EDITORIALBOARD Dr. Lester Andrews University of Virginia Charlottesville, Virginia USA
Dr. J.A. Koningstein Carleton University Ottawa, Ontario CANADA
Dr. John E. Bertie University of Alberta Edmonton, Alberta CANADA
Dr. George E. Leroi Michigan State University East Lansing, Michigan USA
Dr. A.R.H. Cole University of Western Australia Nedlands WESTERN AUSTRALIA
Dr. S.S. Mitra University of Rhode Island Kingston, Rhode Island USA
Dr. William G. Fateley Kansas State University Manhattan, Kansas USA
Dr. A. MiJller Universit~it Bielefeld Bielefeld WEST GERMANY
Dr. H. Hs. G~inthard Eidg. Technische Hochschule Zurich SWITZERLAND
Dr. Mitsuo Tasumi University of Tokyo Tokyo JAPAN
Dr. P.J.Hendra University of Southampton Southampton ENGLAND
Dr. Herbert L. Strauss University of California Berkeley, California USA
PREFACE TO THE SERIES
It appears that one of the greatest needs of science today is for competent people to critically review the recent literature in conveniently small areas and to evaluate the real progress that has been made, as well as to suggest fruitful avenues for future work. It is even more important that such reviewers clearly indicate the areas where little progress is being made and where the chances of a significant contribution are minuscule either because of faulty theory, inadequate experimentation, or just because the area is steeped in unprovable yet irrefutable hypotheses. Thus, it is hoped that these volumes will contain critical summaries of recent work, as well as review the fields of current interest. Vibrational spectroscopy has been used to make significant contributions in many areas of chemistry and physics as well as in other areas of science. However, the main applications can be characterized as the study of intramolecular forces acting between the atoms of a molecule; the intermolecular forces or degree of association in condensed phases; the determination of molecular symmetries; molecular dynamics; the identification of functional groups, or compound identification; the nature of the chemical bond; and the calculation of thermodynamic properties. Current plans are for the reviews to vary, from the application of vibrational spectroscopy to a specific set of compounds, to more general topics, such as force-constant calculations. It is hoped that many of the articles will be sufficiently general to be of interest to other scientists as well as to the vibrational spectroscopist. As the series has progressed, we have provided more volumes on topical issues and, in some cases, single author(s) volumes. This flexibility has made it possible for us to diversify the series. Therefore, the course of the series has been dictated by the workers in the field. The editor not only welcomes suggestions from the readers, but also eagerlv solicits your advice and contributions.
James R. Durig Kansas City, Missouri
PREFACE TO VOLUME 24
The current volume in the series, Vibrational Spectra and Structure, is a single topic volume on gas phase structural parameters. The title of the volume, Equilibrium Structural Parameters, covers the two most common techniques for obtaining gas phase structural parameters: microwave spectroscopy and the electron diffraction technique. Since the quantum chemical method provides equilibrium geometries, the volume is an attempt to provide a connection between the experimental and theoretical parameters. In chapter 1, Professor Harmony has provided a review on molecular structure determinations from spectroscopic data using scaled moments of inertia. He has pointed out the limited number of molecules for which equilibrium parameters have been obtained and the requirement of a large amount of microwave data needed to obtain the equilibrium structural parameters. In chapter 2, Dr. Mastryukov reviews the electron diffraction technique. He also describes how the technique can incorporate structural information from microwave spectroscopy, vibrational spectroscopy, or theoretical calculations to improve the determination of the structural parameters by electron diffraction studies. In chapter 3, Dr. Groner has reviewed the theory and methods of microwave spectroscopy, describing in some detail r 0 and r s structures as well as r m structures and corrections based on ab initio calculations. The final chapter by Professor Bell reviews in some detail the accuracy of the molecular geometry predictions by quantum chemical methods. Professor Bell has recently calculated many of the predicted structural parameters and has presented the data in graphic form rather than in tabular form. The graphic presentation makes it possible to readily note the difference in the parameters predicted at the various levels of quantum mechanical calculations. Therefore, it is believed that the four authors have provided a rather coherent description of the various structural parameters obtained experimentally along with treatments needed to extract equilibrium bond distances and angles. The final chapter provides the necessary data for the reader to ascertain the expected quality of the ab initio predicted parameters based upon the level of the calculation. The editor thanks his administrative assistant, Mrs. Linda Smitka, for providing the articles in camera ready copy form and enduring some of the various tasks associated with the completion of the volume. He also thanks Mr. Richard Hester for typing the subject index and his wife, Marlene, for preparation of the pages with figures as well as some t f the proofreading.
James R. Durig Kansas City, Missouri vii
CONTRIBUTORS TO VOLUME 24
STEPHEN BELL Department of Chemistry, University of Dundee, Dundee DD 1 4HN, Scotland, U.K. PETER GRONER Department of Chemistry University of Missouri-Kansas City Kansas City, Missouri 64113, USA MARLIN D. HARMONY Department of Chemistry University of Kansas Lawrence, Kansas 66045, USA VLADIMIR S. MASTRYUKOV Department of Physics University of Texas Austin, Texas 78712, USA
ix
CONTENTS OF OTHER VOLUMES
V O L U M E 10
VIBRATIONAL SPECTROSCOPY USING TUNABLE LASERS, Robin S. McDowell INFRARED AND RAMAN VIBRATIONAL OPTICAL ACTIVITY, L. A. Nafie RAMAN MICROPROBE SPECTROSCOPIC ANALYSIS, John J. Blaha THE LOCAL MODE MODEL, Bryan R. Henry VIBRONIC SPECTRA AND STRUCTURE ASSOCIATED WITH JAHN-TELLER INTERACTIONS IN THE SOLID STATE, M.C.M. O'Brien SUM RULES FOR VIBRATION-ROTATION INTERACTION COEFFICIENTS, L. Nemes
V O L U M E 11
INELASTIC ELECTRON TUNNELING SPECTROSCOPY OF HOMOGENEOUS CLUSTER COMPOUNDS, W. Henry Weinberg
SUPPORTED
VIBRATIONAL SPECTRA OF GASEOUS HYDROGEN-BONDED COMPOUNDS, J. C. Lassegues and J. Lascombe VIBRATIONAL SPECTRA OF SANDWICH COMPLEXES, V. T. Aleksanyan APPLICATION OF VIBRATIONAL SPECTRA TO ENVIRONMENTAL PROBLEMS, Patricia F. Lynch and Chris W. Brown TIME RESOLVED INFRARED INTERFEROMETRY, Part 1, D. E. Honigs, R. M. Hammaker, W. G. Fateley, and J. L. Koenig VIBRATIONAL SPECTROSCOPY OF MOLECULAR SOLIDS-CURRENT TRENDS AND FUTURE DIRECTIONS, Elliot R. Bemstein
xv
xvi
CONTENTS OF OTHER VOLUMES
VOLUME 12 HIGH RESOLUTION INFRARED STUDIES OF SITE STRUCTURE AND DYNAMICS FOR MATRIX ISOLATED MOLECULES, B. I. Swanson and L. H. Jones FORCE FIELDS FOR LARGE MOLECULES, Hiroatsu Matsuura and Mitsuo Tasumi SOME PROBLEMS ON THE STRUCTURE OF MOLECULES IN THE ELECTRONIC EXCITED STATES AS STUDIED BY RESONANCE RAMAN SPECTROSCOPY, Akiko Y. Hirakawa and Masamichi Tsuboi VIBRATIONAL SPECTRA AND CONFORMATIONAL ANALYSIS OF SUBSTITUJED THREE MEMBERED RING COMPOUNDS, Charles J. Wurrey, Jiu E. DeWitt, and Victor F. Kalasinsky VIBRATIONAL SPECTRA OF SMALL MATRIX ISOLATED MOLECULES, Richard L. Redington RAMAN DIFFERENCE SPECTROSCOPY, J. Laane
VOLUME 13 VIBRATIONAL SPECTRA OF ELECTRONICALLY EXCITED STATES, Mark B. Mitchell and William A. Guillory OPTICAL CONSTANTS, INTERNAL FIELDS, AND MOLECULAR PARAMETERS IN CRYSTALS, Roger Frech RECENT ADVANCES IN MODEL CALCULATIONS OF VIBRATIONAL OPTICAL ACTIVITY, P. L. Polavarapu VIBRATIONAL EFFECTS IN SPECTROSCOPIC GEOMETR/ES, L. Nemes APPLICATIONS OF DAVYDOV SPLITTING FOR STUDIES PROPERTIES, G. N. Zhizhin and A. F. Goncharov
OF CRYSTAL
RAMAN SPECTROSCOPY ON MATRIX ISOLATED SPECIES, H. J. Jodl
CONTENTS OF OTHER VOLUMES
xvii
V O L U M E 14 HIGH RESOLUTION LASER SPECTROSCOPY OF SMALL MOLECULES, Eizi Hirota ELECTRONIC SPECTRA OF POLYATOMIC FREE RADICALS, D. A. Ramsay AB INITIO CALCULATION OF FORCE FIELDS AND VIBRATIONAL SPECTRA,
G~za Fogarasi and Peter Pulay FOUR/ER TRANSFORM INFRARED SPECTROSCOPY, John E. Bertie NEW TRENDS IN THE THEORY OF INTENSITIES IN INFRARED SPECTRA, V. T. Aleksanyan and S. Kh. Samvelyan VIBRATIONAL SPECTROSCOPY OF LAYERED MATERIALS, S. Nakashima, M. Hangyo, and A. Mitsuishi
V O L U M E 15 ELECTRONIC SPECTRA IN A SUPERSONIC JET AS A MEANS OF SOLVING VIBRATIONAL PROBLEMS, Mitsuo Ito BAND SHAPES AND DYNAMICS IN LIQUIDS, Waker G. Rothschild RAMAN SPECTROSCOPY IN ENERGY CHEMISTRY, Ralph P. Cooney DYNAMICS OF LAYER CRYSTALS, Pradip N. Ghosh THIOMETALLATO COMPLEXES" VIBRATIONAL SPECTRA AND STRUCTURAL CHEMISTRY, Achim Mtiller ASYMMETRIC TOP INFRARED VAPOR PHASE CONTOURS AND CONFORMATIONAL ANALYSIS, B. J. van der Veken WHAT IS HADAMARD TRANSFORM SPECTROSCOPY?, R. M. Hammaker, J. A. Graham, D. C. Tilotta, and W. G. Fateley
xviii
CONTENTS OF OTHER VOLUMES
VOLUME 16 SPECTRA AND STRUCTURE OF POLYPEPTIDES, Samuel Krimm STRUCTURES OF ION-PAIR SOLVATES FROM MATRIX-ISOLATION/SOLVATION SPECTROSCOPY, J. Paul Devlin LOW FREQUENCY VIBRATIONAL SPECTROSCOPY OF MOLECULAR COMPLEXES, Erich Knozinger and Otto Schrems TRANSIENT AND TIME-RESOLVED RAMAN SPECTROSCOPY OF SHORT-LIVED INTERMEDIATE SPECIES, Hiro-o Hamaguchi INFRARED SPECTRA OF CYCLIC DIMERS OF CARBOXYLIC ACIDS: THE MECHANICS OF H-BONDS AND RELATED PROBLEMS, Yves Marechal VIBRATIONAL SPECTROSCOPY UNDER HIGH PRESSURE, P. T. T. Wong
V O L U M E 17A SOLID STATE APPLICATIONS, R. A. Cowley; M. L. Bansal; Y. S. Jain and P. K. Baipai; M. Couzi; A. L. Verma; A. Jayaraman; V. Chandrasekharan; T. S. Misra; H. D. Bist, B. Darshan and P. K. Khulbe; P. V. Huong, P. Bezdicka and J. C. Grenier SEMICONDUCTOR SUPERLATTICES, M. V. Klein; A. Pinczuk and J. P. Valladares; A. P. Roy; K. P. Jain and R. K. Soni; S. C. Abbi, A. Compaan, H. D. Yao and A. Bhat; A. K. Sood TIME-RESOLVED RAMAN STUDIES, A. Deffontaine; S. S. Jha; R. E. Hester RESONANCE RAMAN AND SURFACE ENHANCED RAMAN SCATTERING, B. Hudson and R. J. Sension; H. Yamada; R. J. H. Clark; K. Machida BIOLOGICAL APPLICATIONS, P. Hildebrandt and M. Stockburger; W. L. Peticolas; A. T. Tu and S. Zheng; P. V. Huong and S. R. Plouvier; B. D. Bhattacharyya; E. Taillandier, J. Liquier, J.-P. Ridoux and M. Ghomi
CONTENTS OF OTHER VOLUMES
xix
V O L U M E 17B STIMULATED AND COHERENT ANTI-STOKES RAMAN SCATTERING, H. W. SchrStter and J. P. Boquillon; G. S. Agarwal; L. A. Rahn and R. L. Farrow; D. Robert; K. A. Nelson; C. M. Bowden and J. C. Englund; J. C. Wright, R. J. Carlson, M. T. Riebe, J. K. Steehler, D. C. N guyen, S. H. Lee, B. B. Price and G. B. Hurst; M. M. Sushchinsky; V. F. Kalasinsky, E. J. Beiting, W. S. Shepard and R. L. Cook RAMAN SOURCES AND RAMAN LASERS, S. Leach; G. C. Baldwin; N. G. Basov, A. Z. Grasiuk and I. G. Zubarev; A. I. Sokolovskaya, G. L. Brekhovskikh and A. D. Kudryavtseva OTHER APPLICATIONS, P. L. Polavarapu; L. D. Barron; M. Kobayashi and T. Ishioka; S. R. Ahmad; S. Singh and M. 1. S. Sastry; K. Kamogawa and T. Kitagawa; V. S. Gorelik; T. Kushida and S. Kinoshita; S. K. Sharma; J. R. Durig, J. F. Sullivan and T. S. Little
V O L U M E 18 ENVIRONMENTAL APPLICATIONS OF GAS CHROMATOGRAPHY/FOURIER TRANSFORM INFRARED SPECTROSCOPY (GC/FT-IR), Charles J. Wurrey and Donald F. Gurka DATA TREATMENT IN PHOTOACOUSTIC Michaelian
FT-IR SPECTROSCOPY,
K. H~
RECENT DEVELOPMENTS IN DEPTH PROFILING FROM SURFACES USING FTIR SPECTROSCOPY, Marek W. Urban and Jack L. Koenig FOURIER TRANSFORM INFRARED SPECTROSCOPY OF MATRIX ISOLATED SPECIES, Lester Andrews VIBRATION AND ROTATION IN SILANE, GERMANE AND STANNANE AND THEIR MONOHALOGEN DERIVATIVES, Hans Biirger and Annette Rahner FAR INFRARED SPECTRA OF GASES, T. S. Little and J. R. Durig
xx
CONTENTS OF OTHER VOLUMES
VOLUME 19 ADVANCES IN INSTRUMENTATION FOR VIBRATIONAL OPTICAL ACTIVITY, M. Diem
THE
OBSERVATION
OF
SURFACE ENHANCED RAMAN SPECTROSCOPY, Ricardo Aroca and Gregory J. Kovacs DETERM/NATION OF METAL IONS AS COMPLEXES I MICELLAR MEDIA BY UV-VIS SPECTROPHOTOMETRY AND FLUORIMETRY, F. Fernandez Lucena, M. L. Marina Alegre and A. R. Rodriguez Fernandez AB INITIO CALCULATIONS OF VIBRATIONAL BAND ORIGINS, Debra J. Searles
and Ellak I. von Nagy-Felsobuki APPLICATION OF INFRARED AND RAMAN SPECTROSCOPY TO THE STUDY OF SURFACE CHEMISTRY, Tohru Takenaka and Junzo Umemura INFRARED SPECTROSCOPY OF SOLUTIONS IN LIQUIFIED SIMPLE GASES, Ya. M. Kimel'ferd VIBRATIONAL SPECTRA AND STRUCTURE OF CONJUGATED CONDUCTING POLYMERS, Issei Harada and Yukio Furukawa
AND
V O L U M E 20 APPLICATIONS OF MATRIX INFRARED SPECTROSCOPY TO MAPPING OF BIMOLECULAR REACTION PATHS, Heinz Frei VIBRATIONAL LINE PROFILE AND FREQUENCY SHIFT STUDIES BY RAMAN SPECTROSCOPY, B. P. Asthana and W. Kiefer MICROWAVE FOUR/ER TRANSFORM SPECTROSCOPY, Alfred Bauder AB INITIO QUALITY OF SCMEH-MO CALCULATIONS OF COMPLEX INORGANIC
SYSTEMS, Edward A. Boudreaux CALCULATED AND EXPERIMENTAL VIBRATIONAL SPECTRA AND FORCE FIELDS OF ISOLATED PYRIMIDINE BASES, Willis B. Person and Krystyna Szczepaniak
CONTENTS OF OTHER VOLUMES
xxi
V O L U M E 21 OPTICAL SPECTRA AND LATTICE DYNAMICS OF MOLECULAR CRYSTALS; % N. Zhizhin and E. I, Mukhtarov
VOLUME 22 VIBRATIONAL INTENSITIES, B. Galabov and T. Dudev
VOLUME 23 MOLECULAR APPROACH TO SOLIDS, A. N. Lazarev
CHAPTER MOLECULAR SPECTROSCOPIC
STRUCTURE DATA USING
1
DETERMINATION SCALED
MOMENTS
FROM OF INERTIA
Marlin D. H a r m o n y Departmem of Chemistry University o f K a n s a s L a w r e n c e , KS 66045, U S A
I.
INTRODUCTION
II.
REVIEW
..................................................................................................... 2
O F B A S I C S .............................................................................................. 6
III. M O L E C U L A R
STRUCTURE
METHODS
......................................................... 14
A.
r 0 and r e Structures ...................................................................................................... .14
B.
r s Structures .................................................................................................................. 24
C.
r m Structures ................................................ ................................................................ 31
D.
Miscellaneous Methods ............................................................................................... 40
E.
rPm Structures .............................................................................................................. 46 1. Equilibrium structures ofpolyatomic organic molecules ...................................... 46 2. Some pivotal observations ..................................................................................... 48 3. rPm Heavy-atom structures .................................................................................... 53 4. rPm Structures of hydrogen-containing molecules ................................................. 66 5. rPm Applications .................................................................................................... 75
IV.
CONCLUSION
........................................................................................................ 77
ACKNOWLEDGMENTS
REFERENCES
............................................................................................... 79
................................................................................................................ 80
2
I.
HARMONY
INTRODUCTION A principal end result of the high-resolution spectroscopic study of gasphase molecules has been the determination of molecular structures. The concept of structure, or geometrical arrangement of atoms or atomic nuclei in molecules, has been a natural one at the heart of molecular sciences throughout the 20 th century. Its theoretical origin traces to the early work in quantum mechanics and is embodied most clearly in the Bom-Oppenheimer (B-O) approximation [ 1]. This work leads to the concept of the Bom-Oppenheimer potential surface, which maps out the energy of the N atoms (or nuclei) in a molecule as they undergo their vibrational motions in a 3N-6 (or 5) dimensional space. The molecular structure defined by the positions of the nuclei at the global minimum of the 3N--6 (or 5) dimensional surface is commonly known as the e q u i l i b r i u m structure and is usually denoted as an r e structure. A molecule usually has distinctly different B-O potential surfaces (and, hence, r e structures) in its ground and excited electronic states [2]. The discussion in this work will focus attention implicitly upon the ground state potential surface and molecular structure. Based upon the Born-Oppenheimer separation of electronic and nuclear energies, the B-O potential surface should be isotopically invariant and the r e structure of a molecule should also be isotopically invariant. Most of the tightly bound molecules of interest to chemists conform very well to the B-O approximation, although all molecules deviate when examined at sufficient precision. Weakly bound hydrogen-bonded species or van der Waals molecules such as Ar'..HCI are exceptions that require special considerations [3 ].
MOLECULAR STRUCTURE DETERMINATION
3
While the r e structure represents the most well-defined molecular geometry, it is not, unfortunately, one that exists in nature.
Real molecules exist in the
quantum states of the 3N-6 (or 5) vibrational states with quantum numbers (v l, v2,...V3N-6 (or 5)), vi = 0, 1, 2 . . . . .
Even in the lowest (ground) (0,0...0)
vibrational state, the N atoms of the molecule undergo their zero point vibrational motions, oscillating about the equilibrium positions defined by the B-O potential energy surface. It is necessary then to speak of some type of average or effective structures, and to account for the vibrational motions, which vary with vibrational state and isotopic composition.
In spectroscopy, a molecule's
structural
information is carried most straightforwardly by its molecular moments of inertia (or their inverses, the rotational constants), which are determined by analysis of the pure rotational spectrum or the resolved rotational structure of vibrationrotation bonds. Thus, the spectroscopic determination of molecular structure boils down to how one uses the rotational constants of a molecule
Av lV2- 9 9V3N-6 (or 5) Bv lV2 9 9 9V3N-6 (or 5)
(1)
CVlV2 9 9 9V3N-6 (or 5)
in one or more specific states (Vl . . . .
V3N-6 (or 5)) to extract the geometry. Of
particular importance will be the ground state rotational constants (symbolized simply by A 0, B 0, CO), which are those most easily accessible to experiment. What emerges then is that there is a plethora of structural types, depending upon precisely what data are used and precisely how they are used.
Using
spectroscopically determined rotational constants (and, thus, moments of inertia),
4
HARMONY
one can compute and distinguish several types of structures, including re, r 0, (rv), rs, rm, rz, rgi, and map (see Table 1). We will review the characteristics of these various structural types or methods, concentrating eventually our principal attention upon the rm o structure [4-10], which has recently been developed as an excellent approximation to the r e structure of relatively large polyatomic molecules. There is, of course, another important structural method for gas-phase molecules, viz., electron diffraction. Here one does not probe the properties of particular quantum states by spectroscopy; rather, the scattering of electrons from the atoms of a thermal distribution of molecules is studied in a fashion analogous to x-ray diffraction of crystalline solids. This leads to entirely different types of structures, for example, rg. Vibrational motions are, of course, once again the complicating factor. With appropriate corrections, electron diffraction data have led to r e structures of high accuracy. The abundant literature should be consulted for the methods of electron diffraction [ 11 ]. We shall see shortly that it is really quite easy to obtain the r e structures of diatomic molecules to very high accuracy. Also, for a relatively minor number of small polyatomic molecules (such as OCS, SO2, CO 2, CH4), it has been possible to determine r e structures. Certainly, when possible, this is the desired result. As we shall shortly describe, the task is formidable once the molecule reaches 4 or 5 atoms or more, in general. Indeed, for an important prototype organic molecule such
as
ethane
(C2H6) ,
a
true
re
structure
(as
described
below)
MOLECULAR STRUCTURE DETERMINATION
5
T A B L E 1. Types of structures
Symbol
Name
Procedure
ro
Effective
Structure obtained by fitting directly ground state moments of inertia
re
Equilibrium
Structure obtained by fitting equilibrium moments of inertia
rs
Substitution
Structure obtained with coordinates computed from Kraitchman's equations with ground state moments of inertia of several isotopomers
rm
rOm
rz
Mass-dependence
Structure obtained using Im values computed from
or Watson' s
isotopic I0 values by Watson's equations
Scaled moment of
Structure computed from I Pm values obtained by
inertia
scaling I 0 values
Average
Structure computed from moments of inertia of the molecule when atoms are in their average positions Approximate corrections for vibration-rotation
re,I
contribution to I0; resulting structure closely equivalent to r s rg
Thermal average
Thermal average of distances and angles from electron diffraction data
has never been determined. Our interest was, in fact, to attack the problem of r e structures for polyatomic molecules such as ethane, propane and formic acid. This led to the development of the rOm structural method, which provides an excellent approximation to the r e structure for molecules larger even than ethane.
6
HARMONY
Over the past decade or so, the capabilities of ab initio quantum theory have advanced remarkably due to improvements in computer speeds. It is now possible to compute r e structures by ab initio methods for relatively large molecules at the Hartree-Fock level using reasonably sophisticated basis sets [12]. With the inclusion of electron correlation by CI or perturbation methods [13], computed ab initio bond distances are now becoming competitive in quality with experimental r e values. However, since experimental r e values have been largely unavailable for organic molecules of even modest size, it has not been possible to do careful benchmarking. With the addition of the rm ~ (scaled moment of inertia) method, it is now possible to make legitimate comparisons. As an example, we recently determined the near-equilibrium (rm~ ) CC distance in ethane to be 1.522 A from available spectroscopic data [8], providing a firm benchmark for this fundamental prototype CC single bond. Recent ab initio calculations [14] at the HF level with a 6-31G** basis set yield a value of 1.527 A, while addition of MP2 electron correlation gave 1.524 A.
This is really quite excellent quality and
demonstrates the convergence of experiment and theory.
II.
REVIEW OF BASICS In the principal axis system (PAS) of a general polyatomic molecule with the origin of the molecule-fixed axes at the center of mass (COM), the principal moments of inertia are related to the coordinates of the atoms of mass m i by
I a = Iaa = Z m i (b 2 + c 2) i
(2a)
M O L E C U L A R STRUCTURE D E T E R M I N A T I O N
I b =Ibb = ~ m i ( a 2 + c 2)
7
(2b)
i
2 I c = Ice = E m i ( a 2 + b i ) i
(2c)
In the PAS centered at the COM, the off-diagonal components of the inertial tensor I vanish:
lab = - ~ miaib i = 0 i
Iac = - ~ m i a i c
i=0
(3)
i
Ibc = - ~ m i b i c
i= 0
i
Also, the COM condition leads to
~mig i =0 (4)
i
gi = a i , b i , c i From spectroscopy, the rotational constants A, B, C are actually measured"
A = h/87t2I a
(5a)
B = h/8zr2I b
(5b)
C = h/8zr2I c
(5c)
where, for practical purposes h/87t 2 = 505379 amu A 2 MHz. By convention, we choose
A>_B_>C
or
Ia < Ib < Ic
For prolate symmetric rotor molecules,
(6)
8
HARMONY
A>B=C
(7a)
while for oblate rotors
A=B>C
(7b)
In the linear molecule case we have I a = 0 and
I=I b-I c
orB=C
(7c)
Finally, a spherical rotor is defined by
A=B=C
(7d)
In all cases, by virtue of Eqs. (2), it is clear that the molecular moments of inertia of a particular set of atomic isotopes is entirely determined by the atomic coordinates, i.e., the structure. It is also obvious that the moments are isotopomerdependent, since they depend upon atomic mass. It is always possible (in principle) to express Ig (g = a, b, c) in terms of internal parameters, such as bond lengths, bond angles and dihedral angles. These
will be no more than 3N--6 (or 5 for linear molecules) in number, depending upon symmetry restrictions.
Thus, for linear X-X, X-Y, Y-X-Y, X - Y - Z and non-
linear Y-X-Y, X-Y-Z, the number of internal parameters is 1, 1, 1, 2, 2, 3,
respectively. For larger organic molecules, the number of independent parameters grows rapidly; thus, for formic acid (HCOOH, the smallest organic acid) in either of its two planar conformations, there are seven internal structural parameters.
MOLECULAR STRUCTURE DETERMINATION
9
In a few simple cases, the formulas in terms of internal structural parameters can be written easily. Thus, for a diatomic molecule with interatomic distance r and atomic masses m l and m 2,
Ib = i = ~2
where
(8)
(9)
mlm2 ml m2
while for an X - Y - Z linear molecule (such as OCS) with masses m x, my, m z (total mass M) and distances r(XY) and r(YZ),
1 {mxmyr(Xy)2 + mymzr(Zy)2 + mxmz[r(XY ) + r(YZ)]2 } (10) I = I b = ~-
In most cases, analytical expressions are not conveniently written, but in all cases the problem is easily tractable on a computer. The key point to make here is that moments of inertia and, hence, rotational constants (via Eqs. (5)) contain all the information about structure. It should be stressed here also that all the moment of inertia equations above deal with a collection of point masses with fixed positions.
They also
specify properties of a vibrating molecule (a collection of point mass atoms) at some instant in time, or at some point (perhaps fictitious) in 3N-6 (or 5) dimensional space at which the atoms are motionless, such as the equilibrium position. There is a set of very important equations which deal with the solution of Eqs. (2) under the assumption that the atomic positions do not change under isotopic substitution.
If we substitute a particular atom with an isotope that
10
HARMONY
changes the atomic mass by Am, the total mass of the new molecule will change from M to M + Am and the moments of inertia will, in general, change by amounts AIg (g = a, b, c). Then, as shown by Kraitchman [15], the magnitude of the a coordinate of the substituted atom in the PAS of the original (unsubstituted or
parent) molecule is given by
I APa (1
lal- / ~t'
and [b[ and
+
APb Pb - Pa
)(1 +
APc Pc - Pa
)
] 1/2
(11)
[el are obtained by cyclic permutations of a, b and c. The Pg are the
planar secondmoments, for example,
and
1 Pa = 2 (Ib + Ie - Ia)'
(12)
~t, = Mam M+Am
(13)
The other two principal planar moments and expressions for
Ibl and Icl are given
by cyclic permutations of a, b, c. Equations of the form of Eq. (11) are known as the "Kraitchman" equations. symmetry elements.
Special cases arise for molecules with various
For substitution of an atom in a linear molecule, the
magnitude of the a coordinate of the substituted atom is
lal = (AIb/~') ~
(14)
More complete discussions of Kraitchman's equations can be found in standard treatises [16]. In addition, analogous equations are available for multiple isotopic substitution of equivalent atoms, e.g., H20 to D20 [ 17].
MOLECULAR STRUCTURE DETERMINATION
11
The details of spectral analysis need not concem us here; standard sources should be consulted [ 16,18]. All we need to know for our purposes is that analysis of rotational spectra or vibration-rotation bands leads to what are called the
effective
rotational constants as symbolized in Eqs. (1). For simplicity we shall
often write simply A v, B v and Cv, but it is imperative to remember that the simple subscript signifies all the vibrational quantum numbers. The constants are related to the
equilibrium rotational
effective
rotational
constants by
di
Av = A e - ~( V t~A ii
+ -~-) ...+
(15a)
Be- )-"( VOtiBi
di + -~--)...+
(15b)
di Cv --- Ce - )-"i~c (v i + -~-) + ...
(15c)
Bv=
in which the sums are over all the unique vibrations of degeneracy d i and the vibration-rotation constants ai are, in general, different for the three axes and the various modes.
For many purposes, the neglected higher-order terms are
negligible. From measured A v, Bv, Cv values, effective moments of inertia Iav, Ibv, Icv are computed from Eqs. (5). If the ot's are also determined experimentally (see below), then Ae, B e and C e can be obtained, which then leads to the equilibrium moments Iae, Ibe and Ice. Sometimes it is more convenient to define the vibration-rotation contributions in terms of the moments of inertia.
For
example, Igv and Ige (g = a, b, c) are related by
di Igv = Ige + EiI~g(vi + y ) ...+
If all higher terms in Eqs. (15) and (16) are neglected, then
(6)
12
HARMONY
(17)
~ig ~ (-~-~)I2e (zgi
Note that for a diatomic molecule (with I b - I and gb = ec = ~;), Eq. (16) becomes
I v = I e + e(v + 89
(18)
I 0 = I e + 89 e
(19)
or
in the ground vibrational state. Thus, it is seen clearly that even the ground state m o m e n t o f inertia suffers a vibration-rotation defect so that it differs from the equilibrium value. It is worth noting the experimental task involved in evaluating equilibrium rotational constants. For the diatomic molecule, it is necessary to measure B in a m i n i m u m o f two states, e.g., v = 0 and v = 1. Then Eq. (15) yields B0 = Be
mc 2
B1 = Be
3me 2
The experimental values o f B 0 and B 1 yield the desired B e and me values.
(Here
we note for the diatomic molecule that mb = mc = m is usually written me to identify it as an equilibrium property.
W e have avoided the subscript e in Eqs. (15) and
(17) to simplify notation.) For the linear XYZ molecule, with two non-degenerate and one doubly degenerate vibrations, one B e value and three ot values must be determined: B v = B e - m1(v 1 + 89 - m2(v2 + 1) - m3(v 3 + 89
M O L E C U L A R STRUCTURE DETERMINATION
13
Thus, if the (000), (100), (010) and (001) states are studied, the following four equations can be solved to obtain Be, orl, or2, and ot3" B(000) = B e - 89 ct1 - c t 2 - 89 ot3 B(100) = B e - 3/2tXl - c t 2 - 89 ot3 B(010) = B e - 89 ~ 1 - 2ct2 - 89 t~3 B(001) = B e - 89 ct 1 - a2 - 3/2 a3 Non-linear polyatomic molecules are treated similarly except now there are three rotational constants in general.
Thus, for non-linear XYZ (such as C1NO)
molecules, experimental values of A v, B v and C v in the non-degenerate states (000), (100), (010), (001) lead to values of Ae, B e and C e and the nine agvalues. It should be clear then that for even modest-sized polyatomic molecules, the determination of me, B e and C e requires a formidable amount of data from excited vibrational states. Specifically, for an N-atom polyatomic asymmetric rotor (all di values = 1), a total of 3N-6 vibration modes exist and thus 3N-6 excited vibrational states must be probed in addition to the ground state. It turns out then that equilibrium rotational constants (and equilibrium moments of inertia) are not available for very many non-linear polyatomic molecules. It should be mentioned here finally that it is assumed that the rotational spectral analysis has properly accounted for centrifugal distortions in all cases and for other possible factors such as Coriolis interactions, Fermi resonances, internal
rotation, etc., so that the rotational constants A v, B v and C v of Eqs. (15) have these influences removed [16,18]. Moreover, special additional care is needed for
14
HARMONY
those relatively few gas-phase molecules having either electronic spin or orbital angular momentum. With this very brief summary, it is possible now to move on to a discussion of structure. Although our principal aim is to describe and utilize a very recently developed procedure for obtaining what we call rm ~ structures, it is useful and important to review briefly the most common spectroscopic structural procedures.
III.
MOLECULAR STRUCTURE METHODS
A.
r 0 and r e Structures The simplest structural procedure is to use ground state (v = O) effective moments via equations of the form of Eqs. (2), (8) or (10). Such structures are commonly known as r 0 or effective structures since they utilize only A 0. B0, C O values.
For most polyatomic molecules it is customarily
assumed that the interatomic distances are isotopically invariant, even though this is true only for a rigid, non-vibrating or equilibrium molecule. Therefore, data from several isotopic species can be invoked. Thus, for OCS, ifB 0 is determined for each of the isotopes 16-12-32 and 16-13-32, the two B 0 values permit determination of the C-O and C-S distances from two equations with the form of Eq. (10).
Indeed, various isotopomer
combinations are possible, as summarized in Table 2. It is seen that the r 0 structure parameters vary rather widely (far outside experimental error), depending upon which pair is selected for the calculation.
This is a clear
example of the deleterious effects of uncorrected zero-point vibration terms (a or e), along with the assumption of isotopic invariance.
M O L E C U L A R STRUCTURE DETERMINATION
15
T A B L E 2. Effective (r0) structure of OCS a
Isotopomers used
C - O (A)
C-S (A)
16-12-32, 16-12-34
1.1647
1.5576
16-12-32, 16-13-32
1.1629
1.5591
16-12-34, 16-13-34
1.1625
1.5594
16-12-32, 18-12-32
1.1552
1.5653
Mean
1.1614
1.5604
Range
0.0095
0.0077
aSee Ref. [ 19].
Consider the simplest diatomic case. Here it is possible to obtainB v and I v in several excited states v as briefly summarized in Table 3 for the molecules CsC1 and HC1. Then from Eq. (8) we obtain rv:
rv = (Iv/~t)l/2
(20)
Note the regular variation in r v as v changes; indeed, note that r shrinks as the vibrational quantum number decreases. It is the well-known non-harmonic (anharmonic), potential curve that leads to this characteristic variation. Indeed, quantum mechanics shows to a high approximation that r v is a welldefined average
rv=
v 1/2
(21)
16
HARMONY
T A B L E 3. Rotational constants and bond lengths for 133Cs35C1 and H35Cla.
v
Bv (MHz)
rv (A)
CsCI 3
2125.955
2.9303
2
2136.013
2.9234
1
2146.092
2.9165
0
2156.191
2.9097
B e = 2161.189
r e = 2.9063
HCI 3
285847.5
1.3434
2
294835.9
1.3228
1
303876.4
1.3030
0
312990.9
1.2839
B e = 317582.6
r e = 1.2746
aSee Ref. [20] for more extensive data.
On the other hand, for a polyatomic molecule, no well-defined average
distance exists.
For example, for the general polyatomic molecule, the
experimental Igv value is given by an average of I corrected for Coriolis terms [21 ]:
1 = ( 1 ) v + Coriolis terms Igv lg
(22)
It is clear that no simple relation can be written for the distances even if the Coriolis contributions are evaluated. (Imagine averaging the inverse of Eq. (10)).
MOLECULAR STRUCTURE DETERMINATION
17
Now, for diatomic molecules such as CsCI or HC1, the B e, I e and, hence, r e values are easily measured because B 0, B 1, B2 9 9 9 data are usually readily obtained. (Table 3 contains merely a sample of measured B v values [20].) Using any two B v values (such as B 0 and B 1) or doing a least-squares fit, we obtained directly the B e and ~e values and thus r e is obtained as
re = (Ie/~t)l/2
(23)
Table 3 summarizes these results for the two example molecules. Note that r e is obtained alternatively by extrapolating the r v values to v = - 8 9 the hypothetical vibrationless state. Data for the isotopomers of HC1 provide insight into typical mass-dependent structural properties.
From the
isotopomer data of Table 4, it is immediately evident that the r e bond distance of HC1 is invariant (within 1/104) to isotopic substitution, in accord with the expectations of the B-O approximation. It is clear also from the HCI, DCI, TC1 data that the effective bond distance shrinks substantially as the H-atom is substituted by a heavier isotope. It is seen, in contrast, that substitution of a heavy atom (37C1) for a light atom (35C1) has no effect (for HC1, at least) upon the bond length to within 10-4 A. Generally, heavy atom substitutions lead to changes on the order of 0.0001 A, with the trend in the direction identical to the H-atom case. Thus, the r 0 value of 13CO is 0.0001 A shorter than for 12CO [20]. Qualitatively, this mass dependence is easily understood as arising from the heavier isotopomers falling more deeply into the potential
18
HARMONY
TABLE 4. Structural isotope effects for HC1a. r0
re
HC1
1.2839 A
1.2746 A
DCI
1.2812
1.2746
TCI
1.2801
1.2746
H37C1
1.2839
1.2746
a35c1 isotope unless noted. See Ref. [20] for more extensive data.
well, which always leads to shrinkage. It is significant to note that the bond length mass dependence is an order of magnitude greater for the H-atom than for the heavy atoms.
In principle, the r e distance of HC1 might also be
considered to be the value that would be obtained by extrapolating the HC1, DC1, TCI r 0 data to the point at which the hydrogen atom mass goes to infinity. In summary then, diatomic molecule r e structural data are relatively readily available because only one vibrational mode exists and, hence, only one vibration-rotation constant (a) needs to be determined.
Table 5
summarizes data for just a few of the many diatomics that have been investigated.
Note that the experimental precision is sufficiently high in
general to permit the r e length to be specified with an accuracy better than 1 • 10-4 A. Extensive data are available in the literature [20,22].
MOLECULAR STRUCTURE DETERMINATION
TABLE 5.
19
Equilibrium rotational and structural parameters of diatomic molecules a.
HC1
Be
me
re
(MHz)
(MHz)
(A)
317582.7
9209
1.2746
KC1
3856.38
23.68
2.6667
BaO
9371.93
41.73
1.9397
SnO
10664.19
64.24
1.8325
CO
57898.35
OH
566932
524.6 2.171 • 104
1.1283 0.9697
NO
50123.8
513
1.1507
CN
56953
520.7
1.1718
H2
1.8243 • 106
9.180 x 104
0.7414
N2
59905.8
519.2
1.0977
02
43337.9
474.8
1.2075
C2
54557
529
1.2425
C12
7314.6
44.7
1.9879
aAll species are 12g except for OH, NO, CN, and 02, which are 2II, 21I, 2s
and
3 Xg, respectively. More extensive tabulations are given by Huber and Herzberg [22] and by Lovas and Tiemann [20]. Data refer to the most abundant isotopic species.
Returning to the linear triatomic molecule case, we note now that the unsatisfactory situation summarized in Table 2 can be remedied for many triatomics, since the existence of only three (at most) ot values for each principal axis makes the r e structure determination still relatively tractable. Table 6 summarizes some of the key parameters for several triatomic
20
HARMONY
T A B L E 6. Equilibrium rotational and structural parameters for triatomic molecules a ota
ctb
ore
Rotational
Equilibrium
(MHz)
(MHz)
(MHz)
constant (MHz)
structure
Mode
SO 2
1
33.60
50.42
42.75
A e = 60502.05
r e = 1.431 A
(C2v)
2
-1127.41
-2.39
15.98
B e = 10359.24
0 e = 119.3 ~
3
612.44
34.24
32.03
C e = 8845.57
F20
1
-430.95
72.08
39.32
A e = 58744.90
r e = 1.405 A
(C2v)
2
--699.02
42.36
53.28
B e = 10985.28
0 e = 103.1 ~
3
585.01
69.56
115.25
C e - 9255.27
H20
1
0.750*
0.238*
0.2018"
A 0 = 835833
r e = 0.959 A
(C2v)
2
-2.941"
-0.160"
0.1392"
B0=435094
0 e = 103.9 ~
3
1.253"
0.078*
0.1445"
C o = 278372
OCS
1
18.13
(Coov)
2
-10.59
3
36.43
CO 2
1
0.00126*
(Dooh)
2
0.00076*
3
0.00309*
(CO)e = 1.154 A B e = 6099.22
(CS)e = 1.563/~
B e = 0.391625*
r e = 1.160 A
aAsterisks indicate units of cm -1. For more extensive data, see Lovas [23], Herzberg [2,18], Harmony et al. [24], and Landolt-Bomstein [25].
Data refer to the most abundant isotopic
species.
molecules for illustrative purposes.
For the triatomics, the r e precisions are
usually in the range 1 x 10 -4 - 1 • 10-3 A, so we list the distances r o u n d e d to 1 • 10-3 A. Angles are listed with comparable relative precision. M o r e generally, for large polyatomic molecules, it is often possible to obtain sufficient data for an r o structure but not for an r e structure. A simple e x a m p l e can illustrate the problem. Consider ethylene oxide (C2v s y m m e t r y ,
MOLECULAR STRUCTURE DETERMINATION
21
see Fig. 1), which has five independent structural parameters. By studying the microwave spectra of simply the normal isotopomer and one additional species (such as 13C in natural abundance, or perhaps even 180 in natural abundance, or the D 4 species by isotopic enrichment via the compound synthesis), six independent rotational constants are obtained in the ground state (A0, B0, C O for two isotopomers).
Thus, these data, along with the
assumption of isotopic invariance, lead via a least-squares fit to a conventional r 0 structure.
Just as was observed for OCS in Table 2, the
derived parameters will vary, depending upon which isotopic data are utilized. Berry and Harmony [7] have reported an r 0 structure for ethylene oxide using a least-squares fit of the 12 10 values from four isotopomers. On the other hand, obtaining a true r e structure for ethylene oxide requires that 3N--6 = 15 ct values must be determined for each axis; i.e., 45 otgparameters must be evaluated to obtain a set of Ae, Be, Ce values for a single isotopomer. This requires complete analysis of the ground vibrational state and 15 excited vibrational states, a total of 16 complete rotational spectral analyses,
Then, the process must be repeated for an additional
isotopomer in order to obtain a sufficient number of equilibrium rotational constants to compute the five structural parameters.
Thus, the task of
obtaining an r e structure is truly formidable for even a relatively small polyatomic molecule. Of course, for a few polyatomics with high symmetry, r e structures
22
HARMONY
O
,/
H" HI
0
FIG. 1. Structural sketch of ethylene oxide.
have been obtained. Methane (CH4) is a prime example. With T d symmetry, there exist only four unique vibrational modes (with degeneracies d 1 = 1, d 2 = 2, and d 3 = d 4 = 3) and one structural parameter, which has been determined to be re(CH ) = 1.0858 A [26]. A second example is acetylene (Dooh) with five modes (three with d i = 1, two with d i = 2) and two parameters whose r e values are r e (C-C) = 1.2026 A and re(C-H ) = 1.0622 A [27]. Finally, we close this discussion of r e and r 0 structures by presenting in Table 7 the r 0 structure of the bona fide organic polyatomic molecule propane (C3H8) , with a C2v shape, as shown in Fig. 2. The nine independent structural parameters were determined by a least-squares fit of the groundstate moments of inertia of six isotopomers, or a total of 18 independent I 0 values [9]. (As usual, the structure is assumed to be isotopically invariant.)
MOLECULAR STRUCTURE DETERMINATION
23
TABLE 7. Molecular structure of propane a
rgm
ro
rs
CC
1.5209 (9)
1.5337 (10)
1.5279
CH
1.0929 (20)
1.0952 (23)
1.0944
CH s
1.0877 (35)
1.0911 (41)
1.0864
CH a
1.0907 (19)
1.0945 (22)
1.0937
LCCC
112.35 (11)
112.14 (13)
112.23
ZHCH
106.13 (32)
106.54 (37)
106.31
ZHaCH a
107.04 (28)
107.50 (32)
107.32
LCCH s
111.60 (31)
111.30 (35)
111.89
LCCH a
110.62 (10)
110.50 (12)
110.63
aAll distances in ,~ and angles in degrees. Subscripts s and a identify methyl group hydrogens lying in and out of the molecular symmetry plane, respectively. Unlabeled hydrogens occupy the methylene group. See Ref. [28] for original data. Reprinted from Ref. [9] with permission.
H
/
H
/ /
H CJ C c/H \
/
H
H
FIG. 2. Structural sketch of propane.
"H
24
HARMONY
Note that as few as three isotopomers might have sufficed (nine I 0 values), so that a whole range of r 0 structures (as for OCS in Table 2) might have been calculated. The use of all the available data leads to the best overall average r 0 structure. Treating the deviations as being merely statistical, we list the standard deviation (a) in parentheses as some kind of representation of the quality of the fit.
The other structures in the table (rgm and rs) will be
discussed shortly. We note that no true, complete, spectroscopic r e structure is available or even ever likely to become available. For each of a minimum of three isotopomers, a total of 27 excited states (3N--6) plus the ground state would need to be fully spectrally analyzed to obtain the necessary nine Ie values!
B~
r s structures The most extensively used spectroscopic structural method for polyatomic molecules is known as the substitution (rs) method and was proposed by Costain [29] some 40 years ago. In this method, the equations of Kraitchman (Eq. (11)) are used with ground state (I0) moments of inertia for a parent (or normal) molecule and an appropriate collection of isotopomers. It has been generally supposed that the resulting r s coordinates and structural parameters are better approximations to the equilibrium (re) parameters than the simpler r 0 effective parameters. This idea arises from the following simple argument.
Consider a diatomic parent molecule with
moment I 0 and one of the isotopomers with moment I0'. From the diatomic
M O L E C U L A R STRUCTURE D E T E R M I N A T I O N
25
molecule Kraitchman equation (14), calling the coordinate z s and writing I 0 for the Ib0, the square of the coordinate is
z 2 = ~ . I ~ I~
(24)
s
If now Eq. (19) is inserted
2 - " ( I . ' - I ) +e 8 9
e)
Z$
(25)
Then it is clear that if e is a slowly varying function of atomic mass, the vibration-rotation contributions to the moments of inertia will at least partially cancel out, leaving a result closer to the equilibrium result. Indeed, if c z E', then
2 ~ -I e'-Ie , ~ =
Z s
Z
2
(26)
e
so z s ~-, Ze, which shows that the resulting interatomic distance will satisfy r s -~ r e. Costain showed that for the diatomic molecule [29]
r s ~ 89 (r o + re)
to a first-approximation,
(27)
so that the r s distance is, indeed,
a better
approximation to r e than r 0. Because r e < r 0 as described earlier,
re < rs < ro
will always be true for diatomics.
(28)
Moreover, according to Eq. (27), the r s
procedure is expected to correct for about 50% of the r 0 distance error. As an
26
HARMONY
example, if the I 0 values of H35C1 (the parent), D35C1 and H37C1 are used with Eq. (24) to obtain the H and 35C1 z-coordinates, the H-CI distance is found to be r s = 1.2783 A. Comparing this value to the r 0 and r e values in Table 3, it is clear that Eq. (28) is satisfied. Moreover, the r s procedure has removed about 60% of the r 0 discrepancy. As the latter example has illustrated, for the diatomic molecule more data are required to obtain an r s structure than are needed
for an re
structure.
Thus, there is no advantage to be gained generally in obtaining an r s structure for a diatomic molecule, especially since the result is still inferior to the r e value
because
of the
residual
uncancelled
vibration-rotation
terms.
Consequently, r s values of diatomic molecules are seldom computed or tabulated. For polyatomic molecules, even the simplest linear ones, there is no proof that the r s structure should be closer to equilibrium, but it has nevertheless been assumed to be the case. Because the Kraitchman equations rely most heavily on AI values upon isotopic substitution, it has seemed reasonable to suppose that even an incomplete cancellation of vibrationrotation terms would lead to a structure that was closer to equilibrium than the r 0 structures.
For this reason, and because r e structures are simply not
possible, the r s method has been used extensively to obtain structures for polyatomic molecules.
Unfortunately, although the early work of Costain
[29] suggested that the r s structure would be a fairly reliable representation of
MOLECULAR STRUCTURE DETERMINATION
27
the equilibrium structure, the passage of time has abundantly illustrated [410] that this was an overly optimistic view. There are a few obvious and not-so-obvious problems with the r s method. First, it is easy to show that small coordinates are determined very poorly (i.e., with large uncertainties) by Kraitchman's equations [23j. Indeed, an extreme case of this occurs when the value of the quantity in square brackets on the right side of Eq. (11) becomes negative, which frequently occurs due to the residual vibration-rotation terms exceeding in magnitude (but with opposite sign) the contribution from the equilibrium moments of inertia. What results then is an imaginary coordinate, which is, of course, physically impossible. In some cases, for example when the atom in question lies in a symmetry plane (say, the ab plane), it is evident that the value of the c coordinate of the atom can be physically set equal to zero. (We should mention that special symmetry-specific forms of Kraitchman's equations can be used for molecules with planes of symmetry that invoke the zero coordinates at the outset.
As one simple example, planar molecules
have c i = 0 for all atoms i, which results also in I a + I b - I e = 0 for any rigid planar molecule or instantaneously planar molecule.) On the other hand, the imaginary coordinate may arise accidentally (not symmetry related) simply because the atom lies near to a principal axis.
In this case, some other
procedure or assumption will obviously be needed to use the coordinate in obtaining a bond length, which, incidentally, is generally obtained from the coordinates of two atoms m and n by
r(mn) = {(am - an)2 + (b m - bn) 2 + (c m - Cn)2} 89
(29)
28
HARMONY
The most common procedure to solve the small coordinate problem, when only one small coordinate exists along a given axis, is to use the firstmoment (or COM) Equation (4).
Then if the problem lies with the c
coordinate of atom j, its value is obtained from
-~mic i
cj =
~ mj
(30)
where the sum omits atom j but includes all other atoms whose r s coordinates c i are determined from Kraitchman's equations. In this regard, it should be noted that the first-moment equations are not satisfied perfectly by complete sets of r s coordinates [29].
For example, in the PAS of the 16012C32S
molecule, the value o f E miz i is 0.0330 amu A using the r s coordinates [19]. This discrepancy is relatively small but nevertheless shows that the zeropoint vibration effects have not been entirely eliminated. It is evident that the r s method requires strictly that I 0 be obtained for all necessary isotopomers, that is, one for each symmetry-unique isotopic substitution.
This means that either the isotopomers must be observed in
natural abundance (frequently possible for 13C or 15N and sometimes possible for 180 and other less abundant isotopes) or they must be obtained by enrichment through chemical synthesis. In either case, the labor may be non-trivial, but has been accomplished for many molecules [24]. Of course, atoms without more than one stable isotope, such as fluorine (~9F) are excluded from the substitution method in its strict application.
However,
when only a single F atom is present, as in HC-CF, its coordinate can be
MOLECULAR STRUCTURE DETERMINATION
29
found from the first moment or COM relation as described above for the ccoordinate case (Eq. (30)). We will present numerous r s structural results later, but it is useful to discuss the r s structure of propane as a typical example. Table 7 presents the results along with those from the r 0 structure described previously and the rm 9 structure to be described shortly. It was mentioned that the r 0 structure was obtained from the 18 I 0 values of six isotopomers. These include the parent (or normal) molecule with all 12C and 1H isotopes and five more isotopomers for which each unique atom has been replaced by a heavier isotope (2H =- D). The six molecules are 1.
CH3CH2CH 3
2.
13CH3CH2CH 3
3.
CH313CH2CH 3
4.
CDsH2CH2CH 3
5.
CHsHaDaCH2CH 3
6.
CH3CHDCH3,
where for the methyl group s represents an in-plane H or D while a represents an out-of-plane H or D atom.
These six isotopomers and their
rotational constants (or moments of inertia, I0a, I0b, I0c) constitute what we call a substitution data set (SDS). For any molecule, an SDS consists of the minimal set of isotopic species needed to compute a complete r s structure. The data set outlined here was, in fact, used to obtain the r s coordinates and, hence, the r s structure parameters of Table 7. It might be noted at this point
30
HARMONY
that different SDSs are frequently possible for any given molecule.
For
example, in the case of propane, in place of molecule 6 the CH3CD2CH 3 doubly-substituted molecule would be suitable along with the equations of Chutjian [17].
(In fact, in this case, no other data are available, but the
particular example mentioned is not unusual because double-deuterium methylene substitutions are synthetically not uncommon.) Turning attention now to the r 0 and r s distances in Table 7, it is seen that the r s value for the CC distance is 0.008 A smaller, while the r s values for the CH distances are smaller by as much as 0.005 A. The magnitudes of these variations are typical, although it is easy to find examples for which the r 0 - r s differences are either larger or smaller.
Moreover, although in this
case the parameters are in accord with the diatomic result of Eq. (28), viz., r s < r 0, it turns out that when data are reviewed for many molecules, it is about as likely for r s to exceed r 0 as the converse. And there appears to be no way of knowing how the r s values compare to the theoretically well-defined r e value. In our extensive review [4-7] of polyatomic molecules whose r e and r s structures are both known (ignoring X - H distances, X = C, N, O, etc.), r s r e tends mostly to be greater than zero, with r s - r e _--_0.006 being the largest difference. On the other hand, about one-third of the parameters exhibited r s - r e < 0, with relatively small magnitude deviations of___- 0.002 A. The point we are making here is that there is really no simple a priori way of knowing how close (and in which direction) the r s structure is to the r e structure for polyatomic molecules. For this reason, although rs structures continued to be
M O L E C U L A R STRUCTURE DETERMINATION
31
the standard for polyatomic molecules for many years [24], it was clear that r s results were not going to provide the necessary precision and certainty in the future. Thus, a number of efforts began in the 1970s to find more reliable spectroscopic structural methods that might lead to distance parameters within 0.001 A of the unknown but desired r e value.
C.
r m structures In 1973, J. K. G. Watson [30] presented a new theory based upon isotopic mass effects to obtain a structure (known as rm) that was potentially a good approximation to r e. Consider for simplicity a linear molecule, and compute the s u b s t i t u t i o n
coordinate z i of each of the N atoms using an
appropriate SDS as previous described. Eq. (24) for the diatomic molecule applies for each of the N atoms of any linear molecule:
2
AI
Zsi = ~ gi
i- 1 ...N
(31)
Then define the moment of inertia I s for the molecule whose atoms have the coordinates Zsi:
I s = Zmiz2i
i = 1...N
(32)
Note that I s is not a directly measured quantity, but is perfectly well-defined and computable.
For simplicity also, define a vibration-rotation parameter
g g, which relates to the e g of Eq. (16) by (for v i = 0)
g = '~ t;g(di / 2) + .. i
(33)
32
HARMONY
Then for the linear molecule, with only a single axis of concern,
I0 = Ie + g = I0 _ Ie
or (34)
Then g measures the entire discrepancy (or error) between the measured ground state moment of inertia and the equilibrium moment. Now, Watson showed that if a new moment I m is defined by
I m = 2 1s - I 0
(35)
then I m is related to I e by
1 Ie = I m - ~ -
02(M~) miam i 0m 2 +...
Im 8
i =I...N
(36)
That is, I m is the zeroth-order approximation to I e and the sum (divided by M) represents the error to the first-order in the isotopic mass changes Am i. If this quantity and the higher order terms (all of which we define as 5) are sufficiently small, then I m will be a good approximation to I e and might then be expected to yield a structure close to r e. As usual, the first test of the theory used a diatomic molecule, CO [30]. For this simple molecule, I 0 data are available [31 ] for four SDSs as follows: 1.
12-16, 13-16, 12-18
2.
13-16, 12-16, 13-18
3.
12-18, 13-18, 12-16
4.
13-18, 12-18, 13-16
MOLECULAR STRUCTURE DETERMINATION
33
where the parent is underlined in each case. For each of these parents, the I s values can be computed using Eqs. (31) and (32), and then I m is obtained from Eq. (35). These various moments are summarized in Table 8 along with the experimental I e values [32]. From Table 8 it is apparent that I m is a very good approximation to I e in all four cases, especially compared to either the I 0 or I s values. In particular, for the
12C160 parent, we see for illustrative
purposes that = I 0 - I e = 0.0397 amu A 2 and
5 = I m - I e = - 0 . 0 0 3 6 amu A 2
Thus, whereas I 0 is in error by 0.45%, the magnitude of the error in I m is only 0.04 %.
It is interesting to note for CO that Watson's method has
overcorrected the original I 0 data, that is, 8 < 0. Nevertheless, the small I m error of 0.04% guarantees that the r m distances will be very close to r e. Table 9 summarizes the r0, r m and r e distances computed from the Table 8 values. It is evident that r m is, indeed, for this diatomic molecule, a very good, even an excellent, approximation to r e .
For our purposes, the number of
significant figures given in Table 9 is entirely suitable.
Watson [30, 32]
should be consulted for finer details involved with still smaller B-O corrections. The result for CO is essentially typical for diatomic molecules, i.e., the r m distance is expected to be within approximately 0.0001 A of the r e distance (except in the case of H-atom-containing molecules).
Note, of
34
HARMONY
T A B L E 8. Carbon monoxide moments of inertia a. Parent molecule
I0
Is
Im
Ie
12C16 0
8.76846
8.74680
8.72514
8.72873
13C160
9.17186
9.14999
9.12811
9.13122
12C180
9.20688
9.18492
9.16295
9.16617
13C180
9.65274
9.63059
9.60843
9.61105
aAll values in units of arnu A 2. See Ref. [31 ] for extensive ground state data and Refs. [30] and [32] for data used for the Is, Im, and Ie values.
T A B L E 9. Carbon monoxide interatomic distances a. Parent molecule
r0
rm
re
12C160
1.1309
1.1281
1.1283
13C 160
1.1308
1.1281
1.1283
12C180
1.1308
1.1281
1.1283
13C18 0
1.1308
1.1282
1.1283
aAll distances in A. Computed from data in Table 8.
course, that the I m structure, like the r s structure, requires more moment of inertia data than the r e structure, so once again the I m method has no advantage in general for diatomic molecules. The question then is how does the r m method work for polyatomic molecules?
The general answer is that although the procedure always
MOLECULAR STRUCTURE DETERMINATION
35
produces I m values that are relatively close to equilibrium, the resulting_ structures are often of uncertain quality. Why that occurs can be seen by a few examples. Consider SO 2 (C2v symmetry), for which extensive data are available. Using the most common SDS (32S1602, 34S1602, 32S180160), Im can be evaluated for both the a and b axes (and via the inertia defect the c axis as well [30]), yielding Ima and Imb values that differ from Iea and Ieb by 0.055% and 0.0040%, respectively.
Then, since SO 2 has only two structural
parameters, the two I m moment of inertia values permit rm(SO ) and 0m(OSO ) to be computed, yielding r m = 1.4307 A and 0 m = 119.33 ~ which are nearly identical to the equilibrium values = 1.4308 A and 0 e = 119.33. Similarly, excellent I m results will occur for 0 3 (C2v symmetry) and CO 2 (Dooh symmetry) or any molecule whose structure can be evaluated using a single SDS, and which does not contain hydrogen. The problem with hydrogen is that the substitution H--+D (or H---~T, D---~H, etc.) leads to 5 values (see Eq. (36)), which are too large to be neglected in comparison to the leading term I m in the expansion in powers of Ami. When data from more than one SDS are required, slight distortions in the I m values may be introduced by the neglected terms in 5 that fluctuate markedly in magnitude and sign due to the finite changes in Ami (see Eq. (36)). Two examples from Watson [30] and one from Kuchitsu [33] as reported by Harmony et al. [4] are presented in Table 10.
36
HARMONY
TABLE 10. Some r m structures a. re
rm N20b NN
1.1281 (8)
1.1284 (3)
NO
1.1842 (8)
1.1841 (3) OCS b
CO
1.1587 (13)
1.1543 (2)
CS
1.5593 (10)
1.5628 (4) COC12e
CO
1.1808 (62)
1.1756 (23)
CC1
1.7351 (28)
1.7381 (19)
zCICCI
112.08 (28)
111.79 (24)
aAll distances in A. bReported in Ref. [30]. eReported in Ref. [4] with data from Ref. [33].
For linear molecule N20, a minimum of two independent I m values and thus two SDSs with unique parents are needed. In fact, eight uniqueparent I m values were available, so the complete set of data was used in a least-squares fit to give the tabulated values.
In this case, the agreement
between r m and r e values is very good, so apparently the effects of the neglected terms lead to no significant distortions. N20 is an interesting case (as mentioned earlier), because it has a very small coordinate for the central N atom. Indeed, when 15N14N160 is used as the parent, the substitution of 15N for 14N leads to AI < 0, which means the coordinate is imaginary by Eq.
MOLECULAR STRUCTURE DETERMINATION
37
(31). In Watson's treatment, these small or imaginary coordinates are not dropped, set equal to zero, or recomputed by some other method (such as COM relation); in fact, the values from Kraitchman's equations (Eq. (31) for the linear molecule) provide the essential mass-dependent vibration-rotation contributions to the moments of inertia. Now, while the N20 results are very good, those for the entirely analogous OCS case (using again eight unique parents) are really rather poor, with CO and CS variations of +0.0044 and-0.0035 A, respectively. Here, the small distortions in the I m values produced by the neglected terms in 5 lead to substantial structural distortions, even though all I m values are within 0.02% of I e. This discouraging state of affairs was investigated theoretically by Smith and Watson [34], who showed by model force-field calculations that the trouble was an unavoidable consequence of the neglected finite masschange terms Am i in 5. Indeed, if one extrapolated to Ami = 0 (by using two different substitution patterns for each atom, i.e., 160 ~ 170 and 180, 12C 13C and 14C and 32S ~ 33S and 34S), so that 5 ~ 0, then I m ~ I e and the r m structure approaches very closely the r e structure! Thus, Smith and Watson [34] concluded that one could not be confident in general that r m parameters would be even within 0.001 A of the equilibrium values unless the results could be obtained entirely from the I m value(s) of a single SDS. As a final example of a still more complicated molecule, Table 10 shows the results for COC12. Here there is much data available, so that eight parent-molecule SDSs are fully available, yielding
38
HARMONY
eight unique sets of Ima, Imb and Imc values (of which only two of the three are considered independent for the planar molecule). Again, a least-squares fit yields the r m values of the two distances and the bond angle. While a complete spectroscopic determination of r e is not available, Kuchitsu [33 ] has obtained a reliable structure (although with rather large uncertainties) by combining spectroscopic and electron diffraction data. Note that the r m and r e distance values differ in the range 0.003-0.005 A.
The deviations are
within the listed errors (least-squares a for rm), of course, but the point is that the rm structure has such large uncertainties that it is not useful as an estimate o f r e. When the r m procedure is applied to molecules involving hydrogen, the H ~ D substitutions that are required may lead to rather large changes in the mass-dependent contributions and, hence, the neglected terms ~5 may be especially significant. Even for the diatomic molecule HCI, the r m distance computed for the H35C1 parent is ~0.0015 A smaller than r e, a deviation ten times greater than for the CO example described earlier. Of course, for Hcontaining polyatomics the problem is doubly troubled if more than one SDS is necessary. Thus for HCN, Watson [30] found that the r m values of the CH and CN distances were in error (compared to re) by -0.0051 and +0.0012 A. Thus, although the r m method had great promise, it led, unfortunately, to no real advances in distance determinations, especially for polyatomic molecules.
Still, because the method undeniably yields Im values that are
much nearer to I e than either I 0 or Is, the method continued to attract attention
MOLECULAR STRUCTURE DETERMINATION
39
and one wondered whether somehow the method might be "fixed up" in some way. In closing this section, it is perhaps worthwhile to note that the isotopic data requirements of the r m method are rather extreme, so much so that the method would probably not be considered for most polyatomic molecules, even if the resulting structure were reliable. Consider chloroacetylene, H C=C-C1.
For this species, with three structural parameters, three unique-
parent SDSs would then be required to obtain the necessary three I m values. This would require, in tum, that the spectra of a total of ten isotopomers be analyzed to obtain their B 0 and, hence, I 0 values. Suitable SDSs would be, for example: 1.
1-12-12-35, 1-13-12-35, 1-12-13-35, 1-12-12-37, 2-12-12-35
2.
2-12-12-35, 2-13-12-35, 2-12-13-35, 2-12-12-37, 1-12-12-35
3.
1-12-12-37, 1-13-12-37, 1-12-13-37, 1-12-12-35, 2-12-12-37,
where the unique isotopomers are underlined. Note that each SDS yields an I s value and thus an I m value for the parent molecule (always the first listed). If the r m structure were desired for the most stable planar conformer of formic acid (Fig. 3), the simplest organic carboxylic acid with seven independent structural parameters, a total of four unique-parent SDSs would be necessary in order to yield four pairs of (Ima, Imb) values. If four suitable SDSs are written down, one finds that a total of 16 isotopomers must be fully rotationally analyzed in the ground vibrational state. The impracticability of this procedure is only exceeded by the task of obtaining an r e structure,
40
HARMONY
O
H
/
O
H FIG. 3. Structural sketch of most stable conformation of formic acid.
namely, the ground and nine excited vibrational states must be fully rotationally analyzed for four isotopomers, a total of 40 rotational spectral analyses! D.
Miscellaneous Methods The next section will present the major emphasis of the chapter, viz., the scaled moment of inertia or rm 0 structure method. We should mention just a few additional spectroscopic structure procedures that have been used in the past.
First, the double-substitution method of Pierce [35] was
developed to handle the small-coordinate problem of the r s method. In this procedure, essentially, the small coordinate is determined in two different axis systems using two different pairs of isotopomers.
Then, upon
subtracting a second time (to yield AAI-type terms) and relating the two
MOLECULAR STRUCTURE DETERMINATION
41
independent axis systems, the small coordinate is obtained with reduced influence of the vibration-rotation terms.
The method leads to much
improved values for the troublesome small r s coordinates at the expense of more experimental work (assignment and high-precision analysis of more isotopomer spectral data). It does not change the overall quality of the r s method, but certainly provides a way of solving the small coordinate problem. Often, especially for polyatomic organic molecules, it is not feasible or practical to obtain a complete set of substitution structure data, or perhaps there are atoms such as fluorine that cannot be substituted. In these cases, one might nevertheless have data for several isotopomers.
For a general
asymmetric rotor, if the number of independent moments of inertia exceed by at least three the number of independent internal structure parameters, then a least-squares fit of AIg data leads to a structure that has the qualities of the r s structure. Such a method, which might be labeled as rAl , has been proposed and described by N/3sberger et al. [36] and other workers [37]. For example, consider ethylene oxide of Fig. 1.
(C2H40), the C2D40 isotopomer
I f the spectra of the normal species and the
12C13CH40 isotopomer
are
analyzed, a total of nine independent moments of inertia are obtained. Then six AIg values can be obtained (where AIg = Ig (isotopomers) - Ig (parent)), which may be used in a least-squares fit to obtain the "best" values of the five independent structure parameters.
Because only AI values are used, it is
reasonable to expect that the vibration-rotation terms will cancel out to first order as in the r s method. In fact, if a complete SDS is used, the rAI method
42
HARMONY
will lead to a structure very similar (but not identical) to the r s structure. Rudolph [38] has discussed this and a related method in great detail. In the related case, he suggests taking the ~g of Eq. (33) to be isotope independent (this is not true), so that for a given molecule only three values (~ a, g b, ~ c) exist, no matter how many isotopic Ig0 values exist. Then Rudolph proposes to do a least-squares fit of the experimental Ig0 values to obtain the best values of the independent structural parameters plus the three ~ g values. Thus, for ethylene oxide again, the same set of data as discussed in the previous paragraph (nine Ig0 values) could be used to obtain the best values of the five structure parameters and the three ~ parameters. This structure method, which might be called re, I , is intrinsically appealing because it explicitly includes (even if incorrectly) the offending vibration-rotation terms as fitting parameters.
Interestingly, as described by Rudolph [38], the re,I
method leads to structural results that are identical to those of the rAi method for the same input set of data. Neither Pierce's double-substitution method nor the rAI or re,I procedures provide results that are particularly good approximations of the equilibrium structure, basically being variations of the r s procedure. Nakata et al. [39] have pointed out the importance of using complementary data sets when applying Watson's r m procedure. Here, it was noted (actually, Watson [34] made the same observation) that with proper selection of data the effects of the neglected 5 term in Eq. (36) could be minimized.
Consider OCS as an example.
Then the I m value for
MOLECULAR STRUCTURE DETERMINATION
43
16012C32S has zxmi terms in 5 of +2, +1, +2. Now consider the I m value for 18013C34S. In this case, the Ami terms of 5 have the values - 2 , - 1 , -2. Now the other factors in 8, in particular the second derivative, should not change much from one isotopomer to another. Therefore, it is expected that ~5(16012C32S) ~ - 5(18013C34S) These two isotopomers are "complementary" pairs.
Similarly, 16013C32S
and 18012C34S are complementary pairs. Nakata et al. [39] propose using sets of complementary-pair data to evaluate the results, which in effect leads to a cancellation of the 5 values to first order. The procedure, known as the r e structure method, has very limited applicability because of the increased data demands, but has been found to be useful in a few cases such as C120 [39] and COCI 2 [33]. However, the still higher-order terms in ;5 are potentially troublesome and even in simple cases are apparently important. In a very thorough analysis of OCSe, Le Guennec et al. [40] have shown that the r c structure is not of high quality (i.e., is not very near to re) unless additional procedures are utilized to cancel out the higher order terms. Finally, we should mention a procedure whose aim is not to approximate the equilibrium structure but, rather, to evaluate the average structure rz in the ground vibrational state by correcting the measured I0s. It has been shown, correct to terms of order (v + 89 that the moment of inertia about the g-axis of a molecule with all atoms in their average ground state positions, I'g, is given by [41-44]
44
HARMONY
and
I*g = Ig0 - ~ g (harm)
(37)
I*g = Ige + ~ g (anharm)
(38)
where
g = ~ g (harm) + g g (anharm)
(39)
is defined in the v = 0 state by Eqs. (33) and (16). Here the key feature is that the vibration-rotation terms ~ can be broken up into a portion containing only the harmonic contribution to the vibrational potential function and a part containing everything else (the anharmonic part). Moreover, Eq. (37) shows that only the harmonic part of g must be known to correct the experimental effective ground state moments of inertia to give the moments of the average configuration.
While not trivial, these harmonic correction terms (which
include the Coriolis terms) can be made in a straightforward manner from the usually well-known harmonic force field (e.g., Wilson's F matrix [45]). Then the moments I*g can be used to obtain the atomic coordinates in the average configuration and, hence, the average values of the structural parameters designated r z. For the diatomic molecule, it is perhaps useful to summarize the definitions [ 16, 41 ]: I 0 = lxro2 ~, ~t
I* o =
= ,<0o
In order to see the behavior of the r0, rz and r e coordinates for the diatomic molecule, Table 11 presents a few results quoted by Laurie and Herschbach [42]. The key object of the data is to show, in agreement with theory [16, 41], that
re < ro < rz
(40)
MOLECULAR STRUCTURE DETERMINATION
45
TABLE 11. Diatomic distances a. Molecule
rz
r0
re
HF
0.9326
0.9257
0.9170
DF
0.9284
0.9234
0.9171
12C160
1.1323
1.1309
1.1282
13C160
1.1322
1.1308
1.1282
aData from Ref. [42]. All distances in A.
for diatomic molecules.
Also, r z shows clearly the same type of isotope
effects shown by r 0, namely, a marked shrinkage when H is replaced by D and a much smaller shrinkage for the heavy atom 12C ~ 13C substitution. Finally, it should be noted that with improved quality ab initio calculations it has been possible to compute the vibration-rotation parameters a~ in Eqs. (15) for small molecules.
This requires evaluation of the
molecular (vibrational) anharmonic force field as well as the harmonic portion. (See Eq. (39)). Then, with ab initio c~s in hand, the experimental B0s can be transformed to B e values and thus used for the r e structure calculation.
A recent example is that of Botschwina et al. [46], who
determined the r e structure of FC-CH in this manner.
Such combined
theory/experiment equilibrium structure determinations certainly provide an important new procedure for obtaining high quality results.
46
HARMONY
E~
rOm structures
1.
Equilibrium structures ofpolyatomic organic molecules The preceding, somewhat historical, review of spectroscopic structural methods is more-or-less complete up to the 1980s.
Of
course, many new experimental tools had appeared by this time and were leading to additional high-quality data, often with unprecedented precision. These included Fourier-transform microwave spectroscopy [47] and numerous laser-based high-resolution infrared and UVvisible spectral methods [48, 49]. Much of the new effort has been aimed at the investigation of atomic and molecular clusters [50] and unstable or transient molecules (often three-or four-atom systems) including excited state species [51].
And while the equilibrium
structures of many diatomic molecules, numerous triatomic molecules and a few tetratomic or pentatomic molecules were known from spectroscopy, the equilibrium structures of organic polyatomic molecules (even small ones such as those of Figs. 1-3) remained largely unattainable from experimental spectroscopic data. It remained, in fact, a great disappointment that the eighteen experimental, very high-quality I 0 values of six propane isotopomers could not yield a more dependable structure than that represented by either the r 0 or r s structures of Table 7. Moreover, since Watson had shown clearly that the mass-dependence of the I 0 values of isotopomers carried information about the equilibrium structure [30],
MOLECULAR STRUCTURE DETERMINATION
47
it seemed worthwhile to investigate whether it might be possible to salvage, at least, near-equilibrium structures from such data. This led us to the scaled moment of inertia procedure outlined in the following sections. It is important to note that our goal was the development of a procedure for determining reliable, high-quality, near-equilibrium structures of polyatomic organic molecules. This effort led us into the treatment of numerous non-organic triatomic species Oust as in Watson's work [30]) for which the availability of bona fide r e structures provides necessary and important comparison tests.
The
new rPm procedure is not meant as a replacement for a bona fide r e structure when available; rather, it is meant to provide a reliable estimate of the equilibrium structure (a near-r e structure) for the vast majority of molecules (especially organic) for which a bona fide r e structure does not exist. In fact, as will be shown, the r Om structure is expected to provide one of the best estimates of the equilibrium structure of a molecule and should provide valid comparisons to estimates from other procedures, such as pure ab initio quantum mechanics or electron diffraction.
As will be seen shortly, the r pm
structure is obtained from the same data set (typically from microwave spectroscopy) that is used for the calculation of a complete r s structure (every unique atom determined by isotopic substitution, no use of first-moment relations, etc.). Such data are very commonly available for polyatomic molecules, and thus the r ~ structure turns out to be
48
HARMONY
rather easily determined.
It will be demonstrated in the following
sections that the r Pm heavy-atom
parameters
(non-hydrogen-
containing) are always likely to be as good as or better estimates of r e than the r s parameters.
Thus, the r0rn structure should always be
reported in addition to, if not in preference to, the r s structure. Moreover, as in Watson's method [30], small coordinates neither present special problems nor require any special methods in the r pm procedure.
2.
Some Pivotal Observations Because
Watson's
[30] mass-dependence
structure
method
invariably leads to I m moments of inertia that are close to equilibrium, we felt that the procedure merited further consideration, even though data demands are too great for polyatomic organic molecules and even though the fluctuations in 5 from one parent isotopomer to another cause serious problems in real applications.
Upon reviewing the
available data for several small heavy-atom molecules (we use the term to mean non-hydrogen-containing), it was apparent that the Is/I 0 ratio for any particular axis of any molecule was independent, to a first approximation, of which isotopomer was chosen as the parent for the substitution calculations. Table 12 shows these data for OCS while Table 13 presents the results for COC12 for the a- and b-axes. For OCS it is seen that the ratios deviate from the normal isotopomer
MOLECULAR STRUCTURE DETERMINATION
49
TABLE 12. Values of ls/I 0 for OCS isotopic species a.
Isotope
Is/I0
Isotope
Isfl0
16-12-32
0.998502
18-13-32
0.998547
16-13-32
0.998483
16-13-34
0.998520
18-12-32
0.998546
18-12-34
0.998587
16-12-34
0.998527
18-13-34
0.998599
aComputed from Table VII of Ref. [30]. Reprinted from Ref. [4] with permission.
TABLE 13. Values of ls/I 0 for COC12a. Isotope
[Is/I0]a
[Is/I0]b
16-12-35
0.9982778
0.9980422
16-12-37
0.9982626
0.9980840
18-12-35
0.9983278
0.9980648
18-12-37
0.9983057
0.9980866
16-13-35
0.9982212
0.9979832
16-13-37
0.9981986
0.9980114
18-13-35
0.9982470
0.9979820
18-13-37
0.9982228
0.9979872
aReprinted from Ref. [10] with permission.
value by only approximately + 1 part per ten thousand, i.e., +1/104. Similarly, for COC12 the a-axis values deviate from that of the normal isotopomer by somewhat less than +1/10 4, while the b-axis values behave in a like manner. (But note that the ratios are different for the
50
HARMONY
a- and b-axes.)
Entirely analogous results have been observed for
other small molecules. Thus, to within about +1/104, for any axis of any heavy-atom molecule, the Is/I 0 ratios are independent of which molecule is taken as the parent. But this means that the Im/I 0 ratio (for each principal axis) is constant to within approximately + 1/104 for any isotopomer of a given molecule. From Eq. (25), for the g = a, b, c, principal axis,
Img/Iog = 2 Isg/IOg- 1
(41)
So, now let us stress and enlarge upon what this observation says. If one merely computes Img using the single complete SDS for any one convenient isotopomer, say, for example, the normal isotopic species with the most abundant isotopes, the computed ratio Img/I0g could be assumed to be valid to within 1 part in 104 for any other isotopomer. Thus, Irng could be predicted for any other isotopomers (for which the Iogs are known experimentally) to an accuracy of about 1 part in 104. But, as described in Section IIIC, I m is expected to be within a few (to several) parts in 104 of the equilibrium moment I e. Thus, a determination of the Img values for some particular isotopomer leads to predicted values of Img for any other isotopomers that are expected to be excellent approximations (within a few parts in 104) of the desired equilibrium values. For example, if only the I 0 values of the SDS of the normal isotopomer of OCS were experimentally known
MOLECULAR STRUCTURE DETERMINATION
51
(namely, the SDS consisting of the 16-12-32 parent plus the additional three species 18-12-32, 16-13-32 and 16-12-34), the I m for this isotopomer could be used to predict I m for the 18-12-32 species without the need for obtaining any additional I 0 values. (In order to compute the true Watson I m of the 18-12-32 molecule, all the l0 s of the SDS consisting of the 18-12-32, 16-12-32, 18-13-32 and 18-12-34 isotopomers must be determined experimentally.) Whether or not these approximate Watson Im s are useful for structural computations will be addressed in the next section. However, it might be observed here that Watson's [30, 32] r m structure tended to be very good when only the Img values of a single SDS were needed. The observations noted above permit one to obtain near-equilibrium Ig values using only a single SDS, so perhaps one should not be surprised that the resulting structures are of rather high quality. Before moving on, we should address the issue of hydrogencontaining molecules. All of the conclusions above deal with heavyatom systems. Consider now the data of Table 14 for HCN. For this molecule, data are available for some tritium as well as deuterium species and, because of this, one can obtain I s (or Ira) values for some isotopomers in more than one way, i.e., using different SDSs. Thus note, for example, that the I s (or Im) value of the 2-12-14 parent molecule can be determined using either of the SDSs:
52
HARMONY
TABLE 14. Values of ls/I 0 for HCN isotopomersa, b. Species
H---~D
H---~T
Species
D---~H
D---~T
1-12-14
0.9979301
0.9980224
2-12-14
0.9979619
0.9982530
1-13-14
0.9979501
0.9980456
2-13-14
0.9979597
0.9982635
1-12-15
0.9979704
0.9980646
2-12-15
0.9979984
0.9982962
1-13-15
0.9979924
.... c
2-13-15
0.9979895
.... c
T-}H 3-12-14
0.9982330
T--~D 0.9984132
aEntries are identified by hydrogen atom substitution used in determining Is. Heavy atom substitutions are 12 ~ 13 and 14 <-->15 in all eases. bComputed from ground state data tabulated in Ref. [23]. Reprinted from Ref. [6] with permission. CData for TI3CISN molecule not available.
2-12-14, 1-12-14, 2-13-14, 2-12-15 or 2-12-14, 3-12-14, 2-13-14, 2-12-15. Overall, the variations of the Is/I 0 values are several times greater than those observed for the heavy-atom molecules in Tables 12 and 13 where, incidentally, the O, S and C1 substitutions involve Am = +2 mass units.
Actually, if one limits the observations to the species
evaluated with H --~ D substitution and D --+ H substitution, the range of variation is only about 6/105.
In any case, Is/I 0 is still fairly
constant so that, even for the hydrogen atom species, I m can be approximated to within a few parts/104 from the results for, say, just the normal isotopomer. The problem is that the true Watson Im values
MOLECULAR STRUCTURE DETERMINATION
53
themselves are deficient because of the neglected terms in ~5and, thus, are rather discordant approximations of the I e values. For example, Watson's Table XI data [30] show that I m for the HCN molecule is greater than I e by 0.03%, while for the DCN species the I m value is less than I e by 0.08%.
This type of sign change is, as mentioned
earlier, caused by the neglected Ami terms in Eq. (36). We will find that a special treatment will be necessary to obtain reliable I values for the isotopomers of hydrogen-containing molecules.
3.
rgm Heavy-atom Structures Based upon the results described in the previous section, we define a scaling factor pg for each of the three (maximum) principal axes by
pg = [Isg]l/[IOg]l
(42)
where the subscript 1 represents the parent (usually the normal isotopic species) molecule of a complete substitution data set (SDS). Isg is computed from a complete r s (Kraitchman) determination. For a linear molecule, Eqs. (31) and (32) describe the calculation, in which case no subscript g is needed (g = b here, with only B 0 (and Ib0 - I0) existing experimentally). For a polyatomic asymmetric rotor with N atoms,
Isa = E m i ( b 2 + c 2) i
i = 1...N
(43)
54
HARMONY
and analogous expressions for Isb and Isc by cyclic permutation of a, b, c. Then, based upon Eq. (41) and the related discussion, a moment of inertia I0m is defined for each of the L members of the SDS as an approximation to I m and, hence, to I e.
[IPmg]a = (2pg - 1)[Iog ]cz
ot = 1 . . . L
(44)
Note that [I 0mg]1 is an exact Watson I m as can be seen from Eqs. (42), (41) and (35). We explicitly limit our data set to the L isotopic species of an SDS needed for a complete substitution structure. The resulting IPm values are expected for heavy-atom molecules to be excellent approximations of I e for each of the L members of the SDS, perhaps with errors limp - Ie[ o f ~ 0.01--0.05%. Given now the I0m values for an SDS of a particular molecule, the moments are then used in a least-squares fit to determine the structural parameters (distances and angles). Note that the L I Pm values of any SDS will always be sufficient to determine all structure parameters. (After all, the L members of the SDS were selected to give the r s structure!)
The resulting structure is termed an rPm structure.
The
logic of using a least-squares fit rather than a Kraitchman-type computation using I Pm values has been described previously [4-10, 30, 34]. Basically, the least-squares fit provides a balanced, unbiased use of all the data.
M O L E C U L A R STRUCTURE DETERMINATION
55
Table 15 summarizes the key structural data for five linear triatomic molecules reported by our group and others. In all cases we choose the normal isotopomer as the parent (nuclei = 12C, 160, 32S, 14N, 78Se, 79Br) and the substitutions are 13C, 180, 34S, 8~ 81Br.
and
It is easily observed that the r Om distances are excellent
approximations to the r e values, including BrCN, which suggests in this latter case that the reported r e values are correct even though they have large uncertainty. In this regard, our procedure is to list the leastsquares standard deviation as the uncertainty on r Om. For r e, we list the original author's values while for r s we list no uncertainty since ~iae precision of r s is not our concern here. Overall, the rPm distances are within 0.001 A of r e except for one case in which the difference is 0.0011 A.
For the nine distance
parameters of Table 15, the average I m - r e [ deviation is 0.0006 A. CO 2 has been included just to show that the I ~m values (SDS = CO2, 13CO 2, C1802) are all very consistent (o = 2 x 10-5 A for the r Om). Of course, Watson's r m value would be essentially identical to the rOm value because the r m method does fine when only one isotopomer is needed. It is also important to dispel any notion that r s structures are reliable approximations to r e. They may or may not be. For OCS and OCSe, the results are quite poor, while for CO2, even, the deviation is
56
HARMONY
TABLE 15. rPm structures for some heavy-atom linear molecules a.
rs
rm p
re
OCS b CO
1.1605
1.1551 (11)
1.1543(2)
CS
1.5596
1.5621(9)
1.5628(4)
N20b NN
1.1466
1.1277(12)
1.1284(3)
NO
1.1695
1.1846(11)
1.1841(3)
CO2 c CO
1.16133
1.15995(2)
1.15979(10)
OCSe c CO
1.1566
1.1524(14)
1.1535(1)
CSe
1.7092
1.7107(11)
1.7098(1)
BrCN d BrC
1.7891
1.7891 (4)
1.7888(48)
CN
1.1586
1.1571 (6)
1.1577(69)
aDistances in A, angles in degrees. bSee Ref. [4]. cSee Ref. [5]. dSee Ref. [52].
0.0015 A (compared to the rPm deviation of 0.00016 A). For N20, the small coordinate of the central nitrogen leads to a terrible pure r s structure. If the usual expedient of recomputing the central nitrogen
MOLECULAR STRUCTURE DETERMINATION
57
atom coordinate with the first moment equation is taken, NN = 1.1286 and NO = 1.1876 A, so the r s NN value is very close to r e, but the r s NO distance is still 0.0035 A too long. Accepting the modified N20 values, the average I r s - r e l deviation of the nine distances in Table 15 is 0.0022 A, nearly 4x greater than the rgm deviation. Le Guennec et al. [40] have re-examined OCSe in great detail with the inclusion of much new, high-quality data.
They find that if the 18012C78Se
isotopomer is replaced with the 17012C78Se species (data not available when we first reported the results of Table 15) the rgm values of CO and CSe become 1.1537 and 1.7095 A, respectively, reducing the errors to even smaller values. The latter point raises an interesting and important issue. We will discuss the matter further at a later time, but here it is worth noting that the rPm structure Oust as for the r m structure [30, 34]) is expected to be of the highest quality when the isotopic substitution mass changes are minimized. In Watson's theory, this means that the Ami terms in 5 should be held at _1 if possible, such as 12C ~ 13C, 14N <-~ 15N, 160 o
170 (rather than 160 o
180) . Of course, for C1, S and
Br, the most abundant isotopes differ by 2 mass units, so one must take what is available or practical. In addition to minimizing Ami, the most precise structures are likely to arise when all the isotopic mass changes Am i are of the same sign. Thus, ideally, one should choose the parent of the SDS such that each of the nuclei have either the
58
HARMONY
smallest or the largest possible mass. For many common nuclei (C, N, O, H, S, C1), the most abundant isotope is also that of the lowest mass. Therefore, the so-called normal isotopomer is often the natural choice for structure determinations. Before trying to develop some insight into why the r0m structures seem to be such good approximations to r e, let us see if the method carries over to heavy-atom polyatomic non-linear molecules. Table 16 summarizes
some
results
for
a
few
heavy-atom
non-linear
polyatomics. Not very many of these species exist for which reliable r e structures are known.
Two we have omitted are 03 and C120,
which are entirely analogous to SO 2 in that either the r m or r pm method works perfectly well. A third molecule that we mentioned in our earlier paper (NOCI, [5]) has been found to have some data ambiguities that make it impossible to obtain a valid comparison [53]. All of the computations for the planar asymmetric rotors of Table 16 have been made using the ground-state rotational constants for the aand b-axes, i.e., I0a and I0b.
These values, when used with
Kraitchman's planar asymmetric rotor equations [15], or, in the case of COC12, Chutjian's equations [ 17], yield Isa and Isb values that lead to values for Pa and Pb. These, in turn, lead via Eq. (44) to the I oma and I Pmb values for all members of the SDS and then to the structural parameters in Table 16.
It is important to stress that although
MOLECULAR STRUCTURE DETERMINATION
59
TABLE 16. rOm structures for some heavy-atom non-linear molecules a.
rs
rm p
re
COC12b CO
1.1836
1.1767(31)
1.1756(23)
CC1
1.7346
1.7371(14)
1.7381(19)
LC1CC1
112.24
111.89(14)
111.79(24)
SO2c SO zoso
1.4315 119.44
1.4308(0) 119.33(0)
1.4308(1) 119.33(1)
$20d SS
1.8840
1.8840(2)
1.8842(1)
SO
1.4578
1.4554(3)
1.4562(1)
ZSSO
118.00
117.89(1)
117.88(0)
aDistances in A, angles in degrees. bSee Ref. [4]. eSee Ref. [5] and [54]. dSee Ref. [5].
planarity is assumed, Kraitchman's planar molecule equations do not require assumption of an axis of symmetry. Thus, small non-zero and imaginary r s coordinates that may arise for any atom on the symmetry axis are carried along in the calculation of Isg. It may be of interest for calibration purposes to note that the (Pa, Pb) values of COC12, SO 2
60
HARMONY
and
$20
are, respectively,
(0.9982778,
0.9980421),
(1.002009,
0.997988), and (1.000434, 0.998256). The SDSs are" for COC12:
12-16-35-35
13-16-35-35 12-18-35-35 12-16-37-37 for SO 2"
32-16-16
34-16-16 32-18-16 for $20:
32-32-16
34-32-16 32-34-16 32-32-18 Now, it is easily seen that the r0m structures of all three molecules are in very good agreement with the r e values. Indeed, for SO 2 and $20 the results are excellent. In the former case, of course, Watson's r m structure is essentially identical to r0m . As for CO2, we include this example to illustrate that the r Ore structure is perfectly consistent with r m for the few types of molecules that yield valid r m structures. Note again, also, that in each case the r s structure is a poorer representation of r e than is r0m, although for several parameters the results are not bad. COC12 is clearly a severe example of the problems sometimes encountered with r s structures. It is noteworthy that all r Om distances in Table 16 are within approximately 0.001 .h of the r e value with an
MOLECULAR STRUCTURE DETERMINATION
61
average deviation of 0.0006 A compared to the r s average deviation of 0.0025 A. Thus, the r0m scaling procedure appears to be effective for nonlinear (planar) polyatomic species as well as for linear molecules. This is especially interesting since now two parameters (Pa and Pb, typically) must be used in the scaling and, as shown in the examples, the scaling is different along the various axes. Interestingly, for SO 2 and $20 , the scaling factors 2 P a - 1 are greater than one, while the factors 2 P b - 1 are less than one. Thus, the ground, state molecules must be stretched along one axis (a) and compressed along the other axis (b) to obtain the equilibrium molecules. Molecules such as SO2 with extensive high-quality data provide the opportunity for testing various data options. In particular, if other isotopic substitutions are used, the changes found are generally quite small. Thus, if the S160170 species is used in place of S160180, r pm (SO) increases by 0.0001 A; while if the S180180 or S170170 species is used with Chutjian's equations in place of S160180, there is essentially no change in rmP(SO). On the other hand, the use of 3381602 in place of 34S1602 leads to a bond distance decrease of 0.0003 A. Kraitchman's planar equations [15] can, of course, be written in terms of (Ia, Ib) , (Ia, Ie) or (Ib, Ie) moments. Thus, data inputs other than our choice (Ia, Ib) might be made. If (Ib, Ic) ground state data are
62
HARMONY
chosen, then it is best to compute Pb and Pc directly from Isb and Isc, and then to proceed to compute I Pmb and I Pc. That is, choose an axis pair (g, g') and maintain it throughout the entire calculation.
This
minimizes structure distortions that may otherwise arise due to the
inertial defect. Thus, if (I0a, I0b) data are used to compute (Isb, Isc ), which then yield (Pb, Pc) and thus (I Pmb,I Pmc), structure distortions may arise unless the inertial defect As upon isotopic substitution is properly accounted for. This can be done by computing Isc from Isc = Isa + Isb + As, where As is computed as described by Watson [30]. However, based upon our experience, the more complicated procedure has no advantages; we find it best to simply maintain a pair of (g, g') axes throughout the evaluation of (I 9rag, 19rag,). All input data being of similar quality, the choice should make little difference, but in this case we prefer to use (I a, Ib) data. For SO2, the consistent use of (Ia, Ic) or (I b, Ie) data leads to changes of no more than 0.0002 A in r om (SO) [54]. For $20, the largest distance change found is for (I a, Ic) data, which leads to a 0.0006 A change. For heavy-atom molecules such as illustrated in Tables 15 and 16, it is evident that the rPm procedure leads to structures of very high quality, i.e., the rPm structures are very close (within 0.001 A, typically) to the r e structures. The results show that the r Pm structure is not only always as good as, or substantially better than, the
MOLECULAR STRUCTURE DETERMINATION
63
corresponding r s structure, but also appears to avoid the problems associated with the r m structure as described by Watson [34]. Recall for the linear molecule that the problems of the r m structure arose from the leading Am i terms in 5 = I m - I e (see Eq. (36)), which tend to fluctuate in sign from one isotopomer to another. Thus, I m for 16012C32S uses Am c = +1, while I m for 16013C12S uses Am c = - 1 , which leads to terms in ~5that fluctuate in sign. The result, as discussed in detail by Watson [34], is that substantial structural distortions are likely to occur when I m data from more than one isotopomer are needed, even though the I m values are essentially equally close to the equilibrium values for all isotopomers. For the rPm method it has been shown (for the linear molecule) that the error in IPm for any number of the selected SDS is given by
I0 E=IOm - I e = ~ - - - ( ~ ' - 5 ' ) I 0'
(45)
where the primes refer to molecule 1, i.e., the parent isotopomer of the SDS [4, 5]. The right-hand side of Eq. (45) is not easily evaluated, but note that the error no longer depends upon 6 for each molecule but only 5 for the parent molecule (i.e., 5').
Instead, the error now
depends upon g for each isotopomer, which has its own complex mass dependence. Still, it is possible that the errors in I Pm values may
64
HARMONY
undergo smoother changes from one isotopomer to another than do the errors in the I m values. Consider an idealized case for which the errors for the various isotopomers have the constant fractional value (I Pm - Ie)/ie = E/Ie - ?. Then each I Pm is related to I e by I Pm = (1 + ?)I e, and a least-squares fit of the L I Pm values will lead to distance parameters R = (1 + ?)89 e. For a constant fractional error o f ? = 5 x 10-4 (which is typical for I om values), the error in R would then be ~2 x 10-4 A for R e ~1 A. On the other hand, for a real situation the fractional errors may not be constant and, even though the errors may be small (~10-4), variations (especially in sign) may lead to unusual distortions in the structure. This problem occurs when Watson's Im values from different isotopomers [34] are used and leads to the unreliability of r m structures. Based upon the results of Tables 15 and 16 and other examples, it appears that the rPm structural procedure avoids this problem, which suggests that the I Pm values form a more homogeneous (less fluctuating) data set than do the I m values. In order to gain insight into this matter, calculations of the vibration-rotation parameters [5, 6] were performed for several linear XYZ and YXY molecules using model anharmonic force fields and conventional first-order theory [41, 42, 55, 56]. With this procedure, all parameters of interest (I0, Ie, Is, Im, I Pm, ~, 8, E) could be
MOLECULAR STRUCTURE DETERMINATION
65
computed using an assumed re structure and complete (through cubic terms) anharmonic force field. The details can be obtained from the original work [5, 6]. The key features that result are the following: a)
Watson's Im data sets are indeed likely to be unsuitable due to the fluctuating terms in 5.
(Watson showed this in earlier
computations [34].) b)
p I m data sets exhibit relatively smooth fluctuations (errors) when the parent
isotopomer
is chosen
such that all required
substitutions lead to either increases in all isotopic masses or decreases in all isotopic masses. Thus, for OCS, either the 16012C32S-parent SDS or the 18013C34S-parent SDS lead to r ~ distances within 0.0002 A of the r e distances, whereas other parent choices may be unreliable.
For
example, the 16013C32S-parent SDS leads to results that are not too bad (Ir~ - rel = 0.0007 and 0.0005 A), while the 18012C32S-parent SDS gives a seriously distorted structure with distance errors of approximately 0.0025 A. The net result is that both model computations and bona fide equilibrium structure determinations show that the r Om structure is a good representation of the equilibrium structure for heavy-atom polyatomic molecules.
Next we investigate the application to
molecules containing hydrogen atoms, which are known to have large vibration-rotation effects upon substitution of deuterium for hydrogen.
66
HARMONY
4.
rPm Structures of Hydrogen-containing Molecules It is reasonable to apply the rPm procedure to hydrogen-containing molecules. If the procedure is applied directly as described for HCN and HCCH, the results of Table 17 arise. It is seen that the rPrn CH distance values are very poor, with deviations as large as 0.0075 A. But also, the heavy-atom distances are even of reduced accuracy, with deviations of 0.0012 ,A, for CN in HCN and 0.0024 A for CC in acetylene.
Clearly, rPm CH distances are not likely to be accurate
using the theory as presented, but perhaps the heavy-atom distances have some usefulness. Actually, of course, the results of Table 17 are not surprising. We have previously pointed out in Table 14 that Is/I 0 is not constant for H -~ D, D --> H or H --~ T substitutions. Generally, what is found is that the scaling procedure yields reliable rPm values for all isotopomers involving heavy-atom substitutions but unreliable and inaccurate values for isotopomers involving H ~ D substitutions. In fact, one way of approaching the problem is to simply eliminate the data point involving hydrogen isotope substitution. Thus, for HCN, if the SDS is a)
H12C14N
b)
D12C14N
c)
H13C14N
d)
H12C15N
MOLECULAR STRUCTURE DETERMINATION
67
TABLE 17. r~ structures of hydrogen-containing molecules with no correction a.
rs
re
rm p
HCN b CH
1.0631
1.0655 (2)
1.0604 (2)
CN
1.1553
1.1532 (0)
1.1544 (1)
HCCH c CH
1.0586
1.0622 (2)
1.0547 (5)
CC
1.2058
1.2026 (1)
1.2050 (2)
aAll distances in A. bSee Ref. [6]. cSee Ref. [7].
it is found that the I Pm value for species (2) is discordant with the remaining I Om values. Thus, one might try a least-squares fit using only the IOm values for species (1), (3) and (4). If this is done, the resulting distance values and uncertainties are CH = 1.0650 (46) and CN = 1.1535 (9) A. Note now that the CH value is greatly improved while the CN value improves just slightly.
Unfortunately, the
uncertainty of the CH value is unsatisfactorily large due to the omission of the I Om data most sensitive to the CH length. We have found that the I Om values for all the heavy-atom isotopomers are accurate and reliable while those for the H-atom isotopomers are inaccurate and unreliable.
68
HARMONY
Based upon results of this type, which were also supported by model force-field computations [6], it was clear that a special procedure was required to correct the I Pm values for the isotopomers that involved H --> D (or other H-atom) substitutions. The necessary procedure has been described and discussed in detail in the paper by Berry and Harmony [7]. What is required is that the I0m values for those isotopomers of the SDS involving H ~
D substitutions be
corrected by an amount that is equivalent to a translation of the D atom by a distance of 8~D along the C-D axis as shown in Fig. 4 for the linear and for any general molecule. In the linear molecule case, for the substitution of H by D (as in HCN, say), the correction takes the form
(iPm)corrD (IPm)D = + 2 mDZDSZ D
(46)
where (I0m)D is the IPm value calculated by Eqs. ( 4 2 ) - (44). m D is the deuterium atom mass, z D its z-coordinate and 5z D the necessary correctional shift. [z D is the D-coordinate in the principal axis system of the D isotopomer. An estimate is used as a first approximation, which can then be refined iteratively if necessary.] The sign selection is such that + , - is chosen depending upon whether p < 1, p > 1.
More generally, for a non-linear molecule, the particular axes must be specified. For the a-axis,
MOLECULAR STRUCTURE DETERMINATION
69
8ZD
g
.=____ =,...._
(a)
8.bD
c3aD ~CD
(b)
FIG. 4.
Bond elongation vector for X-D bond in (a) a general linear polyatomic molecule and (b) a general non-linear polyatomic molecule.
Components along principal axes are shown in
latter case. Reprinted from Ref. [7] with permission.
n
(iPma )corr D - (IPma)D = + 2 mD ~ ( b i S b i + ciSci)
(47)
i
gives the correction generally when n symmetry-equivalent H atoms are substituted by D atoms, b i and atoms and 6bi,
6C i a r e
ci
are the coordinates of the D
components of 5~D , which is always along the
C-D bond direction [7, 8]. Using data for several well-characterized
70
HARMONY
molecules, we have selected 15~D] = 0.0028 A [7] as a reliable average value for X-H bonds. One expects in principle that the value IS~DI might depend upon the heavy atom X. However, it appears that a single average value is suitable in practice. [Please note that Eq. (12) in Ref. 7 is not a general expression, applying for n = 2 (for example, H20 --~ D20 substitutions) but not for n = 3. The more general result given above as Eq. (47) was first presented in Ref. 8.] We will look at results shortly, but it is worth mentioning that the empirical procedure involving a shift of the D atom along the X-D bond by 0.0028 .A is intuitively pleasing because the shitt is about the amount that occurs upon H ~ D substitution. (See r 0 data in Table 4 and r 0 and rz data in Table 11.) Note, however, that the sign of the shit~ (i.e., the direction of 5f D) depends upon whether pg is greater or less than unity. When pg < 1, which is most common, the X-D bond is elongated as depicted in Fig. 4, while if pg > 1, the X-D bond is shortened by the same amount. It is best to avoid giving too much physical interpretation (or reality) to the shift correction. Rather, Eqs. (46) and (47) should be simply looked upon as being the empirical procedure needed to correct the I0m values of those isotopomers that involve replacement of H by D. Finally, the results for the b- and caxes are written from Eq. (47) by cyclic permutations of a, b, c and, if the SDS involves D ~ H substitution, m D in Eq. (47) is replaced by m H and 5~H is used in place of 5~D with a value ofSr H =-0.0056 A.
MOLECULAR STRUCTURE DETERMINATION
71
For more detailed discussions of the form of Eqs. (46) and (47), see Ref. 7 (especially the Appendix) and Ref. 8. In Table 18 we present r0m structure computations that utilize the corrections given by Eqs. (46) and (47).
In all cases reasonably
reliable r e structures are available for comparison. It is first interesting to compare the HCN and HCCH results of Table 18 with those given in Table 17, where no corrections to the basic scaling procedure were made. Note that in both cases the heavy-atom distances (C-N and C-C, respectively) have improved remarkably, with nearly exact agreement with the r e values.
Naturally, the C-H distances have
improved also, but it is clear that the correction procedure still does not yield highly accurate H-atom distances. Next it is worthwhile looking at the entire set of distances in the table.
The first conclusion we can make is that, for all seven
molecules, the r s distances are inferior to the r Om distances. Second, it is clear that the heavy-atom distances are all of high quality, with only two cases (HNC and CH2C12) leading to deviations greater than 0.001 A. If we compute the average deviations, we obtain X-H distances:
(Irmp - r e 1 ) = 0 . 0 0 1 7 A ( I r s - r e 1) = 0.0045 A
heavy-atom distances:
(Irmp -re I)=0.0005 h ( [ r s - r e 1) = 0.0023 A
72
HARMONY
TABLE 18. rOm structures of hydrogen-containing molecules with corrections for H --~ D substitutions a.
rs HCN
HNC
HN2 +
HCCH
HCCCI
H2CO
CH2C12
re
rm p
Ref. b,c
CH
1.0631
1.0655 (2)
1.0668 (2)
CN
1.1553
1.1532 (0)
1.1531 (1)
NH
0.9862
0.9970
0.9923 (3)
NC
1.1719
1.1684
1.1701 (1)
NH
1.0319
1.0336 (4)
1.0347 (1)
NN
1.0950
1.0928 (1)
1.0929 (0)
CH
1.0586
1.0622 (2)
1.0631 (0)
C=C
1.2058
1.2026 (1)
1.2026 (0)
CH
1.0550
1.0605
1.0599 (4)
C-C
1.2036
1.2030
1.2032 (8)
C-CI
1.6369
1.6353
1.6357 (6)
CH
1.1042
1.1005 (20)
1.1012 (2)
C=O
1.2049
1.2033 (10)
1.2031 (1)
LHCH
116.55
116.30 (25)
116.25 (4)
CH
1.0836
1.0874
1.0851 (11) c
C-C1
1.7687
1.7648
1.7636
LHCH
112.26
111.51
111.90 (17)
LCICC1
112.06
112.03
112.25 (3)
b,c
b,c
c
c,d
c
(3)
aDistances in A, angles in degrees. bSee Ref. [6] in text. cSee Ref. [7] in text. dEquilibrimn structure obtained by high quality ab initio calculations of ~. See Ref. [57].
Thus, it is evident that the heavy-atom distances for H-containing molecules are of the same high quality as for the strictly heavy-atom
MOLECULAR STRUCTURE DETERMINATION
73
species when the corrected (I0m)D values are used. It is important to stress that if the H-atom corrections are not made, the deficient (I 0rn)D values cause the heavy-atom structural parameters to deteriorate markedly. The poorest heavy-atom result of Table 18 is the N - C distance in HNC. This species has been looked at in some detail by model forcefield calculations, which show that the low-frequency bending mode (032 = 477 cm -1) is probably the origin of the somewhat large
discrepancy [6].
We have experienced only one r0m case that
apparently leads to an unacceptably large discrepancy compared to r e . For ethylene, we reported [7] r0m (C=C) = 1.3297 (5) A, while the reported r e value [58] is r e (C=C) = 1.334 (2). Even accounting for the rather low quality re structure, it appears that the r0m value is somewhat low.
Based upon detailed considerations [9], and by
analogy to HNC, it appears that this may be caused by the relatively low-frequency out-of-plane wagging mode of the CH 2 groups. We have proposed that the true equilibrium C=C distance in ethylene is most probably 1.332 + 0.002 A [9]. Now it should be noted that the r0m procedure has not been applied to molecules such as H20, NH 3 and CH 4. For these species the changes in vibration-rotation contributions to the moments of inertia are dominated by the H ~ D substitutions and, thus, the scaling
74
HARMONY
procedure proposed here will be inadequate even with the H ~ D correction term.
Fortunately, species of the type XH n represent a
negligible fraction of organic polyatomic molecules and, therefore, failure of the method here is of little consequence. To summarize where matters now stand, the results presented in Sections E.2 to E.4, plus other examples and supporting theoretical work [4-10], lead us to the following conclusions concerning r0m structures:
a)
Heavy-atom distances are expected to be within 0.001-0.002 A of r e if all necessary H ~ D corrections are made via Eqs. (46) or (47).
b)
Hydrogen atom distances are not given with high reliability by the procedure even after making the H ~ D corrections. Still, it seems likely that X-H distances will be within 0.002-0.003 A of r e .
c)
The r0m structure is a much more reliable representation of the r e structure than is the r s structure, especially for heavy atoms. A word about precision is perhaps in order. We typically list the least-squares o value that results from the least-squares fit in our tabulated data to show the degree to which the scaled moments of inertia model the presumed rigid, equilibrium molecule. Of course, experimental uncertainty in the I0's will also contribute to o.
It is not our intention that one should
MOLECULAR STRUCTURE DETERMINATION
75
automatically use the o value as a bona fide measure of the uncertainty in the parameter. Note from Table 18 that the r~m o values are commonly (but not always) smaller than the deviation from r e. Our conservative view is that when the uncertainty o is less than 0.001 A (or 0.002 A), the parameter uncertainty should be stated as 0.001 A (or still more conservatively as •
A).
When the parameter uncertainty o exceeds 0.001 A (or 0.002 A), then the parameter uncertainty is best stated as •
Finally,
we add one more conclusion (or caveat) to the above list of 3:
d)
Structural distortions may occur that cause the accuracy to decline from the normal 0.001-0.002 A range if especially lowfrequency bending vibrations of hydrogen atoms exist.
5.
rOm applications Here we mention just a few rOm structures of organic polyatomic molecules of particular interest.
First, our rgm structure of ethane
H3C-CH 3 [8] provided the best available spectroscopic estimate of the r e value of the prototypical C-C single-bond length.
The reported
value of rOm (C-C) = 1.522 + 0.002 A is in very good agreement with the estimated r e value from electron diffraction work (1.524 + 0.003 A) [59]. It is interesting next to look at the results for propane (CH 3CH2-CH3) that were presented in Table 7. Note for propane that the rOm (C-C) value is 1.521 A, nearly identical to the ethane value as
76
HARMONY
expected for these very-low polarity alkane molecules. Note further that if one were to select either the r 0 or r s value for the propane C-C distance, one would really be in a different realm since the deviations from rm p are quite substantial. The propane results for C-C (and three out of four of the C-H distances) also illustrate a common, but not absolute observation, viz.,
r0 > rs > rPm
(48)
Recall that for diatomic molecules, Eq. (28) ordered the distances as r 0 > r s > r e. There is no proof of Eq. (28) for polyatomic molecules, but our observation in Eq. (48) for polyatomic molecules is equivalent to Eq. (28) if rPm ~ r e, which is our hypothesis. The rPm structures of symmetric rotors such as the methyl halides are troublesome unless the A rotational constant is available. Using A values from IR data, Le Guennec et al [60] have reported the r o 9
m
structures of CH3CI and CH3Br. For CH3C1, the rPm values of the CC1 and CH distances deviate from the r e values by +0.0017 a n d 0.0019 A, respectively.
For CH3Br, the deviations for the CBr and
CH distances are +0.0013 and -0.0007 A, respectively. The results are satisfactorily in accord with our expectations based upon nonsymmetric rotor molecules. The principal impetus for developing the rPm structure method was the desire to establish near-r e structures for polyatomic organic
MOLECULAR STRUCTURE DETERMINATION
77
molecules. Table 19 presents a summary of results for a variety of small molecules treated by the rOm procedure (including all H --~ D corrections).
In all cases the accuracy (with respect to the true r e
value) is expected to be within +0.001 A or +0.002 A. In a few cases, bona fide experimental r e structures exist (such as for C2H 2 , H2CO and CH3C1), but we list our rPm values for consistency. (In any case, the r gin and r e values are in agreement for these cases as previously discussed.) The various molecules provide interesting comparisons of some key bonds in various bonding situations, including the two cyclic molecules ethylene oxide and cyclopropene. data
provide
reliable
benchmarks
for
ab
In addition, the initio
theoretical
computations.
IV.
CONCLUSION
The scaled moment of inertia (rOm) structure method has been demonstrated to yield an excellent approximation to the theoretically significant equilibrium (re) structure. A practical variant of Watson's r m procedure [30], the method requires only the spectroscopic data necessary to perform a complete ground state substitution (rs) structure determination.
The resulting rOm structure appears to be invariably a more
reliable estimate of the true r e structure than is the r s structure.
For polyatomic
molecules, the heavy-atom distance parameters are expected to be within 0.001-0.002 A of the r e value in most cases. For hydrogen-containing molecules, a special empirical
78
HARMONY
TABLE 19. Selected values of near-equilibrium bond distances (in A). C-C
C=C
C=C
C - - O C=O
C--C1 Ref.
Acetylene
HCCH
1.203
[7]
Chloroacetylene
C1CCH
1.203
1.636 [7]
Formaldehyde
H2CO
Methyl chloride
CH3C1
1.778 [60]
Methylene
CH2C12
1.764
1.203
[7]
[7]
chloride Formic acid
HCO2H
Ethylene oxide
C2H40
Ethylene
H2CCH 2
Ethane
H3CCH 3
1.522
Ethyl chloride
CH2C1CH3
1.510
Vinyl chloride
C2H3C1
Propene
CH3CHCH 2
1 . 4 9 6 1.334
[9]
Cyclopropene
C3H4
1.505 1.293
[7]
Propane
CH3CH2CH 3
1.522
[9]
1.459
1.340 1.196
[9]
1.425
[7]
1.332
[7] [8] 1.789
1.328
[9]
1.726 [61 ]
correction is needed to handle the data for the hydrogen isotopomers. This correction ensures that the quality of the heavy-atom parameters remains high, and leads to H-atom parameters of reduced, but still perhaps useful, accuracy (0.002-0.004 A). The new procedure permits for the first time the practical estimation of reliable near-r e structures for polyatomic organic molecules from gas-phase spectroscopic ground state rotational constants. Along with gas electron diffraction and modem high-quality
MOLECULAR STRUCTURE DETERMINATION
79
ab initio computations, the rOm methodology should provide a reliable new tool for the
high-precision structure determination of modest-sized (6-8 heavy atoms or so) polyatomic organic molecules. ACKNOWLEDGMENTS Over the many years of this structural research, the support of the National Science Foundation, the Petroleum Research fund, and the Research Corporation has been greatly appreciated.
This particular manuscript would not have been possible
without the technical assistance and encouragement of Nancy M. Harmony.
80
HARMONY
REFERENCES 1. ,
M. Born and J. R. Oppenheimer, Ann. Phys., 84, 457 (1927). G. Herzberg, Electronic Spectra of Polyatomic Molecules, Van Nostrand, New York, 1966.
3.
J.M. Hutson and B. J, Howard, Mol. Phys., 45, 791 (1982).
4.
M.D. Harmony and W. H. Taylor, J. Mol. Speetrose., 118, 163 (1986).
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M.D. Harmony, R. J. Berry and W. H. Taylor, J. Moi. Speetrose., 127, 324 (1988).
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R.J. Berry and M. D. Harmony, J. Mol. Speetrose., 128, 176 (1988).
7.
R.J. Berry and M. D. Harmony, Struet. Chem., I, 49 (1990).
8.
M.D. Harmony, J. Chem. Phys., 93, 7522 (1990).
9.
H.S. Tam, J.-I. Choe and M. D. Harmony, J. Phys. Chem., 95, 9267 (1991).
10. M.D. Harmony, Aeets. Chem. Res., 25, 321 (1992). 11. K. Kuchitsu and S. J. Cyvin, Molecular Structure and Vibrations (S. J. Cyvin, ed.), Elsevier, Amsterdam, 1972, p. 183. 12. W. J. Hehre, L. Radom, P. v. R. Schleyer and J. A. Pople, Ab Initio Molecular Orbital Theory, Wiley, New York, 1986. 13.
S. Saebo and P. Pulay, Ann. Rev. Phys. Chem., 44, 213 (1993).
14. K.B. Wiberg, private communication. 15. J. Kraitchman, Am. J. Phys., 21, 17 (1953). 16. W. Gordy and R. L. Cook, Microwave Molecular Spectra, Wiley-Interscience, New York, 1984. 17. A. Chutjian, J. Mol. Spectrose., 14, 361 (1964). 18. G. Herzberg, Spectra of Diatomic Molecules, Van Nostrand, Princeton, NJ, 1950.
MOLECULAR STRUCTURE DETERMINATION
81
19. C.H. Townes, A. N. Holden and F. R. Merritt, Phys. Rev., 74, 1113 (1948). 20.
F.J. Lovas and E. Tiemann, J. Phys. Chem. Ref. Data, 3, 609 (1974).
21.
V. W. Laurie, Critical Evaluation of Chemical and Physical Structural Information (D. R. Lide and M. A. Paul, eds.), National Academy of Sciences, Washington DC, 1974, p. 67.
22.
K. P. Huber and G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinholt, New York, 1979.
23.
F.J. Lovas, J. Phys. Chem. Ref. Data, 7, 1445 (1978).
24.
M. D. Harmony, V. W. Laurie, R. L. Kuczkowski, R. H. Schwendeman, D. A. Ramsay, F. J. Lovas, W. J. Lafferty and A. G. Maki, J. Phys. Chem. Ref. Data, 8, 619(1979).
25. Landolt-Bornstein Numerical Data and Functional Relationships in Science and Technology, New Series, Group II, Springer, Berlin, Vol. 6, 1972. 26.
D.L. Gray and A. G. Robiette, Mol. Phys., 37, 1901 (1979).
27.
A. Baldacci, S. Ghersetti, S. C. Hurlock and K. N. Rao, J. Mol. Speetrosc., 59, 116 (1976).
28.
D.R. Lide, J. Chem. Phys., 33, 1514 (1960).
29.
C.C. Costain, J. Chem. Phys., 29, 864 (1958).
30.
J.K.G. Watson, J. Mol. Spectrosc., 48, 479 (1973).
31.
B. Rosenblum, A. H. Nethercot, Jr. and C. H. Townes, Phys. Rev., 109, 400 (1958).
32.
J.K.G. Watson, J. Mol. Spectrosc., 45, 99 (1973).
33.
M. Nakata, T. Fukuyama and K. Kuchitsu, J. Mol. Spectrose., 83, 118 (1980).
34.
J.G. Smith and J. K. G. Watson, J. Mol. Speetrose., 69, 47 (1978).
35.
L. Pierce, J. Mol. Spectrosc., 3, 575 (1959).
36.
P. N6sberger, A. Bauder and H. Giinthard, Chem. Phys., 1, 418 (1973).
82
HARMONY
37. R. H. Schwendeman, Critical Evaluation of Chemical and Physical Information (D. R. Lide and M. A. Paul, Eds.), National Academy of Sciences, Washington, DC, 1974, p. 94. 38.
H.D. Rudolph, Struet. Chem., 2, 581 (1991).
39.
M. Nakata, M. Sugie, H. Takeo, C. Matsumura, T. Fukuyama and K. Kuchitsu, J. Mol. Speetrose., 86, 241 (1981).
40.
M. Le Guennec, G. Wlodarczak, and J. Demaison, J. Mol. Speetrosc., 157, 419 (1993).
41.
D.R. Herschbach and V. W. Laurie, J. Chem. Phys., 37, 1668 (1962).
42.
V.W. Laurie and D. R. Herschbach, J. Chem. Phys., 37, 1687 (1962).
43.
T. Oka, J. Phys. Soe. Japan, 15, 2274 (1960).
44.
T. Oka and Y. Morino, J. Mol. Speetrose., 8, 300 (1962).
45.
E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, McGraw-Hill, New York. 1955.
46.
P. Botschwina, N. Oswald, J. Fltigge, A. Heyl and R. Oswald, Chem. Phys. Letters, 209, 117 (1993).
47.
T.J. Balle and W. H. Flygare, Rev. Sci. Instru., 52, 33 (1981).
48.
D.J. Clouthier and J. Karolczak, J. Chem. Phys., 94, 1 (1991).
49.
R.J. Saykally, Ace. Chem. Res., 22., 295 (1989).
50. A.C. Legon, Ann. Rev. Phys. Chem., 34, 275 (1983). 51.
T.A. Miller, Ann. Rev. Phys. Chem., 33, 257 (1982).
52. M. Le Guennec, G. Wlodarczak, W. D. Chen, R. Bocquet and J. Demaison, J. Mol. Speetrose., 153, 117 (1992). 53.
G. Cazzoli, C. D. Esposti, P. Palmieri and S. Simeone, J. Mol. Spectrose., 97, 165 (1983).
MOLECULAR STRUCTURE DETERMINATION
54.
83
W.H. Taylor, Ph.D. Dissertation, University of Kansas, Lawrence, KS, 1983.
55. H.H. Nielsen, Rev. Mol. Phys., 23, 90 (1951). 56.
I. M. Mills, in Molecular Spectroscopy: Modern Research (K. N. Rao and C. W. Mathews, Eds.), Vol. 1., Academic Press, New York, 1986, pp. 115-140.
57.
P. Botschwina, private communication.
58.
J.L. Duncan, Mol. Phys., 28, 1177 (1974).
59.
J.L. Duncan, D. C. McKean and A. J. Bruce, J. Mol. Spectrosc., 74, 361 (1979).
60.
M. Le Guennec, G. Wlodarczak, J. Burie and J. Demaison, J. Mol. Spectrosc., 154, 305 (1992).
61.
I. Merke, L. Poteau, G. Wlodarczak, A. Bouddou and J. Demaison, J. Mol. Speetrose., 177, 232 (1996).
CHAPTER ELECTRON
DIFFRACTION:
WITH OTHER
2 A COMBINATION
TECHNIQUES
Vladimir S. Mastryukov Department o f Physics University o f Texas Austin, T X 78712, U S A
I.
I N T R O D U C T I O N ...................................................................................................
86
II.
T H E E L E C T R O N D I F F R A C T I O N T E C H N I Q U E .... , ....................................... 88 A.
Experimental Equipment ............................................................................................. 88
B.
Basic Relations ............................................................................................................ 92
C.
Structure Analysis ........................................................................................................ 97
III. C O M B I N E D A P P L I C A T I O N O F E X P E R I M E N T A L T E C H N I Q U E S ......... 100 A.
Supersonic Nozzle ..................................................................................................... 101
B.
Mass Spectrometer ..................................................................................................... 102
C.
Laser .......................................................................................................................... 103
IV. T E C H N I Q U E S I N C O R P O R A T E D IN S T R U C T U R A L A N A L Y S I S ............. 109
V.
A.
Vibrational Spectroscopy ........................................................................................... 110
B.
Rotational Constants .................................................................................................. 115
C.
Theoretical Calculations ............................................................................................ 121
D.
Liquid Crystal NMR Spectroscopy ........................................................................... 134
CONCLUDING REMARKS AND ACKNOWLEDGEMENTS
...................... 137
R E F E R E N C E S ..............................................................................................................
85
141
86
MASTRYUKOV
"There is no more basic enterprise in chemistry than the determination of the geometrical structure of a molecule. Such a determination, when it is well done, 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" R. Hoffmann, 1983
I.
INTRODUCTION Gas-phase electron diffraction (GED) will be 70 years of age in the year 2000. It is about 20 years older than microwave spectroscopy, the other of just two methods currently in use to determine the geometry of free molecules. Other structural data come from studies of crystals done by X-ray and neutron diffraction. The purpose of this chapter is to briefly describe the major milestones of the evolution of GED during the seven decades of its history focusing
exclusively at
the interactions of this method with other techniques. In 1988, all
these individual interactions were reviewed by a large international group of electron diffractionists (see Part A of Ref. [ 1]) and, therefore, our goal is to briefly characterize them and to give a good representation of the literature that has appeared after 1988; comprehensive coverage is not intended. GED, like any other method, has its own inherent weaknesses as well as unique strengths.
Its data are otten insufficient to provide all the necessary
information for a full evaluation of the structure of any but the smallest molecule. Therefore, interactions with other methods have been found useful to increase the power of GED and extend its range of application. These interactions are of two distinctly different kinds. The first type of interaction occurs at the experimental
ELECTRON DIFFRACTION
87
level and implies some modifications of standard GED equipment; this will be characterized in Section III. The second type of interaction refers to the way the experimental GED data are treated and it implies the inclusion of some additional information from independent sources of both experimental and computational nature; this will be described in Section IV. These two sections are preceded by Section II, which serves as an introduction to both of them, describing the standard GED equipment and standard structure analysis. The first GED experiment was reported in 1930 by Herman Mark and Raimund Wierl in Germany. Many interesting details related to this event can be found in Mark's personal recollections (see Introduction in Ref. [1 ] and Ref. [2]). However, an argument can be made that Lawrence Brockway, more than anyone else may be said to be the father of the GED method [3 ]. Brockway, as a graduate student of Linus Pauling, initiated GED in the United States and as early as in 1936 he was already able to write a review article in Review of Modern Physics which contained structural information for 146 molecules. Currently, there are about 20 groups, in the world, doing GED and they are in the following countries: Belgium, Germany, Hungary, Japan, Norway, Russia, and the United States. Their scientific production reaches about 150 papers per year. With this publication rate it is important to keep a record of all relevant literature and The Sektion flir Spektren- und Strukturdokumentation at the Universit~it Ulm, Germany does this job. This group currently headed by Jiargen Vogt created a structural chemical database for gas-phase compounds [4] and molecular spectroscopy [5] and also participates in publication of LandoltB6mstein Tables, a unique source of structural information [6-10]. In 1973, The Chemical Society in Great Britain began to publish an annual series of
88
MASTRYUKOV
comprehensive reports on molecular structure by diffraction methods. A survey of earlier electron diffraction studies has been written by B. Beagley [11 ]. The last volume in this series was published in 1978 and in a contribution by Sch~ifer [12] one can find all the previous references. In recent years Rankin and Robertson have continued this work and we mention here only the latest contributions [ 13 ]. Finally, a useful source of structural information for selected classes of compounds was included in a book written by Vilkov, Mastryukov and Sadova* [ 14] and Part B of a book edited by Hargittai and Hargittai [ 1].
II.
THE ELECTRON DIFFRACTION TECHNIQUE Before we discuss a combination of GED with other techniques it is convenient to give a brief overview of the electron diffraction method itself. Therefore, in subsection A we describe the experimental equipment currently in use, followed by the basic equations of GED (B) and structure analysis (C). As a logical consequence, Section III describes the changes in running the GED experiment, while Section IV characterizes those modifications, which are used in a combined structure analysis. A.
Experimental Equipment The electron diffraction method is based on measuring the intensity of electrons scattered from a gas jet injected into a high vacuum.
A highly
simplified schematic arrangement of a conventional GED unit is shown in Fig. 1; it includes the following three sections:
(1) An electron optical
The epigraph chosen for this chapter is a quotation from a Foreword written by R. Hoffmann for this book.
ELECTRON DIFFRACTION
89
system producing a well-collimated electron beam; (2) A diffraction chamber equipped with the specimen inlet system (nozzle); and (3) The detector; photographic data recording is usually used in most GED units, although counting techniques (see below) have also been applied successfully. The scattering picture consists of a series of concentric rings superimposed upon a steeply descending background, falling off from the diffraction center towards higher scattering angles.
Since the scattering
intensity decreases so rapidly, a rotating sector made of metallic sheet is used to screen the photographic plate. The sector, a spiral- or heart-shaped cam, is placed immediately above the photographic plate with its axis of rotation coinciding with the incoming beam. Sectored pictures can easily be record zd by a microphotometer.
Typical experimental conditions are as follows:
electron accelerating voltage about 40KV, which corresponds to electron wavelength, E, about 0.06A, vacuum about 10-5 Torr and sample pressure at the container about 20 Torr. Most of the GED apparatuses used all over the world are laboratory built and their characteristics were reviewed by Hilderbrandt [15] and Oberhammer [16] in 1975 and 1976, respectively.
A more recent
contribution by Tremmel and Hargittai [ 17] contains many details about the GED unit used in Budapest, Hungary. Finally, the Antwerp GED unit has been described, in 1997 for the first time [ 18]. A very interesting and promising development of the registration technique is reported by Iijima, Suzuki and Yano in 1998 [19]. The authors
90
MASTRYUKOV
photograhic plate rotating sector
/I
electron gun
9
electron beam
nozzle~l I vacuum chamber
FIG. 1
scattered electrons
gas
Scheme of the gas electron diffraction experiment.
suggest the use of imaging plates (IP) instead of the photographic plates used so far. Due to the wide dynamic range of IP, data of similar quality were obtained both with and without a rotating sector.
Another improvement
consists of the use of a read-out system of a stand-alone type ivstead of a microdensitometer. All this allows the authors to conclude their paper by a very strong statement: "The experimental technique of gas-phase electron diffraction has been characterized by the sector and the microphotometer and is thus called the sector microphotometer method. Both of these are going to be obsolete by virtue of the wide dynamic range of the imaging plates." Although the photographic registration of the scattered electrons is undoubtedly the most widely used technique in conventional GED units, it is not the only one. There were several successful attempts to eliminate the photographic intermediary and sector device.
The first unit using counting
ELECTRON DIFFRACTION
91
technique was built in 1970 at Indiana University by Fink and Bonham [20,21 ]. Later, a similar apparatus has been build by Fink and coworkers at the University of Texas [22]. The high sensitivity of the counting technique allows one to study compounds of low volatility: a sample pressure of 10-2 Torr is sufficient to run an experiment instead of the conventional 20 Torr. In the same Austin group, a unique diffraction unit was built equipped with a M611enstedt analyzer, which allows the measurement of both elastic and inelastic cross sections. It was applied to a study of correlation effects in the Ne atom.
This study represents a major step forward in the details with
which calculated wave functions can be examined experimentally [23 ]. Another non-photographic instrument for GED studies was build by Sch~ifer and coworkers at the University of Arkansas in 1984 [24]. In this machine the scattered electrons are detected by a fluorescent screen, which is optically coupled to a custom multichannel analyzer.
Later several
improvements were introduced into the original design [25-27] and the possibility of using real-time GED as a detector in gas chromatography was demonstrated [28]. Some recent studies of SF 6 [29] and SeF 6 [30] show the application of this procedure. As a continuation of achievements in non-photographic techniques in the 1970s and 1980s, in the 1990s three more GED units were build resembling one of the two types mentioned above. In 1992 BSwering and coworkers at the University of Bielefeld, Germany reported their results [31 ] and a more detailed account was given later [32].
A similar advance
occurred with a machine built in 1992 by Zewail and coworkers at the
92
MASTRYUKOV
California Institute of Technology [33,34].
In this apparatus a newly
designed computerized two-dimensional charge-coupled device (CCD) was used as a detection system. This permitted immediate visualization of the diffraction pattern; a color image is shown in Ref. [33] while a black and white is reported in Ref. [34]. The authors called this technique Ultrafast Electron Diffraction (UED) [33,34].
Finally, in 1995 the experimental
apparatus was described by Geiser and Weber at Brown University, Rhode Island [35]. Therefore, we can conclude that the counting technique started in 1970 lead to significant progress in recent years which is related to progress in technology. As can be seen in later sections, most of these machines were designed for special purposes where the photographic technique is not applicable.
B.
Basic Relations The basis of the use of GED as a structural tool lies in the relationship between the intensity of scattered electrons as a function of scattering angle, 8, and P(r), the probability function which expresses the distribution in internuclear separation.
The molecular parameters determined by this
relationship are rij, the effective internuclear distance between atoms i and j, and lij, the amplitude of vibration (sometimes also denoted as uij). The total scattered intensity, I T, is usually expressed as a function of the variable s, instead of the scattering angle, 8, where s = (4n/~.) sin (0/2)
ELECTRON DIFFRACTION
93
and ~. is the electron wavelength.
When a smooth background, I B, is
subtracted from the total intensity, the molecular term, IM, is obtained. I M = I T - I B. The background itself does not contain information conceming the molecular structure; it is mainly determined by the charge distribution in the atoms of which the molecule it built, and by inelastic and extraneous scattering.
Theoretically, the molecular term can be expressed to an
approximation sufficient for most structure work as follows (for more details of theory see Refs. [ 14, 21, 36-40]): IM -
~ l f i [ I f j l cos (rli _ i~j
rlj)exp(-ll/j
s:)sin S(ra/ . -
9
(1)
The complex atomic scattering factor,
f i(s) = f i(s)] exp[i ~Ti(s)] is usually taken from existing tables. The most recent scattering factors were calculated by Ross, Hilderbrandt and Fink in 1992 [41].
The asymmetry
parameter, K, represents a slight deviation of the argument from a linear function of s and to a good approximation is equal to
al4/6
where a is the
Morse constant. The molecular term slM, or sometimes the ratio S(IM/IB) = sM(s), is analyzed by a least-squares method and the bond distances r a, bond angles and some of the amplitudes of vibration are determined (see more about this in the next section).
The r a distance, sometimes called an operational
parameter, is an ill-defined parameter since it refers to the maximum position of any peak of the
P(r)/r
function.
The better parameter, the rg distance,
94
MASTRYUKOV
which corresponds to the center of gravity of the P(r), peak and is the thermal average of internuclear distance. The relation between these two parameters and the definitions of other parameters, which are of special importance in GED, are given in Table 1. Normally, every GED paper reports the original intensities I(s) and sM(s) curve as recommended by the Commission on Electron Diffraction of the Intemational Union of Crystallography [42]. Such an example from the electron diffraction study of dichlorodimethylsilane [43] is shown in Fig. 2. The experimental background shown in the upper part of Fig. 2 can be either hand-drawn or can be generated computationally using a series of polynomials satisfying a number of criteria. The initial background can be improved in the course of structure analysis [44]. Another curve that is much used in GED study is the Fourier transform of the molecular intensity function, the radial distribution curve.
It is
customarily defined by: Smax
fE(r) =
sM E(s)
exp (-ks 2) sin (sr) ds.
(2)
Smi n
The integration limits Smin and Smax are limited by the experimentally available s range. The constant k is an artificial damping constant that helps to suppress error ripples in the radial distribution curve. The radial distribution curve has the advantage of being more understandable than the intensity curve since it has a peak for each intemuclear distance. The peak is rather narrow and is centered around every
ELECTRON DIFFRACTION
95
TABLE 1. The most important internuclear distance parameters.
ra
effective internuclear distance in the expression of the molecular contribution to electron scattering intensities, Eq. (1), equal of the center of gravity of the P(r)/r distribution function for the experimental temperature.
rg
average value of interatomic distance (for a particular temperature), equal to the center of gravity of the probability distribution P(r) for each pair of atoms in the molecule. It is related to r a by rg = r a -12/r.
rt~
distance between average nuclear positions in the thermal equilibrium at temperature T; calculated from rg or r a
ra = r g - K = r a -12/r- K.
K is the perpendicular amplitude correction (IV.A) K = ( + )/2r e. value of ra extrapolated to temperature of 0 K. rO
limT~ 0 (r~).
It is related to r a by
r T - r 0 = 3//22a I ( l 2) T_ (12) 0 ] + 5r T + K 0 _ (/2)T
/r0
a is the Morse anharmonicity constant, l an rms amplitude of vibration, 8r the change due to centrifugal distortion, K the perpendicular amplitude corrections. r0
effective
intemuclear
parameter
which
reproduces
ground-state
rotational constants A 0, B0, C O(section IV.B). rZ
distance between mean positions of atoms in the ground vibrational state; represents same physical quantity as rO but differs in origin since it is calculated from spectroscopic data.
re
distance between equilibrium positions related to rg by r e = r g - 3al2/2.
96
MASTRYUKOV
l(s)
(CH 3 )2 SiCI2
| r
sM(s)
IV
VVUVVVVV" 2A
. . . .
ilk
A
l
.
.
l
5 FIG. 2
.
.
I
15
_-----_ _ _ A
l
l
25
I
s,/~ -1
Electron-diffraction intensity curves. The two uppermost curves are the total intensities, I(s), obtained at two different "nozzle-toplate" distances.
Empirical background is shown.
Below is a
composite molecular intensity curve sM(s). The bottom curve is the difference between experiment and theory multiplied by a factor of two. distance r encountered in a molecule; it has an approximately Gaussian shape unless several imemuclear distances contribute to the same peak. The radial distribution curve for (CH3)2SiCI 2 [43] shown in Fig. 3 gives examples of
ELECTRON DIFFRACTION
97
f r)
f
C-H
l
Si-C Si-CI Si...H C...C C...CI CI...CI (]...H
A I 0
1 FIG. 3
3
r,A
Experimental radial distribution curve for dimethyldichlorosilane. All the peaks are assigned according to the molecular model shown in the upper right comer; the difference curve shows how well the final model corresponds to the experiment [43 ].
both of these possibilities: the first and the last peaks are very well separated and have almost Gaussian shapes while all the others are composite peaks having either two or three components. C.
Structure Analysis An electron-diffraction structure determination is nowadays based exclusively on comparing experimental, sME(s), and theoretical, sMT(s), molecular intensity curves. The structure parameters are adjusted until the
98
MASTRYUKOV
best fit is obtained and a least-square procedure is always used for this purpose. From the form of Eq. (1) it is clear that the least-squares procedure is nonlinear in the structural parameters. The usual linearizing method based on Taylor's series expansion is used so that the parameters actually adjusted are the shitts in the internal coordinates. The algorithm was formulated a long time ago by Hedberg and Iwasaki [45] and it was implemented in many programs like the one used by the Norwegian school [46]. Structure analysis begins with building a trial model [47]. The radial distribution curve can be very useful for this purpose. From the positions of the peaks one has trial values of rij and from their half widths one can estimate trial values of lij (see Fig. 3). Earlier practice was also to take the initial values for the amplitudes of vibration from similar molecules [48-52], however, at the present time the amplitudes are calculated from the available force fields (see more in Section IV.A.). When the trial model is completed one can calculate sMT(s) using Eq. (1) and compare it with its experimental counterpart, sME(s). This time the structure refinement begins [47]. The least-squares program changes all r's and l's until the difference
k=l reaches its minimum value (n is the number of observed points and Wk is a weight function).
The number of adjustable parameters depends on the
accuracy and on the amount of experimental data and on the complexity of molecule as well. The analysis often requires assumptions about molecular
ELECTRON DIFFRACTION
99
symmetry and those parameter values on which the molecular intensity depends only weakly. These "weak" parameters are constrained to calculated values or to values known from previous experience. These constraints of the amplitudes of vibration and the geometry parameters are discussed in more detail in Sections IV.A and IV.C, respectively. The quality of fit is characterized by the R factor, which is also used in X-ray crystallography:
1/2 m
_
_
k=l
n
. . . .
(4)
Note that the numerator in this expression is identical to Eq. (3). In our example of (CH3)2SiC12 molecule (see Figs. 2 and 3) an R factor was 7.6% [43]. It is not rare in the practice of GED that several models are found which have very similar R factors. It is important to recognize that all these models are solutions of the problem in the mathematical sense and to make a good decision between them is not at all easy. One such example is given by the study of thionlytetrafluoride, SOF4, by Gundersen and Hedberg [53]. These authors found four models in excellent agreement with experiment. Multiple solutions are a serious problem of the GED method and the most general way to solve this problem is by addition of information from independent sources.
In the case of SOF 4, this was done by using three
rotational constants from microwave spectroscopy as constraints [54]. Other examples will be considered in Section IV.
100
MASTRYUKOV
We conclude this Section by mentioning a very important modification of the common least-squares procedure suggested in 1975 by Bartell, Romenesko and Wong
[55] and called the "Method
of Predicate
Observation." In this method, a set of reasonable values of parameters based on similar measurements or experience is incorporated into the least-squares procedure as additional observations.
This addition helps to remove the
natural or accidental linear dependence among the parameters which interfere with their reliable determination. The important feature of this approach is that a guessed value used as a predicate observation does not rigidly constrain the optimized parameters to the a priori estimate. Therefore, the systematic error introduced could be much less critical than if the parameters are frozen.
This powerful method is perhaps the most objective way of
combining GED with molecular mechanics, ab initio calculations and X-ray analysis. A promising example of the application of this procedure known as SARACEN is discussed in Section IV.C.
III.
COMBINED APPLICATION OF EXPERIMENTAL TECHNIQUES In this section we describe those changes in the standard GED apparatus (see Section II.A and Fig. 1, in particular) which have been designed to expand the application of the method. The presentation in this section, as well as in the next one, follows the chronological development. In 1967, a supersonic nozzle replaced the regular one in a GED unit and, in Subsection A, we will see what kind of new information this alteration may produce.
ELECTRON DIFFRACTION
101
In 1975, a quadrupole mass spectrometer was coupled with the GED unit. This type of combination unit is sometimes referred to as ED + MS (see Subsection B). Finally, in 1979 a laser was used for the first time in the GED experiment and it is described in Subsection C. A.
Supersonic Nozzle As early as the late 1960s the Orsay electron diffraction group in France successfully used a supersonic nozzle for production of large cluster~ of rare gas atoms, carbon dioxide and water [56-61]. Bartell and his coworkers at the University of Michigan [62-65] continued this important research in the early 1980s. First, formation of benzene clusters was studied [63] followed by a similar study of n-butane [64]. In the course of the latter study, an interesting phenomenon called "conformational cooling" [65] was observed. A detailed description of these and other findings was given in a review paper written by Bartell in 1986 [66]. The interested reader will find this paper a good introduction to the field and can learn more about the supersonic nozzle system, statistical modelling of clusters, solid and liquid like clusters of polyatomic molecules and trends in nucleation and cluster growth. In the subsequent years large water clusters were studies in detail [67-70] together with nanocrystals of transition-metal hexafluorides, MoF 6 and WF 6 [71 ]. In 1992, after almost three decades of using the supersonic nozzles exclusively for cluster production, a group of German physicists at the University of Bielefeld found another exciting application for this inlet system.
It was suggested to use a supersonic molecular beam to study
102
MASTRYUKOV
Spatially Oriented Molecules [72-77]. A continuous beam of CH3C1 or CH3I molecules is oriented via the hexapole technique before their scattered intensities are measured in the GED unit described before [31,32].
The
molecules are oriented preferentially parallel or antiparallel to the electron beam and show oscillating deviations from the scattering pattern of unoriented molecules up to 4% as a function of momentum transfer, quite similar to previous calculations using the independent-atom model (IAM) [78,79]. In 1994, Fink and co-workers introduced a pulsed supersonic quartz nozzle, which might find a future application in GED [80]. B.
Mass Spectrometer Every GED study is done with the assumption that the composition of the gas phase is known. This assumption is good enough if we deal with stable compounds, which have no tendency to self-association like benzene, CCI4, NH 3, etc. On the other hand, there are cases when a researcher has doubts about what kind of species are present under the experimental conditions. Two typical examples include" 9 there is a dynamic equilibrium between monomeric (M) and dimeric (D) species 2M # D, 9 the compound in question is unstable and is prepared in the course of the GED experiment. Note that both processes are temperature dependent. Normally, when there are doubts about the vapor composition a common practice is to determine it from an independent mass spectrometric
ELECTRON DIFFRACTION
103
measurement. This approach is exemplified by a recent work by Konaka et al. [81] who studied CH3-N=CH2, a product of thermal decomposition of trimethylamine. In this case, the mass spectra were measured to determine the optimum heating conditions.
However, there are situations when the
structural problem can be solved only by simultaneous mass spectrometric and GED measurements. The first attempt for such a combination was made by Kohl and Kennerly at the University of Texas in a study of the thermal isomerization of cyclopropane.
Unfortunately, the results for this pioneer
study remained unpublished and are described only in the dissertation by Kennerly in 1975 [82]. In their apparatus, the quadrupole mass spectrometer was located outside the diffraction chamber on the flange opposite the nozzle.
It had its own high-vacuum system and was connected to the
diffraction chamber via a valve. Slightly later, in 1977, a similar combination was developed in Budapest, Hungary by Hargittai and co-workers [83]. For more details, see also Ref. [17].
Still later, another group in Ivanovo, Russia headed by
Girichev built analogous equipment.
Table 2 shows several molecules
studied in these two groups by a combined approach GED + MS. C.
Laser In 1979, Arvedson and Kohl were the first to use a laser to study vibrationally pumped molecules by GED [84]. The basis for the use of this combination lies in the ability of GED to measure the amplitudes of vibration. This is how the authors describe their reasoning:
104
MASTRYUKOV
TABLE 2. Molecules studied in a combined electron diffraction/ quadrupole mass spectrometer experiment. No.
Molecule
85
8
CI2Gee
84
A12Br6
85
9
Cl2Sid
89
3
BeCI 2
86
10
C13NbO
90
4
Be2CI 4
86
11
F3NbO
91
5
BeF2
87
12
GeI2 e
92
6
Br2Gea
88
13
C22H38CuO4
93
7
Br2Sib
89
No.
Molecule
1
AIBr 3
2
Ref.
Bis(dipivaloylmethanato)copper(II) 620~
aproduced by reaction Ge(s) + GeBr4 (v) bproduced by reaction Si(s) + SiBr4 (v)
1200~ 660~
eProduced by reaction Ge(s) + GeCI4 (v)
1200~
dproduced by reaction Si(s) + Si2CI6 (v) eproduced by reaction GeI4(g) + Ge(s) .
Ref.
653K
> 2 GeBr2 (v). >2 SiBr2 (v). > 2 GeCI2. >3 SiCI2 (v).
' 2 GeI 2 (g).
Since the mean amplitudes of vibration, the l/j's, are functions of the vibrational temperature of the molecule, their relative change with respect to a change in the molecular vibrational energy will depend in the distribution of energy among the modes.
Therefore, information
concerning intramolecular energy transfer among modes can be obtained from a study of the relative changes in the mean amplitudes of all atom pairs of a molecule as a result of the deposition of energy into a particular vibrational mode. [94] The authors studied the absorption by room temperature SF 6 of the 10~t radiation emitted by a CO 2 laser. An electron diffraction unit was modified
ELECTRON DIFFRACTION
105
to allow the intersection at right angles of an electron beam, a gas jet of SF6, and focused 40W c w C O 2 laser beam. The focal spot was positioned just off the gas inlet nozzle, which was equipped with a thermocouple, to monitor its temperature. They found that the relative rise in the vibrational amplitudes indicated that the distribution of absorbed energy was not thermal. Therefore, this work has shown that it is possible to observe the effects of laser excitation of molecules by GED. The same molecule was studied in more detail by Bartell and coworkers [95-98] soon after the first study by Averdson and Kohl. Over 200 diffraction plates were taken at SF 6 sample pressures ranging from 50 to 600 Torr, and at various laser powers and wavelengths. Conclusion reached by the authors can be summarized as follows: Vibrational excitation corresponding to the absorption of up to 3.6 photons/molecule was deduced from the increased amplitudes of vibration of the SF, FFci s, and FFtran s atom pairs and the lengthening of the SF bond.
At high
excitations, electron diffraction intensities were accounted for best if it was assumed that two subsets of molecules were produced, one much hotter than the other.
Vibration-
vibration relaxation from v 3 to the other stretching modes was too fast to be followed.
Relaxation of stretching to
bending could be monitored, crudely, at lower pressures where approximately 30 collisions were needed at the depressed temperatures in the jet. At higher pressures and excitations V-T/R relaxation was observed, corresponding to a transfer of perhaps one-tenth of the vibrational excitation in the course of 103 collisions. [98]
106
MASTRYUKOV
In the course of these studies of laser-pumped SF 6, a number of diffraction patterns were recorded of vibrationally hot molecules that had been excited by accidental irradiation of the nozzle tip. These patterns, when analyzed, were found to yield information about vibrational anharmonicity that is difficult to derive from spectroscopy.
This accident helped to
introduce a new method in which gas molecules can be heated from room temperature to well over 1500 K in the order of a microsecond. Using this procedure the vibrations and thermal expansions of SF 6, CF4, SiF4, and CF3CI have been investigated up to 1200-1700 K [99-102]. Various models proposed for the treatment of increases in bond length have been assessed, among which an anharmonic Urey-Bradley field accounted well for results [99l. In all GED studies described so far, a continuous electron beam was used. In 1983, Ischenko and co-workers at the University of Moscow, Russia introduced a new technique where exciting optical pulses of the laser were synchronized with the pulsed electron beam [103,104]. In this method, the GED intensities are recorded at a well-defined time interval following excitation of the scattering molecules, and repeated electron pulses all diffract from molecules that have the same age relative to the time of excitation.
Using this method it should be possible to characterize short-
lived species or states, to study time-resolved energy absorption, distribution and redistribution, and to measure structural changes.
Clearly, for kinetic
studies it is crucial to choose the time duration of the electronic pulse and the delay time between pumping and probing shorter than the life-times of the
ELECTRON DIFFRACTION
107
species or the duration of the phenomena under investigation. This technique was called Stroboscopic Gas Electron Diffraction. Using the experimental arrangel~ent described in Refs. [ 103 ] and [ 104] a study of the IR multiphoton dissociation of the CF3I molecule, according to the reaction CF3I
hv >CF 3 + I, was initiated with the hope of recording
diffraction pattems of the short-lived CF 3 free radical. However, the authors were successful only in demonstrating changes in scattering intensities of this molecule induced by irradiation; the data have not been analyzed, and it was impossible to judge the success of the experiment from the information given. Later, Mastryukov [105] critically analyzed the data and found that the conclusions reached by the authors [103,104] were too optimistic. This criticism was never answered since the equipment in the University of Moscow was dismantled. A quantitative study of the photodissociation of CF3I was reported by Zewail and co-workers in 1994 who used UED [34]. Here we can find all the documentation lacking in the original publication by Ischenko et al. [ 103,104]. Earlier experiments in Moscow were of limited scientific value because the requisite detection schemes did not exist at the time but they opened the way for similar studies. Later, these studies continued at the University of Arkansas using the on-line data recording techniques mentioned in Section II.A [106].
In the early 1990's Sch~ifer and co-workers studied two
interesting systems using laser irradiation coupled with their equipment. They called their method Time-Resolved Electron Diffraction (TRED) and
108
MASTRYUKOV
proposed it as a tool that can be used to investigate the evolution of internuclear distances in reacting molecular systems. First, the 193-nm photodissociation of isomeric 1,2-dichloroethens was studied [ 107]. The authors conclude: When the time delay between excitation and diagnostic electron scattering is 15 ns, only fragmentation and no cistrans isomerization is observed.
When the time delay is
several milliseconds, cis-trans isomerization is prevalent. This finding confirms the thesis that, because of faster intervening dissociative processes, cis-trans isomerization is not
a
primary
photochemically
driven
unimolecular
rearrangement of 193-nm irradiated dichloroethenes but hinges upon intermolecular collisions. [ 107] The second example was the 193-nm photodissociation of CS 2 [ 108]. This is how this study was characterized in the abstract to this paper: A novel data analysis procedure is described, based on a variational solution of the SchrSdinger equation, that can be used to analyze gas electron diffraction (GED) data obtained from
molecular
ensembles
in
nonequilibrium
(non-
Boltzmann) vibrational distributions. The method replaces the conventional expression used in GED studies, which is restricted to molecules with small-amplitude vibrations in equilibrium distributions, and is important in time-resolved (stroboscopic) GED, a new tool developed to study the nuclear dynamics of laser-excited molecules.
As an
example, the new formalism has been used to investigate the structural
and
vibrational
kinetics
of
C-S,
using
stroboscopic GED data recorded during the first 120 ns following the 193 nm photodissociation of CS 2. Temporal changes of vibrational population are observed, which can
ELECTRON DIFFRACTION
109
be rationalized by collision-induced electronic-to-vibrational energy transfer from excited S(1D) atoms to ground state C-S and CS 2. The time-evolution of the energy transfer is modeled by determining the vibrational distributions and mean internuclear distances (ra, g) of C-S as functions of delay time.
Inverted (non-Boltzmann) distributions are
observed, and the refined parameters of model distributions are presented. [ 108] We conclude our description of experiments involving the photon excitation by mentioning a single contribution by British researchers who combined GED with flash-photolysis [109].
These authors conducted
diffraction studies on the decomposition of chlorine dioxide and biacetyl using electron pulses. Study
of the
photodissociation
dynamics
and
nonequilibrium
vibrational distribution of laser excited species requires significant changes in the theory used earlier for structure determination and described in Section II.B. Therefore, it is not surprising that a number of theoretical papers were published by different groups in an attempt to give a better description of the phenomena involved [ 110-116].
IV.
T E C H N I Q U E S I N C O R P O R A T E D IN S T R U C T U R A L A N A L Y S I S In this section we come back to the use of our regular GED equipment (See Section II. A) and we will see how the usual structure analysis described in Section II. C can be modified in order to get more reliable and more precise structural information.
So far, four different and independent sources of
information were proved to be particularly useful for incorporation into the regular structure analysis. Curiously enough, these techniques were introduced to GED
110
MASTRYUKOV
during four consecutive decades showing rather accurately their chronological sequence. 9 Vibrational Spectra (symbolized as IR) 1950s Rotational Constants, mostly taken from microwave spectroscopy (MW) 1960s 9 Theoretical methods, (Molecular Mechanics, ab initio, DFT) 1970s 9 Liquid crystal NMR (LC NMR) 1980s. These combinations will be reviewed in this order in subsections A, B, C and D. Each of these subsections contains tables listing molecules studied by these particular combinations with GED.
However, there are many cases in which
several techniques were applied simultaneously, then the molecule is mentioned only once in the corresponding table giving the priority to the same sequence of the methods: IR, MW and Theory. The only exception is made for LC NMR, otherwise these cases would have been absorbed by the previous combinations. The ultimate goal of a combined structure analysis is to obtain a self-consistent molecular model by the process schematically depicted in Fig. 4.
This is a
modified version of Fig. 10-2 taken from a review written by Geise and Pyckhout [ 117]. Such a combination became possible when the physical significance of the structural parameters produced by any individual method was clarified and they were made compatible with electron diffraction. A.
Vibrational Spectroscopy Historically, vibrational spectroscopy was the first method which found a very useful application in the GED structure analysis. As early as in 1949, the Karles [ 118] were able to carry out an electron diffraction experiment of
ELECTRON DIFFRACTION
111
Theory Geometry, Energy and Force Field
1 ED
IR
MW
LC NMR
Self-Consistent Molecular Model
FIG. 4.
Components in the determination of a self-consistent molecular model.
sufficient precision to allow a determination not only of the interatomic distances, but also of mean amplitudes of vibrations. The studies reported in this paper were of CO 2 and CC14. The experimental amplitudes of vibration were compared with those calculated by the authors for CO 2 from formulas given by Debye [119] and with those calculated by James [120] for CC14. It had been shown by Debye and James that the amplitudes of vibration could be calculated from molecular vibration frequencies and atomic masses, together with assumptions about the form of the harmonic force field of the molecule. This is how J. and I. L. Karle describe this period of time in their recollections [ 121 ]: Confidence in the increased accuracy with which interatomic distances could be determined, as well as the possibility of
112
MASTRYUKOV
obtaining reliable values for root mean squared amplitudes of vibration, was enhanced by satisfactory comparisons with results from infrared and microwave spectroscopy.
An
incident that occurred soon after our first work using the new
analytical techniques
was
published provided
a
poignant example of the value of such comparisons. One of the first molecules investigated by the new analytical techniques was CO 2 (Karle and Karle, 1949). We reported a value of 0.040+0.007 A for the rms amplitude for the O...O distance. However we had calculated a value of 0.029 A from spectroscopic data which indicated a discrepancy somewhat larger than our reported limit of error for the electron diffraction experiment.
It was soon shown by
Yonezo Morino (1950) that the formula that we were using was in error and that atter correction the computed value for spectroscopic data was 0.041 A in very good agreement with the result from the electron diffraction experiment. [ 121 ] (The authors refer to their own paper, which we already mentioned [118], while the contribution by Morino can be found under Ref. [ 122]). In the early 1950s Morino and his co-workers in
Japan established a
systematic treatment for the calculated of amplitudes of vibration from harmonic force fields, based on the widely used Wilson GF matrix method [ 123], which is still the foundation for the majority of such calculations. The following quotation taken from Morino's personal recollection [ 124] describes the next milestone of this story: In 1959 I visited Trondheim and had a chance to talk with Otto Bastiansen about the future prospects of gas electron diffraction.
My presentation at a colloquium there on the
method of calculation of mean square amplitudes apparently
ELECTRON DIFFRACTION
113
stimulated Sven S. Cyvin, who later became an expert in the field (Cyvin, 1968). Bastiansen took me to a ski-house on the top of the Holmenkollen Olympic Stadium near Oslo, where he told me about his new discovery that distance between two nonbonded atoms in a linear molecule, such as allene or dimethylacetylene, deviates appreciably from the sum of their bond lengths (Almenningen, Bastiansen and MuntheKaas, 1956; Almenningen, Bastiansen and Traetteberg, 1959). An idea came to me that this systematic difference would be attributable to the bending vibrations of linear molecules.
Upon my return to Tokyo, we made a
calculation and found that the observed magnitudes of the 'shrinkages'
were what we expected from the force
constants obtained by spectroscopic measurements (Morino, Nakamura and Moore, 1962). [124] As we will see, the conversation in Holmenkollen had a very important conceptual impact on the GED method. This phenomenon is known as the "Bastiansen-Morino shrinkage effect" or more simply as the "shrinkage effect". Shrinkage effect is a direct consequence of the molecular vibrations and it shows that the bonded and non-bonded interatomic distances measured by GED are not self-consistent, i.e., they do not correspond to a set of distances calculated from a rigid geometrical model.
This phenomenon is
illustrated on a simplified diagram below for a linear triatomic molecule A B 2 (simplification in this case means that for this type of vibration while two B atoms move up, atom A moves down which is ignored in this diagram).
114
MASTRYUKOV
rg
-,~
r e
,~
The B...B distance observed by GED (rg) is shorter than that obtained by doubling the A-B distance as we expect for a linear system. This effect arises because at every instantaneous position in the bending mode, except precisely at the linear (r e equilibrium) configuration, the B...B distance must be less than twice the A-B distance. Usually the shrinkage effect is routinely included in electrondiffraction least-squares refinement.
In order to do so, it has been found
appropriate to introduce a third distance type ra defined as the distance between mean positions of atoms at a particular temperature [125,126].
If
the harmonic force field is known, ra may be calculated from r a according to
Eq. (5)" l2 ra = r a + - - - K r
whereK
2r
(Ax 2) and (Ay2) are the mean square perpendicular vibrational amplitudes.
(5)
ELECTRON DIFFRACTION
115
In Section IV. B will see that this new ra structure plays an important role to make the different methods compatible. Several programs were published to calculate amplitudes of vibration and shrinkage corrections [127-129]. In order to do so, one needs to know the force field of a molecule. There are typically three options for this: 1) An approximate force field is built by transferring the elements, from similar molecules (See, for example, [130, 131 ]). 2) The force field can be calculated theoretically using either molecular mechanics [132, 133] or ab initio calculations. 3) The force field is obtained from the normal coordinate analysis of the vibrational spectrum of a molecule. Data collected in Table 3 illustrates these options showing how it is practically done in different electron-diffraction groups [ 134-165].
B.
Rotational Constants According to Robiette [ 166], there are three major topics that comprise the main areas of interplay between spectroscopy and GED: the amplitudes of vibration, the shrinkage effect and the inter-relation of molecular structures obtained from spectroscopic and electron-diffraction data.
The
first two topics were discussed in the previous subsection while this subsection concentrates on the third topic. In the 1950s it was noticed that bond lengths (ro) derived from spectroscopic measurements of rotational constants (Bo) may differ from GED bond lengths (ra, rg) by an amount which is clearly outside
116
MASTRYUKOV
TABLE 3. Molecules studied by joint analysis of electron diffraction and vibrational spectroscopy.
No.
Molecular formula
Name
Other methods useda
Reference
BBr 3
Boron tribromide
134
B2GaH9
Hydridogallium bis(tetrahydroborate)
BiF 3
Bismuth trifluoride
136
BiI3
Bismuth triiodide
137
Br6H2Si3
Hexabromotrisilane
CdC12
Cadmium dichloride
139
CdI2
Cadmium diiodide
140
CeI 3
Cerium triiodide
141
CI2Mg
Magnesium dichloride
142
10.
C13Ga
Gallium trichloride
143
11.
C13In
Indium trichloride
143
12.
CI4Mg2
Magnesium dichloride dimer
142
13.
C14U
Uranium tetrachloride
14.
F3Sb
Antimony trifluoride
136
15.
F4U
Uranium tetrafluoride
145
16.
I3Sb
Antimony triiodide
137
17.
CFNO 5
Fluorocarbonyl peroxynitrate
Th
146
18.
CFN30
Fluorocarbonyl azide
Th
147
19.
CF202
Fluoroformyl hypofluorite
Th
148
20.
CH2C12
Dichloromethane
MW, Th
149
Th
Th
Th
135
138
144
ELECTRON DIFFRACTION
117
TABLE 3. (continued).
No.
Molecular formula
Name
Other methods useda
Reference
21.
C2FNO2
Fluorocarbonyl isocyanate
Th
147
22.
C2F202S 2
Bis(fluorocarbonyl) disulfane
Th
150
23.
C2F6S
Trifluoroethylidynesulfur Trifluoride
Th
151
24.
C2H203
Formic anhydride
MW, Th
152
25.
C2H6C12Ti Dimethyltitanium dichloride
Th
153
26.
C3H3CIO
2-Chloroacrolein
Th
154
27.
C3H403
Acetic formic anhydride
Th
155
28.
C3H6C12Si Dichloromethylvinylsilane
29.
C3H60 2
Ethyl formate
MW, Th
157
30.
C4H60
Methacrolein
MW, Th
158
31.
C4H60
N-Chloro-N-ethylethanamine
Th
159
32.
C4H100
Diethyl ether
MW, Th
160
33.
CsHloO 2
Isopropyl acetate
Th
161
34.
C6H140
tert-Butyl ethyl ether
Th
162
35.
CTHTCIO
Chloromethyl phenyl
Th
163
36.
C7H7CIO
Chloromethyl phenyl ether
Th
164
37.
C8H22Si2
1,2-Di-tert-butyldisilane
Th
165
156
aThe abbreviations have the following meaning: Th - theoretical calculations (Molecular Mechanics, ab initio and density functional theory); and MW- microwave spectroscopy
experimental error (for definitions of these parameters see Table 1). Critical examinations were made and it became clear that this arises because the spectroscopic and electron-diffraction bond lengths are derived from
118
MASTRYUKOV
observed quantities in which the effects of molecular vibrations are averaged in quite different ways.
In 1961, Bartell, Kuchitsu, and de Neui [167]
published a paper on mean and equilibrium structures of CH 4 and CD 4 where for the first time the precise relationship of electron-diffraction bond lengths to spectroscopic bond lengths was determined. Furthermore, the relation of both types of these structures to the equilibrium structure, r e, was found. This confirmed the essential equivalence of molecular information derived from two different methods if suitable corrections are applied to each. The seminal contribution by Bartell, Kuchitsu and de Neui [167] opened the gate for a stronger interaction between the two previously conflicting methods and Morino [124] stresses the importance of an open discussion" Probably the strongest evidence for the success of the collaboration between electron diffraction and spectroscopy may be found in the prosperity of the Austin Symposium on Gas-Phase Molecular Structure organized by James E. Boggs.
The central theme of the Symposium is the
discussion of the fundamental problems in molecular structural studies by electron diffraction, spectroscopy and other techniques including ab initio calculations.
The
symposium has been held every other year since 1966, with increasing numbers of participants and reports. According to Boggs
[168], the history of the interaction between
experimental structure determinations by microwave spectroscopy and by GED has clearly three different eras: "(1) competition and antagonism, (2) comparison and correction, and (3) integration of analysis" If we agree that
ELECTRON DIFFRACTION
119
the second period began in 1961 with the work by Bartell, Kuchitsu and de Neui [167] mentioned above, then there is no doubt we should assign the contribution by Kuchitsu, Fukuyama, and Morino in 1968 [169] as the beginning of the third period. In order to combine the outcomes of spectroscopy and GED in a consistent manner, we need to bring the results of both methods to the same geometrical basis.
In 1960, Oka [170] proposed a new type of structural
parameter, which he termed r z or the zero-point average structure. The same type of structure can be obtained from GED if we extrapolate r a to absolute zero: r0 = limT~ 0 (ra)
(6)
Thus, the r a structure introduced to account for the shrinkage effect, Eq. (5), being appropriately corrected, helps to relate spectroscopic and electrondiffraction distances. As already mentioned in Table 1, r O and r z represent the same physical quantity and it also puts an end to the variety of different structural types currently in use in the GED structure analysis although there are some more kinds of structure obtained from the ground state rotational constants [ 171 ]. The relation between different kinds of structures derived from spectroscopy and GED is shown in Fig. 5 using as an example a classical case of CH 4 and CD 4 molecules mentioned before [167].
(Fig. 5 is a
modified version of a figure taken from the review paper by Kuchitsu [172]). An examination of Fig. 5 shows that not only primary information taken from
120
MASTRYUKOV
CD4
CH4
rg
GED
ro re
rz
SP ro I
I
1.08
1.09
I 1.10
I
l.ll
r(C-X) X = H , D A FIG. 5
The C-H (open circles) and the C-D (black circles) distances of CH 4 and CD4 in electron diffraction (GED) and spectroscopic (SP) representations together with their equilibrium value, r e.
both methods (rg and r0) is sensitive to isotopic substitution but even r 0 and rz clearly show difference too. This happens because of the mass dependency of anharmonicity and it is only the equilibrium structure r e that is free of the isotope effects. The general strategy of the combination of spectroscopic and GED data is as follows. The rotational constants B 0 first are converted to B z using a harmonic force field B z = B 0 + 0.5 ~ - ( Z sharm
ELECTRON DIFFRACTION
121
while the electron diffraction r a or rg distances are converted to r o (see Table 1).
Then both sets of data are simultaneously treated by least-squares
procedure [ 169]:
II k=l
j=a,b,c
where the first term is identical with Eq. (3) and Ia, Ib and I c are principal moments of inertia calculated from the Az, B z, Cz rotational constants, respectively. In 1988 Kuchitsu, Nakata and Yamamoto [173] reviewed the state of the calculations at the time and collected data on 118 molecules studied by a combined (ED + MW) analysis.
Table 4 serves as a continuation of this
effort giving data on 35 molecules studied in the last decade [174-205]. However, this compilation is not exhaustive and serves only for illustrative reasons. It is also important to stress the point that these particular methods are found to be complementary, which means that their combination gives a better structure than any individual method alone.
C.
Theoretical Calculations As we saw in Section II C, in order to begin the GED structure analysis one needs a trial model. In early days this model was based on the chemical intuition of the researcher while nowadays it can be calculated using some of the methods of theoretical chemistry.
The role of theory is not limited,
however, to calculation of only the geometrical structure as is shown in Fig. 4: it gives also "Energy" and "Force Fields" (see more about theoretical
122
MASTRYUKOV
TABLE 4. Molecules studied by joint analysis of electron diffraction and microwave spectroscopy.
No.
Molecular formula
Name
Other methods useda Th
Reference 174
1.
B4H 10
Tetraborane(10)
2.
GeH6Si
Silylgermane
3.
CH3AsF2
Methyldifluoroarsine
Th
176
4.
CH3F2N
Methyldifluoroamine
Th
177
5.
CH40
Methanol
178
6.
CH6N2
Methyl hydrazine
179
7.
C2H3C1OS Methylchlorothioformate
8.
C2H3C13
1,1,1-Trichloroethane
C2H3F30
1,1,1-Trifluorodimethylether
10.
C2HsN
N-Methylmethyleneimine
81
11.
C2H6C12Si
Dimethyldichlorosilane
43
12.
C2H602
Ethane-1,2-diol
Th
183
13.
C2HsN2
Ethylenediamine
Th
184
14.
C3H403
Glycolicacid
185
15.
C3H6C1N
N-Chloroazetidine
186
16.
C3H7Br
1-Bromopropane
Th
187
17.
C3H7I
1-Iodopropane
Th
188
18.
C3H7N
Cyclopropylamine
19.
C3H7N
Propyleneimine
175
Th
180 181
Th
182
189 Th
190
ELECTRON DIFFRACTION
123
TABLE 4. (continued).
No.
Molecular formula
Name
Other methods useda
Reference
20.
C3H7NO
E-Propionaldehyde oxime
191
21.
C3H7NO
Z-Propionaldehyde oxime
Th
192
22.
C3HTNO
Propionamide
Th
193
23.
C3H7NO2
Methoxyacetamide
Th
194
24.
C3H9C1Si
Chlorotrimethylsilane
195
25.
C3H9N
Isopropylamin
196
26.
C4F6
Hexafluorocyclobutene
190
27.
C4H602
Methyl acrylate
Th
197
28.
C4H60 2
Cyclopropanecarboxylic acid
Th
198
29.
C4H80
2-Methylpropanal
Th
199
30.
CsHsN
Pyridine
Th
200
31.
C5H6S
2-Methylthiophene
Th
201
32.
C5H6S
3-Methylthiophene
Th
202
33.
C5H7S
N-Methylpyrrole
Th
203
34.
C6H10
Bicyclo[3.1.0]hexane
Th
204
35.
C7H5C10
4-Chloro-benzaldehyde
Th
205
aThe abbreviation Th means theoretical calculations (ab initio).
methods in Refs. [206-208]). In the case of a conformational mixture the calculation of energies provides a guess for a conformational ratio.
A
theoretical force field allows one to calculate the amplitudes of vibration and the shrinkage corrections discussed in Section IV A.
A quotation from a
124
MASTRYUKOV
paper by Oberhammer published in 1998 [209] gives some historical background: About 25 years ago experimental and theoretical studies of molecular structures and conformational properties were done quite separately.
Because theoretical methods have
also become applicable for reasonably sized molecules, experimental investigators started to take advantage of these methods and included molecular mechanics (MM), semiempirical, and later ab initio and/or density functional (DFT) calculations in their experimental analyses. Today, because computer programs are very easy to use and sufficient computer capacity is generally available, most experimental studies of gas phase structures by gas electron diffraction (GED), microwave (MW), or high-resolution infrared spectroscopy are combined with theoretical calculations. This combination of experimental and theoretical methods is to the advantage of both experiment and theory. The same author later, in Table 1, mentions four problems existing in the GED method and states that theory may be very useful in their solution: 1. Closely spaced distances 2. .
Location of hydrogen atoms High correlations between geometric parameters and vibrational amplitudes
4.
Several conformations
We can illustrate the first two problems using vinyl halide molecules. The location of hydrogen atoms in the parent molecule, ethene (I),
ELECTRON DIFFRACTION
125
H
H
\ /
/ C-'-C
H
H
H
X
\
/ H
I
/ C---, C
H
\ X
II
is described by just two parameters, i.e. the C-H bond lengths and the C=C-H bond angle. In vinyl halides (II) due to the reduced symmetry the three C-H bond lengths and the three C=C-H bond angles are no longer identical although the actual non-equivalence may be rather small and very difficult to measure. On the other hand, it is very easy to calculate this structure. A similar problem exists in vibrational spectroscopy. Fogarasi and Pulay [210] have provided a review of the ab initio calculations of force constants and they have shown that the less dominant force constants are usually more accurately determined from theory than from experiment. L. Sch~ifer was the first to begin to use in the late 1970s the results from molecular mechanics and ab initio calculations to supplement GED structure analysis. He also coined the acronym MOCED (Molecular Orbital Constrained Electron Diffraction) for such a combination [211,212].
The
major problem facing the direct incorporation of ab initio results (re structure) into electron-diffraction structure analysis is the same as it was for the (GED + MW) combination (see Sec. IV B): the different physical significance of the parameters involved. There were many different attempts to relate the electron-diffraction structures to equilibrium parameters extracted from ab initio and other
126
MASTRYUKOV
calculations [213-217]. However, in the MOCED analysis another approach is used: instead of operating with the absolute values of the parameters, their differences are constrained. Generally, the practice of constraining parameters or their differences is not ideal.
First, it implies that the fixed value is absolutely correct.
Second, fixing parameters can result in unrealistically low standard deviations for correlated parameters.
In order to solve these problems,
Rankin and his co-workers in 1996 [218] introduced still another approach, called SARACEN (Structure Analysis Restrained by Ab Initio Calculations for Electron DiffractioN). This "method hinges on two points: the use of calculated parameters as flexible restraints instead of rigid restraints, and choosing to refine all geometrical parameters as a matter of principle." In essence, the SARACEN method is the combination of the MOCED [211, 212] and predicate observation [55] (see Sec. II C) methodologies, with ab initio data being used to construct the predicate observations necessary to
complete the refinement. This method seems to be rather general allowing one to use any kind of additional independent information such as X-ray crystallography, which has never been used with GED. So far, SARACEN has been applied to the study of only a few molecules [ 174, 218-221 ] but no doubt it will find many applications in the future. Table 5 lists 24 molecules studied by GED assisted by Molecular Mechanics calculations [222-239]. This table is intended to be a continuation of Table 10-4 in the review by Geise and Pyckhout [ 117].
ELECTRON DIFFRACTION
127
TABLE 5. Examples of electron diffraction analyses assisted by molecular mechanics calculations.
Reference
No.
Formula
Name
1.
C3H602
1,3-dioxolane
222
2.
C4H8Br 2
1,4-Dibromobutane
223
3.
C4H8C12
1,4- Dichlorobutane
223
4.
C4H8F2
1,4- Difluorobutane
224
5.
C4H802
2-Methyl- 1,3-dioxolane
222
6.
C4HloO2
1,4-Butanediole
225
7.
C5HloO2
2,2-Dimethyl- 1,3-dioxolane
222
8.
C6H8
3-Methylene- 1,4-pentadiene
226
9.
C6H10
1,5-Hexadiene
227
10.
C6H140
Dilsopropyl ether
228
11.
C7H12
Norbomane
229
12.
C7H12
Cycloheptene
230
13.
CsH12
Bicyclo [3.3.0]oct- 1,5-ene
231
14.
C8H16
Cyclooctane
232
15.
CsH19N
Di-tert-butylamine
233
16.
C9H 18
Cyclononane
234
17.
C9H 18~
Di-tert-butylketone
233
18.
C9H20
Di-tert-butylmethane
233
19.
C10H18
Bicyclopentyl
235
20.
CloH30Si5
Decamethyl cyclopentasilane
236
128
MASTRYUKOV
TABLE 5. (continued). No.
Formula
Name
Reference
21.
C12H22
Bicyclohexyl
237
22.
C 12H24
Cyclododecane
238
23.
C18H21Sc
Tris(methlcyclopentadienyl)scandium
239
24.
C18H21Yb
Tris(methylcyclopentadienyl)ytterbium
239
Similarly, Table 6 is intended to be a continuation of Table 10-5 by the same authors [ 117]. However, the activity in the field of combination GED + ab initio (DFT) is more impressive than the present author imagined at the
start of work on this review.
The list of references grew during its
preparation beyond expectation. Therefore, it was decided to cover only the literature which appeared in 1995-1997 with the occasional papers published in 1998. Table 6 lists data for 102 molecules [240-313 ], about 60% of which were investigated in just two groups headed by H. Oberhammer and D. W. H. Rankin. The actual number of molecules studied with the incorporation of theoretical methods is larger: theory was used in 46 cases mentioned in Table 3 and 4.
ELECTRON DIFFRACTION
TABLE 6.
129
Examples of electron diffraction analyses assisted by
ab initio or DFT calculations.
No.
Reference
Formula
Name
1.
B4F3HsP
Tetraborane(8)-trifluoropho sphine ( 1/ 1)
240
2.
B6C12H8
1-(Dichloroboryl)pentaborane(9)
241
3.
C1FO3S
Chlorine fluorosulfate
242
4.
C1F5OS
Pentafluorsulfanyl hypochlorite
243
5.
C14H3NSi2
Bis(dichlorosilyl)amine
244
6.
F203S
Fluorine fluorosulfate
242
7.
F3Mn
Manganese trifluoride
245
8.
CFNOS
Sulfinyl cyanide fluoride
246
9.
CFNO3S
Fluorosulfonyl isocyanate
247
10.
CFNO 5
Fluorocarbonyl peroxynitrate
248
11.
CF2N2OS
Difluorosulfenylimine cyanide
249
12.
CF3NOS
Fluoroformylinimosulfur difluoride
250
13.
CF3NOS2
(Trifluoromethyl)sulfanyl sulfinylimine
251
14.
CF3NS
Sulfur cyanide trifluoride
246
15.
CF40 2
Bis(fluorooxy)difluoromethane
252
16.
CH4C1P
Chloromethylphosphine
253
17.
CHsC14NSi2
Bis(dichlorosilyl)methylamine
245
130
MASTRYUKOV
TABLE 6. No.
(continued). Reference
Formula
Name
18.
CH9NOSi2
O-Methyl-N,N-disilylhydroxylamine
254
19.
C2CIF30
Trifluoroacetyl chloride
255
20.
C2C1F3OS
(Chlorocarbonyl)trifluoromethylsulfane
256
21.
C2C1F3OS
Trifluorothioacetate chloride
257
22.
C2C12F20
Chlorodifluoroacetyl chloride
255
23.
C2C1202
Oxalyl chloride
258
24.
C2FNO2S
(Fluorocarbonyl)sulfenyl isocyanate
259
25.
C2F4OS
(Fluorocarbonyl)trifluoromethylsulfane
256
26.
C2F603
Bis(trifluoromethyl) trioxide
260
27.
C2F605S2
Trifluoromethanesulfonic anhydride
261
28.
C2HC12FO
Dichloroacetyl fuoride
262
29.
C2HF3OS
Trifluorothioacetic acid
257
30.
C2H5C13Si
(Chloromethyl)dichloromethylsilane
263
31.
C2H6AsF
Dimethylfluoroarsine
176
32.
C2H6CI2Ti
Dimethyldichlorotitanium(IV)
153
33.
C2H6FO2P
O-Methyl methylphosphonofluoridate
264
34.
C2H7NO
N,N-Dimethylhydroxlamine
265
35.
C2HloSi2
1,4-Disilabutane
266
ELECTRON DIFFRACTION
TABLE 6. No.
131
(continued). Reference
Formula
Name
36.
C3F60
Perfluoromethyl vinyl ether
267
37.
C3F6S
1-Trifluoromethylthiop- 1,2,2-trifluoroethane
268
38.
C3F804S2
Bis(trifluoromethylsulfonyl) difluoromethane
261
39.
C3H2C1202
Malonyl dichloride
269
40.
C3H2F6S2
Bis(trifluoromethylthio)methane
270
41.
C3H3F30
Methyl trifluorovinyl ether
267
42.
C3H3F3OS
Methyl trifluorothioacetate
257
43.
C3H3F3S
Trifluoromethyl vinyl sulfide
271
44.
C3H3F6NO
O-Methyl-N,N-bis(trifluoromethyl)hydroxy-
272
lamine 45.
C3H5NO
3-Aminoacrolein
273
46.
C3H6C12Si
Methyl(vinyl)dichlorosilane
274
47.
C3H6F2Si
Methyl(vinyl)difluorosilane
274
48.
C3H7C10
3-Chloropropane-l-ol
275
49.
C3HTN
N-Methylethylideneimine
276
50.
C3H8C12Si
(Chloromethyl)chlorodimethylsilane
263
51.
C3HsFN
(Fluoromethyl)dimethylamine
277
52.
C3H12Si2
1,4-disilapentane
266
53.
C3H14B4
1,2-Propano-2,4-tetraborane(10)
278
132
MASTRYUKOV
TABLE 6.
(continued). Reference
No.
Formula
Name
54.
C4F6OS2
Bis(trifluoromethylthio)ketene
279
55.
C4H2CI2N2
2,5-Dichloropyrimidine
218
56.
C4H2CI2N2
2,6-Dichloropyridazine
220
57.
C4H2C12N2
3,6-Dichloropyridazine
220
58.
C4H2CI2N2
4,6-Dichloropyridazine
220
59.
C4H6
1-Butyne
280
60.
C4H6Br202S
Trans-3,4-Dibromotetrahydrothiophene-1,1 -
281
dioxide 61.
C4H604Sn
Tin(II) acetate
219
62.
C4HTF3Si
(Trifluorosilylmethyl)cyclopropane
282
63.
C4HsBr2
1,3-Dibromobutane
283
64.
C4H80
(Z)-Methyl- 1-propenyl ether
284
65.
C4H9Br
1-Bromobutane
285
66.
C4H9C1
1-Chlorobutane
285
67.
C4H9I
1-Iodobutane
285
68.
C4H9NO
N,N-Dimethylacetamide
286
69.
C4H10Si
(Silylmethyl)cyclopropane
287
70.
C4H12OSb2
Bis(dimethylstibyl)oxane
288
71.
C4H12Sb2Se
Bis(dimethylstibyl)selane
288
ELECTRON DIFFRACTION
TABLE 6.
133
(continued).
No.
Formula
Name
72.
C4H16B4
2,3-Butano-2,4-tetraborane
278
73.
C5H60
Divinylketone
289
74.
CsH9F3Si
(Trimethylsilyl)trifluoroethene
290
75.
C5HloO
Diethylketone
291
76.
C5H100
Tetrahydrofurfuryl alcohol
292
77.
CsHloSi
Trimethylsilylacetylene
293
78.
C5H12OSi
3,3-Dimethyl-3-silatetrahydrofurane
294
79.
C5H12SSi
3,3-Dimethyl-3-silatetrahydrothiophene
295
80.
C5H12Si
Trimethylvinylsilane
296
81.
C6H602
5-Methylfuan-2-aldehyde
297
82.
C6H8
3-Ethenyl-3-methylcyclopropane
298
83.
C6H120
tert-Butyl vinyl ether
284
84.
C6H18N3P
Tris(dimethylamino)phosphine
299
85.
C6H18N3Sb
Tris(dimethylamino)antimony
299
86.
C7H7C10
2-Chloroanisole
300
87.
C7H7NO2
Methyl isonicotinate
301
88.
C7H7NO2
Methyl Nicotinate
302
89.
C7H7NO2
Methyl picolinate
302
Reference
134
MASTRYUKOV
TABLE 6.
(continued).
No.
Formula
Name
Reference
90.
C7H16Ge
1,1,3,3-Tetramethylgermacyclobutane
303
91.
C7H20N3P
Tris(dimethylamino)methylenphosphorane
304
92.
C8H604
4,6-Dihydroxyisophthanaldehyde
305
93.
CsH10
3,4-Dimethylenehexa1,5-diene
221
94.
C8H12S6
1,3,5,7-Tetramethyl-2,4,6,8,9,10-
306
hexathiaadamantane 95.
CsH16B10
1-Phenyl-1,2-dicarba-closo-dodecaborane(12)
307
96.
CsH18F4Si2
1,2-Di-tert-butyltetrafluorodisilane
308
97.
C8H18Si2
1,2-Bis(trimethylsilyl)acetylene
293
98.
CsH22B4Si2
1,2-Bis(trimethylsilyl)-1,2-dicarba-closo-
309
hexaborane 99.
C9H14Si
Trimethylsilylbenzene
310
100.
C11H23M~
Tris(dimethylamido)cyclopentadienyl-
311
molybdenum
D.
101.
C14H14N203
p-Azoxyanisole
312
102.
C70
Fullerene
313
Liquid Crystal NMR Spectroscopy All the previous combinations of GED with other techniques dealt with free molecules, i.e. molecules in the gas phase. However, as early as 1973 it was shown by Diehl and Niederberger [314] that liquid crystal NMR (LC
ELECTRON DIFFRACTION
13 5
NMR) spectra give structural information compatible with GED. Later, in 1979, a scheme was devised to obtain r~ structures from LC NMR data [315]. In principle, with a good harmonic force field, one should be able to combine ra structural information from both LC NMR and GED and determine a more precise molecular structure.
This idea was brought to
fruition in 1981 by Rankin and his co-workers [316]. Similar combinations of LC NMR with microwave spectroscopy [317] and ab initio calculations [318] are also known. In 1988, Rankin [319] wrote a review paper where he described in detail a combination, which can be symbolized as GED + LC NMR. At the time of writing, only two molecules were studied, difluorophosphine selenide [316] and difluorophosphine sulphide [320], so the major step forward was done only later. Quotations from this review by Ran~k~-[3 i9] can serve as a short introduction to the field. The form of the NMR spectrum of a compound dissolved in a normal isotropic solvent depends primarily on the chemical shifts of the spinning nuclei involved and on the coupling constants Jij between them.
These coupling
constants are indirect in that they depend on interactions between magnetic dipoles transmitted through the bonds and any intervening atoms. There is also a direct coupling Dij between two spinning nuclei, but this is not normally observed in NMR spectra. This coupling is through space, and its magnitude depends inversely on the cube of the distance between the two nuclei. However, it also depends on the angle of the vector joining the nuclei to the applied magnetic field, and under normal conditions the molecules tumble rapidly on the NMR timescale and thereby average
136
MASTRYUKOV
the direct coupling to zero. But in a liquid crystal solvent the solute molecules do not tumble freely, and consequently dipole-dipole couplings may be observed. Besides Rankin's group, there is only one more group headed by S. Konaka at Hokkaido University which is interested in structural applications of LC NMR, both alone [318,321,322] and in its combination with GED [323]. From the NMR spectrum measured in liquid crystal solvent, direct dipolar constants Dij can be obtained as was mentioned above in Rankin's quotation. Dipolar couplings Dij are related to structural parameters as
Dij =
8x 2
(3cos200,) ,
r3
(8)
where 0ij is the angle between the magnetic field and a vector rij connecting two nuclei i and j, ? is the gyromagnetic ratio and the bracket denote the average over intramolecular motion and reorientational molecular motion [317,319]. As it happens, the structural information derived from dipolar couplings is often complementary to that given by electron diffraction data. In the first place, the most accurate and easily measure NMR data relate to hydrogen nuclei, and so it is the hydrogen atom positions that are given most reliably by NMR methods. Thus electron diffraction and NMR spectroscopy together can give a complete structure for a molecule containing both hydrogen and heavier atoms. In a least-squares refinement based on the electron diffraction data alone, the quantity to be minimized is defined by Eq. (3), Sec II. C. When we
ELECTRON DIFFRACTION
137
use rotational constants as additional data, the original expression is extended to include the values of the extra observations as can be seen from Eq. (7), Sec. IV. B. Now we need to extend this expression to include the differences between the observed values of dipolar coupling constants and those calculated using the trial structure by Eq. (8). Table 7 lists 15 molecules [316,320,323-334] studied by a combination GED + LC NMR, which really means GED + IR + LC NMR because the conversion to an ra structure always requires a knowledge of the force field. In the majority of cases, rotational constants were also used and, finally, for perfiuorocyclopropene (No. 5) and v-picoline (No. 15) the highest integration of all methods was achieved: GED + IR +MW + ab initio + LC NMR. Therefore, these two molecules serve as an example of the self-consistent molecular model shown in Fig. 4.
VD
Concluding Remarks and Acknowledgements The development of the GED method in combination with other techniques has been presented.
Let us formally assume that the effectiveness of a
combination of GED with one particular technique is related to a number of molecules in Tables 2-7, which is 13, 37, 35, 24, 102 and 15, respectively (doing this we certainly ignore the fact that many molecules were studied by a combination of several techniques). Now, if we divide all these numbers by 13, which is the lowest number, and put the resulting ratios in the order we arrive at the following sequence: 1 (MS); 1.2 (LC NMR); 1.8 (MM) ; 2.7 (MW) ; 2.8 (IR) and 7.8 (ab initio ). This sequence helps us to realize once again the explosive
138
MASTRYUKOV
TABLE 7. Molecules studied by joint analysis of electron diffraction and liquid crystal NMR spectroscopy.
No.
Formula
Name
Other Methods useda
F2HPS
Difluorophosphine sulphide
F2HPSe
Difluorophosphine selenide
3.
CH3NSi
Silyl cyanide
MW
324
4.
C2H4Si
Silyl acetylene
MW
325
C3F4
Perfluorocyclopropene
IR, MW, ab initio
326
6.
C4H4N2
Pyrazine
MW
327
7.
C4H4N2
Pyrimidine
MW
327
8.
C4H4N2
Pyridazine
MW
328
9.
C4H40
Furan
MW
329
10.
C4H4S
Thiophene
MW
330
11.
C6H4C12
ortho-Dichlorobenzene
MW
331
12.
C6H4C12
meta-Dicheorobenzene
MW
332
13.
C6H4C12 para-Dichlorobenzene
RS
333
14.
C6H5C1
Chlorobenzene
MW
334
15.
C6HTN
y-Picoline
IR, MW, ab initio
323
MW
Ref.
320 316
aThe abbreviations have the following meaning: IR - infrared spectroscopy MW - microwave spectroscopy, and RS - rovibrational spectroscopy
growth of studies using ab initio calculations. The reason for this is simple: theory proved to be helpful in the GED structure analysis and it is easy to use. It leaves
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no doubt that ab initio calculations will remain the major additional tool in the hands of the electron diffractionists and those who use microwave spectroscopy [ 171 ] and LC NMR [318]. We also think that particularly the SARACEN method will find many applications in the future. In conclusion, the gas electron diffraction method was chosen as a specfic example in this chapter. However, the problem of combination of several different techniques seems to be a quite general concern. From what we have seen above, one can infer that there are just two basic conditions that determine how successful a particular combination of techniques may be.
These conditions are:
compatibility and complementarity. I would like to thank Professor James R. Durig for his invitation to write this Review.
I thank Professor Lev V. Vilkov for initiating my interest in a
combination of GED with various techniques. Professor James E. Boggs read and corrected the manuscript, which is much better thanks to his comments and strong opinions. Finally, I would like to acknowledge the people who have helped me write this Review.
First my wife, Patricia Hakes, for her constant encouragement.
Collecting the literature data was very much facilitated due to the GEDIS Letters, 1993-1997 prepared by Drs. Jtirgen and Natalja Vogt. I enjoyed discussions with Professors Manfred Fink and Denis A. Kohl, which clarified my ignorance in both scientific and historical matters. I thank N. B6wering, I. Hargittai, K. Hedberg, T. Iijima, S. Konaka, D. W. H. Rankin, L. Sch~ifer, P. Weber and K. R. Wilson for helpful correspondence.
I thank the Robert A. Welch Foundation for support.
Last, but no means least, my gratitude is extended to Linda Smitka and Conrie
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Powell, in the UMKC Department of Chemistry Office, for typing and formatting this manuscript. NOTE A D D E D IN P R O O F
In section II.A, an apparatus built by Zewail and coworkers and using ultrafast electron diffraction (UED) was described and later in Section III.C a successful application of this technique to the photodissociation of CF3I was mentioned [34]. More recently, the same technique was applied to a study of the photodissociation of CH2I2 [335]. Related to ultrafast diffraction measurements in general, the many papers on ultrafast X-ray diffraction published by Wilson and coworkers [336-340] are of imerest, the X-ray and electron diffraction cases being very similar.
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166
I.
GRONER
INTRODUCTION The determination of molecular structures has been one of the objectives of high resolution spectroscopy for a long time. This technique, together with gas phase electron diffraction, is the only practical method available to study structures of molecules in the gaseous state. In that state at sufficiently low pressures (when collisional lifetimes are sufficiently long), molecules are free from interactions with other species and can therefore be studied isolated from the environment. Modem high resolution molecular spectroscopy has a tremendous resolving power
and precision.
Conventional
Stark effect modulated
microwave
spectroscopy has a typical resolution of 1 in 105 and precision of 1 in 106. A similar precision is achieved by high resolution Fourier transform infrared spectroscopy. Fourier transform microwave spectroscopy and laser based infrared spectroscopy have a precision close to 1 in 108. If these spectroscopic methods are coupled with the supersonic molecular beam technique, the resolution approaches 1 in 107.
Moreover, the theoretical models for fairly rigid molecules are
reasonably well understood.
With these models, the spectra observed for such
molecules can be reproduced within the experimental precision.
Many fitting
parameters of these models - the spectroscopic constants - are obtained with high precision. Among them are the rotational constants, which sometimes have 8 or more significant digits. Most models used to fit the observed spectra are based (in principle) on a geometrical model for the molecule in question.
Thus, one might come to the
conclusion that structural parameters of molecules should be obtainable with a
QUEST FOR EQUILIBRIUM STRUCTURE
167
precision that matches the precision of the spectroscopic constants. However, it has been known for a long time that the relation between the model parameters and the geometrical structural parameters is a complicated one because the rotating molecules are not rigid. Even in the vibrational ground state, molecules vibrate. They execute many vibrational periods during one rotation. This is the cause of vibration-rotation interactions, which make the extraction of precise structu~:al parameters from the spectroscopic constants quite difficult. The goal of this article is to present a review of the methods and techniques used to determine gas phase molecular structures from high resolution spectroscopic data. The review covers the period from about 1980 to 1998 with the emphasis on the structures of fairly rigid molecules. The particular problems of determining structures of floppy molecules with low-frequency large-amplitude intemal motions or of Van der Waals complexes are not addressed. Likewise, the determination of structures by a combined analysis of spectroscopic and diffraction data is not a topic of this review. The other principal technique to obtain structures of molecules in the gas phase, electron diffraction, is reviewed in another article in this volume [ 1]. Significant progress has been made during the last two decades in the determination of molecular structures in the gas phase.
The theory and the
development of new methods will be reviewed in Section II. However, the big advances have really been made in the experimental field. In the "old days", the determination of molecular structures in the gas phase from spectroscopic data used to be a problem for a small band of microwave spectroscopists.
The
development of new experimental methods has moved the problem of determining
168
GRONER
accurate and precise molecular structures into the mainstream. Today, rotationally resolved spectral data are collected not only in the microwave region, but also in the infrared and UV/VIS regions. In the microwave region, significant progress has been made with the introduction of pulsed excitation of molecules and the subsequent analysis of the free induction decay signal in the time domain by the Fourier transform. Rotational spectra of molecules with extremely small dipole moments such as those induced by isotopic substitution [2] or centrifugal distortion in molecules without permanent dipole moments can now be measured by exciting gases in waveguides with short powerful microwave pulses [3]. The combination of MW Fourier transform spectroscopy with pulsed supersonic molecular beams has reduced the spectral line width to about 1 kHz. Because of the cooling of the species in the molecular beam, the sensitivity has increased and the spectroscopy of molecules at very low number densities is now routine. With the increased sensitivity and the simplified spectrum achieved by the molecular beam technique, it is now possible to study many isotopic modifications of molecules in natural abundance such as those substituted by 13C (1.10%), 33S (0.75%), 15N (0.37%), 180 (0.20%).
Under favorable circumstances, even D
substituted isotopomers can be detected in natural abundance (0.015%) [4]. Also the spectroscopy of many unstable species such as weakly bound Van der Waals complexes of molecules and atoms or unstable reaction intermediates and radicals as well as of species with low volatility (e. g. by laser induced evaporation) has become feasible.
A review of new experimental methods based on Fourier
transform microwave spectroscopy by Dreizler [5] has 100 references to methods and applications. Fourier transform IR, near IR or visible region spectroscopy is
QUEST FOR EQUILIBRIUM STRUCTURE
169
now a viable source of rotationally resolved spectroscopy, at least for small molecular species. The various laser sources make it possible to study rotationally resolved spectra in the IR and optical regions of the spectrum. These methods, particularly when combined with the molecular beam technique, have been very valuable for nonpolar molecules. Thanks to new techniques, rotationally resolved spectroscopy of ions is no longer the big problem it used to be particularly in combination with the generation of ions in electrical discharges [6]. Reviews of the methods to determine molecular structures have been published in the past [7-9]. In section II of this article, the basic theory of the vibration-rotation interactions are summarized and the various methods to determine the structure of molecules are reviewed, beginning with the
rs
method.
The short introduction to the least squares fitting technique and the r 0 and pseudoKraitchman methods is followed by an overview of newer developments based on the mass-dependence ab initio
(rm)
method. Section II concludes with methods employing
derived quantities, methods based on empirical correlations and a short
summary on the
rz
method. In section III, complete structure determinations of
molecules are listed that have been published from 1980 to 1998. I hope that the information presented in this section is fairly complete for the period, although I know that it is impossible not to miss something. Also in this section, a number of interesting examples of problems and their resolutions are illustrated.
170
II.
GRONER
THEORY AND METHODS
A.
Vibration-Rotation Interactions The theory of vibration-rotation interactions has been developed over the last 50 years by many prominent researchers. It has been presented in many texts on the subject e.g. [ 10], among them a rather complete summary by Aliev and Watson [11 ]. It is based on classical perturbation theory in the form of a sequence of contact transformations.
The results relevant to the
rotational constants are summarized here. The effective rotational constant about the fl axis in the vibrational state characterized by the vibrational quantum numbers v=(v 1 ... v k ...) with degeneracies (d 1 ... d k ...), B~, is given by [121 dk
dk
d/
e~ = ~,, - Zk ~l(v~ + T ) + Zk,/ r~(v, + T)(~, + T)...
(1)
Bfl is the rotational constant at the equilibrium geometry. The parameters or,p and 7"~ are called vibration-rotation constants. The indices k and l sum over all distinct vibrational (normal) frequencies. So far, no general equations are available for the constants 7"~. The constants a~contain harmonic and anharmonic components ak~ = Crka (harm) + akp (anh)
(2)
that are defined by [ 13 ] ~z~(harm) = - 2(Bff)2
pr 2
3o~ + (o~
(3)
QUEST FOR EQUILIBRIUM STRUCTURE
cok
Zku,a~a,
171
C03,n/2 "
In these equations, cok is the harmonic frequency of the normal coordinate Qk and ( ~ = - ( ~ is a Coriolis coupling constant. I,r is the equilibrium moment
of inertia about the axis 7 and a~r = \( ~cOl, / or1 is the inertial derivative with
respect to the normal coordinate Qk at equilibrium.
The cubic potential
derivative in dimensionless normal coordinates, kkl m, is related to the cubic force constant Okl m by
r
\ OQkOQtOQme= kklm(YkYlYm)l'2 hc
(5)
where
Yk = hCCOk/ h2"
(6)
The case of strong Coriolis resonance (e. g. if com= cok) must be treated appropriately and the term containing (~ in Eq. (4) be must be modified (see [ 1 1]). It is less obvious from Eq. (5) that a similar procedure must be applied to the anharmonic term in the case of Fermi resonance (if 2cok ~ COrn)[13]. The effective rotational constant for the vibrational ground state is therefore given by (neglecting the 7 terms)
dk - Z
=
B~ -
ct p
--i
(7)
The effective moment of inertia in the vibrational state v defined as h2
I~ - 2hcB~
(8)
172
GRONER
is related to the moment of inertia at equilibrium by dk
I~ ~ Ifl + ~ c~ (v, + -~-)
(9)
k
where c ~ = I a a~
(10)
For the vibrational ground state we have therefore
IoP~, lfl + ~ c~ Tdk = IP +cp Another
complication
(11)
arises
from the
fact that the rotational
constants B~ are usually obtained as effective fitting parameters of a reduced rotational Hamiltonian [14].
Not only do the numerical values of the
rotational constants depend on the exact form of the reduced Hamiltonian, they also contain small contributions from quartic and higher order centrifugal distortion terms. Watson [14] has proposed to always determine the so-called determinable combinations of these constants. The values of these combinations are independent of the form of the reduction, although they still contain small contributions from the distortion terms.
Up to the
quartic centrifugal distortion terms, the determinable combinations of the rotational constants are B p = B p - 2T ~r where
(12)
T ~`r are the quartic terms in the expansion of the rotational
Hamiltonian Hrot = EBPd~ +ETBr(J~ +d2r)+. . . . P
P,r
(13)
QUEST FOR EQUILIBRIUM STRUCTURE
1 73
Molecular structures determined from equilibrium moments of inertia, Ifl, are called
re
structures. Within the Born-Oppenheimer approximation,
they are well defined as the structures associated with local minima of the potential energy surface. As such they have no isotope effect, and the
re
structures are invariant to the particular choice of the isotopic data set. Molecular structures determined directly from the observed ground state moments of inertia, I0a, are called r 0 structures. They depend strongly on the specific set of isotopomer data used because the isotopic dependence of the vibration-rotation interaction terms ~e is different from the dependence of the moments of inertia. This quickly leads to contradictions in the results from different isotopic species.
Alternately, data from many isotopic
molecules can be averaged by the least-squares method.
The results still
depend somewhat on the isotopic data set and they have low relative precision typically in the range of 1%. Most methods to determine molecular structures try to compensate for or correct the effect of the vibration-rotation terms. As a result, there are now multitudes of methods available with as many special notations as methods. Their isotope dependence varies in practice from zero to full (for r 0 structures).
Their error limits are sometimes difficult to determine.
The
goals of some of the newer methods are internal consistency and maximum relative errors (difference to
re
structure) of O. 1% or less.
174
GRONER
B.
re
Structures The
structure of a molecule is obtained from the equilibrium
re
moments of inertia
Ie
or the rotational constants
B e.
Unless the molecule has
the formula XY n and sufficiently high symmetry, the equilibrium moments of inertia of more than one isotopomer are required because the number of independent structure parameters is larger than the number of independent moments of inertia of a single molecule. For each isotopomer, the constants aft have to be determined by analyzing the vibration-rotation spectrum for each vibrational fundamental or the pure rotational spectrum in the corresponding excited state. For asymmetric rotor molecules with more than three atoms, this task requires therefore a tremendous amount of data. Moreover, attention must be paid to the non-trivial problems of Coriolis and Fermi resonances because inadequate treatment of these effects may lead to incorrect vibration-rotation constants and therefore incorrect equilibrium constants [9].
For these reasons, the number of accurate equilibrium
structures for molecules with more than two atoms is still rather small although some progress has been made during the last two decades.
C.
Traditional
rs
method
In the traditional substitution method, the structural parameters are calculated from the Cartesian position coordinates of the nuclei. The squares of these coordinates are obtained from isotopic differences of principal moments of inertia by using equations first formulated by Kraitchman [ 15]. To apply this method, one needs the principal moments of inertia of a
QUEST FOR EQUILIBRIUM STRUCTURE
175
reference molecule, the parent, and those of all species that have just one of the nuclei substituted by an isotope. Chutjian [16] and Nygaard [17] have derived equations that are applicable in the case of substitutions of complete sets of symmetrically equivalent nuclei. 1.
Single substitution The equations formulated by Kraitchman [15] are based on the relations Px =Em~ x2
(14)
i
I,, = Py + P~
(15)
P. =(Iy + I , - / ~ ) / 2
(16)
Em, x, =0
(17)
i
~m,x,y, =0
(18)
i
and those obtained by cyclic permutation of x, y, and z in equations (14)-(18). The first equation defines the planar moments in terms of the atomic masses m i and the principal coordinates of the parent isotopomer, equation (15) defines the principal moment of inertia I x in terms of the planar moments P (also called second moment), whereas equations (17) and (18) are the first moment condition (also called the center-of-mass condition) and the product of inertia condition, respectively. In the general case, the Cartesian coordinates of atom k in the principal axes system of the parent isotopomer are obtained from the planar moments of the parent and the daughter isotopomer,
176
GRONER
where atom k with mass mass
x~ =
mk+Am , as
].,/-l(p;_ Px)
mk
has been substituted by an isotope with
follows:
=]./ AP~ 1+
(P>,-P:)(P:-Px)
1+
P>,-Px
P:-P,,
(19)
where APx = P ' - Px
.
(20)
The primed quantities refer to the daughter isotopomer, and the reduced mass of isotopic substitution,/z, is defined as
/~-
MAm M'
"
(21)
M and M' are the molecular masses of the parent and daughter species, respectively. Kraitchman [15] also derived several expressions for cases where atom k lies on a symmetry axis or in a symmetry plane. In these cases, one or two of the AP vanish exactly for rigid molecules
(r e
structure).
The resulting relations between the A/can be used to formulate several expressions for a single case. Because of the vibrational contributions to ground state moments of inertia, these relations do not hold exactly for the
(vibrating)
molecules
and,
therefore,
the
resulting
rs
coordinates are not identical. Usually, the averages of all possibilities have been reported as
rs
coordinates.
Rudolph [18] has shown that
these averages are almost identical to the results obtained from
QUEST FOR EQUILIBRIUM STRUCTURE
177
Chutjian-type expressions. For the atom k in the symmetry plane axy, z k vanishes by symmetry and x k is obtained from
=/tI'AP~ 1+ pyAPy _ p,, )
~
(22)
with/ax =/~. The coordinate Yk is obtained by interchanging x and y in these equations.
For the atom k on the symmetry axis z, the
components xk and Yk vanish while z k is given by z,2 = / u ; ' ( e " - e , ) =
' aP,
.
(23)
According to Rudolph [19], the coordinates of atom k in the principal axes system of the substituted molecule can be obtained by changing the sign of Am and interchanging the primed and unprimed quantities in equations (14-23). The complete transformation between rk = (XkYk Zk) and
r k = Rr~ + t
rk' = (xk' yk' zk') can be described by and
r~ = R,(rk - t)
(24)
where t and the elements of R are given by [ 19] t
Am
,
k . R~y, =-Am pxxxY------~' _ py,
(25)
(26)
Equations (24)-(26) can be used to transform any vector between the two principal axes systems. Except for the signs of the components of r k and rk', R and t are completely defined by the planar moments P and P' of the parent and the daughter isotopomers, respectively.
178
GRONER
2.
Symmetrical multiple substitution Chutjian [ 16] derived expressions for molecules in which a whole set of symmetrically equivalent atoms are substituted.
These
expressions were reformulated in terms of the planar moments by Nygaard [ 17]. For double substitution of a pair of atoms related by the symmetry plane Sxy, Xk and Yk are obtained from equation (22) with/Ac = ~t2. ktn is defined by
~=n
Mare M'
(27)
where Am is, as before, the isotopic mass difference of a single atom. The coordinate z k is obtained from equation (23) by setting ktz = 2Am. The equations for substitution of a pair of equivalent atoms related to each other by a C2 operation are also obtained from the equations (22-23) above by setting/.tx = 2Am and flz = Ft2. For a substitution of n atoms (n > 2) related by a Cn axis (= z axis), the coordinate Zk is obtained from equation (23) with ~tz = Pn. The radial coordinate x k (= distance from the z axis) is obtained from
x, =~t,-'(e; 2
where
F
- e,) = ~t,-'ae,
(28)
/.tx = nAm.
Because the principal moment Iz (about the unique principal axis in symmetric rotors) cannot usually be obtained from rotation or
QUEST FOR EQUILIBRIUM STRUCTURE
179
vibration-rotation spectra, the expression for the coordinate Zk is usually written as, assuming A/z = 0, z ,2= ~ t ;1 ( I ' - L ) = , , - ' A / ,
.
(29)
For the same reason, the radial coordinate x k cannot be obtained from symmetrical substitution of symmetric rotors. One way to overcome this problem is to use single isotopic substitution of an off-axis atom making the substituted molecule a slightly asymmetric near-prolate or -oblate rotor. For many of these near symmetric rotors it is difficult to determine lz' precisely. An alternate method described by Li et al. [20] circumvents the need for I z and Iz'.
For a molecule of C3v
symmetry, they combined data from the symmetric rotor parent, the di-substituted asymmetric rotor and the tri-substituted symmetric rotor. In fact, data from any three of the four species (symmetric rotor parent, mono-, di- and tri-substituted isotopomers) can be used to determine z k and the radial coordinate x k. The expressions are listed in Table 1. The differences A/are defined as zxi'=l,-L
,
A/,,- I,,-L
,
(30)
A/'"= 1""- I x . The single, double or triple primes attached to the data refer to the mono-, di- or tri-substituted species, respectively. The expressions in Table 1 are valid for the situation where the parent and the heavier tri-
180
GRONER
T A B L E 1. r s Coordinates of C 3 symmetric rotors from multiple substitutions a.
zk2
Combination P, S, D
x,2
- M ' M " + 2 M"AI[,'
4 M ' A I " - 2 M"AI[,'
3 MAm
3 MAm
P, S, T
- 3 M'A/" + 2 MA/t~" M'" 3 MAm 2 M - M'"
3 M'A/" - .A,r . . . I,,, b 2M 3 MAm 2 M - M'"
P, D, T
3M A/~- ~r^1,,,
M'"
- 3 M A/~ +2 .hX'''Af''' ... ~
3 MAm
2 M'" - M
3 MAm
Ft
Fr
,t vat L a u t b
?F
?~
2M 2 M'" - M
ap, S, D and T refer to the parent and the mono-, di- and tri-substituted species, respectively.
substituted isotopomer are prolate rotors. If the tri-substituted species is oblate, A/b'' must be substituted by A/a" in all expressions given in the table; if the parent molecule is an oblate rotor, A/c' must be replaced by A/b' likewise. Similar expressions may be developed for other symmetric rotor molecules. Wilson and Smith [21] derived equations for general multiple substitutions that can be used if the coordinates of only one of several substituted atoms are unknown.
Pierce [22] and others [23,24] have
introduced the double substitution method to locate nuclei near a principal axis by taking second differences of ground state moments of inertia.
This method has not found widespread application maybe
because of the necessity for very precise data for a larger set of isotopomers.
A formula to calculate an internuclear distance in a
linear molecule directly from the moments I 0 of a parent molecule, two monosubstited species and the disubstituted isotopomer ([25],
QUEST FOR EQUILIBRIUM STRUCTURE
181
citing a private communication by J. K. G. Watson) is related to the double substitution method.
3.
Error estimates for r s coordinates The errors of substitution coordinates can be derived from the errors of the principal moments of inertia or of the planar moments [26]. &k -
K /.oq
with K = 0(APx) / 2 .
The error 5(AP) has two different contributions.
(31)
The first kind is
associated with the experimental error of the planar moments; it is usually negligible.
The second kind, more severe and systematic,
originates from model errors.
The equations for the substitution
coordinates are based on the assumption that the molecules are rigid. Since they are not, the experimental ground state constants contain vibrational contributions that have a different mass-dependence. As a consequence, the substitution coordinates do not satisfy the basic relations (17-18). Another aspect of the same problem is the question of how close the
rs
coordinates are to the equilibrium coordinates
r e.
Costain [26] has shown that = (re +
0)/2
(32)
is a good first order approximation for diatomic molecules. However, no simple formula is available for polyatomic molecules because the relations are so complex and the
rs
coordinates must be compared on a
182
GRONER
case by case basis with experimentally determined
re
coordinates.
Costain [27] has advocated that equation (31) be used to estimate the uncertainty of the
(re)
structure of molecules, and he proposed to use
the average value K = 0.0012 u.A 2 (from the data for N20 ) to estimate the
rs/r e
discrepancies.
Schwendeman [28] proposed to use K =
0.0015 u-A 2 instead. It seems doubtful whether equation (31) really tells us anything about the errors of the used to check the consistency of
rs
re
coordinates. It can only be coordinates.
Even if the
substitution coordinates were completely consistent and fulfilled the basic assumptions (17-18) it would not automatically mean that they were
re
coordinates.
van Eijck [29] reported the results of such a check by calculating AP values for substituted molecules where AP should vanish for symmetry reasons, e. g. substitution of atoms in a symmetry plane or on a symmetry axis. By analyzing over 430 observations reported in the literature up to about 1979, he found that 2K= ~/((AP) 2)
(33)
decreased with increasing mass of the substituted atom. Costain's [27] estimate for K proved to be adequate for 10B/liB, 14N/15N and 160/180 substitution but somewhat too large for the substitution of 12C/13C or any of the heavier atoms (S, C1, Br, Se). However, K for H/D substitutions should be increased considerably to about 0.0032
QUEST FOR EQUILIBRIUM STRUCTURE
183
uA 2. Moreover, van Eijck [29] had to exclude the data of 13 H/D substitutions in methyl groups because they did not fit the pattern of the other H/D substituted species. For the methyl group substitutions, K as defined in equation (33) and (AP) were 0.016 u'A 2 and +0.025 u.A 2, respectively, whereas they were 0.0032 u'A 2 and -0.0004 u.A 2, respectively, for the other 131 H/D substituted species.
Van Eijck
found a correlation between AP and the barrier to intemal rotation of the methyl groups.
D.
Least-squares methods Structural parameters are determined usually by fitting them to experimental data by the least-squares method. traditional
rs
The exceptions are the
method and cases in which the number of rotational constants or
moments of inertia equals the number of structural parameters. An excellent introduction to the least-squares method in spectroscopic applications has been written by Albritton, Schmeltekopf and Zare [30]. Least-squares methods have been used to determine molecular structures successfully first by N6sberger et al. [31 ], Schwendeman [28], and Typke [32].
N6sberger et al. fitted structural parameters (internal
coordinates) to isotopic differences of moments of inertia.
To solve the
normal equations, they used the singular value decomposition of real matrices to calculate the pseudo-inverse of such matrices with the option to omit nearzero singular values in illdetermined systems.
Schwendeman [28] fitted
internal coordinates to moments of inertia or isotopic differences of these
184
GRONER
moments.
The latter was called a pseudo-Kraitchman (or pseudo-rs) fit
because the results were almost identical to the traditional
rs
method.
Discrepancies arose only if some parameters had to be kept constant because the available data set was not complete.
Schwendeman also showed that
essentially the same structures were obtained when Cartesian coordinates were used as adjustable parameters.
Typke [32] described a least-squares
program to fit Cartesian coordinates to isotopic differences using the Kraitchman formulas. He specifically designed the program so that data from multiply substituted species could be used together with data from monosubstituted species. The essential definitions and results of the least-squares procedure are summarized in the following equations. The least squares problem is defined as, adopting the notation of Rudolph [33], y = X 13+ r
(34)
| = a2M
(35)
were y is the vector of the observables, 13is the vector of the parameters to be determined, the Jacobian X contains the partial derivatives of the observables with respect to the parameters, and r is the vector of the residuals. | is the covariance matrix of the observations, a the standard deviation and M the inverse of the generalized weight matrix.
The unknowns are 13, r and a,
whereas y, M and X are given. The solution of this problem is described by [331 --" (X~/1-1 X ) -1
~'vI-ly
(36)
QUEST FOR EQUILIBRIUM STRUCTURE
185
O~ = 62 (~,~-'X)-~
(37)
c 2 = r ~ l - ' r ( n - m)-'
(38)
r=y-X[3
(39)
In these last equations, [3, cr and r are now the expectation values (solutions) of the quantities defined above, and 013 is the covariance matrix of the parameters, n is the number of observables and m the number of parameters. Because the functional dependence of the observables (rotational constants, principal moments of inertia or planar moments) on the structural parameters is strongly non-linear in most cases, an iterative process is essential. Typically, one begins with an assumed structure and expands the moment of inertia functions in terms of the parameters of this structure in a Taylor series up to the linear term.
a/, ) ~ z(Pko Xpk
I~ (obs) = I~ (Pko) + k
OPko
(40)
with Apk = P k - PkO. The differences Ii(obs)- li([30) become the observables y and the changes in the parameters Z$pkcorrespond to the parameters [3. The next approximation for the parameters is then obtained as PkO'= PkO + APk.
(41)
The procedure is repeated with the new parameter set until the least-squares fit converges. A systematic presentation of least-squares methods applied to the determination of molecular structures has been published by Rudolph [33].
186
GRONER
Investigating a number of related cases, he distinguished between structural parameters fitted to rotational constants, moments of inertia or planar moments or differences thereof. A survey of common problems occurring during least-squares fits of molecular structures and possible remedies has been given by Demaison et al. [9].
E.
r 0 and r s
Structures
According to Rudolph [33], structures determined by fitting structural parameters directly to the experimentally determined rotational constants B 0, the moments of inertia I 0 or the planar moments P0 of a series of isotopomers are called r 0 structures. If the covariances of the observables are transformed correctly, the structures resulting from any of these fits are identical. However, if covariances are not transformed, these structures are not identical nor equivalent (for examples, see [34]).
Therefore, one should always
specify what kind of covariance matrix is used with a particular fit. PseudoKraitchman (p-Kr) or pseudo-r s fits are those in which the structures are fit to differences of rotational constants, moments of inertia or planar moments. The structures called
rM
and rap again are equivalent to each other if the
covariances are transformed correctly. On the other hand, the structure rAB is theoretically different from the rA/and rap structures because of a lack of a linear relation between the AB and A/. In practice, rAB structures are very close to other p-Kr structures. The rle structure is obtained when the isotopic moments I 0 are fitted to structural parameters and three e parameters. It is
QUEST FOR EQUILIBR/UM STRUCTURE
187
based on the assumption that the vibrational contributions to the moments of inertia (one for each axis), e, are the same for all isotopic species. The rlr structure is identical to the ra/structure. Its big advantage is the fact that it gives much better predictions of moments of inertia (or rotational constants) than the rA/structure because of the explicit inclusion of the vibrational contributions. A much more detailed account of the particular problems of leastsquares methods in the determination of molecular structures has been given by Rudolph [35]. In this review, Rudolph calls all methods mentioned in the preceding paragraph r 0 or r0-derived methods.
He reserves the term
rs
method strictly to least-squares methods based on the equations of Kraitchman [15], Chutjian [16] and Nygaard [17] and their extensions to multiply substituted species [32,35,21 ] despite the fact that the p-Kr methods rA/, rap, rAB and ric result in structures much closer to the traditional
rs
method than to the r 0 methods. It is a little disconcerting that the notation tbr some of these structures is not uniform. named
re, I
Originally, the tic method was
by Rudolph [33]; it has been sometimes called
rl, ~,
even by the
original author [35]. In this review, the latest notation [9] tie is used.
188
GRONER
F.
r m Structures and beyond
1.
r m Structures
Watson [36] introduced the mass-dependent coordinates r m. If the vibration-rotation constant for a substituted species, 6', is expanded as a function of Amk
~'=~+
Amk+
Omk2 (Amk +...
,
the square of the substitution coordinate becomes for a linear molecule Io - Io
2
- zZ~ +
+ --~
+-~ Omk2
Am k +....
(43)
Retaining only the first two terms in this expansion, the substitution moment of inertia defined as Is
=
E m i z i s2
(44)
i
is given by
I s = I, + ~ m,
.
(45)
Using Euler's theorem on homogeneous functions, Watson [36] showed that the sum in this equation is equal to d2 and, therefore, Is = (le + I0)/2
.
(46)
He defined the mass-dependence estimate of the equilibrium moment of inertia, Ira, as I m = 2Is - 10
(47)
QUEST FOR EQUILIBRIUM STRUCTURE
189
and called the structural parameters fitting these estimates
rm
parameters. Analogous arguments lead to identical definitions of the mass-dependence estimates
Im
for all principal moments of inertia for
nonlinear molecules. To apply the
rm
method, one needs to determine the moments
for a sufficient number of isotopomers. To calculate
I s,
Is
data from the
first moment equations or product of inertia conditions cannot be used, and negative squares of the position coordinates, should they arise, have to be retained. The big advantage of the traditional
rs
rm
method over the
method is the complete cancellation of the first order
term of the mass dependence of e. Therefore, the moments
Im
are
expected to be much better approximations of I e than I s. However, the rm
method suffers from several disadvantages: (i) It can only be
applied if all atoms of the molecule can be substituted by isotopes. (ii) It is not applicable to hydrogen containing molecules because the large relative mass changes involved in H atom substitution render the approximations invalid. (iii) The ground state moments of inertia of many isotopic species of a molecule must be available.
(iv) The
neglected terms in the expansion of c are not as small as expected. Watson [36] obtained good agreement between the
rm
and
re
structures for the test molecules CO, SO 2 and N20. As expected, the results were not so good for HCN because of the large relative mass changes associated with H/D substitutions. Rather unexpectedly, the
190
GRONER
results for OCS were somewhat disappointing because rather large discrepancies between the
r e and
rm
parameters were obtained. In a
follow-up paper, Smith and Watson [37] tested the procedure on synthetic data for OCS. They concluded that neglected higher order terms in the expansion of c as function of the changes in mass (third term in Eq. (42)) were responsible for the unsatisfactory result for OCS and that the good results for N20 were fortuitous.
The test
calculations for OCS suggested one way to salvage the original method: The extrapolation of the provides
I s and
rs
coordinates to Amk = 0 (Eq. (43))
moments that are virtually free of the dependence
Im
on the finite mass changes. The resulting 4.10 -5 A with the
rm
re
rm
parameters agree within
parameters. However, this route is definitely not
practical in most cases because every atom would have to be substituted by two different isotopes.
2.
rc
Structures Kuchitsu and coworkers [38,39] introduced the
an extension of the
rm
rc
method that is
procedure. They recognized that the influence
of the third term in Eq. (43) could be eliminated almost completely by averaging "complementary sets" of isotopic data. dichlorine monoxide, they obtained the rm
parameters obtained for
rc
In the case of
structure by averaging the
35C12160 and 37C12180.
The isotopic
mass changes are positive for all atoms in the first of these species but
QUEST FOR EQUILIBRIUM STRUCTURE
191
negative in the second. Almost identical parameters were obtained by averaging the
rm
parameters from
35C12180and 37C12160. These
parameters agreed very well with the best
re
rc
structure which was
obtained by a careful analysis of microwave data [40].
The same
method has also been applied to phosgene, COCI 2 [39]. Even though at least some of the parameters are indeed very close to the
re
parameters, this method demands even more isotopic data than the
rm
method.
It can therefore be applied only in very special cases for
small and, preferably, symmetrical molecules. The potential range of applications of the
rc
method has been extended a little further by
Nakata and Kuchitsu who introduced an additivity rule for isotopic data [41].
For different types of small molecules, they defined F
functions from the moments of inertia and, using additivity rules discovered for isotopic changes for
rz
distances, they derived
additivity rules for these functions such as 2
3
F12=F+~(F~-F) i=1
and F123=F+ff'~(F~-F)
.
(48)
i=1
F12 and F123 are the F functions for the isotopomers where atoms 1 and 2 or atoms 1 through 3 have been substituted, respectively, whereas F and
F i
are the F functions for the parent molecule and the
molecule where atom i has been substituted by an isotope, respectively.
That rule allows it to estimate the F functions and,
192
GRONER
therefore, the moments of inertia of multiply substituted species from the moments of monosubstituted molecules.
3.
rmP
Structm'es Another, more versatile approximation to the
rm
method has been
introduced by Harmony and coworkers [42-46]. To reduce the number of isotopomers that need to be investigated to derive the
rm
structure,
they used the fact that the ratio of the substitution moment of inertia, I s,
to the respective ground state moment, I0, p = Is/I0
,
(49)
is essentially isotope invariant. They used ,o (one for each principal axis) as determined from moments
Is
I s and I 0
and, therefore,
this procedure the
Im
for other isotopic species. They called
method.
rmP
for one isotopomer to estimate the
The structure is determined by the
least-squares method from a sufficient number of moments
ImO
defined as /m0 = ( 2 ; -
As in the
rm
1)/0
9
(50)
method, first or second moment conditions cannot be used
to evaluate the moments
I s.
Like the
rm
method, it is therefore
restricted to molecules in which all atoms can be substituted. advantage compared to the
rm
method is that it can be used whenever a
complete substitution structure is available. significant
reduction
of
Its
the
necessary
This results in a number
of
isotopic
QUEST FOR EQUILIBRIUM STRUCTURE
193
modifications. Molecules containing hydrogen atoms have to be dealt with by a special procedure since even the r m method gives unsatisfactory results in these cases. A detailed review of the theory of the rm9 method and its applications has been prepared by Harmony for this volume [47].
He recommended that rmO structures should be
determined whenever all the necessary isotopic data are available to determine a complete r s structure. Particularly the error limits of the rmO method have been criticized by Demaison et al. [48]. They claimed that the error estimates were far too optimistic because the errors of the parameters p were not taken into consideration. Furthermore, the fact that the moments 1rap are all multiplied by the factor (2p - 1) generates a significant amount of correlation between these moments that affects not only the error limits but also the numerical values of the rmP parameters. The rmP method has also been criticized by Cooksy et al. [49] because the correction by the factor (2p - 1) does not allow for the proper mass dependence of the quantities e.
4.
rm(1) and
rm(2) Structures
In the paper proposing the r m structure, Watson [36] showed that the vibrational contribution to the ground state moment of inertia, e, is a homogeneous function of degree 89 in the masses and he used this argument to derive the formula for the r m structure (section II.F.1).
194
GRONER
Demaison and Nemes [50] proposed that the relationship between log c and log Ie should be nearly linear with a slope of 89 for all diatomic molecules.
The proposition is based on the equation for diatomic
molecules [ 10] 3B, 3h ro = r~(1--~-(1 +a~)) = re --~-(1 +al)(r2f/O -v2
(51)
where co is the harmonic vibrational frequency, a 1 the first Dunham coefficient, f the harmonic force constant and/.t the reduced mass. Since a 1 and the product re2f are nearly invariant among diatomic molecules [50], c should depend linearly on /fl/2.
To test this
proposition, they investigated the correlation between 6 and I 0 for diatomic and polyatomic molecules. For 76 diatomic molecules, they obtained by weighted least-squares fitting the equation
(52)
log 6 = 0.611 (1 O) log I 0 - 1.776(25)
with a correlation coefficient of 0.93. This correlation held for a range of almost three orders of magnitude in I 0.
Considering the
approximations and the small difference between 10 and I e, the deviation of the slope from 89 is perfectly acceptable.
For 61
polyatomic molecules, they report a similar correlation as log 6 = 1.247(5) log/0 - 2.651(13)
(53)
with a correlation coefficient of 0.98. The much stronger dependence of c on I 0 for the polyatomic molecules is rather surprising. The straight line shown in Fig. 2 of ref.
QUEST FOR EQUILIBRIUM STRUCTURE
195
[50] in relation to the log r versus log I 0 data points seems to be the line described by the equation but it does not look like it were the line fitting the data.
Of course, looks are sometimes deceptive and
unreliable, particularly for a weighted fit if the weights are unknown. If the data for the H2X molecules were excluded, the slope of the line would likely be much less than 1 instead of more than 1. Le Guennec et al. determined equilibrium rotational constants and moments of inertia for FC10 3 [51] and 74GeH3F [52] and calculated the vibration-rotation contributions r for each principal axis. They used them to estimate e' for other isotopomers of FCIO 3 and 74GeH3F using the relation d = e ( I ' / 1 ) 1/2
(54)
.
They corrected the isotopic ground state moments of inertia with these estimates to obtain equilibrium moments from which they determined the approximate
re
structure for both molecules.
A near equilibrium structure for CH3CN was derived by Le Guennec et al. [53] using the same approximation in a different way. They approximated I e in effect with
(55)
le ~ 10--C(10) ''~-
and fitted
re
structural parameters and the proportionality constants c
(one for each axis) to these estimates. They compared the results with those obtained by other methods (r 0, r s,
r i c , rmO ).
196
GRONER
More recently, a method almost identical to this last one by Le Guennec [53] has been used by Watson and coworkers [49,54]. They approximated the ground state principal moment I 0 as I 0 = I m + C(Im)l/2
where
Im
(56)
,
is calculated in the manner of a rigid rotor from the structural
parameters. The parameters and the proportionality constants c (one for each axis) were determined by least-squares fitting. parameters were called
rm(1) because
The
the method is mass-dependent
and because one extra parameter per axis was used. In the case of the NO dimer [54], the rm(1) NO distance was reported to be within 0.0007 A of the
re
distance of the NO monomer.
The details of the rm(1) method have been published only very recently [55] where Watson et al. have given a detailed account of numerous calculations with a variety of models to approximate I 0. All methods tested are based on the correct mass dependence of G [36]. The general method is based on the approximation I 0 = I m + C(Im) 1/2 + d ( m 1 m 2 ...
mN/M)I/(2N-2) + 8L
where I 0 is the effective ground state moment of inertia and moment of inertia calculated from the
rm
parameters,
(57)
. Im
the
c and d are
fitting parameters (one each for each axis), M is the total molecular mass, and representing
mk
are the atomic masses,
Laurie
corrections
for H
sL is a symbolic term atom
distances
(Laurie
QUEST FOR EQUILIBRIUM STRUCTURE
197
corrections are applied in a different form, however, see below). N is the number of atoms. The first correction term (with the coefficient c) essentially scales the moment of inertia with the correct mass dependence. It corrects for the fact that e tends to be positive, making r 0 bond lengths generally longer than the r e distances.
The second
term with the scaling coefficient d corrects for the "small coordinate anomaly" which is so troublesome in the r s method. It is based on the fact that, in model calculations, the contribution to 6 from most atoms with small principal coordinates tends to be negative.
Models that
include only the c-terms are called rm(1), those with both the c- and dterms rm(2). If the Laurie corrections are applied, the models become rm(1L) and rm(2L) , respectively. According to Watson et al. [55], the
structural parameters as well as c and d tend to be strongly correlated if the molecule does not have atoms with small principal coordinates. If Laurie corrections have to be applied, parameters are also likely correlated, particularly in the rm(2L) model. The Laurie corrections are applied by defining the effective XH distance rmeff(XH ) as M rmeff(XH) = rm(XH ) + 6 . m. ( M - m.
)/1/2
(58)
The parameter 5 H is a fitting parameter, the expression in parentheses is the inverse of the reduced mass of the H atom vibrating against the rest of the molecule.
198
GRONER
Watson et al. [55] also described a procedure to take into account the axis rotation induced by isotopic substitution. Instead of three c parameters (one for each axis), six c parameters are necessary to correct the moments of inertia of a general asymmetric rotor molecule. Watson et al. [55] applied these methods to a number of molecules. The rm(2) internuclear distances determined for the linear molecules N20 , CO2, OCS, OCSe, CICN, and BrCN agree within 0.001 A or better with the reported r e distances. The rm(2) structures for C1BO and CIBS can be compared only with r m a n d r s structures, respectively. Excellent agreement between the r e and rm(2) structures was obtained for the nonlinear molecules SO 2 and 0 3. The agreement is not quite as good for nonlinear C1NO and COC12. In the case of CINO, experimental ot constants were available for only two isotopomers to determine the equilibrium rotational constants.
The
discrepancy in the case of COCI 2 is suspected by the authors to arise from inconsistencies in the rotational constants because of the quadrupole coupling.
For these twelve molecules not containing
hydrogen, the standard deviations of the rm(2) fits are significantly smaller than o(ro)/O(rm(2) )
the
standard
deviations
of r 0 fits,
with
ranging from as low as 20 to several hundreds.
ratios with the standard deviations from r s fits,
ratios The
O(rs)/O(rm(2)), range
from 2 to 30. r m ( 1 L ) structures were determined for the linear hydrides
QUEST FOR EQUILIBRIUM STRUCTURE
199
HCN, HCO +, HBO, HBS, HNC, HCNO, HCCCI, HCCCN because none of them (except HCNO) has an atom near the center of mass. The differences of the internuclear distances between the rm(1L) and r e structures are smaller than 0.002 A for the first five molecules. No r e structure is yet available for HCNO. For the last two linear hydrides, the rm(1L) distances between non-hydrogen atoms agree with the r e values within 0.003 A, the HC distances within 0.007 A. The ratios of the standard deviations for different kinds of fits are much less spectacular.
The ratios o(ro)/O(rm(1L)) for these molecules were
between 14 and 50 for the first four linear hydrides, but only between 1.5 and 8 for the second group of four. The ratios even smaller.
a(rs)/a(rm(1L)) were
Because data sets were rather limited, only rm(2)
structures were determined for the nonlinear hydrides HOCI, HNCO, HCOOH, H2CCC12, H2CS and H2CCC. The rotation coefficient Cab was included for H2CCO for a rm(2r) structure while Cab and the Laurie correction were used for H2CO resulting in a rm(2Lr) structure. The ratios a(ro)/a(rm(2)) ranged from 2.5 (for H2CCC12) and 12 (for HCOOH) to 300 whereas a(rs)/a(rm(2)) H2CCC12 and HCOOH) and 46.
was between 1.3 (for
For this group of molecules, the
results are of mixed quality. For HNCO and H2CO, the agreement with the r e structural parameters is quite good although the fit for formaldehyde is only moderately good. H2CS gives a better fit but a
200
GRONER
rather large difference to the
re
CS bond length. For H2CCC12 and
HCOOH, there is agreement with the error limits.
Differences to the
re
re
structure within rather large
parameters are rather large
compared to the uncertainties for HOCI but the error limits of the
re
parameters are also large. The fits for H2CCO and H2CCC are less good but the data sets for these molecules are rather limited. The big advantage of the rm(1) and rm(2) methods is the fact that they are applicable also to molecules containing elements with only a single stable isotope.
Watson et al. [55] observed that the rm(1)
method requires the same amount of isotopic data as the a n d rA1
Corrections based on Ab
methods tic
and should therefore be adopted systematically instead. The
same is true, of course, for the
G.
rs
initio
ab initio
rmP
method.
calculations
calculations have reached a stage where they become
extremely useful to spectroscopists [56-63]. They can be used not only to predict
re
structures but also to provide reasonably accurate force fields and
dipole moments. Results of
ab initio
calculations are used now in a number
of ways to determine molecular structures from spectroscopic data. review by Demaison et al. [9] contains an assessment of the modem methods that are relevant to molecular structure research.
The
ab &itio
QUEST FOR EQUILIBRIUM STRUCTURE
1.
201
Empirical corrections One type of application involves the estimate of empirical offsets by comparing
ab
initio
re
distances with experimental r 0 or
re
distances and to use these offsets as constraints in fits of structural parameters for similar molecules. Le Guennec et al. [64] used two different basis sets, a double-zeta basis with a single set of polarization functions (DZP) and a triple-zeta basis with two sets of polarization functions on C and N atoms (TZ2P), at three different levels of theory (SCF, MP2 and QCISD). For each type of calculation, they derived offsets for C-H, C-C, C-C and C-=N bonds with which they predicted the
re
structure of propyne, HC-CCH 3. Similar offsets determined for
smaller basis sets (6-31G(d), 6-311G(d,p) and 6-31 l+G(d,p)) and electron correlation at the MP2 level have been used for acrylonitrile [65] and a series of dicyanides [48]. van der Veken et al. [66] used a scaling method to adjust structural parameters from
ab
initio
calculations (e. g. MP2/6-
31 l++G(d,p)) to fit experimental rotational constants. By optimizing scaling factors (one for a group of similar parameters, e. g. all C-H distances), they were able to maintain the ratios of parameters at the values predicted by the
ab
initio
calculations.
This new method
promises to be very useful for the determination of the structures of conformer pairs because (1) complete structures for conformer pairs
202
GRONER
are rare (see Section III.B.6), and (2) conformer ratios of parameters can be maintained.
2.
Vibration-rotation constants from a b
initio
calculations
In another application, the quadratic and cubic force constants are calculated by
ab initio
methods. From the force field, the
re
geometry
and the atomic masses, it is possible to calculate the vibrational frequencies and the vibration rotation constants aft (Eq. (2-4)). If they are sufficiently accurate, they can be used in place of the experimental aft constants to estimate the equilibrium constants Be"o from the observed constants in the vibrational ground state, Bg.
This
procedure has been followed by Bailleux et al. [67] for silaethene, CH2SiH 2.
They obtained the vibration-rotation constants by a
CCSD(T) calculation with the triple-zeta basis TZ2Pf. They corrected the experimental ground state constants for seven isotopomers (expressed in terms of Watson's determinable parameters) by the initio a f t
constants and obtained an
well with their rather sophisticated pV(Q,T)Z).
re
ab
structure that agreed remarkably
a b initio r e
structure (CCSD(T)/cc-
The differences were at most 0.0005 A for the bond
lengths and 0.10 ~ for the HCSi and HSiC angles.
This excellent
agreement is evidence that this level of calculation produces good
re
structures and that quadratic and cubic force fields at somewhat lower levels yield good vibration rotation constants.
QUEST FOR EQUILIBRIUM STRUCTURE
H.
203
Other methods
1.
rz
Structures Although this article deals primarily with experimental structures
approximating
re
structures, the
rz
or average structure in the
vibrational ground state is mentioned here for completeness. It is the structure calculated from the averaged positions of the nuclei. It is equivalent to the r O structure that can be obtained from electron diffraction data. The
rz
structural parameters are obtained from the
rotational constants Bzfl defined as dk
(59)
By = BoP+ Z a / ( h a r m ) T = B ] - Z a[(ar~) k
In practice,
rz
k
structures are obtained from spectroscopic data by
correcting the rotational constants Bg with aft(harm)calculated from a suitable harmonic force field.
The analysis of electron
diffraction data also requires a complete normal coordinate treatment to calculate the mean vibrational amplitudes from the harmonic force field. Therefore, the results of joint analyses of electron diffraction data and microwave data are often reported as Isotopic differences of
rz
rz
structures.
bond lengths, 5r, are usually estimated
by a formula derived by Kuchitsu [68,69] 5r = (3/2) a 5(u 2) - 5/s
(60)
204
GRONER
where (u 2) and K are the zero-point mean-square amplitudes of a given bond and its perpendicular amplitude (both obtained from the force
field),
and
a
is the
Morse
anharmonicity parameter,
approximated from the corresponding bond in the respective diatomic molecule. The r z bond length and the mean-square amplitudes are otten used to estimate the r e distance by Kuchitsu's equation [68,69] re = r z - ( 3 / 2 ) a ( u 2 ) + K
2.
.
(61)
Isolated C-H stretching frequencies McKean has demonstrated that there is an excellent correlation between isolated C-H stretching frequencies and r 0 C-H bond lengths [70] resulting in the relationship [71] ro(CH)/A = 1.3982 - 0.0001023 * vis(cH)/cm-1
.
(62)
The correlation has been corroborated by low level ab initio calculations (HF/4-21G) [71].
Although the theoretical (re) bond
lengths are systematically off, the variations of unequal C-H distances in similar environments agree with those determined by the bond length- frequency relationship (Eq. 62). Demaison and Wlodarczak have evaluated the correlation between experimental r e distances and vis(cH) and obtained [72] re(CH)/A = 1.3009(53)- 7.175(173)'10 -5 * vis(cH)/cm-1
form 26 data points.
(63)
They also pointed out correlations between
re(CH) distances and deuterium quadrupole coupling constants,
QUEST FOR EQUILIBRIUM STRUCTURE
205
rg(CH) distances from electron diffraction experiments, and NMR coupling constants j(13C-H) measured in liquid crystal solvents. However, the data sets for these other correlations are not very large nor of sufficient quality to allow them to be used for predictions. Nevertheless, they are useful indicators.
III.
M O L E C U L A R S T R U C T U R E S F R O M S P E C T R O S C O P Y SINCE 1980
A.
Introduction to Tables 2-6 Molecular structures determined since 1980 are listed in the following five tables. As a rule, only molecules are listed for which complete structures have been determined by rotationally resolved spectroscopy.
With a few
exceptions, structures from joint analyses of rotational and diffraction data are omitted. Likewise, no data for diatomic molecules and ions or Van der Waals complexes are included.
For most molecules, complete structures
have been determined for the first time. However, the structures of a number of molecules have been redetermined with more precise rotational constants, by different methods, from different isotopic species or from larger isotopic data sets. In some cases, particularly since about 1990, papers cited provide only the most recent isotopic data, which made it possible to determine a complete structure. In all tables, the species are arranged by the sum formula according to the Hill system (alphabetical order of element symbols except for species containing carbon in which case C followed by H precede the other elements).
The tables contain the sum formula, the structure formula, a
206
GRONER
name, the types of structures determined, the principal experimental method and the literature citation. Table 2 lists the information for molecular ions (without names). Table 3 contains the data for nonpolar molecules. Table 4 lists the information for molecules for which partial or complete structures have been determined for more than one conformation.
In addition, the
number of isotopic modifications studied for each conformer is given (molecules where only one isotopomer was used were excluded). Table 5 contains the data for radicals, carbenes and other exotic species (again without names). Table 6 is devoted to closed shell polar molecules. In the columns identifying the experimental method, "MW" stands for any method studying the pure rotational spectrum of a molecule except for rotational Raman spectroscopy marked by the "rot. Raman" entry. "FTIR" stands for Fourier transform infrared spectroscopy, "IR laser" for any infrared laser system (diode laser, difference frequency laser or other).
"LIF"
indicates laser induced fluorescence usually in the visible or ultraviolet region of the spectrum. "joint" marks a few selected cases where spectroscopic and diffraction data were used to determine the molecular structure. A method enclosed in parentheses means that the structure has been derived from data that were collected by this method in earlier publications.
The type of
structure determined is shown by the symbols identifying the various methods discussed in section II.
"rs" refers
to determinations using the
Kraitchman/Chutjian expressions or least squares methods fitting only isotopic differences of principal or planar moments (with or without first
QUEST FOR EQUILIBRIUM STRUCTURE
207
TABLE 2. Ions.
Sum formula
Structure formula
BFH +
HBF +
CBrN +
BrCN +
CC1N+
C1CN+
CHO + CHO +
Structure
Experimental method
Reference
MW
[73,74]
rs
LIF
[75]
rs
LIF
[76]
HCO +
MW
[77]
HOC +
MW
[78]
rs
MW
[79]
rs/r 0
MW
[8o]
MW
[81]
rs
IR laser
[82]
re*
IR laser
[83]
CHO2 +
HOCO +
CHS +
HCS +
CH2N+
HCNH +
CH 3+
CH 3+
r 0 rz
IR laser
[84]
C2HN2 +
HNCCN +
ro rs
MW
[85]
C2H2+
C2H2+
r o rz
IR laser
[86]
C4H2 +
HCCCCH +
rs
LIF
[87]
C12H+
H2C1+
re
IR laser
[88]
HN2 +
HN2 +
rs
MW
[89]
re
IR laser
[90]
HOSi +
HOSi +
r0
IR laser
[91,92]
H2F+
H2F+
r0
IR laser
[93]
H2 O+
H2 O+
r0
MW
[94]
H3 O+
H3 O+
r0
IR laser
[95]
H3 S+
SH3 +
re
IR laser
[96]
r o rz
MW
[97]
208
GRONER
TABLE 2. (continued).
Sum formula
Structure formula
Structure
Experimental method
Reference
H4N+
NH4 +
ro
IR laser
[98]
CI2H-
C1HCI-
r 0 re*
IR laser
[99l
F2H-
FHF-
re
IR laser
[100,101]
HO-
OH- (OD-)
re
IR laser
[102]
H2N-
N H 2-
r0
IR laser
[103]
TABLE 3. Nonpolar molecules.
Sum formula
Structure formula
Name
Structure
Exp. method Ref.
BH 3
BH 3
borane
r0
FT-IR
[104]
B2H6
B2H6
diborane
ro rs rz re*
FTIR
[105]
CF4
CF4
tetrafluoromethane
r0
IR laser
[106]
CO2
CO2
carbon dioxide
ro rm re
FTIR
[107]
CO2
CO2
carbon dioxide
rc
CSe2
CSe2
carbon diselenide
ro re
FTIR
[108]
C2C12
C2C12
dichloroacetylene
ro rs
FTIR
[109]
C2F2
C2F2
difluoroacetylene
ro re
FTIR
[110]
C2H2F2
CHFCHF
trans- 1,2-difluoroethylene
rslro
FTIR
[111 ]
C2H4
CH2CH2
ethylene
rz
MW
[112]
C2H4F2
CH2FCH2F trans- 1,2-difluoroethane
rslro
FTIR
[ 113]
C2H6
CH3CH 3
rz
MW
[114]
ethane
[41]
[115]
rmP
C2N2
NCCN
cyanogen
rs/ro
rot. Raman
[ 116]
C3H6
C3H6
cyclopropane
ro rz re*
MW
[117]
C4H6
C4H6
s-trans- 1,3-butadiene
r0(P)
MW
[118]
QUEST FOR EQUILIBRIUM STRUCTURE
209
TABLE 3. (continued).
Sum formula
Structure formula
Name
Structure
Exp. method Ref.
C4H8
C4H8
cyclobutane
r0
MW
[1 19,120,121]
C6H6
C6H6
benzene
r0
rot. Raman
[1 22]
r~(p)
MW
[123]
ro
FTIR
[124]
r m re*
FTIR
[125]
C6H12
C6H12
cyclohexane
rsr~
MW
[126]
C8H8
C8H8
cubane
ro
MW
[121]
F2Kr
KrF2
krypton difluoride
re
FTIR
[127]
F2Xe
XeF2
xenon difluoride
re
FTIR
[127,128]
F4Xe
XeF 4
xenon tetrafluoride
r0
FTIR
[129]
F5P
PF 5
phosphorus pentafluoride
r0
MW
[130]
F6S
SF6
sulfur hexafluoride
ro
IR laser
[131]
GeH4
GeH4
gerlTlane
re*
MW
[132]
H2N2
HNNH
trans-diazene
ro rz re*
FTIR
[133]
H4Si
Sill 4
silane
re*
MW
[134]
H4Si
Sill 4
silane
re*
MW
[132]
H4Sn
Sna 4
stannane
re*
MW
[132]
02
02
dioxygen
Fs Fm
MW
[135]
O3S
so3
sulfm" trioxide
ro re
MW
[136]
P4
P4
phosphorus
r0
rot. Raman
[137]
210
o o
o o
n~ 0
U~
r~
0
0
Z
L--~
~ ~ &
~
~
~ ~
U--~
~
~
2 =
0
0
0
~
=
~
0
~
~
0
r~ (N ..
L--~
L--~
~
L--~
GRONER
~
.,-
o
~
~
~~
o
:r162
~
.~0
0
rO
0 ~
0
~
Im ~
0
e',l
...
0
~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
e e ) ) e e e e
0
o ~
0 ..~
0
~ ~.-
0
o ~
0
rn
~~
(N
rO
r~
~
"-'
rO
r~
0
0
~ r ~
0
~
0 0
rO
n~ o
o
0
Z
~9
~
~
~
&
o -
-u
L_~
9
oo
~
~
o
0
o
~
~
~
9~
2
&&
~
~
o
N
o~
oo
r
~
~
&&
~
.,.~
..
~
.~
~
~&
o
-u
,_~
Z
r
~n
o~
0
o
oo
0
N
m
o.~
r
-~
&&&&&
L__a
QUEST FOR EQUILIBRIUM STRUCTURE
n~
o o
0
P....
N
0
o~
Z
o o o o ,.o
211
212
GRONER
TABLE 5. Radicals and carbenes.
Sum formula
Structure formula
Structure
Experimental method
Ref.
AsH2
AsH2
rz
MW
[159]
BaHO
BaOH
r0
LIF
[160]
CCaN
CaNC
rs
LIF
[161]
CC12
CCI2
r0
MW
[162]
CF3
CF3
r0
IR laser
[163]
CFeO
FeCO
rs
MW
[164]
CHTi
TiCH
r0
LIF
[165]
CH30
CH30
rs
MW
[166]
CMgN
MgNC
r0
MW
[167]
C2H
C2H
rs
MW
[168]
C2HN
HCCN
rs
MW
[169]
CEHESi
SiC2H2 (cyclic)
rs
MW
[170]
C3H
HCCC
rs
MW
[171]
C3H
C3H (cyclic)
rs
MW
[172]
C3HO
HCCCO
rors rmP rm(1) MW
C3H2
H2CCC
ro rs re*
MW
[173]
C3H2
C3H2 (cyclic)
rs
MW
[174]
C3N
CCCN
ro rs
MW
[175]
C30
C30
rs
MW
[176]
C3S
C3S
rs
MW
[177]
C4H
HCCCC
ro rs
MW
[175]
C4H2
H2CCCC
rors
MW
[178]
C5H5
C5H5
ro
LIF
[179]
C102
C102
ro rs re
MW/IR laser
[180]
C102
C102
re
MW
[181]
C1S2
CIS2
rs/ro
MW
[182]
C12Ge
GeC12
re
MW
[183]
C12Si
SiCI2
re
MW
[184]
C12Si
SiC12
re
[49]
[41]
QUEST FOR EQUILIBRIUM STRUCTURE
213
TABLE 5. (continued).
Experimental
Sum formula
Structure formula
Structure
F2P
PF2
r0
MW
[185]
H2Si2
SiH2Si
r0
MW
[186]
rs
MW
[1871
method
Ref.
H2Si2
SiHSiH
r0(P)
MW
[188]
NO2
NO 2
re
MW
[189]
O2P
PO 2
r0
MW
[190]
moment or product of inertia conditions).
Procedures where rotational
constants, moments of inertia or second moments were fitted are referred to as "r0".
"rs/rO"
refers to cases where the majority of the structural parameters
have been determined by the
rs
method but that a few parameters have been
obtained by the r 0 method or from second moment equations. In very few case, mostly in Table 3 (conformers), partially determined structures are indicated by "(p)". The structures marked by "re*" are from
rz
structures derived
structures by Kuchitsu's estimate (Section II.H) or by using at least
some vibration rotation constants (z obtained from
B.
re
ab initio
calculations.
Selected examples In this section, a few selected examples are used to demonstrate one or another of the newer methods. The first one (dimethyl sulfoxide) illustrates the advantage of using data from (extra) multisubstituted isotopomers. The
214
0
Ob
o
~
o
t___.a
= = =
t___a
t___J
0
~
~
..2o _,o
,.=
t__J
o
r
~
r
= = = = = =
~
"
~
0
r
=
=
t___a
=
t___J
=
t___a
m
=
~
I=I
~ I,,,~
m
=
~
o
~
o
o
=
L._J
I,.
t-----a
o
GRONER
o.
o
0
m
~
m
z
u
0
o
~
"
~~&z o
=
~
r..q
Z
1-1
o ~
O
~ o
~ O
1~
~ O
I~
~~~
O O O
11~
~~
.
~
~:~, ~
QUEST FOR EQUILIBRIUM STRUCTURE
O
O
~ .~
r.D r,D r,.D r,D r,D r,D r,D r.,D r,.D r,D r,D r,D ~
~
ll,ml
ll,-i ~
~
l.-i
~,~
~
w
O~~~~
O
"~
r,D r.,D r,.D r.,D r,D r,D ~
215
216
o o
r,r.l
[-.
0
Z
o
o
~
r
~ ~ ~ "
o~ ~
o
~ ~ "~
~
~
N-~ ~ ~ ~.~
N
~
~~
N
~
O
N
O
"
~
~
-
Z
o
~ Z
o
o
~ . - r.~ r.D ~
r~
GRONER
MO
r,.D
_
o~~ ~= o~~N~
~ff ~~
~
~ ~ ~ ~ ~
,.~ b~ ~
r..D r,D r,D r,D r,D r,D r..D r,D r,D r,D r,D U
~
O
r,D
~~~ Z
r..D r..D U
~~,., U
r
0
Z
~
"~
-~
0
~
e'q
~
o
"~"
~
~ ~
e'q
~ ~
O
~
0 0 ~ U ~
0
r
~
~ ~
~
r ~
~
e'q
r
~
.~
~
~
C',l ~
r
r i~1
e'q L~
,..,
e'q
~
e'q L~J
e'q
.? .~
~
~
r~ r~
~'~'~
0
N
N
~
N
~
r~ N
~
r~ ~ ~
0 ~
~0
~
~ ~
~
~
~
QUEST FOR EQUILIBRIUM STRUCTURE
,.-t,
o o
0
e'q ~
~
C',,I
~
L___J
e'q
f~ "~ 0 o - -
r~
~
L-.----I
rj
r~
e'q
f~ o
r
~
~
r~
~
0,1
r
~
0
r
~
rj ~
~
~
e',,I L.__J
" 0
-
r
217
218
o~ 0 0
e'q
9~ ~
~
~-B~_
ro
Oq
C~
_~
F~ F~
ro
ro
~
~ =
eXl
~
.9~
... - ~
~, ~
ro
ro
0,1
ro
C'q
C'q
Oq
t"q
o
~
~ o o -,~
o _ _
~
~, o
ro
C~
o
e'q
~
0,1
~
ff
OR
rO ro
0,1
~, . . . .
0,1
ro
eq
~,
-~
r
ro
ffffffffffffffffffffffffffff
ro
Oq
e'q
~
C'q
e'q
~
~~ _~.
0,1
~
C'q
.-~
Ol
rO ro
GRONER
Oq
ro
0,1
-~
0,1
~
~
O,t
ro
_'~
ffff
0,1
ro
~
O,,I
ro
r
Z
0
(",1 ~0 ~0 i'M t",l I____.a ~
~ ~0 ~P ' I
~ ~0
I"~
~
~0 ~0 ~,1 P L-----I ~,I
~
~
~
~0 ~0 ~ ~ ~ I--------I i-------I ~P q
~
(:~
~
O.
I~ P',I L---J
1~
O.
o~,,,1
t"'. P',I ~
~,~
O.
0
0 0
~0
~
r,.)
~
~.
~z
r,.)
,~
~
r,.)
r,.)
r,.)
r,.)
~
,~
z~ o ~ ~ ~ z
z
r,.)
r
0 r,D r,D
zz
~ -':,,~" .~.~- ~. ~ ~' ~ .~ ~
0
0
tCh
~
O.
r,D r,.)
o~
~
t "~ I" ~ I"-. t". ~0 ~1 ~ ~ ~ L----I L----J L----J L--...I ~' ~ "
~
~
r,D ~
r,.)
~ ~.~.
t'. t". I~ r t1______1 'N t N_ J I___...a L
~ ~
~
o
~
~
r,.)
o ~ ~ o ~
.~ ~
t"t",l L__.I
QUEST FOR EQUILIBRIUM STRUCTURE
,,--k ,'1::1
o e~
0
r,D r,.)
219
220
,,.4,
o~.~ 0 0
.1
o
0
Z
~o
ff
N
exl
~
e'q
e'q
e~l
e'q
e'q
e'q
0,1
~
"~ ~
e~
~
e~
e~
e~
e~
"
r~
~ ~
o~
~,
~
o ~ ~ SS SS ~
~o
~
~
rj
=
r...)
~ ~
= =
~
rj
GRONER
e~
"~
0
=
r~ 0 SS
r,.) r~
0 ~
=
~
e'q
o
r...)
0 N
r~
~
~
~~
r~
0 ~
ffffffffff=
rj
ff
0 ~
r...p r j
rj
~
rj
rj
o0 m ~ ~ 0 ~ ~ SS N ~ r..p ~
~
0 N ~
0 ~
r..P r@ rJ
O
Z
~
-
t" ~
~
~'~
~
~
~
~ a
-~
~ U ~ ~ -~~. ~~
_~.~.~
~
0
Z~ ~_
o~
~
eq
"'~
~1
C
~
~
~
0~0~0 ~~
e~
QUEST FOR EQUILIBRIUM STRUCTURE
,..a,
~162
[,..
~
~ e~
~
~
~
"~"
~,~
t~
~
~ ~162
.~-~ ~,~~ , . y,~ m
~ ~ ~~ ~~ Z
~~~Z
~',
~
~
-
221
222
0
.1 [..
o
o
Z
~
~
~
_ _ - ~
~
41,
~
~
"~ "~
~, o
...~ ..~ ..~
~=
_o
~=
~'~
~
e~
.........
~
~
~
0 0 0 0 0 0 0 0 0 ~
ooo~~~~o~ooo8~~
0 0 0 ~
,,,~
.~
~,
GRONER
.~'~
~0
~~~ 0
o
Z
o
~
~
~::~
~=~
r~
~
~
~
0e,I 0eq ~~ ~e,I ~~'N ~~ ~m ~m 0e,I 0eq 0~ I::I::I I::I::I I:::I::I I::I::I I::I:::I ~ l:X::l I::I::I Z Z Z
~
QUEST FOR EQUILIBRIUM STRUCTURE
n~
o o
~
~
,N
0
o 0
r~
0
~
~____o n=~
0
m
0
m
~ ~ ~r ~ 0 0 0
223
224
GRONER
second example (hydrogen pseudohalides) shows how the use of the substitution moments of inertia (instead of the ground state moments) and the comparison of molecules with similar structures help to overcome the "small coordinate problem".
The next example demonstrates the difficulty of
applying some of the newer methods (rmP and
rm(1)) in the problematic case
of the HC30 radical and the big effect of the H/D correction in the rmP method. In the case of vinyl alcohol and vinyl fluoride, it is shown how lower level ab initio calculations can help to locate a problem in the determination of molecular structure.
The fluoroacetylene example
demonstrates that the determination of accurate vibration-rotation parameters is a formidable challenge and that high level ab initio calculations can help to detect unknown resonances among the vibrational levels. The last section addresses the question of structural differences between conformers of the same molecule.
1.
Dimethyl sulfoxide Dimethyl sulfoxide, (CH3)2SO , represents a case where the single substitution method resulted in a very asymmetric methyl group [348] although the H/D substitution does not take place in a symmetry plane [29]. Typke recently reanalyzed the same data by also considering the data of isotopomers in which one or both methyl groups had been deuterated [272]. He found that, in addition to 4 overdetermined sets, 10 different "minimal" sets of isotopic data could be constructed from which a complete substitution structure could be determined. For each
QUEST FOR EQUILIBRIUM STRUCTURE
set, he determined the
rs
225
coordinates by a least-squares fit and then
calculated the rotational constants of all isotopomers for which data were available. All data sets that contained the rotational constants of one particular monodeuterated species gave significantly worse predictions for the remaining isotopic species. He could not determine the cause of the abnormal results particularly since the assignment and the fitting of the spectrum of that one isotopomer had been carefully double-checked [349].
2.
The hydrogen pseudohalides HNCO, HNCS and HN 3 The b coordinates of all three non-hydrogen atoms were imaginary when the Kraitchman equations were used to locate the position of the nuclei in isothiocyanic acid, HNCS [218].
To
determine a reasonable structure for this planar molecule, the authors used the first momem condition for the b coordinates and the product of inertia condition. The second moment condition was used also but with the substitution moment of inertia moment I0. reliable
rs
Is
instead of the ground state
The authors argued that this procedure gives a more
structure instead of a
rs/r 0
hybrid, particularly since Watson
[36] demonstrated the physical significance of the substitution momem Is.
However, it can only be applied if two or more atoms have to be
located by means of auxiliary conditions. The structure of isocyanic acid, HNCO, was determined by the same method, except that the first moment condition had to be invoked also for the a coordinates [217].
226
GRONER
The same procedure was not applicable for hydrazoic acid, HN3, because of missing data for HN15NN [334]. Because the Kraitchman values for the b coordinate of the N atom in both HNCS [218] and HNCO [217] agree very well with the results obtained by the procedure described above, it was assumed that the b coordinate for N 1 (bound to H) in HN 3 can be obtained with sufficient accuracy from the Kraitchman equations, particularly since the magnitude for the coordinate is even larger in HN 3 [334]. The resulting slightly transbent r s structures for the three molecules agree nicely with each other, with N-H distances between 0.993 and 1.015 A and angles at the central atom of the near linear chain between 171.3 and 173.8 ~
3.
HC30 radical The structure of the HC30 radical has been determined by Cooksy et al. [49] by the r O, r s (more precisely: rIe), rmP and rm(1) methods from the data of the parent molecule and of all singly substituted species. The authors concluded that the rmP method was not suitable for this molecule but they did not include the recommended correction for the H/D substitution [45]. Harmony included the correction in a new rmP analysis for the preferred "a" form but excluded the illdetermined rotational constants A of the isotopic modifications [350]. The resulting structure is shown in Table 7 together with the
QUEST FOR EQUILIBRIUM STRUCTURE
227
TABLE 7. Structure of the HCCCO radical ("a" shape) a. r(HC l)
r(C 1C2)
r(C2C3)
r(C3O)
0(HCC)
0(CCC)
0(CCO)
rmp b
1.052(18) 1.230(36) 1.363(36) 1.220(21) 172(1)
165(1)
136.5(6)
rm(1) b
1.060(18) 1.219(3)
1.387(5)
1.192(2)
168(7)
163(2)
136.5(6)
rmp c
1.055
1.222
1.376
1.186
179.0
175.0
140.6
ab initio d
1.061
1.190
1.447
1.165
179.1
175.0
129.8
aDistances r in A, angles 0 in degrees. bRef. [49]. cWith the H/D correction [350]. dMP2/6-311G(2p,2d) [49].
results by Cooksy et al. [49]. The bond distances are much closer to the rm(1) structure than the original rmO analysis but the bond angles are not. Two of the rmP bond angles are very close to values predicted by the ab initio calculation reported in ref. [49].
According to
Harmony [350], the nearly linear structure of the HCCC part of the molecule almost guarantees low frequency anharmonic bending motions, which are troublesome in the rm9 method, particularly if H atoms are involved. Therefore, the error limits on the heavy atom rmP distances could be as large as 0.005 A instead of the more common 0.001 to 0.002 A.
4.
Vinyl alcohol and vinyl fluoride A complete r s structure of syn-vinyl alcohol has been published by Rodler and Bauder [263].
The authors noted the difficulties of
determining an accurate structure because of the close proximity of the
228
GRONER
central C atom and the attached H atom to the b principal axis. They also noted a similar difficulty in the case of vinyl fluoride.
Their
remark about the surprisingly large discrepancy between the experimental CCH angle and earlier ab initio predictions prompted Radom and coworkers [34,351,352] to undertake a systematic ab initio study of substituted ethenes. For a number of related molecules, they compared experimental r 0 structures with ab initio predictions for different basis sets and levels of electron correlation and established empirical correction terms. These predicted r 0 structures agreed well with the experimental structures for vinyl chloride and bromide [352] but not for vinyl fluoride and vinyl alcohol.
Because the many
experimental structures proposed for vinyl fluoride differed sometimes substantially from each other, they also redetermined the experimental structures exploring different methods (r 0, r 1, r B, rAl, rAB ) and weighting schemes.
They found sometimes very large differences
between the methods and weighting schemes for vinyl fluoride and alcohol [351 ] but not for vinyl chloride or bromide. They concluded that the predicted r 0 structures agree with the experiment for vinyl chloride and bromide and that they are the best estimates of the r 0 structure for vinyl fluoride and alcohol.
The problem with the
structure of the latter two molecules is caused entirely by the two atoms close to the b axis.
QUEST FOR EQUILIBRIUM STRUCTURE
5.
229
Fluoroacetylene Borro and Mills [244] have determined the equilibrium structure of fluoroacetylene using experimental B e constants for HCCF and DCCF. B e constants for the 13C substituted species were estimated by scaling the differences Be-B 0 with the factor obtained from ab initio calculations [353].
While the CC and CF distances agreed within
0.0006 and 0.0016 A, respectively, with the ab initio based predictions [353], the CH distance was 0.0036 A shorter. Botschwina and Seeger [245] pointed out that this discrepancy arose from an incorrect value for the vibration-rotation constant ct 1 for the HCCF isotopomer (the difference tXl(theor)-c~l(exp) was only 515 % for DCCF but 31.2 % for HCCF). They also referred to the relation between the C-H stretching frequency and the equilibrium bond length [72] to support their value of 1.0591 A. Borro and Mills discovered ([244] note added in proof) that a quartic term in the potential not considered before brings the vibration-rotation constant a l and therefore the equilibrium structure more in line with the ab initio predictions.
This incidence
demonstrates that determining sufficiently accurate vibration rotation constants is not an easy job and that good ab initio predictions are able to provide valuable support in this endeavor.
230
GRONER
6.
Conformers It has been recognized some time ago [354,355] that the rotational constants of different conformers of one molecule sometimes can only be explained if structural parameters other than torsional dihedral angles have significant conformational differences. In these and other early studies [356], only a limited number of isotopomers have been studied and definitive confirmation of such differences has been published only in the last two decades.
It seems that Hirota's
investigation of 3-fluoropropene [357] was the first almost complete study of the structures Unfortunately, complete
of both conformers rs
of a molecule.
or r 0 structural parameters for both
conformers are available only for a relatively small number of molecules ethanethiol,
such
as
3-fluoropropene,
CH3CH2SH
[358-360],
CH2CHCH2F ethaneselenol,
[148,357],
CH3CH2SeH
[142], ethylmethyl sulfide, CH3CH2SCH 3 [152,153], 1-fluoropropane, CH3CH2CH2F [150], ethylphosphine, CH3CH2PH 2 [146], and 1,2difluoroethane, FCH2CH2F [113,139]. In other cases, the structure is complete
for
only
one
conformer
as
for
ethylfluorosilane,
CH3CH2SiH2F [143,150], thiohydroxylamine, NH2SH [158], sulfur diimide, HNSNH [ 156], propanal, CH3CH2CHO [ 149], and hydrogen trisulfide, HSSSH [157]. Most often however, only partial structures have been determined for both conformations as for difluoroacetic acid, CH2FCOOH [138], 2-propaneselenol, (CH3)2CHSeH [154],
231
QUEST FOR EQUILIBRIUM STRUCTURE
ethanamine,
CH3CH2NH 2
2-methylpropanal,
[144,145],
(CH3)2CHCHO [155], ethylphosphonic difluoride, CH3CH2PSF 2 [141 ], isopropyl difluorophosphine, (CH3)2CHPF2 [151 ], and ethyl difluorophosphine, CH3CH2PF 2 [140]. Two papers have compared structural parameters between different conformers of a number of molecules [150,361].
The most significant differences so far have
been associated with skeletal bond angles. It seems that 1,4- or 1,5interactions are responsible for sizable structural changes.
IV.
CONCLUSION The quest for the equilibrium structures of molecules has been advanced significantly by new methods. The rlc and related methods offer a more versatile and better balanced way to determine structures equivalent to the traditional structure.
The
rmP
rs
method provides a somewhat better modeling of vibration-
rotation interaction; it can however only be used if all nuclei can be substituted by isotopes. The
rc
method seems to give the best results but its applications are
restricted to relatively small molecules.
The
rm(1)/rm(2) method looks very
promising; because it is a new method, only few applications have been published. Ab initio
methods have become extremely useful to structural research. The
CCSD(T)/cc-pV(Q,T)Z calculations seem to give very good molecules.
re
structures for small
If the budget allows it, it is easier to obtain sufficiently accurate
vibration-rotation constants from force fields calculated by the CCSD(T)/TZ2Pf method than by the traditional method of analyzing all vibrational fundamentals of
232
GRONER
numerous isotopomers.
Even lower level calculations (MP2/6-31G(d,p)) are
sufficiently reliable to establish trends and correlations and to estimate corrections for the structural parameters. New experimental techniques have increased the capacity of rotationally resolved spectroscopy tremendously. They are generating experimental data at an unprecedented rate and with excellent precision.
As a result of all these
developments, we know a lot more with much greater precision and accuracy about many more molecules. This includes equilibrium structures of polyatomic molecules.
ACKNOWLEDGEMENT The author wishes to thank Dr. M. D. Harmony for valuable comments and the permission to use unpublished data and to Dr. J. K. G. Watson for sending me the manuscript on the
rm(1) and rm(2) methods prior to publication.
QUEST FOR EQUILIBRIUM STRUCTURE
233
REFERENCES
V. S. Mastryukov, in Vibrational Spectra and Structure, (J. R. Durig, Ed.) Vol. 24, Amsterdam: Elsevier, 1999. ~
~
~
5. ~
0
~
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CHAPTER 4
THE ACCURACY
OF MOLECULAR
GEOMETRY
BY QUANTUM CHEMICAL
PREDICTIONS
METHODS
Stephen Bell Department of Chemistry, University of Dundee, Dundee DD 1 4HN, Scotland, U.K.
I.
I N T R O D U C T I O N ................................................................................................. 254
II.
C O M P U T A T I O N A L M E T H O D S ....................................................................... 256
III. P R E S E N T A T I O N O F R E S U L T S .......................................................................
IV.
258
S M A L L M O L E C U L E S ........................................................................................ 260
A~ Bond Lengths .................................................................................................. 260 B.
V.
Bond Angles .................................................................................................... 270
H Y D R O C A R B O N S .............................................................................................. 273 A.
C - C Single Bonds Lengths in Alkanes and Alkyl Moieties ........................... 273
B.
C - C Bonds Adjacent to C=C, C - C , Oxygen and Fluorine Atoms ................. 277
C.
C=C Double Bonds Lengths ........................................................................... 283
D.
C - C Triple Bonds Lengths ............................................................................. 288
E.
C - H Bond Lengths ......................................................................................... 291
F.
Bond Angles in Hydrocarbons ........................................................................ 302
VI. M O L E C U L E S C O N T A I N I N G N I T R O G E N , O X Y G E N O R F L U O R I N E .... 313 A.
CN, C - O and C - F Bond Lengths ................................................................... 313
B.
C=O Bond Lengths and Associated Bond Angles .......................................... 322
C.
Molecules with Central Oxygen Atoms .......................................................... 333
VII. T O R S I O N A L A N G L E S ....................................................................................... 336 VIII. C O N C L U S I O N S ................................................................................................... 344 R E F E R E N C E S ..............................................................................................................
253
348
254
I.
BELL
INTRODUCTION There are relatively few experimental techniques for the determination of accurate molecular structures of free molecules in their ground states in the gas phase, even considering the subdivisions of spectroscopy: microwave, infrared and Raman. The common techniques come under either spectroscopic or electron diffraction methods. In contrast there are several quantum chemical (QC) methods and very many basis sets and even more different combinations of these. Relatively few combinations are used in this study and an attempt is made to treat several molecules with the same standard method/basis set combinations. Because of this, little attempt is made to make a comprehensive review of the extensive literature. There have been numerous studies of the effects of different methods and different basis sets. General studies coveting a wide range of molecules of special note relevant to this study are those by B. Johnson et al. [ 1] and B. Ma et al. [2], although the former does not include one of the methods used extensively in this study. Many papers study these effects for single molecules individually or for single types of molecule.
The effects of basis sets/methods on small hydride
molecules were considered many years ago and displayed in a graphical manner [3 ] in plots which are two-dimensional and relate two internal coordinates. In QC methods, energy minimization is used usually without constraints and hence the structures obtained are predictions of r e structures in the spectroscopic sense.
The distinction between different types of experimental
structures are given in the Landold-B6rnstein review series [4] and also in other chapters of this publication. However, experimental r e structures are not available
ACCURACY OF MOLECULAR GEOMETRY
255
for many molecules (many more than stated by Ma et al. [2]). In principle, r s, r 0, rg, etc. structures could be found by QC methods. In other words, the r e structures obtained theoretically could be corrected for zero-point and vibrational averaging. In spectroscopy studies, rg and r z are not of much value in analysis of spectra. The differences between r e, r s, and r 0 structures may be fairly large for X-H bonds but are sometimes within experimental error for bonds between first-row atoms and certainly within the differences obtained by different methods/basis sets. The geometrical structures of hydrocarbons are easy to predict if no multiple bonds are present apart for some effects on the C-H bond lengths. By "easy to predict" we mean small calculations using small basis sets and simple methods so that no great expenditure of computer time is required. The greatest difficulties arise in predicting the structures of molecules with electronegative atoms and/or multiple bonds, i.e. molecules with lone pairs and 7t bonds. In this study, the results are categorized by BOND TYPE for several molecules possessing the same type of bond rather than giving the full structure, i.e. all the geometric parameters, of each molecule in a class.
Most of the
molecules considered are organic with only a few small molecules containing no carbon atom. Hence, almost all the bond types included in the study have a carbon at one end of the bond or both ends. Additionally, the breadth of the study is restricted by considering only a few QC methods and a few standard sets of basis functions. Since Gaussian 94 [5] and Gaussian 98 [6] have been used, the Gaussian basis sets normally used by
256
BELL
exponents of these programs will be considered. There may be better methods and basis sets for the purpose of predicting structures but these are commonly used. It would be informative for this study to include the effects of method/basis set on molecular vibrational frequencies and the accuracy of prediction of frequencies. It is clear that structural parameters, at least bond lengths, are related to stretching vibrational frequencies through a Badger type of relationship. Therefore, overshort or overlong bond lengths as obtained by some QC methods lead to stretching vibrational frequencies that are either ridiculously high or low respectively. Also cubic anharmonicity should be calculated if a structure type other than r e were to be considered. Consideration of these matters for a large number of molecules, as in this study, has not been done as it would lead to a much larger project. The torsional angles of included in this study.
gauche conformers of a number of molecules are
However, the prediction of torsional potential energy
functions for these molecules are not included although geometrica.l structures at conformation minima are indeed related to torsional potentials. Further systematic study of these problems is required.
II.
C O M P U T A T I O N A L METHODS All the calculations have been made with the Gaussian 94 [5] or the Gaussian 98 [6] program. Molecular structures for all molecules studied here have been calculated by geometry optimization within the procedures of the restricted Hartree-Fock method (HF) but good agreement with experimental structures is obtained only accidentally with HF when using small basis sets, so that "Post-SCF
ACCURACY OF MOLECULAR GEOMETRY
257
methods" are required that include at least some of the correlation energy. Of these, the Moller-Plesset perturbation method to second order (MP2) is used for all molecules and all basis sets employed. In recent year, methods based on Density Functional Theory (DFT) have become very popular and in particular the hybrid DFT method employing Becke's three-parameter exchange functional B3LYP is the most successful. B3LYP is also applied for all molecules and all basis sets employed. Since some of these are genuine ab initio electronic structure methods and some are parameterized density functional theory methods, they will be referred to collectively as quantum chemical or quantum computational (QC) methods. The basis sets used are the simple basis sets without polarization functions 3-21G and 6-31G, but these give molecular structures in good agreement with experiment only accidentally and in particular give poor agreement for Post-SCF methods. The 3-21G basis set is only included because it still may be necessary to calculate very large molecules with this basis. The extended sets with polarization functions 6-31G(d) and 6-31G(d,p) are used for most molecules.
For smaller
molecules, the much more extended triple-split and polarized basis set 6311+(3df,2p) is used in combination with the MP2 and B3LYP methods only. Other basis set types, especially the Huzinaga-Dunning DZ [7] and TZ [8] Gaussian bases, may be better for some of the molecular types considered but the emphasis of this work is more on method than on basis. All of the MP2 calculations on smaller molecules are done with all electron excitations considered, i.e. MP2(Full). For molecules with more than four firstrow atoms, electron excitations are done with "frozen core", that is electron
258
BELL
excitation from the inner-shell l s-like MOs is not included; some small molecules are also done with frozen core for the very extended basis set used. In any case, the shortening of bond distances on changing from "frozen core" (FC) to "Full" is very precisely predictable for each basis set. The amount of shortening will be given in some cases. Some other ab initio methods of recovering correlation energy are also considered; these are the methods MP3, MP4(SDTQ), CISD, QCISD, and CCSD. These are all done with "Full" inclusion of electron correlation unless otherwise indicated.
Bond lengths are obtained with these methods using the 6-31G(d,p)
basis set only or, if no hydrogens are present, the equivalent 6-31G(d) basis. Only one other hybrid DFT method (B1LYP) using Becke's one-parameter exchange functional is considered but other DFT methods are not included in view of the poor geometrical predictions with some of them as shown by Johnson et al. [ 1]. The geometrical structures obtained by other researchers have not been included because of the need for a consistency of treatment for all the molecules; only a few specific cases will be cited in the self-imposed constraint of considering the effects of method/basis set only. Certain aspects of the geometrical structures of about 40 molecules or conformers are included with at least the same three methods and the same three basis sets.
III.
PRESENTATION OF RESULTS Concerning the presentation of results, even with the idea of restricting the study to tabulations by BOND TYPES, there is still the choice as to whether to list in tables either by METHOD first, as the outer do-loop, then by BASIS SET
ACCURACY OF MOLECULAR GEOMETRY
259
secondly, as the inner do-loop, or by BASIS SET first, in the outer do-loop, then by METHOD secondly, in the inner do-loop.
There is both a need to make
comparisons among basis sets with ONE method, e.g. to study shortening of bonds with increased size of basis set, and also to make comparison among methods for ONE basis set, e.g. to draw attention to the lengthening of bonds with MP2 as against HF or to bonds of certain types being shorter with B3LYP than with MP2. The latter arrangement has been chosen in order to emphasize the superiority of one method against another, especially of B3LYP against MP2 for some bond types. The results of energy minimization by optimizing geometrical parameters, mainly bond lengths and bond angles, are presented in tables in the order described. In many cases, it is difficult to make comparison between digits in tables, especially if they are not adjacent, and therefore for most of the molecules the optimal parameter values are also plotted diagrammatically when comparisons can be made at a glance. These diagrams are effectively only one-dimensional plots; each component diagram is for one parameter of one molecule only as ordinate while the abscissa intervals are used to separate different basis sets and computational methods spaced for labeling purposes.
To avoid an excessive
number of diagram labels, all of the points are strict!y in the same order as in the tables with only "3" for HF/3-21G, "H" for HF, "M" for MP2, and "B" for B3LYP actually labeled. The largest basis set used, 6-31 l+G(3df,2p), is identified as EBS (extended basis set) in the diagrams. Each diagram has a horizontal reference line which is drawn at the value of the parameter for the best available experimental structure. The diagams also
260
BELL
make evident the convergence of the computed value of a molecular parameter on to the experimental equilibrium parameter value, when that is well determined, as the basis set is increased and better theoretical treatments are used. Since there are large numbers of tables and figures, a method of numbering is required. Although for some tables it has been necessary to have more than one figure and in one case there is more than one table for a particular bond type, for ease of reference the table numbers and figure numbers have been "synchronized". Where more than one table or figure is required the extra one has an added letter A~
IV.
SMALL MOLECULES
A.
Bond Lengths The re, r s and/or r 0 structures known for the four small hydride molecules HF, H 2 0 , NH3, CH 4 [9-17], and values for all these are given in Table 1 and Fig. 1 which illustrate that for bond lengths often the relationship r e < r s < r 0 is valid.
The differences in bond lengths are significant for
hydrides (approx. 0.01 A for HF where there is no uncertainty about derivation of structural parameter from observation, and 0.007 A for CH4). The bond length difference between re and r 0 for water is small but this may be due to the derivation treatment for the r 0 value using two rotational constants only.
A C C U R A C Y OF M O L E C U L A R G E O M E T R Y
261
T A B L E 1. Experimental and computed bond lengths of small hydrides molecules. Basis
Method
Expt
re
HF 0.91681 a
H20 0.95781 b
NH 3 1.0124 d
CH4 1.0870g
1.0138 e
rs r0
0.92563 a
0.9581 c
1.0162 f
1.0940 h
3-21G
HF
0.9374
0.9667
1.0026
1.0829
6-31G
HF
0.9209
0.9496
0.9913
1.0821
MP2
0.9470
0.9746
1.0100
1.0953
B3LYP
0.9493
0.9759
1.0059
1.0932
CISD 6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p)
0.9727
HF
0.9109
0.9476
1.0025
1.0837
MP2
0.9339
0.9686
1.0168
1.0897
B3LYP
0.9338
0.9687
1.0194
1.0933
CISD
0.9317
0.9662
HF
0.9005
0.9431
1.0009
1.0835
MP2
0.9210
0.9608
1.0115
1.0843
B3LYP
0.9254
0.9653
1.0180
1.0919
MP3
0.9179
0.9580
1.0107
1.0843
MP4
0.9206
0.9609
1.0138
1.0867
B 1LYP
0.9239
0.9639
1.0167
1.0910
CISD
0.9177
0.9574
1.0102
1.0846
QCISD
0.9202
0.9604
1.0131
1.0865
CCSD
0.9200
0.9603
1.0130
1.0863
HF
0.8970
0.9398
MP2
0.9171
0.9578
1.0092
1.0839
B3LYP
0.9220
0.9608
1.0134
1.0881
CISD
0.9105
0.9517
aRef. [9].
CRef. [10].
eRef. [13].
gR f.
bRef. [ 11 ].
dRef. [12].
fRef. [141.
hRef. [ 15].
262
BELL
Bond Length
Bond Length o o
o co
o --~
, !
.o co
p to 0
.o to "~
!
i
,
I
Expt
.o to tO
i
I
Expt ~
!
.o to CO
o co ~
!
1
@ to 01 "
'
~~ i=
o ,
co
6-31 G
.........~
co
6-31G
4r/
I
6-31G(d)
6-31G(d)
w
O r El (/1 ..o ul
!
~r o
w
i
CD 6-31G(d,p)
6-31G(d,p)
E~ EBS
EBS
Bond Length
Bond Length _~ o o0 . . .
o 4~
._= o to i
.
o o3 i
.o to ol I
o Co
,
Expt
,x0,
o -,,4 I,
co
6-31G
6-31G
6-31G(d)
~
6-31G(d)
=
i33
o E m~
I/l
=
6-31G(d,p)
~"
>.
6-31G(d,p)
, ~ : ~ -r.
(3
O
4"
EBS
~, ............
w
EBS
00
.......
FIG. 1. Experimental and computed bond lengths of small hydride molecules. In all the figures, "3" stands for HF/3-21G, "H" for HF, "M" for MP2, and "B" for B3LYP, and EBS stands for 6-311 +G(3df,2p).
ACCURACY OF MOLECULAR GEOMETRY
263
However, for bonds not involving hydrogen, the differences should be small because of mass effects. If some differences are large for non-hydrides it is probably because of improper treatment. All types of ab initio calculations of these small molecules have been made previously times without number [e.g. 1,2]. The bond lengths obtained by the treatments used in this chapter given in Table 1 and Fig. 1 should be compared with r e values rather than any of the other experimental structural types and especially in cases where the difference between structural determinations is significant. The following are some well-known general observations. The normal effect for most molecules of increasing basis set size, especially by the addition of polarization functions, is usually the shortening of bonds. This is seen clearly in the two-parameter diagrams by Bell [3 ]. In the current diagrams, this effect is seen clearly for Hartree-Fock calculations on the HF molecule and H20 as the points are lower than for other methods and so are isolated in the diagrams, but NH 3 and CH 4 aIe exceptions. Additionally, the spread of bond lengths decreases from HF to CH4; this would be even more obvious if all these diagrams were on the same scale.
The inclusion of correlation energy by whatever method normally
lengthens bonds.
"Full" correlation always gives bond lengths slightly
shorter than "frozen core", for example, for H20, MP2(FuI1)/6-31G(d) gives the O-H bond length as 0.9696 A in comparison with the MP2(FC) value of 0.9697 A, or for NH3, the MP2(Full)/6-31G(d) procedure yields 1.0168 A
264
BELL
and MP2(FC) yields 1.0171 A for the N-H distance.
Because these
differences are small and sometimes also the differences of particular bond types in different molecules are also small, bond lengths have been converged to 0.0001 A and are quoted thus in the tables. It is clear from Fig. 1 that the predicted bond lengths using MP2 and B3LYP converge asymptotically toward the experiment r e value as the basis set increases, with B3LYP usually a little longer than MP2 which is very near to experimental r e value for these molecules. For the more electronegative first-row central atoms, F and O, bond lengths calculated with MP4 are very like those with MP2, but those with MP3 are a little shorter. Also in these molecules, the bond lengths obtained with the CISD method are like those from MP3 calculations and those from QCISD calculations are similar to those from applying the MP4 (and so MP2) method. These conclusions are not really true for CH 4. In the component diagram for CH4 in Fig. 1, the bond length does not exhibit the large excursions of the other molecules with more electronegative atoms. The bond length from B 1LYP calculations are always very like those obtained from B3LYP and, at least for small molecules, there is not a large difference in computing time. Since, as far as geometrical structure is concerned, these two methods give such similar results, it might be just as good to use B 1LYP only. Only a few molecules containing second-row atoms are included in this study and results for HC1 and H2S are shown in Table 2 and Fig. 2. Excellent
ACCURACY OF MOLECULAR GEOMETRY
265
TABLE 2. Bond lengths of small second-row hydrides. Basis
Method
Expt
re
HC1
H2S
1.27455 a
1.3356 b
rs r0
1.3376 c 1.28392 a
rav
1.3372 d 1.3518 c
3-21G
HF
1.2935
1.3505
6-31G
HF
1.2953
1.3531
MP2
1.3175
1.3749
B3LYP
1.3206
1.3787
HF
1.2662
1.3264
MP2
1.2800
1.3395
B3LYP
1.2895
1.3496
HF
1.2656
1.3273
MP2
1.2682
1.3291
B3LYP
1.2861
1.3478
MP3
1.2691
1.3304
B1LYP
1.2846
1.3462
CISD
1.2694
1.3308
MP2
1.2713
1.3318
B3LYP
1.2800
1.3414
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p) aRef. [9]. bRef. [18].
CRef. [19]. dRef. [20].
experimental structures are of course known and especially for a diatomic molecules [9,18-20]. Of course, calculations made without d functions in the basis set give ridiculous bond lengths but they are only included for consistency of treatment. For hydrides with second-row central atoms and for larger basis sets, MP2 calculations give bond lengths shorter than
266
BELL
B M h
HCI
O 3
~._ 03
'(~H
~
O T-r
B
o
?
re
H~
~.
,
~ M , ,
9M
I
Method/Basis
B M
H2S t'~
3 I
I re
~
t
LU r
rO
"t~ v
"I:3 v
O
(.9
r
CO
cb o
~
cb =
B
=
LU
--
Method/Basis
FIG. 2.
Bond lengths of small second-row hydride molecules.
ACCURACY OF MOLECULAR GEOMETRY
267
experimental and those by B3LYP calculation longer {by 0.01 A with 631G(d,p)}, but the MP2 estimate is more like the experimental r e value as noted by Ma et al. [2]. However, the B3LYP value appears to approach the MP2 value as the basis set is increased further to 6-31 l+G(3df,2p). Bond lengths obtained by MP3 and CISD calculations are again similar and also near to those by the MP2 method and the results from the B 1LYP method are again close to those from B3LYP calculation. A few small molecules, F2, H202, and N2H 4, involving single bonds between first-row atoms other than carbon are included here (Table 3 and Fig. 3). Only the X-X bond lengths are given here and not the X-H bond lengths or the HXX angles which are considered later with others of similar bond type (see Tables 24 and 25). These X-X bond lengths exhibit even large swings on changing computational method than the X-H bonds above (Notice that the scale interval of the diagrams here is 0.04 A in contrast to the previous 0.01 A).
For F 2 and H20 2, MP2 with unpolarized basis set is
particularly poor (error 0.085 A, 6%) relative to the r e or r 0 structural values [9,21,22].
However, there is little to choose between the bond lengths
yielded by the MP2 and B3LYP methods in terms of agreement with experimental r e or r 0 parameters for polarized basis sets. The N-N bond length of N2H 4 does not vary so much with basis set and method as for the molecules with more electronegative atoms, and for polarized basis sets the bond lengths by the two major methods are very similar to each other and near to the experimental value [23,24].
268
BELL
TABLE 3. Bond lengths of other small molecules.
Basis
Method
F2
Expt
re
1.41193 a
r0
1.41745 a
H20 2
N2H4
1.463 b
1.447 e 1.449 d
rg 3-21G
HF
1.4025
1.4731
1.4496
6-31G
HF
1.4125
1.4623
1.3964
MP2
1.5034
1.5685
1.4378
B3LYP
1.4678
1.5313
1.4035
HF
1.3449
1.3965
1.4134
MP2
1.4206
1.4681
1.4379
B3LYP
1.4035
1.4558
1.4367
HF
1.3957
1.4112
MP2
1.4665
1.4349
B3LYP
1.4558
1.4360
6-31G(d)
6-31G(d,p)
6-311 +G(3 df,2p)
aRef. [9]. bRef. [22].
MP3
1.4142 e
1.4502
B1LYP
1.4002 e
1.4525
CISD
1.4045 e
1.4406
MP2
1.3956
1.4409
1.4242
B3LYP
1.3947
1.4464
1.4321
MP4 r [23]. dRef. [24].
1.4572 edone as 6-31G(d).
For all three molecules in Table 3, the optimized bond lengths obtained using the extended basis set 6-31 l+G(3df,2p), labeled EBS for short, yields curiously short bond lengths in comparison with other basis sets but especially in comparison with experimental. It appears that in these and some other molecules below employing such larger basis sets may not provide one
A C C U R A C Y OF M O L E C U L A R G E O M E T R Y
1.50
J
M
269
,
F2
1.46
C --I 'ID
M 1.42
0
rn
M
re
3 B
X ILl
1.38
(.9 03
1.34 Method/Basis 1.58
H202 1.54 --
i
B
01 r -,I "0
1.50 --
m
1.46'
"-"
(3
03
C 0
M v
rO
9
M
v
H
0
M x LLI
1.42 --
H 1.38 = MethodiBasis 1.58
....
N2H4 1.54 --
'4...'
Q. 1.50 --
.J "O
C
O Q3 1.46 --
z,
X LU
(.9
(.9
03
03
m
3
,
M
Z
M
B
rO 1.42
M
H
H
H 1.38 Method/Basis
FIG. 3.
Bond lengths of other small molecules.
270
BELL
with improved molecular structures and may, therefore, be a waste of computer resources.
B.
Bond Angles For molecules with electronegative central atoms, bond angles are even more sensitive to basis set and method than bond length [3]. For H20 and NH3, the optimized angles given in Table 4 and Fig. 4 with HF/6-31G method/basis differ from the experimental values by 7 ~ (Compare this 7% error with the largest bond length error found in H20 of 1.8%). For small hydrides with only two internal parameters, the combination of bond length predictions and band angle predictions can be seen together in plots like those by Bell [3]. However, all the procedures currently used, except those with unpolarized basis sets, give good comparison with experimental angles. The differences between the angles calculated with the MP2(FulI) and MP2(FC) models is usually quite small, in the cases here only 0.03 ~ Another way of presenting the NH 3 angles is as out-of-plane angles since these vary over a wider range (33 ~ to 63 ~ for the calculations in Table 3) and hence are a more sensitive measure of deviation from planarity. One second row hydride, H2S, is included with first row hydrides in Table 4 and Fig. 4. The bond angle in H2S does not undergo such large excursions in the diagrammatic presentation as those of H20 or NH3. The B3LYP method appears to give slightly better angles than the MP2 method but both give excellent bond angles with the 6-31G(d,p) basis set.
Once
ACCURACY OF MOLECULAR GEOMETRY
271
TABLE 4. Experimental and computed bond angles of small hydrides. Basis
Method
Expt
re
H 20 104.478 a
rs
r0
105.28 b
NH3
H 2S
106.67 c
92.11 f
107.23 d
91.6g
107.8 e
92.31 h 92.13g
rav
3-21G
HF
107.68
112.40
95.81
6-31G
HF
111.55
116.13
96.16
MP2
109.29
114.26
94.89
B3LYP
108.30
116.18
94.05
CISD
109.37
HF
105.50
107.18
94.37
MP2
104.00
106.36
93.33
B3LYP
103.65
105.76
92.78
CISD
104.26
HF
105.97
107.58
94.38
MP2
103.87
106.12
92.84
B3LYP
103.74
105.75
92.66
MP3
104.26
106.12
92.76
MP4
103.88
105.76
B1LYP
103.86
105.84
92.79
CISD
104.34
106.21
92.92
QCISD
104.08
105.94
CCSD
104.09
105.96
HF
106.39
MP2
104.55
107.29
92.14
B3LYP
105.20
107.25
92.45
CISD
105.08
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p)
92.92
aRef. [11].
CRef. [12].
eRef. [14].
gRef. [19].
bRef. [10].
dRef. [13].
fRef. [18].
hRef. [20].
272
BELL
112.0
H20 angles 110.0
o.
108.o C < "O C O m
e
v
(9 ~" 03 I r
(9 1-
3
03
m LU H
H
106.0 r0
rJ
M
1 04.0 o. x UJ
M
e
102.0
B
Method/Basis
~
116.0
114.0
B
NH3 angles
112.0 01 c ,< "o o El
CL 110.0 ..
108.0
v"o (9
,..., Q. X UJ
(9 ,_ 03 ! (D
r| (O
i
09 rn UJ
H
H
B
r0 106.0 .: re B 104.0
Method/Basis
97.0
3
95.0
H~ angles
H
M
=, O1
B H
(,.9 ~
H
(.9
< "O C O El
e~ X
93.0 W
B rO
re
~
B
B
,,,,* v
rs
91.0 ,
FIG. 4.
Method/Basis
Experimental and computed bond angles of small hydride molecules.
ACCURACY OF MOLECULAR GEOMETRY
273
again the 6-31 l+G(3df,2p) or EBS in the diagram yields angles a little out of line with the angles predicted by the other basis sets.
V.
HYDROCARBONS A.
C-C Single Bonds Lengths in Alkanes and Alkyl Moieties Good experimental structures are available only for a few simple alkanes.
Among the references for ethane [25-29], the r e structure
determination by Duncan et al. [27] is most relevant as the QC methods make predictions of equilibrium structures,
r z [28] and rmP [29] structures for
ethane are also given in Table 5 and Fig. 5. An early r s structure [30] and an rmP [29] structure of propane are available for propane, but the best structure available for butane appears to be an rg structure [31-33] which may not be very relevant to spectroscopic determinations considered here.
By alkyl
moiety is meant a saturated carbon chain of at least two C atoms, i.e. CH 3CH 2- or longer, so that it is sufficiently isolated from groups that influence adjacent bonds as in section 4B. The isolated C-C bond lengths of 1-butene by Kondo et al. [34] and Bouchy et al. [35] are not in disagreement for this bond. The pattern of bond lengths obtained by quantum chemical (QC) methods is almost identical for all of the alkanes and alkyl moieties given. The structures of alkanes are always easily predicted (see CH 4 above) i.e. the vertical spread of points in the diagrams is always less than for the X-X
274
BELL
T A B L E 5.
B o n d lengths o f C - C single bonds in alkanes or alkyl moieties.
Basis
Method
Expt
re
ethane
t-butane
1.526 d 1.522 b
1.521 b 1.536g rgl.533 e
r z 1.5351 e HF
1-butyne
1.530 f
r0
3-21G
c-l-butene
1.528 a
rs rm~
propane
1.5424
1.5408
1.5403
1.5382
1.5467
1.5290
1.5363
1.5445
1.5520
1.5346
1.5457
1.5264
1.5324
1.5227
1.5307
1.5299
1.5402
1.5408 6-31G
HF
1.5299
1.5313
MP2
1.5446
1.5454
1.5333 1.5310 1.5466 1.5452
B3LYP
1.5353
1.5375
1.5401 1.5370
6-31G(d)
HF
1.5274
1.5283
MP2
1.5244
1.5245
1.5297 1.5282 1.5250 1.5245
B3LYP
1.5305
1.5322
1.5341 1.5319
6-31G(d,p)
HF
1.5268
1.5278
MP2
1.5218
1.5225
1.5235
B3LYP
1.5300
1.5317
1.5337
1.5294
1.5258
1.5277 1.5208
1.5287
1.5292
1.5397
1.5226 1.5314
6-311 +G(3df,2p)
MP3
1.5243
MP4
1.5265
B 1LYP
1.5302
CISD
1.5225
MP2
1.5195
B3LYP
1.5277
aRef. [27].
eRef. [28].
eRef. [32].
bRef. [29].
dRef. [30].
fRef. [341.
gRef. [35].
ACCURACY OF MOLECULAR GEOMETRY
275
C-C Bond Length
C-C Bond Length ..=
..~
01 to !
"
..=
01 o3
01 4~
I
'
..=
-.=
01 to
01 (.~
I
'
,
Expt w Expt
--~ 01 01 ,,
--.
~' I
,s
,
o0
~
6-31G
6-31G
6-alG(d) -r-
6-31G(d,p)
"~
EBS
130
l~ C-C Bond Length
C-C Bond Length 01 to
01 ~
I
01 4~
"
O1 !,,3
01 (.~
I
I
t
O1 4~ '
I
01 01 '"
E..ty
Expt
3
6-31G
6-31G co~1~, ~
6.31G
~ 1:o .=,,
6-31G(d)~ ~co
6-31G(d,p) --...
FIG. 5.
=
Bond lengths of C-C single bonds in alkanes or alkyl moieties.
--
276
BELL
bonds in Table 3 and Fig. 3 so that all basis sets with polarization functions give similar bond lengths, although as above, using the MP2 method with unpolarized bases gives C--C lengths which are long.
The addition of p
functions to hydrogen atoms makes very little difference to HF, MP2, B3LYP bond lengths and this will be seen to be true also for other CC bond types. For ethane, other calculation models have been used including MP3, MP4, CISD, and B1LYP, and the MP2 method with the large basis 631 l+G(3df,2p). The calculated C-C bond lengths straddle the experimental r e value with the MP2 and B3LYP predictions of r e on each side and with the B3LYP optimal value nearer [2] for all basis sets up to the largest. The MP2 values show a wider variation with change of basis set. The rmP bond length is shorter than the r e structure value and is more like some MP2 values. Since there is an almost identical pattern in these diagrams, it may be possible for this particular bond type to use the computational result to assess the experimental. The r s bond length in propane seems a little too short and that in cis-l-butene is a little too long. The middle and end C-C bonds in butane are not identical by QC methods although assumed to be so experimentally probably because of the lack of electron diffraction data [33]. The full lines in Fig. 5 are for the end C-C bonds and the dotted lines for the middle bond; it is clear that even theoretically they are very similar and all follow the usual pattern. By comparison with ethane and propane, an r e or r s C-C bond length for butane should probably be around 1.525 A.
ACCURACY OF MOLECULAR GEOMETRY
277
In table column headings and in the diagrams, the identification of conformers is kept as short as possible to avoid clutter in small diagrams or crowding in tables: italic c stands for s-cis or synplanar, italic t for s-trans or antiplanar, and italic g for gauche which may be synclinal or anticlinal depending on the rotational isomerism exhibited. The C-C single bond two CC bonds away from the triple bond in 1butyne (or 1 pentyne) is not in the diagram because of the few points calculated and because the pattem is like other alkyl moieties except that the C-C bond is slightly longer by about 0.01 A than in ordinary alkanes.
C
B.
C
C
C
C
C
C-C Bonds Adjacent to C=C, C-=C, Oxygen and Fluorine Atoms The C-C bonds which are called single bonds according to Lewis-theory but which are adjacent to double bonds, triple bonds, fluorine atoms, or oxygen atoms including the carbonyl group, are all affected by the neighboring bonds or groups and are significantly shorter than the usual alkyl C-C bond. Adjacent C--O bonds and C-F bonds show the smallest effect but the shortening appears to be systematic in all the fluorides and the alcohol listed.
278
BELL
Adjacent C=C double bonds and carbonyls show a slightly larger and similar effect, but the largest effect of all seems to be shown with C-C triple bonds. Experimemal r s or r 0 structures are available for all the molecules with double bonds in Table 6, viz. propene [36-39]; cis-l-butene [34,35]; and cis2-butene [40-42].
The experimental and calculated C-C bond lengths are
displayed in Fig. 6 for propene and 1-butene and the patterns of bond length variation with method/basis is closely parallel. The plot for 2-butene is not given because of the complete similarity of the 2-butene diagram to the 1butene pattern except for the numbers being consistently smaller by about 0.004 .A for 2-butene. However, since the C-C bond length in the r 0 structure of 1-butene by Bouchy et al. [35] of 1.519 A is not realistic, it is not shown in the table or diagram either. The experimental C--C bond lengths in ethanal [43,44] and propanal [45] are quite similar to those of the alkenes and the pattern of variation of the calculated bond lengths is also quite similar. Some extra methods have been applied to ethanal. For two of these molecules, the MP2 method gives bond lengths closer to the r s or r 0 values and for the other two the B3LYP model appears to be better. In Table 7, the bond lengths of the C-C bonds adjacem to the C-F or C-O bonds of some fluorohydrocarbons and ethanol are given. Since r e, r s or r 0 structures are available for 1,2-difluoroethane [46-49], 1-fluoropropane
ACCURACY OF MOLECULAR GEOMETRY
279
T A B L E 6. Bond lengths of C-C single bonds adjacent to double bonds. Basis
Method
Expt
rs
propene 1.501 a
r0
c-l-butene
c-2-butene
ethanal
propanal
1.501 c 1.507 c
rg 1.506b
ravl.51 ld
1.497 e
1.5005 f
1.509h
r z 1.515g
3-21G
HF
1.5095
1.5137
1.5104
1.5069
1.5083
6-31G
HF
1.5015
1.5074
1.5028
1.4946
1.5001
MP2
1.5163
1.5219
1.5176
1.5135
1.5183
B3LYP
1.5049
1.5117
1.5057
1.5031
1.5087
HF
1.5026
1.5078
1.5043
1.5042
1.5077
MP2
1.4976
1.5022
1.4987
1.5016
1.5047
B3LYP
1.5021
1.5080
1.5034
1.5083
1.5121
HF
1.5019
1.5073
1.5030
1.5068
MP2
1.4960
1.5011
1.5005
1.5041
B3LYP
1.5013
1.5074
1.5072
1.5112
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p)
1.4972
MP3
1.5028
MP4
1.5072
CISD
1.5001
MP2
1.4965
B3LYP
1.5010
aRefs. [37,38]. bRef. [39].
CRef. [34]. dRef. [4].
eRef. [40,42]. fRef. [43].
gRef. [44]. hRef. [45].
[50], and trans ethanol [51 ], the usual type of diagram is draw for only these in Fig. 7.
The patterns of variation of calculated C-C bond length with
method and basis are very like those for the aldehydes in Fig. 6.
For
difluoroethane, the MP2 value for the 6-31G(d) and 6-31G(d,p) bases are very close to the r s value. The C-C bond in gauche difluoroethane is shorter
280
BELL
C-C Bond Length ..=
..=
...=
01 o
oi ~
oi bo
-,
r
, ;
C-C Bond Length _=
r
Ol
.Ix
w i,,,
"
I
=
Expt
Expt r
fh
~
6-31G Z-~
6-31G
6-31G(d)
=, , . ~ -I-
E <
6-31G(d,p)
= 6-31G(d)
G0
.<
4 EBS
I) oo _
/
~i)
3:7
6-31G(d,p) l,m
_.
C-C Bond Length oi o
oi --~
I. . . .
I
C-C Bond Length oi I',O
ol o
ol --~ 9-
',I
I
01 bd '
J
I Expt
Expt
j'~
6-31G
6-31G
o.
= .....
-r"
6-31G(d) E ~ , ~
-...=
"O ,.,I
6-31G(d) =r
I
0 "0 :3
6-31G(d~). ~ ) < =
6-31G(d.p)~:~
'I
I
FIG. 6. Bondlengths of C--Csingle bonds adjacentto double bonds.
ACCURACY OF M O L E C U L A R GEOMETRY
281
TABLE 7. Bond lengths of C - C single bonds adjacent to C - F and C - O bonds. Basis
Expt
Method
re
FFeth trans
gauche
1.514 a
1.500 a
rs
1.491 e
Fprop
ethanol trans
1.501 d
1.5157 e
1.506 b
1.493 b
3-21G
HF
1.5145
1.5068
1.5206
6-31G
HF
1.5070
1.4974
1.5130
MP2
1.5227
1.5123
B3LYP
1.5161
HF
6-31G(d,p)
3FFbut
trans
r0
6-31G(d)
3Fbut
1.5079
1.5245
1.5172
1.5049
1.5134
1.5269
1.5311fc
1.5168 fe
1.5271
1.5048
1.5197
1.5254
1.5119
1.5196
1.5118
1.5036
1.5148
1.5177
1.5090
1.5156
MP2
1.5111
1.5009
1.5111
1.5163 fe
1.5063 fe
1.5119
B3LYP
1.5202
1.5096
1.5197
1.5252
1.5165
1.5196
1.5027
1.5142
1.5152
1.4999
1.5101
1.5104
1.5086
1.5192
HF MP2
1.5098
B3LYP
1.5192
aRef. [48].
CRef. [46].
eRef. [51].
bRef. [49].
dRef. [50].
fCfrozen core.
Molecular Abbreviations: FFeth = 1,2 difluoroethane; Fprop = 1-fluoropropane; 3Fbut = 3-fluoro- 1-butyne; 3FFbut = 3,3-difluoro- 1-butyne.
than in the monofluoro compounds but their patterns of variation are closely parallel. The C - C bond next to the C - F bond in 1-fluoropropane in the r s structure seems short being lower than any calculated value by 0.01 A. Some MP2 calculations in Table 7 have been made with frozen core option, but the shortening of C - C bonds on going from frozen core to full is very predictable being 0.0003 A with the 6-31G basis and 0.0017 A with the 6-31G(d) and 631G(d,p) bases.
282
BELL
1.52
M
g-difluoroethane B B
1.51 t-, 01 r= r ,J "13 1.50 t-, O El
re A
M
o,
m
v
(9
~
tO
(5
L) 1.49
rs
|
e~
tO
x uJ 1.48 Method/Basis 1.53
M 1-fluoropropane B
1.52 O1 C
_1
H .51
M
E3.
O El
x LU
o,
(3
fJ
CL
1 .50
(.9 tO
.49 Method/Basis 1.54 ~-, ,
t-ethanol 1.53
M
01
...I "O 0 m
B
B
1.52
. H 1.51
(5 ,,"
% M~
M (5.9
(5
|
x tu
t
i
r
|
to
1.50
9
,
,
MethodlBasis
FIG. 7.
Bond lengths of C-C single bonds adjacent to C-F and C-O bonds.
ACCURACY OF MOLECULAR GEOMETRY
283
All the molecules in Tables 6 and 7 exhibit a similar pattern of variation of alkyl C-C bonds as is evident by comparing the components shown in the diagrams of Figs. 6 and 7 with those of Fig. 5. However, the pattem for C-C single bonds adjacent to the C-C triple bonds in the alkynes is a little different as seen in Fig. 8. For the alkynes, re, r s and/or r 0 structural determinations have been made for propyne [52-55], 2-butyne [56,57] and 1pentyne [58], and values for the C-C bond are given in Table 8 with the QC predictions. Apart from the excessively long C-C bond length for MP2/631 G, the difference from the other molecules in this group is that the MP2 bond lengths are slightly longer than those with B3LYP rather than somewhat shorter; the problems associated with MP2 calculations of molecules with triple bonds will be discussed later.
C.
C=C Double Bonds Lengths Experimental structures are available for each of the five alkene molecules included here; Table 9 includes r 0, r e and r z structures for ethene [59-62], r s structures for propene [36-39], r s and r 0 structures for cis 1-butene [34,35], and r 0 and r a structures for the two conformers of 2-butene [40-42]. The pattem of C=C bond length variation with QC method and basis set is shown in Fig. 9 (except trans 2-butene is not shown) and it is clearly almost identical for all cases. For the 6-31G(d) and 6-31G(d,p) basis sets, the C=C bond length by MP2 calculation is slightly longer and that with B3LYP slightly shorter than r e or r s where they are available. Hence it becomes
284
BELL
1.49
propyne M 1.48
1.47
H
3N
UJ 0
H
1.46 re
B
rO
1.45
Method/Basis 1.49
2-butyne
M
1.48
e-
,~
a:.
(.9
(.9
,J "0 c 0 ID
H
1.47 g
0
M
B
1.46 rO
M
B
~
B
x W I~K
Method/Basis
FIG. 8.
Bond lengths of C-C single bonds adjacent to C-C triple bonds.
ACCURACY OF M O L E C U L A R GEOMETRY
285
T A B L E 8. Bond lengths of C-C single bonds adjacent to C - C triple bonds. Basis
Method
propyne
Expt
re
1.458 a
rs
1.4586 a
r0
1.4596 b
1-butyne
2-butyne
-
rg
1.460 c
1-pentyne
3Fbut
3FFbut
1.4588
1.460 e
1.4675 d
3-21G
HF
1.4659
1.4679
1.4671
1.4682
6-31G
HF
1.4629
1.4674
1.4647
1.4672
1.4624
MP2
1.4806
1.4837
1.4817
1.4827 fc
1.4761 fc
B3LYP
1.4611
1.4662
1.4624
1.4652
1.4601
HF
1.4681
1.4692
1.4738 1.4665 f 1.4695
6-31G(d)
6-31G(d,p)
MP2
1.4611
1.4632
1.4619
1.4644 fc
1.4651 fc
B3LYP
1.4603
1.4643
1.4615
1.4635
1.4656
CISD
1.4654
HF
1.4673
MP2
1.4603
1.4627
1.4608
B3LYP
1.4593
1.4636
1.4605
MP3
1.4649
MP4
1.4685
B1LYP
1.4602
CISD
1.4629
aRef. [55].
CRef. [56].
eRef. [58].
bRef. [53].
dRef. [57].
fCfrozen core.
Molecular Abbreviations: 3Fbut = 3-fluoro- 1-butyne; 3FFbut = 3,3-difluoro- 1-butyne.
obvious that the experimental C=C bond length of 2-butene in both cis and trans conformers (both the assumed value by Kondo et al. [40] and that by
Guyon et al. [42]) is too long by 0.01 A in comparison with the bond length in all the other molecules and all the QC calculations.
286
BELL
T A B L E 9. Bond lengths of C=C double bonds Basis
Method
ethene
Expt
re
1.334 a
rs
propene
1.336 d
r0
1.339 b
c-l-butene
t-2-butene
c-2-butene
1.350 f
1.348 h
ral.347g
ral.346g
1.335e 1.336 e
rzl.3391 c 3-21G
HF
1.3149
1.3158
1.3163
1.3164
1.3188
6-31G
HF
1.3220
1.3236
1.3240
1.3247
1.3272
MP2
1.3507
1.3521
1.3527
1.3536
1.3567
B3LYP
1.3357
1.3381
1.3385
1.3404
1.3433
HF
1.3169
1.3184
1.3191
1.3196
1.3221
MP2
1.3349
1.3367
1.3374
1.3386
1.3416
B3LYP
1.3309
1.3333
1.3338
1.3354
1.3384
HF
1.3165
1.3181
1.3188
MP2
1.3338
1.3358
1.3366
1.3379
1.3408
B3LYP
1.3301
1.3327
1.3333
MP3
1.3313
MP4
1.3394
B 1LYP
1.3292
CISD
1.3277
QCISD
1.3356
MP2
1.3293
B3LYP
1.3248
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p)
aRef. [61 ].
CRef. [62].
eRef. [34].
gRef. [41].
bRef. [59].
dRef. [37].
fRef. [40].
hRef. [42].
A number of calculations have been made for ethene over a wider range of methods and basis sets up to 6-311 +G(3df,2p) in order to check the effect of method thoroughly for the archetypal case. The variation with these extra methods is quite large (the difference of longest to shortest is 0.008 A,
ACCURACY OF MOLECULAR GEOMETRY
287
C=C Bond Length
C=C Bond Length CO --=
CO 1',3 I
"
CO CO
C~ J:=
I
'
GO 01
CO O)
..=
-.~
.-=
-.=
_.~
co 1',3
~ (.o
co J~
r ol
co o3
t
i
,,
I
I'
,
"
'
Expt
Expt co
6-31G
w
"1"
6-31G
b~
6 - 3 ~
6-31G(d, 6-31G(d)
6-31G(d,p)
EBS
=./
.
.
.
.
C=C Bond Length
C=C Bond Length co !',3
t .......
co Co
Co r
~,,.
~
,
Co ol
be O~
co 1',5
co
.....
co co
I
Expt
co J~
,,, I
Expt
,I,
co Crl ,
:
;
(.,o
6-31G
6-31G(d) " ~ ~ ~
6-31G
~
6-31G(d) ~
~ z:
g
"uo.., "O
I 6-31G(d,p)
FIG. 9.
z:~,
6-31
Bond lengths of C=C double bonds.
~ z:
288
BELL
c.f. H20 0.007 A). As in the small molecules in section IV.A, bond lengths calculated by the MP3 method are shorter than those by MP2 and those obtained using the MP4 method are longer.
Also as before, the B1LYP
method gives results very close to those using B3LYP. In this case the CISD method gives a shorter bond length (like B1LYP) and the QCISD model yields a longer bond length which is like the MP2 value. The large basis set, EBS, predicts curiously short C=C bond lengths for ethene.
D.
C-C Triple Bonds Lengths For C-C triple bonds, there are the greatest variations of bond length with changes in QC methods for any bond type considered here. Calculated and experimental bond lengths [52-57,63,64] are given in Table 10 for four alkynes. The diagrammatic presentation in Fig. 10 shows that calculations at the HF level give bond lengths excessively short relative to the chosen experimental reference lines shown and the MP2 model gives bond lengths excessively long with differences between the two methods of 0.032 - 0.038 A. Notice that the diagram scale range is 0.06 A, in comparison with a range of 0.05 A in ethene (Fig. 9), and a range of 0.04 A in ethane (Fig. 5). The variation with basis set is also rather pronounced with bond lengths from HF calculations becoming even longer with increase in size of the basis set. For all basis sets, the B3LYP bond lengths lie between the extremes of HF and MP2 and hence are nearer to experimental for all basis sets. The calculated lengths of the C-C triple bond at the MP2 level are always much
ACCURACY OF MOLECULAR GEOMETRY
289
TABLE 10. Bond lengths of C-C triple bonds.
Basis
Method
ethyne
Expt
re
1.20241 a
1-butyne
2-butyne
1.204 e 1.2066 e
rs
r0
propyne
1.2067 b
1.207 b
1.2073 d
1.2135 e
rg 3-21G
HF
1.1875
1.1884
1.1890
1.1889
6-31G
HF
1.1941
1.1950
1.1956
1.1956
MP2
1.2321
1.2330
1.2343
1.2338
B3LYP
1.2110
1.2128
1.2135
1.2144
HF
1.1854
1.1869
1.1874
1.1879
MP2
1.2164
1.2182
1.2192
1.2199
B3LYP
1.2050
1.2072
1.2078
1.2092
HF
1.1861
1.1872
MP2
1.2171
1.2184
1.2195
1.2197
B3LYP
1.2053
1.2074
1.2080
1.2092
MP3
1.2055
1.2057
MP4
1.2189
1.2202
B1LYP
1.2038
1.2057
CISD
1.2036
1.2012
QCISD
1.2118
HF
1.1800
MP2
1.2085
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p)
aRef. [64]. bRef. [56].
B3LYP 1.1962 CRef. [55]. dRef. [53]
eRef. [57].
too long by approximately 0.010A. Perhaps the only exception to this rule is obtained with 6-31 l+G(3df,2p) where the MP2 value is too large by only 0.008A and the B3LYP bond length is curiously short. It appears therefore
290
BELL
1.24 M
ethyne
1.23
~ } 1.22
M
M
"~ 1.21
1.20
re X
1.19
U.I
~
~" H
1.18
.
.
.
tO
.
.
H
'
.
,,
Method/Basis 1.24
' M propyne
1.23
== 1.22. 0,
M
1.21-
~
B
rs 1.20 -
1.19-
UJ H
H
to
1.18
Method/Basis 1.2
M 2-butyne 1.2 M
M
== 0,
1.2
"~c 1.2
o
v
I~
B
B
rO 1.2
Q. v
1.1
x u.I
(.9
(9 T-
cO i to
H 1.1 Method/Basis
FIG. 10. Bond lengths of C-C triple bonds.
ACCURACY OF MOLECULAR GEOMETRY
291
that the supposed triple-zeta basis set may give spuriously short CC bond lengths. In any event, the C-C triple bond is always too long with the MP2 method. In this case, the bond length by the MP3 treatment is much shorter than that by MP2 and similar to that by B3LYP, but the MP4 bond length is close to that by MP2 as for other molecules. Once again, a bond distance by the B 1LYP method is hardly distinguishable from that by B3LYP and the former should obviously be used if it results in a saving in computing time; for small molecules like most of those considered here the saving is not significant. As usual the QCISD result is similar to those by the MP2 and MP4 methods. The C-C triple bonds of other alkynes closely follow the same pattem as ethyne including the effect of the extra QC methods. Since QC results have been collected for a significant number of compounds with C-C triple bonds and since they all show that MP2 calculations give rather poor bond lengths while the B3LYP model consistently gives bond lengths closer to experimental whether re, rs, r 0 or even r z, these results are presented in a different format in Table 10A for only a few combinations of method and basis set.
E.
C-H Bond Lengths Experimental and theoretical C-H bond lengths for methyl groups in alkanes or alkyl moieties are given in Table 11, for C-H bonds adjacent to the C-C bonds in alkenes in Table 12, and for C-H adjacent to the C-C bonds in
292
BELL
TABLE 10A.
C-C bond lengths by various ab initio and Hybrid DFT methods
and basis sets for a number of substituted acetylenes. Molecule
RHF
RHF
MP2
B3LYP
Expt
6-31G
6-31G(d)
6-31G(d)
6-31G(d)
MW
acetylene
1.1941
1.1854
1.2164
1.2050
1.2024 a
propyne
1.1950
1.1869
1.2182
1.2072
1.204 b
1-butyne
1.1956
1.2192
1.2078
-
2-butyne
1.1956
1.2199
1.2092
1.207 e
1-pentyne trans
1.1956
1.2207
1.2078
1.207 d
gauche
1.1958
1.2211
1.2081
1.207 d
2-pentyne
1.1961
1.2226
1.2098
1.210 d
3-fluoro-l-butyne
1.1932
1.2201
1.2067
4-fluoro- 1-butyne trans
1.1947
1.2202
1.2073
gauche
1.1942
1.2199
1.2069
1-fluoro-2-butyne
1.1942
1.2200
1.2090
3,3-difluoro- 1-butyne
1.1906
1.2192
1.2052
4-fluoro-2-pentyne
1.1943
1.1844
1.2201
1.2092
2-methyl- 1-buten-3-yne
1.1966
1.1885
1.2233
1.2105
4-methyl- 1-pentyne gauche
1.1958
1.2211
1.2081
trans
1.1959
1.2214
1.2083
aRef. [64]. bRef. [55].
CRef. [56]. dRef. [58].
1.1879
1.1858
1.1872
eRef. [65].
1.207 e
ACCURACY OF M O L E C U L A R GEOMETRY
293
T A B L E 11. Bond lengths of C - H bonds in CH3 groups in alkanes or alkyl moieties. Basis
Method
ethane
Expt
rm0
1.089 a
propane
6-31G
1.089 e
1.094 e (1.077) f
op
1.094 c
1.096 e (1.102) f
HF
1.0841
HF
1.0844
MP2
1.0986
B3LYP
6-31G(d)
HF
1.0929
B3LYP
6-311 +G(3df,2p)
1.0965
1.0856
MP2
6-31G(d,p)
HF
1.0962
1.0858
MP2
1.0878
B3LYP
1.0951
MP3
1.0875
MP4
1.0901
B1LYP
1.0941
CISD
1.0861
MP2
1.0873
B3LYP
1.0910
rgl.108 d 1.0845
1.0845
1.0840
1.0850
1.0850
1.0838
1.0845
1.0845
1.0839
1.0854
1.0853
1.0841
1.0989
1.0990
1.0985
1.0998
1.0998
1.0987
1.0964
1.0962
1.0954
1.0975
1.0974
1.0962
1.0857
1.0857
1.0851
1.0865
1.0864
1.0852
1.0935
1.0936
1.0929
1.0942
1.0941
1.0931
1.0962
1.0960
1.0952
1.0972
1.0971
1.0959
1.0858
1.0858
1.0852
1.0858
1.0866
1.0854
1.0882
1.0883
1.0876
1.0891
1.0891
1.0881
1.0950
1.0949
1.0941
1.0961
1.0960
1.0949
[29].
CRef. [30].
eRef. [341.
bRef. [28].
dRef. [32].
fRef. [35].
aRef.
c-l-butene
r s ip rzl.0940 b 3-21G
t-butane
294
BELL
TABLE 12. Bond lengths of C-H bonds adjacent to the C=C bond in alkenes. Basis
Method
ethene
Expt
re
1.081 a
ro
1.085 b
propene
c-2-butene
1.090 f 1.091 c
rsC1H1
c- 1-butene
1.0919 d (1.106) e
1.081
C1H2
1.0931 (1.081)
1.090
C2H3
1.0903 (1.085)
3-21G
6-31G
HF
HF
MP2
B3LYP
6-31G(d)
HF
MP2
B3LYP
1.0736
1.0740
1.0892
1.0868
1.0760
1.0851
1.0875
1.0752
1.0726
1.0733
1.0734
1.0752
1.0769
1.0757
1.0732
1.0734
1.0735
1.0766
1.0780
1.0914
1.0892
1.0889
1.0891
1.0927
1.0943
1.0882
1.0859
1.0859
1.0859
1.0900
1.0913
1.0777
1.0753
1.0755
1.0756
1.0780
1.0794
1.0868
1.0844
1.0845
1.0848
1.0882
1.0900
1.0889
1.0867
1.0867
1.0867
1.0901
1.0915
1.0765
1.0775
1.0938
1.0906
1.0789
1.0893
1.0908
ACCURACY OF MOLECULAR GEOMETRY
295
TABLE 12. (continued).
Basis
Method
6-31G(d,p)
HF
1.0764
MP2
1.0807
B3LYP
6-311 +G(3df,2p)
ethene
1.0867
MP3
1.0791
MP4
1.0817
B1LYP
1.0856
CISD
1.0779
QCISD
1.0811
MP2
1.0795
B3LYP aRef. [61]. bRef. [59].
propene
c-l-butene
1.0780
1.0756
1.0758
1.0759
1.0786
1.0801
1.0816
1.0794
1.0793
1.0795
1.0832
1.0848
1.0881
1.0859
1.0858
1.0858
1.0895
1.0908
1.0826 CRef. [37]. dRef. [34].
c-2-butene
1.0842
eRef. [35]. fRef. [40].
alkynes in Table 13. Good spectroscopic r e or rs structures are available for the C 2 and C 3 molecules" ethane [25-29], ethene [59-62], ethyne [56,63,64], propane [29,30], propene [36-39] and propyne [52-55] and those of other molecules in this section will be discussed. The C - H bond lengths obtained by the various QC models are given in Table 11 for
methyl C-H distances only but methylene C-H distances follow
the same trends as will be summarized below. The pattern of increases and decreases in C-H bond lengths with QC method and basis set is almost
296
BELL
T A B L E 13. Bond lengths of C - H bonds adjacent to the C - C bond in alkynes. Basis
Method
Expt
re
ethyne 1.06250 a
rs
propyne
1-butyne
1-pentyne
3Fbut
FFbut
1.061r 1.0561 e
r0
1.061 b
1.0548 r
-
1.056 d
3-21G
HF
1.0509
1.0503
1.0503
1.0528
1.0532
1.0509
6-31G
HF
1.0531
1.0527
1.0527
1.0568
1.0571
1.0537
MP2
1.0691
1.0692
1.0693
1.0694 fc
1.0697 fe
B3LYP
1.0647
1.0641
1.0641
1.0641
1.0646
HF
1.0570
1.0567
1.0567
1.0568 1.0664 fe
1.0668 fc
1.0672 fc
1.0661
1.0666
1.0670
6-31G(d)
6-31G(d,p)
6-311 + G(3df,2p)
MP2
1.0659
1.0658
1.0660
B3LYP
1.0666
1.0661
1.0661
HF
1.0569
1.0564
MP2
1.0617
1.0612
1.0614
B3LYP
1.0657
1.0651
1.0652
MP3
1.0606
1.0602
MP4
1.0634
1.0631
B1LYP
1.0647
1.0641
CISD
1.0599
1.0582
QCISD
1.0629
HF
1.0547
MP2
1.0611
B3LYP
1.0622
aRef. [64].
CRef. [55].
bRef. [56].
dRef. [58].
fCfrozen core.
Molecular Abbreviations: 3Fbut = 3-fluoro- 1-butyne; 3FFbut = 3,3-difluoro- 1-butyne.
identical for alkanes, alkenes, and alkynes except for the shortening of the bond as the CC distance shortens.
Apart from small shifts along the bond
length axis, Figs. 11, 12 and 13 are almost superposable on each other. The bond lengths obtained by HF calculation are always short and inclusion of
ACCURACY OF MOLECULAR GEOMETRY
297
C-H Bond Length 0
C-H
0
0
_x
I
I
I
Length
Bond o ~o I
--~
Expt r 9
Expt
if_ 6-31G
6-31G i
6-31G(d)
m
" ~
m
!
==,,,, 6-31G(d)
6-31G(d,p) ~: "I-
6-31G(d,p) EBS
~ I : : D
%= C-H Bond Length
C-H Bond Length o o0 1
Expt.. ~"'
Expt
~
6-31G
E
O ~o I
-"r
O "O
6-31G
6-31G(d)
6-31G(d)
6.
6-31G(d,p)
.'L,
r
6-31G(d,p)
FIG. 11. Bond lengths of C-H bonds in CH 3 groups in alkanes or alkyl moieties.
298
BELL
=C-H Bond Length
=C-H Bond Length o -.4
o (3o
I~
l
o ~o I
Expt
_.L o
O (30
o
o r .!
~
Ji
0
Expt
~ o,
r
6-31G
6-31G
""[~ ( 6-31G(d)
6-31 G(d)
""e
~'~
6-31G(d,p)
='c
=
~ =
5. 6-31G(d,p
"'o
EBS
=C-H Bond Length
=C-H Bond Length o
~ 4K,
._=
..=
o (30
o ~o
0
0
0
-'~
O0 ~0 ~.__,,1,1..,1 ,~ll~l'l--'l~--"
.~
0
m
Expt
Expt
~ []
o, o "T"
-r
6-31G
o0
6-31G(d)
o I
I~
o -r
|
6-31G
E
6-31G(d)
-o "O
s r
6-31G(d,p)
* K
6-31G(d p)
"~~'O
FIG. 12. Bond lengths of C-H bonds adjacent to the C=C bond in alkenes.
co
ACCURACY OF MOLECULAR GEOMETRY
299
=C-H Bond Length
=C-H Bond Length ...= O 4~
O 01
O 03
'
i
'
O ",4 '
!
O (30
o 03
o o1
o "4
i
""
Expt o
Expt
6-31G 6-31G 6-31G(d)
,,,,,
6-31G(d) 6-31G(d,p) ,< "-1
<:
E z4,..
6-31G(d,p)
EBS
i0j _C-H Bond Length
=C-H Bond Length Q 01 !
......
0 O3
0
I
!
o (3o
Q .1~
0 U1
Expt
Expt
0 03
~~~o
6-31G 6-31G 6-31G(d)
z 4,...~.~ w
6-31G(d)
6-31G(d,p)
~
o0
"O r
FIG. 13. Bond lengths of C-H bonds adjacent to the C=C bond in alkynes.
300
BELL
correlation whether by MP2 or B3LYP methods always causes an increase in bond length and for bases without polarization functions they are too long. Although the addition p polarization functions to H does not affect CC bond lengths whether single, double or triple bonds (see Figs. 5-10), such additional functions do cause the C-H distances to shorten, particularly those calculated by the MP2 method. For all three archetypal molecules ethane, etbene, and ethyne, and also for propane,
the
MP2/6-31G(d,p)
experimental r s or rmO geometries.
results
are
particularly
close
to
The rmP or r s structures of these four
molecules appear to indicate that for larger basis sets with polarization functions, that MP2 is to be preferred slightly over B3LYP although they are never farther apart than 0.006 A. Other QC methods with the 6-31G(d,p) basis are also always within this range. Two C-H bonds lengths are determined for some alkanes since the conformers studied have a symmetry plane and the C-H bonds on the symmetric plane are indicated as in-plane (ip) and the other pair of C-H bonds above and below the symmetry plane are called out-of-plane (op). For some alkenes considered, three points are given in Fig. 12 as there are three different C-H bonds adjacent to the double bond.
For the QC methods
employed, the ip and op C-H bonds of methyl groups are shown as almost identical. However, the r 0 bond lengths derived by Bouchy et al. [35] differ widely (0.025 A).
ACCURACY OF MOLECULAR GEOMETRY
301
The =C-H bonds in alkenes also do not differ by more than 0.004 A in the QC results and by only 0.002 A in the r s structure of 1-butene [34], but the r 0 structure by Bouchy et al. [35] once again shows results inconsistent with those having =C-H bond lengths varying from 1.081 to 1.106 A. The rather short r 0 value for =C-H bond length of cis 2-butene (1.0798 A) is also not given in Table 12 and Fig. 12. For CH 2 groups in alkanes, the C-H bond lengths average out as follows: for HF calculations 1.0866+0.0016 A, 1.0875+0.0017 A, 1.0884+ 0.0012 A, for MP2 calculations 1.0965+0.0046 A, 1.0978+0.0046 A, 1.0987+ 0.0046 A, and for B3LYP calculations 1.0985+0.0008 A, 1.0995+0.0008 A, 1.1010+0.0008 A for propane, trans butane, and cis 1-butene respectively. These bond lengths get progressively longer by 0.001 A from propane to butane to 1-butene. They are also remarkable consistent across the different basis sets and the inclusion or exclusion of polarization functions has little effect except for the MP2 model as indicated by the + values. In view of the agreement and precision of all the QC results for all these types of C-H bond, the significance or accuracy of a number of experimental structures must be considered and questioned. For example, the differences among the different C-H bonds in propene (~0.01 A) [38] are too large, in fact the shortest one is more nearly in agreement with the r s structure of ethene. The calculated ip and op distances in propane are also smaller than the experimentally determined difference (0.005 A) [30]. The C-H bond lengths of rg structure determinations are not of much value for the
302
BELL
comparisons between QC predictions of r e geometries and experimental values made here so that the rg structure value for butane is far off the diagram scale (Fig. 11), and the older C-H determination [32] of 1.108 ,& is included instead of the more recent value [33] of 1.117 A. Similarly, the rg structure of propene [39] exhibits both methyl and vinyl C-H bond distances far off scale.
F.
Bond Angles in Hydrocarbons The structures of alkanes are easily calculated with inexpensive treatments and this includes the C--C-C angles. Apart from the poor HF/321G angles given in Table 14 and plotted in Fig. 14, there is only a small variation with changing QC method or basis set. The differences between angles found by MP2 calculation and the corresponding ones by B3LYP are not large, being not more than 0.4 ~ The MP2 angles are close to the value in the r s structure [30] for propane.
As for the CH lengths of butane, it is
probable that the bond angles obtained by the MP2 method for butane with be closer to an r e or r s structure value rather than the value of the rg experimental structures, which are either smaller (112.4 ~ [32] or larger (113.8 ~ [33] but they have already been seen to have long CH bonds. Hence the reference line in Fig. 14 is put at the MP2/6-31G(d,p). The pattern of the variations of the C-C-C angle in c/s 1-butene is a little different from those of propane and butane, and in fact is much like the pattern for the C-C=C angle of c/s 1-butene itself and the other alkenes given in Table 15 and in Fig. 15. The difference for any basis set between the
ACCURACY OF MOLECULAR GEOMETRY
303
TABLE 14. C - C - C bond angles in alkanes or alkyl moieties. Basis
Method
propane
Expt
rs
112.4 a
t-butane
c- 1-butene 114.5 c
r0
114.8 e
rg
112.4 b
3-21G
HF
111.62
112.12
114.90
6-31G
HF
112.70
112.99
115.72
MP2
112.66
113.03
115.09
B3LYP
112.79
113.14
115.61
HF
112.79
113.08
115.85
MP2
112.39
112.87
115.15
B3LYP
112.94
113.28
115.83
HF
112.87
113.14
115.91
MP2
112.42
112.92
115.08
B3LYP
112.97
113.31
115.83
6-31G(d)
6-31G(d,p)
aRef. [30].
bRef. [32].
CRef. [34].
TABLE 15. C-C=C bond angles in alkenes. Basis
Method
propene
c-l-butene
Expt
rs
124.3 a
126.1b
r0
126.7 b
c-2-butene
126.7 c 126.94 d
3-21G
HF
124.74
126.62
127.76
6-31G
HF
125.25
127.06
128.13
6-31G(d)
6-31G(d,p)
MP2
124.73
126.31
127.69
B3LYP
125.19
126.60
127.71
HF
125.24
127.19
128.35
MP2
124.56
126.21
127.66
B3LYP
125.26
126.78
127.92
HF
125.26
127.20
MP2
124.47
126.05
B3LYP
125.27
126.77
aRef. [37,38].
CRef. [40].
bRef. [34].
dRef. [42].
127.53
304
BELL
114.0
propane
(!.)
B
B
113.0
B
H
rs
U
~
i
A
O m
112.0 x UJ
v
v
M
M 1o v (3 ,e03 ! co
3
1o
(3 o3
111.0 Method/Basis 114.0
t-butane B
B
-~ 113.0
v M
"41 M
c
Ar
(J
r
112.0
vl o (3
lo v (9
3
o3 ! co
o3 i r
x W
111.0 Method/Basis 117,0
c -1 - b u t e n e
116.o
H
B
H
B
H
C
"U C
0 El
Q. x UJ
L)
~
115.0
M
M
3 (9 rs
o3
(9 ,-o3
v1o ~ (9
114.0 Method/Basis
FIG. 14. C-C-C bond angles in alkanes or alkyl moieties.
ACCURACY OF MOLECULAR GEOMETRY
305
126.0
'V'
propene
H
B
~t " ~ 125.0
3
H
H
B
M
O m O II O O
B
M
M
124.0 03 X W
03
03
123.0 Method/Basis
128.0 c-l-butene H
H H
~t
127.0
/
"o o m O II (-}
126.0
M M
v
(9 03
~C
(9 ~-
(9 03
x LU 125.0 ' Method/Basis 129.0 c-2-butene H H
128.0 c
3
M
M
o
m r II tO O
127.0
O v-,-
/
O ,r03
v
rs
(6 Q. x LU
126.0,
Method/Basis'
FIG. 15. C-C-C bond angles in alkenes.
306
BELL
different QC methods is larger for C-C=C angles than for C-C-C angles being up to 0.7 ~ but again the MP2 method gives better agreement with the experimental r s structure values [34] for all basis sets. However in cis 2butene, although the MP2 results are nearer to experimental, the angle obtained by the MP2 treatment is quite far (0.6 ~ from the r 0 value [40,42], which may need revision. Finally in hydrocarbons, the CCH angles are listed in Table 16 for methyl groups and in Table 17 for C=C-H angles of alkenes. These angles for hydrocarbons are very predictable in that they vary very little between different electronic structure methods and different basis sets. The scale in Fig. 16 has a range of only 2 ~ The B3LYP angles differ by not more than 0.4 ~ from the exactly corresponding MP2 angles, but the methyl ip and op CCH angles differ much more from each other. This is in agreement with the experimental r s structures of propane and cis-l-butene, but the difference between the r s structural ip and op angles in propane [30] is approximately two times that calculated, while the difference in the r s structure in 1-butene [34] is approximately half of that calculated.
The reference lines in the
diagrams in Fig. 16 are drawn at the average of the ip and op r s angles. Similar differences between ip and op CCH angles are calculated for butane but the methyl CCH angles are constrained to be equal in the rg structure [32,33]. Since the diagram for butane is almost identical to that for propane, including absolute magnitudes, the reference line is drawn at the propane CCH average.
A C C U R A C Y OF M O L E C U L A R G E O M E T R Y
307
T A B L E 16. C - C - H bond angles in CH 3 groups. Basis _
Method
Expt
rmO
ethane
propane
t-butane
r sip
111.8 c
op
110.6
HF
110.79
HF
111.19
MP2
111.09
B3LYP
6-31G(d)
HF
111.20
B3LYP
6-311 +
111.29
111.21
MP2
6-31G(d,p)
111.0e
111.2 f
111.3
110.7
110.4 e rzlll.17 b
6-31G
propene
c-2-butene
111.2 a
r0
3-21G
c-l-butene
HF
111.35
111.22
MP2
111.19
B3LYP
111.38
MP3
111.16
MP4
111.16
B1LYP
111.36
CISD
111.19
MP2
111.27
G(3df,2p) B3LYP
112.15g
~110.5 d 111.19
111.12
110.19
111.15
112.65
110.50
110.56
110.84
110.65
110.23
111.36
111.31
110.41
111.49
113.02
111.00
111.06
111.32
111.01
110.61
111.32
111.31
110.40
110.62
112.20
110.69
110.70
111.05
111.23
110.78
111.51
111.47
110.69
111.50
112.83
110.98
111.05
111.23
111.22
110.91
111.34
111.29
110.32
111.43
113.05
111.09
111.15
111.37
110.90
110.47
111.57
111.54
110.56
110.98
112.65
110.82
i10.85
111.10
111.12
110.66
111.55
111.49
110.66
111.54
112.93
111.11
111.18
111.32
111.18
110.85
111.34
111.28
110.32
111.44
111.09
111.15
111.37
110.87
111.58
111.54
110.61
110.97
112.62 110.63
110.79
110.81
111.04
111.08
111.59
111.53
110.71
111.56
111.13
111.20
111.32
111.16
111.38
aRef. [29].
CRef. [30].
eRef. [34].
bRef. [28].
dRef. [33].
fRef. [37].
gRef. [42].
308
BELL
T A B L E 17.
C = C - H b o n d angles in alkenes.
Basis
Method
Expt
re
121.32 a
r0
121.08 b
ethene
propene
c-l-butene
c-2-butene
117.3 e 118.66 f
3-21G
6-31G
rsH1C1C2
120.5 e
121.1 d
H2C1C2
121.5
119.8
H3C2C1
119.0
119.0
121.86
122.70
121.80
121.24
119.57
118.91
121.90
122.76
121.78
121.22
119.04
118.48
121.65
122.59
121.92
121.32
119.18
118.65
121.73
122.48
121.94
121.45
119.07
118.72
121.86
122.75
121.64
121.05
118.89
118.26
121.50
122.48
121.74
121.06
118.96
118.39
121.66
122.42
121.83
121.32
118.86
118.45
121.78
122.67
121.55
120.95
118.86
118.23
HF
HF
MP2
B3LYP
6-31G(d)
HF
MP2
B3LYP
6-31G(d,p)
HF
121.91
121.93
121.88
121.98
121.81
121.70
121.87
121.74
117.88
117.39
117.56
117.55
117.16
117.25
117.30
ACCURACY OF MOLECULAR GEOMETRY
309
TABLE 17. (continued). Basis
Method MP2
B3LYP
6-311+G(3df,2p)
ethene 121.57
121.82
MP3
121.70
MP4
121.66
B1LYP
121.81
CISD
121.71
QCISD
121.71
MP2
111.43
B3LYP
111.77
propene
c-l-butene
c-2-butene
121.32
122.24
117.22
121.61
120.96
118.93
118.40
121.61
122.36
121.77
121.26
118.82
118.43
aRef. [61].
CRef. [37].
eRef. [40].
bRef. [59].
dRef. [34].
fRef. [42].
The diagrams for propene and cis 2-butene, shown in Fig. 16A, are a little different from those already considered but similar in some regards to each other.
However, the two types of CCH angle in cis 2-butene are
calculated to differ by 2 ~ while the r 0 structures makes them the same whether in the derived structure [42] or in the partially assumed structure [40], but the assumed angle of 109.47 ~ is clearly wrong. The angles involving the H atoms adjacent to double bonds are given systematically relative to the C=C bond in Table 17 and Fig. 17, i.e. they are all C=C-H angles. In ethene and 2-butene, there is only one kind of angle
310
BELL
CCH Bond Angle
CCH Bond Angle
.,==
I
"'
Expt
I
--.= _..=
~
~)~._
Expt
0
6-31G 6-31G
I, " r "
r
6-31G(d)
O
%=
w
I~"
f~
J
>~
6-31G(d,p)
6-31G(~~o o e~
,( 6-31G(~,p)~=
EBS
CCH Bond Angle
CCH Bond Angle .._=
~
,.,=
I.'
u
>.
'
I
,,
I .......
Expt
Expt
m.
o '1o
]
Bk. I
/
d
~a
,
"i-
6-31G
6-31G
~3:lv O
/
;
6-31G(d) I:&/""~ ""
6-31G(d)
%n
6-31G(d,p) I:=~
6-31G(d,p) "0
~II,
FIG. 16. C - C - H bond angles in CH 3 groups.
~Q
o
ACCURACY OF MOLECULAR GEOMETRY
311
C..C-H Bond Angle
C-C-H Bond Angle _.L
.ix
o
(3
:
......
b
b
{
I
b ....
o ,
'
-
,
b I
"'
O
o I
"
|
'
Expt O "t~
"=" 10
Expt
151
-r"
"m
-r 1
6-31G
[! 6-31G
g E!
-1-
q
,,= 6-31G(d)
6-31G(d) I:i
"O r r r
D
"O
ui /*" 6-31G(d,p)
6-31G(d,p)
FIG. 16A.
C-C-H bond angles in CH 3 groups.
312
BELL
C=C-H Bond Angle
C=C-H Bond Angle b
b
o
I
I
I
i
--=
O
01
D
O C) "r
._=
9
Expt
Expt
..L
-r"
!
6-31G 6-31G
=
D o
6-31G(d)
W
t
W
===~
6-31G(d)
=
l
I~
=
6-31G(d,pl
I
I,
6-alG(d,p) EBS
C=C-H Bond Angle
C=C-H Bond Angle O
b 4
o I
,
""~o
"T" t~
CJ -r-
_,.
...=
i,,a
..=.
Expt
....=
,.,=
b
i.n
I
1
Expt
-r"
6-31G
6-31G
6-31G(d)
-=-
6-31G(d)
!
133
r
u 6-31G(d,p)
I:
tD
6-31G(d,p)
FIG. 17. C = C - H bond angles in alkenes.
.--=
b '
'
O1
ACCURACY OF MOLECULAR GEOMETRY
313
but in propene and 1-butene there are three angles, which are given with the following atom numbering:
HE
\
/
/ C1 ~
H3
C2
\
H1 In ethene, the angle given by MP2 calculations appear to be consistently nearer to the r s value while in the others listed there is nothing to choose between the MP2 and B3LYP results. The different C=C-H angles predicted are in remarkable agreement with the r s values.
Vl.
M O L E C U L E S CONTAINING NITROGEN, OXYGEN OR FLUORINE
A.
CN, C--O and C-F Bond Lengths Bond lengths for carbon-nitrogen single, double and triple bonds are considered first and experimental bond lengths and QC results are given in Table 18 and Fig. 18. r 0 and rg structures have been determined for methyl amine [66,67] and r s and r a structures have been derived for dimethyl amine [68,69]. The pattem of variation of C-N bond length with QC method and basis set is more like Fig. 9 for C=C alkenes, except for results with the simple 6-31G basis set which is more like Fig. 5 for alkanes or other C-C in Figs. 6-7. Fig. 18 is also similar to Fig. 9 in that the B3LYP bond lengths are very close to the corresponding value from MP2 calculations for the polarized basis sets. However, the predicted r e values are shorter than the values in the r 0 or r s structures by about 0.008 A.
314
BELL
TABLE 18. CN bond lengths in amines, imines and cyanides. Basis
Method
Expt
re
CH3NH 2
(CH3)2NH
CH2=NH
CH3CH=NH CH3C-N 1.1557g
1.462 c
rs
1.273 e
1.4714 a
r0
rg 1.4652 b
1.1571 h 1.273 f
r a 1.455 d
1.1583g rg 1.159 i
3-21G
HF
1.4714
1.4655
1.2565
1.2572
1.1389
6-31G
HF
1.4507
1.4494
1.2614
1.2635
1.1464
MP2
1.4775
1.4766
1.3029
1.1997
B3LYP
1.4599
1.4604
1.2815
1.1716
HF
1.4532
1.2505
1.1347
MP2
1.4638
1.4563
1.2808
1.2829
1.1784
B3LYP
1.4651
1.4572
1.2704
1.2736
1.1603
HF
1.4517
MP2
1.4609
B3LYP
1.4644
6-31G(d)
6-31G(d,p)
1.2500 1.4544
1.2799
1.1348 1.2821
1.2700
aRef. [66].
CRef. [68].
eRef. [70].
gRef. [72].
bRef. [67,4].
dRef. [69].
fRef. [71].
hRef. [52].
1.1782 1.1603 iRef. [73].
Two examples of molecules with C=N double bonds, i.e. imines, are also included in Table 18 but only one in Fig. 18 because of similarity. In spite of its small size, only an r s structure seems to be available for methanimine [70] and the r 0 bond length for the ethanimine [71] is given as the same value as the r s value in methanimine. The plot of QC bond lengths given in the second section of Fig. 18 is very like C=C in Table 9 except that the B3LYP bond lengths are shorter than those by the MP2 method by 0.012 A and nearer to experimental. This outcome is rather like Fig. 9 for alkenes or even Fig. 10 for C-C bonds.
ACCURACY OF MOLECULAR GEOMETRY
315
1.49
methyl 1.46
amine
M
eA
1.47 B
,,J "0 0 El Z
rg
B
M~
3
1.46
!
EL X LU
no (,9 ,t.-03| t,c)
H
1.45 03
H (_9 ,t-03
1.44 Method/Basis 1.31 M methanimine 1.30
'*" 1.29 e,, M
,,,=1 "~ e,, 1.28
M
B
0
m Z II r
# 1.27
rs
B (.9 03
EL X LU
1.26
,r
3
H
H
1.25
v
Method/Basis 1.20
methyl
1.19
~.
cyanide
M
1.18
M
Q,
c
_~ 1.17
B
0
El 1.16 Z III ('1 1.15
1,14
,4)
rs
7" rz
rg ,r(.9 03
~. x LU 3
H~
'
H
1.13 Method/Basis
FIG. 18. CN bond lengths in amines, imines and cyanides.
316
BELL
Acetonitrile is a well studied molecule and good structural parameters have been determined for most structural types [52,72-75]. The pattern of changes of predicted equilibrium C-N length with different QC method and basis set shown in Fig. 18 is very similar to Fig. 10 for the alkynes. As well as the similarity in range of variation (0.07 A), the C-N bond lengths from MP2 calculations similarly miss the experimental reference line by a large margin (0.022 A) which is even larger than in the alkynes and B3LYP gives a much better prediction with both 6-31G(d) and 6-31G(d,p). In general, the MP2 method predicts multiple bonds longer than experimental and also those by the B3LYP method. In order to consider C--O single bond lengths, two alcohols and an ether are included for which significant experimental structures are known [51,7680].
Bonds for carbonyl compounds, C=O double bonds, are considered
separately in the next section in view of the number of aldehydes and ketones studied over many years. The variations of predicted bond lengths obtained by different QC methods are given in Table 19 and Fig. 19. For methanol, a number of extra QC methods have been used, MP3, MP4, CISD, etc. as for small molecules such as ethane, ethene, ethyne, and ethanal above.
The
results are also similar, with bond lengths calculated by the MP3 method shorter than those by MP2 and those obtained using the MP4 method longer. Also the B 1LYP method gives results very close to those using B3LYP and the CISD method gives a shorter bond length (than B 1LYP) and the QCISD and CCSD models yields a longer bond length which is like the MP2 value.
ACCURACY OF MOLECULAR GEOMETRY
317
TABLE 19. Some C-O single bond lengths. Basis
Method
Expt
methanol
CH3OCH 3
1.4214 a
1.410 e
r z 1.4285 b
rg 1.415 d
rs
t-ethanol 1.4354 e
r0 3-21G
HF
1.4447
1.4326
1.4443
6-31G
HF
1.4305
1.4234
1.4364
MP2
1.4709
1.4653
1.4771
B3LYP
1.4523
1.4464
1.4593
HF
1.3996
1.3912
1.4045
MP2
1.4229
1.4137
1.4277
B3LYP
1.4186
1.4099
1.4249
HF
1.3985
1.3915
1.4035
MP2
1.4198
1.4124
1.4249
B3LYP
1.4181
1.4104
1.4243
MP3
1.4166
MP4
1.4227
B1LYP
1.4175
CISD
1.4126
QCISD
1.4200
CCSD
1.4194
MP2
1.4147
1.4073 fc
B3LYP
1.4209
1.4092
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p) aRef. [77].
CRef. [79].
eRef. [51].
bRef. [78].
dRef. [80].
fCfrozen core.
For the main methods studied, the variations in Fig. 19 correspond very well with the diagram for C-N single bonds in Fig. 18 and, in turn, were seen to be similar to Figs. 5-7 for C-C bonds in alkyl moieties except that the variation is larger as seen by the increase in scale range (0.09 A instead of 0.06 A and 0.04 A). Again in view of these diagrams, there is little to choose
318
BELL
1.47
M methanol
~ 1.45
~ 1.4.3 mO 0 d
H
M
# rs
M B
1.41
~,
~,
UJ
d/
sis
M
CH3OCHs
1.46
1.44
e
~ mO
1.42
rO
H
03
(9
(9
N
N
M
M
0 rs 1.40
B
B
,-, E2. x LU
M
H
H
1.38 Method/Basis .48 M
t-ethanol
.47 .46
N
.45
9-~
,44
~ m O
r
ro
1.43
1.42 1.41
H
(9 ~-
M
03 ,.,,., c~
r
M
B
B
x
UJ
1.40
H
1.39 Method/Basis
FIG. 19. Some C - O bond lengths.
~-03 t
H
~-03
ACCURACY OF MOLECULAR GEOMETRY
319
between the MP2 and B3LYP methods as one is nearer to experimental for one molecule and the other for another molecule. Perhaps this outcome has more to do with the experimental determination rather than the theoretical method. Lastly in this sub-section, we consider fluorine-containing molecules and in particular the length of the C-F bond; some of these have been considered in section V.B or V.D above in terms of C-C bonds next to C-F bonds, or C-C bonds. The molecules not discussed before are CH3F [52,8184] and 3-fluoropropene [85-88] for which excellent r e or rs structures are available and the C-F bond lengths are given in Table 20 and used as reference lines in the diagrams in Fig. 20. These diagrams are more like the diagrams for C=C double bonds in Fig. 9 rather than those for C-C bonds in Fig. 5 or even the single bonds Figs. 19 and 20 for C-N and C-O bond lengths, except again the scale range has now increased to 0.10 A. Additionally, ignoring the results of the unpolarized basis sets, in all cases shown the C-F bond lengths by the B3LYP method are shorter than those by the MP2 model and very near to experimental re or r s values. (1fluoropropane is not shown and may be an exception.) In fact, the C-F bond appears to be rather like the C-C bond (Fig. 10) with respect to the relative merits of the MP2 or B3LYP methods.
320
rA
o
r,.) eq
r,.)
ZZ r,j
o
o
ZZ r,.)
o ,.1=
Z~
r ~
.
t"-.
9
.
~ t' ~
o
q..,
.
.
9
.
~ ~
~
.
o
o
~ ~
~
~
o
,
9
o
~ ~ o
9
9
o
t-,q
O
o
.
o
~ ~
~
~
.
o
o
o
o
o
o
.
t' ~ ~ o
o
o
,~.
~
o
o
.
o
o
o
o
o
~ t"'-
~
I' ~
.
o
.
t' ~ t' ~ o
,
o
o
.
~,~
~
o
.
O ~
~
~
o
o
o
o
o
o
o
O o
~,~
~ I~
~'~
o
I-.-. o
o
o
o
t"-.
o
~
o
o
o
o
t' ~
9
o
,n- ~
II
o
R
II
!
R
II
!
R
o
!
R
"7
r II
<6
BELL
ACCURACY OF MOLECULAR GEOMETRY
321
C-F Bond Length
C-F Bond Length
03
o
[
. . . . .
.....
I . . . . . . .
I,
;
Expt
Expt
~4
!
-r 6-3
6-3 0 e~
-1-
~~
6-31G(d)
6-31G(d)
"1"
--r r
"~~
6-31G(d,p)
6-31G(d,p)
C-F Bond Length
C-CI Bond Length ,
,,
O0
(30 O
(30 I~O
CO J~
O0 03
]
I
I
I
l
03
(30
O
FO
J~
I
I
:
I
03
Expt
c~
..h
6-31G
6-31G(d)
O
=2
o
i
-1"
6-31G(d,p)
6-31G(d,p) .
FIG. 20. C-F bond lengths.
.
.
.
.
.
.
.
.
.
322
BELL
Only a few molecules with second-row atoms have been included in this study. The C-CI bond length of CH3C1 is included in Table 20 and Fig. 20 where it shows a quite different variation from the C-F bond lengths already discussed.
In particular, without polarization functions the C-CI
bond length is ridiculously long but also for molecules with second-row atoms MP2 is always better than B3LYP as seen in Table 2 and Fig. 2 for HCI and H2S and as noted by Ma et al. [2].
B.
C=O Bond Lengths and Associated Bond Angles Carbonyl compounds have been thoroughly studied experimentally by spectroscopic methods and theoretically by electronic structure methods, r e structures are known for CO [9] and methanal [90-92,61 ] and an r s structure is reported for acetone [93,94], with less reliable structures for ethanal [43,44] and propanal [45]. Carbon monoxide is included with the four basic aldehydes and ketones in Table 21 and Figs. 21 and 21A because its bond length follows a very similar variation with QC treatment and basis set in spite of the shorter bond. In the case of multiple CO bonds, the pattern of variation of bond length with method and basis set is very similar to that shown by the C-=C bond in alkynes seen in Tables 10 and 10A and Fig. 10 except for the HF/6-31G bond lengths being accidentally near to the experimental values. The most significant similarity is that the C=O bond lengths in the MP2 model are always too long like the C-C bonds in alkynes and even at the best are still long by 0.008 A, while the B3LYP model gives
ACCURACY OF M O L E C U L A R GEOMETRY
323
T A B L E 21. Bond lengths of multiple CO bonds. Basis
Method
Expt
re
CO 1.128323 a
methanal
ethanal
propanal
1.2033 b 1.207 h
rs r0
acetone
1.1309 a
1.2063 c r zl.2078 d
1.2155 e
1.215g rg 1.214 i
rz 1.210 f
3-21G
HF
1.1289
1.2069
1.2085
1.2088
1.2109
6-31G
HF
1.1307
1.2101
1.2139
1.2140
1.2187
MP2
1.1767
1.2577
1.2597
1.2601
1.2638
B3LYP
1.1562
1.2316
1.2360
1.2365
1.2414
HF
1.1138
1.1843
1.1877
1.1881
1.1921
MP2
1.1502
1.2200
1.2225
1.2231
1.2265
B3LYP
1.1379
1.2065
1.2106
1.2113
1.2157
HF
1.1844
1.1878
1.1882
1.1923
MP2
1.2194
1.2218
1.2225
1.2258
B3LYP
1.2067
1.2107
1.2115
1.2090
6-31G(d)
6-31G(d,p)
6-311+ G(3df,2p)
MP3
1.1340 x
1.2081
1.2100
MP4
1.1580 x
1.2221
1.2221
B1LYP
1.1362 x
1.2052
1.2092
CISD
1.1357 x
1.2059
1.2054
QCISD
1.1445 x
1.2152
CCSD
1.1412 x
1.2134
HF
1.1026
1.1780
MP2
1.1334
1.2068
1.2116 fc
B3LYP
1.1245
1.1986
1.2029
aRef. [9].
dRef. [91].
gRef. [45].
fefrozen core.
bRef. [61].
eRef. [44].
hRef. [94].
Xdone as 6-31G(d).
CRef. [90].
fRef. [43].
iRef. [93].
324
BELL
1.18 M
CO 1.16 .C 01 r. -J
1 .14
M
~--
rO
"
re
B
1.12 1--
O3
CL
x uJ 1.10 =
(9 I.-O3
H
,
Method/Basis
methanal 1.25
c
7"
1.23
I
F L i
-I "0 m
e
o ~iiI
1.21
M
'~
M
l'
o
II r
g
re
1.19
8
x UJ H
to
H
W
,/) H
1.17
Method/Basis
ethanal 1.25
~r
1.23
M
M
mO 1.21
1.19
1.17 t
U.!
,
Method/Basis
FIG. 21. Bond lengths of multiple CO bonds.
UJ
ACCURACY OF MOLECULAR GEOMETRY
325
1.26
propanal M
B
124
5
M
M
"~ 122
g~
~-,
3
1 20
B
B
0
tu H
H
~
118
Method/Basis
1.26
M
I ~ 1.24
~j
~, ~
g~ 0
M
M
1.22
1.20
rs
(9 too
" ~a
UJ 1.18
.
H .
.
.
.
.
.
.
r,~
H
.
Method/Basis
FIG. 21A. Bond lengths of C=O bonds in propanal and acetone.
326
BELL
much better predictions for the basis sets with polarization functions. Also once again the 6-311+G(3df,2p) basis or EBS gives curiously short bond lengths which in smaller molecules was traced to inclusion of the triple-split 6-311G basis. The usual treatment is given of the aldehydic C-H bond length in Table 22 and Fig. 22. The patterns of variation with QC treatment are similar to the pattems seen in Fig. 13 for the - C - H bonds of alkynes or perhaps even more like the =C-H of alkenes, except for the longer bonds in the present case and the greater range of variation. The C-H bond lengths obtained by the MP2 method are very close to r e value in methanal. On the other hand, the results from B3LYP calculations appear to be in better agreement with the experimental structures in ethanal and propanal, but these are r 0 structures. The bond angles involving the carbonyl functional group of aldehydes and ketones are given in Table 23, the three molecules with H-C=O angles being in Fig. 23 and those with C-C=O angles in Fig. 23A. For ethanal, the r z structure value is used as a reference line in both figures, for obvious reasons. As observed with hydrocarbons, the variation of bond angles with QC treatment is quite small, although bond angles obtained by the MP2 method are less consistent than angles from B3LYP calculations, as also seen for many structural parameters. The results with the 6-31 l+G(3df,2p) basis for the H--C=O angles of both methanal and ethanal are out of line with the asymptotic convergence shown by the other basis sets; increasing the basis
ACCURACY OF MOLECULAR GEOMETRY
327
TABLE 22. Aldehydic C-H bond lengths. Basis
Method
methanal
ethanal
propanal
Expt
re
1.1005 a
ro
1.1174 b
1.114 d
1.115 f
rzl.ll61 e
rz 1.128 e
3-21G
HF
1.0832
1.0866
1.0873
6-31G
HF
1.0815
1.0856
1.0866
MP2
1.1002
1.1051
1.1063
B3LYP
1.1026
1.1067
1.1073
HF
1.0916
1.0953
1.0960
MP2
1.1039
1.1089
1.1098
B3LYP
1.1104
1.1142
1.1146
HF
1.0933
1.0971
1.0976
MP2
1.0993
1.1045
1.1050
B3LYP
1.1105
1.1144
1.1145
MP3
1.0992
1.1019
MP4
1.1024
1.1075
B1LYP
1.1087
1.1126
CISD
1.0961
1.0992
QCISD
1.1003
CCSD
1.0995
HF
1.0920
MP2
1.0991
1.1063 ~
B3LYP
1.0991
1.1063
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p)
aRef. [61 ].
dRef. [44].
bRef. [90].
eRef. [43].
CRef. [91 ].
fRef. [451.
fefrozen core.
328
~J
r~
II
9
~4 eq
.--;
t",l
r
o
r.4
~
~
~
I' ~
I~
~
~
r
~'~
~
r
t",,I
,ff
t"q
,n:
~
r
~
t' ~
t-,,i e,i r
e,i r
r
,,n: ,n:
r
r
t.,,i e,i r
~
r
e,,i
~
r
e,i
~
r
e,i
,n:
r
r
,-.; e,i r
,n: ,4
r
~
'.::1"
,4
~
r
9,/
t' ~
r
~
~
~
'~
~
~'~
t"'..
~
t',l
,4
"m
r
r
~
,4
,4
~
r
t",l
,n:
t",,I
r
r
,~/
,n: ,4
r
r
~
,4
r
r
t"q
.-..; ,-;
t",l
r
r
r
e,,i ,.-;
r
!
BELL
tt3
~
~
o t",,I
t"-r
O'~
r
t"q
c5
C'q
O'~
o c'q
~
,4,4
r
r
C'q
r
r
,--;
r
.-;
O,I
r q.~-,
,-;
r
eq
ee~
eq eq ei r
,4,4
c; c'q t___a
,--h
A C C U R A C Y OF M O L E C U L A R G E O M E T R Y
,,--k
== O
0~,,~
eq
+
~b
t,q
,--h
q.4 ,.o
329
330
BELL
1.12
methanal B
B
1.11
B 01 C --I v
"~ 1.10 C 0 m "1-
re
|
0
cL x uJ
1.09
i ~03 3
(9 ~-03
03 ,
~
(0
~k,4, H
1.08
Method/Basis 1.12 B
B
M/ A
A
v
rO 1.11
-
e-
--I "0 r 0 m
/
Mr:
J~ 01
1.10-
=,
31o
r
1.09 -
ethanal
6
M
M
Q.
rn
-
LLI
v"0
H v (9 x-03 | r
x--
03| r
H 1.08 Method/Basis 1.12 B
propanal
A
v
rO 1.11
/
B
Jr ol c _J "o c o m
1.10 H
=,
H
x UJ
0
(9 I-03
1.09 3~H
rj:)
1.08 Method/Basis
FIG. 22. Aldehydic C-H bond lengths.
B
cL "o v (9 03
ACCURACY OF MOLECULAR GEOMETRY
331
123,0 methanal 122.5
< C
122.0
0
B
B
rO
"U
0 rn
B
B
A
.
V
,.
Q.
M
rZ
121.5 03 | to
~. X
(.9
(.9
03 | tO
03 ! tO
LU 121.0 Method/Basis 121.5
3
ethanal
M
rz
B
120,5 B B
0
II1
M
i 119.5 w
(.9
(9
0
118.5 Method/Basis 122.0 propanal
3
121.0
B
B A v
g~ 0 r• 120.0
ro
B
H
H
M
Q. "O v
•
LU
(.9 03
tb 119.0 Method/Basis
FIG. 23. H--C=O bond angles in aldehydes.
(9 03
332
BELL
125.5
ethanal 125.0 3
m., O1 t"a to
m
/
B
<
B 124.5
,.,-,
M
X
O
uJ
II
M CL
r 124.0
H
M
"O v
v
r0
(9
(.9 T--
03 t CO
03 | CO
123.5
Method/Basis 125.5
propanal 125.0 -w O1 e"
H
B
H
B
B "t3 C
0 m
124.5 .A v
O
ro
II
3 ~ / ~ i
r 1 2 4 . 0 --
M (9
X
03 |
uJ
M
"~ (9 9r
t.~ "t3" (9 .r
03
r
o3 i
~
r
123.5
MethodlBasis 123.0
acetone 122.5 r ,..,. O1 < "ID tO
m
122.0
(9
M
B
0 II
r
uJ
H
121.5
Q. v ro
H
M
v
(9
(9
03
03 |
121.0
Method/Basis
FIG. 23A. C--C=O bond angles in aldehydes and acetone.
B
ACCURACY OF MOLECULAR GEOMETRY
333
set beyond 6-31G(d,p) does not necessarily give better structural parameters although that should be valuable for other purposes, especially energetics.
C.
Molecules with Central Oxygen Atoms It is well known that there is a large increase in the bond length of water on inclusion of correlation energy and the large decrease in bond angle on addition of polarization functions to the basis set [3]. This appears to correlate with the electronegativity of the central atom. In this penultimate section, we return to these observations by considering other molecules each with a central oxygen atom. In Table 24 and Fig. 24, the variations of the O-H bond lengths with theoretical treatment for hydrogen peroxide (the O-O bond length is considered in Table 3 and Fig. 3), methanol, and ethanol are compared with those seen in water (from Fig. 1). The jumps in bond dimensions from the HF method to either the MP2 or the B3LYP method are even larger in H20 2 than in water, but the "accepted" experimental r 0 value [22] is in good agreement with both the MP2 or B3LYP results for large bases.
The
methanol diagram compares particularly well with that for water, even for the extra QC methods where the B 1LYP bond length is, as usual, hardly different from B3LYP but here MP3 and MP4 are both longer than MP2.
The
experimental reference line is a little high for ethanol [51 ] in comparison with both H20 [11 ] and methanol [77].
334
BELL
TABLE 24. O-H bond lengths in H20, H202 and alcohols. Basis
Method
Expt
re
H20
H20 2
methanol
t-ethanol
0.9710 d
0.9710 e
0.9578 a
rs r0
0.9581 b
0.967 c
3-21G
HF
0.9667
0.9703
0.9637
0.9661
6-31G
HF
0.9496
0.9544
0.9502
0.9505
MP2
0.9746
0.9868
0.9787
0.9796
B3LYP
0.9759
0.9836
0.9778
0.9780
CISD
0.9727
HF
0.9476
0.9492
0.9463
0.9466
MP2
0.9686
0.9756
0.9700
0.9712
B3LYP
0.9687
0.9737
0.9687
0.9690
CISD
0.9662
HF
0.9431
0.9457
0.9423
0.9424
MP2
0.9608
0.9679
0.9622
0.9631
B3LYP
0.9653
0.9707
0.9652
0.9654
MP3
0.9580
0.9630
0.9584
MP4
0.9609
0.9676
0.9621
B 1LYP
0.9639
0.9689
0.9636
CISD
0.9574
0.9597
0.9553
QCISD
0.9604
0.9610
CCSD
0.9603
0.9609
HF
0.9398
MP2
0.9578
0.9635
0.9576
B3LYP
0.9608
0.9659
0.9598
6-31G(d)
6-31G(d,p)
6-31 l+G(3df,2p)
MP4 CISD
0.9656 0.9517
aRef. [11 ].
eRef. [22].
bRef. [ 10].
dRef. [77].
eRef. [51 ].
ACCURACY OF MOLECULAR GEOMETRY
335
Bond Length
O-H Bond Length o
o
o
o
0'}
",4
OO
I
l
i
I
Expt
p
'
o
o
o
o
oi
o)
,,.4
00
I
I
,
I
Expt
~ ~,
~c0
6-31G
--------7:
6-31G
,- e----.._. 6-31G(d)
--7="
6-31G(d)
="---------4 ~
O
6-31G(d,p)
oo
~ =
-I-
6-31G(d,p)
"I
.% EBS
EBS ~=~,
o
=0
,
O-H Bond Length
O-H Bond Length o
o
01
o)
i
I
Expt
o I
~
o
o
00
o1
o~
i
1
I
o
o
o -,4
7
I
Expt
6-31G 6-31G
= ~-~.~._...
6-31G(d) =#--_...._ -r.
6-31G(d,p)
-r"
o
~o
6-31G~ EBS
~w
FIG. 24. O-H bond lengths in H20 , H202, and alcohols.
336
BELL
Table 25 and Fig. 25 show the variation of angles involving a central oxygen atom, i.e. the HOH angle in water, the COH angle in methanol and ethanol, and the COC angle in dimethyl ether with different treatments. For all such molecules and for basis sets without polarization functions, the bond angles are always unacceptably large, particularly at the HF level.
The
experimental r 0 angle in ethanol [51] is lower than the best MP2 value by 1.5 ~ The angles for water and methanol with the EBS basis set are again suspiciously out of line with other basis sets. Hydrogen peroxide is also included in Table 25 and it is seen that the OOH angle increases much (from 96.5 ~ to 100 ~ for MP2 calculations) with increasing basis set size. Although the O--H bond length behaves like the others in Fig. 24, the equilibrium structure of this molecule is difficult to determine, both experimentally and theoretically, especially its torsional angle and this will be discussed in the next section.
VII.
TORSIONAL ANGLES The only structural parameters considered thus far have been the bond lengths and bond angles of well-behaved molecules with relatively high vibrational frequencies and with no low barriers to inversion or torsion. The calculation of molecular parameters associated with low-frequency vibrational modes pose a more severe test of QC methods. Consideration is given here only to torsional problems of gauche or skew conformers where the torsional angle does not take a specific angle by symmetry (i.e. not cis 0 ~ nor trans 180~
Consideration of the
ACCURACY OF MOLECULAR GEOMETRY
337
TABLE 25. HOH, COH, COC, OOH bond angles. Basis
Method
Expt
re
H20
methanol
t-ethanol
CH3OCH3
H202
108.03 e
105.43 d
111.73 e
99.3 f
104.48 a
rs r0
105.28 b
3-21G
HF
107.68
111.14
110.86
113.98
99.45
6-31G
HF
111.55
113.41
113.70
116.06
101.18
MP2
109.29
110.11
110.29
112.32
96.54
B3LYP
108.30
110.03
110.30
113.11
97.92
CISD
109.37
HF
105.50
109.45
109.66
113.81
102.08
MP2
104.00
107.37
107.71
111.08
98.66
B3LYP
103.65
107.65
107.88
112.31
99.68
6-31G(d)
B1LYP
6-31G(d,p)
6-31 l+G(3df,2p)
107.75
CISD
104.36
HF
105.97
109.65
109.89
113.87
102.31
MP2
103.87
107.34
107.66
110.99
98.65
B3LYP
103.74
107.72
107.99
112.33
99.78
MP3
104.26
107.74
99.86
MP4
103.88
107.26
98.66
B1LYP
103.86
107.83
99.94
CISD
104.34
108.09
100.24
QCISD
104.08
107.60
CCSD
104.09
107.63
HF
106.39
MP2
104.55
108.59
111.19 fe
100.11
B3LYP
105.20
109.12
112.89
100.89
MP4 CISD
99.83 105.08
aRef. [11].
CRef. [77].
eRef. [79].
bRef. [10].
dRef. [51 ].
fRef. [22].
fCfrozen core.
338
BELL
HOH Bond Angle
COH Bond Angle
.r,
.r,
o
o
J
I
1
_.1.
_.=
o
o !
I
I
Expt Expt i
i
6-31G i
~
i
'-r
~"
=<
6-31G(d)
6-31G(d)
0
"l"
.,,,,,,,,,,,,,,,,,,O
i.
:~'"
6-31G(d,p)
6-31G(d,p)
-.e
o
~:~ ~=~ e ~
EBS
EBS I
COC Bond Angle ..=
a
~
o
z
o
-
9
,
a
b
I
I
b
I
'
i
I
I
a ~, Expt
E~
...=
b
o
'
COH Bond Angle .=o .~ ~
o J
o
i
I
Expt
= 6-31G
6-31G O
=
6-31G(d)
~::~ ~i~ =
6-31G(d,p)
m u~
!
~' 6-31G(d) ~:~ "
E
6-31G(d,p)! ~:= EBS
FIG. 25.
HOH, COH, COC, OOH bond angles.
J J
,,,I
...=
_..
.~
.~
o I
I,,
o I
,,,
~
ACCURACY OF MOLECULAR GEOMETRY
339
prediction of the corresponding potential energy functions by different C,C methods is made elsewhere but now we restrict our considerations to geometrical structure parameters only. The first three molecules in Table 26 all show rotational isomerism involving H atoms and they each have a rotational isomer, which has no plane of symmetry. The other molecules contain internal rotating tops with heavier groups. In spite of their small size, the experimental determinations of the structures of the first three are apparently difficult. Although listed in the order of the number of atoms, we consider ethanol first as the variation with torsional angle shows some convergence to an agreed value with both MP2 and B3LYP methods with increasing basis set size. Although the trans and gauche conformers of ethanol have only a small conformational energy difference and a structure is available for the trans conformer [51 ], no experimental structure has been found for the gauche conformer and 62 ~ is chosen for the reference line in the diagram in Fig. 26 as it is the nearest integer value to the angles predicted by the B3LYP and MP2 methods. The pattern of dihedral angle variations for N2H 4 is even more well behaved with no large changes in angle with basis set apart from 3-21G and 6-31G. The scale interval in most of the diagrams for torsional angles is 2 ~ The diagram for N2H 4 is a little like that for the C=C-H angle of ethene (Fig. 17) and here also the predicted angles converge well but not to either the r 0 [23 ] or rg value [24]. Hydrogen peroxide is an archetypal molecule for internal rotational, but in spite of its small size its structure has been difficult tu determine. The torsional
340
O
O
O
o~,,r
O r~
I::I O r
O r~
t'q
,d
ID
~D
O
~D S=
!
O
ID
Z
O ~D
9 ~,,,i r.~
cq
9 ~,,,i
cq oo cq
O~
O
eel9
I'~
O o
o
9
eel
r
~
~
9
.--; .~
t",l
O~
t"~
o
t"..,
0o
,::5 ~
O
I'~
o o9
cq Oo
c,,I
~
C',l
O'~
9
cq
9
.--; ,A r.z t-,i ~
I'...
C',l
o
~D
O'~
~
9
~
t"..-
~
I' ~
O
e,i t",i
,..-4
t""-
oo
e,i ,--; ~
O
~
~
C',l
e,i ~
,i,..,,4
c',l
,.6
~,D
~
C'4
,.6
O
t' ~
O
~
oO
O
'~D
O
t'-,i ~
O
~
,A ,,6
,--4
tr
c'q
,:5 t"q
O~
oO
O~
t",l
t'...
O~
~
,,6
cq
o,,,~
.1,.,i
O
II
.,-,4
BELL
,.--k
9
cq
t.__._.a
.,,.,4
o
II
,..z,
O
'---' .___, .~
tD
9
ACCURACY OF MOLECULAR GEOMETRY
341
M 121.0
H202
M re
119.0 e., < ,,.,.,
e,, 117.0 o
.,,,.
o
LLI
9"r' 0 0 -r
H
H
115.0
Q..
v
(3
(9
I--
r.o
113.0 M B e~e 111.0 .... M e t h o d / B a s i s 94.0 3 N2H4
.== Q.
(3 ,z,,-
r 92.0 <1: ....,
(3
! r.o
rg
o
r
B
,m
M
t.,.
o I.IZ 90.0 Z -r-
B Q,.
H M
v
88.0 Method/Basis 66.0 g-ethanol
H
H
H e,, r
64.0
8 0
62.0
,-.
0 "1-
(3 r
B
B (3 eo
~b 60.0
Methor
FIG. 26.
Dihedral angles of gauche conformers of 4-atom groups with h atoms.
342
BELL
angle of hydrogen peroxide obtained in an older structure determination was 120 ~ [96] but recent studies have gradually reduced that value through 119.1 ~ for a supposed r e value [95] to a much lower value of 111.83 ~ [21 ] or 111.9 ~ [22]. The most recent structure [97] reported has a dihedral angle of 113.7 ~ but it also contains the improbable OOH angle (see Table 25) of 102.32 ~ In parallel with the experimental difficulty and the decrease in the value of the experimental dihedral angle, the theoretical determinations also vary widely. If no polarization functions are included in the basis set, the torsional angle obtained by all methods is 180 ~ but the predicted equilibrium angle also decreases as larger basis sets with polarization functions are used and angles between 121 ~ to 117.6 ~ are found with the 6-31G(d,p). However, the further increase in the basis set to 6-311 +G(3df,2p) with the MP2 method produces a predicted dihedral angle of 111.8 ~ [98]. There is also good agreement between the results of MP2 and B3LYP calculations and with some experimental values [21,22]. The dihedral angle predictions for molecules with heavy atom skeletons are given in Table 26 and diagrams are in Fig. 26A. Butane has a
gauche conformer
for which rg structures have been estimated [32,33] with dihedral angles of 63.0 ~ and 64.9 ~
A potential energy function obtained by fitting asymmetric torsion
transitions from infrared and Raman spectra [99] is found to have a
gauche
minimum at 62.8 ~ in agreement with the lower of these two values. The dihedral angles found by the MP2 method are all smaller than those by the B3LYP method with a difference as large as 2 ~ and the B3LYP value is nearer to the more recent rg value.
ACCURACY OF MOLECULAR GEOMETRY
343
CCCCTorsional Angle
CCC=OTorsional Angle 03 O
b
b
o
o
O) (30 O
o
Expt
Expt
.i~-r
6-31G
6-31G
w
~;~
6-31G(d) 6-31G(d) tQ
6-31G(d,p)
~Q 6t-,
~r
~ 1 3 D
6-31G~~
:3
EBS FCCF Torsional Angle
CCC--CTorsional Angle t,d o o
o ....
I
'
I
I
b
N o
o ......
I
,
i
b ...... '
o ,,
,
t
Expt
Expt
I
6-31G
6-31G(d)
..~.....,~
6-31G
E
z
6-31G(d)
r
& 6-31G(d,p)
~
"-
ee_ O
r
O o :3 m
E <
m6"31G(d'P) tD
FIG. 26A. Dihedral angles of gauche conformers of heavy-atom groups.
344
BELL
For the
gauche conformer
of 1,2-difluoroethane, the r 0 torsional angle [46]
of 71.0 ~ is in agreement with the value found by fitting a potential function to torsional transition frequencies [47]. The MP2 and B3LYP results are both lower than this but again the B3LYP values for basis sets with polarization functions are nearer. In 1-butene, the angles from MP2 calculations are much lower than those from B3LYP calculations and indeed the B3LYP results are very near to the experimental r 0 dihedral angle [34].
In propanal, the difference between the
results of the two main methods employed increases further (scale interval in the propanal section of Fig. 26A is 4 ~ and also increases with increase in size of basis set until the equilibrium angle from an MP2 calculation is 5 ~ lower than that using the B3LYP method. One surprising result shown by the diagrams in Fig. 26A is that the Hartree-Fock results are much like those by the hybrid DFT method. In general, for all of these types of molecules with
gauche conformers
involving heavy-atom skeletons, the B3LYP results are in better agreement with experimental but these are all r 0 or rg structures. For torsional angles involving H atoms, there is little to choose between the methods for the examples selected. The equilibrium angles from MP2 and B3LYP are more or less in agreement with each other, if not with the current accepted experimental values.
VIII. CONCLUSIONS What general rules can be drawn from these calculations and observations? It is well known that bond lengths at the Hartree-Fock level are always too short relative to experimental values but it is surprising how well HF/3-21G and HF/631G match experimental bond lengths if not bond angles. There is nearly always a
ACCURACY OF MOLECULAR GEOMETRY
345
large increase in bond lengths going from the HF method to the MP2 method with the same basis. This is particularly true for bonds with electronegative atoms or multiple bonds and some increases with the 6-31G basis are as large as 0.04 fit. Exceptions to this rule are C-C or C-H bonds with larger basis sets and also bonds with second-row atoms. For basis sets with polarization functions, single bond lengths at the MP2 level are usually shorter than corresponding B3LYP lengths with same basis set, but the opposite is true for all multiple bonds C=C, C-=C, C=N, C--N, C=O and some single bonds, such as C--O, C-F, for most of which the MP2 method makes poorer predictions. It is true even for small hydrides that bond lengths with the MP2 treatment are shorter than corresponding B3LYP values with same basis set but the difference between them is small. Equilibrium bond lengths from MP2 calculations are much more sensitive to basis set size and always shorten greatly on addition of more polarization functions but do show an asymptotic approach to what is often experimental. The NH 3 molecule appears to be the sole exception found. Large extended basis sets, such as 6-31 l+G(3df,2p), do not necessarily give better structural parameters, and in fact in some cases appear to give misleadingly short bonds. Calculations with these are worth while for energetics but probably not for obtaining structural parameters, except in the special case of H20 2. Bond angles obtained by the MP2 procedure, including torsional angles, are also usually smaller than corresponding angles found by the B3LYP method. There are a few exceptions, e.g. the H20 2 torsion angle. The C-C-C and C-C=C
346
BELL
bond angles of hydrocarbons determined at the MP2 level are not very sensitive to basis set size but usually reduce slightly with addition of polarization functions. However, CCH angles seem to regularly "converge from below", i.e. they increase with basis set improvement but angles from 6-31G(d,p) or EBS are only slightly larger than those from 6-31G(d). Most angles with carbon at the centre, XCY, are within +1 o of experimental but angles with electronegative atoms at the centre vary widely and in particular HXH angles show the largest variation with basis set and method of any parameter. It is obvious that the MP2 and B3LYP methods each make good predictions for some structural features and poor for others. The MP2 method always does well for C-C and C-H single bonds if polarization functions are in the basis set. For C=C double bonds, MP2 and B3LYP calculations usually give points in the diagrams lying near but on opposite sides of experimental values. This appears to be true also for F-F, O--O, and N-N but only a small test has been done. The B3LYP model is probably better for C-N, C--O, and C-F bonds or in general for all bonds of C atoms to electronegative atoms, i.e. atoms with lone-pairs (as one might describe N, O and F.) For other multiple bonds apart from C=C, and particularly C-C, C=N, C-N, C=O, bond lengths with the MP2 method are always too long. The HartreeFock method, of course, yields bonds which are always much too short. Vibrational frequencies for modes involving such multiple bonds are always much too high with the HF method and perhaps too low with MP2. The most significant outcome from applying the B3LYP method are its improved results with these multiple bonds. Since the bond lengths are excellent, the vibrational frequencies
ACCURACY OF MOLECULAR GEOMETRY
347
for multiple bonds are very good, and not only for stretching modes but also for bending modes.
In the harmonic approximation, frequencies from B3LYP
calculations are still higher than experiment due to neglect of cubic anharmonicity. An assessment of the quality of predictions of molecular vibration frequencies could constitute another study of this sort. It appears from Section VII that torsional angles obtained at the B3LYP level are also nearer to experimental values.
A study of the correctness of
calculated torsional potential energy functions is related to the problem of predicting gauche conformers but is not part of this study. A number of other QC methods have been considered for the smaller molecules in this study. As usually there are only small differences between MP2 or B3LYP parameter values and those obtained by these other methods, and as they are more expensive in computer resources, they need not be done for geometrical reasons. This study has been mostly a comparison between QC methods and how their results measure up to structural parameters from experimental determinations, mainly those from microwave spectroscopy.
However, in view of the great
consistency of the many predictions by QC methods, attention has been drawn to the inconsistencies of some experimental structural parameter determinations.
348
BELL
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ACCURACY OF MOLECULAR GEOMETRY
7. o
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T. Iijima, Bull. Chem. Soc. Japan, 46, 2311 (1973).
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BELL
27.
J.L. Duncan, D. C. McKean, A. J. Bruce, J. Mol. Spectrosc., 74, 361 (1979).
28.
E. Hirota, Y. Endo, S. Saito, J. L Duncan, J. Mol. Spectrosc., 89, 285 (1981).
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45.
J. Randell, A. P. Cox, K. W. Hillig, M. Imachi, M. S. LaBarge, R. L. Kuczkowski, Z. Naturforseh., 43a, 271 (1988).
46.
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ACCURACY OF MOLECULAR GEOMETRY
351
49. N. C. Craig, A. Chen, K. H. Suh, S. Klee, G. C. Mellau, B. P Winnewisser, M. Winnewisser, J. Phys. Chem., 101, 9302 (1997). 50.
M. Hayashi, M. Fujitake, J. Mol. Struet., 146, 9 (1986).
51.
Y. Sasada, J. Mol. Struct., 190, 93 (1988).
52.
C.C. Costain, J. Chem. Phys., 29, 864 (1958).
53.
J. L. Duncan, D. C. McKean, P. D. Mallinson, R. D. McCulloch, J. Mol. Speetrosc., 46, 232 (1973).
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J.L. Duncan, I. J. Wright, D. van Lerberghe, J. Mol. Speetrosc., 42, 463 (1972).
60.
D. van Lerberghe, I. J. Wright, J. L. Duncan, J. Mol. Spectrosc., 42, 251 (1972).
61.
J.L. Duncan, Mol. Phys., 28, 1177 (1974).
62.
E. Hirota, Y. Endo, S. Saito, K. Yoshida, I. Yamaguchi, K. Machida, J. Mol. Spectrosc., 89, 223 (1981).
63.
W.J. Lafferty, R. J. Thibault, J. Mol. Spectrose., 14, 79 (1964).
64.
E. Kostyk, H. L. Welsh, Can. J. Phys., 58, 912 (1980).
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T. Iijima, H. Jimbo, M. Taguchi, J. Mol. Struet., 144, 381 (1986).
68.
J.E. Wolrab, V. W. Laurie, J. Chem. Phys., 48, 5058 (1968).
352
69.
BELL
B. Beagley, T. G. Hewitt, Trans. Far. Soe., 64, 2561 (1968).
70. R. Pearson, F. J. Lovas, J. Chem. Phys., 66, 4149 (1977). 71. F. J. Lovas, R. D. Suenram, D. R. Johnston, F. O. Clark, E. Tiemann, J. Chem. Phys., 72, 4964 (1980). 72. M. Le Guennec, G. Wlodarczak, J. Burie, J. Demaison, J. Mol. Speetrose., 154, 305 (1992). 73.
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ACCURACY OF MOLECULAR GEOMETRY
353
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A U T H O R INDEX Numbers in brackets are reference numbers and indicate that an author's work is referred to although his name is not cited in the text. Underlined numbers give the page on which the complete reference is listed.
Allinger, N. L., 115 [133]; 123 [208]; 126 [216, 217, 229, 230, 232, 233, 234, 235, 237, 238]; 127 [229, 230, 232, 233, 234, 235]; 128 [237, 238]; 149; 154; 155; 156; 254 [2]; 255 [2]; 263 [2]; 267 [2]; 276 [2]; 322 [2]; 348
A
Aarset, K., 121 [199]; 123 [199]; 126 [223]; 127 [223]; 128 [283, 285]; 132 [283,285]; 153; 155; 159 Abdo, B. T., 137 [326]; 138 [326]; 162
Almenningen, A., 126 [226, 230, 231, 232, 234, 237, 238]; 127 [226, 230, 231, 232, 234]; 128 [237, 238]; 155; 15___66;278 [41 ]; 283 [411; 286 [41 ]; 350
Abiog, O. P., 208 [ 111 ]; 239 Ackermann, O. S., 128 [303]; 134 [303 ]; 160
Amano, T., 207 [82, 88, 91, 92, 96];
Adachi, M., 211 [152, 153]; 220 [305]; 221 [305]; 230 [152, 153]; 241; 250
23__27;238 Andersen, B., 98 [46]; 144
Adamo, C., 255 [6]; 256 [6]; 348,
Anderson, D. G., 137 [332]; 138 [332]; 162
Agopovich, J. W., 217 [249]; 219 [284]; 246; 248
Anderson, M. A., 212 [167]; 242.
Akhmanov, A. S., 106 [103, 104]; 107 [103, 104]; 147
Anderson, T. G., 207 [77, 89]; 237; 238
Alberts, I. L., 121 [176]; 122 [176]; 130 [176]; 137 [326]; 138 [326]; 152; 162
Anderson, W. D., 222 [326]; 25 ! Andres, J. L., 255 [5, 6]; 256 [5, 6]; 348
Albritton, D. L., 183; 234
Andresen, U., 168 [4]; 217 [247]; 233; 246
Aliev, M. R., 170 [11, 13]; 171 [11, 13];233
Apponi, A. J., 216 [229]; 245
AI-Laham, M. A., 255 [5, 6]; 256 [5, 6]; 348
Araki, M., 207 [97]; 23____88
Allen, G., 119 [172]; 152
Arcas, P., 223 [338, 339]; 251
Allen, M. D., 216 [229]; 245
Archipova, E. Y., 126 [231 ]; 127 [231 ]; 155 355
356
AUTHOR INDEX
Arguello, G. A., 115 [ 148]; 116 [ 148]; 150
Balle, T. J., 46 [47]; 82
Banks, R. E., 137 [326]; 138 [326]; 162 Arvedson, M. M., 103; 104 [84]; 146 Barnes, M., 212 [ 165]; 242 Atavin, E. G., 126 [238]; 128 [238]; 156 Attfield, C. J., 137 [326]; 138 [326]; 162; 222 [335]; 251 Attia, G. M., 216 [228]; 218 [269]; 245; 248 Aubke, F., 128 [242]; 129 [242]; 156 Audit, P., 101 [56, 57, 59, 60]; 126 [561; 145 Aumayr, F., 102 [77]; 146 Awata, R., 213 [189]; 243
Barone, V., 255 [6]; 256 [6]; 348 Bartell, L. S., 94 [42]; 100; 101 [62, 63, 64, 65, 66, 67, 68, 69, 70, 71]; 105; 106 [99, 100, 101,102]; 118; 119; 144; 145; 147; 152; 273 [25, 32, 33]; 274 [32]; 276 [33]; 293 [32]; 295 [25]; 302 [32, 33]; 303 [32]; 306 [32, 33]; 307 [33]; 340 [32, 33]; 342 [32, 33]; 349; 350 Bar~, C. P. J., 140 [339, 340]; 163 Baskakova, P. E., 128 [299]; 133 [299]; 160 Bastiansen, O., 128 [280]; 132 [280]; 159
Ayabe, M., 191 [40]; 222 [40]; 235 Ayala, P. Y., 255 [5, 6]; 256 [5, 6]; 348
------
B
Bauder, A., 41 [36]; 81; 168 [3]; 183 [311; 208 [1181; 209 [119, 120, 123, 126, 130]; 216 [224]; 218 [263, 265]; 220 [289, 291]; 221 [320]; 227; 233; 234; 239; 240; 245; 247; 249; 250
Badawi, H. M., 220 [296]; 221 [312]; 249; 250
Baughcum, S. L., 220 [290]; 249
Bailleux, S., 202; 216 [67]; 236
Beagley, B., 88; 121 [185]; 122 [185]; 141; 153; 313 [69]; 314 [69]; 352
Bailly, D., 208 [107]; 239
Beauchamp, R. N., 219 [279, 284]; 248
Baker, J., 255 [5]; 256 [5]; 278 [48]; 281 [48]; 320 [48]; 348; 350
Beckers, H., 216 [228]; 245
Baker, J. G., 316 [76]; 352 Bakhtiar, L., 91 [26]; 143 Bakken, P., 128 [280, 293, 298]; 132 [280]; 133 [293, 298]; 134 [293]; 159; 160 Baldacci, A., 22 [27]; 81
Begemann, M. H., 207 [95]; 238 Begley, R. F., 208 [106]; 238 Bell, S., 221 [307]; 250; 254 [3]; 263; 270; 333 [3]; 348 Belova, I. N., 104 [91, 92]; 147 Belova, N. V., 104 [94]; 147
AUTHOR INDEX
Belyakov, A. V., 94 [43]; 96 [43]; 97 [43]; 99 [43]; 122 [43]; 128 [299]; 133 [299]; 144; 160 Benedict, W. S., 260 [12]; 261 [12]; 271 [12]; 349 Ben-Nun, M., 109 [115]; 140 [337, 339]; 148; 163 Benston, O. J., 91 [24]; 142
357
Blake, A. J., 126 [218]; 128 [254, 266, 281]; 130 [254, 266]; 131 [266]; 132 [218, 281]; 137 [324, 326]; 138 [324, 326]; 155; 157; 158; 159; 162 Blom, C. E., 218 [265]; 220 [289]; 247; 249 Blom, R., 126 [239]; 128 [239]; 156 Blukis, U., 316 [79]; 317 [79]; 337 [79]; 352
Berecz, I., 103 [83 ]; 104 [90]; 146; 147 Bobin, B., 209 [131 ]; 240 Bergmann, L., 93 [40]; 144 Bernath, P. F., 212 [160]; 241 Berry, R. J., 4 [5, 6, 7]; 7; 27 [5, 6, 7]; 30 [5, 6, 7]; 52 [6]; 54 [5, 6, 7]; 56 [5]; 58 [5]; 59 [5]; 63 [5]; 64 [5, 6]; 65 [5, 6]; 67 [6, 7]; 68 [6, 7]; 69 [7]; 70 [7]; 72 [6, 7]; 73 [6, 7]; 74 [5, 6, 7]; 78 [7]; 80; 192 [43, 44, 45]; 226 [45]; 23____55 Bethune, D. S., 128 [313]; 134 [313]; 161
Bocquet, R., 201 [65]; 214 [204]; 236; 244 Bogey, M., 202 [67]; 207 [79, 80]; 212 [168, 174]; 213 [186, 187, 188]; 216 [67]; 223 [337]; 23___66;237; 242; 243; 251 Boggs, J. E., 118; 121 [204]; 123 [204, 206, 207]; 152; 154; 204 [71]; 237 Boh~tka, S., 103 [83]; 104 [90]; 146; 147
Bickelhaupt, F., 128 [303]; 134 [303]; 160
Bohme, D. K., 207 [91 ]; 238
Bickford, C., 126 [236]; 127 [236]; 156
Bohn, R. K., 219 [277]; 248
Binder, H., 128 [309]; 134 [309]; 161
Bolesov, I., 128 [298]; 133 [298]; 160
Binkley, J. S., 255 [5]; 256 [5]; 348
Bolvin, H., 213 [ 186, 187]; 243
Birk, M., 222 [329]; 251
Bondybey, V. E., 169 [6]; 233
Bizzocchi, L., 216 [238]; 246
Bonham, R. A., 91; 93 [21]; 142
Bjamov, E., 216 [223]; 245
Booij, A. S., 115 [137, 144, 145]; 116 [137, 144, 145]; 149; 150
Blair, P. D., 135 [320]; 137 [320, 324]; 138 [320, 324]; 161; 162
Bord6, J., 209 [131]; 240 Bord6, J. C., 209 [131 ]; 240
358
Borisenko, K. B., 128 [292, 295, 305]; 133 [292, 295]; 134 [305]; 159; 160 Bom, M., 2 [ 1]; 80
AUTHOR INDEX
278, 281, 307]; 129 [240, 241, 253]; 130 [176, 262]; 131 [278]; 132 [218, 281]; 133 [278]; 134 [221, 307]; 137 [326]; 138 [326]; 149; 152; 155; 156; 157; 158; 159; 161; 162
Borrmann, H., 128 [309]; 134 [309]; 161
Brassington, N. J., 209 [137]; 240
Borro, A. F., 217 [244]; 229; 246
Brazier, C. R., 212 [ 160]; 241
Borseth, D. G., 221 [312]; 250
Br6ant, C., 209 [ 131 ]; 240
Bos, Y., 215 [213]; 244
Breidung, J., 202 [67]; 209 [127, 129]; 216 [67]; 236; 240
B6siger, H., 135 [315]; 161 Breuning, E., 128 [254]; 130 [254]; 15___7_7 Botham, R. A., 273 [32]; 274 [32]; 293 [32]; 302 [32]; 303 [32]; 306 [32]; 340 [32]; 342 [32]; 350
Breuning, H. J., 128 [288]; 132 [288]; 159
Botschwina, P., 45; 72; 82; 83; 200 [59]; 212 [173, 175]; 217 [245]; 229 [245, 353]; 236; 242; 246; 252
Briggs, P. M., 221 [314]; 250
Botskor, I., 210 [144, 145]; 219 [283]; 231 [ 144, 145]; 240; 241; 248
Brookman, C. A., 137 [325]; 138 [325]; 162
Boucher, D., 283 [54]; 288 [54]; 295 [54]; 316 [75]; 351; 352
Brown, C. A., 128 [313]; 134 [313]; 161
Bouchy, A., 273; 274 [35]; 278 [35, 42]; 279 [42]; 283 [35, 42]; 285 [42]; 286 [42]; 293 [35]; 295 [35]; 300; 301; 303 [42]; 306 [42]; 307 [42]; 309 [42]; 350
Brown, F. X., 212 [ 169]; 242
Broderson, S., 320 [89]; 35____33
Brown, J. M., 212 [180]; 222 [333]; 242; 251
Bouddou, A., 78 [61]; 83; 217 [256]; 247
Brown, R. D., 212 [176]; 216 [226, 233]; 217 [246, 253]; 219 [281]; 242; 245; 246; 247; 248
B6wering, N., 91; 102 [31, 32, 72, 73, 74, 75, 76, 77]; 143; 145; 146
Bruce, A. J., 75 [59]; 83; 273 [27]; 274 [27]; 295 [27]; 350
Boyd, A. S. F., 135 [316]; 137 [316]; 138 [316]; 161
Brupbacher, T., 209 [126]; 219 [277]; 240; 248
Brain, P. T., 115 [135]; 116 [135]; 121 [174, 176]; 122 [174, 176]; 126 [174, 218, 221]; 128 [240, 241, 253, 262,
Bt~hl, M., 115 [165]; 117 [165]; 128 [240, 244, 262, 278, 281, 313]; 129 [240, 244]; 130 [262]; 131 [278]; 132
AUTHOR INDEX
[281]; 133 [278]; 134 [313]; 151; 156; 158; 159; 161
359
Camy-Peyret, C., 223 [338]; 251; 267 [22]; 268 [22]; 333 [22]; 334 [22]; 337 [22]; 340 [22]; 342 [22]; 349
Bunker, P. R., 260 [ 11 ]; 261 [ 11 ]; 271 [11]; 333 [11]; 334 [11]; 337 [11]; 349
Cao, J., 140 [335,337, 338]; 162; 163
Burant, J. C., 255 [6]; 256 [6]; 348
Carballo, N., 222 [330]; 251
Burczyk, K., 195 [51]; 222 [51, 322]; 235; 250
Carli, B., 267 [22]; 268 [221; 333 [221; 334 [22]; 337 [221; 340 [22]; 342 [22]; 349
BUrger, H., 195 [51, 52]; 202 [67]; 208 [108, 110]; 209 [127, 128, 129, 133]; 214 [191,203]; 216 [67, 228, 231]; 217 [241,243]; 222 [51, 52, 322, 325, 332]; 223 [338, 339]; 235; 236; 239; 240; 243; 244; 245; 246; 250; 251 Burie, J., 76 [60]; 78 [60]; 83; 195 [53]; 196 [53]; 218 [53]; 220 [295]; 222 [325]; 236; 249; 251; 283 [54]; 288 [54]; 295 [54]; 314 [72]; 316 [72, 75]; 351;352 Butcher, S. S., 230 [354]; 252
Carter, S., 200 [60]; 215 [216]; 236; 245 Casper, B., 115 [148]; 116 [148]; 128 [242, 272]; 129 [242]; 131 [272]; 150; 156; 158 Cassak, P. A., 221 [314]; 250 Cavalleri, A., 140 [340]; 163 Cazzoli, G., 58 [53]; 82; 207 [74, 85]; 209 [135]; 214 [205]; 215 [211]; 222 [328]; 237; 240; 244; 251
Buxton, L. W., 180 [24]; 234 Centeno, S., 128 [255]; 130 [255]; 157
C Callomon, J. H., 254 [4]; 279 [4]; 283 [56]; 285 [56]; 288 [56]; 289 [56]; 292 [56]; 295 [56]; 296 [56]; 314 [4]; 348; 351 Caminati, W., 208 [118]; 209 [119, 120]; 215 [221]; 239; 245 Cammi, R., 255 [6]; 256 [6]; 348 Campanelli, A. R., 128 [310]; 134 [310]; 161
Challacombe, M., 255 [5, 6]; 256 [5, 6]; 348 Champion, R., 219 [281 ]; 248 Chang, M. C., 208 [117]; 239 Chatterjee, K. K., 221 [307]; 250 Cheeseman, J. R., 255 [5, 6]; 256 [5, 6]; 348 Chen, A., 208 [113]; 210 [113]; 230 [113]; 239; 278 [49]; 281 [49]; 320 [49]; 340 [49]; 351
Campbell, D., 102 [76]; 146 Chen, K., 126 [235]; 127 [235]; 156
360
AUTHOR INDEX
Chen, W., 212 [178]; 242; 255 [5, 6]; 256 [5, 6]; 348
Coffey, D., 186 [34]; 218 [34]; 228 [34, 352]; 234; 252
Chen, W. D., 56 [52]; 82; 195 [52]; 214 [204]; 222 [52]; 236; 244
Cohen, E. A., 214 [202]; 222 [329]; 223 [341]; 244; 251; 252; 260 [14]; 261 [14]; 271 [14]; 349
Cheng, M.-S., 211 [151]; 231 [151]; 241
Colmont, J. M., 219 [285]; 223 [347]; 248; 252
Chickos, J. S., 209 [ 121 ]; 239 Chiu, N. S., 126 [212]; 154 Choe, J.-I., 4 [9]; 22 [9]; 23 [9]; 27 [9]; 54 [9]; 73 [9]; 74 [9]; 78 [9]; 80; 192 [46]; 221 [319]; 235; 250 Christen, D., 115 [128, 151]; 117 [151]; 121 [182]; 122 [182]; 128 [277]; 131 [277]; 149; 150; 153 159; 220 [288]; 249 Christensen, D., 278 [37]; 279 [37]; 283 [37]; 286 [37]; 295 [37]; 303 [37]; 307 [37]; 309 [37]; 350
Cook, R. L, 10 [16]; 11 [16]; 13 [16]; 44 [16]; 80; 170 [10]; 194 [10]; 233; 260 [10]; 261 [10]; 265 [19]; 271 [10, 19]; 334 [10]; 337 [10]; 34 Cooksy, A. L., 193; 196 [49]; 212 [49]; 226 [49]; 227 [49]; 235 Cooper, T. A., 214 [192]; 243 Cordonnier, M., 213 [187, 188]; 243 Cosl6ou, J., 201 [65]; 214 [203]; 216 [231 ]; 223 [347]; 236; 244; 245; 252 Cossi, M., 255 [6]; 256 [6]; 348
Christiansen, J. J., 216 [222]; 245 Chung-Phillips, A., 342 [98]; 353 Chusova, T. P., 104 [93]; 147 Chutjian, A., 10 [17]; 30; 58; 80; 175; 178; 187; 234
Costain, C. C., 24; 25; 26; 27 [29]; 28 [29]; 81; 181; 182; 234; 283 [52]; 288 [52]; 295 [52]; 314 [52]; 316 [52]; 319 [52]; 320 [52]; 351 Costner, T. G., 319 [86]; 352 C6t6, C. Y., 140 [336]; 162
Cioslowski, J., 255 [5, 6]; 256 [5, 6]; 348 Clark, F. O., 314 [71]; 352 Clark, W. C., 319 [83]; 320 [83]; 352 Clifford, S., 255 [6]; 256 [6]; 348 Clouthier, D. J., 46 [48]; 82 Cockman, R. W., 137 [324]; 138 [324]; 162
Coucke, I., 115 [158]; 117 [158]; 151 Cox, A. P., 137 [326]; 138 [326]; 162; 211 [149]; 216 [225]; 218 [260]; 219 [280]; 222 [335]; 230 [149]; 241; 245; 247; 248; 251; 278 [45]; 279 [45]; 322 [45]; 323 [45]; 327 [45]; 329 [45]; 340 [45]; 350 Cradock, S., 135 [320]; 137 [320, 324, 325, 327, 328, 332, 333, 334]; 138 [320, 324, 325, 327, 328, 332, 333,
AUTHOR INDEX
334]; 161; 162; 216 [234]; 218 [269]; 246; 248
361
Davis, M., 93 [36]; 143 Davis, M. J., 126 [221]; 134 [221]; 155
Craig, N. C., 208 [111, 113]; 210 [113]; 219 [279]; 230 [113]; 239; 248; 278 [49]; 281 [49]; 320 [49]; 340 [49]; 351
Davis, R. W., 215 [214]; 216 [223]; 222 [326, 331 ]; 244; 245; 251
Cramb, D. T., 215 [213]; 244
de Meijere, A., 128 [298]; 133 [298]; 160
Cremer, D., 220 [299]; 249 De Natale, P., 223 [342]; 252 Crofton, M. W., 207 [84, 86]; 208 [98, 102]; 237; 238 Cross, P. C., 44 [45]; 82; 112 [123]; 149 Cs~sz~r, A. G., 200 [61]; 213 [187]; 236; 243 Cui, Q., 255 [6]; 256 [6]; 348 Cutin, E. H., 128 [247, 249, 250]; 129 [247, 249, 250]; 157 Cyvin, S. J., 4 [11]; 80; 98 [48, 49, 50]; 114 [125, 126]; 115 [127]; 144; 149
de Neui, R. J., 118; 119; 152 de Smedt, J., 115 [149, 158, 163]; 116 [149]; 117 [158, 163]; 126 [215]; 150; 151; 154 Debye, P., 111; 148 Decius, J. C., 44 [45]; 82; 112 [123]; 149 Deeley, C. M., 222 [327]; 251 Defrees, D. J., 255 [5]; 256 [5]; 348 Delcroix, B., 223 [337]; 251
D Dakkouri, M., 128 [282, 287]; 132 [282, 2871; 159 Dang-Nhu, M., 260 [ 15]; 261 [ 15]; 349 Daniels, A. D., 255 [6]; 256 [6]; 348 Danielson, D. D., 128 [258]; 130 [258]; 157 Dantus, M., 92 [33, 34]; 107 [34]; 140 [34]; 143 Dapprieh, S., 255 [6]; 256 [6]; 348 Dateo, C. E., 200 [58]; 236
Della V6dova, C. O., 115 [147, 150]; 116 [147]; 117 [147, 150]; 128 [247, 249, 250, 251,252, 255, 256, 257, 259, 269, 279]; 129 [247, 249, 250, 251, 252, ]; 130 [255, 256, 257, 259]; 131 [257, 269]; 132 [279]; 150; 157; 158; 159 Della, E. W., 209 [ 121 ]; 239 DeLucia, F. C., 260 [ 10, 13 ]; 261 [ 10, 13]; 265 [19]; 271 [10, 13, 19]; 319 [83]; 320 [83]; 334 [10]; 337 [10]; 349; 352 Demaison, J., 43 [40]; 56 [52]; 57 [40]; 76 [60]; 78 [60, 61]; 82; 83; 119 [171]; 139 [171]; 152; 169 [9]; 174 [9]; 186;
362
187 [9]; 193; 194; 195 [50, 51, 52, 53]; 196 [53]; 200 [9, 64]; 201 [48, 65]; 202 [67]; 204 [72]; 209 [133]; 214 [191, 203, 204]; 216 [67, 231]; 217 [241, 243, 256]; 218 [53]; 219 [48]; 220 [64, 295]; 221 [48]; 222 [51, 52, 325]; 223 [347]; 229 [72]; 233; 235; 236; 237; 240; 243; 244; 245; 246; 247; 249; 251; 252 283 [54, 55]; 285 [55]; 288 [54, 55]; 289 [55]; 292 [55]; 295 [54, 55]; 296 [55]; 314 [72]; 316 [72, 75]; 351; 352 Demuynck, C., 207 [79, 80]; 212 [168, 174]; 213 [186, 187, 188]; 237; 242; 243 Denker, M., 128 [288]; 132 [288]; 159 DesMarteau, D. D., 215 [208, 215]; 244; 245 Destombes, J. L., 207 [79, 80]; 212 [168, 174]; 213 [186, 187, 188]; 237; 242; 243 Di Lonardo, G., 223 [342]; 252 Diehl, P., 134; 135 [315, 317]; 136 [317]; 161 Dixon, T. A., 207 [77, 89]; 237; 238 Domenicano, A., 93 [39]; 123 [207, 208]; 128 [310]; 134 [310]; 135 [317]; 136 [317]; 144; 154; 161 Dommen, J., 209 [ 126]; 240 Donohoe, D., 128 [307]; 134 [307]; 161 Dore, L., 207 [74]; 214 [205]; 216 [238]; 237; 244; 246 Dorofeeva, O. V., 126 [232, 234, 237]; 127 [232, 234]; 128 [237]; 156 Dou, Q., 91 [27]; 143
AUTHOR INDEX
Downs, A. J., 115 [135, 153]; 116 [135]; 117 [153]; 121 [176]; 122 [176]; 128 [241, 253]; 129 [241, 253]; 130 [153, 176]; 149; 151; 152; 156; 157 Dr6an, P., 214 [191]; 243 Dreier, R., 102 [74, 75]; 146 Dreihupl, K.-H., 128 [304]; 134 [304]; 160 Dreizler, H., 168 [4, 5]; 209 [136]; 217 [247]; 219 [280]; 220 [301]; 224 [348]; 233; 240; 246; 248; 249; 252 Drouin, B. J., 221 [314]; 250 Dubrulle, A., 283 [54]; 288 [54]; 295 [54]; 316 [75]; 351; 352 Dubus, H., 212 [ 174]; 242 Duckett, J. A., 216 [230]; 245 Duerst, R. W., 220 [290]; 249 Duncan, J. L, 73 [58]; 75 [59]; 83; 208 [105, 114]; 238; 239; 273 [27, 28]; 274 [27, 281; 283 [53, 59, 60, 611; 285 [531; 286 [59, 61]; 288 [53]; 289 [53]; 293 [28]; 295 [27, 28, 53, 59, 60, 61]; 307 [28]; 309 [59, 61]; 319 [81]; 320 [53, 81 ]; 322 [61 ]; 323 [61 ]; 327 [61 ]; 329 [61]; 350; 351; 352 Dunning, T. H., 257 [7, 8]; 349 Durig, D. T., 201 [66]; 236; 319 [86, 87]; 320 [87]; 352 Durig, J. R., 125 [210]; 154; 167 [1]; 168 [3]; 169 [8]; 172 [14]; 179 [20]; 193 [47]; 201 [66]; 210 [140, 141,146, 148]; 211 [151]; 214 [201]; 215 [208, 215]; 216 [228, 234]; 218 [258, 269]; 219 [287]; 220 [293]; 221 [307]; 230 [146, 148]; 231 [140, 141, 151, 361];
AUTHOR INDEX
233; 234; 235; 236; 240; 241; 244; 245; 246; 247; 24__._88;249; 250; 252; 278 [47]; 319 [86, 87]; 320 [87]; 340 [47]; 342 [99]; 344 [47]; 350; 352; 353 Dymanus, A., 216 [239]; 246
363
Enmi, J., 136 [321]; 16__22 Epple, K. J., 220 [304]; 250 Ermolaeva, L. I., 126 [230]; 127 [230]; 155 Ernst, M., 115 [138]; 116 [138]; 15___00
E Ebizuka, M., 128 [312]; 134 [312]; 161 Ebsworth, E. A. V., 137 [324]; 138 [324]; 162
Esposti, C. D., 58 [53]; 8__22;207 [74, 85]; 209 [135]; 214 [205]; 215 [211]; 216 [238]; 222 [328]; 237; 240; 24___44; 246; 251 Eujen, R., 195 [52]; 222 [52]; 236
Edwards, H. G. M., 208 [116]; 209 [122, 137]; 239; 240 Edwards, T. H., 265 [18]; 271 [18]; 349 Egawa, T., 103 [81]; 115 [159, 160, 161, 162]; 117 [159, 160, 161, 162]; 121 [179, 186, 190, 197]; 122 [81, 179, 186, 190]; 123 [190, 197]; 128 [276, 291, 301, 302, 312]; 131 [276]; 133 [291, 301, 302]; 134 [312]; 136 [321, 323]; 137 [323]; 138 [323]; 14___66;151; 152; 153; 15__99;160; 161; 162; 319 [84]; 320 [84]; 352
Ewbank, J. D., 91 [24, 25, 26, 27, 28, 29, 30]; 107 [106]; 108 [107, 108]; 109 [108, 110, 111, 113, 114]; 126 [212]; 142; 143; 148; 154
-----F~ Fajgar, R., 202 [67]; 216 [67]; 23___66 Faksnes, L. G., 121 [199]; 123 [199]; 153
Farges, J., 101 [58, 61]; 145
Eggers, D. F., 265 [20]; 271 [20]; 319 [82]; 320 [82]; 349; 352
Farkas, O., 255 [6]; 256 [6]; 348
Ellis, M. C., 222 [335]; 251
Faust, W. L., 91 [25, 28]; 108 [107]; 142; 143; 148
Elmes, P. S., 212 [176]; 219 [281]; 242; 248
Favero, L. B., 215 [221 ]; 245
Endo, Y., 208 [112, 114, 117]; 209 [121, 132, 134]; 212 [164, 166, 177]; 213 [185]; 214 [194, 196, 198]; 239; 240; 242; 243; 244; 273 [28]; 274 [28]; 283 [62]; 286 [62]; 293 [28]; 295 [28, 62]; 307 [28]; 350; 351 English, J. T., 91 [27]; 108 [107]; 143; 148
Favero, P. G., 207 [74]; 209 [135]; 215 [211 ]; 237; 240; 244 Fayt, A., 216 [240]; 246 Feder, W., 224 [348]; 252 Fender, R. S., 115 [165]; 117 [165]; 151
364
AUTHOR INDEX
Ferris, A. C., 222 [335]; 251
Fujii, M., 126 [228]; 127 [228]; 155
Fink, A. J., 102 [78]; 146
Fujitake, M., 209 [121]; 211 [150]; 212 [162, 182, 184]; 216 [235, 237]; 218 [267, 270]; 220 [298]; 230 [150]; 231 [150]; 239; 241; 243; 246; 247; 248; 249; 279 [50]; 281 [50]; 320 [50]; 340 [50]; 351
Fink, M., 91 [20, 21, 22, 23, 31, 32]; 93 [21, 40, 41]; 102 [31, 32, 72, 73, 75, 78, 79, 80]; 142; 143; 144; 145; 146 Firth, S., 214 [192]; 215 [209]; 222 [331]; 243; 244; 251 Fischer, E., 210 [144, 145]; 231 [144, 145]; 240; 241 Flaud, J-M., 223 [338, 339, 346]; 251; 252; 267 [22]; 268 [22]; 333 [22]; 334 [22]; 337 [22]; 340 [22]; 342 [22]; 349 Fleisher, H., 128 [244]; 129 [244]; 156 Fletcher, D. A., 212 [ 161 ]; 241 Fliigge, J., 45 [46]; 82; 212 [173]; 229 [353]; 242; 252 Flygare, W. H., 46 [47]; 82; 292 [65]; 351 Fogarasi, G., 125; 128 [312]; 134 [312]; 154; 161 Foresman, J. B., 255 [5, 6]; 256 [5, 6]; 348 Fox, A., 207 [91 ]; 238 Fox, D. J., 255 [5, 6]; 256 [5, 6]; 348 Fox, M. A., 128 [240, 278]; 129 [240]; 131 [278]; 133 [278]; 156; 159 Frey, H., 140 [335]; 162 Friedl, R. R., 222 [329]; 251 Frisch, M. J., 255 [5, 6]; 256 [5, 6]; 348 Fujii, K., 219 [282]; 248
Fujiwara, H., 103 [81]; 121 [186, 190, 197, 201]; 122 [81, 186, 190]; 123 [190, 197, 201]; 128 [276, 291, 301, 312]; 131 [276]; 133 [291, 301]; 134 [312]; 146; 153; 154; 159; 160; 161; 2121159];241 Fukuyama, T., 35 [33]; 36 [33]; 38 [33]; 42 [39]; 43 [33, 39]; 81; 82; 119; 121 [169]; 152; 190 [38, 39]; 191 [39]; 215 [39]; 222 [38]; 235; 267 [24]; 268 [24]; 278 [39]; 279 [39]; 283 [39]; 295 [39]; 302 [39]; 314 [73]; 316 [73]; 339 [24]; 340 [24]; 349; 350; 352 Furuya, K., 218 [261 ]; 247 Fusina, L., 223 [342]; 252
G Gabrys, C. M., 207 [86]; 237 Galabov, B., 210 [ 140]; 231 [ 140]; 240 Galbraith, H. W., 208 [ 106]; 238 Gallucci, J. C., 126 [229]; 127 [229]; 155 Gard, G. L., 128 [261]; 130 [261]; 158 Gatial, A., 126 [226]; 127 [226]; 155 Geise, H. J., 89 [18]; 110; 115 [149, 152, 155, 157, 158, 163]; 116 [149]; 117 [152, 155, 157, 158, 163]; 121 [200]; 123 [200]; 126 [ 117, 215, 229];
AUTHOR INDEX
365
127 [229]; 128 [117]; 142; 148; 150; 151; 153; 154; 155
Golubkov, V. V., 106 [ 103, 104]; 107 [103, 104]; 147
Geiser, J. D., 92; 109 [116]; 143; 148
Gomperts, R., 255 [5, 6]; 256 [5, 6]; 348
Gerke, C., 214 [191]; 243 Gonzalez, C., 255 [5, 6]; 256 [5, 6]; 348 Gerry, M. C. L., 214 [197]; 215 [206, 207, 213, 214, 218]; 216 [223, 230]; 219 [276, 277, 285]; 221 [321]; 222 [323, 324, 326]; 223 [341]; 225 [218]; 226 [218]; 243; 244; 245; 248; 250; 251; 252; 316 [77]; 317 [77]; 333 [77]; 334 [77]; 337 [77]; 352 Ghersetti, S., 22 [27]; 81 Gill, P. M. W., 254 [1]; 255 [5, 6]; 256 [5, 6]; 258 [1 ]; 263 [1 ]; 348; Gillies' C. W., 217 [249]; 219 [279, 284]; 246; 248
Goodman, L., 209 [124, 125]; 239 Goodman, P., 87 [2, 3 ]; 111 [ 121]; 112 [121, 124]; 113 [124]; 118 [124]; 141; 148; 149 Gordy, W., 10 [16]; 11 [16]; 13 [16]; 44 [16]; 80; 170 [10]; 194 [10]; 233; 260 [13]; 261 [13]; 271 [13]; 342 [96]; 349; 353 Gosney, I., 126 [221]; 128 [281]; 132 [281]; 134 [221]; 155; 159
Girichev, A. G., 104 [86, 87]; 146
Gottlieb, C. A., 193 [49]; 196 [49]; 2i2 [49, 173, 175, 178]; 226 [49]; 227 [49]; 235; 242
Girichev, G. V., 104 [86, 87, 91, 93, 94]; 115 [145]; 116 [145]; 146; 147; 150
Gould, R, O., 128 [281]; 132 [281]; 159
Grabow, J.-U., 168 [4]; 220 [301]; 233; Giricheva, N. I., 104 [86, 91, 92, 93, 94]; 115 [145]; 116 [145]; 146; 147; 150
249 Grace, D. S. B., 126 [226]; 127 [226]; 155
Glatzer, D., 102 [80]; 146 Graner, G., 208 [107]; 222 [322]; 239 Goates, S. R., 105 [95, 96]; 106 [99, 100, 101]; 147 Gobbato, K. I., 128 [252, 255, 256, 257, 259, 260]; 129 [252]; 130 [255, 256, 257,259,260]; 131 [257]; 157; 158 Godfrey, P. D., 212 [176]; 216 [226, 233]; 217 [246, 253]; 219 [281]; 242; 245; 246; 247; 248 GSdtirhkzy, L., 104 [85]; 146
Grassi, G., 208 [118]; 209 [120, 123, 126]; 220 [289]; 221 [320]; 239; 240; 249; 250 Gray, D. L., 22 [26]; 81; 260 [17]; 349 Gray, T. G., 128 [309]; 134 [309]; 161 Greatrex, R., 128 [240, 278]; 129 [240]; 131 [278]; 133 [278]; 156; 159
366
Green, J. C., 128 [311]; 134 [31i]; 161 Greene, T. M., 115 [135]; 116 [135]; 121 [176]; 122 [176]; 128 [241, 253]; 129 [241, 253]; 130 [176]; 149; 152; 156; 157
AUTHOR INDEX
I-I Ha, T.-K., 216 [224]; 245
Griffiths, J. H., 220 [288]; 249
Haaland, A., 115 [139, 143, 144, 153]; 116 [139, 143, 144]; 117 [153]; 126 [239]; 128 [239, 288, 299, 303, 311]; 130 [153]; 132 [288]; 133 [299]; 134 [303, 311]; 150; 151; 156; 159; 160; 161; 278 [41]; 283 [41]; 286 [41]; 350
Griffiths, L. E., 126 [220]; 132 [220]; 155
Haas, A., 128 [268, 270]; 131 [268, 270]; 158
Gromov, A. Yu., 128 [294]; 133 [294]; 160
Haeger, J., 102 [80]; 146
Gregory, D., 91 [22]; 142
Groner, P., 210 [140, 146, 148]; 211 [151]; 215 [208, 215]; 216 [228]; 218 [258]; 219 [287]; 230 [146, 148]; 231 [140, 151, 361]; 240; 241; 244; 245; 247; 249; 252
Hagen, K., 115 [138]; 116 [138]; 121 [177, 180, 187, 188, 199]; 122 [177, 180, 187, 188]; 123 [199]; 126 [223, 224]; 127 [223, 224]; 128 [258, 263, 274, 283, 285, 296]; 130 [258, 263]; 131 [274]; 132 [283, 285]; 133 [296]; 150; 152; 153; 155; 157; 158;159; 160
Grotjahn, D. B., 216 [229]; 245 Hahn, J., 211 [ 157]; 230 [157]; 241 Gudeman, C. S., 207 [78, 90]; 237; 238 Haist, R., 128 [261]; 130 [261]; 158 Guelachvili, G., 320 [89]; 353 Guillemin, J.-C., 223 [337]; 251
Halonen, L., 214 [191]; 222 [332]; 223 [339]; 243; 251; 316 [74]; 352
Guirgis, G. A., 220 [293]; 249
Hammel, A., 126 [239]; 128 [239]; 156
Gundersen, G., 99; 144
Handy, N. C., 215 [216]; 245
Gundersen, S., 121 [205]; 123 [205]; 126 [235]; 127 [235]; 154; 156
Hanratty, M. A., 207 [75]; 237 Harder, H., 214 [ 191 ]; 243
Gtinthard, H. H., 41 [36]; 81; 183 [31 ]; 234 Guo, T., 140 [339, 340]; 163 Guyon, P., 278 [42]; 279 [42]; 283 [42]; 285 [42]; 286 [42]; 303 [42]; 306 [42]; 307 [42]; 309 [42]; 350
Hargittai, I., 87 [1 ]; 88; 89; 93 [38, 39]; 103 [17, 83]; 104 [88, 89, 90]; 110 [117]; 115 [140]; 116 [140]; 121 [173]; 123 [206, 207, 208]; 126 [117, 212, 227]; 127 [227]; 128 [117, 297, 305, 306, 310]; 133 [297]; 134 [305, 306, 310]; 135 [317, 319]; 136 [317, 319]; 141; 142; 143; 144; 146; 147; 148; 150; 152; 154; 155; 160; 161
AUTHOR INDEX
Hargittai, M., 87 [1]; 88; 89 [17]; 93 [38] 103 [17]; 104 [85]; 110 [117]; 115 [136, 137, 140, 141, 142]; 116 [136, 137, 140, 141, 142]; 121 [173]; 123 [206]; 126 [117, 212]; 128 [117, 245]; 129 [245]; 135 [319]; 136 [319]; 141; 142; 143; 146; 148; 149; 150; 152; 154; 156; 161 Harman, L. A., 115 [135]; 116 [135]; 149
367
[142, 143, 150, 152, 153, 154, 358, 360]; 231 [150]; 240; 241; 24___66;247; 248; 249; 250; 252; 279 [50]; 281 [50]; 320 [50]; 340 [50]; 351 Head-Gordon, M., 255 [5, 6]; 256 [5, 6]; 348 Hedberg, K., 87 [3]; 98 [45, 47]; 99 [53, 54]; 121 [177, 183, 184, 196]; 122 [177, 183, 184]; 123 [196]; 126 [224, 225]; 127 [224, 225]; 128 [258, 273, 275, 313]; 130 [258, ]; 131 [273, 275]; 134 [313]; 141; 144; 145; 152; 153; 155; 157; 158; 161
Harmony, M. D., 4 [4, 5, 6, 7, 8, 9, 10]; 6 [8]; 7; 20 [24]; 22 [9]; 23 [9]; 27 [4, 5, 6, 7, 8, 9, 10]; 28 [24]; 30 [4, 5, 6, 7]; 31 [24] 35; 36 [4]; 49 [4, 10]; 52 [6]; 54 [4, 5, 6, 7, 8, 9, 10]; 56 [4, 5]; 58 [5]; 59 [4, 5]; 63 [4, 5]; 64 [5, 6]; 65 [5, 6]; 67 [6, 7]; 68; 68 [6]; 69 [7, 8]; 70 [7, 8]; 72 [6, 7]; 73 [6, 7, 9]; 74 [4, 5, 6, 7, 8, 9, 10]; 75 [8]; 78 [7, 8, 9] 80; 169 [7]; 192; 193; 208 [115] 221 [319]; 226 [45, 350]; 227 [350]; 233; 235; 239; 25__00; 252; 273 [29]; 274 [29]; 293 [29]; 295 [29]; 307 [29]; 350
Hedberg, L., 94 [44]; 99 [129]; 121 [183, 184, 196]; 184]; 123 [196]; 128 [258, [258]; 134 [313]; 144; 145; 157; 161
Harper, J., 208 [ 105]; 238
Hegelund, F., 209 [133]; 240
Hartfeld, N., 128 [243]; 129 [243]; 156
Hehre, W. J., 6 [ 12]; 80
Hasegawa, A., 219 [274]; 248
Heineking, N., 215 [207]; 217 [247]; 244; 246
Hassler, K., 115 [138, 165]; 116 [138]; 117 [165]; 128 [274, 308]; 131 [274]; 134 [308]; 150; 151; 158; 161
Helliwell, J., 140 [339]; 163
Hattori, H., 121 [195]; 123 [195]; 153 Hay, P. J., 257 [8], 349 Hayashi, M., 210 [142, 143]; 211 [150, 152, 153, 154]; 216 [235, 236, 237]; 217 [254, 255]; 218 [254, 255, 257, 259, 266, 267, 268; 270]; 219 [271, 273, 274, 275]; 220 [298, 300, 302, 303, 305]; 221 [305, 306, 313]; 230
[54]; 115 122 [183, 313]; 130 149; 153;
Heenan, R. K., 101 [62, 63, 64, 65]; 145; 273 [33]; 276 [33]; 302 [33]; 306 [33]; 307 [33]; 340 [33]; 342 [33]; 350
Hellwege, A. M., 87 [6, 7]; 141 HeUwege, K.-H., 87 [6, 7]; 141 Helminger, P., 260 [10, 13]; 261 [10, 13]; 265 [19]; 271 [10, 13, 19]; 334 [10]; 337 [10]; 349 Henry, W. J., 126 [221 ]; 134 [221 ]; 155 Hensel, K. D., 215 [206]; 244
368
AUTHOR INDEX
Heppeler, A., 137 [326]; 138 [326]; 162 Hermann, T., 128 [287]; 132 [287]; 159 Herrebout, W. A., 201 [66]; 236; 319 [87]; 320 [87]; 342 [99]; 352; 353 Herschbach, D. R., 43 [41, 42]; 44 [41, 42]; 45 [42]; 64 [41, 42]; 82
[28, 34]; 295 [28, 34, 38, 40, 62]; 301 [34, 38]; 303 [34, 38, 40]; 306 [34, 40]; 307 [28, 34]; 309 [34, 40]; 314 [4]; 319 [85, 88]; 320 [85]; 340 [34]; 344 [34]; 348; 349; 350; 351; 352; 353 Hirota, F., 219 [282]; 248 Hizer, T. J., 210 [141]; 231 [141]; 240
Herzberg, G., 2 [2]; 11 [ 18]; 13 [ 18]; 18 [22]; 19 [22]; 20 [2, 18]; 80; 81; 260 [9]; 261 [9]; 265 [9]; 267 [9]; 268 [9]; 322 [9]; 323 [9]; 349
Hnyk, D., 115 [165]; 117 [165]; 128 [244, 307]; 129 [244]; 134 [307]; 151; 156; 161
Heully, J. L., 128 [245]; 129 [245]; 156
Hofmann, M., 115 [135]; 116 [135]; 128 [241, 253, 307]; 129 [241, 253]; 134 [307]; 149; 156; 157; 161
Heusel, H. L., 210 [148]; 230 [148]; 241
Holden, A. N., 15 [19]; 28 [19]; 81
Hewitt, T. G., 313 [69]; 314 [69]; 352
Honjou, H., 212 [183]; 243
oo
Heyl, A., 45 [46]; 82; 200 [59]; 229 [353]; 236; 252 Higginbothan, H. K., 273 [25]; 295 [25]; 349
Hopf, H., 126 [226]; 127 [226]; 155 Horemans, N., 121 [200]; 123 [200]; 153 Horn, A., 126 [237]; 128 [237]; 156
Hilderbrandt, R. L., 89; 93; 115 [132]; 126 [213, 214, 222]; 127 [222]; 142; 144; 149; 154; 155 Hillig, K. W., 211 [149]; 220 [296]; 221 [312]; 230 [149]; 241; 249; 250; 278 [45]; 279 [45]; 322 [45]; 323 [45]; 327 [45]; 329 [45]; 340 [45]; 350
Horn, M., 212 [ 175]; 242 Hosmane, N. S., 128 [309]; 134 [309]; 161 Hovic, J. W., 101 [71]; 145 Howard, A. E., 221 [319]; 250
Hirano, T., 200 [59]; 236 Hirota, E., 168 [2]; 208 [100, 101, 112, 114, 117]; 209 [121, 132, 134]; 212 [162, 163, 166, 182, 184]; 213 [185, 189, 190]; 214 [193, 194, 196, 198]; 230; 233; 238; 239; 240; 241; 242; 243; 244; 252; 254 [4]; 260 [16]; 261 [16]; 273 [28, 34]; 274 [28, 34]; 278 [34, 38, 40]; 279 [4, 34, 38, 40]; 283 [34, 38, 40, 62]; 285 [40]; 286 [34, 40, 62]; 293
Howard, B. J., 2 [3]; 80; 196 [54]; 223 [54]; 236 Hoy, A. R., 260 [11]; 261 [11]; 271 [11]; 333 [11]; 334 [11]; 337 [11]; 349 Hsu, S. L., 292 [65]; 351 Hu, B., 208 [ 111 ]; 239 Huang, J., 101 [67, 69]; 145
AUTHOR INDEX
369
Hubbard, S. D., 216 [225]; 245
Imaishi, H., 230 [358]; 252
Huber, K. P., 18 [22]; 19 [22]; 81; 260 [9]; 261 [9]; 265 [9]; 267 [9]; 268 [9]; 322 [9]; 323 [9]; 349
Inada, N., 218 [257]; 219 [271]; 247; 248
Hulce, M., 128 [293]; 133 [293]; 134 [293]; 160 Hunter, A. N., 216 [233 ]; 246
Inagusa, T., 217 [254, 255]; 218 [254, 255, 259, 266, 267, 270]; 220 [300]; 247; 248; 249 Infinsen, I. M., 278 [41]; 283 [41 ]; 286 [41 ]; 350
Hurlock, S. C., 22 [27]; 81 Hutson, J. M., 2 [3 ]; 80; Hwang, S.-H., 215 [208]; 244
~
Inoue, K., 136 [323]; 136 [373]; 138 [323 ]; 162 Ischenko, A. A., 91 [29, 30]; 106; 107 [103, 104, 106]; 108 [108]; 109 [108, 110, 111, 113, 114]; 143; 147; 148
I------
Ibushi, N., 219 [275]; 248 Ihee, H., 140 [335]; 162 Iijima, K., 115 [130, 131]; 121 [185, 191, 192]; 122 [185]; 123 [191, 192]; 149; 153 Iijima, T., 89; 121 [178, 181,189, 195]; 122 [178, 181, 189]; 123 [195]; 142; 152; 153; 254 [4]; 273 [26]; 278 [44]; 279 [4, 44]; 295 [26]; 313 [67]; 314 [4, 67]; 316 [78]; 317 [78]; 322 [44, 94]; 323 [44, 94]; 327 [44]; 329 [44, 94]; 348; 349; 350; 351; 352; 353
Ishii, H., 221 [318]; 250 Iwasaki, M., 98; 144 Izuha, M., 212 [ 170]; 242
J Jacobs, J., 128 [246]; 129 [246]; 157 J~iger, W., 215 [207]; 219 [277]; 244; 248 Jagod, M.-F., 207 [84, 86]; 237 Jakob, J., 128 [272]; 131 [272]; 158
Ikeda, C., 217 [254]; 218 [254]; 220 [300, 302]; 247; 249
James, R. W., 111; 148
Ikhlef, A., 140 [336, 339]; 162; 163
Janssen, G., 210 [138]; 230 [138]; 240
Imachi, M., 210 [143]; 211 [149]; 220 [296, 297]; 221 [306]; 230 [143, 149]; 240; 241; 249; 250; 278 [45]; 279 [45]; 319 [88]; 322 [45]; 323 [45]; 327 [45]; 329 [45]; 340 [45]; 350; 353
Jebber, K. A., 342 [98]; 353 Jemson, H. M., 215 [213]; 244 Jensen, P., 320 [89]; 353
370
AUTHOR INDEX
Jeon, I., 128 [306]; 134 [306]; 160
Kalasinsky, V. F., 278 [47]; 340 [47]; 344 [47]; 350
Jerzembeck, W., 216 [231 ]; 245 Kaminaka, S., 216 [235]; 246 Jiang, Z., 140 [336, 339]; 162; 163 Kanada, M., 212 [ 171 ]; 242 Jimbo, H., 313 [67]; 314 [67]; 351 Kaneko, K., 218 [261 ]; 247 Jimenez, R., 140 [340]; 163 Karakida, K., 314 [73]; 316 [73]; 352 Johansen, T. H., 115 [138]; 116 [138]; 128 [263, 274]; 130 [263]; 131 [274]; 150; 158
Karle, I. L., 110; 111; 112 [118, 121]; 148
John, E. O., 121 [177]; 122 [177]; 152
Karle, J., 110; 111; 112 [118, 121]; 148
Johns, J. W. C., 209 [124, 125]; 239; 267 [22]; 268 [22]; 333 [22]; 334 [22]; 337 [22]; 340 [22]; 342 [22]; 349
Karlsson, F., 221 [310]; 250
Johnson, B., 255 [6]; 256 [6]; 348
Kasai, P. H., 316 [79]; 317 [79]; 337 [79]; 352
Karolczak, J., 46 [48]; 82
Johnson, B. G., 254; 255 [5]; 256 [5]; 258; 263 [1 ]; 348
Kasai, Y., 212 [ 164]; 242
Johnson, R. D., 210 [140, 141, 146]; 230 [146]; 231 [140, 141]; 240; 241
Kato, H., 216 [225]; 218 [268]; 220 [303]; 221 [313]; 245; 247; 249; 250
Johnston, D. R., 314 [71 ]; 352
Kato, M., 121 [185]; 122 [185]; 15__33; 218 [262]; 247
,
Jones, H., 212 [180]; 223 [344]; 242; 252 Joseph, P. J., 210 [148]; 230 [148]; 241
Jtilicher, B., 115 [ 148]; 116 [ 148 ]; 150
-----
K------
Kacner, M. A., 105 [95, 96, 97, 98]; 147
Kagi, E., 223 [336]; 251
Kattija-Ari, M., 221 [319]; 250 Kawaguchi, K., 207 [73, 88]; 208 [99, 100, 101, 104]; 212 [164]; 213 [190]; 214 [194, 196, 198]; 223 [336]; 237; 238; 242; 243; 244; 251 Kawashima, Y., 214 [193, 196, 198]; 243; 244 Kazerouni, M. R., 121 [ 183, 184]; 122 [183, 1841; 153 Keith, T., 255 [6]; 256 [6]; 348
Kagramanov, N. D., 104 [88, 89, 90]; 146; 147
Keith, T. A., 255 [5]; 256 [5]; 348
Kakar, R. K., 230 [356]; 252
Kennerly, R. E., 103; 146
AUTHOR INDEX
371
Keul, H., 220 [299]; 249 Khachkuruzov, G. A., 340 [95]; 342 [95]; 353 Khalaf, S., 215 [209]; 244
Klapdor, M. F., 128 [260]; 130 [260]; 158 Klee, S., 208 [113]; 210 [113]; 230 [113]; 239; 278 [49]; 281 [49]; 320 [49]; 340 [49]; 351
Khristenko, L. V., 115 [154]; 117 [154]; 151
Kleinpoppen, H., 102 [76]; 146
Kieffer, J. C., 140 [336, 339]; 162; 163
Kloster-Jensen, E., 128 [280]; 132 [280]; 159
Kijima, K., 223 [340]; 251
Knuchel, G., 221 [320]; 250
Kilb, R. W., 278 [43]; 279 [43]; 322 [43]; 323 [43]; 327 [43]; 329 [43]; 350
Koehler, K. F., 128 [264]; 130 [264]; 158
Killian, T. C., 212 [173]; 242
Koga, Y., 214 [202]; 244
Kim, S. B., 92 [33, 34]; 107 [34]; 140 [34]; 143
Kohata, K., 267 [24]; 268 [24]; 339 [24]; 340 [24]; 349
Kimura, M., 126 [228]; 127 [228]; 155; 316 [80]; 317 [80]; 352
Kohl, D. A., 103; 104 [84]; 126 [213, 214]; 146; 154
Kinsey-Nielsen, S., 212 [ 160]; 241
Kojima, E., 128 [291]; 133 [291]; 159
Kirby, C., 180 [25]; 214 [25]; 216 [227]; 234; 245
Kojima, T., 313 [66]; 314 [66]; 351
Kirchmeier, R. L., 121 [177]; 122 [177]; 128 [290]; 133 [290]; 152; 159
Kolonits, M., 104 [85]; 115 [136, 137, 140, 141]; 116 [136, 137, 140, 141]; 128 [245]; 129 [245]; 146; 14____99;150; 156
.,
Kirpichenko, S. V., 128 [294]; 133 [294]; 160 Kishioka, K., 216 [237]; 246 Kisiel, Z., 215 [221 ]; 245 .
.
.
.
Komaromi, I., 255 [6]; 256 [6]; 348 Kon, M., 135 [318]; 136 [318]; 139 [318]; 161
.
Kiyono, H., 128 [301, 302]; 133 [301, 302]; 160 Kizer, K. L., 179 [20]; 234 Klaeboe, P., 126 [226, 237]; 127 [226]; 128 [237]; 155; 156
Konaka, S., 103; 115 [159, 160, 161, 162]; 117 [159, 160, 161, 162]; 121 [186, 190, 197, 201,202, 203]; 122 [81, 186, 190]; 123 [190, 197, 201, 202, 203]; 126 [228]; 127 [228]; 128 [276, 291, 301, 302, 312]; 131 [276]; 133 [291, 301, 302]; 134 [312]; 135 [318]; 136 [318, 321, 322, 323]; 137 [323];
372
AUTHOR INDEX
138 [323]; 139 [318]; 146; 151; 153; 154; 155; 159; 160; 161; 162; 316 [80];
Krisak, R., 121 [ 180]; 122 [ 180]; 152
317 [80]; 352
Krisher, L. C., 180 [23 ]; 234
Kondo, S., 214 [202]; 244; 273; [34]; 278 [34, 40]; 279 [34, 40]; [34, 40]; 285 [40]; 286 [34, 40]; [34]; 295 [34, 40]; 301 [34]; 303 40]; 306 [34, 40]; 307 [34]; 309 40]; 340 [34]; 344 [34]; 350
274 283 293 [34, [34,
Kondou, T., 121 [ 189]; 122 [ 189]; 153 Konikowski, D., 121 [ 182]; 122 [ 182]; 153 Konings, R. J. M., 115 [137, 141, 144, 145]; 116 [137, 141, 144, 145]; 149; 150 Koput, J., 200 [60]; 236; 267 [21]; 342 [21]; 349 Kom, M., 128 [270, 279]; 131 [270]; 132 [279]; 158; 159 Komath, A., 128 [243]; 129 [243 ]; 156 Kostyanovsky, R. G., 128 [264]; 130 [264]; 158 Kostyk, E., 288 [64]; 289 [64]; 292 [64]; 295 [64]; 296 [64]; 351 Kovfics, A., 115 [145]; 116 [145]; 128 [306]; 134 [306]; 150; 160
Kraitchman, J., 10; 58; 61; 80; 174; 175; 176; 187; 234 Krannich, L. K., 128 [299]; 133 [299]; 160
Krohn, B. J., 208 [ 106]; 238 Kronberg, M., 128 [246]; 129 [246]; 157 Krosley, K., 126 [224]; 127 [224]; 155 Kroto, H. W., 180 [25]; 214 [25, 192]; 215 [209, 212, 220]; 216 [227, 232]; 219 [286]; 234; 243; 244; 245; 246; 249 Krupnov, A., 207 [80]; 237 Kuchitsu, K., 4 [11]; 35 [33]; 36 [33]; 38; 42 [39]; 43 [33, 39]; 80; 81; 82; 87 [8, 9, 10]; 94 [42]; 114 [126]; 118; 119; 121 [169, 173, 179]; 122 [179]; 141; 144; 149; 152; 190; 191 [39, 41]; 203; 204; 208 [41]; 210 [147]; 212 [41]; 214 [41]; 215 [39, 41,210]; 217 [41,242]; 222 [38, 41]; 223 [41]; 235; 236; 237; 241; 244; 246; 254 [4]; 267 [24]; 268 [24]; 273 [31]; 278 [39]; 279 [4, 39]; 283 [39, 57]; 285 [57]; 288 [57]; 289 [57]; 295 [39]; 302 [39]; 314 [4]; 319 [84]; 320 [84]; 322 [92]; 339 [24]; 340 [24]; 348; 349; 350; 351; 352; 353 Kuchitsu, T., 314 [73 ]; 316 [73 ]; 352 Kuczkowski, R. L., 20 [24]; 28 [24]; 31 [24]; 81; 169 [7]; 210 [149]; 220 [296, 297, 299]; 221 [312]; 230 [149]; 233; 241; 249; 250; 278 [45]; 279 [45]; 322 [45]; 323 [45]; 327 [45]; 329 [45]; ?:~0 [45]; 350 Kudin, K. N., 255 [6]; 256 [6]; 348
Krasnova, O. G., 115 [145]; 116 [145]; 150
Kiahn, R., 121 [ 182]; 122 [ 182]; 153
Kreiner, W. A., 207 [84]; 237
Kukolich, S. G., 221 [309, 314]; 223 [343]; 250; 252
AUTHOR INDEX
Kuna, R., 209 [ 127]; 240
373
[41]; 45 [42]; 64 [41, 42]; 81; 82; 169 [7]; 233; 313 [68]; 314 [68]; 351
Kurai, N., 121 [203]; 123 [203]; 154
Kuroki, N., 115 [160]; 117 [160]; 151
Le Guennec, M., 43; 56 [52]; 57; 76; 78 [60]; 82; 83; 195 [51, 52]; 196; 200; 214 [204]; 217 [241, 243]; 218 [53]; 220 [64]; 222 [51, 52]; 235; 236; 244; 246; 283 [55]; 285 [55]; 288 [55]; 289 [55]; 292 [55]; 295 [55]; 296 [55]; 314 [72]; 316 [72]; 351; 352
Kutchuk, I., 128 [298]; 133 [298]; 160
Lecoultre, J., 207 [87]; 237
Kuwada, K., 230 [358,~360]; 252
Lee, S. K., 207 [88]; 238
Kuze, N., 115 [159, 160]; 117 [159, 160]; 121 [201, 202]; 123 [201, 202]; 128 [291, 301, 312]; 133 [291, 301]; 134 [312]; 136 [323]; 137 [323]; 138 [323]; 151; 154; 159; 160; 161; 162
Lee, S.-T., 207 [83]; 237
Kurai, T., 128 [302]; 133 [302]; 160 Kurokawa, H., 135 [318]; 136 [318]; 139 [318]; 161
L LaBarge, M. S., 211 [149]; 220 [299]; 230 [149]; 241; 249; 278 [45]; 279 [45]; 322 [45]; 323 [45]; 327 [45]; 329 [45]; 340 [45]; 350
Lee, T. J., 200 [58]; 236 Lees, R. M., 316 [76, 77]; 317 [77]; 333 [77]; 334 [77]; 337 [77]; 352 Legon, A. C., 46 [50]; 82 Lehrich, F., 126 [226]; 127 [226]; 155 Leibold, C., 128 [250, 255, 267, 284]; 129 [250]; 130 [255]; 131 [267]; 132 [284]; 157; 158; 159
LaBrecque, G., 217 [249]; 246 Lennox, J. C., 91 [24]; 142 Lafferty, W. J., 20 [24]; 28 [24]; 31 [24]; 81; 169 [7]; 208 [111 ]; 223 [346]; 233; 239; 252; 254 [4]; 279 [4]; 288 [63]; 295 [63]; 314 [4]; 348; 351
Leroi, G. E., 265 [20]; 271 [20]; 349 Lesarri, A. G., 201 [65]; 236
Lakin, N. M., 222 [333]; 251
Li, Y. S., 179; 211 [151]; 214 [201]; 216 [234]; 218 [269]; 231 [151]; 234; 241; 244; 246; 248
Lam, M. E., 215 [206]; 244
Liashenko, A., 255 [6]; 256 [6]; 348
Laurenson, G. S., 135 [316]; 137 [316]; 138 [316]; 161
Lide, D. R., 16 [21 ]; 18 [21 ]; 23 [28]; 41 [37]; 81; 82; 98 [47]; 144; 182 [28]; 183 [28]; 217 [28]; 234; 273 [30]; 274 [30]; 278 [36, 37]; 279 [37]; 283 [36, 37]; 286 [37]; 293 [30]; 295 [30, 36,
Lahaye, J.-G., 216 [240]; 246
Laurie, V. W., 16 [21 ]; 18 [21 ]; 20 [24]; 28 [24]; 31 [24]; 43 [41, 42]; 44; 44
374
37]; 301 [30]; 302 [30]; 303 [30, 37]; 306 [30]; 307 [30, 37]; 309 [37]; 350 Lieb, M., 128 [271]; 131 [271]; 158 Liebman, J. F., 128 [293]; 133 [293]; 134 [293]; 160 Liedle, S., 126 [233]; 127 [233]; !56 Liedtke, M., 211 [ 157]; 230 [ 157]; 241
AUTHOR INDEX
Long, D. A., 209 [ 137]; 240 Lorencak, P., 220 [296]; 221 [312]; 249; 250 Lovas, F. J., 16 [20]; 17 [20]; 18 [20]; 19 [20]; 20 [23, 24]; 28 [24]; 31 [24]; 52 [23]; 81; 169 [7]; 211 [156, 158]; 214 [200]; 230 [156, 158]; 23___33;241; 244; 314 [70, 71]; 352 Lu, K. J., 128 [309]; 134 [309]; 161
Liescheski, P. B., 137 [327, 329, 330, 331, 332, 333]; 138 [327, 329, 330, 331,332, 333]; 162
Lieschke, J., 91 [32]; 102 [32, 72, 73, 74, 75]; 143; 145; 146 Lii, J.-H., 126 [216]; 155; 254 [2]; 255 [2]; 263 [2]; 267 [2]; 276 [2]; 322 [2]; 348 Lin, C. C., 278 [43]; 279 [43]; 322 [43]; 323 [43]; 327 [43]; 329 [43]; 350
Luo, J. Y., 91 [27]; 108 [ 107, 108]; 109 [108]; 143; 148 Lyutsarev, V. S., 115 [134]; 116 [134]; 149
M
....
Ma, B., 126 [216, 217]; 155; 254; 255; 263 [2]; 267; 276 [2]; 322; 348 Ma, S., 209 [127, 128, 129]; 240
Lin, J., 219 [287]; 249 , Lindenmayer, J., 223 [344, 345]; 252 Little, T. S., 210 [148]; 230 [148]; 24___11; 278 [47]; 319 [86]; 340 [47]; 344 [47]; 3.50;352
Maagdenberg, A. A. J., 210 [138]; 230 [138];240 Machida, K., 208 [ 112]; 23___99;283 [62]; 286 [62]; 295 [62]; 351
Liu, J., 278 [47]; 340 [47]; 344 [47]; 350
Mack, H.-G., 115 [146, 147, 148, 150, 151]; 116 [146, 147, 148]; 117 [147, 150, 151]; 121 [182]; 122 [182]; 128 [246, 247, 248, 249, 250, 251,252, 255, 256, 257, 259, 261,269, 277, 279, 286]; 129 [246, 247, 248, 249, 250, 251, 252]; 130 [255, 256, 257, 259, 261]; 131 [257, 269, 277]; 132 [279, 286]; 150; 153; 157; 158; 159
Liu, R., 108 [107]; 148
Mader, H., 214 [191]; 243
Lobastov, V. A., 91 [30]; 109 [114]; 143; 148
Maekawa, S., 121 [197]; 123 [197]; 153
Litz, M., 217 [241 ]; 246 Liu, D.-J., 207 [83]; 237 Liu, G., 255 [6]; 256 [6]; 348
Lobreyer, T., 121 [ 175]; 122 [175]; 152
Maggard, P., 91 [30]; 143
AUTHOR INDEX
375
Maki, A. G., 20 [24]; 28 [24]; 31 [24]; 81; 169 [7]; 233; 254 [4]; 279 [4]; 314 [4]; 348
Mastryukov, V. S., 88; 93 [14]; 94 [43]; 96 [43]; 97 [43]; 98 [48, 49, 50, 51, 52]; 99 [43]; 107; 115 [133]; 1211204]; 122 [43]; 123 [204]; 126 [230, 231, 232, 234, 235, 237, 238]; 127 [230, 231, 232, 234, 235]; 128 [237, 238, 298]; 133 [298]; 142; 144; 147; 149; 154; 155; 156; 160; 167 [1]; 233
Malick, D. K., 255 [6]; 256 [6]; 348
Mata, F., 222 [330]; 251
Mallinson, P. D., 283 [53]; 285 [53]; 288 [53]; 289 [53]; 295 [53]; 320 [53]; 351
Mathers, T. L., 126 [222]; 127 [222]; 155
Maguire, J. A., 128 [309]; 134 [309]; 161 Maier, J. P., 207 [75, 76, 87]; 237
Maltsev, A. K., 104 [88, 89, 90]; 146; 147
Mathews, C. W., 64 [56]; 83; 170 [12]; 233 Mathur, S. N., 221 [319]; 250
Mann, D. E., 278 [36]; 283 [36]; 295 [36]; 350 Mansour, H. R., 208 [ 116]; 239 Mark, H., 87 [2]; 141 Marsden, C. J., 115 [142, 151]; 116 [142]; 117 [151]; 128 [245]; 129 [245]; 150; 156; 200 [64]; 215 [213]; 220 [64]; 236; 244; 283 [55]; 285 [55]; 288 [55]; 289 [55]; 292 [55]; 295 [55]; 296
Matsumura, C., 42 [39]; 43 [39]; 82; 190 [38, 39]; 191 [39, 40]; 210 [139, 147]; 193 [193]; 214 [195, 199, 202]; 215 [39]; 217 [248]; 218 [262]; 220 [294]; 222 [38, 40]; 230 [139]; 2~5; 240; 241; 243; 244; 246; 247; 249; 278 [46]; 281 [46]; 340 [46]; 344 [46]; 350 .
Matsuoka, M., 121 [ 191]; 123 [ 191 ]; 153
[55];351
Matsuura, H., 209 [ 132, 134]; 240
Marstokk, K.-M., 121 [193, 194, 198]; 123 [193, 194, 198]; 153
Matsuyoshi, T., 115 [ 130, 131 ]; 149
Martin, J. M. L., 200 [56]; 236
McCarthy, M. C., 212 [ 175]; 242 McClelland, J. J., 91 [23]; 142
Martin, R. L., 255 [5, 6]; 256 [5, 6]; 348 Martinez, T. J., 109 [ 115]; 148 Martinsen, K.-G., 115 [134, 139, 143, 144]; 116 [134, 139, 143, 144]; 149; 150 Martner, C. C., 207 [90]; 238
McCulloch, R. D., 283 [53]; 285 [53]; 288 [53]; 289 [53]; 295 [53]; 320 [53]; 351 McDowell, R. S., 208 [106]; 238 McGrady, G. S., 115 [153]; 117 [153]; 121 [176]; 122 [176]; 130 [153, 176]; 151; 152
376
AUTHOR INDEX
McKean, D. C., 75 [59]; 83; 115 [153]; 117 [153]; 130 [153]; 151; 204 [70, 71]; 237; 273 [27]; 274 [27]; 283 [53]; 285 [53]; 288 [53]; 289 [53]; 295 [27, 53]; 320 [53]; 350; 351
Mihill, A., 91 [31, 32]; 102 [3 l, 32, 79]; 143; 146 Millam, J. M., 255 [6]; 256 [6]; 348 Milledge, J., 109 [ 109]; 148
McKellar, A. R. W., 196 [54]; 207 [92]; 223 [54]; 236; 238 McNab, H., 126 [218, 219]; 131 [219]; 132 [218]; 155 McNaughton, D., 208 [ 109]; 216 [226]; 217 [253]; 219 [286]; 239; 245; 247; 249 _
McRae, G. A., 214 [202]; 223 [341]; 244; 252 Meerts, W. L., 216 [239]; 246 Meguellati, F., 222 [322]; 250 Meier, C., 91 [31, 32]; 102 [31, 32, 72, 73, 74, 75]; 143; 145; 146
Miller, G. P., 128 [306]; 134 [306]; 160 Miller, J., 126 [218]; 132 [218]; 155 Miller, R. E., 265 [20]; 271 [20]; 349 Miller, T. A., 46 [51]; 82; 169 [6]; 212 [179]; 233; 242 Mills, I. M., 64 [56]; 83; 115 [129]; 149; 170 [12]; 200 [61]; 215 [216]; 217 [244]; 229; 233; 236; 24__55;246; 316 [74]; 352 Minkwitz, R., 121 [182]; 122 [182]; 128 [267, 272]; 131 [267, 272]; 153; 158 .,
,
Mitrushenkov, A. O., 200 [63 ]; 236 Mellau, G. C., 208 [ 113]; 210 [113 ]; 230 [113]; 239; 278 [49]; 281 [49]; 320 [49]; 340 [49]; 351 Mennucci, B., 255 [6]; 256 [6]; 348
Mitzel, N. W., 126 [220]; 128 [254, 265, 266, 304, 308]; 130 [254, 265, 266]; 131 [266]; 132 [220]; 134 [304, 308]; 155; 157; 158; 160; 161
Merer, A. J., 212 [ 165]; 242
Miwa, T., 115 [130]; 149
Merke, I., 78 [61 ]; 83; 216 [231]; 217 [256]; 219 [280]; 245; 247; 248
Miyazaki, K., 212 [ 181 ]; 24___33
Merritt, F. R., 15 [19]; 28 [19]; 81
Miyazaki, S., 216 [235]; 218 [267]; 2!9 [273]; 246; 247; 248
Metha, G. F., 212 [ 165]; 242
Mkadmi, E. B., 214 [203 ]; 244
Mews, R., 128 [250]; 129 [250]; 157
Moazzen-Ahmadi, N., 207 [92]; 238
Meyer, M., 220 [301 ]; 249
Mohamad, A. B., 214 [201]; 216 [234]; 218 [269]; 24___44;246; 248
- - - . _ . . _ .
Meyer, R., 209 [ 120]; 239 Meyer, V., 209 [ 136]; 240
MSllendal, H., 121 [193, 194, 198, 205]; 123 [193, 194, 198, 205]; 153; 154
AUTHOR INDEX
377
MSller, K., 216 [222]; 245
Muir, J. M., 137 [334]; 138 [334]; 162
Molnfir, J., 115 [136, 137, 141, 142]; 116 [136, 137, 141,142]; 149; 150
Muir, M., 278 [48]; 281 [48]; 320 [48]; 350
Momose, T., 212 [ 166]; 242
MUller, H. S. P., 128 [242]; 129 [242]; 156; 219 [285]; 221 [321]; 222 [322, 323]; 248; 250; 251
Moncur, N. K., 265 [18]; 271 [18]; 349 Montgomery, J. A., 255 [5]; 256 [5]; 348
Murase, N., 121 [179]; 122 [179]; 152 Murtagh, V., 137 [326]; 138 [326]; 162
Montgomery, Jr., J. A. 255 [6]; 256 [6]; 348
Myers, R. J., 316 [79]; 317 [79]; 337 [79]; 352
Monts, D. L., 91 [25, 27, 28]; 142; 143 Mootz, D., 128 [260]; 130 [260]; 158 Morgan, H. W., 260 [13]; 261 [13]; 271 [ 13]; 349 Morimoto, Y., 219 [271 ]; 248 Morino, Y., 43 [44]; 8__22; 112; 113 [124]; 118; 119; 121 [169]; 148; 149; 152; 210 [139]; 213 [189]; 230 [139]; 240; 243; 273 [34]; 274 [34]; 278 [34, 38, 40, 46]; 279 [34, 38, 40]; 281 [46]; 283 [34, 38, 40, 57]; 285 [40, 57]; 286 [34, 40]; 288 [57]; 289 [57]; 293 [34]; 295 [34, 38, 40]; 301 [34, 38]; 303 [34, 38, 40]; 306 [34, 40]; 307 [34]; 309 [34, 40]; 322 [92]; 340 [34, 46]; 344 [34, 46];350;351;353 Moriyama, H., 115 [162]; 117 [162]; 151 Morokuma, K., 255 [6]; 256 [6]; 348 Morrison, C. A., 121 [174]; 122 [174]; 126 [174, 218, 219]; 131 [219]; 132 [218]; 137 [326]; 138 [326]; 152; 155; 162 Mueller, R. P., 207 [94]; 238
~ N Nagashima, M., 101 [62]; 145 Nagayama, A., 211 [154]; 230 [154]; 241 Najib, H., 222 [332]; 251 Nakagawa, J., 210 [142]; 211 [152, 153, 154]; 218 [268]; 219 [275]; 221 [306, 313]; 230 [142, 152, 153, 154, 360]; 240; 241; 247; 248; 250; 252 Nakagawa, T., 322 [92]; 353 Nakanaga, T., 207 [96]; 238 Nakata, M., 35 [33]; 36 [33]; 38 [33]; 42; 43; 43 [33, 39]; 8__1_1;8__22;121; 152; 190 [38, 39]; 191 [39, 41]; 208 [41]; 210 [147]; 212 [41]; 214 [41]; 215 [39, 41, 210]; 217 [41, 242]; 222 [38, 41 ]; 223 [41]; 235; 241; 244; 246; 319 [84]; 320 [84]; 352 Nakata, N., 219 [273]; 248 Nakata, T., 210 [147]; 241
378
Nanaie, H., 210 [ 141 ]; 220 [293]; 231 [ 141 ]; 240; 249
AUTHOR INDEX
Nygaard, L., 175; 178; 187; 234 Nyg~rd, I., 121 [199]; 123 [199]; 153
Nanayakkara, A., 255 [5, 6]; 256 [5, 6]; 348 Naumov, V. A., 115 [156, 163]; 117 [156, 163]; 151 Nefedov, O. M., 104 [88, 89, 90]; 146; 147 Nelson, R., 322 [93]; 323 [93]; 353 Nemes, L., 169 [8]; 194; 195 [50]; 233; 235 Nereson, N. G., 208 [ 106]; 238 Nethercot, Jr., A. H., 32 [31 ]; 34 [31 ]; 81 Newton, S. Q., 115 [159]; 117 [159]; 151 Niederberger, W., 134; 161 Nielsen, C. J., 126 [226]; 127 [226]; 155 Nielsen, H. H., 64 [55]; 83 Niide, Y., 218 [257]; 220 [292]; 247; 249
.O Oberhammer, H., 89; 115 [146, 148, 150, 151]; 116 [146, 148]; 117 [150, 151]; 121 [175, 182]; 122 [175, 182]; 124; 126 [233]; 127 [233]; 128 [242, 243, 246, 247, 248, 249, 250, 251,252, 255, 256, 257, 259, 260, 261,267, 268, 270, 271,272, 277, 279, 284, 286, 290, 309]; 129 [242, 243, 246, 247, 248, 249, 250, 251, 252, ]; 130 [255, 256, 257, 259, 260, 261]; 131 [257, 267, 268, 270, 271, 272, 277]; 132 [279, 284, 286]; 133 [290]; 134 [309]; 142; 150; 152; 153; 154; 156; 157; 158; 159; 161 Obi, K., 212 [ 164]; 242
Ochterski, J., 255 [6]; 256 [6]; 348 Oelfke, W. C., 342 [96]; 353 Ogata, T., 219 [282]; 221 [311]; 248; 250 Ohashi, N., 213 [190]; 243
Nikrahi, A., 128 [240]; 129 [240]; 156
Ohashi, O., 115 [130]; 121 [191, 192]; 123 [191, 192]; !49; 153; 215 [220]; 218 [261]; 245; 247
Nixon, J. F., 215 [212, 220]; 216 [232]; 244; 245; 246
Ohno, K., 209 [132, 134]; 215 [220]; 216 [232]; 240; 245; 246
N6sberger, P., 41; 81; 183; 234
Ohshima, Y., 212 [ 164, 177]; 242
Novick, S. E., 212 [178]; 242
Oka, T., 43 [43, 44]; 82; 119; 152; 207 [83, 84, 86]; 208 [98, 102]; 237; 238; 322 [90, 91 ]; 323 [90, 91 ]; 327 [90, 9 ~]; 329 [90, 91]; 353
Novikov, N. P., 128 [300]; 133 [300]; 160 Novikov, V. P., 115 [ 164]; 117 [ 164]; 151
Okabayashi, T., 219 [278]; 248
AUTHOR INDEX
Okutani, H., 210 [142]; 230 [142]; 240 Oldani, M., 207 [88]; 209 [123]; 216 [224]; 238; 239; 245
379
Parkin, R. P. G, 128 [311 ]; 134 [311 ]; 161
Oleta, A., 128 [268]; 131 [268]; 158
Parsons, S., 121 [174]; 122 [174]; 126 [174, 218, 219]; 128 [265]; 130 [265]; 131 [219]; 132 [218]; 152; 155; 158
Oppenheimer, J. R., 2 [ 1]; 8__.00;
Pasinszki, T., 219 [277]; 248
Ortiz, J. V., 255 [5, 6]; 256 [5, 6]; 348
Patterson, C. W., 208 [ 106]; 238
Osamura, Y., 212 [171]; 242
Paul, D. W., 91 [24, 25, 26, 27, 28]; 142; 143
Osina, E. L., 98 [50, 52]; 144 Osman, O. I., 219 [286]; 249 Oswald, M., 200 [59]; 212 [173]; 229 [353]; 236; 242; 252
Paul, M. A., 16 [21 ]; 18 [21 ]; 41 [37]; 81; 82; 98 [47]; 144; 182 [28]; 183 [28]; 217 [28]; 234 Pearson, R., 314 [70]; 352
Oswald, N., 45 [46]; 82
Peau, E. W., 221 [308]; 250
Oswald, R., 45 [46]; 82; 229 [353]; 252
Pedersen, E. J., 221 [315]; 250
Owrutsky, J. C., 207 [90]; 208 [103]; 238
Pelz, G., 342 [97]; 353 Peng, C. Y., 255 [5, 6]; 256 [5, 6]; 348
Oyamada, M., 210 [143]; 216 [236]; 219 [275]; 220 [303]; 230 [143]; 240; 246; 248; 249
Peng, Z. H., 115 [157]; 117 [157]; 151 Penn, R. E., 180 [24]; 234
Ozeki, H., 207 [97]; 238 Pesch, T. C., 216 [229]; 245
P Page, E. M., 128 [296]; 133 [296]; 160 Palmieri, P., 58 [53]; 8__22;200 [62, 63]; 222 [328]; 236; 251 Papou~ek, D., 170 [13]; 171 [13]; 233 Paquette, L. A., 126 [229]; 127 [229]; 155
Petersen, K. A., 200 [57]; 236 Petersson, G. A., 255 [5, 6]; 256 [5, 6]; 348 Petrova, N. V., 104 [91 ]; 147 Pickett, H. M., 222 [329]; 251; 260 [14]; 261 [14]; 271 [14]; 349 Picton, M. J., 128 [278]; 131 [278]; 133 [278]; 159
380
AUTHOR INDEX
Pierce, L., 40; 811; 180; 180 [23]; 234; 322 [93]; 323 [93]; 353
Przhevalskii, I. N., 340 [95]; 342 [95]; 353
Pierlot, A. P., 217 [253 ]; 247
Pszcz61kowski, L., 215 [221 ]; 245
Pigou, P. E., 209 [ 121 ]; 239
Pulay, P., 6 [ 13 ]; 8__.00;125; 154
Piltch, N. D., 207 [89]; 238
Pulham, C. R., 115 [135]; 116 [135]; 121 [176]; 122 [176]; 126 [220]; 128 [241]; 129 [241]; 130 [176]; 132 [220]; 137 [326]; 138 [326]; 149; 152; 155; 156; 162
Piskorz, P., 255 [6]; 256 [6]; 348 Pliva, J., 209 [124, 125]; 239 Plyler, E. K., 260 [12]; 261 [12]; 271 [12];349
Purves, C., 137 [328]; 138 [328]; 162 Puzzarini, C., 200 [62, 63 ]; 236
Pola, J., 202 [67]; 216 [67]; 236 Polanz, O., 217 [243]; 223 [339]; 246; 251
Pyckhout, W., 110; 121 [200]; 123 [200]; 126 [117, 229]; 127 [229]; 128 [117]; 148; 153; 155
Poll, W., 128 [260]; 130 [260]; 158 Pomelli, C., 255 [6]; 256 [6]; 348
___
Q___
Quade, C. R., 230 [356, 359]; 252 Popik, M. V., 115 [164]; 117 [164]; 128 [300]; 133 [300]; 151; 160 Pople, J. A., 6 [12]; 8_00;254 [1]; 255 [5, 61; 256 [5, 6]; 258 [1 ]; 263 [1 ]; 348 Pote, C. S., 254 [4]; 279 [4]; 314 [4]; 348 Poteau, L., 78 [61 ]; 8__33;217 [256]; 247
R
~
Rablen, P. R., 128 [273]; 131 [273]; 158 Rabuck, A. D., 255 [6]; 256 [6]; 348 Radom, L., 6 [12]; 80; 186 [34]; 218 [34]; 228; 234; 252
Poussigue, G., 260 [ 15]; 261 [ 15]; 349 Radtke, J., 214 [191]; 243 Powell, D. L., 126 [226]; 127 [226]; 155 Praas, H. W., 128 [279]; 132 [279]; 159
Raeker, T., 126 [222]; 127 [222]; 155 Raghavachari, K., 255 [5, 6]; 256 [5, 6]; 348
Preusser, J., 222 [324]; 251 Rahner, A., 222 [332]; 251 Priem, D., 216 [231]; 219 [285]; 245; 248
R~iksi, F., 140 [336, 339, 340]; 162; 163 Ramondo, F., 128 [310]; 134 [310],~1~61
AUTHOR INDEX
Ramsay, D. A., 20 [24]; 28 [24]; 31 [24]; 81; 169 [7]; 233; 254 [4]; 279 [4]; 314 [4]; 348 Randell, J., 211 [149]; 219 [280]; 230 [149]; 241; 248; 278 [45]; 279 [45]; 322 [45]; 323 [45]; 327 [45]; 329 [45]; 340 [45]; 350 Rankin, D. W. H., 88; 115 [135, 156, 165]; 116 [135]; 117 [156, 165]; 121 [174, 176, 200]; 122 [174, 176]; 123 [200]; 126 [174, 218, 219, 220, 221]; 128 [240, 241,244, 253,254, 262, 265, 266, 278, 281, 304, 307, 308]; 129 [240, 241, 244, 253]; 130 [176, 254, 262, 265, 266]; 131 [219, 266, 278]; 132 [218, 220, 281]; 133 [278]; 134 [221, 304, 307, 308]; 135 [316, 319, 320]; 136 [319]; 137 [316, 320, 324, 325, 326, 327, 328, 329, 330, 331,332, 333, 334]; 138 [316, 320, 324, 325, 326, 327, 328, 329, 330, 331,332, 333, 334]; 142; 149; 151; 152; 153; 155; 156; 157; 158; 159; 160; 161; 162 Rao, K. N., 22 [27]; 64 [56]; 81; 83; 168 [2]; 170 [11, 12]; 171 [11]; 183 [30]; 233; 234 Raoult, B., 101 [58, 61]; 145 Rauk, A., 128 [264]; 130 [264]; 158 Raw, T. T., 217 [249]; 246 Reed, D., 128 [307]; 134 [307]; 161 R6ffy, B., 128 [245]; 129 [245]; 156 Rego, C. A., 218 [260]; 247 Rehfuss, B. D., 207 [84, 86]; 208 [102]; 237;238 Reinenmann, S., 128 [267]; 131 [267]; 158
381
Rentzepis, P. M., 92 [35]; 140 [339]; 143; 163 Replogle, E. S., 255 [5, 6]; 256 [5, 6]; 348 Resnik, P. R., 128 [267]; 131 [267]; 158 Rice, D. A., 128 [296]; 133 [296]; 160 Richards, R. J., 219 [285]; 248 Richardson, A. D., 128 [273, 275]; 131 [273,275]; 158 Ritter, S. K., 128 [290]; 133 [290]; 159 Riveros, J. M., 230 [355]; 252 Robb, M. A., 255 [5, 6]; 256 [5, 6]; 348 Robertson, H. E., 88; 115 [135, 156, 165]; 116 [135]; 117 [156, 165]; 121 [176]; 122 [176]; 126 [218, 219, 220, 221]; 128 [240, 241, 244, 253, 254, 262, 265, 266, 278, 281,307, 308]; 129 [240, 241, 244, 253]; 130 [176, 254, 262, 265, 266]; 131 [219, 266, 278]; 132 [218, 220, 281]; 133 [278]; 134 [221, 307, 308]; 137 [325, 326, 327, 331]; 138 [325, 326, 327, 331]; 142; 149; 151; 152; 155; 156; 157; 158; 159; 161; 162 Robiette, A. G., 22 [26]; 81; 115; 152; 214 [197]; 216 [223, 230]; 243; 245; 260 [17]; 349 Robinson, J. S., 216 [229]; 245 Rodler, M., 212 [ 176]; 218 [263]; 220 [291]; 227; 242; 247; 249 Romano, R. M., 128 [251]; 129 [251]; 157
382
AUTHOR INDEX
Romenesko, D. J., 100; 145
S
Rood, A. P., 109 [ 109]; 148
Sadova, N. I., 88; 93 [14]; 142
Rosenbaum, N. H., 207 [90]; 208 [ 103 ]; 238
Saebo, S., 6 [ 13 ]; 80
Rosenblum, B., 32 [31 ]; 34 [31 ]; 81 Rose-Petruck, C., 140 [339, 340]; 163 RSsler, R., 128 [288]; 132 [288]; 159 Ross, A. W., 93; 102 [78]; 144; 146
Saeboe, P. C., 121 [188]; 122 [188]; 128 [285]; 132 [285]; 153; 159 Saito, S., 207 [73, 81, 97]; 208 [112, 114]; 212 [159, 169, 170, 171, 172]; 213 [185, 189, 190]; 237; 238; 239; 241; 242; 243; 273 [28]; 274 [28]; 283 [62]; 286 [62]; 293 [28]; 295 [28, 62]; 307 [28]; 350; 351
Rossetti, C., 208 [ 107]; 239 RSsslein, M., 207 [75, 76, 86, 87]; 237
Rouault, M., 101 [56, 57, 58]; 126 [56]; 145 Roussy, G., 273 [35]; 274 [35]; [35, 42]; 279 [42]; 283 [35, 42]; [42]; 286 [42]; 293 [35]; 295 [35]; [35]; 301 [35]; 303 [42]; 306 [42]; [42]; 309 [42]; 350
278 285 300 307
Rowe, W. F., 220 [290]; 249 Roytburg, A., 196 [55]; 197 [55]; 198 [55]; 200 [55]; 236
Sakaizumi, T., 115 [130]; 121 [191]; 123 [191]; 149; 153; 218 [261]; 221 [318]; 247; 250 Sakurai, Y., 278 [40]; 279 [40]; 283 [40]; 285 [40]; 286 [40]; 295 [40]; 303 [40]; 306 [40]; 309 [40]; 350 Samdal, S., 115 [164]; 117 [164]; 121 [193, 194, 198]; 123 [193, 194, 198]; 126 [235]; 127 [235]; 128 [280, 292, 295, 300, 303]; 132 [280]; 133 [292, 295, 300]; 134 [303]; 151; 153; 156; 159; 160 Sander, S. P., 222 [329]; 251
Riick, H., 193 [48]; 201 [48]; 219 [48]; 221 [48]; 235
Sari-Zizi, N. Ben, 222 [332]; 251
Rtidiger, S., 128 [277]; 131 [277]; 159
Sasada, Y., 279 [51 ]; 281 [51 ]; 316 [51]; 317 [51]; 333 [51]; 334 [51]; 336 [51]; 337 [51]; 339 [51]; 351
Rudolph, H. D., 42; 82; 119 [171]; 139 [171]; 152 169 [9]; 174 [9]; 176; 177; 184; 185; 186 [9, 33]; 187 [9, 33]; 193 [48]; 200 [9]; 201 [48]; 217 [251]; 219 [48, 283]; 220 [301,304]; 221 [48]; 223 [344]; 224 [348]; 233; 234; 235; 247; 24___88;249; 250; 252 Ruland, H., 214 [191]; 243
Saykally, R. J., 46 [49]; 82; 207 [77, 89, 90, 93, 94, 95]; 208 [103]; 237; 238 Scappini, F., 207 [85, 86]; 237 Schaefer, C., 93 [40]; 144 Schaefer, H. F., 126 [216]; 155; 254 [2]; 255 [2]; 257 [8]; 263 [2]; 267 [2]; 276 [2]; 322 [2]; 348; 349
AUTHOR INDEX
383
Schiifer, E., 207 [93 ]; 238
Scuseria, G. E., 255 [6]; 256 [6]; 348
Schiifer, L., 88; 91 [24, 25, 26, 27, 28, 29, 30]; 107; 108 [107, 108]; 109 [108, 110, 111, 113, 114]; 115 [159, 162]; 117 [159, 162]; 125; 126 [211, 212, 215]; 141; 142; 143; 148; 151; 154; 204 [71 ]; 237
Seeger, S., 217 [245]; 229; 246
Schaper, I., 115 [146]; 116 [146]; 128 [248]; 129 [248]; 150; 157
Sekishita, K., 218 [261 ]; 247
Schatte, G., 115 [151]; 117 [151]; 150 Scheffler, D., 115 [146]; 116 [146]; 128 [248]; 129 [248]; 150; 157 Schenzel, K., 115 [165]; 117 [165]; 151
Seip, H. M., 93 [37]; 94 [42]; 98 [46]; 115 [127]; 143; 144; 149 Seip, R., 126 [238]; 128 [238]; 156
Sellers, H. L., 126 [212]; 154 Senzlober, M., 202 [67]; 208 [110]; 216 [67]; 236; 239
Seppelt, K., 115 [151]; 117 [151]; 150 Serenellini, S., 215 [211]; 244
Scherer, W., 115 [153]; 117 [153]; 128 [311]; 130 [153]; 134 [311]; 151; 161
Severi, G., 209 [135]; 240
Schlegel, H. B., 255 [5, 6]; 256 [5, 6]; 348
Shen, Q., 121 [180, 204]; 122 [180]; 123 [204]; 126 [222, 236]; 127 [222, 236]; 152; 154; 155; 156
Schleyer, P. v. R., 6 [12]; 80; 115 [135]; 116 [135]; 128 [241,253, 307]; 129 [241, 253]; 134 [307]; 149; 156; 157; 161
Sheridan, J., 220 [288]; 249 Shida, T., 212 [ 166]; 242
Schmeltekopf, A. L., 183; 234
Shiki, Y., 216 [236]; 219 [274, 275]; 246; 248
Schmidbaur, H., 128 [304]; 134 [304]; 160
Shimoda, T., 121 [195]; 123 [195]; 153
Schmidt, R. E., 230 [359]; 252 Schmiedekamp, C. W., 91 [22]; 142 Schultz, G., 104 [88, 89, 90]; 126 [227]; 127 [227]; 146; 147; 155 Schwendeman, R. H., 20 [24]; 28 [24]; 31 [24]; 41 [37]; 81; 82; 169 [7]; 182; 183; 217 [28]; 233; 234 Scurlock, C. T., 212 [ 161 ]; 241
Shishkov, I. F., 126 [235]; 127 292, 294, 295, [289, 292, 294, 158; 159; 160
115 [ 154]; 117 [ 154]; [235]; 128 [264, 289, 297]; 130 [264]; 133 295, 297]; 151; 156;
Shlykov, S. A., 89 [18]; 104 [91, 92, 93, 94]; 115 [152, 155]; 117 [152, 155]; 142; 147; 151 Shreeve, J. M., 121 [177]; 122 [177]; 128 [290]; 133 [290]; 152; 159
384
AUTHOR INDEX
Siam, K., 91 [28]; 115 [162]; 117 [162]; 126 [212]; 143; 151; 154
Sokolov, V. I., 128 [288]; 132 [288]; 159
Sickafoose, S. M., 221 [309]; 250
Sommer, S., 208 [ 110]; 239
Sim, G. A., 88 [11]; 89 [15, 16]; 93 [37]; 100 [55]; 115 [166]; 141; 142; 143; 145; 152
Sorensen, G. O., 221 [315]; 250
Simeone, G., 222 [328]; 251
Spiridonov, V. P., 106 [ 103, 104]; 107 [103, 104, 106]; 147; 148
Souter, P. F., 115 [135]; 116 [135]; 149
Simeone, S., 58 [53]; 82 Squier, J. A., 140 [339, 340]; 163 Simmons, N. P. C., 215 [212, 220]; 244; 245
Staats, P. A., 260 [13]; 261 [13]; 271 [13];349
Simon, A., 128 [309]; 134 [309]; 161 Stahl, W., 216 [231 ]; 245 Sipachev, V. A., 128 [295]; 133 [295]; 160
Staley, S. W., 221 [319]; 250
Skancke, A., 128 [294]; 133 [294]; 160
Stanley, A. E., 211 [151]; 231 [151]; 241
Skancke, P. N., 115 [ 154]; 117 [154]; 128 [289]; 133 [289]; 151; 159
Stanton, J. F., 106 [102]; 147
Sluyts, E., 115 [155]; 117 [155]; 151 Smart, B. A., 121 [176]; 122 [176]; 126 [218, 219, 220, 221 ]; 128 [254, 265, 266, 304, 308]; 130 [176, 254, 265, 266]; 131 [219, 266]; 132 [218, 220]; 134 [221, 304, 308]; 152; 155; 157; 158; 160; 161
Stefanov, B. B., 255 [5, 6]; 256 [5, 6]; 348 Steimle, T. C., 212 [161]; 241 Stevens, W. J., 211 [156, 158]; 230 [156, 158]; 241 Stewart, J. P., 255 [5]; 256 [5]; 348
Smith, B. J., 186 [34]; 218 [35]; 228 [34, 351,352]; 234; 252 Smith, J. G., 37; 42 [34]; 54 [34]; 57 [34]; 63 [34]; 64 [34]; 65 [34]; 81; 190; 235 Smith, Z., 180; 187 [21]; 220 [290]; 221 [310]; 234; 249; 250 Snyder, L. E., 265 [18]; 271 [18]; 349
Stiefvater, O. L., 211 [155]; 217 [250, 251, 252]; 221 [316, 317]; 231 [155]; 241; 247; 250 Stoicheff, B. P., 283 [56]; 285 [56]; 288 [56]; 289 [56]; 292 [56]; 295 [56]; 296 [56]; 351 Stolevik, R., 98 [46]; 115 [127, 138]; 116 [138]; 121 [187, 188]; 122 [187, 188]; 126 [223]; 127 [223]; 128 [263, 274, 283, 285]; 130 [263]; 131 [274];
AUTHOR INDEX
385
132 [283, 285]; 144; !49; 150; 153; 155; 158; 159
Sutter, D. H., 168 [4]; 209 [136]; 233; 240
Stone, S. C., 208 [ 111 ]; 239 Strahan, S. E., 207 [94]; 238
Sutton, L. E., 88 [11, 12]; 89 [15, 16]; 93 [37]; 100 [55]; 115 [166]; 141; 142; 143; 145; 152
Strain, M. C., 255 [6]; 256 [6]; 348
Suzuki, W., 89; 142
Strand, T. G., 98 [46]; 128 [303]; 134 [303]; 144; 160
Swang, O., 115 [144]; 116 [144]; 128 [299]; 133 [299]; 150; 160
Stratmann, R. E., 255 [6]; 256 [6]; 348
Sykora, S., 135 [315]; 161
Stroh, F., 219 [276]; 248
Szalanski, L. B., 215 [218]; 225 [218]; 226 [218]; 245
Struchkov, Yu. T., 126 [239]; 128 [239]; 156 Styger, C., 209 [130]; 240
Szanto, P. G., 207 [77, 89]; 237; 238 Sz~sz, G., 128 [306]; 134 [306]; 160
Suenram, R. D., 211 [156, 158]; 214 [200]; 230 [156, 158]; 241; 244; 314 [71];352 Sugie, M., 42 [39]; 43 [39]; 82; 190 [38]; 191 [40]; 210 [147]; 214 [193, 195, 199, 202]; 217 [248]; 218 [262]; 220 [294]; 222 [38, 40]; 235; 241; 243; 24___44;24__66;247; 249
------
T
Tack, L. M., 207 [90]; 208 [103]; 212 [176]; 238; 242 Tafipolsky, M. A., 115 [164]; 117 [164]; 121 [205]; 123 [205]; 128 [300, 303, 311]; 133 [300]; 134 [303, 311]; 151; 154; 160; 161
Sugino, M., 115 [161]; 117 [161]; 151 Suh, K. H., 208 [113]; 210 [113]; 230 [113]; 239; 278 [49]; 281 [49]; 320 [49]; 340 [49]; 351
Tagaki, K., 313 [66]; 314 [66]; 322 [91]; 323 [91]; 327 [91]; 329 [91]; 351; 353 ,
Taguchi, M., 313 [67]; 314 [67]; 351 Sullivan, J. F., 216 [234]; 231 [361]; 246; 252
Takemura, M., 316 [80]; 317 [80]; 352
Sundermeyer, W., 121 [175]; [175]; 152
122
Takenaka, T., 121 [189]; 122 [189]; 153
Suslova, E. N., 128 [295]; 133 [295]; 160
Takeo, H., 42 [39]; 43 [39]; 82; 1O0 [38, 39]; 191 [39, 40]; 210 [139, 147]; 214 [193, 195, 199, 202]; 215 [39]; 217 [248]; 218 [262]; 220 [294]; 222 [38,
386
40]; 230 [139]; 235; 240; 241; 243; 244; 246; 247; 249; 278 [46]; 281 [46]; 340 [46]; 344 [46]; 350 Takeuchi, H., 115 [159, 160, 161, 162]; 117 [159, 160, 161, 162]; 121 [186, 190, 197, 201, 202, 203]; 122 [186, 190]; 123 [190, 197, 201, 202, 203]; 126 [228]; 127 [228]; 128 [291, 301, 302, 312]; 133 [291, 301, 302]; 134 [312]; 135 [318]; 136 [318, 321, 322, 323]; 137 [323]; 138 [323]; 139 [318] 151; 153; 154; 155; 159; 160; 161; 162 Tam, H. S., 4 [9]; 22 [9]; 23 [91; 27 [9]; 54 [9]; 73 [9]; 74 [9]; 78 [9]; 80; 192 [46]; 235 Tamagawa, K., 316 [80]; 317 [80]; 352
AUTHOR INDEX
Taylor, P. R., 200 [56]; 216 [226]; 236; 245 Taylor, W. H., 4 [4, 5]; 27 [4, 5]; 30 [4, 5]; 35 [4]; 36 [4]; 49 [4]; 54 [4, 5]; 56 [4, 5]; 58 [5]; 59 [4, 5, 54]; 62 [54] 63 [4, 5]; 64 [5]; 65 [5]; 74 [4, 5]; 80; 83; 192 [42, 43]; 217 [253]; 235; 247 Tetsuda, Y., 218 [261 ]; 247 Thaddeus, P., 193 [49]; 196 [49]; 212 [49, 173, 175, 178]; 226 [49]; 227 [49]; 235; 242 Thibault, R. J., 288 [63 ]; 295 [63 ]; 351 Thiel, W., 128 [244]; 129 [244]; 156; 202 [67]; 209 [127, 129]; 216 [67]; 236; 240
Tamassia, F., 214 [205]; 244 Tanabe, M., 121 [201,202]; 123 [201, 202]; 136 [323]; 137 [323]; 138 [323]; 15___44;140; 106; 162
Thomas, C. J., 128 [309]; 134 [309]; 161 Thome, L. R., 214 [200]; 244
Tanaka, K., 207 [82]; 212 [181, 183]; 219 [278]; 237; 243; 248
Tiemann, E., 16 [20]; 17 [20]; 18 [20]; 19 [20]; 81; 314 [71]; 352
Tanaka, T., 212 [181, 183]; 213 [189]; 223 [340]; 243; 251; 319 [88]; 353
Tien, E. H., 219 [285]; 248
Tang, J., 207 [81 ]; 237
Timofeeva, T. V., 126 [239]; 128 [239]; 156
Tang-Pedersen, A., 221 [315]; 250
Titov, V. A., 104 [93 ]; 147
Tanimoto, M., 213 [189]; 219 [278]; 243; 248; 283 [57]; 285 [57]; 288 [57]; 289 [57]; 351
Tokue, I., 278 [39]; 279 [39]; 283 [39]; 295 [39]; 302 [39]; 350
Tanoura, M., 212 [ 181 ]; 243 Tarrago, G., 260 [ 15]; 261 [ 15]; 349
Tolley, C. L., 218 [258]; 247
Tomasi, J., 255 [6]; 256 [6]; 348
Tarroni, R., 200 [62, 63 ]; 236
Townes, C. H., 15 [19]; 28 [19]; 32 [31 ]; 34 [31 ]; 81
Tatsunami, R., 128 [302]; 133 [302]; 160
Townson, N., 121 [176]; 122 [176]; 130 [ 176]; 152
AUTHOR INDEX
387
Traetteberg, M., 126 [225, 226, 231]; 127 [225, 226, 231]; 128 [258, 280, 293, 298]; 130 [258]; 132 [280]; 133 [293,298]; 155; 157; 159; 160
Vabischevich, M. G., 106 [103]; 107 [103]; 147
Travers, M. J., 212 [ 178]; 242
Vacherand, J. M., 220 [295]; 24___99
Tremmel, J., 89 [17, 18]; 103 [17, 83]; 104 [88, 89, 90]; 115 [149, 157]; 116 [149]; 117 [157]; 142; 146; 147; 150; 151
Valente, E. J., 101 [63 ]; 145
Trickey, P., 128 [281]; 132 [281]; 159 Trowell, P. L., 214 [201 ]; 244 Trucks, G. W., 255 [5, 6]; 256 [5, 6]; 348 Truter, M. R., 88 [ 12]; 141 Tsuchiya, M. J., 212 [183]; 243
V
van Alsenoy, C., 115 [149, 152, 155, 157, 158, 163]; 116 [149]; 117 [152, 155, 157, 158, 163]; 121 [200]; 123 [200]; 126 [215]; 150; 151; 153; 154 van de Heisteeg, B. J. J., 128 [303]; 134 [303]; 160 van der Veken, B. J., 115 [ 149, 155]; 116 [149]; 117 [155]; 150; 151; 201; 219 [287]; 236; 249; 319 [87]; 320 [87]; 342 [99]; 352; 353
Tsuchiya, S., 278 [44]; 279 [44]; 322 [44]; 323 [44]; 327 [44]; 329 [44]; 350
van Eijck, B. P., 182; 183; 210 [138]; 218 [264]; 220 [295]; 224 [29]; 230 [138]; 234; 240; 247; 249
Tsunekawa, S., 267 [23]; 268 [23]; 339 [23 ]; 340 [23 ]; 349
van Goethem-Wiersma, T. J., 210 [138]; 230 [138]; 240
Tuzova, L. L., 115 [163]; 117 [163]; 151
van Lerberghe, D., 283 [59, 60]; 286 [59]; 295 [59, 60]; 309 [59]; 351
Typke, V., 183; 184; 187 [32]; 219 [272]; 224 [272, 348]; 225 [349]; 234; 248; 252; 316 [75]; 352
van Loock, J., 89 [18]; 142 van Opheusden, J., 218 [264]; 247 van Schaik, M. M. M., 218 [264]; 247
U
van Vaals, J. J., 216 [239]; 246
Ulic, S. E., 115 [148]; 116 [148]; 128 [256, 259, 260]; 130 [256, 259, 260]; 150; 157; 158
van Zoeren, E., 218 [264]; 247
Ulrich, W., 196 [55]; 197 [55]; 198 [55]; 200 [55]; 236
Vandenhaute, R., 216 [240]; 246
Vance, W. N., 106 [101]; 147
Vanhouteghem, F., 126 [215 ]; 154
388
AUTHOR INDEX
Varberg, T. D., 222 [333]; 251
Waiters, A., 223 [337]; 251
Vefghi, P., 137 [325]; 138 [325]; 162
Waltner, H., 102 [80]; 146
Verma, U. P., 216 [222]6; 245
Wang, A., 342 [99]; 353
Veme, H. P., 115 [153]; 117 [153]; 130 [153]; 151
Wang, Y., 115 [149, 158]; 116 [149]; 117 [158]; 128 [309]; 134 [309]; 150; 151; 161
Vilkov, L. V., 88; 93 [14]; 98 [50]; 115 [154, 164]; 117 [154, 164]; 128 [264, 289, 292, 294, 295, 297, 300]; 130 [264]; 133 [289, 292, 294, 295, 297, 300]; 142; 144; 151; 158; 159; 160 Vogelsanger, B., 209 [119, 120]; 221 [320]; 239; 250
Vogt, J., 87 [4, 5]; 104 [86, 87]; 115 [134, 139, 143]; 116 [134, 139, 143]; 135 [315]; 141; 146; 149; 150; 161; 215 [219]; 216 [222]; 245 Vogt, N., 104 [86, 87]; 115 [134, 139, 140, 143]; 116 [134, 139, 140, 143]; 146; 149; 150 Volden, H. V., 115 [134, 144, 153]; 116 [134, 144]; 117 [153]; 121 [205]; 123 [205]; 126 [235]; 127 [235]; 128 [288, 303]; 130 [153]; 132 [288]; 134 [303]; 149; 150; 151; 154; 156; 159; 160
Wanitschek, M., 128 [309]; 134 [309]; 161 Warner, H. E., 207 [91, 92]; 238 Watanabe, R., 136 [321]; 162 Waterfeld, A., 128 [261]; 130 [261]; 158 Watson, J. K. G., 31; 32 [30]; 33; 33 [32]; 34 [30, 32]; 35 [30]; 36 [30]; 37; 38; 42; 46; 47; 48; 49 [30]; 51; 53; 54 [30, 34]; 57 [30, 34]; 62; 63; 64; 65; 77; 81; 170; 171 [11]; 172; 181; 188; 189; 190; 193 [36, 49]; 196; 196 [36]; 197; 198; 200; 212 [49]; 223 [54]; 225; 226 [49]; 227 [49]; 233; 235; 236 Weber, P. M., 92; 109 [ 115, 116]; 140 [339]; 143; 148; 163 Weidlein, J., 126 [239]; 128 [239]; 156
Volkmer, M., 91 [31, 32]; 102 [31, 32, 72, 73, 74, 75]; 143; 145; 146
Welch, A. J., 128 [307]; 134 [307]; 161
Vrtilek, J. M., 212 [178]; 242
Welsh, H. L., 288 [64]; 289 [64]; 292 [64]; 295 [64]; 296 [64]; 351
W Wada, R.-I., 121 [181]; 122 [181]; 152
Westwood, N. P. C., 219 [277]; 248 Wiberg, K. B., 6 [14]; 8___00;128 [273]; 131 [273]; 158
Walker, B. C., 140 [340]; 163 Widers, C. W., 140 [340]; 163 Wallasch, M., 220 [299]; 249 Widmer, R., 209 [ 123]; 239 Walsh, R., 128 [296]; 133 [296]; 160
AUTHOR INDEX
Wiedenmann, K. H., 193 [48]; 201 [48]; 217 [246]; 219 [48, 283]; 221 [48]; 235; 246; 248 Williamson, J. C., 92 [33, 34]; 107 [34]; 109 [112]; 140 [34, 335]; 143; 148; 162 Williamson, J. M., 212 [179]; 242 Willner, H., 115 [146, 147, 148, 151 ]; 116 [146, 147, 148]; 117 [147, 151]; 128 [242, 246, 248, 256, 260]; 129 [242, 246, 248]; 130 [256, 260]; 150; 156; 157; 158; 208 [108]; 215 [206]; 217 [241]; 222 [322]; 223 [338]; 239; 244; 246; 250; 251
389
Winter, H. P., 102 [77]; 146 Wlodarczak, G., 43 [40]; 56 [52]; 57 [40]; 76 [60]; 78 [60, 61]; 82; 83; 119 [171]; 139 [171]; 152; 169 [9]; 17419]; 186 [9]; 187 [9]; 193 [48]; 195 [51, 52, 53]; 196 [53]; 200 [9, 64]; 201 [48]; 204 [72]; 214 [204]; 217 [241, 243, 256]; 218 [53]; 219 [48, 285]; 220 [64]; 221 [48]; 222 [51, 52, 325]; 229 [72]; 233; 235; 236; 237; 244; 246; 247; 248; 251; 283 [55]; 285 [55]; 288 [55]; 289 [55]; 292 [55]; 295 [55]; 296 [55]; 314 [72]; 316 [72]; 351; 352 Wodarczyk, F., 283 [58]; 285 [58]; 292 [58]; 296 [58]; 340 [58]; 351
Wilson, A. J. C., 93 [41 ]; 144 Wold, K. H., 128 [263]; 130 [263]; 158 Wilson, E. B., 44; 82; 112; 149; 180; 187 [21]; 220 [290]; 230 [354, 355]; 234; 249; 252; 278 [43]; 279 [43]; 283 [58]; 285 [58]; 292 [58]; 296 [58]; 322 [43]; 323 [43]; 327 [43]; 329 [43]; 340 [58]; 350; 351
Wolrab, J. E., 313 [68 ]; 314 [68 ]; 351 Wong, M. W., 255 [5, 6]; 256 [5, 6]; 348 Wong, T. C., 100; 145
Wilson, K. R., 109 [115]; 140 [336, 337, 338, 339, 340]; 148; 162; 163 Winnewisser, B. P, 208 [113]; 210 [113]; 222 [334]; 226 [334]; 230 [113]; 239; 251; 278 [49]; 281 [49]; 320 [49]; 340 [49]; 351 Winnewisser, G., 211 [ 157]; 215 [218]; 216 [223]; 225 [218]; 226 [218]; 230 [157]; 241; 245; 316 [77]; 317 [77]; 333 [77]; 334 [77]; 337 [77]; 342 [97]; 352;353 Winnewisser, M., 208 [ 113 ]; 210 [113]; 215 [218, 219]; 216 [222]; 219 [276]; 221 [308]; 225 [218]; 226 [218]; 230 [113]; 239; 245; 248; 250; 278 [49]; 281 [49]; 320 [49]; 340 [49]; 351
Woods, R. C., 207 [77, 78, 89]; 237; 238 Wright, I. J., 283 [59, 60]; 286 [59]; 295 [59, 60]; 309 [59]; 351 Wu, G., 115 [152, 155, 157]; 117 [152, 155, 157]; 151
-----
X-----
Xu, L.-H., 208 [ 111 ]; 239
.y
Yakovlev, V. V., 140 [339]; 163 Yamada, C., 212 [163]; 241
390
AUTHOR INDEX
Yamada, K., 215 [217, 218]; 225 [217, 218]; 226 [217, 218]; 245; 322 [92]; 353 Yamada, K. M. T., 211 [157]; 230 [157]; 241; 342 [97]; 353 Yamaguchi, I., 208 [ 112]; 218 [261]; 239; 247; 283 [62]; 286 [62]; 295 [62]; 351 Yamamoto, S., 121; 152; 207 [73 ]; 212 [169, 170, 171, 172]; 215 [210]; 237; 242; 244; 319 [84]; 320 [84]; 352 Yamanouchi, K., 121 [ 179]; 122 [ 179]; 152; 210 [147]; 241 Yan, X., 128 [311 ]; 134 [311 ]; 161 Yang, L., 126 [237]; 128 [237]; 156 Yano, Y. F., 89; 142 Yesipenko, A. V., 128 [294]; 133 [294]; 160 Yoshida, K., 208 [112]; 239; 283 [62]; 286 [62]; 295 [62]; 351 Yoshikawa, M., 219 [282]; 248 Yu, L., 212 [179]; 242
9
Z
Zakrzewski, V. G., 255 [5, 6]; 256 [5, 6]; 348 Zare, R. N., 183; 234 Zauer, K., 128 [305]; 134 [305]; 160 Zavgorodnii, V. S., 94 [43]; 96 [43]; 97 [43 ]; 99 [43 ]; 122 [43 ]; 144 Zech, S., 102 [80]; 146
Zewail, A. H., 91; 92 [33, 34]; 107; 109 [112]; 140 [34, 335]; 143; 148; 162 Zgurskii, A. V., 106 [103, 104]; 107 [103, 104]; 147 Zhao, W., 201 [66]; 236; 319 [87]; 320 [87]; 352 Zhen, M., 210 [148]; 230 [148]; 241 Zink, R., 128 [308]; 134 [308]; 161 Ziurys, L. M., 212 [167]; 216 [229]; 242;245 Zuev, M. B., 115 [156]; 117 [156]; 15i
SUBJECT INDEX
----
B6CI2H8, 129
A-----
ab initio methods, 6
BaO, 19
acetic acid, 218
BaOH, 212
acetic formic anhydride, 117
BBr3, 116
acetone, 220, 323,328
Be2C14, 104
acetonitrile, 316
BeCI2, 104
acetyl chloride, 218
BeF 2, 104
acetylene, 78, 292
benzene, 209
acrolein (2 isomers), 220
BH2CI, 214
A12Br6, 104
BH2NH2, 214
A1Br3, 104
BH 3, 208
alkanes, 273 alkyl moieties, 273
BH3NH3, 214
allyl alcohol, 220
BHCI2, 214
allylsilane, 221
bicyclo[3.1.0]hexane, 123
3-aminoacrolein, 131
bicyclo[3.3.0]oct-1,5-ene, 127
aminoborane, 214
bicyclohexyl, 128
antimony trifluoride, 116
bicyclopentyl, 127
antimony triiodide, 116
BiF 3, 116
AsF 3, 214
BiI3, 116
AsH2, 212
bis(dichlorosilyl)amine, 129
A v, Bv, Cv values, 11
bis(dichlorosilyl)methylamine, 129 bis(dimethylstibyl)oxane, 132
B
bis(dimethylstibyl)selane, 132
B2GaH 9, 116
bis(fluorocarbonyl)disulfane, 117
B2H6, 208
bis(fluorooxy)difluoromethane, 129
B4F3HsP, 129
bis(trifluoromethyl)trioxide, 130
B4H10, 122
bis(trifluoromethylsulfonyl)difluoromethane, 131 391
392
SUBJECT INDEX
bis(trifluoromethylthio)ketene, 132 bis(trifluoromethylthio)methane, 131 bismuth trifluoride, 116 bismuth triiodide, 116 1,2-bis(trimethylsilyl)acetylene, 134
1,2-bis(trimethylsilyl)- 1,2-dicarbacloso-hexaborane, 134 borane, 208 borane monoammoniate, 214
----C c-2-butene, 279, 286, 294, 303,307, 308 c-1-butene, 274, 279, 286, 293,294, 303,307, 308 C2, 19 C2C12, 208 C2C12F20, 130 C2C1202, 130
Born-Oppenheimer approximation, 2, 17
C2C1F30, 130
boron tribromide, 116
C2CIF3OS, 130
Br2Gea, 104
C2F2, 208
Br2Sib, 104
C2F202S2, 117
Br6H2Si3, 116
C2F4OS, 130
BrBS, 214
C2F603, 130
BrCN, 56, 198, 214
C2F605S2, 130
BrCN+, 207
C2F6S, 117
BrNCO, 215
C2FNO2, 117
bromine isocyanate, 215
1-bromobutane, 132 1-bromopropane, 122
C2FNO2S, 130 C2H, 212
2-bromopropane, 220
C2H!oSi2, 130
3-bromopropene, 220
C2H2+, 207
bromosulfidoborane, 214
C2H2F2O, 217
butane, 340 1,4-butanediole, 127 2,3-butano-2,4-tetraborane, 133 1-butene, 306, 340
C2H2N2O, 217 C2H2N2S, 217 C2H203, 117
1-butyne, 132, 274, 285, 289, 292, 296
C2H3C13, 122
2-butyne, 283,285, 289, 292
C2H3CIOS, 122 C2H3F30, 122 C2H40, 41
SUBJECT INDEX
393
C2H4Si, 138
C3F804S2, 131
C2HsBr, 218
C3H (cyclic), 212
C2HsCHO, 211
C3H12Si2, 131
C2H5C13Si, 130
C3H14B4, 131
C2HsF, 218
C3H2 (cyclic), 212
C2H5I, 218
C3H2C1202, 131
C2HsN, 122
C3H2F4, 219
C2HsNH2, 210
C3H2F6S2, 131
C2HsOCH3, 220
C3H3CIO, 117
C2HsPF2, 210
C3H3F3, 219
C2HsPH2, 210
C3H3F30, 131
C2HsPSF2, 210
C3H3F3OS, 131
C2HsSCH3, 211
C3H3F3S, 131
C2HsSeH, 210
C3H3F6NO, 131
C2HsSiH2F, 210
C3H4N2, 220
(C2H5)20, 221
C3H403, 117, 122
C2H6AsF, 130
C3H4S, 220
C2H6C12Si, 122
C3HsNO, 131
C2H6C12Ti, 117, 130
C3H6, 208
C2H6FO2P, 130
C3H6C12Si, 117, 131
C2H602, 122
C3H6C1N, 122
C2H7NO, 130
C3H6F2Si, 131
C2HsN2, 122
C3H60 2, 117, 127
C2HC12FO, 130
C3H604, 220
C2HF3OS, 130
C3H7Br, 122
C3F4, 138
C3H7C10, 131
C3F60, 131
C3H7F, 211
C3F6S, 131
C3H7I, 122
394
SUBJECT INDEX
C3H7N, 122, 131
C4H7F3Si, 132
C3H7NO, 123
C4H8, 209
C3H7NO2, 123
C4HsBr2, 127, 132
C3H7OCH3, 221
C4H8C12, 127
C3H8C12Si, 131
C4H8F2, 127
C3H8FN, 131
C4H80, 123, 132
C3H9CISi, 123
C4H802, 127
C3H9N, 123
C4H9Br, 132
C30, 212
C4H9C1, 132
C3OS, 221
C4H9I, 132
C3S, 212
CnH9NO, 132
C4F6, 123
CsH100, 133
C4F6OS2, 132
C5H1002, 117, 127
C4H10O, 117
CsH10Si, 133
C4H1002, 127
CsH12OSi, 133
C4H10Si, 132
CsH12Si, 133
C4H12OSb2, 132
CsH12SSi, 133
C4H12Sb2Se, 132
CsH4CN, 221
C4H16B4, 133
CsH5, 212
C4HECI2N2, 132
CsHsIn, 221
C4H4N2, 138
CsHsN, 123
C4H40, 138
CsHsNO, 221
C4H4S, 138
CsH5TI, 221
C4H6, 132,208
C5H60, 133
C4H6Br202S, 132
C5H6S, 123
C4H60, 117
C5H7S, 123
C4H602, 123
CsH9F3Si, 133
C4H603, 221
C6H10, 123, 127
C4H604Sn, 132
C6H12, 209
SUBJECT INDEX
395
C6H120, 133
CsH22B4Si2, 134
C6H140, 117, 127
C8H22Si2, 117
C6H18N3P, 133
C8H604, 134
C6H18N3Sb, 133
CsH 8, 209
C6H4C12, 138
C9H14Si, 134
C6H4F2, 221
C9H18, 127
C6H5C1, 138
C9H180, 127
C6H6, 209
C9H20, 127
C6H602, 133
C10H18, 127
C6H7N, 138
C10H30Sis, 127
C6H8, 127, 133
C11H23MoN3, 134
C70, 134
C12H22, 128
C7H12, 127
C12H24, 128
C7H16Ge, 134
C 14H14N203 , 134
C7HEoN3P, 134
C18H21Sc, 128
C7H5C10, 123
C 18H21Yb, 128
C7H7CIO, 117
cadmium dichloride, 116
C7H7C10, 133
cadmium diiodide, 116
CTH7NO2, 133 C7H8, 221
CaNC, 212 carbon dioxide, 208 carbon diselenide, 208
CsHlo, 134
carbon monoxide, 34
C8H12, 127
carbonyl fluoride chloride, 215
C8H12S6, 134
carbonyl selenide, 217
C8H16, 127
carbonyl sulfide, 216, 217
CsH16B10, 134
CsH18F4Si2, 134 CsH18Si2, 134
C-C single bonds, 273 CCCN, 212 CC12, 212 CC14, 111
CsH19N, 127 CD4, 120
396
SUBJECT INDEX
CdC12, 116
CH2CHCH2OH, 220
CdI2, 116
CH2CHCH2SiH3, 221
CeI3, 116
CH2CHCHO, 220
centrifugal distortions, 13
CH2CHC1, 217
cerium triiodide, 116
CH2CHCN, 219
CF2N2OS, 129 CF2NC1, 215 CF2NH, 215 CF202, 116 CF3, 212 CF3I, 107 CF3NOS, 129 CF3NOS2, 129 CF3NS, 129 CF3SiH3, 216 CF4, 106, 208 CF402 , 129 CFN30, 116 CFNO3S, 129 CFNO5, 116, 129 CFNOS, 129 C-H stretching frequencies, 204 CH2=NH, 314 CH2CCHF, 219 CH2CCI2, 217 CH2(C)CN2, 221 CH2CH2, 208 CH2CHBr, 217 CH2CHCH2Br, 220 CH2CHCH2F, 210
CH2CHF, 218 CH2CHI, 218 CH2CHOH, 218 CH2CHSiH3, 219 CH2C12, 71, 72, 116 CH2CICH2C1, 218 CHE(CN)2, 219 CH2CO, 217 CH2FCH2F, 208, 210 CH2FCOOH, 210, 230 CH2I2, 215 CH2N2, 216 CH2OHCOOH, 218 CH2PCI, 215 CH2PH, 216 CH2SiH2, 202, 216 CH3+, 207 (CH3)ECF2, 220 (CHa)2CHBr, 220 (CHa)ECHCHO, 211, 231 (CH3)ECHCI, 220 (CH3)ECHF, 220 (CH3)ECHI, 220 (CHa)ECHPF2, 211, 231
SUBJECT INDEX
397
(CH3)2CHSeH, 211,230
CH3CN, 195,218
(CH3)2CO, 220
CH3COCI, 218
(CH3)2NH, 314
CH3COOH, 218
(CH3)2S, 219
CH3F, 319, 320
(CH3)2SiC12, 96, 99
CH3F2N, 122
(CH3)2SiH2, 219
CH3GeH2Br, 216
(CH3)2SO, 219,224
CH3GeH2CN, 218
(CH3)3P, 221
CH3GeH2F, 216
CH3AsF2, 122
CH3HgCN, 218
CH3Br, 77
CH3-N=CH2, 103
CH3BS, 216
CH3NH2, 314
CH3CEN, 314
CH3NHNHCH3, 210
CH3C2H30, 220
CH3NSi, 138
CH3CCH, 220
CH30, 212
CH3CH=NH, 314
CH3OCC10, 218
CH3CH2CH2F, 230
CH3OCH2I, 219
CH3CH2CHO, 230
CH3OCH2SiH3, 219
CH3CH2CI, 218
CH3OCH3, 317, 337
CH3CH2NH2, 231
CH3OCHCH2, 220
CH3CH2PF2, 231
CH3OCN, 218
CH3CH2PH2, 230
CH3OCOCN, 219
CH3CH2PSF2, 231
CH3SiH2F, 216
CH3CH2SCH3, 230
CH3SiH2I, 216
CH3CH2SiH2F, 230
CH4, 22, 118, 120, 260
CH3CH3, 208
CH4C1P, 129
CH3CHCCH2, 221
CH40, 122
CH3CHC12, 218
CH5C14NSi2, 129
CH3C1, 77
CH6N2, 122
398
SUBJECT INDEX
CH9NOSi2, 130
chlorosulfidoborane, 214
CHCI(CH)ECH2, 221
chlorotrimethylsilane, 123
CHC1CHC1, 217
CI2Mg, 116
CHCIF2, 215
cis,trans- 1,2,3-trifluorocyclopropane, 219
CHFCHF, 208 CHFCO, 217 (CHO)2CH2, 220 chloride, 78 chlorine fluorosulfate, 129 chlorine peroxide, 222 chloroacetylene, 39, 78, 217 2-chloroacrolein, 117
cis- 1,2-dichloroethene, 217 cis- 1-butene, 278
cis- 1-chloro- 1,3-butadiene, 221 cis-2-butene, 278
cis-methylvinylether, 220 C12, 19 CI2GeC, 104
2-chloroanisole, 133
C120, 43, 58, 222
(chlorocarbonyl)trifluoromethylsulfane, 130
Cl2Sid, 104
4-chloro-benzaldehyde, 123 chlorobenzene, 138
CI3Ga, 116 Cl3In, 116
chloroborane, 214
C13NbO, 104
1-chlorobutane, 132
CI4H3NSi2, 129
chlorodifluoroacetyl chloride, 130
C14Mg2, 116
chlorodifluoromethane, 215
C14U, 116
(chloromethyl)chlorodimethylsilane, 131
C1BO, 214
(chloromethyl)dichloromethylsilane, 130
CIBS, 214 CIBSe, 214
chloromethyl phenyl, 117
C1CN, 198
chloromethyl phenyl ether, 117
C1CN+, 207
chloromethylphosphine, 129
C1CP, 215
chlorooxoborane, 214
CIFsOS, 129
1-chlorophosphaethene, 215
CIFO3S, 129
chlorophosphaethyne, 215 2-chloropropane, 220 3-chloropropane- 1-ol, 131 chloroselenoborane, 214
C1HCI-, 208 C1NO, 222 135C12160, 190
SUBJECT INDEX
399
CIO2, 212
cyclopropylamine, 122
C1OOCI, 222
cyclopropylgermane, 220
CIS2, 212 C1SF, 222 CN, 19 CNCN, 219 CO, 19,323 CO 2, 20, 55, 111, 198, 208
D 1-(dichloroboryl)pentaborane(9), 129 DCI, 17, 18 decamethyl cyclopentasilane, 127 diazirine, 216 diborane, 208
cobalt tetracarbonyl hydride, 221
1,4-dibromobutane, 127
COC12, 36, 43, 48, 215
1,3-dibromobutane, 132
Coriolis interactions, 13
dichlorine monoxide, 222
CS 2, 109
dichloroacetyl fluoride, 130
133Cs35C1, 16
dichloroacetylene, 208
CSe 2, 208
dichloroborane, 214
cubane, 209
1,4-dichlorobutane, 127
cyanogen, 208
1,2-dichloroethane, 218
cyanogen bromide, 214
1,1-dichloroethene, 217, 218
cyanogen fluoride, 215
1,2-dichloroethens, 109
cyanogen iodide, 216
dichloromethane, 116
cyanogen N-oxide, 219
dichloromethylvinylsilane, 117
cyclobutane, 209
2,6-dichloropyridazine, 132
cyclobutene ozonide, 221
3,6-dichloropyridazine, 132
cyclododecane, 128
4,6-dichloropyridazine, 132
cycloheptene, 127
2,5-dichloropyrimidine, 132
cyclohexane, 209
dicyanomethane, 219
cyclononane, 127
diethyl ether, 117
cyclooctane, 127
diethylether, 221
1,3-cyclopentadiene- 1-carbonitrile, 221
diethylketone, 133
cyclopentadienyl indium, 221
3,3-difluoro- 1-butyne, 281,285, 292, 296, 320
cyclopentadienyl thallium, 221 cyclopropane, 78, 208 cyclopropanecarboxylic acid, 123
difluoroacetic acid, 210, 230 difluoroacetylene, 208
400
SUBJECT INDEX
1,2-difluorobenzene, 221 1,3-difluorobenzene, 221
di-tert-butylmethane, 127
difluoroborane, 214
divinylketone, 133
1,2-difluoroethane, 210, 230, 278, 281, 320, 340, 344
double-substitution method, 40, 42
1,2-di-tert-butyltetrafluorodisilane, 134
E
difluoromethanimine, 215 difluorophosphine selenide, 138
3-ethenyl-3-methylcyclopropane, 133
difluorophosphine sulphide, 138
E-propionaldehyde oxime, 123
2,2-difluoropropane, 220
equilibrium rotational constants, 11
difluorosulfenylimine cyanide, 129
equilibrium structures, 46
4,6-dihydroxyisophthanaldehyde, 134
ethanal, 279, 323,327, 328
diiodomethane, 215
ethanamine, 231
dilsopropyl ether, 127
ethane, 6, 78, 208
dimethyl sulfoxide, 219, 224
ethane, 274, 293,307
2,2-dimethyl- 1,3-dioxolane, 127
ethane- 1,2-diol, 122
3,3-dimethyl-3-silatetrahydrofurane, 133
ethaneselenol, 210
3,3-dimethyl-3-silatetrahydrothiophene, 133
ethene, 124
dimethyldichlorosilane, 122 dimethyldichlorotitanium(IV), 130 3,4-dimethylenehexa- 1,5-diene, 134 dimethylfluoroarsine, 130 1,2-dimethylhydrazine, 210 dimethylsilane, 219 dimethylsulfide, 219 dimethyltitanium dichloride, 117 1,3-dioxolane, 127 dioxygen, 209 1,4-disilabutane, 130 1,4-disilapentane, 131 disulfur monoxide, 223 di-tert-butylamine, 127 1,2-di-tert-butyldisilane, 117 di-tert-butylketone, 127
ethanol, 340 ethene, 286, 294, 308 ethyl bromide, 218 ethyl chloride, 78, 218 ethyl difluorophosphine, 210, 231 ethyl fluoride, 218 ethyl formate, 117 ethyl iodide, 218 ethyl methylether, 220 ethylamine, 210 ethylene, 78, 208 ethylene oxide, 21, 78 ethylenediamine, 122 ethylfluorosilane, 210 ethylmethylsulfide, 211,230 ethylphosphine, 210, 230 ethylphosphonic difluoride, 231
SUBJECT INDEX
ethylphosphonothioic difluoride, 210 ethyne, 289, 296
401
3-fluoro-l-butyne, 281,285, 292, 296, 320 4-fluoro-l-butyne trans, 292
F ~----
F 2,
267
fluorocarbonyl azide, 116 fluorocarbonyl isocyanate, 117
F2HPS, 138
fluorocarbonyl peroxynitrate, 129
F2HPSe, 138
fluorocarbonyl peroxynitrate, 116
F20, 20
(fluorocarbonyl)sulfenyl isocyanate, 130
F203S, 129 F3Mn, 129 F3NbO, 104 F3Sb, 116
(fluorocarbonyl)trifluoromethylsulfane, 130 (fluoromethyl)dimethylamine, 131 fluoroformyl hypofluorite, 116
F4U, 116
fluoroformylinimosul fur difluoride, 129
FBO, 214
fluoroketene, 217
FC-CH, 45
fluoromethyl methylether, 218
FCCCN, 219
fluoromethylethylether, 220
FCH2CH2F, 230
fluorooxoborane, 214
FCH2OC2H5, 220 FCH2OCH3, 218 FC103, 195,222
4-fluoro-2-pentyne, 292 fluorophosphaethyne, 215 3-fluoropropene, 210, 219, 320
FCN, 215
1-fluoropropane, 211,230, 278, 281, 320, 340
FCOC1, 215
2-fluoropropane, 220
FCP, 215
fluoropropynenitrile, 219
FeCO, 212
fluorosulfonyl isocyanate, 129
Fermi resonances, 13
formaldehyde, 78
FHF-, 208
formic acid, 40, 78, 216
fluorine fluorosulfate, 129
formic anhydride, 117
fluoroacetylene, 217, 229
formyl chloride, 215
fluoroallene, 219
FuUerene, 134
1-fluoro-2-butyne, 292
furan, 138
402
SUBJECT INDEX
-----
G
H20, 270
74GeH3F, 195
H2O+, 207
gallium trichloride, 116
H20 2, 267, 334, 337, 340
Gas-phase electron diffraction (GED), 86
H2S, 223
GeCI2, 212 GeH3Br, 214 GeH3C3H5, 220 GeH3C1, 222 GeH3F, 222 GeH4, 209
H2S, 265 H2S, 270 H2Se, 223 H2SiO, 223 H2Te, 223 H3C-CH3, 75
GeH6Si, 122
H30 +, 207
GeI2e, 104
HBF+, 207
germane, 209
HBF2, 214
germyl bromide, 214
HBO, 199, 214
germyl chloride, 222
HBS, 199, 214
germyl fluoride, 222
HC-CCH 3, 201
glycolic acid, 122, 218
HC-CF, 28 HC30, 226
H2, 19 H2CCC, 199, 212
HCCC, 212 HCCCC, 212 HCCCCH+, 207
H2CCCC, 212
HCCCH2F, 219
H2CCCI2, 199, 200
HCCCHNH, 219
H2CCCO, 219
HCCCI, 72, 199, 217
H2C1+, 207
HCCCN, 199
H2CO, 72
HCCCO, 212
H2CS, 199, 216 H2CSe, 216
HCCCO radical, 227 HCCF, 217 HCCH, 67, 72
H2F+, 207
HCCI, 217
H20, 20, 223,260, 334
HCCN, 212
SUBJECT INDEX
403
HC1, 17, 18, 19
HSSSH, 211,230
HC1, 265
hydrazoic acid, 222
HCN, 52, 66, 72, 199, 215
hydridogallium bis(tetrahydroborate), 116
HCNH+, 207 HCNO, 199 HCo(CO)4, 221
hydrogen cyanide, 215 hydrogen isocyanide, 215 hydrogen pseudohalides, 225
HCO+, 199,207 HCOCI, 215 HCOOH, 199, 216 HCOOOH, 216
hydrogen selenide, 223 hydrogen sulfide, 223 hydrogen telluride, 223 hydrogen trisulfide, 211
HCS+, 207
hypobromous acid, 214
hexabromotrisilane, 116
hypochlorous acid, 222
1,5-hexadiene, 127 hexafluorocyclobutene, 123
----
I-------
HF, 45, 260
I3Sb, 116
HN2+, 72, 207
ICN, 216
HN3, 222, 225
1-iodobutane, 132
HNC, 71, 72, 199, 215
1-iodopropane, 122
HNCCN+, 207 HNCO, 199, 215, 225 HNCS, 215,225 HNCSe, 215 HNNH, 209 HNO2, 222
2-iodopropane, 220 imidazole (1,3-diazole), 220 indium hydroxide, 222 indium trichloride, 116 InOH, 222 iodoacetylene, 217 iodomethyl methylether, 219
I-rNO3, 223
HNSNH, 211,230 HOBr, 214
isocyanic acid, 215 isocyanogen, 219 isopropyl acetate, 117
HOC +, 207
isopropyl difluorophosphine, 211, 231
HOC1, 199,222
isopropylamin, 123
HOCO +, 207
isoselenocyanic acid, 215
HOSi +, 207
isothiocyanic acid, 215
404
SUBJECT INDEX
------
methyl Nicotinate, 133
K----
KC1, 19
3-methylene- 1,4-pentadiene, 127
KCN, 216
4-methyl- 1-pentyne gauche, 292
ketene, 217
methyl picolinate, 133
Kraitchman's equations, 10, 25, 26
methyl silylmethylether, 219
KrF2, 209
methyl trifluorothioacetate, 131
krypton difluoride, 209
methyl trifluorovinyl ether, 131 methyl(vinyl)dichlorosilane, 131
"
'
L
'
methyl(vinyl)difluorosilane, 131
'
Landolt-B6mstein Tables, 87
methylallene, 221
LiCH 3, 216
methylbromogermane, 216 methylchloroformate, 218
9
'~I
'
methyldifluoroamine, 122
magnesium dichloride, 116
methyldifluoroarsine, 122
magnesium dichloride dimer, 116
methylene, 78
malonaldehyde, 220
methylfluorogermane, 216
malonyl dichloride, 131
methylfluorosilane, 216
manganese trifluoride, 129
5-methylfuan-2-aldehyde, 133
meta-dicheorobenzene, 138
methylgermyl cyanide, 218
methacrolein, 117
methyliodosilane, 216
methanal, 323,327, 328
methyllithium, 216
methanol, 122, 317, 334, 337
methylmercury cyanide, 218
methoxyacetamide, 123
methyloxirane, 220
2-methyl- 1-buten-3-yne, 292
2-methylpropanal, 123, 211, 231
methoxyethylene ozonide, 220
2-methylthiophene, 123
methyl acrylate, 123
3-methylthiophene, 123
methyl chloride, 78
methylpropylether, 221
methyl chlorothioformate, 122
methylpropylether, 221
methyl cyanate, 218
methylsulfidoborane, 216
methyl cyanide, 218
MgNC, 212
methyl cyanoformate, 219 2-methyl- 1,3-dioxolane, 127
N
methyl hydrazine, 122
N,N-dimethylacetamide, 132
methyl isonicotinate, 133
N,N-dimethylhydroxlamine, 130
SUBJECT INDEX
405
N 2, 19
-----
O - - - -
N2H4, 267, 339, 340
02, 19,209
N20, 36, 57, 198
0 3, 35, 58, 223
NaSH, 223 NCCN, 208
OCS, 15, 20, 42, 48, 55, 65, 198, 216, 217
NCCNO, 219
OCSe, 55, 198, 217
N-chloroazetidine, 122
OH, 19
N-chloro-difluoromethanimine, 215
OH- (OD-), 208
N-chloro-N-ethylethanamine, 117
O-methyl methylphosphonofluoridate, 130
NH2- , 208 (NH2)2CO ,
216
NH2SH, 211,230
O-methyl-N,Nbis(trifluoromethyl)hydroxylamine, 131
NH 3, 260
O-methyl-N,N-disilylhydroxylamine, 130
NH 3, 270
ortho-Dichlorobenzene, 138
NH4 +, 208
1,2,5-oxadiazole, 217
nitric acid, 223 nitric oxide dimer, 223 nitrosyl chloride, 222
oxalyl chloride, 130 oxoborane, 214 ozone, 223
nitrous acid, 222 nitrous oxide, 223 N-methylethylideneimine, 131 N-methylmethyleneimine, 122 N-methylpyrrole, 123 NNO, 223 NO, 19, 57 (NO)2, 223 NO 2, 213
______
p
._~_.
P4, 209 para-dichlorobenzene, 138 p-Azoxyanisole, 134 pentafluorsulfanyl hypochlorite, 129 1-pentyne, 283,285,296, 340 2-pentyne, 292 1-pentyne trans, 292 perchloryl fluoride, 222
NO dimer, 196
perfluorocyclopropene, 138
NOCI, 58
perfluoromethyl vinyl ether, 131
norbornadiene, 221
peroxyformic acid, 216
norbornane, 127
PF 2, 213
406
SUBJECT INDEX
PF 3, 222
rPm structures, 14
PF 5, 209
(rv), 4
PH 3, 223
ro, 4, 5
1-phenyl- 1,2-dicarba-closododecaborane(12), 134
r 0 parameter, 95 r a distance, 95
phosgene, 215 phosphaethene, 216
re, 4,5,17
phosphine, 223
r e distance, 95, 114
phosphorus, 209
r e structure, 3, 46, 174
phosphorus pentafluoride, 209
rg distance, 95, 114
PO 2, 213
rm, 4, 5
potassium cyanide, 216
r m structures, 31, 188, 193
propadienone, 219 r o and r s structures, 186
propanal, 211 propanal, 279, 323,327, 328, 340
rotational constant A, 7
propane, 22, 23, 78
rotational constant B, 7
propane, 274, 293, 303,306, 307
rotational constant C, 7
2-propaneselenol, 211,230
r s, 4, 5, 174
1,2-propano-2,4-tetraborane(10),
131
r s coordinates, 181
propene, 78, 294
r s structures, 24
propene, 275,279, 286, 303,307, 308
r v, 15
propionamide, 123
rz, 4, 5
propyleneimine, 122 propyne, 201,220
r z distance, 95
propyne, 292
ra distance, 95
propyne, 285, 289, 296
re,I, 4, 5
pyrazine, 138
S
pyridazine, 138 pyridine, 123
S(CN)2, 219
pyridine-N-oxide, 221
$20, 59, 62, 223
pyrimidine, 138
SbH 3, 223 SeF 6, 91 selenoformaldehyde, 216
rmP, 4, 5,6
SUBJECT INDEX
407
SF 6, 209
SOF4, 99
SF6, 91, 104, 106
SPCI3, 222
SH3+, 207
spiro[2.4]hepta-4,6-diene, 221
SiC2H 2 (cyclic), 212
stannane, 209
SiC12, 212 SiF 4, 106 SiH2Si, 213 SiH3Br, 214
stibine, 223 s-trans-l,3-butadiene, 208 structure formula, 212 substitution (rs) method, 24 sulfidoborane, 214
SiH3CN, 216
sulfinyl cyanide fluoride, 129
SiH3NCO, 216
sulfur chloride fluoride, 222
Sill 4, 209
sulfur cyanide trifluoride, 129
SiHSiH, 213
sulfur dicyanide, 219
silaethene, 202
sulfur diimide, 211
silane, 209
sulfur dioxide, 223
silanone, 223
sulfur hexafluoride, 209
silene, 216
sulfur trioxide, 209
silyl acetylene, 138
sulfuryl chloride fluoride, 221
silyl bromide, 214
syn-vinyl alcohol, 218
silyl cyanide, 138, 216 silyl isocyanate, 216 silylgermane, 122 (silylmethyl)cyclopropane, 132 single substitution, 175 SnH4, 209 SnO, 19
T t-2-butene, 286 t-butane, 274, 293,303,307 TC1, 17 tea-butyl ethyl ether, 117
tert-butyl vinyl ether, 133 t-ethanol, 317, 334, 337
SO2, 20, 35, 59, 223
tetraborane (10), 122
SO2C1F, 221
tetraborane(8)-trifluorophosphine ( 1/ 1) 129
SO 3, 209 SOCI2, 222 sodium hydrogen sulfide, 223
1,1,2,2-tetrafluorocyclopropane, 219 tetrafluoromethane, 208 tetrahydrofurfuryl alcohol, 133
408
SUBJECT INDEX
1,1,3,3-tetramethylgermacyclobutane, 134
1,3,5,7-tetramethyl-2,4,6, 8,9,10hexathiaadamantane, 134
trifluoromethyl vinyl sulfide, 131 trifluorophosphine, 222 (trifluorosilylmethyl)cyclopropane, 132
1,3,4-thiadiazole, 217
trifluorothioacetate chloride, 130
thiete, 220
trifluorothioacetic acid, 130
thioformaldehyde, 216
trimethylphosphine, 221
thiohydroxylamine, 211
trimethylsilylacetylene, 133
thionyl chloride, 222
trimethylsilylbenzene, 134
thiophene, 138
(trimethylsilyl)trifluoroethene, 133
thiophosphoryl chloride, 222
trimethylvinylsilane, 133
TiCH, 212 tin(II) acetate, 132
tris(dimethylamido)cyclopentadienylmolybdenum, 134
torsional angles, 336
tris(dimethylamino)antimony, 133
trans 2-butene, 283,
tris(dimethylamino)methylenphosphorane, 134
trans ethanol, 279
trans- 1,2-difluoroethane, 208
tris(dimethylamino)phosphine, 133
trans- 1,2-difluoroethylene, 208
tris(rnethlcyclopentadienyl)scandium, 128
trans-2,3-difluorooxirane, 217 trans-3 ,4-dibromotetrahydrothiophene-
1,1-dioxide, 132
tris(methylcyclopentadienyl)ytterbium, 128
trans-diazene, 209
U
tricarbon oxide sulfide, 221
uranium tetrachloride, 116
1,1,1-trichloroethane, 122
uranium tetrafluoride, 116
trifluoroacetyl chloride, 130
urea, 216
trifluoroarsine, 214
1,1,1-trifluorodimethyl ether, 122 trifluoroethylidynesul fur trifluoride, 117
vinyl alcohol, 224
trifluoromethanesulfonic anhydride, 130
vinyl chloride, 78, 217, 228
trifluoromethyl silane, 216 (trifluoromethyl)sulfanyl sulfinylimine, 129 1-trifluoromethylthiop- 1,2,2trifluoroethane, 131
vinyl bromide, 217, 228 vinyl cyanide, 219 vinyl dicyanide, 221 vinyl fluoride, 218, 224 vinyl halides, 125 vinyl iodide, 218
SUBJECT INDEX
vinylsilane, 219 ------
W
-----
X
--~--
water, 223 ------
XeF 2, 209 XeF4, 209 xenon difluoride, 209 xenon tetrafluoride, 209 u
y-picoline, 138 "'
Z
(Z)-methyl- 1-propenyl ether, 132 Z-prop-2-ynylideneamine, 219 Z-propionaldehyde oxime, 123
409
