of general organic compounds and the distribution of proteins (Figure 12). In fact, even in hydrocarbons the electrostatic contribution to their intermolecular interaction may be substantial, even if these molecules are among the least polar of heteroatomic substances. This fact may help to explain the differences in the space-group distributions. We already pointed out that it is very difficult to have a center of symmetry in proteins. The frequent presence of a screw axis may be due to the existence of dipole moments perpendicular to the protein axis. Indeed, Warshel proposed [ 147] that the

!

o,

j

~\~ \ o

/

(a)

/~

) (b)

Figure 12. Venn diagrams (a) for hydrocarbons and (b) for molecules containing carbonyl or nitrile groups.

Order in Space

333

single most important element of structure-function correlation in protein interactions was the electrostatic energy, which probably also has a strong influence on the packing of these molecules [117].

D. PackingSymmetryof InorganicCompounds A recent paper by Baur and Kassner [148] reported a statistical analysis of the frequency distribution of space groups of inorganic compounds and compared it with the corresponding frequency distribution for the structures of organic and organometallic substances [144]. An interesting result of this work derived from their taking into account the recently made corrections from false low-symmetry to higher true symmetry groups, leading to an important reshaping of the frequency distribution and possibly eliminating one cause of bias in this kind of statistical analysis. Interesting studies were performed by MUller [149] on the space symmetry groups of tetraphenylphosphonium and arsonium salts with a cation-to-union ratio of 1"1. These salts crystallize by forming parallel columns of cations and by accommodating the anions in holes between the columns. Preferred space groups are 14 for tetrahedral anions and P4/n for anions having a fourfold rotation axis. Anions with lower symmetry cause a symmetry reduction of the space group that can be traced by group-subgroup relationships. These relationships for a number of known structures can be helpful for structure determination, since once the lattice constants and extinction rules for a given crystal are known, comparison with the "family tree" can reveal the space group and the structure type. Moreover, as shown

1/4

+

0 (

I-i: P,i

sc 1/4

inve+rsi~ -centre

P4/n

Figure 13. Symmetry of space groups oftetraphenylphosphonium and arsonium salts [41], crystallizing by forming parallel columns of cations and accommodating the

anions in holes between the columns. The two frequent space groups in this family can be obtained by adding either a screw axis (group I-4) or a center of symmetry (group P4/n) to the point group of the cation.

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in Figure 13, the two frequent space groups in this family can be obtained by adding either a screw axis (space group I4) or a center of symmetry (space group P4/n) to the point group of the cation. The best packing is obtained by one of these two symmetry operations as it is for molecular structures. The study of packing in inorganic structures has not been performed as yet in depth, but it is very promising. Also, possibly interesting correlations could be found by considering the distribution of the symmetry operations active in optimizing the packing. This kind of study may, in general, be useful in recognizing similar structures in different space groups. Indeed, small changes in the structure may produce dramatic changes in the space group symmetry, while leaving the crystal structure and chemistry almost unchanged. ACKNOWLEDGMENTS I am grateful to Marcello Zocchi and Christopher Viney for having offered their insight in many useful discussions.

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INDEX calculations chemical shifts in small gaseous molecules, 117 comparison with X-ray diffraction results, 267 electrostatic representations, 237 hydration effects, 242-256 limitations, 298 modification of experimentalbased model, 204, 212-215 packing of atoms and molecules, 305, 306 partial atomic charges for molecular mechanics force-fields, 236-237 peptides, 242-243 phosphoric acid, 267-270 photochemistry of chlorine dioxide, 192-193 porphyrins, 282 pseudopeptides, 271-277 rationalization of unexpected experimental geometries, 202, 206-209 reproduction of experimental data, 205, 209-212 rovibrational contributions to inertial moments, 65

A b initio

self-consistent reaction field theory, 237 simple molecules, 243-248 supermolecules, 236-237, 248249 verification of experimental results, 215-223 Absorption spectroscopy, 179-180, 187-193 Acetaldoxime, Z-E isomerization equilibria, 118 Acetylacetone, keto-enol tautomerism, 118, 119 N-Acetyl- 1-tryptophane methylamide, electrostatic potential, 288-289 Adamantane, C-C bond lengths, 47 Adjustment variances, 89, 94 Alkanes, internal rotation, 122-124 Alkyl nitrites, gas-phase conformers, 119-120, 129-130, 137-141 Alkylidenecyclopropanes, X-ray structure, 216-217 Alternant conjugated hydrocarbons, 317 Amides; see N,N-Disubstituted amides; Peptides; Proteins Amino acids, hydration, 228 339

340

INDEX

Ammoniaborane, X-ray structure determination and ab initio calculations, 209-212 Ammonium halides, structure, 319320 Arsonium salts, space symmetry, 333-334 Atom-atom potential method, packing of atoms and molecules, 305, 311-316 Atomic basin, definition, 298 Aziridine, pressure dependence of nitrogen inversion, 141-142

1-Bromo-2-chloroethane, gas-phase study of rotational isomerism, 124 Buckingham potential, interaction potential for pair of atoms, 313 Buckminsterfullerenes, structure, 35, 36 Bullvalene, Cope rearrangement, 132-133, 142-144 t-ButylCOproline-histidinemethylamide, electrostatic potential, 288-291,293

Basis set extension, effect on ab initio calculations on pseudopeptide, 276-277 Basis set superposition errors (BSSE) corrections, interaction energies of formic acid and water, 248-249 Bending potential functions, metal halides, 44-46 Benzene crystal structure, 317 symmetry analysis, 2 Berry pseudorotation of sulfur tetrafluoride, 132, 144 Bicyclopropylidene, X-ray structure, 216-217 Bimolecular rate constants and collision efficiencies, 145-151 Biotin, electron density, 295-298 Bis(trisyl)oxadiborirane, X-ray structure determination and ab initio calculations, 213-215 Body center cubic (bcc) structures, geometrical model of packing, 307 Bond lengths assumption of equality for parent species and isotopomers, 90 changes with substitutions, 46-47

Cambridge Structural Database biases, 328-329 data on X-C-C bond angles, 208, 209 data source for study of geometrical packing, 3 l0 prediction of structures, 55 Caps and spheres method, calculation of the volume of the molecule, 309-3 l0 Carbon dioxide, crystal structure, 308 Carbon monoxide, Fourier transform spectroscopy at 298 K and 20 K, 174, 175 Carbon-cage molecules, intramolecular interactions, 52-53 Cartesian coordinates and internal coordinates, 89-90, 98-99, 100 Chain molecules, 318-319 Charge density analysis by crystallographic methods, 262 Charge-transfer interactions, pDiisocyanobenzene, 55 Chemical shape energy costs of changes, 54-55 influence on molecular reactivity, 46

Index

341

Chiramers, 12-14 Circular dichroism, proteins, 238 Collision efficiencies and bimolecular rate constants, 145-151 Computer codes AM1,233, 235 AMPAC, 142-143 DNMR5, 125 GAUSSIAN, 237 GEOM, 93 MINDO/3, 233, 234 MNDO, 233, 234, 243 MNDO-PM3, 233, 234-235 MOPAC, 233-235, 243 STRFIT, 79 STRFTQ, 93 Condition number, definition, 91 Conformational processes ab initio calculations, 237 early research, 39-40 energy requirements, 54 gas-phase NMR studies, 116-152 hydration effects, 252-254 Connes' accuracy, Fourier transform spectroscopy, 170 Continuous symmetry measure

(CSM) applications, 11-20, 24-27 definition, 3-6, 27-28 Cope rearrangement of bullvalene, 132-133, 142-144 Costain's errors, 88-89, 92, 111 Covariance matrices, 88, 89, 90, 101, 103, 104 general least squares, 72, 73, 75, 77 ro-method, 93-94 rrfit method, 83-85, 93 Crystal-field effects, energy contents, 54 Crystallographic methods; see X-ray diffraction

Crystals comparison to gas-phase structures, 53-55 density correlation with packing coefficient, 310 engineering, 55-57 molecular self-assembly, 58 packing, 55, 307 symmetry analysis, 2-3 Cubic close packed (ccp) structures, geometrical model of packing, 307 Cyclohexane, ring inversion, 134, 141,145 Cyclohexene, ring inversion, 133-134 1-Cyclopropylidenedispiro[2.0.2.1 ]heptane, X-ray structure and ab initio calculations, 217-218, 219, 221-223 7-Cyclopropylidenedispiro[2.0.2.1 ]heptane, X-ray structure and ab initio calculations, 217-219, 221-223 Cyclopropylidenespiropentane, Xray structure and ab initio calculations, 217-218, 220223 Databases; see also Cambridge Structural Database; Molecular Gas Phase Documentation (MOGADOC) comparison of structures, 40 Deformation density deconvolution of parameters between thermal motions, 267-270 definition, 263 dynamic maps, 266 electronic excited state, 299 experimental compared to theoretical, 270-277 peptides, 280-281

342

phosphoric acid, 268-269 porphyrins, 282-283 p-Dicyanobenzene, crystal structure, 55-56 2,6-Difluorobenzenamine, lack of hydrogen bonds, 49 2,6-Difluorophenol, hydrogen bonds, 48-49 1,2-Dihaloethanes, bond angle changes on internal rotation, 52 p-Diisocyanobenzene, crystal structure, 55-56 N, N-Diisopropylacetamide, barrier to internal rotation, 127-128 B-Diketone, keto-enol isomerization studied by gas-phase NMR spectroscopy, 118-119 1,2-Dimethoxyethane, internal rotation and gas-phase NMR vicinal coupling constants, 123 N, N-Dimethylformamide chemical shift differences in gas and solvent, 127 difference in bond angles, 47-48 N,N-Dimethylnitrosamine, barrier to internal rotation in gasphase, 130-131 N, N-Dimethylthioacetamide, hindered internal rotation, 127 Dipole moments and symmetry, 331333 Discharge sources, radical generation, 176 Disordered phase, 323-324 N,N-Disubstituted amides conformer equilibria studied by gas-phase NMR spectroscopy, 121 gas-phase NMR studies of rotational barriers, 127-129

INDEX

N-Disubstituted trifluoroacetamides, N-substituent size effect on rotational barrier, 128-129, 130 DNA helices, packing, 310-311 Drug design, crystal engineering, 57 Electron density distributions electrostatic potential calculations, 284-293 molecular structure aspect, 37, 54 multipole analysis, 267-282 topological analysis, 294-298 transition metal compounds, 282284 from X-ray diffraction, 262-267 Electron diffraction bond lengths, 54 of gases, 67 Electrostatic interactions and crystal structure, 331-333 Electrostatic potentials, calculation from X-ray diffraction data, 284-293 Ellipticity of bonds, definition, 295 Emission spectroscopy, 186-187, 193-194 Environmental concerns, design of new materials, 57 Ethanes internal rotation, 52, 122-124 symmetry of rotation of tetrahedra, 12-16 Ethylacetoacetate, keto-enol isomerization, 118-119 Ethylene [2+21 reactions, orbital symmetry, 3 Euler's theorem, 106, 107 Fellgett's advantage, Fourier transform spectroscopy, 169-170 Fenestrane, intramolecular interactions, 52-53

Index

First moment relations, molecular rotational resonance (MRR) spectroscopy, 69, 71, 78, 80, 92 2-Fluorobenzenamine, lack of hydrogen bonds, 49 2-Fluorophenol, hydrogen bonds, 48-49 Fock matrix, modification to include solvation effects, 235-236 Folding method in symmetry analysis occluded shape, 19-20 proof, 8-10 tetrahedricity of phosphate tetrahedron, 10-11 Formic acid, potential energy distributions in aqueous solutions and vapor, 245-247 Formic acid/water supermolecule, hydration stages, 244-245 Fourier transform spectroscopy absorption spectroscopy, 179-180, 187-193 aliasing, 166, 167 apodization, 164-165 emission spectroscopy, 186-187, 193-194 gas-phase radicals, 157-195 infrared spectroscopy, 170-171, 174, 179-186, 228, 239 jet spectroscopy, 173-176, 179-195 Michelson interferometer, 159-163 molecular rotational resonance (MRR) spectroscopy, 111112 proteins, 237 ultraviolet/visible spectroscopy, 171-173, 179-180, 186-195 Gas-phase NMR spectroscopy, 116152 alkyl nitrites, 129-130, 137-141

343

amides, 127-129 aziridine, 141-142 bandshape analysis, 125 bimolecular rate constants and collision efficiencies, 145-151 bullvalene, 132-133, 142-144 conformational processes, 116-152 conformer equilibria, 118-121, 122-124 cyclohexane, 131,141 cyclohexene, 133-134 N, N-Dimethylnitrosamine, 130131 pressure dependence, 117, 135-144 ring inversion in six-membered tings, 131-132, 133-134, 141 sulfur tetrafluoride, 132, 144 temperature dependence, 126-134 Gas-phase structures compared to crystal structures, 53-55 General least squares, molecular rotational resonance (MRR) spectroscopy, 68, 72-77, 88 Glycine supermolecule, hydration, 252-255 Glycouryl, electron density, 295-298 Harmonic force constants, calculation, 241 Hellmann-Feynman theorem, 82, 85 Heptacyclotetradecane, intramolecular nonbonded interactions, 52-53 Hessian matrix, definition of elements, 294 Heterocyclopropenes, structure, 212213 History of molecular geometry, 37-40 Hydration, 228-256; see also Solvent effects computational background, 229237, 240-242

344

glycine, 252-255 N-methylacetamide, 251,252-254 peptides, 242-243, 251-252, 254, 255-256 simple molecules, 243-248 spectroscopic background, 237240 supermolecules, 248-250 Hydrogen bonds crystal structure, 310, 319-322 N,N-disubstituted amides, 121 electrostatic potential, 286 glycouryl and biotin, 297 intermolecular, 54, 55, 57 intramolecular, 48-52 peptide-like molecules, 287-293 peptides, 242, 320 resonance-assisted, 49-50 semiempirical molecular orbital calculations, 234 Hyperconjugation effects, bond angle changes, 209 Icosahedral structures, truncated, 3435 Imidazole, electrostatic potential, 286 Inertial defects definition, 71-72 estimate of Costain's error, 89 re-structure, 108-109 Inertial moment tensors, 66, 68-72 Inertial moments, 64, 65, 93-94 Infrared spectroscopy; see also Fourier transform spectroscopy formic acid vapor, 246 protein secondary structure, 238239 Inorganic compounds packing coefficients, 311 packing symmetry, 333-334 Interaction energy, geometrical model of packing, 308-309

INDEX

Intercalates geometrical model of packing, 308 hydrogen bonding, 321-322 Interferograms, 160-163 sampling of points, 166 sampling rate in IR and UV/VIS, 172-173 Intermolecular interactions, 53-55 p-dicyanobenzene and pdiisocyanobenzene crystals, 55-56 Internal coordinates and Cartesian coordinates, 89-90, 98-99, 100 in computer code, 93 Laurie's corrections, 104 and planar moments, 101 symmetry preservation, 102-103 Internal rotation N,N-disubstituted amides, 121 geometrical changes during, 52 substituted alkanes, 122-124 Internal vibrational redistribution (IVR), 134, 135, 136, 139, 141,144, 145 Internuclear distances anharmonic correction, 43-44 and centroids of electron density distributions, 54 differences in parameters, 42-46 distance-average, 42, 43 effective, 41 equilibrium, 41-42, 43 floppy system differences, 43-44 harmonic approximation, 43-44 Intramolecular interactions, 46-52 nonbonded, 52-53 Intramolecular motion, aspect of molecular structure, 37 Ionic compounds gaseous molecular structures and crystal structures, 55 packing, 307

Index

Iron II tetraphenylporphyrin, ground state, 282 Isomerization studies, gas-phase NMR spectroscopy, 117-119 Isotopomers of chlorine dioxide, 189 molecular rotational resonance (MRR) spectroscopy, 65, 66 peptides, 249-252 Isotropy subgroup, definition, 29 J Method for studying rotamers, temperature dependence of long-range coupling constants, 122 Jacquinot advantage, Fourier transform spectroscopy, 170 Jet spectroscopy, 173-176, 179-195 nozzle geometry, 176 x formalism, electrostatic properties, 265, 286, 289, 292 Kepler, J., studies of packing in snow crystals, 37-38 Keto-enol isomerization, 118-119 Kitaigorodsky categorization, space groups, 327, 328 Kraitchman's equations computer code, 92 limitation to singly substituted or symmetrically disubstituted isotopomers, 93 rs method, 66-67, 78-81 rs-fit method comparison, 91 Laplacian of electron density, 294, 295 Laser spectroscopy of gas-phase radicals, 158 Laurie's corrections, 90-91, 98, 99, 110 molecular rotational resonance (MRR) spectroscopy, 104

345

LCAO approximation ab initio calculations, 236-237 semiempircal molecular orbital calculations, 234 Least squares method, molecular rotational resonance (MRR) spectroscopy, 68, 72-77, 88 Lennard-Jones potential, molecular mechanics, 231 Lewis, G.N., discovery of electronpair covalent bond, 40 Lindemann mechanism, pressure dependence of bimolecular region unimolecular rate constants, 146 Liquid crystalline state, 323-324 Lone pairs (LP), packing, 324-326 Metal halide molecules, floppy system example, 43-44 Methyl isocyanide, 148-151 Methyl nitrite, conformer conversion, 145, 147-151 N-Methylacetamide (NMA) hydration states, 251,252-254 model of vibrations in polypeptide backbone, 238 scaled quantum mechanical methods (SQM), 247-248 Methylacetoacetate, keto-enol isomerization studied by gasphase NMR spectroscopy, 118-119 2-Methylaziridine, pressure dependence of nitrogen inversion, 141-142 Methylenecyclopropane, X-ray structure, 216-217 Methylisocyanide, bimolecular kinetics of isomerization, 145 1-Methyl- 1-silabicycloheptane, symmetry analysis, 11-12

346

Michelson interferometer and Fourier transforms, 159-163 Microwave rotational spectroscopy; see Molecular rotational resonance (MRR) spectroscopy Mirror symmetry continuous symmetry measure (CSM) of water, 17 most probable shape, 30 symmetry transform evaluation, 78 Molecular beam, Fourier-transform molecular rotational resonance (MRR) spectroscopy, 111 Molecular dipole moments and symmetry, 331-333 Molecular dynamics, 232 Molecular Gas Phase Documentation (MOGADOC), 67 Molecular geometry changes due to internal rotation, 52 consideration of motion, 44-46 importance, 35-37 representations, 40-42 Molecular globularity, definition, 316 Molecular mechanics background, 230-232 hydration of proteins, 231-322 Molecular orbital contours, point selection for symmetry selection, 16-17 Molecular radio astronomy, MOGADOC database, 67 Molecular recognition in crystals, 55 energy costs of changes, 54-55 intermolecular interactions, 43 storage and retrieval of molecular structural information, 58

INDEX

Molecular rotational resonance (MRR) spectroscopy, 63113 average-structure (rz or r~) method, 67 centrifugal distortion, 66 complementary (r~) method, 67, 104-105, 108-109 double resonance technique, 111 Eckart conditions, 100 effective (ro) structure, 66-67, 92104 equilibrium (re) structure approximated by other structures, 78, 79, 105, 106, 109, 110 general least squares, 68, 72-77, 88 inertial moment tensor, 68-72 Laurie's corrections, 90, 104 mass dependence (rm) method, 6768, 104-108 ro-derived structure, 96-104 substitution (rs) method, 66-67, 78-92 Molecular salts, packing coefficients, 311 Molecular self-assembly, 58 Molecular shape of single molecules, influence on packing, 316318 Monte Carlo techniques, simulations of aqueous solutions of small molecules, 232 Multiplex advantage, Fourier transform spectroscopy, 169-170 Multipole analysis of electron density d orbitals occupancies, 282-284 deconvolution between thermal motion parameters and deformation density parametrization, 267-270 electrostatic properties calculations, 299

Index

reproduction of deformation density, 281-282 size of molecules tractable, 277279 transferability of multipole parameters, 277-281 Nitrobenzene, geometry compared to 2-nitrophenol and 2nitroresorcinol, 49-52 2-Nitrophenol, hydrogen bonds, 4951 2-Nitroresorcinol, hydrogen bonds, 49-52 NMR spectroscopy; see also Gasphase NMR spectroscopy advantages for conformational studies, 116 ammoniaborane, 211 oxadiboriranes, 214 protein hydration, 231-322 Normal coordinate analysis (NCA) basic equation, 240 vibrational spectra analysis, 238 Nyquist criterion, data points taken in Fourier transform spectroscopy, 166, 167, 173 Occluded shapes, symmetry analysis, 18-20 Operational effects on structure determinations, 54 Operational parameters, definitions, 41 Orbital symmetry, ethylene [2+2] reaction, 3 Orbits and symmetry groups, 28-29 Orientation in 2D, optima location, 29-30 Oxadiboriranes, X-ray structure determination and ab initio calculations, 213-215

347

Packing coefficient, geometrical model of packing, 309-311 Packing of atoms and molecules, 303-337 atom-atom potential method, 311316 close packing principle, 306-309 covalent solids, 307 dipole moments and symmetry, 331-333 early research, 37-38 effects on molecular geometry, 203-204 entropy effects, 322-324 geometrical model, 306-311 helices and rods, 319 hydrogen bond effects, 319-322 ideal packing, 307 lone pair effects, 324-326 molecular shape effects, 316-319 packing symmetry of inorganic compounds, 333-334 space group statistics, 328-331 symmetry of structural unit and crystal symmetry, 326-328 Packing of crystals, coordination number (CN), 307 Packing of spheres, dense random, 314 Paterno, E., reference to conformational isomers, 39-40 Pattern recognition, Raman spectra of proteins, 239-240 Pauling, L., The Nature o f the Chemical Bond, 40 Peptides; see also Proteins alanine helix, 255, 256 amide vibrational modes, 238-239 hydration, 254, 255-256 Raman spectra of isotopomers, 250-252 scaled quantum mechanical methods (SQM), 250, 251252, 255-256

348

transferability of multipole parameters, 280-281 Perfluoroadamantane, C-C bond lengths, 47 Perturbation methods, solvation, 233 Phenol, geometry compared to 2nitrophenol and 2nitroresorcinol, 49-52 Phosphoric acid, ab initio calculations, 267-270 Photochemistry chlorine dioxide, 192-193 Fourier transform absorption spectroscopy of photoreactive radicals, 179-180 radical generation by photolysis, 178 Planar moment tensors, 68-72, 79-80 Planar moments, 93, 94 Point-symmetry group centroid of orbit, 28-29 in proof of folding method, 8-9 Polo's vectors, 100 Polymers, packing, 308, 318-319 Polymorphic modifications for study of intermolecular interactions, 54 Polypropylene, exception to packing rules, 318 Polywater, lack of molecular structure, 35 Porphyrins, deformation density maps, 282-283 Potential energy and Laplacian of electron density, 295 packing of atoms and molecules, 315 solvation effects, 230 Potentials, molecular mechanics, 231 Principal axis system (PAS) calculation of approximations, 99

INDEX

derivative with respect to internal coordinates, 100 molecular rotational resonance (MRR) spectroscopy theory, 70, 71 r~ structure, 106-107 ro computer program, 79 r, method, 78 Principal inertial moments, free rotation of rigid body, 64 Principal planar moments derivative with respect to principal axis system, 99 r, method, 78 Probability distributions of symmetry values, 24-27 Proteins; see also Peptides crystal structure, 332-333 hydrogen bonds, 320 packing coefficients of crystals, 310 secondary structure, 228, 230, 237240 space group, 330-331 Pseudoatom multipole modeling; see Multipole analysis of electron density Pseudo-Kraitchman (p-Kr) method, 85, 93, 95, 97 convergence with other fits, 111 Pseudopeptides ab initio calculations compared to experiment, 271-277 electrostatic potential, 287-293 Pyrazine, interaction energy from Xray diffraction experiments, 286 Pyrolysis, radical generation, 176, 178 Radial modeling of phosphorus, 269 Radical generation controlled chemical reaction, 176, 177, 193-194

Index

photolysis, 178 pyrolysis, 176, 178 Radicals atmospheric, 158 Fourier transform spectroscopy, 157-195 photoreactive, 179-180 Raman spectroscopy, 228 of Ala-X peptides, 242-243 proteins, 238, 239-240 Reflection axis angle, 30 point groups, 29 Repulsion potential, variation with symmetry, 13-16 Residual density maps, 266 Residuals, general least squares, 73 Rice-Ramsperger-Kassel-Marcus's (RRKM) theory Berry pseudorotation of sulfur tetrafluoride, 132, 144 Cope rearrangement of bullvalene, 142-144 gas phase conformational processes, 134, 135-137, 139-141 pressure-dependent gas-phase rate constants, 144-145 Ring inversion cyclohexane, 134, 141,145 cyclohexene, 133-134 six-membered rings, 131-132 Rotamers of ethane, 13-14 gas-phase DNMR spectroscopy, 122-124 Rotane, X-ray structure, 216 Rotational averaging approach, vicinal coupling constants and internal rotation of alkanes, 123-124 Rotational constants definitions, 68

349

ro-method, 93-94 rs-fit method, 84 Rotational crystalline state, 323-324 Rotational isomeric state (RIS) model, vicinal coupling and internal rotation, 122-123 Rotational symmetry most probable symmetric shape, 21-24 point groups, 29 symmetry transform evaluation, 67 Rovibrational contributions inertial moments, 65 linear regression study, 95-96 ro-method, 96-97 r,-method, 78, 86 Scaled force constants, peptides, 228 Scaled quantum mechanical methods (SQM) formic acid/water supermolecules, 243-249 hydration of larger molecules, 254, 255-256 hydration states of NMethylacetamide (NMA), 252-254 isotopic substitution, 249-252 peptides, 242-243 principles, 228, 230, 240-242 Scattering factor of iron atom, 263 Second moment relations molecular rotational resonance (MRR) spectroscopy theory, 70, 71 rs method, 78, 80 r,-fit method, 92 Selection-by-angle, point selection for representation of a shape, 18-19 Selenides, Se-C bond lengths and carbon valence states, 46-47

350

Self-consistent reaction field theory, ab initio calculations, 237 Semiempirical molecular orbital calculations, 233-236 Silicon carbide, crystal structure, 308 Silicon carbide nozzle for pyrolysis, 178 Silicon dioxide, crystal structure, 308 Singular value decomposition, 74, 93 Slater-Condon parameters, semiempircal molecular orbital calculations, 234 Smectic liquid crystal, 323-324 Solvation; see Hydration; Solvent effects Solvent effects; see also Hydration Cope rearrangement of bullvalene, 132-133 fluorine-fluorine exchange in sulfur tetrafluoride, 132 internal rotation of alkyl nitrites, 120, 130 internal rotation of 1,2dimethoxyethane, 123 internal rotation of N,Ndisubstituted amides, 121, 127-129 keto-enol tautomer equilibria of /3-diketones, 119 proton coupling constants for 1,1,2-trichloroethane, 124 ring inversion in six-membered rings, 131-132, 133-134 theory, 230 Space groups Kitaigorodsky categorization, 327, 328 statistics, 328-331 Spiro compounds, X-ray structure determination and ab initio calculations, 215-223 Spiropentane, X-ray structure, 216

INDEX

Structure factors, processing of accurate crystallographic data, 262 Structures, comparison, 40-46 Substitution data set (SDS) definition, 68 ro structure, 95 rs method, 93 rs-fit method, 83 Sulfides, S-C bond lengths and carbon valence states, 46-47 Sulfones, S-C bond length change upon CH3/CF3 substitution, 46-47 Sulfur tetrafluoride, Berry pseudorotation, 132, 144 Supermolecules ab initio calculations, 236-237, 248-249 formic acid in different hydration stages, 244 glycine, 252-255 Supersonic expansion, 173-176, 179195 Supramolecular structures, 58 Symmetry bounds of symmetry measure, 2728 continuous symmetry measure (CSM), definition, 3-6, 2728 dipole moments, 331-333 folding method proof, 8-10 limitations of analysis, 2-3 local symmetry, 204 1-methyl-l-silabicycloheptane, 1112 molecular rotational resonance (MRR) spectroscopy, 71, 79, 81, 83, 86-87, 98, 102103 occluded shapes, 18-20 packing, 326-334

Index

351

point selection for contour representation, 16-17 points with uncertain locations, 21-27 probability distributions of values, 24-27 rotating tetrahedra of ethane, 1216 symmetry transform, 4, 6-8 tetrahedricity of phosphates, 10-12 Syn/anti conformational exchange process, 137, 139-141

Ultraviolet/visible absorption spectroscopy of radicals, 171173, 179-180, 186-195 Uncertain locations, symmetry of points, 21-27 Unimolecular kinetics conformational processes, 135-137 Cope rearrangement of bullvalene, 142 Universal potentials in atom-atom potential method, 313 Urea, electrostatic potential, 286-287

Tautomers

Variance, general least squares, 73, 75 Venn diagrams, space group statistics, 329-330, 332 Vicinal coupling, dependence on internal rotation, 122-124 Vinylcyclopropane, X-ray structure, 216,218 Virial theorem, Laplacian of electron density and total energy, 295 Volume of the molecule in geometrical model of packing, 309

ab initio calculations, 237

gas-phase NMR spectroscopy, 118-119 solvent effects, 236 Tetraethylborate, X-ray structure determination and ab initio calculations, 206-208 Tetrafluorohydroquinone, hydrogen bonds, 4849 Tetrahedra of ethane, symmetry of rotation, 12-16 Tetrahedricity of a phosphate tetrahedron, folding/unfolding method, 10-11 Tetraphenylphosphonium salts, space symmetry, 333-334 Topological analysis charge densities, 299 electron density, 294-298 Transition dipole, peptides, 228 1,1,2-Trichloroethane, solvent effect on proton coupling constants, 124 Triethylborane, X-ray structure determination and ab initio calculations, 206-208 Triethylboroxin, structure determination, 206

Water continuous symmetry measure (CSM) on lone-pair orbital, 16-17 quasi-symmetry, 3 symmetry analysis, 2 transition dipoles, 228 Wilson's vectors, 90, 100 X-ray diffraction ab initio calculations, 202-224 aspherical pseudoatoms, 264 bond lengths, 54 boron compounds, 206-209 deformation density maps, 266

352

distribution of symmetry values, 27 electrostatic properties, 262-299 formalism, 265, 286, 289, 292 multipole parameters, 265-266 residual density maps, 266

INDEX

scattering factor of iron atom, 263 size of molecules tractable, 277279 valence pseudoatom density, 264 Zero flux surface, definition, 298

Advances in Molecular Modeling Edited by Dennis Liotta, Department of Chemistry Emory University "... as a result of the revolution in computer technology, both the hardware and the software required to do many types of molecular modeling have become readily accessible to most experimental chemists." Because the field of molecular modeling is so diverse and is evolving so rapidly, we felt from the outset that it would be impossible to deal adequately with all its different facets in a single volume. Thus, we opted for a continuing series containing articles which are of a fundamental nature and emphasize the interplay between computational and experimental results." -- From the Preface to Volume 1 REVIEWS: 'q'he first volume of Advances in Molecular Modeling bodes well for an exciting and provocative series in the future."

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