Computational Chemistry: Reviews of Current Trends Vol. 10

Volume 10

editor

World Scientific

Computational Chemistry Reviews of Current Trends Volume 10

Computational Chemistry: Reviews of Current Trends Editor-in-Charge:

Jerzy Leszczynski, Dept. of Chemistry, Jackson State University, USA

Published Vol. 1: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 2: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 3: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 4: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 5: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 6: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 7: Computational Chemistry. Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 8: Computational Chemistry. Reviews of Current Trends Edited by Jerzy Leszczynski Vol. 9: Computational Chemistry: Reviews of Current Trends Edited by Jerzy Leszczynski

Computational Chemistry Reviews of Current Trends Volume 10

editor

Jerzy Leszczynski Department of Chemistry Jackson State University USA

\jp World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • S H A N G H A I • H O N G K O N G • TAIPEI • CHENNAI

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COMPUTATIONAL CHEMISTRY: REVIEWS OF CURRENT TRENDS Volume 10 Copyright © 2006 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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PREFACE When ten years ago the World Scientific Publishing Company invited me to edit a volume on Computational Chemistry, I was very thrilled about the project. A few years earlier, in 1992, we started an annual conference series "Conferences on Current Trends in Computational Chemistry" (CCTCC) that has been attended by a broad group of advanced professionals and students interested in this area of research. From my interactions with the speakers and participants of these series of meeting I gained an impression that the books focusing on reviews of current computational techniques and their applications, that are delivered in a timely manner and are affordable for budget-oriented graduate students, would be a good addition to the published literature. Some colleagues shared my enthusiasm and provided their reviews for the first volume of this series that was published in 1996. Ten books and ten years later, we are delighted to present you the decennial volume of the series. This volume will be released just few months prior to the 15th CCTCC conference that is scheduled for November 2006. There have been important developments during this decade — computers are more powerful and faster, code features are enhanced and more efficient, and studied molecules are larger and can be studied not only in vacuum but also in an environment of a solvent or in crystal. In addition, the chemical and physical community celebrated the 1998 Nobel Prize that was awarded to two leading computational experts. It was not by chance that Walter Kohn and John A. Pople have been recognized by the Nobel Committee for their contributions to the development of efficient computational methods for quantum chemistry. The messages that the community is receiving from the administration are also very encouraging. This year John H. Marburger III, the director of the White House Office of Science, announced that "computational science — the use of advanced computing capabilities to understand and solve complex problems — is now critical to scientific leadership, economic competitiveness and national security". The US President's Information Technology Advisory Committee in their recent report consider computational science a key technology field of the 21 st century because it allows researchers to study complex processes and phenomena that are experimentally difficult to characterize. As a result, they recommend that advances in computational sciences and the development of supercomputer infrastructures be supported.

vi

Preface

These optimistic news indicate that our volumes are well-timed and not only present the reviews of current advances in computational methodologies and applications for advanced specialists already involved in these research areas but will also contribute towards the education and training of a new generation of professionals necessary to cover these fast developing areas of computational science. Since each volume covers not just one subject but a broad range of methods and applications, the series provides a balanced mix of general topics and reviews of new findings. As expectations on the outcomes of applications of current computational chemistry methods are rising, the researchers use such techniques to study larger systems and to anticipate more accurate results. This is impetus for the development of more efficient methods based on the first-principle multi-level simulations appropriate for complex species. Among such methods, the Density Functional Theory (DFT) has received the most attention in the last decade. The first chapter of this volume written by T. Wesolowski reveals an alternative formalism of the DFT method based also on Hohenberg-Kohn theorems. In this formalism, the orthogonal orbitals are used not for the whole system but only for subsystems. The orbitals are derived from the KohnSham type of one-electron equations in which all terms representing the interactions between the subsystems are expressed as universal functionals of electron density. The chapter reviews the formal aspects of the density functional approaches, the development and testing of the density functionals and the possible use of the orbital-free embedding in multi-scale modeling. It also contains numerous examples, including applications for condensed phase species. The next chapter presented by M. R. Pederson, K. Park and T. Baruah covers the applications of Density Functional Theory in the investigations of molecular magnets. A class of molecules containing collections of spin-polarized ions is often referred to as molecular magnets. There are two groups in these systems that behave very differently when exposed to magnetic fields. One group relates to molecules containing metal centers for which the spin-orbit interaction is quite small. The second group corresponds to molecules containing strongly coupled metal centers and non negligible spin-orbit interactions. The authors describe how density-functional theory may be used to study this class of molecules which has received significant experimental attention lately, because it exhibits the phenomenon of

Preface

vii

resonant tunneling of magnetization. This phenomenon is fundamentally interesting, and has the potential to be technologically useful. The examples provided in the chapter show that the magnetism in these molecules can be influenced by applied magnetic fields, by the addition of extra electrons and by changes in hydrogen bonding. The discussion includes numerical details in addition to the underlying theoretical framework. Another example of the theory development is given in the chapter by P. Carsky and R. Curik. Their focus is the development of a simple but rigorous ab initio computational method that can be applied to vibrationally inelastic electron scattering by polyatomic molecules. Vibrational electron energy loss spectroscopy (EELS) is becoming a useful tool in providing information concerning the excited states of molecules and the electronic structure of negative ions and their unimolecular decomposition. Since the theoretical basis of the EELS are much less developed than those of vibrational or rotational spectroscopy, this chapter, which presents a discussion of the theory and examples of the applications, provides necessary information for those interested in the development and applications of EELS. The next chapter reviews the reactions of free atoms and radicals which play an important role in the modeling of complex processes occurring in the polluted atmosphere and in combustion chemistry. J. Jodkowski discusses the computational models of the reaction rate theory most frequently used in the theoretical analysis of gas-phase reaction kinetics and presents examples of the reactions of reactive components of the polluted atmosphere, such as O2, NOx, OH, NH2, alkyl radicals, and halogen atoms. Kinetic parameters of the reactions under investigation are provided in an analytical form convenient for kinetic modeling studies. The presented expressions allow for a successful description of the kinetics of the reaction systems in a wide temperature range and could be used in kinetic studies of related species. Hydrogen bonding is a unique interaction with crucial importance for chemical and bio-chemical reactions including life processes. The number of studied aspects of H-bonding is large and continuously increases due to newly revealed features. One of the current aspects of hydrogen bonding is associated with the experimental NMR studies of this phenomenon. There are three spin-spin couplings that can arise between the atoms which form the X-H-Y hydrogen bond. Coupling constants have signs associated with them which have been determined

viii

Preface

experimentally for a few hydrogen-bonded complexes. Until recently, the only model used to predict these signs was the Dirac Vector Model that often fails, even for simple molecules. J. E. Del Bene and J. Elguero, based on the results of rigorous calculations carried out using the ab initio equation-of-motion coupled cluster singles and doubles (EOMCCSD) method, provide generalizations concerning the signs of the oneand two-bond-reduced coupling constants. These generalizations make possible the prediction of the signs of the corresponding spin-spin coupling constants (J) measured experimentally, taking into account the magnetogyric ratios of the pair of coupled atoms. The chapter provides discussion of the theoretical methods and obtained results and covers the details of the newly-proposed Nuclear Magnetic Resonance Triplet Wavefunction Model (NMRTWM) that offers insight into the signs of the reduced coupling constants. In addition to H-bonding there are many interesting aspects of biological reactions that benefit from computational studies. In the last chapter of this volume, J. V. Burda, J. Sponer, and J. Leszczynski present a review of a biologically important process: activation of cisplatin and its analogs. Cisplatin (cis-dichlorodiammineplatinum cisDDP) is one of the most frequently explored platinum compounds since Rosenberg's discovery of its anticancer activities. The mechanisms of the function of this drug are not fully understand. The chapter covers a review of the theoretical studies that are devoted to the first step of such activation. The complex nature of such a process that includes ligand replacement by water molecules is discussed. I would like to thank all of the authors for the their excellent contributions and fruitful collaboration on this book and all volumes of the series. The very efficient technical assistance of Dr. Manoj K. Shukla in putting together this volume is greatly appreciated. I have had many opportunities to discuss various aspects of this project with the most important part of the community — the readers. As always, I value your feedback. Your comments are very important to me. So please feel free to e-mail your suggestions to [email protected]. Jerzy Leszczynski Jackson, MS September 2005

CONTENTS Preface

v

1. One-Electron Equations for Embedded Electron Density: Challenge for Theory and Practical Payoffs in Multi-Level Modelling of Complex Polyatomic Systems TomaszA. Wesotowski

1

2. Density-Functional Based Investigation of Molecular Magnets Mark R. Pederson, Kyungwha Park and Tunna Baruah

83

3. Vibrational Spectra by Electron Impact: Theoretical Models for Intensities Petr Cdrsky and Roman Curik

121

4. Theoretical Description of the Kinetics of Gas-Phase Reactions Important in Atmospheric Chemistry Jerzy T. Jodkowski

139

5. Predicting and Understanding the Signs of One- and Two-Bond Spin-Spin Coupling Constants across X-H-Y Hydrogen Bonds Janet E. Del Bene and Jose Elguero

229

6. Towards the Elucidation of the Activation of Cisplatin in Anticancer Treatment Jaroslav V. Burda, Jin Sponer and Jerzy Leszczynski

265

Index

323

Content Index

331

Chapter 1: One-Electron Equations for Embedded Electron Density: Challenge for Theory and Practical Payoffs in Multi-Level Modelling of Complex Polyatomic Systems

Tomasz A. Wesolowski Universite de Geneve, Departement de Chimie Physique 30, quai Ernest-Ansermet, CH-1211 Geneve 4, Switzerland E-mail: [email protected]

Abstract The idea of describing a many-electron system using only its electron density, i.e. without constructing its wavefunction, was initiated in the works of Thomas and Fermi. Hohenberg-Kohn theorems of modern density functional theory transformed this idea into an exact theory. The Kohn-Sham formalism, widely used in computer simulations of polyatomic systems today, is based on these theorems but is not orbital-free. It reintroduces orbitals to minimize errors in approximating the total energy. The present review concerns an alternative formalism based also on Hohenberg-Kohn theorems, in which orthogonal orbitals are used not for the whole system but only for subsystems [Cortona, Phys. Rev. B, 44 (1991) 8454]. These orbitals are derived from Kohn-Sham-like one-electron equations, called here Kohn-Sham Equations with Constrained Electron Density (KSCED), in which all terms representing t h e interactions between t h e subsystems are expressed as universal functionals of electron density. This formulation provides the formal basis for the orbital-free embedding in first-principles based multi-level simulations of complex systems, in which the orbital-level is retained for a selected subsystem, whereas its environment is described at the orbital-free level [Wesolowski and Warshel, J. Phys. Chem., 9 7 (1993) 8050]. The formal aspects, development and testing of relevant approximate density functionals, and the possible use of the orbital-free embedding in multi-level modelling are covered in detail in this review. Examples of applications, especially those concerning the electronic structure of embedded systems in the condensed phase are provided. 1.

Introduction

C o m p u t e r s i m u l a t i o n s of c o m p l e x p o l y a t o m i c s y s t e m s b e c a m e a n i m p o r t a n t t o o l s u p p l e m e n t i n g e x p e r i m e n t a l studies a n d c h e m i s t r y a n d m a t e r i a l s science 1 ' 2 . T h e y c a n b e used: i) t o get access t o d a t a n o t available directly in e x p e r i m e n t a l m e a s u r e m e n t s , ii) t o verify h y p o t h e s e s concerning t h e m e c h a n i s m of t h e investigated process, in) a n d t o m a k e p r e d i c t i o n s . At different scales, m e t h o d s b a s e d o n different physical laws a r e used. At m a c r o s c o p i c scale, c o m p u t e r s i m u l a t i o n t e c h n i q u e s a r e

1

2

T. A.

Wesolowski

based on the continuum description of matter. Models using macroscopic laws of electromagnetism, elasticity, mechanics of liquids, statistical physics, etc. have been also applied to micro- and nanoscopic sizes 3>4.5,6,8,9,io,ii There is a certain minimal length at which the macroscopic laws cease to be adequate. Its determination is, however, far from being a trivial issue (see the works concerning the minimal size of a group of atoms in a protein molecule, at which the continuum dielectric description starts being adequate 3>5>12>13 for instance). On the other hand, quantum mechanics provides the basics laws describing matter at the atomistic scale. There is certainly also a characteristic maximal length at which using the laws of quantum mechanics becomes either not possible or even not adequate in describing properties and behavior of large disordered polyatomic systems a . The maximal size of the systems described at the quantum mechanics level in practical computer simulations increases constantly owing to: methodological developments concerning new formalism to approach the acceptably accurate approximate solutions of the Schrodinger equation (emergence of the Kohn-Sham formalism 14 , for instance) better computer implementations of existing formalisms, and the constantly improving performance of available computers. As a result, systems comprising hundreds or even thousands atoms can be reasonably well described today. Not only larger systems can be studied now but the accuracy of the results improves owing to progress in development of good approximations to the exchange-correlation functional 15 which is the key quantity in the Kohn-Sham formalism. It has to be borne in mind, however, that the accuracy of currently known approximations to the exchangecorrelation functional is not sufficient to make Kohn-Sham equations competitive with high-level wave-function based methods such as those using coupled cluster form of the many-body wavefunction 16 ' 17,18 , single 19 - or multi-reference (see Ref.20 for instance) many-body perturbation theory. High-end wave-function based methods, however, can be applied only for even smaller systems. Nevertheless, for a large class of systems and problems the current accuracy of the Kohn-Sham method is adequate. These optimistic perspectives for quantum mechanics based simulation techniques should not overshadow the simple fact that the experimental conditions in materials science, nanoscience, and most of chemistry are such that the investigated objects comprise a huge number of atoms which is orders of magnitude larger than the maximal number of atoms which could be treated in practical computer modelling studies (10 3 -10 4 ). This brings us to the key issue in computer simulations of polyatomic systems: How to bridge the gap between the macroscopic and microscopic two types of description? Bridging this gap is needed if the computer simulations are expected to model realistically the behavior and properties of a chemical system in condensed phase. The present review concerns our work concerning such integrated simulation technique. Our strategy comprises two principal elements: a The phenomena, where long-range correlations in quantum systems are important, are not considered in the present.

One-Electron Equations for Embedded Electron Density

3

• Partitioning t h e electron density in real space: Opposite to other quantum mechanical descriptors such as wavefunction or density-matrix, electron density is a well defined and observable quantity at both subatomic and macroscopic scales. • S y s t e m - i n d e p e n d e n c e : Aiming at universal applicability, the interactions between the subsystems resulting from partitioning the total electron density are described using only system-independent quantities.

The Hohenberg-Kohn-Sham formulation of non-relativistic quantum mechanics 14,21 provides the general formal framework to apply the above guidelines. It uses electron density as basic variable, applies only fundamental constants, and is exact in principle. Therefore, our approach to multi-level modelling is based on Hohenberg-Kohn theorems 2 1 and the subsystem formulation of density functional theory 22 . In this formulation, one-electron functions (embedded orbitals) are used to construct the electron density of each subsystem emerging from a given partitioning of electron density in real space. Opposite to Kohn-Sham orbitals, their embedded counterparts do not have to be orthogonal if they belong to different subsystems. In this review, the case of two subsystems will be considered: one of them representing the investigated system and another one corresponding to the environment - solvated molecule and solvent for instance. It will be shown that the orbitals representing the environment (which can be of the macroscopic size) can be replaced by the electron density belonging to a certain class. Our work concerning the development of approximate functionals specific to this formalism, its computer implementation, and applications, will be comprehensively reviewed in this chapter. It has the following structure. Section 2 provides the introduction to the key mathematical objects and theorems relevant for the subsystem formulation of density functional theory. The differences and similarities to the Kohn-Sham formalism are emphasized. One-electron equations for embedded electron density referred throughout this work as Kohn-Sham Equations with Constrained Electron Density (KSCED) are introduced in this section. All the discussions and analyses are made in the exact functional limit i.e. concerns the exact functionals even the ones which do not have known analytic form. The subsequent section (Section 3) concerns approximating the relevant density functionals: their exact properties, practical approximations, the procedures to test them, and the criteria of applicability. Section 4 provides the key elements of the generalized formalism in which one-electron equations for embedded electron density are used within the general framework of linear-response time-dependent density-functional-theory. Issues relevant to the numerical implementation of KSCED are discussed in Section 5. The applications are reviewed in Section 6. Complementary information is given in the Appendices.

4

T. A.

1.1. Units, • • • • • • • • •

Wesolowski

symbols,

and

conventions

All considerations concern non-relativistic Coulomb many-electron systems. Atomic units are used in all equations. Unless specified, the formulas are given for the spin-compensated case. p is used to denote electron density (charge density is -ep). & is used to denote one-electron functions (Kohn-Sham or embedded KohnSham orbitals). \PS is used to denote single-determinantal wave-functions. 5ij is the Kronecker symbol. The subscript 0 in E0 (ground-state energy) and po (ground-state electron density) indicate that these quantities are exact. Tilda is used to indicate an approximate quantity (as in po, EQ, EHK[p\).

2. T h e formalism We start this section with a short overview of the basic concepts in the HohenbergKohn-Sham formulation of density functional theory. In the second part, the key definitions and equations relevant for the orbital-free embedding formalism will be introduced. The considered formalisms are presented as four possible strategies to reach ground-state energy and electron density. Each of the considered strategies is based on Euler-Lagrange minimization of a different total energy functional: a) the Hohenberg-Kohn functional depending explicitly on the electron density, b) the Kohn-Sham functional which depends on a set of one-electron functions used to construct the electron density, c) the functional depending on two sets of oneelectron functions used to construct two components of the total electron density, and, finally, d) the functional depending explicitly on two types of quantities: a set of one-electron functions for one subsystem and the electron density of the other one. All considerations in Section 2 concern exact functional.

2 . 1 . Orbital-free

energy functional

EHK[p]

The existence of the correspondence between the external potential in which the electrons move and their density (the first Hohenberg-Kohn theorem 2 1 ) makes it possible to define the total energy functional. It is denoted here by EHK[p]. The analytic form of the functional EHK[p] is not known except for its two components: V[p] - the energy of the interaction with external potential vext (the potential generated by nuclear charges is used as an example below) and J[p] - Coulomb

One-Electron Equations for Embedded Electron Density 5 electron-electron repulsion energy, reading:

J ti

\r~Ri\

For a given number of electrons (2N) in an external potential vext, the second Hohenberg-Kohn theorem provides a link between the EHK[p] functional and the ground-state energy (EQ): E0 = min EHK [p] p

>2iV

(3)

L J

v

'

where the electron density minimizing the above expression is the ground-state electron density (p0). If a reasonable approximation to the whole EHK [p] taking the form of an analytic functional of p were available, atomic and molecular system could be described without using orbitals. Such a development would represent the ultimate success of the ideas initiated by Thomas 2 3 and Fermi 24 at the early days of quantum mechanics. The kinetic energy functional (T[p\) is the largest component of EHK[p] of unknown analytic form. The fact that T[p] needs to be approximated makes orbital-free methods based on Euler-Lagrange minimization of EHK [p] less useful in practical computer simulations. Good approximations applicable universally for chemical systems are not known although a significant progress has been made recently (for a review, see Refs. 25>26>27). To our knowledge, no approximation to T[p] has been developed so far which would make orbital-free methods useful in describing such key molecular process as formation of a chemical bond. Recently, Chan et al. used a Thomas-Fermi-Dirac-von Weizsacker type of the orbital-free energy functional in variational calculations for several diatomic molecules 28 . The fact that the obtained bonding energies and interatomic distances were only qualitatively good exemplifies these difficulties. The orbital-free simulations based on approximations to EHK[p\ are, however, attracting increasing interest because of their potential usefulness in studies of solids, liquids, and other materials where the simulations are aimed at describing collective behavior of large disordered systems and relative energies of different bound states not the absolute value of cohesion energy are the primary objects of interest (for representative applications see Refs. 2 9 _ 4 2 . For a comprehensive review, see Ref. 2 6 . For more recent developments, see Ref. 2 7 . 2.2. Energy functional

depending

on one set of orbitals

aKS[{i}}

Kohn and Sham proposed to deal with the kinetic energy component of EHK[p], which is at the origin of difficulties in practical applications of orbital-free methods,

6

T. A. Wesolowski

in an alternative way 14 . In the Kohn-Sham formalism, the major part of T[p] - the kinetic energy of a fictitious system of noninteracting electrons (Ts[p}) - is evaluated exactly but at the expense of introducing orbitals. For a given p, Ts[p] is defined as: Ts[p}=

min < * S | T | * S >

(4)

where tp s are single-determinantal wavefunctions constructed using { 0

(5)

p(r)dr = N

/ IVp(r)1/2 2

\ dr

where N is an integer number


(6)

(7)

An alternative proof of N-representability of such a class of electron densities was given by Harriman 4 6 . Eqs. 5-7 define a large class of electron densities relevant for chemical systems. Unless specified, the electron densities considered throughout the present review are N-representable. The searching procedure of Eq. 4 is a key element of the Levy constrained-search (LCS) formulation of density functional theory 4 7 . Since the orbitals in the {cf>i} set are orthogonal, the corresponding electron density is: N

P= 2 ^ > |

2

(8)

Using the definitions of the functionals V[p], J[p], and T3[p] given in Eqs. 1, 2, and 4, the Hohenberg-Kohn energy functional can be expressed as: EHK [p] = EKS [p] = Ts [p] + V[p} + J[P] + Exc[p]

(9)

where Exc[p] (exchange-correlation functional) is defined as: Exc[p] = EHK[p]

-V[p]HK

J[p] - Ts[p] KS

(10)

For iV-representable p, the functionals E [p] and E [p] are equivalent. The distinction is made in view of their possible practical computer simulations. If oneelectron functions ({0°}) minimizing the right-hand-side of Eq. 4 are known for a given p, the numerical value of Ts [p] can be evaluated exactly and the analytic form of Ts[p] as an explicit functional of p is not needed.

One-Electron

Equations for Embedded Electron Density

For a set of orthogonal one-electron functions {fc} (not related with {$}), generalized functional b can be defined (ELKS [{&}]): N 1 S * ^ ! , ^ , . . . , ^ ] = 2 ^ < +V\p] + J[p) + Exc[p]

7

a

(11)

where p is related with 0, via Eq. 8. The ground-state electron density and energy can be derived using the functional •zKS. Eo=

The

EHK[p]

functionals

min

^r,min

and

^KS[i,4>2,-,4>N]

EKS[p],

can

be

(12)

expressed

by

means

of

KS

E [^,4>2,...,4>N}:

EHK[p]=EKS[p]=

EKS [!, N\

mm

(13)

ii}—»p

Kohn-Sham functional makes the task of approximating the total energy functional easier but also offers a possibility to perform practical calculations. EulerLagrange minimization of the functional EKS[(pi,(f>2,..., 4>N] taking into account the condition of orthogonality: %jd?=5i,j

(14)

/ •

leads to the celebrated Kohn-Sham equations 14 :

> + ^M

i = l,N

(15)

where electron density p is related to the one-electron functions (pi as in Eq. 8, and where the effective potential V^ [p, r\ takes the following form:

VKS

f

-, _ y ^ c

Zi

f

P(f)

,w

SExc{p]

for the external potential generated by a set of atomic nuclei. In the following part, the distinction will be made for a special class of electron densities which are called noninteracting pure-state v-representable^. An electron density belongs to this class if it corresponds to the ground-state in a system of noninteracting electrons in some external potential 4 4 . Obviously, Eqs. 15-16 can only lead to noninteracting pure-state v-representable electron densities. If the groundstate electron density for a given system does not belong to this class, it cannot b

Throughout this work, a distinctive symbol (S instead of E) is used to denote an energy functional which depends explicitly on other quantities than electron density.

8

T. A. Wesolowski

be obtained from the Kohn-Sham equations even at the exact functional limit discussed in this section. Opposite to ./V-representability, which is assured for a large class of densities satisfying criteria given in Eqs. 5-7, noninteracting pure-state vrepresentability cannot be determined beforehand. Nevertheless, Eq. 15 can be used to find this p among all noninteracting pure-state v-representable electron densities which minimizes The Kohn-Sham formalism owes its success as a practical tool in computer modelling polyatomic systems to the fact that the analytic forms of all but the Exc[p] terms in the functional EKs[i, <j>2,..., 4>N] are known. Practical computations rely, therefore, only on approximations to Exc[p\. The currently known ones are sufficiently good to make the Kohn-Sham formalism a tool of inestimable practical value in computer simulations of complex systems 48 .

2.3.

Energy functional depending functions E s [{£}, {4>fB}]

on two sets of

one-electron

Cortona introduced the total energy functional 22 in which the terms of unknown and known analytic dependence on p are partitioned in yet another way - as it is done in neither EHK[p] nor EKS[p\. Compared to the Kohn-Sham case, this functional, depends not on one but on several functions - electron densities of subsystems making up the whole system: p = PA+ PB+Pc-i

(17)

Only the case of two subsystems (A and B), the most relevant for the subject of the present review, will be considered hereafter. The energy functional of Cortona (ES[PA,PB]), can be expressed using the defined before functionals as: ES[PA, pB] = TS[PA] + Ts[pB] + Trd[pA,

PB] + V[pA + PB] + J[pA + PB]

+ EXC[PA+PB]

with the following definition of

(18)

T?ad[pA,PB}-

Trd[PA,PB]=TS[PA+PB}-

TS[pA] - T„[pB]

(19)

The Kohn-Sham and Es functionals yield the same numerical values for any pair of pA and pB (ES[PA,PB] = EKS[pA + PB\) provided pA and pB are both Nrepresentable. This follows from the fact that if two functions satisfy the conditions Eqs. 5-7 their sum does it too. Therefore, if both PA and pB are N-representable their sum is too. These two functionals differ, however, in the way the kinetic energy is evaluated in practical simulations. If the one-electron orthogonal functions

One-Electron Equations for Embedded Electron Density 9 ,A(LCS)^ ,B(LCSh { 5J {{4>i ^°>} and {4>"^°' }))

obeying the conditions, NA

(20) i=l NB

PB =

2^|0;B(LCS) ,2

(21)

i=l

and minimizing the kinetic energy in the Levy constrained-search procedure m i n 4 , o ^ M { < * s j f | * s >}] and m i n ^ < ! ^ p s { < * s | f | * s >}] are known, the numerical value of both T^p^] and Ts[pB] can be evaluated exactly. In such case, Ts [PA + PB] can be expressed in a hybrid way - as a sum of two functionals depending each on one set of one-electron functions and the functional depending on two electron densities (T™ad[pA,PB])-

Ts[PA

+

2Y//^LC^

pB] =

2

NB

E

,B(LCS)

4{LCS))

- ^

-¥>

bf{LCS)\+Trd[pA,PB]

(22)

Using the functional ES\PA,PB] in practical simulations hinges, therefore, not only on approximations to Exc[pA + PB] but also T™ad[pA, PB]- This brings us to the first challenge as far as practical applications of ES[PA,PB] in modelling is concerned - approximating T™ad[pA, PB] as an explicit functional depending on two electron densities. Our research on this subject will be reviewed in Section 3. Similarly as it was done for the Kohn-Sham functional in the previous section, we define here the functional Es[(pf, f,..., <^ B ] depending on two sets of one-electron functions (not necessary the ones minimizing T S [PA] and TS[PB])Within each set, the one-electron functions are assumed to be orthogonal. ES[^,

*A; f, <$, ..., 4>%B] = V[pA + PB] + J[PA + PB] + Exc[pA + PB]

+ 2£(tf

+ *£(*;

+Trd[PA,PB] •

(23)

^

The three functionals: EHK[p], H K S [{>;}], and Es[{
10

T. A.

Wesolowski

density and energy can be obtained as: E0 =

min

min

{<j,A}

PA—2NA PB—>2NB

=

min

>pA{4>f}

>PB

Es[cf>f,

$,..., <j>^A; 4>i , 4>2 > •••>
(24)

ES[PA,PB]

min .2W> »2JV B

pA PB

The first minimization corresponds to searching among all pairs PA and pB and the two innermost minimizations represent the Levy constrained-search procedures for Ts[pA] and Ts[pB], respectively: ES[PA,PB}=

•^SriA iA iA . ,B ,B ±B c •>\ ! V2 '•••' VNA>Vl 1V2 '•••' YNB\

min mm {4>A}^pA{?}^PB

+ PB] + J[PA + PB] + Exc[pA + pB] + T™d[pA,

V[pA

+

min {?}—>PA

V[pA

2V/V fr{

PB]

NB

1.

+

\

min {*f}—PB

2V t{

/

xB

- \ *

+ PB] + J[pA + PB] + Exc[pA + PB] Trd[pA,PB]+Ts[pA}+T3[pB]

+

V[pA

+ PB] + APA + PB] + Exc[pA + pB] + Ts[pA + pB]

EKS[PA+PB]

= EHK[PA

(25)

+ PB]

Eq. 25 provides an useful relation between the functional H s and other energy functional discussed previously. The functional ES[PA, PB] is obviously symmetric: (26)

ES[PA,PB]=ES[PB,PA]

Eq. 18 can be used to evaluate the total energy of the system for any pair of iV-representable electron densities PA and pB. It is, therefore, worthwhile to recall the model proposed by Gordon and Kim 4 9 in 1972 which can be seen as a particular application of Eq. 18. Gordon and Kim used the Hartree-Fock electron densities of isolated rare-gas atoms as p^K and p^K in Eq. 18 to evaluate the energy of a dimer. The interaction energy can thus be expressed as: rGK _

Ei,

ES[pGK^ pGK]

_ EKS^GK} _ j ^ g * ]

+

^ ^nucA,nucB

(27)

where EnuCAtnuCB represents the energy of repulsion between nuclei of the two subsystems. In the above expression, pAK and p^K are assumed functions which are

One-Electron

Equations

for Embedded

Electron

Density

11

plugged into all three functionals in Eq. 27. Owing to the fact that the total electron density of the complex was assumed to be the sum of unperturbed electron densities p^K and p%K, the TS[PA] and T3[PB] components cancel each other in Eq. 27 and do not contribute to EfJ£ at all. The interaction energy calculated from Eq. 27 depends obviously on the choice of p^K and p%K. Using the functionals defined before, the exact expression for the interaction energy reads: mi Es[p mm A >HB ]-E^[pA]-E^\pB]+E,

Et,

nucA,nuCB

(28)

where p™m / pA and p"gm ^ p°B because all these four electron densities are not assumed functions but they are results of the minimization for the respective functionals: Eq. 25 for Es[p%in, pg1™] and Eq. 13 for both EKS[p°A] and EKS\pB). The fact that the electron density of the complex might differ from that of the superposed electron densities of isolated molecules is taken into account in Eq. 28. We turn back to the energy minimization of H s (Eq. 24). Euler-Lagrange minimization taking into account the conditions of orthogonality for each set:

I

t*<j>fdr = 5,1,3

(29)

<j)?*4>fdr = *5., .j

(30)

leads to two coupled sets of one-electron equations 22 : Y72 _1_ vKSCED

r„ , „

-V + Veff

_

Y72

2

V

i \rK3CED

+ Veff

. -*

[PA,pB,rj

r

n

. -*

[PB,PA,r]

bf=eUf

l,j\r

i = l,Nh

bf=ef
(31)

(32)

The effective potential VjjfCED [pA,PB',r\ in Eq. 31 is neither the Kohn-Sham effective potential for the isolated subsystem A at the electron density p = PA'

VeKffCED [PA,PB;

A = V*f [PA + PB; rl +

5Trd{PA,PB]

(33)

5PA

nor the Kohn-Sham effective potential for the total system at the electron density PAB

=

PA

+

PBN„u

vSf^A-

Zi

PAB{?) f PAB(

,W ,

g - ^ J +J \r>j ^ ^

SExc[p]

Sp

(34)

12

T. A.

Wesolowski

To underline these differences, we refer to each of the coupled equations (Eqs. 31 or 32) as Kohn-Sham Equations with Constrained Electron Density (KSCED). For the same reason, the effective potential in these equations is referred to by KSCED effective potential0. The first set of one-electron equations (Eq. 31) leads to the minimum of the functional H s subject to the following constraint: 5pB = 0 which corresponds to freezing the electron density of subsystem B. The second set (Eq. 32) represents a similar constrained minimization for the other subsystem. The KSCED effective potential in Eq. 31 can be simply related with the KohnSham effective potential (Vjjf [PA + pB\A) f ° r the whole system: irKSCED Vejf

r . n .x\— l/KSf„ , _ . ^1 , ' " a " " IP A, PB\ [pA,PB,r\Vejf [PA + PB,r\^ — SPA

(%t.\ (35)

In general, however, STrd[PA,pB] Sp7A

, 5Trd[PA,pB] ^

,„.. ( 3 6 )

SPB

which results in different effective potentials in Eqs. 31 and 32: r . -* K/ -.rKSCED •/effeKffCBD\PA,p B;f\^v e ff [PA,PB\r\tV eff

VKSCED

V

\„

n

. -I

[PB,pA,r\

/o7\

(37)

As a consequence, the two sets of embedded orbitals {4>A} and {4>B} obtained from Eqs. 31 and 32 are not orthogonal in general. In the following, we will denote the electron densities derived from Eqs. 31-32 by pA:SCED and pgSCED. Both of them are obviously noninteracting pure-state v-representable. Eqs. 31-32 can lead to exact ground-state electron density po only if it can be constructed as a sum of two noninteracting pure-state v-representable components po = PA + PB- Due to this observation, four possible cases for a given choice of NA and NB such that N = NA + NB are possible: A) po is noninteracting pure-state v-representable and it can b e d e c o m p o s e d into t w o noninteracting pure-state v-representable c o m p o n e n t s pA and pB. Variational calculations using either EKS or H s lead to the same total electron density which is the ground-state electron density po = pASCED + c

Since 1996, we use t h e notation V^^CED in our publications for t h e following reasons: i) t o avoid confusion with t h e effective potential in K o h n - S h a m equations; ii) t o make evident t h a t it depends on two-electron densities independently a n d n o t on t h e s u m of t h e m (VjffCED =

VjjfCED [pA,PB',rl); Hi) t o indicate t h a t this potential is denned using universal functionals of electron densities. In t h e original paper b y C o r t o n a [Eq. 14 in Cortona, Phys. Rev. B, 4 4 (1991) 8454], t h e effective potential for each subset of embedded orbitals (jk) was referred t o as: Vext + j ^ ~ \

*fj

and Warshel, J. Phys.

1- LL~s^ Chem.,

M

^

• In our original publication (Eqs. 20-21 in [Wesolowski

9 7 (1993) 8050]), it was denoted by V*, ( p ( r ) ) .

One-Electron

Equations

for Embedded Electron Density

13

PB

B) po is noninteracting pure-state v-representable b u t it c a n n o t b e d e c o m p o s e d i n t o t w o noninteracting pure-state v-representable pA a n d pB. Eqs. 31-32 lead to the electron density pQ = pA:sCED + pB:SCED ^ p0 such HK HK that E [p~o] > E [po] as the consequence of the second HohenbergKohn theorem. C) po is n o t noninteracting pure-state v-representable but can be d e c o m p o s e d i n t o t w o noninteracting pure-state v-representable c o m p o n e n t s pA and pB. Eqs. 31-32 provide a better theoretical framework to reach the exact groundstate electron density p0 and energy E0 because p^SCED + pgSCED = po, S whereas Kohn-Sham equations cannot lead to po (PQ i=- po). D) po is n o t noninteracting pure-state v-representable d e c o m p o s e d i n t o t w o noninteracting pure-state c o m p o n e n t s pA and pB:

and cannot b e v-representable

Neither 'B.KS nor 3s can be used to reach the ground-state electron density (Po).

2.4. Energy functional and the embedding 2.4.1. Definition of

depending on one set of embedded electron density: SE[{4>AA};PB]

orbitals

ZE\{
The search for the minimum of S s (Eq. 24) must include all possible pairs pA and pB and its result does not depend on the search order. Eq. 24 can thus be rewritten as: EQ =

min pB^2NB

=

min A

pA^2N

min

min

min

pA—*2AM {<^}—> PJ1 {4>f

min pB^2NB

min {
Es[<j>f ,jfA;cj)f

,<j>B, . . . , < ^ B ]

}^PB

min

Es[f ,<j>B,...,

4>%B]

{(/>B}^pB

(38) Using the functional EE defined as: 2

S

[ ( 0 ; P B ] =

min B

{<£,

}—>PB

Zs[t,BNB]

(39)

14

T. A. Wesolowski

Eq. 24 takes the following form: E0 = =

min pB^2NB

min PA^2N*

min ZE[(frf, <j>£, ...,<J>^A;PB] {PA

min Es \PA,PB\ pA—>2N*

min pB—>2NB

(40)

EE is an explicit functional of the orbitals in one subsystem and the electron density of the other one (ps): H B [ { ^ } ; p B ] = 2 ^ < ?\ - V | # * > +TS[PB]

+Trd[pA,PB]

z

i=i

+ V\p]+J\p]+Exc\p]

(41)

where p=p

B

+ 2 ^ | ^ |

2

(42)

i=l

The embedding functional (EE{{fA}; ps]), which is the key element of the orbitalfree embedding formalism reviewed in this chapter, can be seen according to Eq. 39 as a contraction of the functional Es [<j>£, 2 > • • •, ^A ; i •> 4>2 J • • • i 'PN3 1 constrained search has been performed for one subsystem. If an arbitrarily chosen p1™1 is used in Eq. 40 instead of performing the full search among JV-representable ps the following inequality holds: E0^

min

min

EE[tf,

4>l..., K^PB™1}

(43)

pA—>2NA {4>f}—>pA We note, however, that: Theorem: For a system with the N-representable ground-state electron density (po), any Nrepresentable p%lal, such that p1™1 ^ po, can be used to find ground-state energy in the following search procedure: E0=

min PA

min

>2NA {4>A} >pA

S

B

[^,^

<x;/»n

(44)

Proof: The proof proceeds by constructing the set of orthogonal one-electron functions satisfying Eq. 44. Let us consider the electron density plJ^al defined as: -trial „ Arial PA - Po - PB

( AC\

I 4 5J p^ satisfies the conditions of N-representability (Eqs. 5-7): i) it is non-negative (the assumption), ii) the integral J pt^,aX(f)dr is an integer number (equal to 2N — 10,1

m

whic

One-Electron Equations for Embedded Electron Density

15

2NB), and Hi) p^10"1 being the difference of two functions integrable in the sense of Eq. 7, is also integrable in this sense. Therefore, p ^ l a ; is N-representable and the Levy constrained search can be performed. Let's consider all possible sets of orthogonal functions {4>A} such that:

2

E K A Wl 2 = PAZal (rl

(46) ,Ar"

and denote the ones which minimize the kinetic energy (2 J2iLi < t\~\^2\'t't with {4>t } (Levy constrained search). Ts[p^al)

2V<^|-^V2|0f

min

2

>)

NA ^A(LCS),

£<

_

1V2,^A(LCS)

(47)

i=l

Using {4>t(LCS)}

to evaluate S E [ { ^ f ( L C S ) } ; pgio<] leads to

EE[{
^

=

N ' -

<(hA{LCS)\_ ^LCS^ _ 1V 2 | ^ ( L C 5 ) > +

<

^ , ^ 1

1=1

nnad nad r Jtriala

+ Ts

Arial~\ [p' A - ', p]

mm i=l nnadl atrial] d M ial + Tr [PAatrial , PB ] + V[po] + APO] + Exc[po]

TpSl.trial [PA

-k

= EKb[p0

J,rial] >PB \

- p%tal

J?S r „

- & [Po + p%lal\

trial] trial tr PB >PB

= EKb[Po]

= E0

(48)

which ends the proof. Corollary: For p%%al not satisfying the condition p ^ l o i ^ po, E0<

min

min

triah ZE{<j>A, cj>A,..., 4>NA; p%

(49)

16

T. A. Wesolowski

2.4.2. H [{<j>iA}] PB]

as

a

basis for multi-level

modelling

The above theorem and the corollary are the key elements in our strategy to use electron density partitioning in multi-level modelling of complex materials where P g m i can be identified with the environment of an embedded system. To reach the exact ground-state, p g l o i does not have to be the result of any differential equation but it must belong to a certain class. This class is, however, very large and comprises all 7V-representable functions p g m ! such that po — p1^1 ^ 0. For this class, Eq. 44 holds and the search for the ground-state electron density (p0) can be replaced by the search for the difference po — p%lal. It is worthwhile to underline that p^10,1 might represent a subsystem comprising even a macroscopic number of electrons. Eq. 44 leads thus to a new interpretation of Eq. 31 which was introduced as a partner to be solved simultaneously with the coupled equation for another subsystem (Eq. 32). The functional EE[{4>f};ps] is well defined also for any ps which can be plugged into equation Eq. 31. In particular, it can be an electron density (pg l a ') derived not from Eq. 32 but from some other type of calculations. It can be used, therefore, to perform the search for the ground-state energy (EQ) and density (po) according to Eq. 44 provided the electron density used as p g M i satisfies the condition po — p%lal ^ 0. Let's denote the electron density derived from Eq. 31 for a given p%lal by p™"'. The electron density p™m is obviously noninteracting pure-state v-representable. Therefore, if po — plpal does not belong to the class of noninteracting pure-state vrepresentable electron densities, it cannot be obtained from Eq. 31. As a consequence p™"+p*"^p0

(50)

= EHK[pr2in

(51)

and Es[pTn,PBia']

+ p%ial] > E0

in such a case. Note that the above inequality holds even if p 0 — p%ml is iV-representable. Unfortunately, no simple criteria to determine noninteracting pure-state v-representability are known. This condition is more restrictive than that of 7V-representability of po — p%%al and cannot be established a priori for a given function. Es\pJin,p%ial] as well as p^"™ + p%ial might depend on the choice for p%ial if the condition of noninteracting pure-state v-representability of po — p%lal is violated. The dependence on the results derived from Eq. 31 on the choice for plg%al might be studied by means of full minimization of the functional Es (solving Eqs. 31-32) with respect to variation of both PA and ps • The fact that the exact ground-state electron density can be obtained as a sum of two components: a trial electron density belonging to the class of functions, such that px = Po — p%ml is noninteracting pure-state v-representable, and the electron density derived from Eq. 31 provides a formal basis for the electron-density-partitioning based strategy in multi-level modelling. If subsystem A represents a small part of the

One-Electron Equations for Embedded Electron Density 17 whole mesoscopic or macroscopic system, obtaining p^10,1 from quantum mechanical orbital-based methods is not practical but it can de derived from other approaches adequate to the macroscopic scale. The noninteracting pure-state v-representability of p0~plf^al cannot, however, be assured in such case. The freeze-and-thaw cycle (see Section 5) offers a practical way to verify noninteracting pure-state v-representability of po — p%lal • For nano-, meso-, or macroscopic systems it is not possible but it can be performed for small chemically relevant model systems. Using ptglal obtained not from Eq. 32 but from other methods involves, however, an additional practical restriction as far as the evaluation of the total energy is concerned. To this end, numerical value of Ts [p] is needed. Unfortunately, currently known approximations to Ts[p] are not sufficiently accurate to be used for this purpose. We note, however, that the numerical value of Ts\ptglal] is not needed in such cases as: • If the principal object of interest is the embedded electron density PA, embedded orbitals {{4>i}), o r electronic-structure-dependent properties of the subsystem A, the knowledge of the numerical value of Ts[ptg'lal] is not needed at all. • If the process under investigation involves mainly changes of the geometry and electron density localized in subsystem PA assuming that ps is constant in the process under investigation might be justified. In such case, the numerical value of T s [ps] is not needed because it is constant and all properties depend on the difference Es[pA, PB] — EHK[PB\2.4.3. The orbital-free pB-dependent

embedding effective potential

The KSCED effective potential for the embedded system can also be simply related to the Kohn-Sham effective potential for the isolated system Vjff [PA\ r\:

veKffED [PA, PB-A = veKff [pA; *=] + vgf [pA, PB; A

(52)

where

^w^f-^+Z^*SExc[p] 8p

SEXC [p] P=PA+PB

P

ST d +

r \PA,PB]

(53)

SPA

The embedding potential of Eq. 53 does not depend explicitly on neither 4>A nor (j)B orbitals. It is expressed entirely by means of the functional derivatives of universal system-independent density functionals. Therefore, we refer to it as orbital-free

18

T. A. Wesolowski

embedding potential or KSCED embedding potential. It offers a possibility to introduce efficient computer simulation techniques for large systems. We recall here that electron density is a well defined quantity in both microscopic (atomistic) and macroscopic scales. Therefore, electron density based coupling between the two subsystems given in Eq. 53 provides a general formal framework suited for multi-level modelling allowing for the use of different physical laws for various parts of the system. In particular, the orbital-level of description for the subsystem of the primary interest (say PA) can be maintained with a complete accounting for it environment represented only by its electron density which could be obtained in a simpler way 50 . Our original applications of the orbital-free embedding formalism to study solvated molecules (H2O or CH4) 51 or a model complex for the proton transfer reaction (FHF~) 5 2 can serve here as representative illustrations. In all these applications, Eq. 31 was used to derive electron density of the solvated system, but the electron density of its environment (PB) was not derived from the partner equation (Eq. 32) but it was obtained in a simplified way: as a superposition of electron densities of isolated solvent molecules. Average properties of the solvated systems were obtained using a sample of configurations of solvent molecules. For each configuration, Eq. 31 was solved using pB approximated as a sum of electron densities of all solvent molecules. Generating pg in this simplified way represents thus a negligible computational cost compared to solving simultaneously the two coupled equations (Eq. 32 and Eq. 31). In our recent study of solvatochromic shifts (see the Application section), this simplifying assumption was analyzed in detail showing that such such treatment of solvent molecules surrounding the organic chromophore is a justifiable approximation 53 . It is useful to relate the KSCED embedding potential with the term used in various computational methods following the embedding strategy to represent interactions between the subsystem described at the orbital level with its environment. There is a vast literature concerning the 'embedded molecule approach'. For pioneering papers in chemistry and solid state physics Refs. 54>55. Methods known under such names as QM/MM, QM/MD, hybrid methods, etc. are currently widely used in studies of large systems such as biomolecules, materials and solids, surfaces, solvatation. Refs. 56>57>58>59 provide comprehensive reviews. In these methods, the approximate solutions of the Schrodinger equation are obtained using a modified isolated-subsystem Hamiltonian into which certain terms (coupling terms) are added to account for the presence of the environment. Usually, these additional components of the potential are system-dependent and depend on empirical parameters. The KSCED embedding potential given in Eq. 53 provides thus an universal firstprinciples based form of the coupling terms known in empirical approaches. The simplest and the most commonly used form of the coupling term comprises only electrostatic components (electrostatic embedding). It has been combined with Kohn-Sham calculations as well as with wave-function based methods. In the electrostatic embedding, only the first two components of the whole embedding poten-

One-Electron

Equations for Embedded Electron Density

19

tial are retained. They represent Coulomb interactions between the electrons in the investigated system and the nuclear charges and electrons in its environment. Moreover, these terms are frequently simplified further using the multicenter expansion for the electrostatic potential generated by the electrons and nuclei in the environment. Such a simplified embedding scheme is computationally not expensive but has to be used taking into account that the positive electric charges in the environment attract electrons from the subsystem QA which can make the results numerically unstable. The results of Rothlisberger and collaborators who reported the charge spill-out problem in calculations using plane-waves as the basis set 60 can be used as a good illustration of such difficulties. These authors proposed a practical remedy to this problem - an ad hoc procedure of scaling down electrostatic attractions at close distances 60 . This brings us to more sophisticated type of coupling terms which include not only electrostatic embedding but also other terms to account for exchange repulsion or intermolecular Pauli repulsion effects. For instance, Schaffer and J0rgensen applied such terms in his Angular Overlap Model 61 to account for non-electrostatic effects on the electronic states of transition metal centers in crystal environment. Recently, Wheatley used such empirical terms to study polarizabilities of anions in crystal environment 62 . Such terms can be seen as the approximations to the non-additive kinetic energy component of the KSCED embedding potential given in Eq. 53. The above examples indicate that taking into account non-electrostatic components of the embedding potential is indeed indispensable. The most straightforward and formally sound route to account for nonelectrostatic effects is provided by the use of the pseudopotential concept. As shown by Phillips and Kleinman 63 , the orbitals in a selected subsystem (pseudoorbitals - {
(54)

HPK differs from the all-electron Hamiltonian (H) by an additional dependent term (pseudopotential) which depend explicitly on the all-electron orbitals ({fa}) and the corresponding eigenvalues of the all-electron Hamiltonian ({«i}): Ncore PK

H fa = Hfa + Yl (e* " ei)(^l&)&

(55)

The above equation, although provides an exact relation could be useful in practice only if the pseudopotential can be reasonably well approximated without knowing the all-electron orbitals {fa} and eigenvalues {ej}. Such approximations are available to separate core and valence electrons. The Phillips-Kleinman formal route can also be used to separate electrons in different molecules 64 ' 65 . This route is, however, not orbital-free. The environment needs to be described at the orbital-level. Therefore, this group of methods will not be discussed further here.

20

T. A.

Wesolowski

We underline also the fundamental difference between the orbital-free embedding route (Eq. 53) and that offered by Phillips-Kleinman pseudopotentials. In the calculations based on Eqs. 31-32, the embedded orbitals of two different subsystems are not subject to the orthogonality condition which is explicitly imposed in Eq. 55. 3. A p p r o x i m a t i n g t h e unknown c o m p o n e n t s of t h e H s functional For each functional discussed previously, the role of its components with unknown analytical form is different (see Table 1). Currently, approximating the exchangeTable 1. E

Approximate components of the functionals EHK[p],

EKs[{i}],

Es[{f}; {f}], a n d

A

E [{<j) }\PB] and the corresponding potentials for variational calculations. Unknown components

Energy Functional Energy

Effective potential

EHK[P]

Ts[p] + Exc[p]

ST M S

sKSl{M]

EXc{p]

<5£*cM

s s [{0 l A };{^f}]

TTd[PA,PB]

and

6

Y"]

Sp

6T™d[pA,pB] SpA

+ Exc[p\

'

ST™d{pA,pB] SpB '

and *%w

Sp ^constant HB[{^};PB]

Trd[PA,PB]+

Ts[pB]

+Exc[p\

sTryA,PB]

and

SE.M

correlation functional (and the associated potential) is the object of great interest (for a review see Ref. 15 ) because of its key role in the Kohn-Sham formalism. To apply, Es or and EE in practical computer modelling additional challenges appear: i) development of an useful approximation to T™ad{pA, PB\ and it) assuring the best overall accuracy of the sum of all approximate components of the energy functional. These issues will be discussed below. 3.1.

The bi-functional

of the non-additive

kinetic

energy:

Trd[pA,PB] Application of the functionals 3s and EE in computer simulations hinges on approximations to T™ad[pA,PB}- This functional is explicitly used in the evaluation of the total energy, whereas its functional derivative is one of the components of the KSCED effective potential. Unfortunately, the analytic form of this functional is not known except for some particular cases. Below, our attempts to approximate this functional and its functional derivative will be reviewed. We start with an overview of exact properties of T™ad[pA, PB\-

One-Electron

Equations for Embedded Electron Density

21

3.1.1. Exact properties of T™ad[pA, PB] Zero-overlap case If the electron densities of two interacting subsystems {PA{T) and ps(rO) do not overlap (PA(^) • PB{^) = 0), the kinetic energy of the whole system (Ts[pA + PB]) is just the sum of the kinetic energies of the two isolated subsystems (TS[PA\ and Ts[pB}). Therefore,

Trd[pA,PB}=0

(56)

for non-overlapping pairs of electron densities. A pair of electron densities of two molecules separated by a large distance provided the most relevant chemical system for which the overlap disappears at infinite intermolecular separation. For small overlaps, occurring at intermediate separation T™ad[pA, PB] =£ 0 but its numerical value can be expected to be small. For this reason, pairs of electron densities corresponding to molecules involved in a weakly bound intermolecular complex provide an ideal testing ground for approximations to Trd[pA, PB}L D A limit For a homogeneous electron density in a given volume V, the kinetic energy can be derived form the electron gas theory. For a pair of homogeneous electron densities (PA(f) = PA9 a n d PB(r) = pB9) the analytic form of T™ad[pA, PB] obtained using Thomas-Fermi kinetic energy functional applied to TS\PA + PB], TS[PA],

and

TS[PB]

reads:

TrdlPA9, PhB3} = CTF Jy {[PA9 + PB9f3 - (PA9)*" ~ (PB9 ) 5 / 3 ) df (57) where CTF

= ^(3TT2)2/3.

In our work on development of approximations to T™ad[pA, PB], only such approximations which satisfy the T^ad[p^, pB9] limit were considered. This restriction has to be underlined in view of the fact that it is a common practice in to sacrifice known exact conditions in the process of parametrizing approximate functionals. For instance, a very good approximation to Exc[p] developed by Handy and collaborators 66 does not obey homogeneous electron gas limit. Partitioning electron density in four electron s y s t e m s T™ad[pA,PB] can be expressed analytically for a pair of electron densities (pa and pa ) obtained from the following partitioning of a noninteracting pure-state v-representable electron density {p^) in a spin-compensated four-electron system:

22

T. A. Wesohwski p<,2>=a-2-|fc|2 + ( l - a ) - 2 - | f c | 2

(58)

and p = (1 - a ) - 2 - | 0 ! | 2 + a • 2 - | ^ | 2

(59)

where 0 i an(A 4>2 are two Kohn-Sham orbitals for this system. The orbitals 4>\ and 2 are not uniquely defined because any pair of Kohn-Sham orbitals can be subject to a unitary transformation without changing neither the total density (pg ') nor the kinetic energy {TS[PQ ']). Each of the considered Kohn-Sham orbitals is doubly occupied and generates the corresponding orbital-density (pi = 2 • |(/>i[2 and p 2 = 2 • |2|2). For the pair p£° and p^\ the analytic form of T™d[p{a\p(?>] reads 67 :

n d

r lPi"\pi")} = JlfHdr

(60)

where: f(n\ J[

=

'

P±±P2 a(l-a)lpiVp2-p2Vpil2 P1-P2 ' a ( l - a) (p? + p | ) + P i p 2 ( 2 a 2 - 2 a + 1)

ffm l j

°

The function / ( a ) is non-negative. As a consequence, T™ad[pa ,pa ] is also nonnegative. It equals to zero only for such pairs p\ and p 2 for which p i / p 2 = const which cannot occur here because pi and p 2 are orbital densities corresponding to two orthogonal Kohn-Sham orbitals. t> A B -representable pairs of electron densities Here, we identify a large class of pairs of electron densities with great relevance to chemical systems which will be referred to as w j4B -representable. Definition: The pair of electron densities p& and pB is vAB -representable if there exists an external potential v for which one can choose a subset of corresponding Kohn-Sham orbitals yielding pA, whereas the remaining orbitals yield ps-

One-Electron

Equations

for Embedded Electron Density

23

We use the term "a t> AB -representable pair of electron densities PA and ps" to avoid confusion with a "pair of noninteracting pure-state v-representableAi electron densities". NA + NB Kohn-Sham orbitals for a given system can be used to generate NBtNA\ •y' 4B -representable pairs. According to the above definition, the construction of a -u^-representable pair for a given external potential vext is quite challenging a task because it involves obtaining the exact Kohn-Sham orbitals corresponding to the potential vAB which is the same as vext (vAB = vext). However, t ^ - r e p r e s e n t a b l e pairs PA and PB can be easily constructed from Kohn-Sham orbitals obtained using also an approximate exchange-correlation potential vxc. In such case, the generating external potential vAB cannot be identified with the external potential vext. The two quantities differ by more than just a constant. vAB includes also the difference between the exact (VXC(PA + PB)) and the approximate (VXC(PA + PB)) exchange-correlation potentials calculated at p = PA + PB• Although the analytic form of T™ad[pA, PB] is not known for a u^ B -representable pair of electron densities, the following theorem holds 68 : Theorem: For any vAB-representable

pair of electron densities: Trd[PA,PB]^0

3.1.2. Approximating

T™d[pA, PB} and

(62)

6Tr l P PB] s p *'

Any function of two electron densities (PA and ps), their gradients (VpA and VPB), and higher derivatives (V2PA and V 2 p s , etc.), can be used to approximate T™ad[pA, PB}- Obeying the LDA limit restricts the form the approximate functional to:

frd[PA,PB} = CTF JV ((PA + PB)5/3 - (PA)5/3 - (PBf3) + f f{pA,PB^pA,VpB,^2pA^2pB,-)df Jv

df (63)

where / can be, in principle, any function such that: f(pA,PB,0,Q,0,...)

=0

(64)

i.e. it disappears if the two electron densities are uniform. In our analyses of the accuracy of various approximations, only such functions / were considered which could be obtained as: Trd[pA,

PB} « fs[pA

+ PB} - fs\pA]

- fs[pB]

(65)

24

T. A.

Wesoiowski

where Ts [p\ is used to denote an approximate functional of the kinetic energy. The above restriction, ties the development of the approximations to T™ad[pA, PB] to progress in approximating Ts [p]. The analytic form of the corresponding functional derivatives can be also obtained as: 6Trd[PA,pB] 5pA

_ 8Ts[p] Sp STS[PA

5Ts[p] P=PA+PB

+ PB}

5pA

"

ST8\pA]

SpA

P=PA

5T3{PA+PB}

SpA

Sfs{pA]

5pA

(66)

In principle, testing the accuracy of a given approximation to f™ad[pA, PB\ would require exact reference data for T™ad[pA, PB]- In general, such reference data for a given pair of electron densities pA and PB can be obtained by means of the Levy constrained-search procedure and the definition of T™ad[pA, PB]- In the embedding potential of Eq. 53, however, not T™ad[pA, PB] but its functional derivative is used. It is, therefore, useful to start with the analysis of the accuracy of approximations OPA

A simple numerical procedure allows one to test the accuracy of any approximate —" ^IPA,PB\ ^ rj,j^g p r o c e ( j u r e consists of the following steps 69 : A Obtaining t h e reference electron density (po ) from Kohn-Sham calculations applying a given approximation to the exchange-correlation functional Exc[p\. The reference density pj does not depend on T™ad{pA, PB] at all. B Obtaining t h e —" ^PA'PB'-dependent electron density po = PA + PB where both components pA and p s are derived from Eqs. 31-32 and where the same approximation to the exchange-correlation functional as the one used in the reference Kohn-Sham calculations is used. C Comparing pr^

and po.

In the above procedure, any difference between pg and po can be attributed uniquely to the inaccuracy of the applied —° S[PA'PB' provided all other details of the computer implementation (grids, basis sets, etc.) are the same in these two types of calculations. We recall here the hypothetical case of p^ which cannot be decomposed into two noninteracting pure-state v-representable contributions. In this case, Eqs. 31-32 cannot lead to pTQ . Nevertheless, the exact —" s'PA'PBi would lead to the smallest

One-Electron

deviation between pg

Equations

for Embedded Electron Density

25

and p0 for such case. It can be used, therefore, to identify the

best approximation to —° sf*

• The described testing procedure was used in a

series of our works where various theoretical routes to approximate —• 69 70 71 72

d

[PA,PB

> were

S aT

explored ' ' ' . The tested approximations to T™ \pA,pB] and to ""^'PB] where obtained using Eq. 65 and the following approximations to Ts[p\: i) the gradient-free one proposed by Zhao, Lee, and Parr 7 3 ; ii) the zeroth- (GEAO) and second-order (GEA2) terms in regular gradient expansion of the kinetic energy 74 ; Hi) several gradient-dependent approximations of the generalized gradient approximation (GGA) form. Below, the accuracy of the last two groups of approximations will be discussed in detail. Gradient Expansion A p p r o x i m a t i o n route For the kinetic energy functional, regular gradient expansion approximation (GEA) 7 4 takes the following form:

ra=0

n=0 ^

For atomic and molecular systems, the zeroth-order term derived by Thomas and Fermi 2 3 ' 2 4 , which reads:

P5/3dr

TO[P] = CTFJ

(68)

underestimates T3[p] by about 10%, whereas adding the second-order term:

T2\p] = l-l[F£dr>=±Ty\p] 9 8J p 9

(69)

results in a significantly better approximation. Combining Eq. 65 with GEA for Ts[p] leads to the following approximations to

TrdlPA,PB\: j^GBAO)

J ( (pA

[pAj pB] = CTF

+ pBfS

pf

_ pW}

d?

{7Q)

in the zeroth order and

ir d(GBA2) [PA, PB] = CTF J ((PA + pBf3 - p5f 1

f \pAVpB-

72 J ;

in the second order.

/

•

PAPB(PA

PBVPA\2

-df

+ PB)

PT) d?

(Tl)

26

T. A. Wesolowski

The

ST

corresponding

PB]

" 2T

read:

analytic

5Trd(GEA0)[pA,PB]

expressions

for t h e functional

_ 5 ^ _ _ ((n , ,2/3 2/3 CTFUpA+PBr-pT

$PA

derivatives

(72)

in the zeroth order and STnad{GEA2)[

^

B]

/

5

2 / 3

2 / 3

\

= -3CTF {(PA + PB)' - Pi ) 1

(V2{PA+PB)

36 V

PA

+

V2PA PB

PA

B ) | 2IV ^1 f (\V(PA ' V ( p A ++PPB)\ -^) 2 2

+

(73)

in the second order. The second-order term in the regular gradient expansion multiplied by the factor of 9, known as the von Weizsacker functional 75 (T s w [p]), provides also the exact expression for the kinetic energy in one-electron- or spin-compensated two-electron systems. It can be expected, therefore, that GEA applied in Eq. 65 leads to a good approximation also to T™ad[pA, pg] obtained using Eq. 65. Unfortunately, testing the accuracy approximated using zeroth- and second-order GEA using of ^ r ^ . ^ 1 the testing procedure described in the previous section showed that the second-order contribution does not lead t o any improvement. The second-order contributions does not improve but even worsens the accuracy of —' ^PA'PB> 6 9 . Fig. 1 shows the electron densities derived from Eqs. 31-32 in which zeroth- and second-order GEA w a s for —'' jp'PB used for the H 2 ...NCH complex at the equilibrium geometry. PA and ps correspond to the monomers forming the complex. The PA,PB overlap is small but not negligible in this model system. Moreover, forming a complex affects the electron density of each monomer. Especially, electron density of H2 becomes polarized due to the large dipole moment of NCH. Zeroth-order GEA, leads to a reasonably good approximation to —" S]-PA'PB' as indicated by the similar qualitative behavior of the electron density derived from the reference calculations (Kohn-Sham - not depending on approximations to T™ad[pA, PB}) and Eqs. 31-32. The second-order contribution to T™ad[pA, PB] worsens qualitatively the results. The second-order terms result in a similar effect for in the HF...HF complex for which it was analyzed for several intermolecular distances. Since the PA,PB overlap is different for each geometry, such analysis makes it possible to detect systematic effects. Fig. 2 shows that the presence of the second-order GEA term

One-Electron

Equations

for Embedded

Electron

Density

27

0.03 Kohn-Sham reference o—o Tnad: GEAO - B — a Tnad: GEA2

0.02

0.01

0.00

-0.01

-0.02

-1.5

-1.0

-0.5

0.0

1.0

0.5

X[A] Fig. 1. T h e interaction induced deformation of t h e electron density of t h e H2 molecule resulting from t h e interactions with t h e H C N molecule in t h e H2--.NCH linear intermolecular complex oriented along t h e 2-axis (zn1 = 0 A, z # 2 = - 0 . 7 7 A, zjv=-2.45 A). GEAO and G E A 2 approximations [PA,PB] were used in Eqs. 31-32. Reference d a t a derived from K o h n - S h a m calculations. to SPA Details can be found in [Wesolowski and Weber, Intl. J. Quant. Chera., 6 1 (1997) 303].

worsens systematically the accuracy of

&T™d[pA,pB\ SpA

. At larger intermolecular dis-

tances (d(FF) ^ 3.5 A), where the PA, PB overlap is small, errors of are smaller than that of —a

^

L

•

5T"

j —

[PA,PB]

'. At very small intermolecular separations

(d(FF) Z 2.0 A), the accuracy of both ^

\PA2Rsl OPA

and * r ' ( 7 " W » l

^

opA

teriorates rapidly with increasing PA,PB overlap. The fact that the second-order contributions worsen rather than improve the accuracy of —" ^PA,PB' is an intriguing result emerging from our analyses described in detail in Refs. 69>76. We find the rationale for this by recalling discussed previously formal results concerning nonnegativity of T™d\pA , ps] in some cases. The four-electron density partitioning case is less relevant for the systems on which the accuracy of 5T"ad5[pp^PB] derived using GEA was tested. But the case of a w" 4s -representable pair of electron densities is much more general. It is striking to note that, the second-order GEA contribution to T™ad[pA, PB} is non-positive for all pairs of electron densities (for u A B -representable and not •unrepresentable pairs alike). This follows from its explicit analytic form given here in the second line of Eq. 71. We recall now that the second-order GEA term (T2 in Eq. 69) is closely related to the von Weizsacker functional T^lp}75 v {T^ }p} = 9T2[p\), which is the exact kinetic energy functional for one- and twospin-compensated electron systems. Using T^[p} in approximating T"a }PA,PB}

28

T. A.

Wesolowski

15.0

5.0

|

-5.0

-15.0 G B

OTnad: GEA(O) H Triad: GEA(2)

-25.0

-35.0

1.5

2.0

2.5

3.5

3.0

4.0

4.5

5.0

6.0

d(FF)[A] ad

A,pB] Fig. 2. Accuracy of approximations to ST? [p obtained from gradient expansion approxiSPA mation (zeroth- or second-order GEA) for the HF...HF dimer at various intermolecular separations d (d(FF)). The difference between the Coulomb energy (J) obtained using approximate 6T™ [pA,pB] $PA in Eqs. 31-32 and that (Jref) obtained from supermolecular Kohn-Sham calculations, which do not a of aaccuracy d e p e n d On ^,, 'is ^ used LUSCU aas o umeasure ^ o o u i c VJI . v , i j u i c ^ , j uof i aa ggiven , i v c i i aapproximation . ^ j j j n j A ± i i i a , i i i u i i to \>yj — —IPA.PBI j Details of the calculation are given in [Wesolowski, Phys. Rev. Lett., 88 (2002) 209701] and references therein.

leads, therefore, to a gross violation of the derived inequality. T^ [p] has been proposed by some authors as the starting point for development of better approximations of the general form 77,78 : Ts[p] « T™[p} + small correction

(74)

Eq. 62 indicates that the contribution of the "small correction" to T™ad[pA, PB] must be non-negative and and must overweight the negative contribution due to T^. Therefore, in our searches for approximations to T™ad[pA, PB] neither routes that of Eq. 74 nor GEA was followed further. d

PB) The analyses described above shows that accuracy of 6Tr [PA, deteriorates SpA upon introducing the second-second order GEA terms. In view of possible applications of Eqs. 31-32 for studies of potential energy surface related properties (geometries, interaction energies, harmonic frequencies, etc.) it is useful to compare the energies obtained using a given approximate functional T™ad{pA, PB] to the reference Kohn-Sham results which do not depend on the non-additive kinetic energy functional at all. The differences between the reference supermolecular Kohn-Sham energies and the ones derived from two coupled KSCED equations cannot, however, be analyzed in terms of the accuracy of T"ad[pA, PB] in a straightforward manner.

One-Electron Equations for Embedded Electron Density

29

The accuracy of the energies derived from Eq. 23 using electron densities derived from Eqs 31-32 depends on the accuracy of two approximate quantities: SpA and T™ad[pA,PB}- The discrepancy between the KSCED energy obtained with a given approximation for T™ad[pA, PB] and the corresponding Kohn-Sham reference data can, however, be used to select the most appropriate approximation for applications concerning primarily interaction energies. The smaller are these deviations the better is the approximation. Fig. 3 shows the potential energy curves corresponding to for the HF...HF dimer obtained using GEAO and GEA2 approximations and the Kohn-Sham reference. Results for other hydrogen-bonded intermolecular complexes can be found in Refs. 69>71>72. They lead to the similar conclusion as that one emerging from the studies concerning the accuracy o f

ot,

WA,pBi.

n e i t h e r

G E A 0

nor GEA2 are adequate for studies of the energetics of weakly overlapping pairs of electron densities pA and pB50.0

G B

40.0

TH

Kohn-Sham reference O Triad: GEA(0) B Triad: GEA(2)

30.0

0.0

-10.0

1.5

3.5 d(FF) [A]

5.5

Fig. 3. The interaction energies in the HF...HF complex at various intermolecular separations calculated using Kohn-Sham calculations (reference) and Eqs. 31-32 with GEAO and GEA2 approximations to TJ}ad[pA,pB]- Details of the calculation are given in [Wesolowski, Phys. Rev. Lett., 88 (2002) 209701] and references therein.

Generalized Gradient A p p r o x i m a t i o n route We attribute the failure of the gradient expansion approximation in approximating T"ad[pA,PB] in the case where PA and ps do not overlap significantly as it is the case of two weakly interacting molecules to the 'artificial' attraction arising from the second-order GEA contribution to T"ad[pA,PB}- Interestingly, in the recent ar-

30

T. A. Wesolowski

tide Chai and Weeks found that eliminating the Laplacian-dependent component from the second-order GEA analytic expression for T^p' improves qualitatively the results of variational orbital-free calculations 79 . Locally, the magnitude of this contribution might be larger than that of the zeroth order. This might happen far from nuclear centers. So-called reduced density gradient s(r) defined as: s(f) =

(75)

\Vp\/(2pkF)

1 /S

where, k-p — (37r2p) , is a dimensionless quantity measuring locally the magnitude of electron density gradient. Fig. 4 shows a typical behavior of s. It increases rapidly with the increasing distance from the closest nucleus.

Fig. 4. Variation of the reduced density gradient s(f)

|vp| 2p(37rV) 1 / 3

along the Civ axis in the

CsHg...02 complex. The benzene plane corresponds to z=0.0 A and the O2 molecule to z=3.0 , Data taken from [Wesolowski et al, J. Chem. Phys., 108 (1998) 6078]. It is convenient to rewrite the sum of the first two terms in the regular gradient expansion Eq. 67 as: T?EA*\p]

= CTFjp^(r)(l

= CTF

+

f p5/\ff)FGEA2(s)df

\^~/)d?

(76)

One-Electron Equations for Embedded Electron Density 31 We recall here the adequacy of the Generalized Gradient Approximation (GGA) to exchange-correlation energy functional. It was introduced to overcome the very disappointing accuracy of the exchange-correlation functional derived using Gradient Expansion Approximation 8 0 , 8 1 ' 8 2 . The idea behind GGA is to obtain the analytical form of the gradient-dependence of the functional in question (Exc[p\) not from gradient expansion approximation but from other theoretical- (the B88 8 3 , P W 9 1 8 4 , PBE 8 5 exchange-correlation functionals, for instance) or empirical- (HCTH 6 6 ) considerations. Opposite to its exchange-correlation counterpart, the functional Ts[p] can be very reasonably approximated by means of gradient expansion approximation truncated to the second order 86 . The accuracy of the corresponding T"ad[pA, PB] functional is, however, very disappointing as discussed in the previous section. The encouraging experience with GGA route for Exc[p] suggests a similar solution for

TrdiPA,pB\TsGGA{p] = CTF Ip5^(f)F(s(r-))df

(77)

The above form includes also the Thomas-Fermi functional (LDA) for which F(s) = const = 1. Fig. 5 shows the considered enhancement factors: GEA2 -

Fig. 5. E n h a n c e m e n t factors F{s) (dimensionless) considered in generalized gradient approxim a t i o n s to Ts[p\. T h e second-order gradient approximation is denoted by G E A 2 . O t h e r G G A approximations t o Ts[p] r e p o r t e d in t h e literature are denoted by: LC94 [Lembarki and Cherm e t t e , Phys. Rev. A, 5 0 (1994) 5328] L L P [Lee et al, P h y s . Rev. A. 4 4 , (1991) 768], P W 9 6 [Fuentealba and Reyes, Chem. Phys. Lett. 2 3 2 , (1995) 31], P B E [Perdew et al, J. Chem. Phys., 1 0 8 (1998) 1522], T K 9 2 [Thakkar, Phys. Rev. A, 4 6 (1992) 6920],

32

T. A.

Wesolowski

second-order gradient expansion approximation, LC94 - of the analytic form used in the Perdew-Wang (PW91) exchange functional 84 reparametrized for kinetic energy by Lembarki and Chermette 8 7 , LLP - of the analytic form used in the Becke (B88) exchange functional reparametrized for kinetic energy by Lee, Lee, and Parr 8 8 , PBE - of the analytic form used in the Perdew-Burke-Ernzerhof (PBE) exchange functional 85 tested also for kinetic energy 89 , PW86 - of the analytic form used in the Perdew-Wang (PW86) exchange functional 82 reparametrized for kinetic energy by Fuentealba and Reyes 90 , and TK92 - proposed by Thakkar 9 1 . In the TK92 approximation, a linear combination of two enhancement factors is used: that of the B88 exchange functional and the one derived by Ou-Yang and Levy 92 taking into account the scaling conditions for higher-than-second-order contributions in GEA. Except for the TK92 9 1 functional, all of them behave similarly at small reduced density gradients: F(s) -> 1

at s -> 0

(78)

0

at s -> 0

(79)

1 36CW

at s -> 0

(80)

dF(s) ds d2F(s) ds2

Their initial growth (up to about s = 1.6) matches that of GEA2. For very large s, their behavior is qualitatively different: FPW91(s)^0 FLC94(s)—0 FPBE(s)

-> 1.804

for

s^oo

(81)

for

s^oo

(82)

for

s -> oo

(83)

whereas other considered enhancement factors diverge at s —> oo but slower than that of GEA2. Despite their similar behavior at small s (see Fig. 5) the corresponding KSCED results vary significantly. Figs. 6-7 show a very strong dependence of the results on the choice of the enhancement factor F(s) in the GGA approximation for —? gfA f° r two hydrogen-bonded intermolecular complexes. Our dedicated studies on various weakly bound systems 6 9 ' 7 1 indicate that the enhancement of the analytic form proposed by Perdew and Wang for the exchange functional 84 and which was reparametrized for kinetic energy by Lembarki an Chermette 8 7 leads Not visible at the scale of the picture is that the TK92 enhancement factor has a sightly negative slope at s —> 0.

One-Electron

Equations

for Embedded Electron Density

33

15.0

5.0

-5.0

E "a 2

-15.0

G — G Triad: GEA(O) Q BTnad:GEA(2) x x Tnad: GGA(LLP) O OTnad: GGA(LC94) A — A Tnad: GGA(PW86) -+Tnad:GGA(OL91) V — V T n a d : GGA(TK92)

i

-25.0

-35.0

-45.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.5

5.0

6.0

6.5

d(F-F) [A] 5T™d[p ,p }

Fig. 6. Accuracy of different gradient-dependent approximations to j A B fortheHF...HF dimer at various intermolecular separations (d(FF)). The considered approximations differ in the analytic form of the enhancement factor used in Eq. 77. The accuracy of each considered approximation approximation ito — ' $ A.PBI j g m e 3 S u r e c j a s the difference between the Coulomb energy derived using it and the exact reference value obtained from supermolecular Kohn-Sham calculations.

invariably to the most accurate —" analytic formula reads:

PA PB S

'

at small PA,PB

overlaps. Its complete

100s2 j s2 1 + 0.093907s arcsinh(76.32s) + (o.26608 - 0.0809615e-

^LC94(s)

=

1 + 0.093907s arcsinh(76.32s) + 0.57767 • 1 0 " 4 s 4

(84)

The good performance of the enhancement factor given in Eq. 84 in approximating T„ [PA,PB\ ^_e t j j e s r r i a n e s t deviations from T" ad [p>i,pB]-independent KS results) was confirmed for other hydrogen-bonded complexes 72 . Fig. 8 shows more directly the quality of KSCED electron densities obtained using the LC94 enhancement for the HF...NCH complex in linfactor in the GGA expression for 8T?<"'[PA,PB] &PA ear arrangement at equilibrium geometry. The complexation-induced deformation of the total electron density derived from KSCED calculations matches quite accurately the reference data (supermolecule Kohn-Sham calculations) indicating thus a satisfactory accuracy of the used approximation to —" sp ' * n * n e vicm^y °f the NCH molecule, the electron densities derived from two types of calculations are indistinguishable. Also the energies obtained using the LC94 enhancement factor in KSCED calculations proved to be the closest to the supermolecule Kohn-Sham reference data (see Figs. 9-10). Therefore, the corresponding approximation was

34

T. A.

Wesolowski

15.0

=r

"50

rM. " 1 5 . 0

5 ...

G — O Triad: GEA(O) Tnad: GGA(LC94) H l-Tnad:GGA(LLP) A — A Tnad: GGA(PW86) V—VTnad:GGA(TK92)

-25.0

-35.0

-45.0

2.3

2.5

2.7

2.9

3.1

3.3

3.5

d(F-N)[A] Fig. 7. Accuracy of different gradient-dependent approximations to —a—^IPA.PBS j o r j n HF...NCH dimer at various intermolecular separations (d(FN)). The considered approximations differ in the analytic form of the enhancement factor used in Eq. 77. The accuracy of each approximation is measured as the difference between the Coulomb energy derived from a given approximation and that obtained from supermolecular Kohn-Sham calculations which do not depend on 6T™d[pA,pB]

chosen for all our subsequent applications of the KSCED formalism. The approximate T™ad\pA,pB\ functional defined in Eqs. 65, 77, and 84 will be denoted by T"a [PA,PB] here. The results shown on Figs. 6-7 indicate also that the behavior of F(s) at small s is an important factor determining the accuracy of T™a [PA,PB] and that the condition that the slope and the curvature of F(s) at s = 0 should be the same as that of GEA(2). Note that the TK92 enhancement factor violates the condition given in Eq. 80 and the corresponding approximation to T™ad[pA, PB] appears to be the worst among the analyzed ones. It is worthwhile to notice, however, that despite a rather good performance of T™a \PAIPB] at large intermolecular distances, the accuracy of this approximation deteriorates with the decreasing intermolecular distance for all complexes studied. Decreasing the intermolecular distance increases the PA,PB overlap which can be expected to increase also the magnitude of T™ad[pA,PB] making thus the KSCED results more susceptible to the possible errors in T™ad[pA, PB] than at larger intermolecular separations. None among considered functionals of the GGA type seems satisfactory at large PA,PB overlaps. Development of approximations applicable at larger pA, PB overlaps represents a challenging objective to extend the range of applicability of KSCED calculations. Currently, we consider the GGA97 approximation to T™ad[pA, PB] the most accurate

e

One-Electron

Equations for Embedded Electron Density

35

reference Tnad: GGA(LC94)

-0.05

-0.10

-0.15

0.0 z[A]

2.0

Fig. 8. The interaction induced deformation of the electron density in the HF...NCH linear intermolecular complex calculated using Kohn-Sham calculations (reference data) and Eqs. 31-32 with the GGA(LC94) approximation to T?ad[pA, PB]- The atoms are on the z-axis: zc=l-i54 A, 2JV=0.0 A, and Z H = 2 - 2 1 7 A for NCH, and zH=-1.8S3 A and zF=-2.80 A for HF. Data taken from [Wesolowski at al, J. Chem.Phys., 105 (1996) 9182].

6.0 5.0

1

0

i

J

1

'

1

•

1

•

— reference 0 ™ OTnad:GGA(LC94) OTnad:GEA(0) ©•-

- \

4.0 3.0

-

i>

•

2.0 1.0

-

ia.

V

-1.0

*

-2.0 -3.0

-

k

0.0

2.9

3.1 3.3 d(F-F) [A]

3.5

5T Fig. 9. Performance of the GEA0 and GGA(LC94) approximations to T^ad [pA, pB] "°'^^^ for the HF...HF dimer at various intermolecular separations. The reference Kohn-Sham results for the dimer do not depend on T"aii[pA, PB]- Details of calculations can be found in [Wesolowski, J. Chem. Phys., 106 (1997) 8516].

36

T. A.

Wesolowski

10.0 9.0 O &

8.0

reference O Triad: GGA(LC94) O Triad: QEAO

7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 -3.0 -4.0 -5.0

2.3

3.1

3.3

d(0-0)[A]

10. Performance of the GEA0 and GGA(LC94) approximations to TrdiPA,pB] for the water dimer at various intermolecular separations. The reference KohnSham results for the dimer do not depend on T™ad[pA,PB] Details of calculations can be found in [Wesolowski, J. Chem. Phys., 106 (1997) 8516].

Fig.

ST™d[pA,pB] SPA

one for small PA,PB overlaps. In the applications of this approximate functional, we use the following rule of thumb: Strongly overlapping cases, where the numerical value of \T"ad[pA,PB}\ is larger than the characteristic interaction energy in the studied system, lie beyond the range .

of applicability of the GGA97 approximation

3.2.

Approximating

zrr\n,o.d. r

to either T™ad[pA, PB\ or—<•

the energy functional

1

[PA PB S

' ',

Hs

Any pair of approximate functional Exc[p] and T™ad[pA, PB] defines an approximate functional Es which can be used in Eqs. 23 and 31-32 for practical calculations. Local Density Approximation offers the simplest and most straightforward way because it can be applied to both the exchange-correlation and non-additive kinetic energy components of Es. Moreover, it is non-empirical. On the other hand, LDA approximation is worse than the gradient-dependent one (GGA97) as far as d

PB} approximating sTr iPA, and T™ad[pA, PB} is concerned. Therefore, finding the SpA most suitable GGA partner for the exchange-correlation energy to approximate H s for weak PA,PB overlaps is an objective worth pursuing. Below, two types of approximations to Es will be considered: the one where all its components are approximated using LDA and the other one where they are approximated using GGA functionals.

One-Electron

Equations

for Embedded

Electron

Density

37

Weakly interacting intermolecular complexes represent an obvious choice for testing the accuracy of any approximations to the functional Es. At infinite separation oi interacting molecules, the overlap between their electron densities disappears and consequently the non-additive kinetic energy disappears as well. Close to the equilibrium geometry the overlap between the electron densities of interacting molecules is small but not negligible. Therefore, we use such complexes as a testing ground for approximations to T™ad[pA,pB\. Unfortunately, the overall accuracy of any approximate functional Es hinges critically on the accuracy of its exchangecorrelation part in such cases. Since the semi-local approximations cannot account for long-range dispersion effects, a proper description of the energetics of such complexes in the region where both the overlap and non-local dispersion effects play a role represents one of the most important challenges 6 for density functional theory 97,98,103,104,105

3.2.1. Local density

approximation

Local density approximation (LDA) provides the analytic expressions for both an T"ad[pA,PB] d Exc{p\ components of the functional 5 s . Unfortunately, the results described in the previous section show clearly that LDA leads to a rather poor approximation to —° gp ' • There is also a large amount of numerical evidence indicating that LDA applied to Exc[p\ leads to a rather unsatisfactory (significantly overestimated usually) interaction energies for weakly interacting intermolecular complexes 98 . Despite these flaws of LDA in approximating each separate component of Es (i.e. Exc and T"ad), it will be considered below. Its use to approximate H s aims at at detecting possible cancellation of errors in the approximated quantities. The LDA form of the non-additive kinetic energy is given in Eq. 70, whereas its exchange 93 ' 94 and correlation 95,96 components in Appendix B. Applying LDA in Eqs. 23, 31, and 32 leads to surprisingly good results for raregas d i m e r s " . Fig. 11 shows the KSCED(LDA) the potential energy curve for the Ar-Ar dimer calculated using the aug-cc-pVQZ basis set. Based on our analysis of the dependence of KSCED energies on the basis sets 1 0 0 , the shown results can be considered as converged. The agreement between the KSCED results and the high-level ab initio reference data taken from Ref. 1 0 1 is remarkable. The position of the energy minimum is, however, shifted by about 0.2 A compared to the ab initio reference curve. For other dimers Ne-Ne and Ne-Ar, the agreement with the reference data also very good 99 . The quality of the above results is similar to that of the interaction energies dee

T h e reader can find a representative overview of currently investigated strategies t o i n c o r p o r a t e dispersion forces into t h e framework of density functional theory in t h e r e p o r t of t h e workshop Van der Waals Forces and Density Functional Theory CECAM, Lyon, France, June 3-17, 2005 available a t h t t p : / / w w w . c e c a m . f r .

38

T. A.

Wesolowski

3.5

4

4.5

5.5

d(Ar-Ar) [A] Fig. 11. Interaction energy as a function of distance in the Ar-Ar dimer (reference data taken from [Slavicek et at, J. Chem. Phys, 119 (2003) 2102).

rived using the Gordon-Kim model 49 . In both cases, Eq. 18 is used to derive interaction energies. A fundamental difference between these two approaches has to be underlined. In KSCED(LDA) calculations the electron densities PA and ps are results of variational calculations (Eqs. 31-32), whereas Gordon and Kim used Hartree-Fock electron densities in Eq. 18. In view of the discussed previously flaws of LDA in approximating T?ad[pA, PB] and —' &p,P and the known inadequacy of E^A[p] for weakly interacting systems, the good agreement between the KSCED (LDA) and ab initio reference data can only be explained by a compensation of errors in four approximated quantities used in in Eqs. 23, 31-32: T s na ^ ' [PA,PB], <5T" id(GSAO)

SEt 1M and — -J— ^ ^ , , E^ JJ XCA[p\ t/yj cum j-— . Non-variational calculations, in which the Kohn-Sham(LDA) electron densities of isolated atoms are used in Eq. 27, lead to the energies which are very similar to the results of variational calculations. This indicate that the errors in the LDA functionals rather than potentials compensate each other for these systems. There probably is no universal reason for such remarkable compensation of errors. Indeed, using LDA in KSCED calculations indicates clearly that the accuracy the interaction energies depends strongly on the system. For 22 weak intermolecular complexes given in Table 2, KSCED(LDA) interaction energies at the equilibrium geometry, were compared with high-level ab initio reference data (considered were only such systems for which CCSD(T) benchmark-quality interaction energies were available) 102 . These comparisons shows clearly that the compensation of errors in 6T?

rf,nad(GEA0)r

Ts

\PA,PB

SPA

Ei

K ^d 2 ^ -

in the case of rare-

One-Electron

Equations

for Embedded Electron Density

39

jas dimers is not universal. It is rather an exception than the rule. Results collected Table 2. Comparison between the KSCED (LDA and GGA97) interaction energies with reference (CCSD(T)) results calculated at the same ab initio determined equilibrium geometry. A positive value indicates the attractive interaction. Energies (well depths) are given in kcal/mol. More details can be found in [Wesolowski and Tran, J. Chem. Phys., 118 (2003) 2072J. Complex

LDA

Ne-Ne F 2 - Ne (linear) F 2 - Ne (T-shaped) N2-N2 N 2 - Ar (linear) Ar - Ar N 2 - Ar (T-shaped) F2 - A r C 3 H 6 - Ne C2H4 — CH4 CH4 - CH4 C2H6 — CH4

0.09 0.09 0.14 0.30 0.20 0.21 0.24 0.09 0.21 0.64 0.46 0.52 0.12 0.42 0.59 1.06 0.73 0.42 0.72 0.87 1.09 1.94

CF 4

-

C3H6 C6H6 C2H4 C6H6 CgH6 C3H8 C6H6 C6H6 C6H6

CF 4 - Ar - Ar — C2H4 - CH4 — C2H6 — C3H8 — C2H4 - C6H6 - C2H2

KSCED GGA97 0.38 0.50 0.55 0.59 0.46 0.49 0.44 0.53 0.74 0.88 0.89 0.98 0.82 0.95 1.15 1.57 1.27 1.30 1.52 1.62 1.86 2.47

CCSD(T) 0.08 0.17 0.17 0.22 0.22 0.26 0.29 0.35 0.36 0.49 0.50 0.71 0.78 0.86 1.11 1.33 1.45 1.82 1.94 2.06 2.46 2.83

in Table 2 and shown in Fig. 12 indicate clearly, that LDA is an acceptable approximation only for the weakest complexes including the Ne — Ne and Ar — Ar rare-gas dimers and complexes formed by atoms and diatomic molecules. For more strongly interacting complexes such as those involving larger molecules, KSCED(LDA) interaction energies are qualitatively wrong. They are underestimated usually but in a rather erratic manner (see Fig. 12).

3.2.2. Generalized Gradient

Approximation

The disappointing results with approximating unknown components of 2 s by means of local density approximation bring us back to T s na [PA,PB] found in our dedicated analysis to have the most accurate associated functional derivalPA PBi tive °-^ ' . The PW91 exchange-correlation functional 84 was chos A sen as the most appropriate partner approximation to be used together with f"ad(-GGAg7)[pA,pB] in Eqs. 23, 31, and 32 due to the following reasons:

40

T. A.

Wesolowski

0.0

0.5 1.0 1.5 2.0 2.5 Reference Interaction Energy [kcal/mol]

Fig. 12. Comparison between the interaction energies derived from KSCED calculations applying local density approximation for 22 intermolecular complexes at equilibrium geometry with accurate ab initio (CCSD(T)) reference data. Details of calculations can be found in [Wesolowski and Tran, J. Chem. Phys., 118 (2003) 2072],

• This functional appears to be the most suitable (least bad) among GGAs to describe weak intermolecular complexes close to the equilibrium geometry 1 0 3 ' 1 0 4 ' 1 0 5 . • It satisfies one of the exact properties of the exchange-correlation functional: it obeys the Lieb-Oxford bound 1 0 6 . • Opposite to empirical functionals, the PW91 functional was derived without fitting to any data concerning van der Waals systems but it is based on firstprinciples. • Almost perfect compensation of errors of LDA in the case of rare-gas dimers suggests that a similar effect might occur for generalized gradient approximation. March and Santamaria showed that, the exchange- and kineticenergy functionals are linked because they can be obtained from the same approximation to the one-particle density matrix 1 0 7 . Lee, Lee, and Parr provided a numerical examples of such link in the GGA case 88 . Therefore, the exchange-correlation GGA partner of T™° '[PA,PB\ was chosen in such a way that both approximate GGA functionals share the same analytical form of their enhancement factors. The above choice of GGA approximations for the exchange, correlation, and nonadditive kinetic energy components of H s will be referred to as the GGA97 set of approximate system-independent functionals hereafter. The interaction energies calculated using the GGA97 set of approximate functionals for the same 22 weak in-

One-Electron

Equations

for Embedded

Electron

Density

41

termolecular complexes shown on Fig. 12 discussed previously 102 compare favorable with ab initio reference data (see Table 2 and Fig. 13). Comparing Figs. 12 and 13 shows clearly that introduction of the gradient-dependence improves qualitatively the overall accuracy of the calculated interaction energies.

2.5

-

2.0

-

•" D

1.5

D

1.0

0.5

/ D

•

D

w-• 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Reference Interaction Energy [kcal/mol] Fig. 13. Comparison between t h e interaction energies derived from K S C E D calculations applying generalized gradient approximation (GGA97) for 22 intermolecular complexes at equilibrium geometry with accurate ab initio ( C C S D ( T ) ) reference d a t a . Details of calculations can b e found in [Wesolowski and Tran, J. Chem. Phys., 1 1 8 (2003) 2072].

3.2.3. Role of approximations in KSCED interaction energies for weak intermolecular complexes The remarkable compensation of errors of all four unknown quantities in KSCED ,nad{GEA0) 6frd(aEA0)[PA,PB\ pLDA [p], and SEt. IM for calculations: T. [PA,PB], Sp &PA rare-gas dimers at the LDA level and the fact that such compensation does not take place for larger intermolecular complexes indicates that a more detailed analysis of the role of approximations is needed. To this end, we analyze a particular case for which: i) The complexation induced changes of the two electron densities are small and that the energetic effect of such changes can be linearized. ii) The complexation induced changes of the electron density are larger than the errors of the electron densities of both monomers resulting from the use

42

T. A. Wesolowski of an approximate Exc[p] functional and can be neglected: ~ 0

(85)

= &P°B « 0

(86)

A

PA-PA^PA-P°A=

pB - PB »

P°A

PB-P°B

The linearized expression for the interaction energy reads: Eint = E [PA, PB] - E = Es[p\

[pA\ - E

\pB\ +

EnuCA,nuCB

+ ApA,P°B + Aps] - EKS[p°A} - EKS[pB]

E»{pA,pB}~E«b[pA]-E«»[pB}+E

I

+ / Ap,

SEs\pi,p2] dpi

+ 0(A pA,

2

A pB,

EnuCAtnuCB

nucA,nucB

5Es[puP2\ I=»°A

d

?

+

Af>B

P2=PB

2

+

8pi P2=P°B

(87)

&pA • A p B )

where A

PA = PA +

PA

PB = P°B + A p s

(89)

A similar linearized formula can be written using approximate quantities: Es, pA, and PB- ES and Es differ only in the exchange-correlation and non-additive kinetic energy parts: Exc[p\ = Exc[p] + AExc[p]

(90)

Ts[p]=fs[p} + ^fs[p]

(91)

The difference between the exact interaction energy obtained from Eq. 87 and that derived using approximate functionals defined as: AEint = Efnt\pA,pB}

-

Eint[pA,PB}

(92)

One-Electron Equations for Embedded Electron Density 43 depends explicitly on the errors in the used functionals and potentials. AEint

= AExc[p°A + ~pB] - AExc[p°A] - AExc{p%] + ATs[pA + pB] - ATs[p°A] - ATs[p°B]

f

ApA (fO

SAExc[p] \

6 Afs[p] dp

Sp

5Afs[p]

df

Sp P=PAJ

+ J&PB(7)

/

5AExc[p]

\

+

Sp

5Afs[p] 8p

SAfs[p] Sp

df

P=P°A+P%

+

0{ApA,Ap2B,ApA-ApB)

(93)

' s *^le e r r o r of the exchange-correlation potential

Noticing that — s i

(Avxc{p](f))

and — j ^ - = — j ^ - = Avts[p]{r)) is the error of the effective potential associated with the kinetic energy, and regrouping similar terms in the above equation leads to: AEint

= AExc\pA - AExc[pA\

+ P%\ + &fs[pA -

+ pB]

&Ts[pA}

-AExc[p°B]~Afs[pl] + J (ApA (r) + ApB (r)) • (Avxc

[p°A + p°B] (f) + Avts [p°A + p°B] (r)) df

- J ApA (r) • Avts [p°A] (f) dr ~ J ApB (f) • Avts [p%] (f) df + 0(ApA,ApB,ApA-ApB)

(94)

Eq. 94 can be used to analyze the effect of errors in the approximate quantities (functionals and potentials) on the interaction energy in weakly interacting complexes. The first six terms represent the zeroth-order contribution to the error i.e. do not account for the complexation induced change of the electron densities of the isolated monomers. Such terms can be expected to be dominant for weak complexes in the absence of strong electrostatic and polarization effects. Terms from the seventh to ninth represent the first-order contribution. Their relative significance increases

44

T. A.

Wesolowski

with the increase of the interaction induced deformation of the electron density (note that they depend on integrals involving ApA and A p s ) . This might be expected to be the case for even rare-gas atoms at at very short intermolecular separations (beyond the energy minimum) or for complexes formed by molecules which are either polar, strongly polarizable or both, for instance. In the zeroth order (the first six terms in Eq. 94), the sum of the exchangecorrelation and functionals AExc[p] + ATs[p] is calculated the same electron density (PA + P°B> PA> o r PB)- Therefore, the zeroth-order contribution to the error of the calculated interaction energy depends on the sum of errors of these two functionals. Any cancellation of errors in Ts[p] and Exc[p] results thus in improvement in the calculated interaction energies. The remarkably good accuracy of the KSCED(LDA) interaction energies for rare-gas dimers can be seen as an illustration that such cancellation takes place at the LDA level. The zeroth-order terms are dominant not only for rare-gas dimers. Table 5 shows that it amounts to more than 95% of the interaction energy for complexes formed by small hydrocarbon molecules. Even for complexes formed by carbazole and Ne, Ar, N2, CO, and CH4 molecules zeroth-order contributions amount to about (80-90%) of the KSCED interaction energy 108 . In the first order, the same considerations cannot be applied. Only the seventh term in Eq. 94 depends on the accuracy of the sum of the exchange-correlation- and kinetic-energy potentials. The two remaining terms depend only on the accuracy of the kinetic-energy potential. We recall here the known difficulties in approximating g\p'. It can be expected, therefore, that for any intermolecular complex for which large complexation induced changes of the electron density are expected to take place, the errors of the KSCED interaction energies would originate from these terms. 3.3. Criteria in chosing the approximations calculations: The GGA97 set

to 3

s

in

practical

We did not find any approximation to T™ad[pA, PB] and —° ^IP^,PB] w n j c n c o m c i be universally applicable to strongly overlapping PA and ps- At weak overlaps, approximating to these quantities by means the GGA97 approximation provides a satisfactory solution. The situation is different, as far as approximating the exchange-correlation energy is concerned. The PW91 approximation is our first choice to be used as the partner for the GGA97 approximation to the kinetic energy component of H s . This combination of approximate functionals leads to a very satisfactory description of non-covalently bound complexes. Moreover, the kinetic- and exchange energy parts are conjoint in the Lee, Lee, and Parr sense 88 . For the sake of brevity, this choice of approximate functionals to the non-additive kinetic and exchange-correlation components of S s will be referred to as the GGA97 set of approximate functionals in this work. The GGA97 label will be used for either the sum of both approximate

One-Electron

Equations for Embedded Electron Density

45

components of H or its non-additive kinetic term. We recall here that the primary target of orbital-free embedding calculations based on Eq. 31 are environment induced shifts. Possible flaws of the PW91 approximation to Exc[p\ might be expected to affect negligibly the shifts. If, however, the PW91 approximation is known to be unacceptable as far as description of isolated subsystem A is concerned, it should be replaced by the most adequate one.

4. E x t e n s i o n - Localized excited states in condensed m a t t e r So far, only ground-state properties of embedded systems were considered. Groundstate density functional theory of Hohenberg, Kohn and Sham has been generalized to time-dependent phenomena by Runge and Gross 109 . Linear response formulation of time-dependent density functional theory (LR-TDDFT) allows one to get information concerning also electronic excitations in molecular systems 110 (for a comprehensive review see Ref. m . A generalization of the subsystem-based formulation of ground-state density functional theory to excited states is provided in Ref. 1 1 3 . Here, only the key elements of this unified formalism are provided concerning its possible application to study localized excitations in condensed phase where it is assumed that the response of the whole system to external time-dependent perturbation is localized in a pre-defined subsystem 114 . The core element of LR-TDDFT is the relation between the dynamic (frequency dependent) polarizability of the system under investigation and the quantities derivable from Kohn-Sham equations. LR-TDDFT has been subject to many reviews (for fundamental aspects see the recent review by Gross and Marquard 1 1 2 , for the original description of the adaptation of LR-TDDFT to molecular systems see the review by Casida 1 1 1 ). Before discussing the response of an embedded electron density, the key elements of LR-TDDFT will be provided here f . In the linear-response range, a time-dependent electric field oriented along the .?-axis:

w(f,t)

= z£z(t)

(95)

induces time-dependent changes of the electron density. Below, a periodic (u>) field will be considered. The frequency-dependent spin-density response (6pa(f,u))) is expressed by means of the unperturbed Kohn-Sham spin-orbitals (ia{r)) i-e. the orbitals obtained in the absence of a time-dependent electric field:

f

The general lines given here follow the more detailed introduction to LR-TDDFT provided in

46

T. A. Wesoiowski AUOrbitals

5pa(r,io)=

Yl

*a(f)

(96)

where SPija is the time-dependent linear response of the coefficients matrix and the indices i and j run over all eigenvectors of the unperturbed Kohn-Sham equations (including the spin-orbitals unoccupied in the ground-state). Note that 5Pija is a function of w because the response of the system under investigation is not the same for all frequencies. The electron density response results in a frequency-dependent induced dipole moment: 5fj,u(u>) = - ^2 uija5Pija(w)

(97)

where «y ff is the matrix element of u (u = x, y, or z) in the basis of the Kohn-Sham orbitals of the unperturbed system. The dynamic polarizability is thus: fi*-fj*>0

i \ o auz(uj) = -2

V^ }_^

.

,

RedPija(uj) uija—£

J

K

(98)

where Re5Pija(u) is the real part of 5Pija{u>). The key equation in LR-TDDFT is the expression linking the dynamic polarizabilities with the quantities derivable from time-independent Kohn-Sham equations 111 : auz(uj) = 2 u t S " 1 / 2 [ n - w 2 l ] _ 1 S " 1 / ^

(99)

where I is the identity matrix and the matrices S and Cl are defined as :

{fkr

~ fir)

( Q r - Efcr)

'

Qija,kiT = Sa,TSitk5j,i (e;T — £fcr)

+ 2\/{fi
— fja)

(CJCT - £ia)Kija,klT\{fkT

— fir)

( Q T ~ Cfcr)

The matrix K relates the changes in the Coulomb and the exchange-correlation parts of the Kohn-Sham effective potential (truncated to the linear terms) and the

One-Electron Equations for Embedded Electron Density 47 changes in the electron density:

KlJaMT

= J

|^(r)0JCT(r)pF^^r(r')C(r-')df'dr

+//^W^^|^^y^^)C(^)^df

(ioi)

The quantity

nw-f^'fX

d02)

6pa(r)6pT(f') will be referred to here as exchange-correlation response kernel. The values of u>, at which the matrix Cl — co2I becomes singular, are very useful because they can be combined with the sum-over-states expression for the mean dynamic polarizability: 3

i=l

I

*

where the poles UJI = Ej — Eo are the vertical excitation energies and the residues:

h = \{Ei-

Eo) ( | ( * 0 1*1 * / ) | 2 + l(*o \y\ * / ) | 2 + |<*o \z\ * / ) | 2 )

(104)

are the oscillator strengths. Therefore, the poles and the residues of the dynamic polarizability obtained from Eq. 99 can be used to obtain excitation energies and oscillator strengths by solving the following equation:

£lFI-u}2F1=0

(105)

Turning back to the embedded systems, we recall that the embedded electron orbitals are derived from Eq. 31 and yield the electron density which minimizes the total energy under the condition that 5p = SpA (i.e. 8pg = 0), where 5p is the variation of electron density. Assuming that the frequency-dependent response of the electron density of the whole system can be expressed by means of the embedded

48

T. A.

Wesolowski

orbitals of the subsystem A:

5p,j (f, id) RJ 5pAa (r, Lj)

(106)

and following the same steps as the ones leading to Eq. 99 allows one to express the dynamic polarizability of an embedded subsystem with the quantities derivable from Eq. 31: c C 6 M = 2u f ( s e m b ) ~ V 2 \uemb

- w 2 l] ~* ( s e m 6 ) ~ 1 / 2 z

(107)

Eqs. 107 and 99 differ by the fact that in the matrices Semb and flemb the embedded orbitals (occupied and unocupied) are used instead of the orbitals for the isolated subsystem A and that the matrix Clemb corresponds to the KSCED effective potential. -iem6

_

^g-,r0i,fc0j,i

^ij^kir ~ 71

7~rTA

TXT >

u

(-1UJV

(/*r - fir) H ~ *&)

^ i j a ^ i T — fio-,T5itk5j,i (e i T — e fcT )

+ 2v / ( / i . - / ^ ) ( ^ - 4 ) ^ , « r

\IUkr-flr)^tr-4r)

The matrix K 6 " 16 relates the linear changes in the Coulomb, the exchangecorrelation, and the non-additive kinetic energy components of the KSCED effective potential with the changes of in the electron density in the subsystem A:

KfT^ '&T

= JI JI =

fa^fai^T^^fai^fai^df^df fa^fai^r^-^fa^fa*^)^

it'if)fa(r)/*"""(r,r)

fa(7)

fa(f)dfdf

(109)

where114 f total (fy)

=

y/ree ^ ^

+

femb

^

^

(11Q)

One-Electron

Equations

for Embedded

Electron

Density

49

and where ffree (f", f1) is the exchange-correlation response kernel for the isolated subsystem A defined in Eq. 102 where the embedding component of the total response kernel reads:

emb

f

(?y)

52EXC [p] (f) 5p (f) Sp (f) P=PA+PB

82EXC [p] (f)

Sp(r)6p(f')

s2Trd[PA,pB](7) 6pA(r)6pA(r*)

[iiL)

The above equation was obtained assuming that 5ps = 0. Lifting this assumption, for all w ^ 0 (not only the w = 0 case) leads to a generalized expression 8 given in Eq. 4.33 of Ref. 1 1 3 .

5. Numerical implementation Eqs. 23 and 31 can be used in two types of studies: i) those following the multi-level strategy, where they are applied to obtain the electron density of a selected subsystem embedded in an environment represented by means of the electron density ps obtained from simpler methods; and ii) those where no approximations are made concerning electron densities of the subsystems A and B. Moreover, the approximations for the relevant functionals have been subject of testing and improvements within the last decade. For these reasons, we developed as much as four different computer implementations which were used for different purposes. Our first computer implementation of Eq. 31 was used to evaluate the difference between the energy of the total system and that of the environment (Es[pA, PB\ — EHK\PB\) at the LDA level 50 . It was developed as an interface for hybrid quantum mechanics/classical molecular dynamics simulations 52 ' 50 ' 51 . The atomic basis sets used in this implementations were restricted to have the form of gaussian lobe functions introduced by Whitten 1 1 5 . A more universal implementation 70 , developed in 1995-2001, is based on the code deMon 116 . Compared to the first implementation, the most important new features include: i) use of gaussian type of atomic orbitals of the s-, p-, and d-type, ii) evaluation of analytic gradient of the electron density (Vp) enabling the use of GGA type of functionals, Hi) efficient grid integration techniques, iv) evaluation of analytic gradients of the KSCED energy with respect to the position of nuclei allowing for geometry minimization and vibrational analysis 117 , v) possibility to solve two KSCED equations (Eqs. 31 and 32). The deMon implementation was applied in our studies aimed at developing and testing approximations to the non-additive kinetic energy functionals 6 9 ' 7 0 ' 7 1 ' 7 2 ' 1 0 2 ' 1 1 9 . It has also been used in several applications g

N o t e t h a t t h e two subsystems are referred to as A a n d C in Eqs. 4.30-4.33 of Ref.

113

.

50

T. A.

Wesolowski

aimed at obtaining the parameters of the potential energy surface for intermolecular complexes and embedded m o i e cules 1 0 8 ' 1 1 7 ' 1 2 0 ' 1 2 1 ' 1 2 2 ' 1 2 3 ' 1 2 4 ' 1 2 5 ' 1 2 6 ' 1 2 7 . Warshel and collaborators modified the deMon implementation of KSCED by including a non-expensive evaluation of frozen electron density pB and interfacing it with their molecular dynamics code to study enzymatic reactions 1 2 8 , 1 2 9 . Since 2004, we have been using two new implementations 1 0 0 ' 1 1 4 which were developed basing on two advanced computer packages solving Kohn-Sham equations: deMon2K 130 and Amsterdam Density Functional (ADF) 1 3 1 , 1 3 2 . Below, the key elements of these implementations will be given. 5.1. Linear combination Basis sets

of atomic

orbitals

for embedded

orbitals:

In our numerical implementations of Eqs. 31-32, the embedded orbitals are constructed as a linear combination of atomic orbitals (xfB)-

? = I24xtB

<^ = E « B

(H2)

(113)

i=i

Gaussian (GTO) or Slater (STO) type of atomic orbitals orbitals (xf) comprising s-, p-, d-, f- and functions with higher orbital momentum are used in the deMon2K (GTO) and ADF (STO) implementations. In Eqs. 112-113, all atoms in the system (comprising the subsystems A and B) are used to construct each set of embedded orbitals. The corresponding calculations are referred to by KSCED(s) where s stands for supermolecule. If the above expansion is limited to a pre-selected group of atoms in one or both of the above equations, the corresponding electron density is localized. The calculations, in which only atoms belonging to one molecule are used in constructing the corresponding electron density, are referred to by KSCED (m) here where m stands for monomer. In this case, the truncated expansion takes the following form:

t = T , 4 4

(114)

<^k=Y.c?kXf

(115)

1=1

The KSCED(s) expansion was used in our research on development of approximations to the non-additive kinetic energy functional and can be used to verify the basis set convergence in any particular application. In practice, KSCED (s) calculations do not offer any advantage over Kohn-Sham calculations as far as speed of computations is concerned because both the number of necessary grid points as well as the size of the matrices to diagonalize are the same in these two cases.

One-Electron

Equations

for Embedded Electron Density

51

For various intermolecular complexes, it has been found that KSCED interaction energies converge rather rapidly with both the number of atomic centers and the atomic basis sets. Table 3 collects the KSCED results (interaction energy and the energy of the highest occupied orbital in one of the monomers) for two series of the basis sets (CC-PVxZ and AUG-CC-PVxZ, for x=2-5) applied in the KSCED(s) or KSCED(m) schemes for the CH4-CH4 complex. Table 3. Dependence of the KSCED(GGA97) results on the basis set for the CH4-CH4 complex. Interaction energy (Eint in kcal/mol) and the energy of the highest occupied orbital (in eV) of the CH4 monomer(£pg M O ). Calculations were made at the CCSD(T) equilibrium geometry taken from [Tsuzuki et al, Chem. Phys. Lett., 287 (1998) 202]. KSCED(m)

KSCED(s)

Eint

e.HOMO CHt

CC-PVDZ

-0.794

-9.29918

-0.799

-9.29894

CC-PVTZ

-0.827

-9.43767

-0.854

-9.43910

CC-PVQZ

-0.845

-9.45539

-0.868

-9.45661

CC-PV5Z

-0.863

-9.46194

-0.863

-9.46288

AUG-CC-PVDZ

-0.828

-9.42979

-0.880

-9.43498

AUG-CC-PVTZ

-0.857

-9.45892

-0.874

-9.45924

AUG-CC-PVQZ

-0.860

-9.46230

-0.876

-9.46284

AUG-CC-PV5Z

-0.871

-9.46205

-0.878

-9.46312

basis set

Eint

e,HOMO CHt

Numerical experience shows, that the KSCED(m) expansion of the electron density of each subsystem is satisfactory as far as interaction energies are concerned 100 ' 133 .

5.2. Numerical

integration

In the self-consistent procedure based on a linear combination basis set functions to solve Eq. 31, the following matrix elements are evaluated numerically on the grid: (VeKffCED)id

=

(x,(rlKKffCED[PA,PB;r-\\X3(rl)

= (Xi(f) \Vilf

[PA + PB-A\XJ(?))

+

(xi(rl

EM

xj(rl (116)

52

T. A. Wesolowski

where Vjjf [PA + PB\ r\ is given in Eq. 34. The exchange-correlation component of the above matrix elements is the same as in conventional Kohn-Sham calculations for the whole system (comprising the subsystems A and B). Therefore, the same supermolecular Kohn-Sham grid is used in KSCED calculations (in deMon2K and ADF implementations). Numerical analyses show that this grid is also adequate for the evaluation of the kinetic-energy dependent matrix elements owing to similarity between the analytic forms of the exchange- and kinetic energy components of Vj^fCED at the LDA and GGA levels (see Appendix A). At the LDA and GGA levels, the matrix elements given in Eq. 116 and the difference ES[PA,PB] — EHK[PB\ disappear at pA — • 0. Therefore, a large part of the supermolecular Kohn-Sham grid can be reduced depending on the extent of PA- In the deMon2K implementation, a special technique known as adaptive grid134 is used in constructing the grid as a possible CPU time-saving option. The grid is constructed taking into account the the magnitude of the contribution of each orbital to the exchange-correlation potential matrix denoted as t. The adaptive grid technique was also implemented for KSCED embedding potential 135 showing that it leads to significant reduction of the size of the grid - especially in the case where the subsystem B is significantly larger than the subsystem A. In KSCED calculations, the parameter t is calculated as: t =

Xi(f)

APA

+ PB} +

5Trd[PA,pB] SpA

Xitf

(H7)

Table 4 collects the results of Eq. 31 for the acetone molecule embedded surrounded by 57 water molecules obtained using different grids: supermolecular Kohn-Sham (coarse, medium, and fine) and several adaptive ones differing in the value of the threshold parameter t. The quantities shown in Table 4 are good indicators of any inaccuracy in the numerical integration: i) Ts [p] - the largest component of the total energy (evaluated analytically using embedded orbitals which depend on the accuracy of the matrix elements evaluated numerically), ii) T™ad[pA, PB] - (evaluated numerically), iii) EHOMO - the e n e r g y of the highest occupied embedded orbitals (which depends directly on the accuracy of the matrix elements evaluated numerically). Indeed, the adaptive grid technique leads to grids which are smaller than the supermolecular ones without loosing accuracy of the calculated properties of the embedded molecule. 5.3.

Freeze-and-thaw

cycle

The two coupled KSCED equations (Eq. 31-32) can be solved using the freeze-andthaw procedure 70 , which consists of the following steps: Step 0 Preparation of the electron density of subsystem B, (Kohn-Sham calculations for the isolated subsystem).

One-Electron

Equations for Embedded Electron Density

53

Table 4. Dependence of the results of Eq. 31 on the grid type: a) supermolecular (coarse, medium, and fine) and b) adaptive (see text for description of the parameter t). Calculations are made at the LDA level for an acetone molecule embedded in a cluster of 57 water molecules. Grid type

Grid Size

Ts\p] [hartree]

TrdiPA,PB] [hartree]

(-HOMO [Hartree]

Adaptive: t = l • 1 0 ~ 3

23759

189.905711

0.063124

-0.22617

Adaptive: t = l • 1CT 4

56220

189.906435

0.063447

-0.22598

Adaptive: t = l • 1 0 - 5

130248

189.906205

0.063501

-0.22601

6

275201

189.906367

0.063467

-0.22599

Adaptive: t = l • 1 0 ~ 7

527083

189.906356

0.063470

-0.22599

Adaptive: t = l • 1 0 ~ 8

912952

189.906359

0.063470

-0.22599

coarse

430657

189.906565

0.063474

-0.22600

medium

1147000

189.906474

0.063468

-0.22599

fine

3254093

189.906367

0.063470

-0.22599

Adaptive: t = l • 1 0 ~

Step 1 Solving Eq. 31 using ps taken from Step 0. S t e p 2 Solving Eq. 32 using PA taken from the previous step. S t e p 3 Solving Eq. 31 using ps taken from the previous step. Convergence check: if YES go to Finish, if NO go back to Step 2. S t e p 4 Solving Eq. 32 using PA taken from S t e p 3 the previous step. Convergence check: if YES go to Finish, if NO go back to Step 3. Finish Evaluation of the total energy (Eq. 23) and other observables.

The difference between the total energy evaluated according to Eq. 23 in two consecutive steps is used as the principal convergency criterion. Typically, the same threshold as the one used used in the self-consistent procedure to solve each of the KSCED equations (Eq. 31 or Eq. 32) is used (10~ 6 hartree typically). Criteria based on other observables such as orbital-energies, dipole moment, etc. can be used depending on the observable of primary interest.

54

T. A.

Wesolowski

Table 5 shows that such quantities as the total energy and the highest occupied orbital energy converge rather rapidly in the case of two intermolecular complexes: C2H4-CH4 and CH4-CH4. Results of non-variational calculation (step 0) are also Table 5. The interaction energy Eint (in kcal/mol) and the energy of the highest occupied orbital (in eV) of the monomer A in the A...B complex calculated at subsequent steps of the freeze-and-thaw cycle calculated at the GGA97 level. Calculations were made at the CCSD(T) equilibrium geometry taken from [Tsuzuki et al, Chem. Phys. Lett., 287 (1998) 202] for CH4-CH4 and [Tsuzuki et al., J. Phys. Chem. A, 103 (1999) 8265] for C2H4-CH4. Results obtained using electron densities PA and ps of the isolated monomers correspond to step 0. Freeze-and-thaw cycle starts (step 1) with solving Eq. 31 in which pa is the electron density of the isolated molecule B in the A...B complex. Step

C2H4-CH4

Eint

CH 4 -CH 4 C2H4

HOMO

Eint

t

CH4 HOMO

t

0

-0.8107

-6.7799

-0.8455

-9.4714

1

-0.8239

-6.8120

-0.8568

-9.4525

2

-0.8552

3

-0.8556

4

-0.8556

5

-0.8556

-0.8677 -6.8288

-0.8677

-9.4554

-0.8677 -6.8290

-0.8677

-9.4554

collected in Table 5. In this case, the interaction energy is calculated by means of Eq. 28 for the electron densities PA and ps of isolated monomers, whereas CHOMO corresponds to that of the isolated molecule A in the A...B complex.

6. Applications The GGA97 set of approximate functionals, used in Eq. 18 to evaluate energy and in Eqs. 31-32 to obtain electron densities, defines a computational method which will be referred to as KSCED(GGA97) for the sake of brevity. Some of the KSCED(GGA97) results were already discussed in the context of testing/developing approximations to EE. This section concerns another type of applications, for which comparing the accuracy of various functionals was not the main issue. The reviewed applications are divided into three groups. The first one concerns structure and energetics of weak intermolecular complexes. Such studies represent a certain deviation from the main subject of this review, i.e. multi-level computer simulations, because

One-Electron

Equations for Embedded Electron Density

55

the subsystems A and B are treated at the same level and no additional assumptions concerning electron density are made. They are reviewed here to illustrate the strengths and weakness of the GGA97 set of approximate functional. The second group concerns the properties of embedded systems which are determined by the dependence of the energy on the positions of atoms in the embedded subsystem (geometry, energetics of stable conformers, barrier heights and vibrational properties). The studies of the electronic structure of embedded molecules (or ions) - our main interest and the most appropriate domain of applicability of Eq. 31 are collected in the third group. Such studies were aimed at the evaluation of the environment induced shifts of the electronic-structure dependent properties (spin-density, orbital levels, electronic excitations).

6.1. Potential energy surface I: Structure intermolecular complexes

and energetics

of weak

Fully variational calculations, in which the two coupled KSCED equations (Eqs. 3132) are solved by means of the freeze-and-thaw cycle, were applied in all cases reviewed below. In variational calculations, no approximations are made concerning the electron densities and the quality of the calculated energies depends only on the accuracy of the applied approximate functionals. The first application of the GGA97 set of approximate system-independent functionals concerned complexes formed by benzene and diatomic molecules such as N2, O2, CO. In each case, the geometry of the stable conformer and the interaction energy was derived from KSCED(GGA97) calculations. The KSCED(GGA97) interaction energies were compared to the experimental dissociation energies. These two quantities differ by the contribution due to zero-point vibrations which estimated using harmonic approximation is small compared to the interaction energies. It was found that these two quantities agree very well (differences not exceeding 0.1 kcal/mol). At the minimum energy geometry of each complex, the diatomic molecule is localized above the ring center in a parallel orientation to the plane of the benzene ring 1 2 0 . Similar studies 1 0 8 were made for complexes involving carbazole (carbazole-X, where X=Ne, Ar, N2, CO, or CH4). For these complexes, dissociation energies were determined experimentally with great precision by Leutwyler and collaborators 136 . They provide, therefore, a very valuable reference set for assessment of the accuracy of the GGA97 set of approximate functionals. Data collected in Table 6 shows that the calculated interaction energies correlate very well with experimental dissociation energies. Except for the Ne case, the calculated well depths are slightly larger (about 0.1-0.15 kcal/mol) than experimental dissociation energies. This overestimation is of the order of magnitude of the effect of the zero-point vibration energies. For the Necarbazole complex, the difference between the KSCED (GGA97) well depth and the experimental dissociation energy (0.88 kcal/mol and 0.48±0.13 kcal/mol, respectively) originates most probably from the inaccuracies in the GGA97 set of approx-

56

T. A.

Wesolowski

imate system-independent functionals. No direct experimental data concerning the geometry of studied complexes is available. The KSCED(GGA97) equilibrium geometries show common trends. In the Ar, Ne, N2, and CH4 complexes, the molecule (or atom) bound to carbazole is localized above the center of its five-member ring. The geometry of the CO-carbazole complex is slightly different because the CO molecule is localized above the C-C bond opposite to nitrogen of the ring. Unlike N2, for which both nitrogen atoms are at almost the same distance from the carbazole plane, the carbon atom is localized significantly closer to the carbazole plane (about 0.6 A) than oxygen (see Fig. 1 in Ref. 1 0 8 ) . The effect of the relaxation of the electron densities on the calculated interaction energies was also investigated in Ref. 108 . In non-variational calculations the electron density of isolated monomers were used to evaluate the interaction energies using Eq. 27. It was found that the energetic effect of the relaxation is rather small. The largest effect occurs for the CO-carbazole complex where it increases the well depth by 0.32 kcal/mol (less than 15% of the total interaction energy). The largest contributions of higher than zeroth-order terms occurs for the CO-carbazole complex i.e. a complex where both monomers posses a permanent dipole moment. For hydrogen-bonded complexes, the effect of relaxation was found to be larger reaching up to 50

Table 6. Interaction energies in the X-carbazole complexes (X=Ne, Ar, N2, CO, or CH4) derived from variational and non-variational KSCED(GGA97) calculations. AE intax = E™[iaUonal - E?°™ariational the contribution to the KSCED(GGA97) interaction energy due to the relaxation of the electron densities px and pcarbazoleEnergies are given (in kcal/mol). Theoretical and experimental data are taken from [Wesolowski, et al, J. Chem. Phys., 116 (2002) 6411] and [Burgi, et al, J. Chem. Phys., 103 (1995) 7228], respectively. X

Ne Ar N2 CH 4 CO

-D 0 (exp)

-0.48±0.13 -1.47 -1.79±0.03 -1.91±0.05 -2.06±0.08

T?variational a int

-0.880 -1.301 -1.948 -1.558 -2.299

T^nonvariational

-0.776 -1.179 -1.800 -1.361 -1.976

A

J^relax int

-0.104 -0.122 -0.148 -0.197 -0.323

Another application of the GGA97 set of approximate functionals concerns adsorption of molecular hydrogen on carbonaceous surfaces. This process is a key element in astrophysical theories aimed at understanding the abundance of hydrogen in interstellar space (for recent review see Ref. 1 3 7 ) . It is also of great importance for the development of hydrogen storage devices based on carbon 1 3 8 , 1 3 9 . KSCED(GGA97) calculations were used to obtain minimum geometries and interaction energies between the molecular hydrogen and polycyclic aromatic hydrocarbons (PAH) of increasing size (from benzene up to ovalene) which were used as models of the infinite

One-Electron Equations for Embedded Electron Density 57 graphite surface (see Fig. 14). In each case, the hydrogen molecule is adsorbed above the ring center. For the largest PAH, the calculated interaction energy of 1.27 kcal/mol is in excellent agreement with available experimental estimates (1.11.2 kcal/mol) 1 2 5 . For the smallest model of the surface (benzene), the equilibrium distance between the surface and the nearest hydrogen atom of the H 2 molecule amounts to 2.51 A (the hydrogen molecule is located on the GQ symmetry axis in the equilibrium geometry). The interaction energy equals -1.18 kcal/mol. These values are in a very good agreement with the corresponding results (2.698 A and -1.17 kcal/mol) of benchmark calculations (MP2-R12/aug-cc-pVQZ') by Hiibner et al. which became recently available 140 . The analysis of smaller energetic effects exposes, however, two failures of the GGA97 set of approximate functionals. The first one concerns the relative energetics of the conformers differing in the orientation of the H2 molecule with respect to the plane of PAH. The second one concerns the dependence of the interaction energy on the size of the PAH molecule. In the benzene-H2 complex, the parallel arrangement is less stable than the perpendicular one by about 0.34 kcal/mol according to benchmark MP2 calculations 140 . The KSCED(GGA97) energies of these two conformers are almost the same (the parallel conformer is only 0.01 kcal/mol less stable). The origin of this discrepancy is probably due to the inaccuracies in the GGA97 set of approximate system-independent functionals in describing the energetics of the overlapping electron densities and merits further investigation. It was also found that the interaction energy is not significantly affected by the size of the PAH representing the surface. For instance, at the parallel arrangement, the interaction energy increases from -1.17 to -1.24 kcal/mol along the benzene to ovalene series. For the perpendicular arrangement, increasing the number of rings is accompanied by a decrease of the interaction energy from -1.17 to -0.90 kcal/mol. This is in a qualitative disagreement with the energies derived from MP2 calculations which can account for attraction between non-overlapping electron densities (Hiibner et al. report the increase of the interaction energy by about 0.1 kcal/mol per each ring attached to the central benzene 1 4 0 ). We recall here the relevant property of the exact exchange-correlation functional: unlike its kinetic energy counterpart, it is not additive even in the case of two non-overlapping electron densities. Semi-local approximate functionals (both E^A[p] and TGGA[p]) are additive in such case. The used approximation in the GGA97 set for T™ad\pA, PB] is adequate because it obeys the exact relation:

Trd[pA,pB] = 0 = Trd(GGA97)[pA,pB]

(us)

but its exchange-correlation partner (PW91) is not. The exact exchange-correlation functional is not additive for non-overlapping electron densities pA and ps whereas, E™91lPA

+ PB] = E™01\PA]

+ CW91[PB]

(H9)

As a consequence, the attraction between non-overlapping electron densities is accounted for in neither Kohn-Sham nor KSCED calculations if semi-local approxi-

58

T. A.

Wesolowski

mations are used for the exchange-correlation functionals which cannot take into account the long-range attraction between non-overlapping electron densities.

<»jtHy

Fig. 14. H2 molecule adsorbed on the largest polycyclic aromatic hydrocarbon (ovalene) used as a model of the graphite surface. For details see [Tran et at, 3. Phys. Chem. B, (2002) 106 8689].

The adequacy of the GG A97 set of approximate functionals in describing relative energetics of conformers of a weakly bound intermolecular complex was tested also for the benzene dimer 124 . The T-shaped conformer was found to be the most stable (Ei n t =-2.09 kcal/mol). The slipped parallel arrangement was found to be slightly less stable ( E , n t = - 1 . 9 5 kcal/mol). The stacked conformers were found to be less stable by about 0.5 kcal/mol, whereas the planar conformers were the least stable ( E i n t =-0.5-0.6 kcal/mol). The studies of the neutral- and cationic complexes involving phenol and such atoms or small molecules as Ne, Ar, N2, H2O, NH3, CH 3 OH, CH 3 F were aimed at verification if the GGA97 set of approximate functionals could describe correctly the energetic balance between the hydrogen-bonded (or planar) and stacked arrangements 1 2 7 . These two effects compete but the experimental identification of the most stable conformer is available only for a few among the investigated complexes. According to the KSCED results, the stacked complex is more stable only in the case of the neutral phenol-Ar complex. Experimental dissociation energies are available for a few among the complexes studied and the KSCED interaction energies compare well with this reference data.

One-Electron

Equations for Embedded Electron Density

59

Fig. 15. The most stable conformer of the (C36HisN)2 complex (top view). For details see [Tran et aL, J. Phys. Chem., 108, (2004) 9155].

The last group of applications concerns the largest systems studied so far - dimers formed by nitrogen-containing polycyclic aromatic hydrocarbons: (C3oHi5N)2 and (C36H 15 N)2. These molecules are targets for organic synthesis as potential building blocks of organic conducting nanocolumns 141 . Calculations were made to predict the conformational preferences of such dimers 1 2 6 . The non-variational variant of calculations, in which the non-relaxed electron densities of the monomers were used in Eq. 18 instead of solving Eqs. 31-32. This choice was made for practical reasons and was justified by the estimation of the effect of the relaxation on the shape of the potential energy surface. Test calculations for the potential energy curves corresponding to horizontal displacement at the equilibrium interplanar distance show that the relaxation leads to an uniform shift of about 1.0 kcal/mol. Increasing the interplanar distance reduces the magnitude of the energetic effects due to relaxation. All energies discussed below were obtained with non-relaxed electron densities. A slipped parallel arrangement was found to be the most stable for both dimers ( E i n t =-8.4 kcal/mol for (C 3 oHi 5 N) 2 and Eint =-9.6 kcal/mol for (C 3 6 Hi 5 N) 2 ). The most favorable geometry of (C36HisN)2 is shown in in Fig. 15. The interplanar distance at the energy minimum equals to 3.29 A and 3.35 A for (C3oHisN)2 and (C36Hi 5 N)2, respectively. It was found that restoring the centered arrangement, in which the nitrogen atoms are localized on top of each other, involves an energetical expense in the range of 1.5 to 2.0 kcal/mol for each considered dimer. Such an

60

T. A.

Wesolowski

arrangement of nitrogen atoms would be the most favorable from the point of view of the electric conductivity of a nanocolumn built of such monomers. The effect of substitution of nitrogen by carbon on interplanar distance was also investigated 126 . To this end, the minimum structures and the selected potential energy surfaces obtained for the two heterocyclic dimers were compared to the corresponding results for the ovalene dimer (C32H;u)2. Ovalene has the intermediate size between that of the considered heterocyclic compounds. Compared to the ovalene dimer (which is nitrogen-free), the presence of nitrogen in the heterocyclic compounds affects only slightly the interplanar distance by increasing it by 0.02-0.04 A. 6.2. Potential energy surface II: Structure, energetics, vibrational properties of embedded systems

and

In the applications considered in the previous section, the two subsystems were described at the same level. This and the following sections concern such applications of Eqs. 23 and 31 in which the electron density (ps) of the environment is obtained using simpler approaches instead of solving Eq. 32. We recall that, if the electron density pB satisfies the condition ps S Po, and Eq. 44 holds, regardless how ps was obtained. Such studies represent a practical implementation of the electron-density partitioning strategy in multi-level computer simulations. This type of applications of Eqs. 23 and 31 includes the studies of: i) the potential energy surface corresponding to the proton transfer between two water molecules in the active center of the carbonic anhydrase enzyme 123 and it) the vibrational properties of the CO molecule trapped inside a porous solid m . Opposite to the other applications of Eqs. 23 and 31 discussed in this review, the two subsystems (A and B) interact very strongly and the condition of the smallness of the PA,PB overlap does not apply. In the first case, subsystem A comprises two water molecules and the Zn 2 + cation. The environment (subsystem B) comprises three histidine molecules. Since histidines are ligands of Zn 2 + , the two subsystem interact very strongly (see Fig. 16). The interactions between Zn 2 + and its histidine ligands cannot be expected to be well described by means of the GG A97 set of approximate functionals for the reasons discussed before. However, the studies concerned the shape of the potential energy curve corresponding to the transfer of proton between two water molecules but not the energetics of the histidine-Zn 2+ interactions. The histidines are localized far from the water molecules involved in the investigated proton-transfer process. It can be expected, therefore, that the possible inaccuracies in the description of the Zn 2+ -histidine interactions do not affect the energetics of the proton transfer. Indeed, it was found that the potential energy curve derived from Kohn-Sham calculations for the whole system shown on Fig. 16 are very similar to the ones derived using Eqs. 18, 31, and 32, (see Figs. 2-3, 5 in Ref. 1 2 3 , note also that the supermolecule Kohn-Sham results are denoted by DFT and the freeze-and-thaw KSCED ones by CDFT2 in these figures). The possibility to apply the multi-level strategy in view of modelling of the whole enzyme was also investigated in Ref. 123 . It was shown that using pg in

One-Electron

Equations for Embedded Electron Density

61

Fig. 16. The system used to model the proton transfer in carbonic anhydrase [Hong, et al, J. Comput. Chem., 21 (2000) 1554]. Subsystem A comprises the Zn 2 + cation and two water molecules, whereas the three histidines are considered as the environment (subsystem B).

Eqs. 18 and 31, which was not derived from the second of the two coupled KSCED equations (Eq. 32) but was assumed to be just the superposition of electron densities of three isolated histidines, does not worsen significantly the KSCED results. This simplification leads obviously to large possible computational savings and can be used as practical alternative to supermolecular Kohn-Sham calculations. Fig. 6 in Ref. 1 2 3 shows that these two approaches' 1 lead to very similar potential energy surfaces (the relative energies at stationary points agree within less than 10%). This way to reduce the computational effort was extensively used in subsequent works by Warshel and collaborators 128 ' 129 . A similar strategy to reduce the computational effort was used in studies of vibrational properties of CO trapped inside the metal exchanged ZSM5 zeolite 117 (see Fig. 17). Subsystem A was chosen to comprise the alkaline cation (Li + , N a + , or K + ) and the CO molecule, whereas the (AlSii502iH32)~ anion representing the framework of the zeolite was considered as environment (subsystem B). The conformation of the CO molecule trapped inside the zeolite was optimized for each alkali-metal exchanged zeolite. The harmonic frequencies corresponding to the degrees of freedom involving embedded CO were compared to that of the free CO molecule. The calculated shifts agree very well with experimental data (see Table 7), which justifies further analysis of these results. For instance, switching off selected components of the KSCED embedding potential and comparing the results obtained using cluster models of the zeolite framework of different sizes, made it possible to h

Supermolecular Kohn-Sham results are denoted by DFT and the ones derived using superposition of electron densities of the molecules in the environment by FDFT4 in Ref. 1 2 3 .

62

T. A.

Wesolowski

Fig. 17. The system used to study vibrational properties of the CO molecule trapped inside the metal exchanged ZSM5 zeolite. Subsystem A comprises the alkaline cation (Li+, Na+, or K+) and the CO molecule, whereas the (AlSiisOaiEte)^ cluster represents the zeolite framework (subsystem B). Hydrogens were used to saturate the cut Si-0 bonds. Details of calculations are given in [Wesolowski et al, J. Chem. Phys., 115 (2001) 4791]. Table 7. The shifts (in c m - 1 ) of the frequency of the CO stretching mode resulting from the interactions in the alkaline-metal exchanged ZSM5 zeolite. Theoretical shifts are the differences between the harmonic frequencies in the presence of interactions and in the free CO molecule (2119 c m - 1 ) . Experimental shifts are calculated with respect to the gas phase experimental value for CO (2143 c m - 1 ) . Data taken from [Wesolowski et al, J. Chem. Phys., 115 (2001) 4791]. Environment

Li+..CO

Na+..CO

K+..CO

none (calc.) ZSM5 (calc.) ZSM5 (exp.)

110 46 45

89 38 29-35

74 17 10-23

attribute the relative role of such factors as: i) the interactions with the nearest Al and its ligands, ii) the interactions between CO and the opposite wall of channel, and Hi) long-range electrostatic interactions, in affecting the structural and vibrational properties of the embedded CO molecule.

One-Electron

Equations for Embedded Electron Density

63

6.3. Electronic structure: Electron density, spin density, orbitals, orbital energies, and responses of atoms and molecules in condensed phase All observables calculated using the embedded orbitals derived from Eq. 31, depend on the accuracy of the non-electrostatic components of the KSCED effective potential (Eq. 53). The first application of the GGA97 approximation to the kinetic energy component of the embedding potential given in Eq. 53, which was aimed at such properties, concerned the effect of noble-gas matrix on the hyperfine tensor of the embedded the M g + cation 119 . The isotropic component of the hyperfine tensor Aiso(Mg) has been measured in neon and argon matrices (222.4 and 211.6 gauss, respectively 142 ). Aiso(Mg) is proportional to the spin-density on top of the Mg nucleus (see Appendix C). The KSCED embedding potential was used to evaluate difference between Aiso(Mg) for embedded in the noble gas matrix and isolated M g + cation. Such an application represents an ideal case for testing the accuracy of the overlap-dependent components of the KSCED embedding potential and the GGA97 set of approximate system-independent functionals in particular. The electrostatic component of VjffCED [PA ,PB',r], which has known analytic form (see Eq. 53), can be expected to play a minor role because of the spherical symmetry of noble gas atoms. Moreover, the embedded system is just an atom which eliminates the need to take into account the effect of the interactions with environment on the geometry of the embedded system 118 . Therefore, the accuracy of the non-electrostatic components (which are approximated) determines the quality of the calculated changes of the spin-density on top of the Mg nucleus resulting from interactions with the matrix. A spin-polarized version of Eq. 31 (see Appendix C) was used to determine Aiso(Mg) located in the center of the cube formed by eight noble gas atoms (either Ne or Ar). In each case, the size of the cube corresponds to the interatomic distances {R,MgNe =2.97 A, RutgAr = 3.39 A) obtained by means of supermolecular Kohn-Sham calculations for the same system taken from the literature 1 4 3 . For each matrix, the electron density of the eight noble gas atoms was used as ps in Eq. 31. The numerical values obtained using the largest considered basis set (fully decontracted 12s6pld set for Mg) agree very well with experimental data (222.9 gauss and 210.4 gauss for Ar and Ne, respectively). Such a good agreement of the absolute values of A, so (Mg) is probably fortuitous. Even for the isolated cation, the spin-density on top of the nucleus depends on the choice of the approximation to the exchange-correlation functional. However, the calculated shift Av4 iso (Mg) = Y4 iso (Mg..8Ne)-/lj SO (Mg..8Ar), which is determined by the approximations used in the KSCED embedding potential, is very stable with respect to the changes of the basis set and agrees very well with experiment. The absolute values of .A,So (Mg) depend much stronger on the basis set because the used orbitals of the gaussian type which do not obey the nuclear cusp condition 144 . The good numerical values of the calculate shifts of AiSO (Mg) indicate that the GGA97 set of approximate system-independent functionals provides a very reason-

64

T. A.

Wesolowski

able approximation to the KSCED effective potential and can be subject to further tests/applications concerning electronic structure of embedded chemical objects. Two types of such applications are reviewed below. 6.3.1. f-levels

of lanthanide cations in chloroelpasolite crystals

The ligand field splitting energies of /-levels derived from optical spectra are available for lanthanide cations in various elpasolite crystals 145 . This data represents

hu

t2u

a

2u

Fig. 18. Splitting of the /-levels in the octahedral environment.

a challenge for first-principles based computational methods. Attempts to use the Kohn-Sham formalism to derive ligand-field splitting energies in a system comprising the lanthanide cation and its nearest neighbors in the crystal lead to very discouraging results 146 . This indicates a serious flaw of the used approximations for the exchange-correlation potential. Atanasov et al.14,6 noticed that the relevant Kohn-Sham orbitals, although of predominantly /-type contain also significant contributions from the orbitals localized on ligands, and considered such mixing as an artifact. The buried /-orbitals of the lanthanide cation are not expected to be involved in covalent bonding. Indeed, a procedure eliminating this mixing improves the results 1 4 7 . Applying Eq. 31 to determine the splitting energies with PA representing the lanthanide cation offers an alternative computational strategy to address this problem. This type of application of Eq. 31 can be seen as the first-principles based realization of such past attempts to refine ligand-field theory which take into account non-electrostatic effects61. In octahedral environment, such as the one occur-

One-Electron

Equations

for Embedded Electron Density

65

ring in the elpasolite crystals, the /-levels are split into three ligand-field induced levels: t2U, t\u, and a^u (see Fig. 18). The splitting energies in chloroelpasolite crystals calculated using the KSCED(GGA97) embedding potential (Eq. 53) show a good agreement with experiment for the whole lanthanide series 148 . The calculated

500

400

300

c

200

4

6

8

10

Number of f—shell electrons

Fig. 19. The differences between the t%u and a2u levels (in c m - 1 ) in octahedral environment for the whole lanthanide series. Squares are used for experimental data. Full and empty circles correspond to KSCED results obtained with relaxed (polarized) or not-relaxed (not polarized) electron density of the ligands, respectively. Lines are used to guide the eye. Theoretical data taken from [Zbiri et at, Chem. Phys. Lett., 397, (2004) 441]. Experimental data taken from [Richardson et al, J. Chem. Phys, 83 (1985) 3813].

t2u — &2U splitting energies agree within 10% with experiment but the differences are not systematic (see Figure 19). Due to the approximations/assumptions made in the computations, which are not related directly to the accuracy of the KSCED embedding potential, such as: i) using of the average-of-configurations electron density in KSCED equations, ii) possible differences in the error of the exchange-correlation functional for different configurations, iii) excluding the possibility of ligand-tocation charge transfer, and iv) assuming the geometry corresponding to the sum of the ionic radii, the origin of the deviation between theory and experiment cannot be identified in a straightforward manner. For comparison, Kohn-Sham calculations lead to qualitatively wrong results 1 4 6 . It is interesting to note that more than half of the splitting energy originates from the polarization of the ligands by the lanthanide cation. At large PA,PB overlaps, the non-electrostatic part of the KSCED embedding potential is strongly repulsive which assures that the two subsystem do not overpolarize each other. The good numerical values of the calculated splitting energies indicate, therefore, that the non-electrostatic part of the KSCED effective

66

T. A.

Wesolowski

potential is well approximated by means of the GGA97 set of approximate systemindependent functionals. Neglecting the non-electrostatic (non-additive kinetic energy and exchange-correlation) components of the KSCED embedding potential results in unacceptable results (wrong order of levels, for instance) 1 4 8 . Similar tendencies occur for the splitting energies between the t\u and a^u levels (see Fig. 20). Their relative deviation from experimental data are larger than the ones for the t2u and a2u levels. The calculated t\u — a2u splitting energies

1200

I

r a

-• •-.

400

"©

0

0

2

4

o-

©

6

o

©•••

8

•o

e

10

e-

•©

12

e

14

Number of f—shell electrons

Fig. 20. The differences between the t\u and a^u levels (in c m - 1 ) in octahedral environment for the whole lanthanide series. Squares are used for experimental data. Full and empty circles correspond to KSCED results obtained with relaxed (polarized) or not-relaxed (not polarized) electron density of the ligands, respectively. Lines are used to guide the eye. Theoretical data taken from [Zbiri et al, Chem. Phys. Lett., 397, (2004) 441]. Experimental data taken from [Richardson et al, J. Chem. Phys, 83 (1985) 3813].

decrease almost monotonically with the number of electrons in the /-shell, whereas experimental data show traces of an oscillatory behavior (higher values for even than for odd numbers). The magnitude of the splitting energies can be correlated with the shape of the embedding potential inside the cage formed by its ligands. Fig. 21 shows the KSCED embedding potential calculated at the LDA level for a trial spherically symmetric electron density inside the cage formed by six Cl~ anions. It increases the fastest along the x, y, and z axes. Therefore, the orbitals oriented along one of these axes can be expected to be the least stabilized by the ligands. Indeed, the highest energy corresponds to the t\u orbitals which have a character and are oriented along the C4 axes (see Figure 18). The a2u orbital is the most compact and its lobes are oriented along the direction of the slowest increase of V^h. As a result, the corresponding energy levels are the lowest. The

One-Electron

\l {#

Equations for Embedded Electron Density

67

/ .' 0.7

0.4

|;<>

Fig. 21. The KSCED embedding potential (Eq. 53 at the LDA level) inside the cage formed by the ligands. The potential is shown in the plane formed by four CI atoms. In the vicinity of each C I - anion, the embedding potential is strongly repulsive. The potential is the most attractive in the center of the cage. The lines are drawn at: 0.4, 1.09, 1.10, 1.11, 1.12, 1.13, 1.14, and 1.15 atomic units.

conclusions emerging from these calculations concerning indispensability of the nonelectrostatic overlap-dependent components of the KSCED effective potential are indispensable in accurate prediction of the /-levels is in line with the reasoning leading to the empirical Angular Overlap Model introduced in the sixties by Schaffer and J0rgensen 61 .

6.3.2. The effect of hydrogen bonding on excited states of a chromophore in condensed phase UV or Visible light absorption spectra of molecules in condensed phase represent a long-standing challenge for computational methods. Accurate description of shapes of absorption lines is possible only if the computer model is capable to describe the electronic structure of the isolated chromophore and to represents the effect of the environment on the chromophore arising from interactions with a large number of solvent molecules. The first objective can be achieved at the expense of using highlevel wave-function based methods. The other one depends on the nature of the chromophore-solvent interactions, the electronic structure of the solvent molecules as well as its the structure and dynamics. Due to this complexity many theoretical approaches are in use. Usually a compromise is made making it possible to take into account the environment as good as possible at the expense of the accuracy of the method in describing electronic structure of the chromophore. The methods can

68

T. A. Wesotowski

vary from such where a very accurate method is used to describe the chromophore and a rather primitive model for the solvent which does not even takes into account the atomic nature of the environment. A recently reported combination of Multireference Configuration Interaction with Singles and Doubles (MR-CISD) method with conductor-like model of the solvent (COSMO) 1 4 9 provides a representative example. Such type of modelling, where the two levels of description are an obvious mismatch, is very useful if a reasonable description of the electronic structure of the chromophore requires high-level wave-function based methods whereas the effects of the environment are nonspecific i.e. can be adequately described by the averaged reaction field of a polarizable medium. Alternatively, methods describing the chromophore and the solvent at the same type of approximate formalism (usually low or medium quality) are also in use. They are especially useful if specific interactions are important and/or the solvent requires to be treated at the statistical level i.e. to be represented as a statistical ensemble. The effect of non-specific interactions on the electronic structure of a chromophore in condensed phase is frequently accurately represented using the continuum-dielectric representation of the environment 150 ' 151 . The specific interactions, cannot be properly accounted for using such a model (see for instance Ref. 1 5 2 ) . Using high-level wave-function based methods to account for the specific interactions is possible only for models comprising the chromophore and a few molecules in its environment 1 5 3 , 1 5 4 ' 1 5 5 because of the rapid increase of the computational effort with the size of the system. Similar size limits concern supermolecule strategy to investigate the excited states a chromophores in condensed phase using the LR-TDDFT formalism. Therefore, the supermolecule treatment of a chromophore in condensed phase can be applied only in absence of long-range effects. In the case of LR-TDDFT calculations, a large size of the investigated systems exposes also non-trivial problems concerning assignment of excitations and identification of artificial excitations arising from errors in the applied exchange-correlation potential 5 3 . For these reasons, the orbital-free embedding strategy based on Eq. 31 generalized for excited states 113 > 114 provides an alternative worth exploring. It is worthwhile to recall at this point that the computational advantages of using Eq. 31 for this purpose offers computational advantages only if one is interested in obtaining the shifts of the electronic excitation energies induced by the interactions with the environment. The quality of the excitation energies of the isolated chromophore depends obviously on the used approximations for the exchange-correlation potential and kernel in the LR-TDDFT formalism. The first application of Eq. 31 in studies of excited states concerned the effect of hydrogen bonding on the lowest excitation energies in the guanine-cytosine and adenine-thymine base pairs 1 1 4 . These studies show that the effect of hydrogen bonding between a chromophore and its environment on the electronic excitations localized in the chromophore be accurately represented by means of the orbital-free embedding potential of the Eq. 53

One-Electron

Equations for Embedded Electron Density

• polarized partner molecule O not-polarized partner molecule

69

M

0.6

e

£.

0.2

o.o

o o9» '^0.2

0.0

0.2 0.4 shift [eV] (supermolecule)

0.6

0.8

Fig. 22. The complexation induced shifts of the energies of the lowest excitations of nucleic acid bases in the dimers. Shifts are calculated with respect to the free molecule (guanine or cytosine in the guanine-cytosine complex and adenine or thymine in the adenine-thymine complex). For each monomer four excitations are given: singlet-singlet A', singlet-singlet A", singlet-triplet A', and singlet-triplet A". Data taken from [Wesolowski, J. Am. Chem. Soc, 126, (2004) 11444].

form. The absolute values of the excitation energies and their assignment are collected in Tables 8 and 9. The deviations between supermolecule and embedding results do not exceed 0.05 eV in the whole range of calculated shifts varying between —0.2 and +0.7 eV (see Fig. 22). For each considered chromophore, the shifts were calculated using two types of the electron density of the partner molecule (ps): either the one derived from freeze-and-thaw calculations (variational - allowing for electronic polarization of the environment) or that corresponding to the isolated partner molecule (without electronic polarization of the environment). Fig. 22 shows clearly that neglecting the electronic polarization of the partner molecule leads to qualitatively good results. They are, however, significantly worse than the ones derived using relaxed (polarized) electron density of the environment. The effect of hydrogen bonding is not uniform for all considered low lying localized excitations. Indeed, the embedding potential generated by cytosine felt by the orbitals of guanine is not uniform (see Fig. 23). A more detailed analysis indicates that the quantitative description of the hydrogen-bonding induced shifts in the excitation energies is due to both the electrostatic and non-electrostatic (overlap dependent) components of the KSCED embedding potential. Fig. 23 illustrates also of a very important property of VjjfCED [PA , PB ', r\ • It depends not only on the electron density of the environment (PA) but also on that of the embedded molecule. At the LDA and GGA levels, the embedding potential VgJJb[pA,PB',r\ given in Eq. 53 depends only on PB at ^ —> 0. Its non-electrostatic component

70

T. A.

Wesolowski

calculated at the LDA and GGA levels is strongly repulsive at large PA,PB overlaps. The most repulsive domains reflect the shape of the cytosine molecule. Around guanine, however, VjjfCED [PA , PB ; r\ reveals more local features. For instance, the embedding potential changes its sign in the middle of the C = 0 bond. This causes different effects on different orbitals (n, TT, and TT*) localized there. The fact that Table 8. The relative energies of the lowest excited states (singlet) in the nucleic base pairs (A-T and G-C) and in the isolated bases. The results of embedding calculations with not-polarized environment are given in parentheses. Data taken from [Wesolowski, J. Am. Chem. Soc, 126, (2004) 11444], Base

T T A A C C G G

Transition Symmetry Qualitative description A' 7T — > TV* A" n —> TV* A' 7T — > TV* A" TV* A' TV — > TV* A" n —> TV* A' TV — > TV* A" TV — • (J*

free base 4.71 4.19 4.73 4.31 4.20 3.71 4.70 3.89

Excitation Energy [eV] base in t h e dimer supermolecule embedding 4.64 4.66 (4.69) 4.29 4.31 (4.33) 4.69 4.68 (4.66) 4.46 4.52 (4.43) 4.52 4.51 (4.42) 4.38 4.39 (4.23) 4.53 4.57 (4.64) 4.12 4.17 (4.16)

Table 9. The relative energies of the lowest excited states (triplet) in the nucleic base pairs (A-T and G-C) and in the isolated bases. The results of embedding calculations with not-polarized environment are given in parentheses. Data taken from [Wesolowski, J. Am. Chem. Soc, 126, (2004) 11444]. Base

T T A A C C G G

Transition Symmetry Qualitative description A' TV — * TV* A" n —> TV* A' TV — * TV* A" <7 —> TV* A' TV —• TV* A" n —> TV* A' TV — + TV* A" TV —* a*

free base 3.62 3.95 3.76 4.16 3.50 3.65 3.87 3.86

Excitation Energy [eV] base in t h e dimer supermolecule embedding 3.62 (3.62) 3.61 4.12 4.12 (4.09) 3.72 3.72 (3.73) 4.30 4.33 (4.29) 3.73 3.72 (3.67) 4.33 (4.14) 4.33 3.75 3.76 (3.80) 4.11 4.17 (4.15)

the effect of hydrogen bonding on localized electronic excitations can be adequately represented by the KSCED embedding potential using the GGA97 set of approximate system-independent functionals brings important practical benefits as far as calculation of the solvatochromic shift is concerned. The equivalence of the LRT D D F T excitation energies derived using either Kohn-Sham orbitals for the whole system or the embedded KSCED orbitals for the chromophore, shown in Ref. 114 and discussed previously, provides the basis for significant computational savings in computer simulations of solvated chromophores. Owing to electron density par-

One-Electron Equations for Embedded Electron Density

71

titioning, it is possible to restrict the responses of the whole system to a predefined part of the whole system (a chromophore).

0.6 "_ 0.5 1 0.4 -

i

( /

0.3 - . 0.2 1

/ f

0.1 I

1! i

0.0 |jj

i.

\

-0.1

•

Fig. 23. The KSCED embedding potential (Eq. 53 at the LDA level) generated by the frozen electron density of cytosine in the guanine-cytosine complex. The lines are drawn in the plane of the molecules at: —0.1, 0.0, 0.2, 0.4 and 0.7 atomic units.

These advantages of the orbital-free embedding strategy were exploited in our recent computer simulation studies of acetone solvated in water 53 . Clusters of varying size (up to 250 water molecules) were used to represent the solvent to study the effect of distant solvent molecule on the lowest excitations (n —> 7r*) for this solvated chromophore. Using V3fCED[pA,PB\r\ incorporated into the general framework of LR-TDDFT does not involve any practical difficulties although supermolecule Kohn-Sham LR-TDDFT calculations are currently not possible for clusters of this size. It was found that water molecules beyond the radius of 8 A influence negligibly the energies of the analyzed excitations. It is worthwhile to underline, that the assignment of excitations is not trivial in LR-TDDFT because all pairs of occupied and unoccupied orbitals can contribute to a given excitation 111 . In the acetone case, however, a single pair corresponding to HOMO (n) and LUMO (7r*) provides the dominant contribution. The shape of the absorption band for the solvated acetone was obtained by means of averaging of the excitation energies for a sample of geometries derived from Car-Parrinello molecular dynamics simulations 156 for the chromophore and about 50 water molecules at 300 K. It was found that interactions with the solvent affect significantly the adsorption band by shifting its maximum

72

T. A.

Wesolowski

by 0.23±0.03 eV towards higher frequencies. The error bar on the calculated shift is due to a rather small number of geometries in the sample (220). The calculated shift compares very well with the experimental value of 0.20±0.01 eV. For each consid-

Fig. 24. The KSCED embedding potential (Eq. 53 at the LDA level) generated by the frozen electron density of the water molecules surrounding acetone shown in the plane comprising the carbon atoms. In the vicinity of each solvent molecule, the embedding potential is strongly repulsive. The lines are drawn at: -0.05, -0.04, -0.03, -0.02, -0.01, 0.0, 0.005, 0.1, 0.04, and 0.07 atomic units.

ered geometry, the corresponding shift is the result of the non-uniform effect of the KSCED embedding potential on the embedded orbitals of acetone. Fig. 24 shows the embedding potential for a one of the geometries used in the averaging procedure. As in the previously analyzed cases, the embedding potential is characterized by strongly repulsive domains corresponding to the solvent molecules. In the space occupied by acetone, however, Vjj^CBD[pA,PB',r\ exhibits more local features. 7. Concluding remarks This review concerned the developments and applications relevant for the electrondensity partitioning strategy in multi-level modelling of complex materials introduced in one of our earlier works 50 . Our interest in electron density partitioning based formalism originates from the fact that electron density is a well defined quantity for systems of sizes ranging from microscopic to macroscopic. Moreover, electron density is a basic quantity in density functional theory of atoms and molecules. It is shown that the search for the ground-state electron density for a system of any size can be replaced by the search for the difference between the total ground-state electron density and some trial electron density (see Eq. 44). The trial electron density might represent a subsystem of any size (even macroscopic). In order to

One-Electron Equations for Embedded Electron Density

73

obtain the exact ground-state electron density in such a search, the trial electron density does not need to be derived from any differential equation but must satisfy a simple local condition: at any point the trial electron density must not be larger than the ground-state electron density. Using the subsystem formulation of density functional theory 22 , it was shown that this difference can be obtained from oneelectron equations for the embedded orbitals (Eq. 31) reffered to in this work as Kohn-Sham Equations with Constrained Electron Density (KSCED). The effective potential in these equations is expressed by means of universal system-independent density-functionals. Practical applications of the KSCED effective potential rely on approximations to its two components. One of them (exchange-correlation) is an object of intensive research relevant for the Kohn-Sham formulation of density functional theory, whereas the other one (non-additive kinetic energy) is specific for the formalism reviewed here. Our attempts to approximate this term represent, therefore, a key step in our long-range objective aimed at using the KSCED equations as an ingredient in multi-level simulations of a complex system. The set of system-independent approximate functionals, called GGA97 in this work, appears to be the most adequate for studies of such embedded systems which do not interact strongly with their microscopic environment. This picture emerges from the analysis of various tests of its accuracy for model systems as well as a representative number of applications reviewed in detail here. New computer implementations of Eqs. 23 and 31 were used in our recent computer simulation studies following the multi-level strategy. In such studies, the electron density of the subsystem of principal interest was obtained using Eq. 31 but that of the environment was derived from other (simpler) methods - assuming it to be a superposition of electron densities of certain number of molecules in the environment for instance. In view of the fact that density functional theory and the approximations to the exchange-correlation functional is the field of intensive research, we expect further progress and intend to keep in pace with these developments. An example of these developments are our recent attempts to approximate the exchange-correlation component of the KSCED effective potential in a hybrid way 53 . The orbital-dependent SAOP potential 1 5 7 was used to approximate Vjff [PA ; r\ in Eq. 52, whereas the exchange-correlation component of the KSCED embedding potential given in Eq. 53 was approximated at the GGA97 level. This type of hybrid approximation applied in Eq. 52 makes it possible to use the orbital-free embedding methodology also in cases where the description of the isolated subsystem of interest at the GGA level is not satisfactory and the orbital-dependent approximations to the exchangecorrelation functional are known to be more adequate. In particular, the Optimized Effective Potential techniques (for recent review see Ref. 1 6 2 ) can be expected to bring new breakthroughs as far as exchange-correlation functional is concerned 163 and any future level of Kohn-Sham theory can be incorporated into the general framework based on the embedding functional EE and one-electron equations for

74

T. A.

Wesotowski

embedded orbitals (KSCED). The improvements of the approximation needed to express the effective potential in Eq. 31 as a functional of two electron densities is our next objective. This involves two types of issues. One is specific to the subsystem formulation of density functional theory, i.e. approximating the non-additive kinetic energy functional especially at larger PA,PB overlaps. One of the possible strategies involves subpartitioning of the electron density 86 . The other one, however, is closely linked to progress in approximating the exchange-correlation functional (and potential). A well balanced, approximation to be used in the subsystem formulation should minimize the overall error of all its approximate components. This can be achieved by trial and error 159 but a deeper understanding of the links between different density functionals would provide the best guidelines 88 ' 107 . This review, concerned mainly our own work on approximating: i) the KSCED effective potential and ii) the total functional Es given in Eq. 23. One-electron equations for embedded orbitals (Eq. 31) were applied also by others. In particular, it was used in fully variational calculations for atoms in solids following the original Cortona's strategy 1 6 5 ' 1 6 6 ' 1 6 7 ' 1 6 8 ' 1 6 9 ' 1 7 0 ' 1 7 1 ' 1 7 2 ' 1 7 3 - 1 7 4 ' 1 7 5 - 1 7 6 - 1 7 7 ' 1 7 8 . Several authors applied Eq. 31 in embedding calculations, where electron density of different subsystems was derived from different methods 128,129,179,180,181,182,183,184,185,186,187,188,189^

A

d o g d y

^

^

c o m p u t a t i o n a l

ap

_

proach was proposed Carter and collaborators, in which the orbital-free embedding potential of the Eq. 53 form was combined with the in wave-function type of description for the subsystem of primary interest 1 9 0 ' 1 9 1 , 1 9 2 , 1 9 3 . Acknowledgments Discussions with Profs. E.-J. Baerends, M.E. Casida, A. Savin and Drs. G.K.L. Chan, O. Gritsenko, K. Pernal concerning selected issues covered in this review are greatly acknowledged. The author is thankful to Mr. M. Dulak for his work on issues discussed in the Numerical Implementation section and Dr. R. Kevorkiants for help in preparing some figures. This work was supported by Swiss National Science Foundation and Swiss Federal Office of Education and Science.

One-Electron Equations for Embedded Electron Density 75 A p p e n d i x A. Functional derivatives: Generalized Gradient Approximation In generalized gradient approximation (GGA), the exchange- and kinetic energy functionals (EGGA[p\ and TGGA[p\) take the following general forms: rpGGA

[p] = CTF Jpsl\r)FT{x(f))dr

(A.l)

^GGA

[p] = -Cx J p4/3(r)Fx(x(r))dr

(A.2)

and

where x = x(r) = '-jj^ and each function Fr{x(r)) or Fx(x{f)) determines the gradient-dependence of the corresponding GGA functional. For any differentiable (up to the second derivative) function Fx(x(f)), the functional derivative of the exchange energy functional can be obtained using the following expression: ^

^

3

3

where D = Y

Yl

= ~\cxp^

(Fx(x) -

xF'x{x)+x*F^x))

dtpdldjpdJp.

For the kinetic energy functional the similar expression reads: 6

^

M

= §CW

+ C

T F

/ 3

D ^

(FT(X)

- xF^x)

4

+

{F>T{x) - xF?{x)) - C

-x^(x]

T

F

^ ^ -

(A.4)

A p p e n d i x B . T h e exchange-correlation functional in Local D e n s i t y Approximation LDA leads to the exchange functional of a simple analytic form given by Dirac 9 3 : 1 /^

ELXDA[P] = -\ ( £ )

JPi/3df = -Cxjp^dr

This functional was used in the computational methods, such as the Xa developed by Slater for chemical applications 94,194 . In LDA, the correlation energy functional takes the following form: E.LDA [p] = Jp(f)-ec(p(f))dr

(B.l) method,

(B.2)

76

T. A. Wesolowski

The analytic expression for ec (p) is not known but the numerical values covering a large range of densities obtained by Ceperley and Alder 95 using quantum Monte Carlo calculations for homogeneous electron gas are available. In this work, we use the label LDA for the analytic fit for this data by Vosko, Wilk, and Nusair 96 which takes the following form: £

. , A /, x2 2b m arctan c (P) = 77 ^FT\ + 7T ; 2x + b 2 V X{x) Q A bxo 2 X(x0)

fln(x-x0r \ X{x)

+

2(b

+

2x0)aict&n^_ Q

(B3)

2x + b

where 1/6

\4/37rp

• X(x) — x2 + bx + c • Q = \/4c - b2 • A = 0.0621814, x0 = -0.409286, b = 13.0720, and c = 42.7198. Other set of coefficients A, XQ, b, and c were fitted for completely polarized electron densities (p-j- ^ 0 and py = 0 ) . A p p e n d i x C. Open-shell s y s t e m s For any considered kinetic-energy functional defined for spin-compensated systems, the Oliver-Perdew spin-density generalization can be applied 195 : Ts[p\pl]

= \ (Ts[2pT] +T s [2p*])

(C.l)

A similar relation holds for the exchange functional 195 : Ex[p\p^=l-{Ex[2p^+Ex[2p^])

(C.2)

The correlation energy functional (Ec[p\p^}) cannot be decomposed in a similar way. For most approximate functionals of the correlation energy, the corresponding spin-density generalization is known. Therefore, the spin-polarized extensions of Eq. 23 for energy and Eqs. 31-32 for potentials are straightforward. The isotropic component of the hyperfine tensor (Aiso - Fermi contact term) was evaluated in open-shell KSCED calculations as: 47T

A i s o (nuc) = —gPgnucPnuc ~

_1

t

\p\{fnuc)

I

- p\(rnucj\

(C.3)

where g and gnuc are electron and nuclear magnetons; /3 and f3nuc are electron and nuclear gyromagnetic ratios, and nuc indicates the nucleus for which Ais0 is calculated.

One-Electron

Equations for Embedded Electron Density

77

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Chapter 2: Density-Functional Based Investigation of Molecular Magnets

Mark R. Pederson3, Kyungwha Parkb and Tunna Baruah ac Center for Computational Materials Science - 6392, Naval Research Laboratory, Washington, D. C. 20375-5345, USA Department of Physics, Virginia Tech University, Blacksburg VA 24061-0435, USA c

Department of Physics, University of Texas - El Paso, El Paso TX 79968, USA

Abstract Molecules containing transition metal centers provide a complexity that leads to many phenomena of import to chemistry, physics and biology. In particular the metal centers may vary their charge and spin states leading to metal centers that may be viewed classically as closed shell unpolarized states or open-shell spin-polarized ions. A large class of molecules containing collections of spin-polarized ions are often referred to as molecular magnets. Even within this somewhat specialized set of molecules, the name molecular magnet is not specific enough as there are two extremes in these systems that behave very differently when exposed to magnetic fields. In this paper we start by discussing the two idealized cases. One idealized class relates to molecules containing metal centers for which the spin-orbit interaction is quite small. The low lying excitations in this class of molecules may be studied from Heisenberg Hamiltonians. Another idealized case corresponds to molecules containing strongly coupled metal centers and

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non negligible spin-orbit interactions. For this case it is the spin-orbit interaction that is responsible for the lowest energy excitations in the molecule. This class of molecules has received significant experimental attention lately because it exhibits the phenomenon of resonant tunneling of magnetization which is fundamentally interesting and potentially technologically useful. We discuss how density-functional theory may be used to study this class of molecules and provide examples to current systems of interest. The examples show the magnetism in these molecules can be influenced by applied magnetic fields, addition of extra electrons and changes in hydrogen bonding. The discussion includes numerical details in addition to the underlying theoretical framework. While most of this chapter is devoted to our work on this class of anisotropic molecular magnets some discussion of the exchange-coupled idealization is included to contrast the two types of behavior.

1. Exchange and Spin-Orbit Coupling in Molecules The general goal of computational chemistry is to predict and/or describe the behavior of a many-electron system. Such investigations may be based on either traditional wavefunction viewpoints or density functional theory. Regardless of whether one is working within the framework of traditional quantum chemistry or density functional theory, the electronic wavefunctions contain both spatial and spin contributions. The introduction of spin into electronic wavefunctions leads to several interesting and different types of behavior that are somewhat clumsily placed under the single umbrella of magnetism. First, because of the requirement of antisymmetry of the electronic wavefunction, the coulomb interaction between electrons leads to an additional attractive term, between electrons of similar spin, which is referred to as the electronic exchange interaction. Because of this interaction, an open shell system tends to maximize its spin especially in cases where the electronic wavefunctions are well localized and orthogonal by symmetry. If they are orthogonal by symmetry there is absolutely no kinetic energy penalty associated with placing them in parallel spin states and there is always a decrease due to the Coulombic exchange interaction. This effect is often referred to a ferromagnetism but for the purpose of introduction it would be best to

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think of this effect as one of two necessary conditions to achieve a ferromagnetic state. If we now consider two open shell atoms or ions that are reasonably well separated, the phenomenon referred to as antiferromagnetic behavior may be observed. In the simplest possible examples, the coulomb exchange for the separated ions has dictated that each atom prefers to have a net moment, or in other parlance, a number of unpaired electrons. However as the distance between the two ions decreases, the Pauli principle requires the no-longer non-overlapping electronic wavefunctions to remain orthogonal. Forgetting about symmetry for the moment this may be accomplished in two ways. First, if the moments on neighboring sites are parallel (ferromagnetically coupled), the electronic wavefunctions must be spatially orthogonal which leads to an increase in the systems kinetic energy and a decrease in the systems Coulomb exchange energy. Second, if the moments on neighboring sites are antiparallel, the spatial part of the wavefunctions on neighboring molecules need not be orthogonal. Indeed the occupied a-spin electrons on site A may even lower their kinetic energy slightly by allowing the tails of their wavefunctions to mix with the unoccupied a-spin electrons on site B. The occupied P-spin electrons on site B may similarly decrease their kinetic energy. This decrease in the system's kinetic energy is accomplished at the expense of a smaller onsite Coulomb exchange energy. In a single-determinental picture the system chooses the option that leads to the lowest total energy even though one of these options allows for a spatially symmetric wavefunction and the other one does not. In the most general case, both of these choices lead to a spin-contaminated wavefunction since a single-determinental wavefunction is usually not an eigenstate of spin. The second case is generally referred to as an antiferromagnetically ordered wavefunction. It has long been recognized that a projection method, first introduced by Lowden, [1] can be used to project out a wavefunction of total spin and/or existing spatial symmetries and that at least one of these projections will lead to a wavefunction with an energy that is lower than the original single determinant. The projection method is at least tedious and possibly not very useful even for relatively simple cases. Instead, when a system has a propensity toward antiferromagnetic behavior one often attempts to construct a Heisenberg Hamiltonian by performing calculations on many different metastable collinear spin-orderings within the molecule which

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lead to broken symmetry states. As Davidson noted in a recent paper on this method, there have been so many studies within this framework that a complete review of the subject of Heisenberg Hamiltonians is at least difficult [2]. The eigenvalues of this Heisenberg Hamiltonian correspond to spin excitations of the system. In recent years, chemistry-based investigations along these lines have been lead by Davidson, [2,3] Noodleman [4-6] and Chen [7] and more recently Krylov [8]. This sort of methodology is appropriately applied to at least two classes of molecules which respectively may be referred to as di- and tri- radicals and as molecular magnets. There are a class of interesting transitionmetal containing molecules for which the broken symmetry approach could reasonably accurately describe the interactions between the molecular system and an applied magnetic field. Examples of this sort of system will be discussed at the end of the paper. However, as already noted above and expounded upon below, referring to the above systems as ferromagnetic or antiferromagnetic behavior is at least imprecise especially in cases where the extracted coupling constants are large. In such a case, which represents an important class of molecular magnets, it is the interaction described below that determines the lowtemperature magnetic behavior of a system. While the following discussion will be repeated, mathematically, in a later section it is useful to consider the following problem. Suppose we have performed a calculation on a molecule that has some number of unpaired electrons and we have satisfied ourselves that the resulting wavefunction is accurate enough to describe many features of the chemical system. Regardless of the level and complexity of theory the shape of this wavefunction has been primarily defined by the electron-electron and nuclei-electron Coulomb interactions and by kinetic energy operator. To a lesser but non negligible extent the moderate Coulomb and kinetic exchange interactions discussed above refine the shape of the electronic wave-functions. Further, the wavefunction will contain products of spatial functions and spinors which, especially for this discussion, are most appropriately referred to as a and f3 spinors [9]. If we start with a nonrelativistic Hamiltonian, which in particular does not contain the spin-orbit interaction, a unitary transformation on these spinors would produce a new set of spinors and the new wavefunction would have the same energy as the original wavefunction. When the spin-orbit interaction is introduced this is no longer the case. Once the spin-orbit interaction is introduced the

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second-order spin-orbit energy depends upon a 2x2 unitary transformation on the original set of spinors. The energy as a function of the no longer ignorable 2x2 unitary transformation is responsible for magnetic reordering energies at very low temperatures. This realization was first explained by van Vleck [10] and represents one contribution to what is known as the magnetic anisotropy.

Fig. 1. Schematic diagram of the geometry of the Mri|2-acetate molecule. The shading scheme of the atoms is as follows: darkness increases from H (white), to Mn, to C, and to O (darkest). For simplicity, in this figure eight CH3COO attached to Mn ions, from above and below, are replaced by eight HCOO. From the above discussion, there is no guarantee that the exchange interactions are necessarily stronger than the spin-orbit interactions. Indeed there are cases where this is probably not the case. Further even if the exchange interactions are indeed large in comparison to the spinorbit induced magnetic anisotropy, one needs to also consider the possibility that dipolar interactions between localized spins in different parts of a molecule could contribute appreciably to the magnetic anisotropy [11]. Such dipolar interactions are thought to be important in organic radicals where distances between localized spins may be significantly reduced in comparison to molecules containing wellseparated metal centers. However for cases where the exchange

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interaction leads to strong couplings between neighboring transition metal ions and where dipolar interactions are negligible, a phenomenon known as resonant tunneling of magnetization (RTM) can be observed. Before describing this phenomenon (RTM) more fundamentally, we discuss the following simplified qualitative picture. Suppose a molecule containing a set of strongly coupled transition-metal ions is placed in a magnetic field that opposes the overall spin projection of the molecule. The molecule would prefer to align its total moment with the applied field. If it is unable to physically rotate then there are two ways this realignment could occur. First a local moment on a given site could flip which would cost some energy that is determined by the exchange coupling parameters and the Ising or Heisenberg Hamiltonian. Some of the energy cost, associated with a localized spin flip, would be regained since the local flipped spin would now be parallel to the field. Once the first flip occurred, a second flip could occur until every ion in the problem had individually performed a summersault. The temperature at which the total moment of molecule would flip through this sort of mechanism would be associated with the largest exchange interaction between the local spins in the problem. An alternative way to achieve moment reversal would be for every electron spin in the problem to simultaneously perform a summersault. If this feat is accomplished the kinetic and coulomb exchange energies never change since the relative orientation of each moment in the problem remains fixed in reference to the others. Assuming a system with no net orbital moment, the only change in energy would be the interaction between the total spin and the magnetic field. If the spin-orbit interaction is neglected we would conclude that the moment orientation follows the magnetic field direction with no energy cost. However, the spin-orbit energy changes as each electron in the problem does a summersault and it is this energy difference that determines the reorientation barrier. As shown experimentally [12,13] the situation becomes even more interesting when one considers the quantum-mechanical analog of this experiment and when the total moment of the molecule is large. Under these conditions rather than jumping over the anisotropy barrier, a tunneling can occur between states of different spin projections when the energies of these levels are aligned by an applied magnetic field. It is a now a remarkable fact that experimentalists have identified a class of seemingly complex and complicated molecules which somehow behave as a single quantum-mechanical spin at reasonably high temperatures. This

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single-spin behavior is achieved in spite of the fact that these molecules are composed of hundreds of atoms and thousands of electrons. A full explanation as to how this collective phenomenon arises presents computational chemists and physicists with a great challenge. In this work, we describe some of our attempts aimed at understanding several aspects of these molecules. First, we show how an all-electron mean-field formalism can be used to reduce a problem containing many-electrons and atoms into a single spin. Second, we show how the presence of extra electrons or changes in weak chemical bonds can produce measurable changes in the magnetic behavior of these systems. We include the best case and worst case theoretical examples as well as one example that exhibits characteristics due to both onsite anisotropy and exchange coupling. While the primary emphasis of this work is on the phenomena associated with anisotropy of the metal centers we close with a significantly abridged review of broken-symmetry methodologies for exchange-mediated molecular magnets and to our applications of these methods.

2. Molecular Magnets The experimental observation of resonant tunneling [12,13] of magnetization in the Mn12-Acetate molecule [14] has focused significant attention on molecular magnets and molecular magnetic materials [15-39]. This molecule represents the anisotropic class of molecules. The molecules are composed of a reasonably small number of transitionmetal centers that are stabilized by oxygen anions and ligands. Each transition metal is six-fold coordinated by an organic atom with secondnearest-neighbor metal centers. The exchange coupling between the metal centers are relatively strong leading to well defined spin-ordering of the metal centers. The lowest electronic excitations are also relatively large in these systems (0.2-1.0 eV). For systems containing a partially occupied shell of d electrons the local symmetry about each metal cation is significantly perturbed from Oh symmetry which ensures an anisotropy in the spin-orbit energy (For an early discussion on this effect, See [10]). However, there are also molecular magnets with d5 configurations where the magnetic anisotropy is still very large [17].

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In addition to the intrinsic interest in quantum phenomena in systems as large as 1-2 nm, these nanomagnetic molecules may form the basis of future molecular scale devices. In principle, they can be used to store information (presently at economically unfeasible temperatures) using similar strategies as in larger classical magnetic storage devices. However, the resonant tunneling of magnetization complicates such behavior at the nanoscale. As discussed in Refs. [40, 41], the magnetic signature of these molecules are also rather sensitive to the presence of excess electrons, changes in hydrogen bonding, temperature and pressure so they could also be useful as in situ diagnostics for early detection of failure in materials. While a very good phenomenological understanding of these molecules exists, a better atomistic understanding can only be garnered by computational investigation. Such investigations are quite challenging since these small molecules are actually rather large from a computational perspective and since the energy scales of interest are smaller than we often expect to be able to quantitatively understand from computational quantum mechanics. In Sec III, we discuss the computational methodology employed for this work. In Sec. IV, we discuss the calculation of magnetic anisotropy energies within mean-field methodologies. We then turn to applications on various molecular magnets in Sees. V-VIII. By way of examples on Fe8, Co4, Mni2-Acetate and the Mat monomer and dimer we show how these systems behave and illustrate that the following hierarchy of interactions to be important. As mentioned earlier, the electronic structure and dynamics of the electrons is determined in entirety by large effects such as the Coulomb and kinetic interactions followed by exchange (both coulombic and kinetic). Once the electronic wavefunctions are determined, the spin-orbit interaction leads to the second-largest energy scale. However, when averaged over all electrons the first-order spin-orbit interaction is exactly zero and it is the second-order contributions to the spin-orbit interaction that are nonzero and also rotationally noninvariant. This effect leads to the anisotropy Hamiltonian and a molecule that behaves almost exactly as a large single spin in an anisotropic field. On a significantly finer energy scale, there are many different interactions which complicate the ideal single-spin behavior. These smaller effects are currently being investigated both experimentally and theoretically. A consensus about which interactions are most important does not currently exist. Here, we review our work on the role of exchange interactions

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between different cations, the effect of excess electrons, and coupling with spin and vibrational degrees of freedom. Further detail can be found in the references. The topics discussed here range from cases that have proven to be challenging to density functional theory, cases where density-functional theory [42] has been successful, and cases where density functional theory has been predictive. It also includes a reasonably large sample of molecules that have been looked at within density-functional theory.

3. DFT: Theoretical and Computational Framework Within the density-functional theory (DFT) [42] and the BornOppenheimer approximation, the energy for a nanoscale system consisting of nuclei and electrons is written according to

ia

In the above equation the spin densities is constructed from Kohn-Sham orbital densities. The only unknown quantity in the above expression is the exchange-correlation energy which is represented by the third term of the above equation. Approximations for this term have been developed by Perdew and coworkers [43-45] and Becke, Li, Yang and Parr [46,47]. The Perdew-Burke-Ernzerhof generalized-gradient approximation (PBE-GGA) [45] to the density-functional theory, which is used in this work, depends on the spin densities, their gradients and their second derivatives. Once this ansatz for the total energy is accepted, an application of the variational principle tells us that that it is necessary to self-consistently solve a Schroedinger-like equation of the form

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M. R. Pederson, K. Park and T.

Baruah

< 5^i(7\ - - V 2 + Vscf(r)

- €i\ipia >= 0.

(2)

In this work the Kohn-Sham orbitals are expressed in terms of a set of gaussian-orbital functions (fj(r)) and spinors (x<, Recording to: ipi8(r) = ZjvCyjj^Xa.

(3)

Given a set of basis functions {fj(r)}, this reduces the problem to determining a set of expansion coefficients rather than the value of each wavefunction at each point in space. Projection of the resulting equations onto the basis set leads to a secular equation. Once the ground-state density and wavefunctions are determined many molecular properties can be investigated [48-56]. These properties include electronic structures, vibrational modes [52] and their infrared and Raman spectra [54], ionization energies, electron affinities, polarizabilities and magnetic properties [57-59]. It is the last effect that will be discussed in greater detail below. Prior to discussing this, a brief overview of some of the numerical methods used for solving the equations will be given. We then discuss the calculation of magnetic anisotropics in greater detail. One aspect of the approximations to the density functional theory is that some of the integrals required to solve the above equation can not be evaluated in closed form even for the gaussian-orbital basis functions used in this work. Therefore some numerical work is required to solve the above equations. An extensive literature on our gaussian-orbital-based methods exists and the interested readers are referred to Refs. [48-56] for a more detailed discussion of the numerical methods used here. Here we provide a brief discussion on some of these algorithms. First, we use a variational integration mesh [49] that allows us to find a set of mesh points {rj} for the precise numerical evaluation of integrals required for solution of the density-functional equations. Each matrix element or integral can be rewritten as:

/ = I dirQ{v) = ^lQ{vi)Vi

(4)

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where V; is the volume associated with point r ; . The integrands Q(r) have the feature that they are either singular (1/r) or strongly peaked at all the positions of the atom centers and that they decay with many different length scales from these atom positions. In Ref. [49], it was shown that for any particular molecule it is possible to decompose space into many non-overlapping space filling regions using three different types of building blocks. The first building block is a sphere with an atom at the center. The second building block is the part of space that is outside a sphere of radius R and inside a cube of dimension 2Rx2Rx2R. The third region is a parallelepiped with no atoms contained within. In Ref. [49] several different quadrature schemes are discussed which are well suited for integrating functions that arise in density-functional calculations on real molecules. These schemes effectively allow one to minimize the number of mesh points, and therefore the computational cost, required to achieve a given level of numerical precision. Table 1. The maximum and minimum exponents of the bare Gaussians, the number of bare Gaussians and the number of contracted Gaussians for each angular momentum for different atoms used in the calculation of Fe8. Atom

amax

a.min

^bare S p d

H C

77.84 0.0745 2 . 2 x l 0 4 0.0772

6 12

2 11 32 1

N

5.1 xlO 4 0.0941

13

3 2 1

O

6.1xl0 4 0.1049

13

3 2 1

Fe Br

3.8xlO 6 0.0452 7.9xlO 6 0.0781

20 21

5 3 2 5 4 2

Since the variational mesh allows any desired numerical precision, the accuracy of a calculation is then limited only by the completeness of the basis set used in Eq. 2. Our approach is to use atomic orbitals that accurately reproduce the exact density-functional total energies for each atom. To this set of functions, we add two or three additional functions of each occupied angular momentum and some polarization functions. One aspect of our basis sets is the algorithm used for determining the decay functions. This algorithm treats the gaussian decay parameters as variational quantities. For a given set of decay parameters the Kohn-Sham orbitals are determined self-consistently. At the end of the self-consistency cycle the derivatives of the total energy with respect to the gaussian decay parameters are determined using the Hellmann-Feynman theorem. Once

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M. R. Pederson, K. Park and T.

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these gradients are determined the conjugate-gradient method is used to find the set of decay parameters which minimize the energy. This method is discussed in Ref. [56]. Also in Ref. [56] we show that uniform accuracy (independent of atomic charge) can be obtained by ensuring that the shortest range function scales roughly as Z10/3. In Table I we give a bit more information about the basis sets used for the Fe8 molecule which is discussed below. A third issue related to both accuracy and computational efficiency is choosing an algorithm that allows for both a scalable solution of Poisson's equation and the use of fast multipole methods for systems larger than 1000 atoms. Given a set of Kohn-Sham orbitals the density is determined according to:

p(r)

= E iCT |^(r)| 2 t-'mwX mn J my-) J n\y)

.

(5)

While general basis sets lead to an 0(N 2 ) pairs of basis-function products, localized basis sets decrease this number to O(NM) with M a characteristic number of neighbors. As discussed in Ref. [49], the number of points at which the density must be calculated is also linear in the number of atoms. However since each product of localized basis functions is also localized, the determination of the density is intrinsically an O(N) problem. Further simplification occurs when the gaussian orbitals are used. Since the product of a pair of gaussian functions can always be expressed as a single gaussian function, it follows that the total and spin densities can always be analytically expressed as an O(N) sum given by: p(r) = S A i i i i m P ( r - A,) e x p ( - A ( r - Ai) 2 )

(6)

where, P(r-Ai) is a polynomial function centered at atomic position Aj. Once the spin densities are analytically decomposed according to the above expression, it is possible to determine the gradients and second derivatives as well as the Coulomb potential of each of these charge distributions analytically in the near field region or by using multipole methods in the far-field regions. Since the Coulomb potential is expressed in terms of a large number of these localized densities it is clear that this analytic decomposition allows for a massively parallel solution to Poisson equation.

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4. Predicting Magnetic Properties of Molecules and Clusters The magnetic molecules of current interest are typically composed of approximately 8-12 transition metal atoms which are locked at their lattice sites by a carefully arranged host consisting of organic molecules and ligands [14]. Without these ligands the transition metal lattice is found to be unstable [57,58]. The magnetic anisotropy of these molecules determines the idealized single-spin behavior of these molecules. For the original discussion of magnetic anisotropy energies in crystals see Ref. [10]. In Figs. 1 and 2, we present pictures of the Mn- and Fe- based magnetic molecules. These molecules are characterized by low symmetry, ferrimagnetic ordering and large overall moments. Both molecules have net moments of 20 uB. As discussed in Ref. [59], both the resonant tunneling magneticfield strengths and the magnetic anisotropy barrier are primarily determined by the second-order spin-orbit coupling interaction. Classically, the spin-orbit coupling interaction is caused by an electron moving with velocity v through the electric field (E) due to the other electrons and the nuclei. Since the electron has a spin, the interaction energy is given by t/(r)P,S) = ~ S . p x V $ ( r ) .

(7)

For further discussion see Refs. [60-62], In the above, E has been replaced by the gradient of the total Coulomb potential, and the electronic velocity (v), has been replaced by the momentum operator (p). The factor of two in the denominator, also derivable from the Dirac equation, is due to the Thomas precession. Given a spherically symmetric Coulomb potential and some simple algebraic reductions, the above expression is usually rewritten according to

While the above equation is exact for spherical systems, an attempt to approximate the spin-orbit coupling in multicenter systems as a superposition of such terms on a lattice could omit nonspherical corrections that may be especially important for anisotropy energies.

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Instead of using the traditional LS-representation for the spin-orbit coupling we return to more fundamental representation (Eq. 7) and note that since all basis-set oriented mean-field methods expand the singleelectron wavefunctions according to Eq. (3), it is only necessary to determine matrix elements of the form: Uja,ka> =< fjX*\U(T, P, S)\fkXa> > = E * T < fj\Vx\fk

>< Xa\Sx\Xv'

>

(9)

with the operator Vx defined according to:

^J>.

ao,

In Ref. [59] it is shown that an integration by parts reduces the above matrix element to the simplified form:

<^>^«K>-<|'*'f» <»> with matrix elements for Vy and Vz determined by cyclic permutations of the coordinate labels in the above equation. As explained in Ref. [59], the above equation is valid for both periodic systems and finite systems. This representation for the spin-orbit coupling matrix offers several advantages over the more usual LS representation. First, it does not require the determination of the electric field and depends only on the ability to accurately determine the Coulomb potential and the gradient of each basis function in the problem. This representation for the spin-orbit-coupling matrix is especially ideal for basis functions constructed from gaussian-type orbitals, Slater-type functions and planewaves. Upon inclusion of spin-orbit coupling, the first-order change in the total energy vanishes under most conditions and the second-order energy for a collection of Kohn-Sham spin-orbitals {§ia} is perturbed according to the following approximation:

Density-Functional

A

M™' = M™'*

sf

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97

= I w ZxyM™ S™ Sy "

_^<

=

MVMia'><

MVy\K

>

(12)

=<x«\sx\-x!a>.

The operators V and S operate on the spatial and spin degrees of freedom respectively and transform like pseudovectors under the symmetry operations. Now, taking a general unitary transformation on a fixed set of spinors, |Xi > =

cos#| T > + e i / 3 s i n 0 | | >

|X2 > = -e~i/3 sin d\ | > + cos 0| | >

(13)

and assuming uniaxial symmetry for the molecule, we find that the second-order spin-orbit coupling energy as a function of spin projection reduces to 4 cos 2 (26)

HMll + M?z+Mll + MH)lvl xx

I"

Mxx)

+{MH + Mll)-A1.

(H)

4 For the uniaxial systems discussed here, we have used the fact that all off-diagonal cartesian matrix elements vanish and that Mxx = Myy. Because of the symmetry there is also no dependence on the phase. As discussed in Ref. [56] the equation can be simplified according to:

A = A + D < Sz >2 with A^MH+MH+M^+MH+M^+M^)^ and D

(15)

= -KN~2 (Mll + Ml2z+Mll + Mll -Ml1-Mf-Mil-Mfz).

(16)

In the above equation, A is the average second-order contribution due to the spin-orbit interaction. If we identify < Sz > with M, we can determine

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M. R. Pederson, K. Park and T. Baruah

that the energy spacing between different M states. For the remainder of the paper we will replace < Sa > by Sa and assume the mean-field energy surface may be requantized in terms of operators rather than expectation values of operators. From the equation for the anisotropy barrier we see that there are several competing factors which are responsible for the formation of the anisotropy barrier. One figure of merit is the size of the energy denominator. A small energy denominator will typically make the magnitude of the M matrices larger. It is also evident that the size of the M-matrices requires some degree of spatial overlap between a pair of occupied and unoccupied electrons. This implies that pairs of occupied and unoccupied states which share the same principal quantum numbers and total angular momentum allow for large numerators in Eq. 14. For lower-symmetry cases, the same sort of derivation leads to an expression for the second-order spin-orbit energy given by: A = A + T,XtyJxySxSy.

(17)

By diagonalizing the y-matrix, the principal magnetic axes can be determined. Ignoring the uninteresting constant term and performing a little bit of algebra, the more common form of the anisotropy Hamiltonian in the rotated coordinate system is given by:

A = DS2z + E(S2x-S2y).

(is)

In the remainder of this chapter we discuss several different calculations which illustrate how barriers can be influenced by different environmental factors, ligands, charge states and transition metal centers.

5. Electronic Structure of the Fe 8 Molecular Magnet Apart from Mni 2 -acetate, the other extensively studied molecular magnet is the Fe8 molecule. It contains eight Fe centers and has the chemical formula [Fe802(OH)i2(tacn)6]8+ where tacn = 1,4,7 triazacyclononane (C6N3Hi5). The Fe(III) ions have a ferrimagnetic ordering leading to a total spin of S = 10. The +8 charge in the crystals are compensated by 8 Br ions. The molecular building block is [Fe802(OH)12(tacn)6]8+ Br 8 \7H 2 0. The approximate D 2 symmetry observed in the molecule [23], is formally broken by the presence of halide atoms and waters of crystallization. The

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central iron atoms are connected by oxo-hydroxo bridges. The Fe(III) ions have a d5 electron configuration. The ferrimagnetic coupling of spins between the eight Fe atoms results in an S = 10 spin ground state [26]. The organic tacn-rings are very important for stabilizing the magnetic core of the molecule because the three pairs of nitrogen dangling bonds complete a quasi six-fold environment for the Fe atoms. Our calculations were done on [Fe802(OH)i2(tacn)6]8+.Br6" and the system has overall 2+ charge. To coax the system toward ferrimagnetic spin-ordering, we follow the same method used in Ref. [59] and start our calculations with overlapping atomic potentials. To allow for the possibility of spin ordering, we add to this potential an empirical starting potential which favors the spin-ordering shown in Fig. 2. After the first iteration, this potential is removed and all the electronic and spin degrees of freedom are optimized variationally. The system may either accept or reject the ferrimagnetic ordering, depending upon the relative energetics of the spin ordering. The geometry of the cluster was optimized until forces on all atoms dropped below 0.002 a.u. The basis set employed is presented in Table 1.

Fig. 2. The optimized geometry of the Fe8Br6 cluster. The large balls show isosurfaces of the spin density at 0.0005. The orange surface shows the majority spin while the purple surface shows the minority spin. The spin charge is seen on some of the bridging oxygens.

100

M. R. Pederson, K. Park and T.

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The spin-ordering was found to be ferrimagnetic, with S = 10, in accord with experiment. The spin density of the Fe8 is shown in Fig. 2. Spin density isosurfaces of both signs are seen on Fe sites as well as the bridging oxygen atoms, indicative of ferrimagnetic ordering. Further, the spherical spin density around the Fe site confirms the spin-polarization expected for a closed-shell d5 electron configuration. Experiments [26,63] confirm not only the ferrimagnetic ordering but find smaller local moments on the two minority-spin Fe atoms than on the six majority-spin Fe sites. In order to ascertain changes in the local moments as a function of atom type, we placed a sphere of 2.19 Bohr around each Fe and calculated the net moment inside each sphere. For the two iron atoms with opposite spins, we obtain a moment of 3.42 |xB, whereas the six ferromagnetically aligned irons have moments ranging from 3.74 (iB to 3.85 UB, which is in qualitative accord with experiment. As discussed in greater detail in Ref. [64] our calculations on local moments in the Fe8 compound indicate six S = 5/2 majority spin Fe sites and two minority spin Fe atoms that are somewhere between S = 2 and S = 5/2. -1—•—I—•—I—'—I—•—r

Oxygen (p)

•"M i i "• i i i T n

i i i

I ' I i I i I Nitrogen (p)

Lil

r^-i ' i1-' I ^ I

' i i"r

H W LL

rV> ,xrv i <• i ' i i i

o >

i < i ' i i i ' i I'I"I'I

t w z LU Q

I i I ' I i"l Carbon (p) 1 ' l'i l"n Bromine (p)

JA

I •' I • I ' I i Iron (p)

^in|, i^rT-fr.i ^^rtf^V j. |

i I ' I ' I i I ' Iron (d)

i n W V A ^ J&. &

h i TOTAL MAJORITY

-14

-12

-10

-8

t*. -6

-4

ENERGY (eV)

Fig. 3. Calculated density of states for majority spin electrons. Figures 3 and 4 display the calculated total and atom-projected spinmajority and spin-minority density of states for Fe8Br62+. The spin of the HOMO and LUMO are different and there is a small gap of 0.15 eV between them. The spin majority channel has a HOMO-LUMO gap of 0.54 eV. This

Density-Functional Based Investigation of Molecular Magnets 101 peak is in qualitative accord with the optical conductivity measurements of Musfeldt et al. who find a peak at 0.9 eV. The spin minority channel shows a gap of 0.23 eV which has been assigned to a peak in the experimental optical conductivity measurements that appears at 0.35 eV. To determine if these gaps would indeed be optically observable we have calculated the optical absorption spectrum using the dipole matrix elements (length formulation) and a reasonable broadening factor similar to the method of Ref. [65]. Oxygen (p) ' I ' I ' I Nitrogen (p) to

f3*t H "

LU

I

l O i

I

• • I • l-"tCarbon (p)

A ^ f t i -i

' I ' r>"1r Bromine
JA

O i • i-

• I • I ' I

CO

Iron (p)

z111

-fwPf'fr-t'T-frltl •

•

• H iN».rfr>rT>oi-/h

-16

-14

-12

-10

Iro (d)

&A-

1A -8

' I • I • I TOTAL MINORITY

-6

ENERGY (eV) Fig. 4. Calculated density of states for minority spin electrons.

Fig. 5. Spin-averaged joint density of states for Fe8Br62+. The theoretical curves are broadened by a Gaussian, with full width half maximum of 0.27 eV, to simulate a realistic spectrum. The two curves show the spectra along two principal axes of the polarizability tensor.

102

M. R. Pederson, K. Park and T. Baruah

Figure 5 shows the calculated optical absorption spectrum, obtained from an analysis of the dipole matrix elements of Fe8Br62+. The two projections correspond to two most different polarization axes which have been determined by diagonalizing the bare polarizability tensor. The calculated and experimental electronic structures were found to be relatively good agreement. Despite this fact the anisotropy parameters calculated for this molecule seem to overestimate the experimental results by about a factor of two.

6. Mni2-Acetate and Its Derivatives The Mn12-Acetate complex [Mni2012(COOCH3)i6 (H 2 0) 4 ]:2 (CH3COOH) 4(H 2 0)] was first synthesized by T. Lis et al. in 1980 [14] and it is a prototype for the class anisotropic SMMs. It had been largely ignored by physicists and chemists until two experimental groups [12,13] independently observed quantum tunneling of the direction of the total magnetic moment in this material in 1996. Since then, numerous theoretical and experimental research on the Mn12-acetate and its derivatives have been performed to understand their magnetic properties, electronic structures, mechanisms of the quantum tunneling, and extremely slow relaxation time of magnetization at low temperatures [18,26,28,59,66-75]. In this chapter, we focus on the electronic and magnetic aspects of the SMM Mni2-acetate and its derivatives as well as some plausible mechanisms of the quantum tunneling. A single Mn12-acetate molecule has S4 symmetry and consists of four Mn4+ ions (S = 3/2) in the inner cubane and eight Mn3+ ions (S = 2) in the outer crown coupled via oxygen anions and surrounded by organic atoms shown in Fig. 1. The ground-state spin configuration was first suggested from the magnetic hysteresis loop measurements, [12,13] electron paramagnetic resonance (EPR) spectroscopy, [28,70] and inelastic neutron scattering [68,76]. Pederson and Khanna [59] calculated the ground state of the SMM within DFT to find good agreement with experiment. Later it was directly confirmed by nuclear magnetic resonance (NMR) measurements [77]. The four inner Mn spins are antiferromagnetically coupled to the eight outer Mn spins so that the ground state has S = 10. The highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap for majority (minority) spin was found to be 0.45 eV

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(2.08 eV). The energy gap between the minority (majority) spin LUMO and the majority (minority) spin HOMO was 0.89 eV (1.63 eV) [59]. From spinorbit coupling within a molecule, the second-order magnetic anisotropy barrier (MAB, or magnetization reversal barrier) was calculated to be 55 K, which is in good agreement with experiment [59].

HEAD

TAIL

Qx

,C—C—H H- O

Fig. 6. Six distinct configurations of a Mn12-acetate molecule with four HAc solvent molecules. The circles represent molecules. N is the number of HAc molecules whose heads point into a Mni2 molecule. From the top leftmost, clockwise, the symmetry of each configuration is S4, C2, Cu S4, C b and Cj. Cited from Ref. [40]. Although many interesting properties have been examined in the groundstate spin manifold S = 10, very little is known in the excited-state spin manifolds and there exist controversies in locations of the excited-state manifolds in the energy space [78-82]. The knowledge of the low-lying excited-state manifolds is important even from the standpoint of understanding the ground-state properties in the case that the low-lying excited-state manifolds are significantly overlapped with the ground-state manifold. In this context, within DFT, we considered low-energy collinear spin excitations. These excitation energies are much smaller than single-electron excitations [35]. To determine the low-energy excitations we have calculated the four exchange coupling constants [83]. Based on the calculated exchange couplings, we constructed an isotropic Heisenberg Hamiltonian and obtained the ground-state manifold S = 10 and the

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three lowest-energy excited-state manifolds, S = 9. We found that the upper limit on the energy gap between the ground-state S = 10 and the firstexcited-state (second-excited-state) manifold S = 9 is 40.5 K (47.5 K) [83]. This implies that the first two excited-state manifolds are highly overlapped with the ground-state manifold. Recent EPR spectroscopy confirmed that our calculated energy gaps are reasonable [84]. We note that there is a significant need to build a generalized spin Hamiltonian which includes the magnetic anisotropy and the exchange interactions. We also found that the MABs for the low-lying excited-state-manifolds S = 9 are approximately the same as that for the ground-state manifold based on the calculated MABs for the low-energy collinear spin excitations [83]. The many-spin Hamiltonian has a dimension of 108 which makes diagonalization difficult. We have used a Lanczos algorithm and also shown that a perturbative approach can be used to exactly determine the ground state. We also attempted to determine the lowest-energy by diagonalizing within a subspace of the lowest M = 10 configurations. Even when we used 100-3000 low lying configurations we found that the energies were not well converged with respect to the Lanzcos treatment. However, it is interesting that once the exact ground state eigenvector was obtained there were only thirteen spinconfigurations with appreciable coefficients in the many-spin wavefunction. These states correspond to the collinear mean-field state (roughly 50 percent) and three sets of four equivalent states with canted moments on the cations. We discuss this further in Sec. IX and discuss some possible reasons why the affects of other configurations do not affect anisotropy for systems with large local moments. The magnetic quantum tunneling arises from mixing between states in the spin-up and spin-down orientations (M and -M) due to some type of transverse magnetic anisotropy. Because of an exact S4 symmetry of a Mn12-acetate molecule, the tunneling should only be allowed between states satisfying the following selection rule: M-M' = 4n where n is an integer. However, tunneling between states violating the selection rule is also observed [12,13], To understand this discrepancy, three different theories have been proposed such as Chudnovsky et a/.'s dislocation-induced tunneling [85], Cornia et a/.'s solvent-disorder model [86], and Pederson et al.'s spinvibron coupling for isotopically impure samples [87]. Magnetic tunneling measurements [88,89] and high-frequency EPR spectroscopy [84,90]

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revealed that their experimental data may not agree quantitatively with what Cornia et al. or Chudnovsky et al.'s suggested. More recent calculations by Pederson, Bernstein and Park on the spinorbit-vibron effect suggest that it is probably too small to account for symmetry breaking or the enhanced fourthorder barrier. 1

.3? CO

O •J)

'c 03 40 O

1

• Mn 1 2 • Fe substitution for Mn • K donor addition

L_

.a >-50 QO

1

• •

"S

t

CO

E

30 -

5 P J

r~

CM " 0

1

2

i 3

4

Number of excess electrons Fig. 7. Calculated second-order MAB vs number of extra electrons for the Fe substituted and K added Mn^-acetate geometries and the Mn^-acetate geometry. Cited from Ref. [41]. To further quantify the Cornia model and to understand other possible disorder induced symmetry breakings [40,41] we considered (1) positional disorder induced by acetic acid (CH3COOH) solvent molecules, (2) orientational disorder induced by methyl groups (CH3) in the Mn^-acetate molecule, and (3) transfer of extra electrons to the Mn^-acetate. Densityfunctional calculations showed that only hydrogen-bonded solvent molecules contribute to the transverse anisotropy among all six possible configurations shown in Fig. 6. The maximum value of the transverse anisotropy E caused by the positional disorder is E = 0.016 K, [40] which will greatly facilitate the low-symmetry tunneling. This calculated E value is 3 times larger than what Cornia et al. predicted [86] and agrees well with experiment [84, 88-90]. The effect of the methyl-induced disorder on E was found to be negligible [40].

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Regarding transfer of extra electrons, we considered (i) replacing a few Mn ions by Fe ions and (ii) adding a few electron donors such as potassium to the Mni2-acetate [41]. Some of the considered cases have been experimentally realized [9,93]. In both cases, we found that the MABs decrease considerably with increasing number of extra electrons (Fig. 7). This is due to lack of Jahn-Teller distortion for the ions where extra electrons sit. The values of E and the easy-axis tilting angles for the geometries with one- and two-extra electrons are substantially larger than those induced by solvent disorder. For the tetra-Fe substituted geometry with S4 symmetry, we found that the total magnetic moment of the lowest-energy state is 8 |XB and accordingly the exchange coupling constant between outer metal ions increased by a factor of 5 compared to the Mn^-acetate. This implies that molecular environments considerably influence the magnetic properties of the molecule.

7. The Mn 4 Monomer and Dimer The Mnt dimer is interesting because it allows one to study a system where the very weak exchange interaction between the monomers affects the magnetic behavior. The SMM [MiuOBCUtOiCd^CHzMCsHjN^] (hereafter Mn4 monomer) was first synthesized by Hendrickson's and Christou's groups [94]. The Mat monomers can form a dimer which is well separated from neighboring dimers. The Mat monomer is comprised of three Mn3+ ions (S = 2) and one Mn4+ ion (S = 3/2) at the corners of the distorted cube shown in Fig. 8. Each Mn ion is in a distorted octahedral environment and is coupled via oxygen anions. Unlike the single molecule magnets, the magnetic quantum tunneling for the Mo* was observed before the external magnetic field reverses and no tunneling was observed at zero external field [94]. To understand the tunneling mechanism, we first calculated the electronic structure and magnetic and vibrational properties of the Mn4 monomer [95,96]. Then the dimeric geometry was constructed by application of an inversion geometry to the optimized monomeric geometry.

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Fig. 8. M114 dimer geometry. The dimer is formed by inversion of the threefold symmetric monomer. Each monomer has a magnetic core consisting of three ferromagnetically coupled Mn'+ spins (S = 2) coupled antifcrromagnetically to one Mn4+ spin (S = 3/2) ion leading to a total spin of S = 9/2. The distance between the two central CI atoms marked as the dotted line was measured to be 3.86 A. Cited from Ref. [95]. Density-functional calculations showed [95] that in the ground state the three Mn3+ spins are antiparallel to one Mn4+ spin so that the total moment is 9 uB which agrees with experiment. The HOMO-LUMO energy gap for majority (minority) spin is calculated to be 1.02 eV (2.42 eV), while the energy gap between the minority (majority) LUMO and the majority (minority) HOMO is calculated to be 1.17 eV (2.28 eV). The magnetic easy axis is along the threefold axis (the bond between Mn4+ and CI in the distorted cube) and the calculated MAB is approximately 11-12 K which is in good agreement with the experimental value of 14 K. We also considered different collinear spin configurations and calculated the exchange coupling constants between Mn ions within the Mai monomer [96]. The calculated exchange constants can be used to construct the Heisenberg exchange Hamiltonian. The diagonalization of the Hamiltonian gives rise to excited-state spin manifolds. Our density-functional calculations overestimated the exchange constants by a factor of 2

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compared to experiment. After reducing the calculated exchange constants by a factor of 2, we found that the first (second) excited-state spin manifold has S = 7/2 (S = 11/2) and is 245 K (418 K) above the ground-state manifold S = 9/2. So there is no overlap between the low-lying excited-state manifolds and the ground-state manifolds. Based on the calculated normal modes [54] for the Mn4 monomer, we predicted infrared absorption spectra and Raman scattering spectra [96] which were confirmed by experiment [97]. All vibrational modes are stable. o i

-1

.

1

.

'

•

1

.

'

•

1

.

'

•

1

-

CD

O _i

-2 -

- 3

'

-1

•

-0.5

0

0.5

'

1

[CI-CI DISTANCE] - Expt. Value (3.86 Angstroms)

Fig. 9. Logarithm of exchange constant J as a function of the monomer-monomer distance relative to the experimental value. The numerical uncertainty in J is 004 K. The slope of the curve is about -2. Cited from Ref. [95]. The binding energy for the dimer is about 0.16-0.78 eV. Densityfunctional calculations confirmed that in the stable dimeric configuration the magnetic moments of the two monomers are antiparallel to each other [95]. The calculated exchange coupling constant between the two monomers is 0.2-0.3 K which is comparable to the experimental value of 0.1 K. If the intermolecular exchange interaction is antiferromagnetic and significant, the first step of the magnetic hysteresis loop depends on the exchange coupling alone not the magnetic anisotropy parameter in contrast to the first

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class of the SMMs. The antiferromagnetic exchange interaction imposes an extra bias to dimers in small external magnetic fields so that the magnetic moments of the monomers can be flipped even before the magnetic field is reversed. Since our calculated exchange coupling constant is quite small, we computed the exchange constant as a function of the monomer-monomer separation. As is expected the calculated exchange constant decreases exponentially with increasing separation (See Fig. 9).

8. The Co 4 Molecular Magnet Cobalt clusters are known for ferromagnetic ordering and from this standpoint Co is a good choice for achieving a high spin magnetic molecule. However, the number of magnetic molecules based on Co is limited. Yang et al. have synthesized a Co based magnetic molecule in 2001. The experiments indicate that this molecule has total spin of 6 indicating ferromagnetic ordering. Their experimental measurement of the D parameter suggested a comparatively large parameter of 100 K. This raised the possibility of achieving molecular magnets with a high anisotropy parameter using Co.

Fig. 10. The molecular structure of the Co4-hmp. The structure of the Co4 molecule is shown in Fig. 10. The four Co and four oxygens from four hydroxy methyl pyridine ligands form a cubane like central Co 4 0 4 structure. The molecule has S4 symmetry. Our NRLMOL calculations confirmed the ferromagnetic ordering indicated by experiment. However, the magnetic anisotropy barrier was found to be 23 K. In the

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case of C04, the experimental determination of the D parameter is not conclusive. Our calculations showed that in this system, the anisotropy barrier has a large pressure gradient [30]. During the optimization cycle, the distance between the hydroxymethyl pyridine ligands were found to increase, which in turn led to a increase of the HOMO-LUMO gap and subsequent reduction of the anisotropy barrier. The barrier during the optimization was found to vary between 60 K and 23 K. While the calculations are carried out for a gas-phase molecule, the experiments are done on the crystalline phase. In the crystalline environment, the separation between the ligands are likely to be smaller than in the gas-phase and that may lead to a higher value of barrier. On the other hand, our calculations showed that when the geometry of the molecule is changed from the lowest-energy staggered geometry to a higher energy eclipsed geometry, the system has triaxial anisotropy and the anisotropy barrier increases to 95 K. Another high-energy isomer which lies between the staggered and the eclipsed structure, has an easy-plane anisotropy with a barrier of 50 K. Therefore, any experiment which can change the orientation of the ligands, can achieve a higher barrier. Within the LCAO framework used here, it is possible to isolate the contribution to the gamma matrix from each of the metal centers [30]. We found that each Co atom has the same local orientation of the local anisotropy axes: the hard-axis lies along the in-plane Co-0 bond where Co and O are attached to two parallel pyridine ligands and the local medium axis lies along the Co-N bond. Such an arrangement in the staggered version leads to orthogonal hard axes resulting in a global easy axis along the molecular axis. For the eclipsed conformer, the local hard axes are parallel resulting in a triaxial system. For the Co 4 molecular magnet synthesized by Yang et al. the comparison between theory and experiment may not be meaningful due to the inconclusive nature of the experimental data, the comparison with another Co based molecular magnet seems to be more promising. This Co based molecular magnet was synthesized by Murrie et al. [39] and contains six Co sites. This molecule also has a Co 4 0 4 central cubane and the other two Co atoms are surrounded by water molecules. The experiments by Murrie et al. have shown that the energy barrier of the Co6 lies between 25-29 K which is close to the theoretical prediction for the Co4. However, the ligands in the Co6 molecule are entirely different.

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9. Exchange Coupling in Molecular Magnets As noted in the introduction, the term Molecular Magnets is used as a classification of a reasonably broad range of molecules which exhibit at least two different idealized and physically different behaviors. The discussion of the previous section has concentrated on that class of molecules where magnetic anisotropy defines the low-energy excitations of the molecular systems and has been based on a density-functional framework. Tacit to this discussion is the proposition that the behavior of the system could be well described by a collection of Kohn-Sham orbitals even though a Slater determinant constructed from such orbitals would form a spincontaminated and possibly broken-symmetry Slater determinant. One must emphasize that one of the primary tenets of the density-functional theory is that the Kohn-Sham orbitals should not necessarily be viewed as single-electron spin-orbitals for the purposes of constructing a zeroth order many-electron wavefunction. From this perspective it is reasonable to evade any discussion about problems related to spin contamination and symmetry breaking. However, there are significant formal and pragmatic efforts within both computational chemistry and condensed matter physics to either artfully or rigorously blend the beneficial aspects of wavefunctionand density-based theories into a more useful tool for the prediction of molecular properties. Moreover, the original postulates of densityfunctional theory were based upon nondegenerate ground states which further conjures up questions regarding rigor when applications of a DFT-based quantum-mechanical method are applied to a multireference system such as the Mn 12 -Acetate molecule. Some of these questions can be addressed by many extensions of the density-functional theory and the reader is referred to the literature for such discussions [98-101]. There are in fact a variety of calculations that suggest densityfunctional-based Slater determinants provide a good starting point to the many electron wavefunction and numerous researchers have adopted broken-symmetry approaches within both DFT and UHF pictures to address the multireference characteristics of a many-spin problem. As exemplified by the Heitler-London picture of the dissociation of molecular hydrogen into two antiparallel localized spins on the well separated atoms, the prospects for variationally describing well separated moments in terms of broken-symmetry single-determinental pictures reaches back to the origins of quantum mechanics. The use of broken-symmetry approaches within quantum chemistry are too numerous to concisely

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review especially since the use of such methods is not limited to systems containing only transition metal centers [2]. For a discussion of the application of such methodologies to purely organic systems the reader is referred to recent work by Krylov [8] in applications to organic triradicals. In reference to systems containing metal centers we note that it has been used, within a density-functional framework, by Noodleman [4-6] to look at biologically interesting materials containing transition metal centers. Early on, Chen and coworkers have used broken symmetry approaches to look at antiferromagnetically ordered Fe and Ni clusters [7]. Within selfinteraction corrected density-functional theory, Pederson et al. [102] have used such approaches to study the dissociation of of Li2 into two well separated antiferromagnetically aligned spin-polarized Li atoms. Barojas et al. [103] have looked at antiferromagnetically ordered Rh clusters. Applications specific to molecular magnetism include work, within a UHF formulation, by Davidson on the M114 monomer and by Park et al. on the Mn4 dimer [95]. Baruah et al. have performed calculations on the exchange parameters of a Co4 based molecular magnet. Ruiz et al. have recently calculated exchange parameters for several molecular magnets [104]. Kortus et al. have calculated exchange parameters for the V| 5 molecular magnet [35] and Park et al. have performed calculations on the Mn12-Acetate molecular magnet. Of the systems discussed above, the V15 molecular magnet, referred to as a spin system in Ref. [35], is a quintessential example of a system where the role of spin-orbit and anisotropy is a tiny perturbation to the low energy spin excitations. The Mn 12 -Acetate molecular magnet lies at the opposite extreme. With some qualification (See Ref. [83]), the kinetic exchange interactions are large enough that spin-orbit and anisotropy play the dominant role in determining the behavior of the system. Within DFT, the Mn 4 monomer has exchange interactions that are large enough to force the monomer to behave as a molecular magnet. However, as discussed in the previous section the dimer exhibits a small exchange coupling which leads to an additional crossing of spin manifolds that appears at an energy on the same scale of resonant tunneling of magnetization. Our calculations on Co4 also suggest that this system has ferromagnetic exchange interactions that are large enough to exhibit RTM. It is still left to future calculation to clearly determine how many of the molecular magnets fit into the two possible idealizations.

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In an attempt to partially clarify the role of exchange interactions in molecular magnetism we discuss in detail the examples that are most clearly on opposite extremes. In Ref. [83], exchange parameters for the Mn 12-Acetate molecule have been calculated. There are three types of Mn ions. Each of the four S = 3/2 Mn ions that form the cubane unit are strongly coupled to one of the outer crown Mn (Jl) and moderately coupled to two of the other type of crown Mn (J2) In addition each crown Mn is weakly coupled to the three other cubane Mn centers (J3). The S = 2 crown ions form a closed network with each crown ion coupled weakly to its two nearest neighbors. Park et al. have performed densityfunctional calculations on a total of twelve low lying spin configurations that have energies in the range of 0.0 to 0.196 eV. They have then determined the four exchange parameters by fitting the resulting total energies to the Ising Hamiltonian. The resulting Hamiltonian leads to a matrix size of dimension 108 which has been diagonalized using the Lanzos method [105] and a parallel implementation due to Hellberg. The exchange parameters c o n f i r m that the lowest lying Ising configuration has the moments on the crown Mn centers antiparallel to those on the Mn cubane. The Lanzos diagonalization predicts three low lying S = 9 manifolds. With respect to the ground state S = 10 manifold, these states are found at 40.5 K, 47.5 K and 80.5 K respectively. The lowest lying S = 10 excited states is at 430 K which is clearly a large energy in comparison to the anisotropy barrier. These two S = 10 configurations would be coupled by the spinorbit operator (not present in the isotropic Heisenberg Hamiltonian). However the energy denominator of 430 K is rather large in comparison to the anisotropy of the ground-state manifold. As discussed earlier the lowest S = 1 0 t o S = 9 spin excitation is in reasonable agreement with experimental estimates due to Hill et al. At first sight, the fact that the exchange-based S = 10 to S = 9 splitting (40.5 K) is actually smaller than the anisotropic-based splitting of the S = 10 levels raises questions about the simplified single-configuration picture discussed throughout. However, as shown in Ref. [83] the low-energy M s = 9 configurations correspond to flipping the spins of three electrons on a cubane Mn and four electrons on one of the crown Mn. This is essentially a seven-electron excitation which is not directly coupled to the ground state by a two-electron operator such as the spin-orbit interaction. Therefore even if this configuration were to appear appreciably in a multiconfiguration

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treatment of the molecule, there would be no off-diagonal contributions to the anisotropy (the diagonal contributions of such states are discussed below). We note that such an argument would break down for anisotropic systems composed of spin 1/2 centers or for systems with spin 1 centers and very small interionic exchange contributions. There are electronic interactions that would be directly coupled by the spinorbit operator however these interactions are already assumed to be unimportant when one uses a Heisenberg treatment. For example, promotion of either one or two electrons from occupied to unoccupied states produces such configurations. The electronic structure calculations suggest that these excitations are on the order of 5000 K so they should not be very important from the standpoint of anisotropy. The arguments above may alleviate questions about the low lying S = 9 states in the Mn12 molecule directly perturbing the anisotropy of the S = 10 ground state manifold. Harmon et al. [81] have also performed many-spin analyses on this molecule using an 8-spin model of Mni2-acetate and exchange parameters that have been extracted from experimental measurements. The DM interaction is included in their work. They also find low-lying S = 9 manifolds and put forth the reasonable suggestion that virtual excitations between the S = 10 and S = 9 manifolds may impact the tunneling rates in this system. We have not yet addressed the problem of spin contamination and how this would affect predicted anisotropics if the Kohn-Sham DFT solution is viewed as a starting point for a multiconfigurational treatment. Park et al. have performed an exact diagonalization of the Ms = 10 manifold and confirmed that the ground state of this system corresponds to an eigenstate of spin with S = 10. Further the dominant term in the expansion (817176 determinants) is indeed the classical ferrimagnetic term and appears with an amplitude of 0.6. There are only 12 other determinants that contribute significantly to the many-spin wavefunction. These terms are constructed by tilting the spin projection on a cubane site (to S = -1/2) and the spin projection on a crown site to S = +1). The details of this work are still in progress and will be reported at a later date. Before continuing we make one more comment about the work of Ref. [83]. For each of the 12 low lying spin ordered determinants studied, the magnetic anisotropy of this state has been recalculated. All of these low lying configurations have a magnetic anisotropy energy in the range of 54.1-55.4 K. These results highlight that localized-ion nature of MAE and shows that changes in

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relative spin ordering do not significantly influence the magnetic anisotropy. It also shows that it is not unreasonable to expect that the sum of the diagonal contributions to MAE could be rather stable. Another example where many-spin excitations have been looked at in transition-metal assembled molecules is in the Vi 5 molecule. This molecule consists of a collection of spin 1/2 V"14 centers that are fivefold coordinated by oxygen atoms. The calculations on this molecule showed that the magnetic anisotropy was indeed insignificant and suggested the only observable anisotropic affect might be in a broadening in the Landau-Zener tunneling experiments. The exchange parameters for this molecule were also calculated within DFT and the resulting spin Hamiltonian of dimension 2 15 was diagonalized to obtain the spin excitation function. The results were in qualitative accord with experiment but overestimated the low-energy excitations by a factor of approximately 2.4.

10. Summary To summarize, we have reviewed recent density-functional-based investigations on molecular magnets that have been performed by our group during the last five years. While the ideal single-spin behavior seems to be well represented by density functional theory a significant amount of additional computational and theoretical work is required to have a predictive and accurate method for understanding the finer details of these systems.

Acknowledgments This work was supported in part by ONR, the DoD HPCMO CHSSI program, NSF and the Keck Foundation. We gratefully acknowledge the collaborative arrangements with Georgetown, Howard and SUNY Stoney Brook Universities. The work here has been influenced by many collaborators. We thank D.V. Porezag, J. Kortus, S.Hellberg, N. Bernstein, S.N. Khanna, and S. Richardson for direct and or indirect contributions to the results reviewed here.

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24.

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25. J. Kortus, M.R. Pederson, C.S. Hellberg and S.N. Khanna, Eur. Phys. Jour. D 16, 177 (2001); M.R. Pederson J. Kortus, and S.N. Khanna, J. App. Phys. 91, 7149(2002). 26. R. Caciuo, G. Amoretti, A. Murani, R. Sessoli, A. Caneschi and D. Gatteschi, Phys. Rev.Lett. 81, 4744 (1998). 27. K. M. Mertes, Y. Suzuki, M. P. Sarachik, Y. Paltiel, H. Shtrikman, E. Zeldov, E. Rumberger, D. N. Hendrickson, and G. Christou, Phys. Rev. Lett. 87, 227205 (2001). 28. L. Barra, D. Gatteschi, and R. Sessoli, Phys. Rev. B 56, 8192 (1997). 29. Mukhin, B. Gorshunov, M. Dressel, C. Sangregorio, and D. Gatteschi, Phys. Rev. B. 63, 214411(2001). 30. J. Kortus, T. Baruah, N. Bernstein, and M. R. Pederson, Phys. Rev. B. 66, 092403 (2002). 31. L. Barra, A. Caneschi, D. Gatteschi, D. P. Goldberg, and R. Sessoli, J. Solid State Chemistry 145, 484 (1999). 32. T. Baruah and M. R. Pederson, Chem. Phys. Lett., 360, 144 (2002). 33. E. C. Yang, D. N. Hendrickson, W. Wernsdorfer, M. Nakano, R. Sommer, A. L. Rheingold, M. Ledezma-Gairaud, cond-mat/0109219. 34. D. C. Bradley, R. G. Copperthwaite, S. A. Cotton, K. D. Sales, J. F. Gibson, J. Chem. Soc, Dalton Trans. 2 (1973) 191. 35. J. Kortus, C. S. Hellberg and M. R. Pederson, Phys. Rev. Lett. 86, 3400 (2001). 36. J. Kortus, M. R. Pederson, C. S. Hellberg, and S. N. Khanna, Euro. Phys. J. D, 16, 177 (2001). 37. M. R. Pederson, J. Kortus, and S. N. Khanna, J. Appl. Phys., 91, 7149 (2002). 38. M. Murrie, S. J. Teat, H. Stoeckli-Evans, and H. U. Gudel, Angew. Chem. Int. Ed. 42, 4633 (2003). 39. J. Kortus, M. R. Pederson, Tunna Baruah, N. Bernstein, and C. S. Hellberg, Polyhedron, 22, 1871 (2003). 40. K. Park, T. Baruah, N. Bernstein, and M. R. Pederson, Phys. Rev. B 69, 144426 (2004). 41. K. Park, T. Baruah, and M. R. Pederson Phys. Rev. B (in press); preprint condmat/0405059. 42. P. Hohenberg and W. Kohn, Phys. Rev. 136, (1964) B864; W. Kohn and L.J. Sham, Phys. Rev. 140, (1965) Al 133. 43. J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). 44. J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. Singh and C. Fiolhais, Phys. Rev. B 45, 6671 (1992). 45. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996). 46. Becke, Phys. Rev. A 38, 3098 (1988);A. D. Becke, J. Chem. Phys. 98 (1993) 13 47. Lee, W. Yang, and R. G. Parr Phys. Rev. B 37, 785-789 (1988). 48. M. R. Pederson and C. C. Lin, Phys. Rev. B 35, 2273 (1987).

118 M. R. Pederson, K. Park and T. Baruah 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64.

65. 66.

67. 68. 69. 70.

71. 72. 73. 74.

M.R. Pederson and K.A. Jackson, Phys. Rev. B 41 7453 (1990). K.A. Jackson and M.R. Pederson, Phys. Rev. B 42, 3276 (1990). M.R. Pederson and K.A. Jackson, Phys. Rev. B 43, 7312 (1991). A.A. Quong, M.R. Pederson and J.L. Feldman, Sol. Stat. Comm. 87, 535 (1993) M.R. Pederson, J.Q. Broughton and B.M.Klein, Phys. Rev. B 38, 3825 (1988). D.V. Porezag and M.R. Pederson, Phys. Rev. B 54, 7830 (1996). Briley, M. R. Pederson, K. A. Jackson, D. C. Patton, and D. V. Porezag, Phys. Rev. B 58, 1786 (1998). D. Porezag and M. R. Pederson, Phys. Rev. A 60, 9566, (1999). M. R. Pederson and S. N. Khanna, Phys. Rev. B 59 R691 (1999). M. R. Pederson and S. N. Khanna, Chem. Phys. Lett. 303 373 (1999). M. R. Pederson and S. N. Khanna, Phys. Rev. B. 60 9572 (1999). H.J.F. Jansen, Phys. Rev. B 38, 8022 (1988). For example see: S. Gasiorowicz, Quantum Physics, John Wiley and Sons, (New York, 1974), page 272. Kittel Introduction to Solid State Physics, John Wiley and Sons, (New York, 1963) See page 181. Y. Pontillon, A. Caneschi, D. Gatteschi, R. Sessoli, E. Ressouche, J. Schweizer and E. Lelievre-Bema, J. Am. Chem. Soc. 121, (1999) 5342 T. Baruah, J. Kortus, M.R. Pederson, R. Wesolowski, J.T. Haraldsen, J.L. Musfeldt, J.M. North, D. Zipse and N.S. Dalai, Phys. Rev. B 70, 214610 (2004). W. Y. Ching, Y. N. Xu, and K. W. Wong, Phys. Rev. B 40, 7684 (1989). E. M. Chudnovsky and J. Tejada, Macroscopic Quantum Tunneling of the Magnetic Moment, Cambridge Studies in Magnetism Vol. 4 (Cambridge University Press, Cambridge, 1998). Garg, Europhys. Lett. 22, 205 (1993). M. Hennion, L. Pardi, I. Mirebeau, E. Suard, R. Sessoli, and A. Caneschi, Phys. Rev. B 56 8819 (1997). N. V. Prokofevand P. C. E. Stamp, Phys. Rev. Lett. 80, 5794 (1998). S. Hill, J. A. A. J. Perenboom, N. S. Dalai, T. Hathaway, T. Stalcup, and J. S. Brooks, Phys. Rev. Lett. 80, 2453 (1998); J. A. A. J. Perenboom, J. S. Brooks, S. Hill, T. Hathaway, and N. S. Dalai, Phys. Rev. B 58, 330 (1998) A. A. Mukhm, V. D. Travkin, A. K. Zvezdin, S. P. Lebedev, A. Caneschi, and D. Gatteschi, Europhys. Lett. 44, 778 (1998). M. Gomes, M. A. Novak, R. Sessoli, A. Caneschi, and D. Gatteschi, Phys. Rev. B 57, 5021(1998). M. I. Katsnelson, V. V. Dobrovitski, and B. N. Harmon, Phys. Rev. B 59, 6919 (1999). Parks, J. Loomis, E. Rumberger, D. N. Hendrickson, G. Christou, Phys. Rev. B 64, 184426(2001).

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Based Investigation

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75. K. Park, M. A. Novotny, N. S. Dalai, S. Hill, and P. A. Rikvold, Phys. Rev. B 65, 014426 (2002); 66, 144409 (2002); S. Hill, S. Maccagnano, K. Park, R. M. Achey, J. M. North, and N. S. Dalai, ibid. 65, 224410 (2002). 76. Mirebeau, M. Hennion, H. Casalta, H. Antres, H. U. Gudel, A. V. Irodova, and A. Caneschi, Phys. Rev. Lett. 83, 628 (1999). 77. T. Goto, T. Koshiba, H.-L. Tsai, A. R. Schake, S. Wang, J. B. Vincent, K. Folting, D. Gatteschi, G. Christou, and D. N. Hendrickson, J. Am. Chem. Soc. 115, 1804(1993). 78. R. Sessolli, H-L. Tsai, A.R. Schake, S. Wang, J.B. Vincent, K. Folting, D. Gatteschi, G. Christou, and D.N. Hendrickson, J. Am. Chem. Soc. 115, 1804 (1993). 79. Raghu, I. Rudra, D. Sen, and S. Ramasesha, Phys. Rev. B 64, 064419 (2001). 80. N. Regnault, Th. Jolicoeur, R. Sessoli, D. Gatteschi, and M. Verdaguer, Phys. Rev. B 66, 054409 (2002). 81. W. Boukhvalov, A. I. Lichtenstein, V. V. Dobrovitski, M. I. Katsnelson, B. N. Harmon, V. V. Mazurenko, V. I. Anisimov, Phys. Rev. B 65, 184435 (2002). 82. G. Chaboussant, A. Sieber, S. Ochsenbein, H.-U. Gudel, and M. Murrie, preprint condmat/0404194. 83. K. Park, M. R. Pederson, and C. S. Hellberg, Phys. Rev. B 69, 014416 (2004). 84. K. Petukhov, S. Hill, N. E. Chakov, K. A. Abboud and G. Christou, Phys. Rev. B 054426 (2004). 85. M. Chudnovsky and D. A. Garanin, Phys. Rev. Lett. 87, 187203 (2001); Phys. Rev. B 65, 094423 (2002). 86. Cornia, R. Sessoli, L. Sorace, D. Gatteschi, A. L. Barra, and C. Daiguebonne, Phys. Rev. Lett. 89, 257201 (2002). 87. M. R. Pederson, N. Bernstein, and J. Kortus, Phys. Rev. Lett. 89, 097202 (2002). 88. del Barco, A. D. Kent, E. M. Rumberger, D. N. Hendrickson, and G. Christou, Phys. Rev. Lett. 91, 047203 (2003). 89. E. del Barco, A. D. Kent, N. E. Chakov, L. N. Zakharov, A. L. Rheingold, D. N. Hendrickson, and G. Christou, Phys. Rev. B 69 020411(R) (2004). 90. S. Hill, R. S. Edwards, S. I. Jones, N. S. Dalai, and J. M. North, Phys. Rev. Lett. 90, 217204 (2003). 91. R. Schake, H.-L. Tsai, R. J. Webb, K. Folting, G. Christou, and D. N. Hendrickson, Inorg. Chem. 33, 6020 (1994). 92. J. Eppley, H.-L. Tsai, N. de Vries, K. Folting, G. Christou, and D. N. Hendrickson, J. Am. Chem. Soc. 117, 301 (1995). 93. M. Soler, W. Wernsdorfer, K. A. Abboud, J. C. Human, E. R. Davidson, D. N. Hendrickson, and G. Christou, J. Am. Chem. Soc. 125, 3576 (2003). 94. W. Wernsdorfer, N. Aliaga-Alcade, D. N. Hendrickson, and G. Christou, Nature (London) 416, 406 (2002).

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95. K. Park, M. R. Pederson, S. L. Richardson, N. Aliaga-Alcalde, and G. Christou, Phys. Rev. B 68, 020405(R) (2003). 96. K. Park, M. R. Pederson, and N. Bernstein, J. Phys. Chem. Sol. 65, 805 (2004). 97. J. M. North, N. S. Dalai, D. Foguet-Albiol, A. Vinslava, and G. Christou Phys. Rev. B 69, 174419 (2004). 98. M.E. Casida, J. Chem. Phys. 122, 054111 (2005); J. Guan, M. Casida and D.R. Salahub, J. Mol. Struct. -Theochem. 527 229 (2000). 99. Wasserman, N.T. Maitra, K. Burke, J. Chem. Phys. 122, 144103 (2005); R. Gaudoin and K. Burke, Phys. Rev. Lett. 94, 029901 (2005), Ibid. 93, 137001 (2004). 100. J.F. Dobson, Phys. Ref. Lett. 73 2244 (1994). 101. C.A. Ullrich and K. Burke, J. Chem. Phys. 121 28 (2004). 102. M.R. Pederson, R.A. Heaton, and C.C. Lin, J. Chem. Phys. 82, 2688 (1985). 103. C.H. Chien, E. Blaisten-Barojas and M.R. Pederson, Phys. Rev. A 58 2196 (1998). 104. E. Ruiz, G. Rajaraman, S. Alvarez, B. Gillon, J. Stride, R. Clerac, J. Larionova, S. Decurtins, Ang. Chem. - Int. Ed. 44 2711 (2005). 105. I. Cullum and R. Willoughby, Lanczos algorithm for large symmetric eigenvalue computations, In Theory, Vol 1 (Boston, MA, 1985).

Chapter 3: Vibrational Spectra by Electron Impact: Theoretical Models for Intensities

Petr Carsky and Roman Curik J. Heyrovsky Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolejskova 3, 18223 Prague 8, Czech Republic

Abstract Besides its valuable applications in surface science and to problems of chemisorption and catalysis, vibrational electron energy loss spectroscopy (EELS) is becoming a useful tool in giving information on excited states of molecules and on the electronic structure of the respective negative ions and their unimolecular decomposition. Need for a deeper chemical insight has been also recognized for a variety of electron-molecule collisional problems that have been traditionally a domain of physics. Theoretical basis for EELS is less developed than for other spectroscopies used in chemistry such as infrared, UV7VIS, or NMR. It is a feeling of the authors that the achievements of the electronic structure theory of molecules can be more exploited in the electron scattering theory. The aim of this paper is to present our attempt to develop a simple but rigorously ab initio computational method that can be applied to vibrationally inelastic electron scattering by polyatomic molecules. Some general problems of ab initio calculations on vibrational inelastic differential cross sections are discussed.

121

122

P. Cdrsky and R. Curik

1. Introduction Vibrational excitation of molecules by electron impact belongs to processes that have been of primary importance for the understanding of technological and environmental problems. This field has been traditionally the domain of physicists, both theorists and experimentalists, and the systems studied were mostly collisions of electrons with diatomics. The state of the art of the theory in this field in 1995 was nicely described by Rescigno and collaborators1 by their opinion that "compared to the large number of high quality ab initio calculations that have been carried out on bound molecular electronic states, ab initio calculations of electron-molecule collisions have been rather rare", and by Winstead and McKoy2 who reported that "there are still many uncertainties with regard to convergence, assessing the results of calculations is complicated by large uncertainties in the experimental data, where data are available at all. This unsatisfactory state of affairs simply reflects the status of the field, both experimentally and theoretically. Though progress in the field has accelerated in recent years, calculating inelastic electron-molecule cross-section is still far of being routine". The progress was slow and still in 1998 Gianturco and Lucchese3 stated that "the difficulties arising from treating the scattering as accurately as possible are still formidable and have therefore been approached with some success only in the last few years". The situation did not change considerably in the last six years. In his recent review Itikawa4 characterizes it as that "a lot of problems are still to be solved" and "compared to the large number of theoretical studies of vibrational excitation of diatomic molecules, the number of theoretical works for polyatomic molecules is very limited". This reflects a growing interest in the theory and experiment on electron scattering by polyatomic molecules. Processes in the atmosphere induced by electron collision and use of plasma for surface etching are closely related to chemistry and it has been realized that a deeper chemical insight into these problems would be profitable. Already in 1994 Allan reported in his review5 that electron impact techniques are useful for chemistry in giving information on excited states of molecules, and the electronic structure and unimolecular decomposition of the respective negative ions. Recently it has been recognized that the DNA damage is not caused directly by radiation but indirectly by secondary low-energy electrons released by radiation6'7. It is therefore a challenge for theory to predict the preferential sites of DNA for electron attack8. In our opinion the progress in the theory of electron scattering by polyatomic molecules could be accelerated by a better linking

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of the achievements of the electronic structure theory with the electron scattering theory. Unfortunately, the two theories have been developed in parallel and they have been using different techniques, computational approaches and even terminology, which makes their combination difficult. Our first task is, therefore, to "translate" in the following section the basic concepts and terms used in the scattering theory to the language of the electronic structure theory. In the next section we outline the essence of the Discrete Momentum Representation (DMR) of the Lippmann-Schwinger equation which is the method developed in our laboratory9'10. It is a rigorous ab initio method but computationally simple enough that it can be applied to moderately large polyatomic molecules. Section 4 presents an illustrative application. The purpose of this paper is to introduce the theory of scattering processes to chemical community and to show that chemists could contribute to the development in this field. Therefore, we did not attempt for a literature search on experimental and theoretical studies on vibrational excitation of polyatomic molecules by electron collisions. An interested reader can find it in special publications (see for example the review articles4'11 and databases1213). Also, no attempt was made for a complete coverage of the theoretical ab initio approaches. Instead we refer the interested reader to a few papers quoted below.

2. Static-exchange approximation In this section we present the simplest conceivable ab initio approach to vibrationally inelastic electron scattering by polyatomic molecules. For simplicity, start with elastic scattering. Assume that the target is a closed shell molecule which is described by a single Slater determinant l(p]a cp^ (p2a tp2p... (Pna 9np I where cp's are occupied molecular orbitals. In the time independent approach solving the scattering problem is equivalent to an eigenfunction problem of the total Hamiltonian which describes the molecule interacting with the scattered electron. In contrast to the bound-state case, scattering solutions are from the continuum spectrum of the total Hamiltonian. One can picture it as scattered electrons occupying virtual orbitals, which is not bound anymore and its wave function does not disappear at large distances. For simplicity, we adopt a seemingly crude "static-exchange" approximation, that is, we do not allow any change of molecular orbitals as the scattering electron is approaching, in other words,

124

P. Cdrsky and R. Curik

we do not allow any polarization of molecular orbitals under the effect of the scattering electron. It can be shown14 that static-exchange approximation is a first order approximation in perturbative expansion of the total energy of system in terms of electron-molecule interaction strength. This single-particle non-local and energy dependent interaction is called, at the first or higher order, optical potential. The second order leads to expressions similar to MP2 corrections of the Hartree-Fock energy. Static-exchange approximation limits the application possibilities, but as shown below, it gives useful results if the energy of the scattering electron is higher than about 8 eV and if resonances do not play any significant role in the scattering process. Hence, the total system can be described by a single Slater determinant ^

= \(fha(P\j}92a(p2P

•••9na(P„pF\

(1)

where F is the continuum wave function for the scattered electron and it may be viewed as a singly occupied molecular pseudoorbital. Schrodinger equation HXV = EXV can be converted by simple manipulations Hartree-Fock equation

15

(2) to a one-electron analog of the

_ I V J + y F(r) = y F ( r ) ,

(3)

where the V operator by its form V = - Y — ^ — + 2J-K

(4)

A\T-RA\

represents the electron-molecule interaction potential in the static-exchange approximation. Having available the wave function, proceed to the quantity which determines the probability of the scattering process. In the photon spectroscopy the critical quantity is the transition moment and the respective operator is the dipole moment operator. In the scattering theory, the analog of the transition moment is the following integral ( 1 ^

=

« -{te)re

Te

(5)

which is called the amplitude. The plane-wave functions represent F(r) far from the target molecule before the collision (incoming electron) and far

Vibrational Spectra by Electron Impact

125

from the target molecule after the collision (outgoing electron). The probability for the elastic electron scattering kin —»kout is given as D

ij=

(6)

T^l/f,!

and in the scattering theory it is referred to as differential cross section. The T operator is given by the fundamental equation of the scattering theory which is called Lippmann-Schwinger equation T = U + UGQT

(7)

where U is defined for convenience as U - 2 V and G is the Green's function for the free system:

This is an analog of propagators known in perturbation expansions in many-body theories such as the M0ller-Plesset theory. Hamiltonian H0 represents asymptotic Hamiltonian, e.g. without the interaction of the molecule with the scattering electron.

3. Discrete Momentum Representation (DMR) of the Lippmann-Schwinger equation 3.1 Lippmann-Schwinger equation In the previous section we noticed that all the scattering information can be obtained from the transition operator T which satisfies Lippmann-Schwinger equation (7). In order to obtain cross sections for vibrationally inelastic processes we project the equation (7) onto electronic space spanned by plane waves and a two-dimensional space for nuclear dynamics defined by a vibrational ground state xo and its first excited state Xi- The resulting equation follows:

Ui*«IT|Z0*,,) - {ZA.IU|Ztfiu.)+L\dk'

2

kl-2E-k

+ie (9)

126

P. Cdrsky and R. Curik

where E0 = 0 and Ej are the energies of the vibrational ground state and first excited state, respectively. Vectors kin and koat represent plane waves for the incoming and outgoing electrons. Numerical discretization of the three-dimensional integral on the r.h.s. of eq. (9) (p and i indices run through abscissas of the radial and angular quadratures, respectively) Nrad

Nang

jd* jdQk g(k) = J > „ 2 > , 8(kpi) p

(10)

•

leads to a set of two coupled matrix equations: (T

^00

T J

01

\

T T V io 1n J

u

=

00

^10

u

01

^11 J

+

^00 ^10

u

01

^11 J


-*oo -'oi

T

T

with the interaction matrix elements U:

V

(12)

Kl- =Uok,-|^|^>One of the computational profits of the chosen basis set of plane waves is that matrix of the free Green function (8) is diagonal. Second advantage is that matrix elements for the corresponding T operator in a given basis set, aside from a constant factor, directly give scattering amplitudes/as can be seen from the equation (5). The technique of equations (9) and (11) coupled over vibrational channels is in the scattering theory often called (vibrational) close-coupling. However, there is a price to be paid for both profits. It is the evaluation of matrix elements for the interaction potential in the basis of plane waves for electronic motion and harmonic oscillator functions for the nuclear degrees of freedom. 3.2 Harmonic approximation and the interaction potential In a manner similar to that used in vibrational photon spectroscopy (IR and Raman) the interaction potential U is expanded along the normal mode coordinate £ as

U(r,Z) = U(r,0) + — 3£

• b + higher-order terms .

(13)

Vibrational Spectra by Electron Impact

127

Now the integration over the normal mode coordinate can be carried out analytically and the interaction matrix elements are easily simplified to the following form * „ ( £ ) * , \U{Ra)\x^)K)

= (*,

\U(Rea)\k0

(14)

and dRr,

V2 p,n

H

(15)

(k^UiRpp,) R:

*RPn

where Ra are the position vectors of nuclei and Rea stands for the equilibrium nuclear geometry of the molecule. So far no approximation for a form of the optical potential U has been made. At this point we assume static-exchange approximation defined by the eq. (4). First two terms of eq. (4) describe electrostatic interaction consisting of nuclear attraction and static repulsion with the bound electrons. Corresponding matrix elements for the static part Us come out as a Fourier transform of the charge density:

1 (*l|tf,<*a)|*o)

2

jdr p{r)t

iK.r

2

7i K

(16) n K

B

flV

and the exchange part (17) where the repulsion integrals with mixed plane-wave and gaussian functions are defined as follows (*„*, l / / v ) s — i - 5 - fdr.dr, e*°« e~^

^ —

gfl(r2,Ra)g;(r2,Ra)

( f c 0 v l / / f c 1 ) ^ - i - ? J dJ r 1 d r 2 e*°" gv{rx,Ra) ^ — gMM(r2,Ra) (IjiY K ~r2\

e ^

(18)

Here PMV denotes density-matrix elements, K = k0 - kj is the momentum transfer vector and Za is the nuclear charge of nucleous a.

128

P. Cdrsky and R. Curik

From eq. (15) it is clear that also the derivatives with respect to coordinates of nuclei of above expressions are required. The technique of evaluation of the hybrid coulombic and exchange integrals (18) and their derivatives can be found in the literature16'17'18 . Matrix of the normal modes dRfi„ /d(f and gradients of the density matrix P^ are calculated by a standard quantum chemistry software for the evaluation of vibrational frequencies of molecules. 3.3 Averaging over molecular orientations All the scattering amplitudes and cross sections mentioned in the previous paragraphs have been derived for a fixed orientation of the molecule. This simple assumption is hidden in equations (14) and (15) where the vibrational degrees of freedom are integrated out but rotational degrees are kept as a fixed parameter throughout the equations. However in most processes of practical relevance such as electronmolecule collisions in industrial plasmas and upper atmosphere, orientations of the molecules seem to be not fixed. On another hand typical interaction times for 1-30 eV of collision energy is 10"14-10"15 s. Timescale for the rotations of the polyatomic molecules at room temperature is 10~12 s and longer. This comparison allows us to assume that scattered electron responds adiabatically to the rotations of the molecules and validates the fixed-nuclei approximation19'20 implicitly assumed in equations (14) and (15). Nevertheless orientation of the molecule with respect to the incoming electron is random and therefore cross sections must be averaged over all the orientations of the molecule. This is carried out by the following technique. Inelastic differential cross sections are obtained from (11) as D, - ^ O f i * , - < - * „ * ; ) = 4^ 4 ^|(A:,.|r i ( ) |A: ; .)| 2 .

(19)

Averaging over molecular orientation can be expressed by the angular integrals:

Dv = —^ [da d/3 Ay sin fi - £ {kt(a, fi, y) <- * {a, fi, y)), off

(20)

ui 2

where incoming vector kj{a,fi,y) and outgoing vectorfc,-(a,/?,y)are fixed in a frame which rotates by Euler angles {a,f},y) with respect to the molecular

Vibrational Spectra by Electron Impact

129

frame. In order to carry out integration in (20) we expand differential cross section Dtj in terms of spherical harmonics as D

ii = Z Z Z Z 4„./v O*,-) YVm.(kj) .

(21)

;=0 m=-//'=0 m'=-/'

This expansion allows us to carry out the integral on r.h.s of (20) analytically leading to (see for example ref. [21]) — D

1

L

« = T Z A P, (cos0)y) , ™

(22)

1=0

where

A = Z AmM •

(23)

m~—/

Pi are Legendre polynomials of the first kind and co,-, is the angle between vectors k{ and kj. In the DMR method the At coefficients are obtained as10 A = ^

Z Z

w W

< J Dij Pi ( c o s w
(24)

with the angular quadrature weight coefficients w, defined by eq. (10).

4. Electron energy loss spectrum of propane In this section we present in Figures 1 and 2 an example of comparison of measured and calculated vibrational Electron Energy Loss Spectra22 (EELS). Scattered electron intensities have been calculated for all 27 normal modes of the propane molecule. The electronic ground state of the molecule was described by the restricted Hartree-Fock wave function within Sadlej's basis set23. Cross sections obtained from DMR calculations were converted to a theoretical band spectrum by assuming Lorentzian shape for all bands with a half-width taken from the width of the observed elastic peak. The bands were centered at the positions of observed vibrational frequencies. As can be seen for collision energy of 10 eV the DMR spectrum compares well with the observed one. The agreement is less satisfactory in Figure 2, where the collision energy is 6 eV. The reason for this behavior will be explained below.

130

P. Cdrsky and R. Curik

0.06

E^meV) Fig. 1. Electron energy loss spectrum of propane (thin line) and the calculated spectrum (thick line) for the incident electron energy of 10 eV and the scattering angle of 100°. The bars at the bottom of the figure stand for the calculated differential cross sections.

0.06

400 EjCmeV) Fig. 2. Electron energy loss spectrum of propane (thin line) and the calculated spectrum (thick line) for the incident electron energy of 6 eV and the scattering angle of 70°. The bars at the bottom of the figure stand for the calculated differential cross sections.

Vibrational Spectra by Electron Impact

131

5. Assessment of the DMR method DMR represents a practical compromise between accuracy and reliability on the one hand and computational feasibility for larger polyatomic molecules on the other. As with any other method employing the static-exchange approximation, applicability of DMR is limited to electron energies higher than about 8 eV. Neglect of polarization effects also prevents to account for resonances. However, at energies around 10 eV, where resonances usually do not occur significantly, DMR performs well and the experimental EEL spectra are reproduced well, as it is documented for propane in the preceding section. The case for collision energy of 6 eV and the scattering angle of 70° is presented here as an example where the static-exchange approximation fails. At lower energies the response of the molecular orbitals to the effect of incoming electron becomes more important since the interaction time is getting longer. Moreover this effect becomes more significant at lower scattering angles, because weakly scattered electrons colliding with the large impact parameters are strongly influenced by polarization of the molecular electronic charge distribution. The lack of polarization effects can be overcome by application of the second-order corrections to the optical potential. They are energy dependent and resemble the M0ller-Plesset second-order energy correction24. However implementation of such ab initio corrections is computationally expensive and to our knowledge it was applied25 only once for the simplest molecule H2. Instead simpler and local form of polarization potential is widely used. It is based on Local Density Approximation of DFT26 and this model provides reliable results for molecules from first- and second-row atoms. Both strategies are considered for the extension of DMR.

6. General remarks on ab initio calculations of vibrationally inelastic electron scattering Obviously, theory of vibrationally inelastic electron scattering is open to all complications that have been encountered in the theory of elastic electron scattering1'2'21. In contrast to the electronic structure theory, we do not have in the theory of electron scattering any standard procedures as configuration interaction, M0ller-Plesset theory or coupled clusters, not speaking about the

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availability of widely used software such as Gaussian, Molpro, Molcas, or Aces programs, to note at least a few. Instead, each laboratory dealing with the theory of electron scattering is using its own software based typically on its own theoretical approach. Basic equations such as equation (3) for the static-exchange approximation have been derived long time ago but no standard procedures are available for expanding the total N + 1 electron wave function into a series of basis functions that would be analogs of configurations or clusters. No standard procedure is available for the construction of these basis set N + 1 electron functions from one-electron functions. Rigorous ab initio approaches require the evaluation of twoelectron repulsion integrals in a mixed Gaussian-plane wave basis set. Formulas for these integrals are known but so far their computation is less effective than evaluation of integrals in a pure Gaussian basis which precludes applications to a somewhat more extended polyatomic molecular targets. An alternative technique applied in R-matrix method27 makes use of very diffuse Gaussian functions to represent the continuum space. However this approach leads to the linear dependency problems for polyatomic molecules. Vibrationally inelastic electron scattering brings some additional theoretical and computational problems. As already noted in Section 2, the simplest conceivable approach must be "two-channel" by its nature which means that all matrices needed in calculations of elastic scattering are of a double size in a vibrationally inelastic treatment. Integration over the vibrational coordinate is usualy done numerically for a particular vibrational motion, which is feasible for diatomics and triatomics, but it becomes impractical for true polyatomic molecules. We believe that the use of the harmonic approximation is inevitable, as in the theory of photon vibrational spectroscopy. The results for the propane molecule presented in Section 4 indicate that harmonic approximation could be acceptable also in the theory of vibrational EELS. Computationally, the harmonic approximation leads to derivatives of integrals over one-electron functions with respect to nuclear coordinates. Again, in DMR calculations these integrals are in a mixed Gaussian-plane wave basis. We derived formulas for analytical evaluation of their derivatives, but their effective evaluation remains still a computational challenge. In the next part of the section we would like to address one of our problems encountered during the study of vibrational excitations of polyatomic molecules by electron impact. The problem is related to the long-

Vibrational Spectra by Electron Impact 133

range nature of the interaction during the excitations of IR active modes where differential cross section (DCS) peaks at forward scattering angles. Electrons contributing to this process are well described by First Born Approximation (FBA) which keeps only the first term (£/) on r.h.s. of eq. (7). As has been mentioned in the static-exchange approximation U consists of the static part Us and the exchange contribution Uex. Exchange part Uex vanishes outside the molecular electron density distribution and therefore does not contribute to long-range part of the potential. To this end we examine behavior of Us for small scattering angles. For small K = \K\ only first two terms in exponential expansion of the Fourier transform in eq. (16) can be kept as follows (*iH*o> = ~4^

ldrp(r)(l+iKr+...).

(25)

Thus for a neutral molecule from eq. (15) we obtain inelastic FBA scattering amplitude fB : K-%

(26)

K^J

where dfi/di; is the derivative of the dipole moment^ with respect to normal mode coordinate
d(TB . .

. .

*-dQ<*A-*A>

jfc, 2 koK

v

4- A -

dfi 2 —

(27)

^

We conclude that for small scattering angles differential cross section behaves as \IK and its value is proportional to the infrared intensity of the particular vibrational mode. Similar problems are encountered in a description of elastic or rotationally inelastic collisions of the electrons with molecules that have permanent dipole moment. However in this case K is never zero because k0 and k] have different norms due to an energy transfer to the vibrational excitation. It is obvious such irregular behavior of the DCS for small scattering angles leads to slowly converging expansion in eq. (21). This problem has been addressed by Itikawa28'29. In order to make convergence of expansion (21) faster Itikawa proposed Born-closure technique widely used for the rotational excitations. However Born-closure approach was primarily designed for the scattering amplitudes described in a basis of partial waves.

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P. Cdrsky and R. Curik

In our case we start already from differential cross sections Dy hence we developed an alternative procedure described below. In the first step we rewrite Dy as »,,

(28)

where

Af = ^Dy and

KS1 =

\k.-k-

(29)

From eq. (27) we can expect D , / behaves in the regular manner and does not peak for small angles any more. Now the averaging procedure (20) is applied. The important point here is that Ky is a constant in the averaging integral on r.h.s. of eq. (20). Hence by means of eq. (28) we extracted irregular behavior of Dy and the matrix elements Di}R converge in angular expansion (21) much faster than Dy, 0.06

0.04

U Q 0.02

30

60

90 120 scattering angle [deg]

150

180

Fig. 3. Comparison of the averaged differential cross sections for the vibrational excitations of IR active mode v4 of methane molecule. Dashed line represents the application of eq. (22) with angular expansion limit L = 23. The solid line is obtained with the procedure employing (29) with L = 8.

Vibrational Spectra by Electron Impact

135

In the formula for averaged differential cross section (22) the irregular term is extracted explicitly,

^ = i7rlA f i ^ C 0 S < y )'

(30)

since the A[ coefficients were obtained this time from regularized cross sections D/ and therefore they are expected to vanish for higher / values. The efficiency of this technique is presented in Fig. 3. As it is seen in Fig. 3 even high angular momenta with / = 23 fail to properly describe the DCS. In contrast, regularized cross section D,/ leads to the fast convergent expansions.

7. Conclusions Recent progress in the theory of vibrationally inelastic electron scattering extended its application possibilities to various problems that are of interest in chemistry. The calculations are not limited to diatomics and triatomics any longer and also the reliability of the calculations is improved by the feasibility of rigorously ab initio calculations on desktop PC's. If plausible approximations are introduced such as the static-exchange approximation and harmonic approximation, molecular targets composed of 10 atoms can be treated. A more efficient evaluation of two-electron integrals and their derivatives with respect to nuclear coordinates and the progress in the construction of computers will certainly open a way to treatments of even larger molecules in the near future. Figure 1 shows that the use of staticexchange and harmonic approximations is justifiable, although these approximations may seem to be crude at the first sight. Good agreement between theory and experiment is only due to the fact that the collision energy was sufficiently high (10 eV in Figure 1). At lower energies (6 eV in Figure 2), the movement of the scattering electron is slow enough to polarize molecular orbitals of the target and the use of the static-exchange approximation is not appropriate any more. Economic improvement of the optical potential for polarization is therefore highly topical. Since a long time attempts were done to use various empirical polarization potentials. The most successful appears to be that based on the density functional theory21. However, even at its present state of the art the theory of electron scattering

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is becoming a useful tool for predictions and better understanding of measured electron energy loss spectra.

Acknowledgments This paper summarizes a part of work done in the projects EPIC of the Research Training Network (No. HPRN-CT-2002-00179) and EIPAM (No. PESC7-20). The support of European Commission and European Science Foundation are gratefully acknowledged. We also acknowledge the support of the Academy of Sciences of the Czech Republic (grants No. A100400501 and No. 1ET400400413).

References 1. T. N. Rescigno, B. H. Lengsfield III and C. W. McCurdy (1995) in Modern Electronic Structure Theory, Part I, D. R. Yarkony (ed.), World Scientific, Singapore, p. 501 2. C. Winstead and V. McKoy (1995) in Modern Electronic Structure Theory, Part II, D. R. Yarkony (ed.), World Scientific, Singapore, p. 1375 3. F. A. Gianturco and R. R. Lucchese (1998) /. Chem. Phys. 108, 6144 4. Y. Itikawa (2004) J. Phys. B 37, R l . 5. M. Allan (1994) Chimia 48, 372. 6. B. Boudaiffa, P. Cloutier, D. Hunting, M.A. 5, L. Sanche (2000) Science 287, 1658 7. S. Ptasinska, S. Denifl, P. Scheier and T. D. Mark (2004) J. Chem. Phys. 120, 8505 and references therein 8. A. Grandi, F.A. Gianturco, N. Sanna (2004) Phys. Rev. Lett. 93, 048103 9. M. Polasek, M. Jufek, M. Ingr, P. Carsky and J. Horacek (2000) Phys. Rev. A 61, 032701 10. R. Curik and P. Carsky (2003) /. Phys. B. 36, 2165 11. L. G. Christophorou and J. K. Olthoff (2002) Appl. Surf. Sci. 192, 309. 12. Y.-K. Kim, K. K. Irikura, M. E. Rudd, M. A. Ali, P. M. Stone, J. S. Coursey, R. A. Dragoset, A. R. Kishore, K. J. Olsen, A. M. Sansonetti, G. G. Wiersma, D. S. Zucker, and M. A. Zucker (2004), Electron-Impact Ionization Cross Section for Ionization and Excitation Database (version 3.0). [Online] Available: http://physics.nist.gov/ionxsec, National Institute of Standards and Technology, Gaithersburg, MD.

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13. Y. Itikawa (2001) Bibliography on Electron Collisions with Molecules (preprint), National Institute for Fusion Science, Japan 14. B.A. Lippmann, M. H. Mittleman and K.M. Watson (1959) Phys. Rev. 116, 920 15. N. F. Lane (1980) Rev. Mod. Phys. 52, 29 16. P. Carsky and M. Polasek (1998) J. Comp. Phys. 143, 266 17. P. Carsky and T. Reschel (1998) Collect. Czech. Chem. Commun. 63, 1264 18. P. Carsky and M. Polasek (2000) J. Comp. Phys. 181, 1 19. D. M. Chase (1956) Phys. Rev. 104, 838 20. 1.1. Fabrikant (1976) Sov. Phys. JETP 44, 77 21. F. A. Gianturco, D. G. Thompson and A. Jain (1995) Computational Methods for Electron-Molecule Scattering, the Plenum, New York 22. D. B. Popovic, D. E. David, J. Michl, R. Curfk and P. Carsky (2004) J. Chem. Phys. 121, 10551 23. A. J. Sadlej (1991) Theor. Chim. Acta 79, 123 24. C. J. Joachain (1975) Quantum Collision Theory, North-Holland Physics Publishing, Chapter 20 25. A. Klonover and U. Kaldor (1978) J. Phys. B 11, 1623 26. N. T. Padial and D. W. Norcross (1984) Phys. Rev. A 24, 1742 27. J. Tennyson and L. A. Morgan (1999) Phil. Trans. R. Soc. Lond. 357, 1161 28. Y. Itikawa (1997) Int. Rev. Phys. Chem. 16, 155 29. Y. Itikawa (2000) Theor. Chem. Ace. 105, 123

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Chapter 4: Theoretical Description of the Kinetics of Gas-Phase Reactions Important in Atmospheric Chemistry

Jerzy T. Jodkowski Department of Physical Chemistry, Wroclaw Medical University, pi. Nankiera 1, 50-140 Wroclaw, Poland

Abstract Some aspects of gas-phase reactions are discussed in a broader scientific context. Special attention is paid to reactions of free atoms and radicals, which play an important role in the modeling of the complex processes occurring in the polluted atmosphere and in combustion chemistry. Hydrogen atom abstraction and recombination reactions is the most abundant category of radical reactions of particular interest from both the theoretical and practical points of view. The computational models of the reaction rate theory most frequently used in the theoretical analysis of gas-phase reaction kinetics are briefly discussed. Selected reactions of reactive components of the polluted atmosphere, such 0 2 , NOx, OH, NH2, alkyl radicals, and halogen atoms, are discussed in detail. Kinetic parameters of the reactions under investigation are shown in an analytical form convenient for kinetic modeling studies. The expressions presented allow a successful description of the kinetics of the reaction systems in a wide temperature range.

139

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J. T. Jodkowski

1. Introduction There are many different types of chemical reactions and a wide variety of theoretical models and experimental techniques has been developed to investigate them.1"3 A particularly great amount of effort has been devoted to investigations of the kinetics and mechanism of reactions in the gas phase. This class of reactions is of the highest importance from both the theoretical and experimental points of view. A great many gas-phase reactions proceed very quickly, and their mechanism is much simpler than that of analogous reactions in other phases. The gas phase can usually be treated as an ideal gas, which makes the introduction of further simplifications into the theoretical description of the reaction kinetics possible. Therefore, the most sophisticated kinetic models only describe the gas-phase reactions directly, because molecular interactions and the course of the reaction can be investigated separately from the influence of the reaction environment. The kinetic parameters of elementary gas-phase reactions can be applied to describe the kinetics of complex chemical reactions, which proceed via a multi-step reaction mechanism. This is of key importance for possible practical applications, including the kinetic modeling of complex reaction systems and optimization of technological processes. Many of the most important elementary reactions that occur in the gas phase are those of very reactive species, such as atoms and free radicals.4"6 Such reactions are often extremely fast compared with the reactions of stable molecules. Two groups of radical reactions are especially prevalent: recombination and atom abstraction reactions. In recombination (association) reactions, stable molecules are formed by the fusion of atoms or radicals. These reactions may be very fast. In the case of atoms or simple radicals, the high-pressure limiting rate constant may reach a very high value, close to that derived from the gas-kinetic frequency. Atoms and radicals may easily react with stable molecules. Atom abstraction reactions are of great interest, especially hydrogen abstraction reactions, in which an attacking atom or radical abstracts a hydrogen from a molecule, giving rise to a new radical and a new molecule. The radicals formed in these reactions may initiate a variety of possible subsequent reactions. These two groups of reactions, recombination and H-abstraction, play an important role in the kinetic description of many processes occurring in the polluted atmosphere and in combustion.4"8 Theoretical analysis based on quantum mechanical ab initio calculations of the potential energy surface of the reaction system provides the information necessary for a better understanding and explanation of experimental findings. The height of the energy barrier is clearly the major factor in determining the rate constant,

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

141

yielding the dominant contribution to the magnitude of the rate constant and its dependence on temperature. Therefore, a realistic description of the reaction energetics is an essential criterion for the choice of method and the level of calculations. Results of ab initio calculations enable us to locate and characterize the critical points (reactants, products, intermediate complexes, and transition states) of the potential energy surface. The calculated properties of the molecular structures taking part in the reaction mechanism can then be used as the starting point for computational methods of the reaction rate theory to evaluate the rate constants expressed in the form A x (T/300)n x exp(-E/T) convenient for kinetic modeling studies over a wide range of temperatures. A large part of H-abstraction reactions is thought to be elementary.4"7 Theoretical analysis is then limited to an ab initio search for the transition state of the potential energy surface. The molecular structure of the transition state corresponds to the saddle point of the potential energy surface and is located at the top of the energy barrier for the minimum energy path between the reactant and product states. The classical transition state theory still remains a major theoretical tool in describing the kinetics of H-abstraction reactions. The rate constant, kTST, obtained in terms of conventional transition-state theory1"3'9 (TST) is given by

where: QA, QB, and Q* are the total partition functions of the reactant molecules (A or B) and the transition state (=£), E0 is the height of the energy barrier for the reaction at 0 K, and a is the reaction path degeneration. The vibrational and rotational contributions to the partition function are usually derived by the classical harmonic-oscillator rigid-rotor approximation. A reaction involving hydrogen atom transfer is usually characterized by a significant tunneling effect, which is represented in the rate constant calculation by the tunneling correction factor Kw as k = KwkTST

(1-2)

The tunneling correction may distinctly improve the values of calculated rate constants for reactions with high energy barriers, especially at low temperatures.1'9 A reasonable correction of the rate constant may be obtained, even in a one-dimensional approximation, using the Eckart type of potential. In this approach, the asymmetric potential is characterized by the forward and backward barrier heights and the imaginary frequency of the transition state.1 A number of different kinds of tunneling corrections have been evaluated by

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J. T. Jodkowski

Garrett and Truhlar,11 who found that the simplest, that due to Wigner,12 was often also the most accurate. A more general way to include the tunneling effect is to use the generalized formalism of the transition state theory called the variational transition state theory (VTST) or canonical variational transition state theory (CVT).9'13 The conventional transition state is located at the saddle point of the potential energy surface. In contrast, in the VTST every point on the reaction coordinate is identified with a generalized transition state (GTS). The molecular structure of the GTS is described parametrically by the value of s, corresponding to the position of the GTS on the reaction coordinate. The reaction coordinate s is defined as the signed distance along the minimum energy path in the isoinertial coordinate system from the saddle point (corresponding to s = 0) to the reactants or products. The reaction coordinate has a negative value on the reactant side (s = -°° corresponding to the reactant state) and positive in the product formation direction (s = +°° for the product state). In this way, the conventional transition state corresponds to GTS (s = 0) and, consequently, the rate constant kTsT = kGTST (s = 0). The rate constant, calculated in terms of the canonical transition state theory, is obtained by variational minimization of the generalized transition state rate constant with respect to the value s of the reaction coordinate k CVT (T) = mink GTST (T,s)

(1.3)

s

The results of the CVT rate constant calculations provide information on details of the potential energy surface useful for incorporation of the tunneling effect. The tunneling correction factor can be calculated at different levels of theory14 by using: the zero-curvature tunneling (ZCT) approximation, the centrifugal-dominant small-curvature tunneling (SCT) approximation, and the large-curvature tunneling (LCT) approximation, which includes tunneling into vibrationally excited states. In the case of a unimolecular reaction, the activated complex is a single molecule which has gained the necessary energy by collision with other molecules.7'15'16 The generally accepted mechanism of the thermal dissociation of a molecule AB AB + M i ^ AB* + M AB*H> A + B

(1,-1) (2)

consists of three elementary processes: excitation of the reactant molecule AB by collision with an energy transfer agent M, collisional deactivation of the vibrationally excited molecule AB by M, and the decomposition of molecules AB having an internal energy greater than the critical energy. The identity of

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

143

the collider M is immaterial; it may be AB itself, one of the products, or some foreign gas. Its role as a collisional energizer and deenergizer is very important, and the rate of reaction depends on the nature and concentration of the bath-gas molecules. Reverse processes, i.e. recombination (association) reactions of atoms and radicals with each other or with stable molecules, present a similar dependence of the reaction kinetics on pressure and temperature. The scheme of an recombination (association) reaction also comprises three elementary steps A + B t > AB*

(-2,2)

(-D

AB* + M -> AB + M

because a third body, M, is needed to stabilize the energized adduct molecule, AB . The kinetics of the recombination reaction is then described by the same rate constants of the elementary processes as in the dissociation reaction. The overall rate constants for dissociation, k ^ , and recombination, krec, reactions show the same pressure dependence, because their values at a given bath-gas pressure are related to each other by the equilibrium constant, which of course is pressure independent. Fig. 1 shows a typical fall-off curve, which describes

10' k

0

/ /

k„/2

c 3

I 101

10l

10-2 [M]C10-1

10°

[M] (bar)

Fig. 1. The fall-off curve for a thermal dissociation/recombination reaction.

144

J. T. Jodkowski

the pressure dependence of the overall rate constant for a dissociation/ recombination reaction given by the Lindemann-Hinshelwood equation ,

,

k0/kM l + k0/k„

with the limiting low-pressure, ko, and high-pressure, k„, rate constants. The indices "diss" and "rec" have been omitted in k, ko, and k^, which refer either to dissociation or recombination (association). At low pressures of M, the rate constant k is a linear function of the bath-gas pressure [M] and tends to the high-pressure limit k<„ as the pressure [M] increases. The center of the fall-off range is defined as the bath-gas pressure [M]c at which the limiting lines intersect, i.e. ko([M]c) = k». The Lindemann-Hinshelwood equation well describes the behavior of the reaction system at relatively low and high pressures, but overestimates the values of the rate constant at pressures in the intermediate region, i.e. not very far from the center of the fall-off curve. 71S16 The most realistic description of the kinetics of a unimolecular reaction is given by the RRKM method,316 which has been successfully used in the investigations of a wide variety of reaction systems. However, the general equation of the RRKM method is quite complex, and values of the rate constant at a given temperature and pressure are obtained by numerical evaluation of a complicated integral expression. This is an important limitation of the RRKM method because kinetic modeling studies require a simple expression, best in an analytical form, convenient for estimating the rate constant under any experimental conditions of pressure and temperature. Simplicity is precisely the essential advantage of the kinetic models developed by Troe and co-workers.17"19 In their approach, the pressure dependence of the rate constant is analyzed in terms of the construction of falloff curves. Kinetics in the intermediate pressure range is mainly determined by the high-pressure (kj) and the low-pressure (ko) limiting rate constants. The proposed overall expression17"'9 for the pressure-dependent rate constant, k, applicable in all pressure regimes, is given by the reduced fall-off equation in logarithmic form: l0g(k/kJ=log U ^ - —

^i + Kn/k^j

4-

- ^ L

+

(L5)

nog(k 0 /k,J

l

N J

with N = 0.75-1.27 log Fc

(1.6)

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

145

The center-broadening factor, Fcent, characterizes the broadening of the fall-off curve at the center of the fall-off range and determines the shape of the fall-off curve.17"19 This fall-off equation enables a shorthand description of the behavior of reaction systems in an analytical form. A direct least squares fit to experimental results at any temperature is the simplest procedure to obtain the fall-off parameters k0, k w and Fcent. This is, however, not the recommended way, because the fitting parameters are usually tightly correlated. The final value of ko is very sensitive to the choice of the input value of k^, and vice versa. Therefore, considerably different sets of fitting parameters may lead to fits of similar quality, especially in the case of distinctly scattered experimental data. More coherent and realistic results can be expected from unimolecular rate theory. The first step of theoretical analysis involves preliminary construction of the theoretical fall-off curves, which are expressed by the limiting low- and high-pressure rate constants, ko and k«,, calculated from the molecular parameters of the reactant(s) and product(s). Both ko and k«, depend on values of the internal method parameters, which are varied to reach the best agreement between the theoretical fall-off curves and experiment. The high-pressure limiting rate constant, k„, can be evaluated using one of the versions of the statistical adiabatic channel model developed by Troe et al.20'21 In this approach, koo is a function of two empirical parameters: a "looseness" parameter, a, and the Morse parameter, p. The value of (3 is usually derived from the force constant and dissociation energy of the breaking bond, whereas a is often used as a fitting parameter. However, Cobos and Troe22 have shown that the best agreement between theoretical and experimental values of k„ for many reactions studied is reached at a value of a corresponding to a ratio of a/p* of 0.46 ± 0.09. The low-pressure limiting rate constant, k0, can be expressed in the form ko = (3cscko

(1.7)

with the collision efficiency (3C and the strong-collision rate constant sc k 0 calculated from molecular parameters in terms of the factorized expression derived by Troe.23 The ko is a linear function of the bath-gas pressure [M], so that ko/[M] should be pressure independent. The collision efficiency factor, /?c, is estimated by a comparison of the calculated value, sc k 0 (corresponding to |3C = 1), with ko estimated experimentally. On the other hand, f5c is related to the average energy transferred per collision, M, and to the energydependence factor for the density of states, F E , by the relationship |3C l-p:

_ /2

M

""FEkBT

(L8)

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J. T. Jodkowski

Under the assumption that M is temperature independent, the above equation describes the temperature dependence of (3C. The temperature dependence of ko for a given bath-gas is then completely determined by the value of M, which is varied to refine the fall-off curve.17"23 Most chemical processes are complex reactions. As knowledge of the mechanisms of gas-phase reactions has advanced, it has become increasingly clear that the formation of the reaction products often proceeds by a few elementary steps. Many gas-phase reactions which were previously considered to be elementary have been shown to be complex.24 The multi-step reaction mechanism is usually manifested by the occurrence of a complicated or negative temperature dependence of the measured overall rate constant. The theoretical treatment of a multi-step reaction depends mainly on the properties of intermediate species formed during the reaction.1,24 An analysis of the reaction kinetics can be made by one of the following approaches assuming that: (i) the concentration of intermediates is described by conjugated multivariable differential equations, (ii) the intermediates are assumed to be in equilibrium with the reactants, and (iii) the steady-state assumption is made for the intermediates. The first approach is rigorous, but the system of differential equations is usually very complex and can be solved by numerical methods only. The equilibrium assumption for intermediates is very simple, but seems to be unrealistic for many reaction systems. The most promising are then methods based on the steady-state assumption.24"31 Mozurkevich and Benson,24 Chen et al. " and, more recently, Jodkowski et al. have developed theoretical models for describing the kinetics of a bimolecular reaction which proceeds via an intermediate adduct/complex A + B ±+ C -» Products

(3)

A general expression taking into account the rotational energy was derived from RRKM theory.29 If the intermediate C is sufficiently short-lived (or the total pressure is sufficiently low) that it is not stabilized by collisions, the rate constant k* for the formation of the product(s) can be written as

k

e X p * = "hQ oA QVB vl d E I Wj A c ( E , J ) W « r 7(E,J) ^ ~ + VW r T (E,J) o AC TS

(

-

E / R T )

(L9)

where QA and QB are the partition functions of the reactants, with the center of mass partition function factored out of the product Q A QB and included in z. The partition functions related to the inactive/adiabatic degrees of freedom which are not taken into account by the integral are also included in z. Wj(E,J) denotes the sum of states with energy less than or equal to E and angular

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

147

momentum quantum number J for the transition state (TS) of the second elementary step and an activated complex (AC) for the formation of the adduct, C, whereas V0 is the threshold energy for the reaction at the angular momentum J = 0. All computational effort is then related to calculation of the sum of the states, Wj(E,J). Jodkowski et al.29 have developed an effective method based on the microcanonical version of the statistical adiabatic channel model approach.32 The computational procedure depends on the level at which the conservation of angular momentum is considered.29"32 The theoretical models discussed above are frequently employed in the description of the kinetics of gas-phase reactions, especially reactions of atoms and free radicals. This class of reactions is of interest in a broader scientific context, and a better understanding of their mechanism is of primary importance for the development of chemical modeling. Free atoms and radicals are very reactive species, which occur in and take part in many different reaction systems. Therefore, a radical reaction usually proceeds in competition with a few parallel or subsequent processes. The kinetic behavior of the reaction system may be very complicated and difficult for quantitative description. Theoretical investigations of the reaction kinetics provide information useful for a better understanding and correct interpretation of experimental findings. Results of ab initio calculations are employed to evaluate the rate constant in terms of the computational methods of the reaction rate theory. A theoretical analysis of some selected radical reactions is discussed in the following sections. Of particular interest are the reactions of highly reactive species, which have been recognized as components of the polluted atmosphere,4"6'8 such as 0 2 , NOx, OH, NH2, alkyl radicals, and halogen atoms. The choice of reactions is, of course, subjective and depended on the author's own experience and research interests.

2. The hydrogen abstraction reaction from methanol by some atoms and radicals Methanol is considered a promising, environmentally friendly, alternative fuel or fuel additive to gasoline.33'34 It is well established that the combustion of methanol leads to less emission of dangerous air pollutants compared with gasoline. It has a higher octane number, a leaner flammability limit, and a higher flame speed. Modeling studies of the pyrolysis and oxidation of methanol35"39 show that hydrogen atom abstraction plays an important role in methanol combustion. The hydrogen abstraction reaction from methanol yields both hydroxymethyl CH2OH and methoxy CH 3 0 radicals due to the presence

148

J. T. Jodkowski

of the non-equivalent hydrogens of the methyl and hydroxyl groups of CH3OH. Let us consider H-abstraction from methanol by atoms and radicals (X = H, F, CI, Br, CH3 and OH). The isomeric radicals CH2OH and CH 3 0 formed in the competitive reaction channels CH3OH + X -^CH 2 OH + HX ^CH30

+HX

kx(CH2OH) k x (CH 3 0)

with the rate constants, kx(CH2OH) and k x (CH 3 0), are characterized by different thermochemistry and reactivity. The highly reactive species formed in the parallel reaction channels may interfere with each other or initiate different subsequent reactions. The relative efficiency of the channel products is described by the branching ratio I\ which is defined as the ratio of the rate constant of a given reaction channel to the sum of the rate constants of all the parallel reaction channels. Knowledge of the branching ratios and their temperature dependence is of fundamental importance for combustion and atmospheric modeling. However, despite their practical and theoretical importance, the kinetics and mechanism of H-abstraction from methanol are not very well known. Results of theoretical and experimental investigations show that the reaction rate distinctly depends on the nature of the attacking atom or radical. The values of the overall rate constant, kx,ov, presented in Table 1 differ at room temperature by ten orders of magnitude," " ' from 4.2 x 10"20 cm3molecule"1s"1 for CH3OH + CH3 to 1.7 x 10"10 cm 3 moleculeV for CH3OH + F. Values of k Xov are well correlated with the reaction enthalpy, with the exception of the chlorine atom reaction. This nearly thermoneutral reaction, CH3OH + CI, has a rate constant of 6 x 1 0 " " cm3molecule"'s"1 at 298 K, which is over sixty times greater than the 9 x 10~13 cm3molecule~V observed in the case of the distinctly exothermic reaction CH3OH + OH. The high value of the overall rate constant for the CH3OH + CI reaction suggests that only a very small or negligible energy barrier can be expected for the dominant reaction channel. Except for CH3OH + F, the values of the estimated branching ratios for the other reactions indicate a distinct dominance of the reaction channel which leads to the formation of hydroxymethyl radicals as the final product. This is in line with the expectation based on thermochemical and statistical considerations. The available thermodynamic data40"42 indicate that the CH2OH radical is by 38 kJmol"1 a more stable molecular structure than CH3O. The hydroxymethyl reaction channels are then 38 kJ mol"' more exothermic (or less endothermic) than the respective methoxy channels. Thus, the hydrogen of the CH3 group should be more easily abstracted than that of the hydroxyl group of

Theoretical Description of the Kinetics of Gas-Phase Reactions 149 Table 1. Thermochemical and kinetic parameters for CH3OH + X. AH

Reaction

r,298

(kJmor 1 ) CH3OH + F CH3OH + CI CH3OH + Br CH3OH + H

-> CH2OH + F

-167 ± 4

-> CH3O

-129 ± 4

-> CH2OH + CI

CH3OH + OH

-28 + 4

+ CI

10 ± 4

-> CH2OH + Br

37 ± 4

-> CH3O

75 ± 4

-»CH3O

+ Br

-> CH2OH + H -^CH30

CH3OH + CH 3

+F

+H

H> CH2OH + CH 3 -> CH3O

+ CH3

-33 + 4 5±4 -36 ± 4 2+4

-» CH2OH + OH

-96 ± 4

-» CH3O

-58 ± 4

+ OH

k

a)

(cm3molecule"'s"')

r(CH 2 OH) b)

(1.1 - 1.7) x 10"10 0.41 ± 0.04 (5.3 -6.3) x 1 0 " 2.4xl0" 1 7 c ) (1.2-2.8) x 10"15

0.93 ± 0.05 =1 >0.99

4.2xl0" 2 0 d )

0.76 ± 0.05

(0.8-1.0)xl0"12

0.89 ± 0.03

the overall rate constant of the decay of methanol, kXi0V = kx(CH2OH) + kx(CH30) at 298 K. the branching ratio of the reaction channel forming CH2OH defined as r(CH2OH) = kx(CH2OH)/kx,ov at 298 K.

CH3OH. The presence of the three hydrogen atoms of the methyl group also speaks in favor of the formation of CH2OH. In consequence, the branching ratio factor r(CH 2 OH) is expected to be no lower than 0.75. The dominance of this reaction channel, which leads to the formation of CH2OH, is experimentally observed in the H-abstraction from methanol by chlorine, bromine, and hydrogen atoms, and by the hydroxyl and methyl radicals.29"31'43 However, the reported values of the branching ratio for CH3OH + F show an anomalously efficient methoxy yield. This suggests a possibly different reaction mechanism for CH3OH + F. The overall rate constant for this reaction is very high, i.e. 1.7 x 10"'° cm3molecule"'s"1 at room temperature, which is close to the gas-kinetic frequency. This unexpectedly high value of k Fov may be interpreted as the result of a different mechanism in the reaction of methanol with fluorine atoms. Amongst the reactions studied, those of methanol with halogen atoms CH3OH + F/Cl/Br are of importance for atmospheric chemistry.4"6'8 The other

150

J. T. Jodkowski

reactions, i.e. CH3OH + H/CH3/OH, play an important role in methanol pyrolysis and combustion.7'43 The kinetic analysis of these reactions comprises a series of papers by Jodkowski et al.29'31 which offer an explanation of the peculiarities of experimental findings.

2.1. The CH3OH + F reaction system The CH3OH + F reaction system has been the object of quite intensive experimental study, using both indirect44"49 and direct50"55 methods. Both reaction channels are highly exothermic, with reaction enthalpies of -167 ± 4 kJmol"1 and -129 ± 4 kjmoi"1 for the hydroxymethyl and methoxy formation channels, respectively. Although the reported values of the branching ratios are very scattered, showing serious experimental uncertainties, almost all estimates indicate an anomalous methoxy branching ratio, r(CH 3 0), higher than 50%. In particular, the measurements of Durant54 lead to a value of r(CH 3 0) of 0.6 + 0.2 at room temperature and to branching ratios for the mixed isotopomers (CD3OH and CH3OD) to be temperature independent. The value obtained by Dobe et al.22 for the methoxy branching fraction of 0.57 ± 0.05 seems to be the most credible one. The theoretical description of the reaction kinetics is based on results of ab initio investigations.29'56'57 The most reliable estimates of the heats for both reaction channels were reached using the Gaussian-2 (G2) method.58"61 The results of ab initio calculations by Jodkowski et al.29 show that the reaction mechanism is considerably more complex than was previously expected. This apparently elementary gas-phase reaction proceeds with the formation of intermediate complexes of reactants or products. The optimized structures of the intermediates important for the reaction kinetics are featured in Fig. 2. Three transition state structures were found for the separate reaction paths. The potential energy profiles (without zero-point vibrational energy, ZPE) obtained at the G2 level are shown in Fig. 3. Attack of the fluorine atom from the hydroxyl group side leads to the formation of the hydrogen-bonding molecular complex MCI a, and then, via the transition state TS1, to the methoxy-HF molecular complex MCI, which finally dissociates into the methoxy channel products. The other reaction pathway, related to an attack of the fluorine atom on the methyl group, also consists of three elementary steps. First the molecular complex MC3 is formed, and then, via the corresponding transition state F..H..CH2OH (TS2), this leads to the molecular complex of hydroxymethyl-HF (MC2), which yields the hydroxymethyl channel products. Both CH3O and CH2OH radicals can isomerize to one another. The cyclic

Theoretical Description

CH3OH..F (MCla)

of the Kinetics

CH30..H..F(TS1)

of Gas-Phase Reactions

151

CH3O..HF (MCI)

F..H..CH2OH(TS2)

CH2OH..HF (MC2)

MC3

TS3

Fig. 2. Molecular structure of the intermediate structures for the CH3OH + F reaction system.

29

152

J. T. Jodkowski

0

~ -50 "5 E -5

o -100 DC UJ

z111 -150

CH 2 OH + HF -155.5

MC2

-200

Fig. 3. Profile of the potential energy surface for the CH3OH + F reaction system obtained at the G2 level (ZPE is not included).29

transition state TS3 is a transition state for that isomerization, catalyzed by HF, i.e. the reaction CH2OH + HF -» CH 3 0 + HF. The total energy of TS3 is very close to that of MC3. The similarity of the molecular structures of MC3 and TS3 suggests that TS3 may support the formation of the methoxy channel products, according to the reaction pathway leading from the reactants to MC3 and then, by TS3 and MCI, to the methoxy channel products. The full reaction mechanism consists of three reaction pathways: CH3OH + F ±5 MCla ± ; MCI -> CH3O

+ HF

CH3OH + F ±» MC3 i ^ MC2 -» CH2OH + HF CH3OH + F ±5 MC3 t ; MCI -> CH2OH + HF In this way, TS3 may play an important role in the reaction mechanism by increasing the yield of the CH 3 0 radicals. In consequence, part of the methoxy radicals are also formed by attack of the fluorine atom at the methyl-side. This may explain the anomalously high value of the methoxy branching ratio observed experimentally54'55 for the reaction of methanol with fluorine.

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

153

The heights of the potential barriers for the second elementary steps are very small. Adding the zero-point vibrational energy may change the location, on the energy scale, of the transition states towards the respective intermediate complexes. Jodkowski et al.29 show that at the G2 level both intermediates, MCla and MC3, are not energetically stable after inclusion of ZPE, i.e. the total energy (included ZPE) of the molecular complex becomes higher than that of the adjacent transition state. The elementary processes which lead from the reactants to the corresponding molecular complexes proceed without any energy barrier. The occurrence of a similar "negative" energy barrier has also been reported by Chen et al.26'2S for the hydrogen abstraction from hydrocarbons by halogen atoms. All the characteristic points of the potential energy surface for the CH3OH + F reaction system are then located below the reactants' energy level at 0 K and very close to each other. Therefore, unimolecular dissociation/ recombination processes play an important role in the description of reaction kinetics. Glauser and Koszykowski56 also came to similar conclusions. However, their overall rate constant, calculated using transition state theory under the zero-energy barrier assumption, is about five times lower than that estimated experimentally.50"55 This disagreement is essentially related to the pre-exponential factor, and indicates that it is not possible to get a higher value of the rate constant on the basis of classical transition state theory. That is why the reaction mechanism must be determined by unimolecular processes. The rate constants for the reaction channels have been calculated by Jodkowski et al.29 under the assumption of a zero energy barrier for the second elementary steps. The results of the calculations are collected in Table 2.

Table 2. Calculated values of the overall rate constant, kFov and the branching ratio of the methoxy radicals, r(CH 3 0).

a)

T (K)

kpo V , ca i c (cm3molecule"'s"')

k F o V?eX p (cm 3 molecule's"')

300 400 500 800 1000

1.69 x l O 1 0 1.92 x l O 1 0 2.11 xlO- 10 2.53 x 10"10 2.73 x 10 10

(1.1-1.7) x 10"10

kF,ov = k F (CH 3 0) + kF(CH2OH).

b)

r(CH 3 0) calc.29 exp.54,55 0.59 0.61 0.63 0.64 064

r(CH 3 0) = k F (CH 3 0)/k Fov .

0.57 ± 0.05 0.62 ±0.05

154

J. T. Jodkowski

The temperature dependence of the calculated overall rate constant is weak and can be expressed in the form k F (CH 3 0) = 1.0 x 10"10 x (T/300) 050 u

kF(CH2OH) = 6.9 x 10" x (T/300)

027

cm 3 molecule 1 s 1 cmWleculeV1

The weak temperature dependence of both k F (CH 3 0) and kF(CH2OH) is responsible for the very weak temperature dependence of the branching ratio factor, r(CH 3 0), which changes from 0.59 at 300 K to 0.64 at 1000 K.29 The results of the calculations by Jodkowski et al.29 are in excellent agreement with experimental findings.50"55 The calculated values of k F (CH 3 0) = 1.0 x 10~10 cnrWoleculeY' and kF(CH2OH) = 7.0 x 10"u cnrWlecule's - 1 lead to the overall rate constant kF?ov = 1.7 x 10~10 cm3molecule"1s"1 at 300 K. Results of measurements of kF>0V at 300 K cover the range of (1.1 - 1.7) x 10"10 cm3molecule"Is"1. However, the theoretical value of Jodkowski et al.29 corresponds to the upper limit of the experimental results because unimolecular processes were analyzed in the high-pressure limit approach. There is no experimental information about a possible pressure dependence of the reaction kinetics of CH3OH + F, but the predicted values of the rate constants k F (CH 3 0) and kF(CH2OH) should be considered as their limiting high-pressure values. The calculated rate constants and the methoxy branching ratio, r(CH 3 0), correctly reproduce experimental results both qualitatively and quantitatively.29 This has significant importance for chemical modeling studies because of the lack of experimental measurements of the rate constants for CH3OH + F at higher than ambient temperature. The values of the rate constants obtained via the equilibrium constants for the reverse reactions CH 3 0 + HF -> CH3OH + F, denoted by k HF (CH 3 0), and CH2OH + HF -> CH3OH + F, denoted by kHF(CH2OH), can be expressed as k HF (CH 3 0) = 1.5 x 1 0 U x exp(-15740/T)

cmWleculeV1

kHF(CH2OH) = 3.6 x 10 12 x exp(-19860/T)

cnrWleculeV 1

Both reactions are strongly endothermic, proceed very slowly, and do not play any role in subsequent rearrangements of the methoxy and hydroxymethyl radicals.

2.2. The CH3OH + CI reaction system The kinetics of the reaction of methanol with chlorine was studied experimentally by using both indirect62,63 and direct51,64"73 methods. Results of measurements show high values of the overall rate constant. The estimates of

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

155

&ci,ov available in the literature are all within the range of (5.3 - 6.3) x 10"n cnrWlecule's" 1 at room temperature.64"67 The distinct dominance of the hydroxymethyl reaction channel is reported. The possible abstraction of the OH group from methanol was found to be negligible in comparison to the abstraction of a hydrogen of methyl group.51'64"67 Results of ab initio calculations30,70,74"76 show that the reaction mechanism is quite complex and the reaction proceeds via the formation of intermediate complexes. The molecular structures of the transition states and intermediate complexes are very similar to those of the corresponding structures occurring in the CH3OH + F reaction system. However, in contrast to the reaction with fluorine atoms, all the molecular complexes are thermally stable structures. The profile of the potential energy surface obtained by Jodkowski et al.30 at the G2 level is shown in Fig. 4 (with intermediates labeled analogously to the CH3OH + F reaction system). Attack of the chlorine atom either towards the methyl side or the hydroxyl side of CH3OH leads to the formation of a loose molecular complex, MC3. This intermediate complex is a key structure in the reaction mechanism and may yield both the hydroxymethyl and methoxy channel products. The next elementary step for the hydroxymethyl reaction channel is the isomerization of MC3 through TS2 (with a very low energy barrier of below 0.5 kJ mol"1) to the

40

20

o E -5

0

>-

(3 DC UJ

ID -20 CH-OH + HCI

/

MC3

-40

Fig. 4. Schematic energy profile of the CH3OH + CI reaction calculated at the G2 level.30

156

J. T. Jodkowski

molecular complex MC2, which dissociates to the final channel products. The methoxy channel products can be formed along two different reaction pathways. However, the formation of CH 3 0 radicals is expected to be a negligible process at low temperatures due to a high energy barrier for this reaction channel. The rate constants for the CH3OH + CI reaction system were derived by Jodkowski et a/.30 using the same method which was first applied to describe the kinetics of the reaction of methanol with fluorine atoms.29 The rate constant calculated for the formation of hydroxymethyl radicals kCi(CH2OH) can be expressed in the temperature range 300 - 1000 K as kci(CH2OH) = 6.6 X 10"11 x exp(23/T)

cnrWlecule's" 1

The relatively high value of the rate constant of 6.6 x 10"11 cm3molecule"1s"1 at room temperature is a result of an almost zero energy barrier for the decomposition of MC3. The weak negative temperature dependence of kci(CH2OH) can be explained in terms of the key importance of the elementary process MC2 —> MC3 with temperature variation. The formation of the methoxy radicals is described by the rate constant k C i(CH 3 0), which is a few orders of magnitude lower than that of the hydroxymethyl channel. The temperature dependence of the calculated value of k a (CH 3 0) can be shown as k c ,(CH 3 0) = 1.5 x 10"12 x (T/300)2 5 x exp(-3660/T)

cm3molecule_1s '

In accordance with the high energy barrier for this reaction channel, k a (CH 3 0) strongly depends on temperature. The calculated branching ratio values indicate that hydrogen abstraction from the hydroxyl site is negligible for temperatures below 1000 K. A comparison of calculated and experimental values of kci(CH2OH) is shown in Fig. 5. The theoretical results are in line with direct measurements, with a kci,0v of (5.3 - 6.3) x 10"11 cm3molecule"1s"1 at room temperature.64"67 The calculated value of 6.6 x 10"11 cm3molecule"1s"1 can be considered as an upper limit of kcl(CH2OH) because the dissociation/recombination were analyzed in the high-pressure limit regime. The temperature dependence of the calculated rate constants cannot be examined because of the lack of direct measurements at temperatures higher than ambient. Much more information is available on the kinetics of the reverse reaction CH2OH + HC1 -> CH3OH + CI, which was studied in the temperature range 500 - 1000 K.67 Realistic values of the reaction enthalpy obtained at the G2 level allows an evaluation of the rate constants

Theoretical Description of the Kinetics of Gas-Phase Reactions 157

-10.0 CD 0) S O O

O

J -10.5

F E 0

£• -11.0

n X 0

0

-11.5

tj> O

-12.0

300

500

700

900

T(K) Fig. 5. Values of the rate constant kci(CH2OH) measured experimentally51" (symbols) and calculated by Jodkowski et a/.30 (line).

kHci(CH2OH) and kHC1(CH30) for the reverse reactions on the basis of theoretically derived equilibrium constants. The temperature dependence of these rate constants can be shown in the form kHCi(CH2OH) = 5.9 x 10"13 x exp(-2760/T) 14

cnrWleculeV

25

kHCi(CH30) = 2.7 x 10" x (T/300) ' x exp(-2230/T) cnr'moleculeV 1 The rate constants for the hydrogen abstraction from HCl by hydroxymethyl and methoxy radicals are close to each other, with values at 300 K of 6.0 xlO"17 cnvVoleculeV 1 and 1.6 xlO"17 cm3molecule~1s"1 for kHci(CH2OH) and kHci(CH30), respectively. The Arrhenius plot for the CH2OH + HCl -> CH3OH + CI reaction in the temperature range of 500 - 1000 K is shown in Fig. 6. The theoretical values of the rate constant &Hci(CH2OH) are in good agreement with experiment.67 Only one experimental point at 1000 K seriously deviates from the theoretical line. However, this experimental value seems to be significantly underestimated compared with the other experiments at lower

158

J. T. Jodkowski -13.5

Fig. 6. Arrhenius plot for the CH2OH + HC1 -> CH3OH + CI reaction comparing the theoretical results of Jodkowski et al.30 (line) with the measurements of Dobe et al.55'61 (dots).

temperatures. The exact value of fcHci(CH2OH) at room temperature is not known. However, an upper limit for £Hci(CH2OH) at 300 K of 5 x l 0 " 1 6 cm 3 molecule"V\ obtained by extrapolation by Dobe et al.55 is in line with that predicted theoretically of 6 x 10"17 cm3molecule"1s"1. This agreement confirms the reliability of the temperature dependence of the rate constant for the primary reaction CH3OH + CI -» CH2OH + HC1 derived by Jodkowski et al.30

2.3. The CH3OH + Br reaction system The kinetics and mechanism of the reaction of methanol with bromine is not well known. The overall rate constant was found to depend either on the chemical composition or the pressure of the system, which may be explained by a change in the nature of the termination reaction involving bromine atoms (hetero- and homogenous recombination with other radicals).77'78 Results of a

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

159

80

60

"I 40 O

20

DC UJ

z m

o -20

-an

Fig. 7. Schematic energy profile of the CH3OH + Br reaction.

recent investigation by Dobe et al.w in the temperature range 439 - 713 K indicate greater values of the rate constant than previously estimated. Extrapolation on the basis of the derived Arrhenius parameters leads to a value of the overall rate constant kBr,0v of 1 x 10"16 cm3molecule"Is"1 at room temperature. This relatively low value of the overall rate constant for CH3OH + Br suggests high energy barriers for both reaction channels. The optimized structures of the bound complexes and transition states are close to those of the analogous species in the CH3OH + F and CH3OH + CI reaction systems, but the contact distances between bromine and other atoms are slightly longer. The occurrence of bound complexes implies a complex mechanism of the kinetics of the methanol reaction with bromine. The profile of the potential energy surface obtained by Jodkowski et al.30 is shown in Fig. 7. The molecular complex denoted by MC3 is the most stable structure in the CH3OH + Br reaction system. The reaction mechanism is somewhat different from that of the reaction with chlorine. For any reaction channel of the CH3OH + Br reaction system there are two possible reaction pathways: with and without formation of MC3 in the first elementary step. The hydroxymethyl radicals may be produced in the reactions

160

J. T. Jodkowski

CH3OH+Br ±5 MC3 ±± MC2 -> CH2OH + HBr CH3OH + Br ±5 MC2 -» CH2OH + HBr whereas methoxy radicals are formed in the reactions CH3OH + Br ±5 MC3 t * MCI -> CH 3 0 + HBr CH3OH + Br ±4 MCI -» CH3O + HBr The kinetics of the reaction CH3OH + Br can be described in terms of the RRKM-like method29 used in the rate constant calculations for CH3OH + CI. Results of the rate constant kBr(CH2OH) calculations by Jodkowski et al.30 are shown in Fig. 8. The temperature dependence of kBr(CH2OH) can be expressed as kBr(CH2OH) = 4.9 x 10 12 x (T/300)25 x exp(-2980/T)

cnrWleculeV 1

-12.0

_0> 3 O

-12.5

^)

o E O E -13.0 u O CM

X

o m •13.5 O

-14.0 1.5

2.0

2.5

1000K/T

Fig. 8. Arrhenius plot for the CH3OH + Br -> CH2OH + HBr reaction 40 , comparing the kinetic measurements by Dobe et al. (symbols) with 30 the theoretical results obtained by Jodkowski et al. (line).

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

161

The theoretically derived values of k Br (CH 3 0) show that the methoxy channel, with a rate constant of k Br (CH 3 0) = 2.7 x 10 12 x (T/300)19 x exp(-9820/T)

cm3molecule"1s"1

is inactive at temperatures below 1000 K.30 This is in line with the experimentally observed occurrence of only hydroxymethyl radicals in the reaction products. Values of the branching ratio for the H-abstraction from methanol show significant differences when the attacking electronegative atom changes from F, CI, and Br. At relatively low temperatures, practically all bromine and almost all chlorine atoms are consumed by the hydroxymethyl reaction channel, while the methoxy radical yield is higher than 50% in the CH3OH + F reaction.2930 Direct studies of the reverse reaction CH2OH + HBr —> CH3OH + Br can also provide valuable additional information on the kinetics of the CH3OH + Br reaction system. The rate constants for the reverse reactions, kHBr(CH2OH) and kHBr(CH30), evaluated by Jodkowski et al.30 can be expressed in the form

r—1

-IU.O

1

1

1

1

1

1

1

1

T

'0

3

-11.0

1

1—

/* 1 -

0

a> 0

E

"E 0 x O

CM

-11-2

s

X O

A1

00

X 2£ CI 0

-11.4

\>

2

3

4

5

1000K/T

Fig. 9. Arrhenius plot for the CH2OH + HBr -» CH3OH + Br reaction comparing kinetic measurements by Dobe et al.40 (symbols) with theoretical results obtained by Jodkowski et al.i0 (line).

162

J. T. Jodkowski

kHBr(CH2OH) = 2.0 x 10 12 x exp(395/T) kHBr(CH30) = 4.1 x 10 14 x (T/300)1'9 x exp(-1305/T)

cr^moleculeV 1 cnrWeculeV1

Both reverse reactions are exothermic, which is why the values of the rate constants kHBr(CH2OH) and kHBr(CH30) are a few orders of magnitude higher than those of the corresponding primary reactions. At room temperature the calculated values of the rate constant are 9.8 x 10"12 cm3molecule"1s"1 and 5.3 x 10"16 cnrWlecule's" 1 for the CH2OH + HBr and CH 3 0 + HBr reactions, respectively. The high value of kHBr(CH2OH) suggests an important role of the hydroxymethyl reaction in atmosphere modeling. However, a significantly larger value of the rate constant kHBr(CH30) has recently been found at room temperature by Szilagyi et al.19 An Arrhenius plot for the CH2OH + HBr reaction is shown in Fig. 9. The calculated values30 of the rate constant kHBr(CH2OH) are in very good agreement with experiment40'77'78 in a wide range of temperature. The complex reaction mechanism is seen in the weak negative temperature dependence of the calculated rate constant, similar to that observed experimentally.

2.4. The CH3OH + H reaction system The reaction of methanol with hydrogen atoms plays an important role in methanol combustion. About 53% of methanol is consumed in reaction with hydrogen atoms under fuel-rich conditions.80 Despite the importance of the reaction CH3OH + H, its kinetics is not very well known. The available information is mainly from investigations on methanol pyrolysis and direct experiments. " However, the results of experiments obtained at lower temperatures are not in agreement with high-temperature shock tube measurements. The value of the overall rate constants, kH,0V, at room temperature of 2.8 x 10"15 cm3molecule"1s"1 recommended by Tsang43 is about five orders of magnitude lower than that of the analogous reaction of methanol with fluorine atoms, and about two orders of magnitude lower than that of the almost thermoneutral reaction with chlorine. Recent theoretical studies by Lendvay et al.sl and Jodkowski et al?1 show that the formation of CH2OH is the dominant reaction channel, contributing to the overall reaction by over 95% below 1200 K and by about 90% at 2000 K. Results of ab initio calculations31'87"89 show that the branching products are then formed directly, because no intermediate molecular complexes were found. However, the reaction between H and CH3OH occurs through sizable energy barriers, which give rise to low rate constants for both the CH 3 0 and

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

163

50

>a cc ui

o

u

-50

Fig. 10. Schematic profile of the potential energy surface for the CH3OH + H reaction obtained at the G2 level.31

CH2OH reaction channels. This is in line with the results of kinetic measurements.80"86 The height of the energy barrier of the reaction channels depends on the level of calculation. Jodkowski et al.31 revealed that the lowest energy barriers for both reaction channels are those predicted by the G2 method. The profile of the potential energy surface is shown in Fig. 10. Formation of CH2OH is favored due to the lower energy barrier in this direction. The rate constants were derived from conventional transition state theory and improved by the tunneling factors calculated for Eckart-type potential.1'10 The results of calculations by Jodkowski et al?1 are shown in Fig. 11. The temperature dependence of kH(CH2OH) and k H (CH 3 0) can be written as kH(CH2OH) = 1.9 x 10~13 x (T/300) 32 x exp(-1755/T)

cm 3 moleculeV 1

k H (CH 3 0) = 3.0 x 10 14 x (T/300) 34 x exp(-3640/T)

cnrWleculeV 1

The relatively large value of the power n at (T/300) reflects the non-Arrhenius behavior of the calculated rate constants due to the strong negative temperature dependence of the derived tunneling factor.

164

J. T. Jodkowski

Spindler and Wagner (1982) Hoyermann et al. (1981) Vandooren and Van Tiggelen (1981) Meagher etal. (1974) Aders(1973)

-10

-11

I'

r -14

Tsang (1987) Warnatz (1988) - Jodkowski etal.(1999)

-15 1

2 1000K/T

3

Fig. 11. Arrhenius plot for the CH3OH + H reaction comparing kinetic measurements81"84'86 (symbols) and results of calculations31 or data evaluations43'85 (lines).

The available kinetic measurements are difficult to analysis because they were obtained in different experimental regimes: direct measurements at 298 - 630 K and high-temperature data (above 1000 K) from shock tube Q 1 QA Of.

experiments. " ' Extrapolation of the high-temperature data leads to an overestimation of the low-temperature rate constants, which suggests lower values of the activation energy at high temperatures. The equation proposed by the kinetic data evaluation of Tsang43 can be considered as the best compromise between the low- and high-temperature measurements. However, Tsang's values43 seem to overestimate the rate constant over 1000 K. It is worth noting that the kH,0V values predicted theoretically by Jodkowski et al.3i are in better agreement with the high-temperature measurements than those estimated from the Tsang equation.43 At 2000 K, the theoretical value of kH,0v of 4.5 x 10"" cm3molecule"1s"1 is two times lower than that recommended by Tsang,43 i.e. 9.5 x 10"" cm3molecule"1s"1, which distinctly overestimates the experimental results of Spindler and Wagner,86 Vandooren, and Van Tiggelen,83 and the kinetic data evaluation by Warnatz.85 In the intermediate

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

165

temperature range, the kH,0v values derived by Jodkowski et al. of 8.9 x 10" cm3molecule's"1 and 2.4 x 10~13 cm3molecule"1s"1 are about three and two times lower than the results of measurements at 600 K and 700 K, respectively. In contrast, Tsang' s equation43 only slightly overestimated the values of the rate constants in this temperature range and reproduces experimental results in the range of 3 0 0 - 5 0 0 K well. In conclusion, theoretical investigations provide valuable information and allow a quite satisfactory description of CH3OH + H kinetics, especially in the high-temperature range. Results of the calculations by Jodkowski et al?1 and Lendvay et al?1 show that the formation of methoxy radicals is negligible at room temperature and does not exceed 10% at 2000 K. There are no experimental data available for the reverse reactions CH 3 0 + H2 -^ CH3OH + H and CH2OH + H2 -> CH3OH + H. The temperature dependence of the rate constants calculated by Jodkowski et al?1 is given by £H 2 (CH 2 OH) = 8.8 x 10"15 x (T/300)35 x exp(-5270/T)

cm3molecule"1s"1

= 1.7 x 10 15 x (T/300) 40 x exp(-2470/T)

crn'moleculeV'

£H2(CH30)

Both reactions proceed slowly and are only of minor importance in the CH3OH + H reaction system.

2.5. The CH3OH + CH3 reaction system Both reaction channels for the reaction of methanol with the methyl radical were studied experimentally.43'90 The reaction channel enthalpies are very close to those of the analogous channel for reaction with hydrogen. However, the value of the overall rate constant kCH3,ov of 3.7 x 10"20 cm3molecule"1s"1 at room temperature is about five orders of magnitude lower than that of the reaction with hydrogen atoms.43 The reaction channel yielding hydroxymethyl radicals is considered dominant, although the hydroxymethyl branching ratio decreases with decreasing temperature. A very large isotope effect was observed.91"93 At low temperatures (77 K) H-abstraction from methanol proceeds almost 1000 times faster than D-abstraction from CD3OD. An explanation of this, in terms of the quantum tunneling effect, was provided in an ab initio study by Tachikawa et al.94 A theoretical description of the reaction kinetics was given in the study by Jodkowski et al.31 No molecular complexes of reactants or products were found in the ab initio calculations of the potential energy surface of the

166

J. T. Jodkowski

60

40 "5 E 20 -j

>a cc 0 w z

CH3O+CH4

UJ

-20

-40

Fig. 12. Schematic energy profile of the potential energy surface for the CH3OH + CH3 reaction obtained at the G2 level.31

reaction system. The branching products (CH2OH and CH 3 0 radicals) are then formed directly in one elementary step. The energy profile obtained at the G2 level is shown in Fig. 12. The high energy barriers imply small values of the overall rate constants. In contrast to the reaction of methanol with hydrogen and halogen atoms, the height of the energy barrier in the methoxy direction is, at the G2 level, lower than that calculated for the hydroxymethyl reaction channel. The results of the rate constant calculations31 carried out on the basis of TST with the tunneling factors calculated for the asymmetric Eckart potential110 lead to the expressions kcH3(CH2OH) = 1.4 x 10"15 x (T/300)4-9 x exp(-3380/T) 16

kcH3(CH30) = 2.7 x 10 x(T/300)

47

xexp(-2910/T)

cnrWeculeV cm 3 molecule"V

The large value of the parameter n is related to the strong negative temperature dependence of the tunneling factor, similar to that observed for the reaction CH3OH + H. The branching ratios depend distinctly on temperature. Only at very low temperatures is the formation of methoxy radicals dominant. At room

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

167

-10 • Spindler and Wagner (1982) • Kerr and Parsonage (1976) - • Tsang (1987) Jodkowski et al. (1999)

-12 !s> | o E m E .o

-14

~X -16

-18

-20 1

2 1000K/T

3

Fig. 13. Arrhenius plot for the CH3OH + CH3 reaction comparing available measurements86'90 (symbols) and the results of calculations31 or kinetic data evaluations43 (lines).

temperature, almost the same methoxy and hydroxymethyl fractions are observed because the slightly lower energy barrier for the methoxy channel is balanced by the statistically favored H-abstraction from the CH3-group at low temperature. At high temperatures, the three hydrogens of the methyl group, compared with the one of the hydroxyl group of methanol, prevail against the difference in the energy barriers, which leads to the dominance of the hydroxymethyl reaction channel. The theoretically derived value of the branching ratio r(CH 2 OH) of 0.7 at 550 K is in excellent agreement with the values of 0.71 and 0.77 recommended by Kerr and Personage,90 and Tsang,43 respectively. Fig. 13 shows a comparison of the values of the overall rate constant calculated by Jodkowski et al.31 with those estimated experimentally by Spindler and Wagner,86 as well as the results of the kinetic data evaluations in

168

J. T. Jodkowski

the range of 350 - 550 K done by Kerr and Personage, and Tsang. The calculated values of the overall rate constant are systematically greater than those recommended by Tsang,43 and distinctly better reproduce the results of the high-temperature measurements of Spindler and Wagner.86 The excellent agreement between the theoretical values of &CH3,OV and the results of low- and high-temperature experiments proves the high quality of the calculated rate constants & C H 3 (CH 2 OH) and £CH 3 (CH 3 0). Attack of methyl radicals on methanol is less selective than the CH3OH + H reaction, which yields practically only hydroxymethyl channel products. The calculated branching ratio for H-abstraction by methyl radicals depends sensitively on temperature, with a possible dominance of CH 3 0 formation at very low temperatures.31 The reverse reactions CH2OH + CH4 -> CH3OH + CH3 and CH 3 0 + CH4 —> CH3OH + CH3 have not been investigated experimentally. The temperature dependence of the rate constants & C H 4 (CH 2 OH) andfcCH4(CH30)is given31 by &CH 4 (CH 2 OH) = 5.2 x 10 15 x (T/300) 50 x exp(-7475/T)

cnrWlecule's" 1

£CH 4 (CH 3 0) = 2.5 x 10 15 x (T/300) 50 x exp(-2810/T)

cm3molecule"V1

The inverse reactions are slow processes. However, the reverse reaction CH3O + CH4 -> CH3OH + CH3 is the fastest process in the CH3OH + CH3 reaction system.31 The rate constant fcCH4(CH30) is one order of magnitude higher than the overall rate constant for the primary reactions at 300 K. Therefore, the methoxy radicals formed in the reaction CH3OH + CH3 —> CH 3 0 + Cftt are quickly converted back to the reactants. This may explain the lack of methoxy radicals in the reaction products.

2.6. The CH3OH + OH reaction system The kinetics of the reaction of methanol with hydroxyl radicals was studied extensively by experimental and theoretical methods.31'37'55'83'95"108 This great interest in the CH3OH + OH reaction is a result of its importance in lean and moderately rich methanol flames. A recent flame modeling study predicted that about 38% of methanol is consumed in the CH3OH + OH reaction.80 The direct kinetic measurements in a wide temperature range were supported by theoretical analysis.31'106 Both reaction channels are exothermic, but the estimated value of the overall rate constant &OH,OV of 9.1 x 10~13 cm3molecule V 1 at room temperature43 is more than one order of magnitude lower than that for the almost thermoneutral reaction of methanol with chlorine. Results of the branching ratio estimated at low temperature are subject to serious uncertainty.

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

169

A compromise value of 0.1 is considered a realistic estimate for the branching ratio of CH 3 0 at room temperature.43 A theoretical analysis of the reaction kinetics was given in the study by Jodkowski et al.3> A hydrogen-bonded molecular complex was found for the hydroxymethyl reaction channel. The formation of this complex may be a ratedetermining process in the two-step reaction mechanism of the H-abstraction from methyl group. On the other hand, the reaction channel which produces methoxy radicals is a simple metathesis reaction. The profile of the potential energy surface obtained by Jodkowski et al.31 at the G2 level is shown in Fig. 14. Both reaction channels are exothermic. It is worth noting that the molecular complex, MC, formed in the reaction possesses an energy excess sufficient for dissociation to the channel products. The energy barrier found for the backward process is, at the G2 level, 95 kJ mol"1 higher than the energy barrier for dissociation into the channel products.31 At relatively low temperatures the reverse process is negligible, and the dominant part of MC undergoes decomposition to the final channel products. The calculated small energy barriers indicate that the CH3OH + OH reaction should be a fast process, with a distinct dominance of the hydroxymethyl channel.

20 0 ~

-20

•5

E 3 -40

> DC

w -60 UJ

-80 -100 -120

Fig. 14. Schematic energy profile of the potential energy surface for the CH3OH + OH reaction obtained at the G2 level.31

170

J. T. Jodkowski

The rate constants calculated by Jodkowski et al. were evaluated on the basis of two methods: conventional TST and the RRKM-like method which has previously been applied to describe the kinetics of the H-abstraction from methanol by halogens.29'30 The derived temperature dependence of the rate constants for both reaction channels can be expressed by kOHCH2OH) = 3.5 x 10 13 x (T/300)28 x exp(210/T)

cm3molecule1s"1

ko H (CH 3 0) = 7.6 x 10~15 x (T/300) 34 x exp(575/T)

cm3molecule 's '

At relatively low temperatures, the values of the rate constant for the hydroxymethyl formation calculated using transition state theory are very close to the "exact" values obtained for the two-step reaction mechanism. The increase in the importance of the back process (MC —> reactants) at high temperatures leads to slightly lower values of the rate constant for the complex mechanism than that for direct H-abstraction. However, even at 3000 K the

-10.0

• • O D A V O O T

\

•'»

, \ -10.5 a

- \

o \

'w

£ 3 -11.0

>

,p

\

X ' ? o\ * \ 0

o o E

T

E

~|- -11.5 o

' \

* > \

CO

_o

i • i Hess and Tully (1989) Walllngton and Kurylo (1987) Greenhiil and O'Grady (1986) Meier et al. (1984,1985) Hageleetal. (1983) Vandooren and Van Tiggelen (1981) Westbrook and Dryer (1979) Bowman (1975) Warnatz (1988) Tsang (1987) Jodkowski et al. (1999)

: \ 3Sfcs ^ss*. • ••

-12.0

^ f e^ ?^| | | *:;

•

•

-12.5 1

2

3

1000K/T

Fig. 15. Arrhenius plot for the CH3OH + OH reaction comparing available measurements83'95"105 (symbols) and results of calculations31 or kinetic data evaluations43'85 (lines).

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

171

difference does not exceed 15%. The results of calculations by Jodkowski et al.31 are compared with experiments in Fig. 15. Pardo et al.106 studied the hydroxymethyl reaction channel at the MP4/6311G**//MP2/6-31G* level. Their values of 4.1 x 10~13, 1.4 x 10~12, and 2.9 x Iff12 cnvWleculeY 1 at 300, 600, and 800 K are, however, considerably lower than those derived by Jodkowski et al.31 of 7.1 x 10~13, 3.5 x 10~12, and 7.3 x 10~12 cnAnolecule'V 1 , respectively. The results of Pardo et al.i06 are also lower than the most credible experimental results, i.e. of Hess and Tully.105 The value of k0H,ov = 7.6 x 10"13 cm3molecule"1s"1 obtained by Jodkowski et al.31 at 300 K is in line with the (0.8 - 1.1) X 10~12 cmVolecule'V 1 from the measurements by Meier et al.101'102, Hagele et al.100, Ravishankara and Davis98, Overend and Paraskevopoulos", and Campbell et al?1 At higher temperatures, the theoretical results of Jodkowski et al?1 overestimate the value of koH.ov by a factor of 2 compared with the experiments of Hess and Tully105 and that recommended by Tsang.43 On the other hand, the high-temperature measurements are distinctly scattered. The results of Bowman95, and Westbrook and Dryer37 are clearly incompatible with the experiments of Vandooren and Van Tiggelen.83 It is worth noting that the temperature dependence of the theoretical rate constant ko Hov of Jodkowski et al?x is very close to that predicted by the most valuable evaluation of kinetic data, i.e. of Tsang43, which is seen by nearly parallel curves in the high-temperature range. The reverse processes CH2OH + H 2 0 -> CH3OH + OH and CH 3 0 + H 2 0 -> CH3OH + OH are strongly endothermic reactions. The rate constants £H 2 OCH 2 OH) and &H2o(CH30) can be expressed as £H 2 O(CH 2 OH) = 4.2 x 10"14 x (T/300) 30 x exp(-10440/T) cm3molecule V 1 fcH2o(CH30) = 1.5 x 10 15 x (T/300) 38 x exp(-5780/T)

cnrWleculeY 1

In accordance with the small values of the rate constants, these reactions do not play any role in the subsequent fate of the methoxy and hydroxymethyl radicals in the CH3OH + OH reaction system.

3. The H-abstraction reactions of CF3OH/CF30 systems in the atmosphere Trifluoromefhanol, CF3OH is the simplest perfluorinated alcohol. The occurrence of trifluoromefhanol in the stratosphere is a result of the atmospheric degradation of hydrofluorocarbons and hydrochlorofluorocarbons containing a CF3 group.109 In the oxygen-rich atmosphere, the CF 3 radicals formed are converted into CF 3 0 2 and then yield CF 3 0 radicals in reaction with

172

J. T. Jodkowski

nitric oxide.110'111 The subsequent atmospheric fate of the trifluoromethoxy radicals is less known, because CF 3 0 radicals are practically inert towards 0 2 and only slowly react with 0 3 . The major process of CF3O radical loss in the lower atmosphere is probably related to the reaction with NO, yielding CF 2 0 and FNO.112'113 Some role may also be played by reactions with hydrocarbons.112'114 The other reaction pathways are the reactions with water and hydrogen halides HX (X = F, CI and Br), which produce CF3OH according to CF3O + H 2 0 -> CF3OH + OH

(1)

CF3O + HX -> CF3OH + X

(2)

In the troposphere, where water is abundant, reaction (1) might be an efficient loss process for CF 3 0 radicals.115'116 CF3OH could be considered as a temporary reservoir for CF 3 0 radicals in the atmosphere.114'117 However, its atmospheric fate is less recognized. The reverse reactions CF3OH + OH -* CF3O + H 2 0

(-1)

CF3OH + X -> CF3O + HX

(-2)

are possible loss processes for CF3OH which, moreover, enable the regeneration of CF 3 0 radicals in the atmosphere. Available information on the kinetics and thermochemistry of the above reactions is highly uncertain and contradictory. The value of the heat of formation of CF3OH was an object of prolonged controversy among some research groups. Results of ab initio calculations118"125 at various levels of theory show that the CF3O-H bond is unusually robust, with an O-H bond strength of 502 kJmol"1. This value is about 40 kJmol 1 stronger than in methanol, and is comparable to or ever greater than that of the HO-H bond in water.126 This is distinctly inconsistent with results obtained using the group additivity approach,127"129 suggesting that the O-H bonds in CF3OH and CH3OH are of similar strength. Recent experiments130131 support the higher, theoretically predicted, value of the O-H bond strength in trifluoromethanol. The unusually large CF3O-H bond strength is explained by the negative hyperconjugation effect.118'124 The role of CF3OH in atmospheric chemistry is not well known. Only one kinetic measurement was made on the kinetics of CF3OH + OH.117 More experimental information is available for the reverse reaction CF 3 0 + H 2 0. 6 The other reactions have not been studied experimentally. Therefore, the kinetic description of this class of reactions is based on results of theoretical 1 1R-19S

investigations.

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

173

3.1. The kinetics of the reactions CF30 + H20 and CF3OH + OH There are a few experimental studies on the kinetics of CF 3 0 + H 2 0, but the values of the estimated rate constant are rather uncertain and have been obtained with unsatisfactory precision.117"119 The CF 3 0 + H 2 0 reaction is slow, and measured values of the rate constant k] are (0.2 - 10) x 10"17 cm3molecule_1 s"1 at room temperature.6 The best compromise value of ki = 1.7 xlO"17 cm3molecule"1s"1 at room temperature is recommended by recent kinetic data evaluations.132133 The value of 7 x 10"17 cnrWlecule'V 1 at 220 K derived by Wallington and Schneider117 seems to be distinctly overestimated. There is no credible information on the kinetics of CF3OH + OH. However, the rate constants for the CF 3 0 + H 2 0 and CF3OH + OH reactions are related to one another by the equilibrium constant. Estimated in this way, a value of the rate constant k_i for CF3OH + OH of 2 x 10"17 cm3molecule"1s1 at 298 K is the most realistic one.132,133 Results of ab initio calculations of the potential energy surface of the CF 3 0 + H 2 0 reaction system show that the mechanism of H-abstraction is not elementary. The reaction proceeds with the formation of loosely bound intermediate complexes of reactants or products. The molecular arrangement of the structures taking part in the reaction mechanism is shown in Fig. 16. The molecular complexes CF3OH..OH and CF3O..OH2 are loosely bonded, and their structural parameters are close to those obtained for the isolated subunits. The occurrence of the intermediates implies a complex, three-step reaction mechanism: CF3OH + OH ±s CF3OH..OH ±* CF3O..OH2 -> CF 3 0 + H 2 0

CF3OH..OH (MCI)

CF3O..H..OH (TS)

CF3O..OH2 (MC2)

Fig. 16. Geometry of the molecular structures of importance in the CF3OH + OH reaction system.125

174

J. T. Jodkowski

60

40

E

20

>(3

cc UJ

0

CJ^OH.OH

i

u£&2i&0

UJ

10.9 -20

-21.6

MC1

-40

Fig. 17. Schematic energy profile of the potential energy surface for the CF3OH + OH reaction obtained at the G2 level.125

The profile of the potential energy surface obtained by Brudnik et al.115 at the G2 level is shown in Fig. 17. When the loosely bound intermediates are not stabilized by collisions, they can be omitted in the reaction mechanism. The kinetics of the reaction can, in a first approximation, be described by the rate constant obtained from classical transition state theory. The rate constant calculations of Brudnik et al}25 show that this approach is realistic at temperatures below 1000 K. The temperature dependence of the rate constants calculated for CF3O + H 2 0 can be expressed as ki = 2.5 x 10"13 x (T/300) 14 x exp(-3130/T)

crn'molecule'V 1

k.i = 1.9 x 10 12 X (T/300) 10 x exp(-3650/T)

cnrWleculeV

and

for the CF3OH + OH reaction.

Theoretical Description of the Kinetics of Gas-Phase Reactions 175 -13 D Turnipseed et al. (1995) O Wallingtonetal. (1993).

-14

'in

•"o> - 1 5 3 U 0>

o E

n

E

.o

-16

O)

o -17

-18 1

2

3

4

1000K/T Fig. 18. Arrhenius plot for the CF 3 0 + H 2 0 reaction comparing measurements115'116 (symbols) with theoretical results125 1^9 1 ^^

(solid line) and kinetic data evaluations

'

(dotted line).

A comparison of the experimental and calculated values of the rate constants for the reaction CF 3 0 + H 2 0 is displayed in Fig. 18. The measurements by Turnipseed et al.U6 and the kinetic evaluations give only an upper limit of the rate constant. Therefore, the theoretical curve can be considered the best compromise between the available estimates. The values of the rate constant obtained theoretically by Brudnik et al.125 enable filling the gap in the experimental data in the intermediate temperature range. Both reactions, CF 3 0 + H 2 0 and CF3OH + OH, proceed slowly, with rate constants lower than 1 x 10"17 cn^molecule'V 1 at room temperature. Therefore, they do not play any important role in the atmospheric chemistry of CF3 radicals.

176

J. T. Jodkowski

3.2. Reactions of halogen atoms with trifluoromethanol The hydrogen abstraction reactions from trifluoromethanol by halogen atoms CF3OH + X -» CF3O + HX

(X = F, CI, Br)

lead to the formation of HX molecules, which are typical reservoirs of halogens in the polluted atmosphere. The reverse reactions CF3O + HX -> CF3OH + X

(X = F, CI, Br)

may be an efficient source of atmospheric CF3OH. Despite their potential importance for atmospheric chemistry, these reactions have not been studied experimentally. The major source of information on the kinetics of these reactions is a theoretical study by Brudnik et al.134 Results of calculations of the potential energy surface predict the existence of weakly bound intermediate complexes. This implies a complex, multi-step reaction mechanism. The profile of the potential energy surface of the reaction systems is shown in Fig. 19. 150

TS1X 1 3 1 - -~rCF,0 + HBr / \122 > —2 / -^^^116

100

92-j I

I

50

/ \

CF.O + HCI \67^-2 62-^-=^

1/

rr 2 4

>

(5 DC 111 lil

T\

CF-OH + X 0

•—

//

vS V -10^^

\

1

-9\

-50 \ ^ . C F ^ + HF •

\.7d

-77 - i — ^ MC1X MC2X

3

-100

Fig. 19. Profile of the potential energy surface for CF3OH + X reaction systems obtained at the G2 level.134

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

177

The reaction pathways are very similar to that postulated for the CF3OH + OH reaction system. The mechanism of H-abstraction from CF3OH by the halogen atom X consists of three elementary steps, i.e. CF3OH + X ±+ CF3OH..X <=> CF3O..HX -» CF3O + HX and CF3O + HX £5 CF3O..HX i * CF3OH..X -> CF3OH + X for the reverse reaction. The kinetics of the reaction is determined by the height of the energy barrier for the second elementary step. This is reflected in the temperature dependence of the calculated rate constant kF = 1.7 x 1 0 " x (T/300) 073 x exp(-2550/T)

cm3molecule V 1

kci - 6.5 x 10"12 x (T/300)123 x exp(-10550/T)

cnrWleculeV 1

kBr = 5.2 x 10 12 x (T/300) 164 x exp(-15150/T)

cm'moleculeV 1

The lowest energy barrier for the CF3OH + X reaction was found for the reaction of trifluoromethanol with fluorine atoms. Chlorine and bromine atoms are considerably more abundant in the atmosphere than fluorine,8 but the energy barriers for the reactions with CI and Br are very high. The calculated values of the rate constants at room temperature are 3.5 x 10~15, 3.7 x 10'27, and 6.3 x 10~34 cm3molecule"1s"1 for kF, kci, and kBr, respectively. Therefore, at the atmospheric concentrations of halogen atoms only a small fraction of CF3OH is converted into CF3O radicals in CF3OH + X reactions. The reverse processes, CF3O + HX, are considerably more important for the kinetic modeling of atmospheric reactions, especially the reactions CF3O + HC1 and CF3O + HBr, which are exothermic and proceed with small energy barriers. The temperature dependence of the rate constants kHx calculated by Brudnik et al.134 is given by the expressions kHF = 7.2 x 10~13 x (T/300)086 x exp(-l 1150/T)

cnvWleculeV 1

kHC1 = 1.6 x 10 13 x (T/300)136 x exp(-2200/T)

cm3molecule1s"1

k HBr = l.Ox 10 1 3 x (T/300)172 xexp(-200/T)

cmWleculeV1

and illustrated in Fig. 20. The reaction CF 3 0 + HBr is the fastest process, with a rate constant kHBr of 5.2 x l O 1 4 cnAnolecule's 1 calculated at 300 K. The rate constant kHCI for CF3O + HC1 is less, with l.Ox 10"16 cm3molecule"1s"1, and the slowest one is

178 J. T. Jodkowski -10

T

•

r

-15

I™ 3 O

21 o E -25 to

E o

€ -30 o

-35

-40 1

2

3

4

1000K/T

Fig. 20. Arrhenius plot of the calculated rate constants for the reactions CF30 + HX.

CF 3 0 + HF, with a kHF of 5.4x If/29 cmVolecule's" 1 at room temperature.134 Reaction rate depends either on the rate constant or on the concentration of the reactants. Both HCl and HBr are considered reservoirs of chlorine and bromine in the atmosphere. Considering the values of kHa and kHBr, the reactions CF3O + HCl/HBr appear of great interest for modeling studies of the polluted atmosphere. However, new experimental investigations are strongly desired to verify the theoretical predictions.

3.3. Radical recombination reactions resulting in the formation of trifluoromethanol and trifluoromethyl hypohalites The successive substitution of halogen atoms into methanol leads to the formation of trifluoromethanol, CF3OH, and the trifluoromethyl hypohalites CF3OF, CF3OCI, and CF3OBr. The appearance of these compounds in the stratosphere suggests their potential importance in the atmospheric ozone budget.8 CF 3 and CF 3 0 radicals are formed in the atmosphere as a result of the

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

179

degradation of hydrofluorocarbons (HFCs) and hydrochlorofluorocarbons (HCFCs).109 Trifluoromethanol and trifluoromethyl hypohalites can be produced by the recombination reactions CF3 + OX -» CF3OX CF3O + X -» CF3OX

(1) (X = H, F, CI and Br) (2)

The atmospheric concentrations of the reactants are high enough to enable an efficient formation of CF3OX molecules. However, despite the possible importance of this class of compounds for atmospheric chemistry, the available kinetic data are very limited.6 The reactivity of CF3OX is determined by the properties of the functional groups -CF3 and -OX. The atmospheric fate of trifluoromethanol is uncertain. Thermal decomposition to COF2 and HF proceeds very slowly because of the high activation barrier.135"137 However, CF3OH may be efficiently decomposed when the reaction proceeds heterogeneously on the surface of water droplets.136 Trifluoromethyl hypofluorite, CF3OF, is a very reactive molecule. ' Its functional group, OF, is relatively weakly bonded and reveals strong oxidizing and fluorinating properties. The most favorable decomposition mechanism of atmospheric CF3OF is probably related to F 2 elimination, resulting in the formation of COF2. Trifluoromethyl hypochlorite, CF3OCI, can be efficiently formed in the recombination reactions due to the high concentrations of CI and CIO in the atmosphere.140141 The chlorine compounds are very effective in destroying stratospheric ozone. They may retard the formation of ozone by sequestering oxygen atoms in the chloride compounds. The kinetics of the atmospheric decomposition of CF3OCl has not been studied experimentally. Trifluoromethyl hypobromite, CFsOBr, is probably an important intermediate in the ozone layer depletion initiated by halones.142 Both COF2 and FBr are seen as major products of the thermal decomposition of CF3OBr.143 The photolysis of CF3OBr mainly produces bromine and various organofluorine compounds. The molecular properties of CF3OX molecules have been studied both experimentally and theoretically. However, the kinetics of CF 3 + OX and CF3O + X is not well known. Except for CF 3 0 + F recombination,144"146 there are no experimental and theoretical kinetic investigations for the other reactions. The kinetics of those reactions was studied theoretically by Brudnik et alP5 in terms of ab initio calculations at different levels of theory. The total energies of the reactants were examined using G2 methodology. The values of the enthalpy of formation at room temperature for CF3OX molecules

180 J. T. Jodkowski

500

CF 3 0+H

•497

400

483-

-CF3+OH

398 392: 384

CF,+BrO 5CF.+FO CF3+CIO

o E 300 -3

CF3O+CI , CF,0+Br = a. 200 " C F , 0 + F "

>

228 219 •194 :

LU

100

CF3OX

Fig. 21. Relative energies of reactants and products obtained at the G2 level at 0 K.

calculated at this level of theory135 are in satisfactory agreement with those estimated experimentally. A schematic energy diagram is shown in Fig. 21. All recombination reactions are exotermic, but the heats of reaction for CF 3 0 + H and CF3 + OH are distinctly predominant. The values of the reaction enthalpies of the other reactions are very similar for the same type of reaction and cover the narrow ranges of 194 - 228 kJ mol"1 and 384 - 398 kJ mol"1 for CF3O + F/Cl/Br and CF3 + OF/ClO/BrO, respectively.135 Because of lack of experimental information, the kinetic behavior of the reaction systems must be limited to the high-pressure range. The theoretical analysis by Brudnik et a/.135 was based on one version of the statistical adiabatic channel model: the maximum free energy method of Quack and Troe.20 By this approach, the most reliable results are usually obtained at values of the internal method parameters a and |3 corresponding to the ratio a/p = 0.46 ± 0.09.22 The values calculated by Brudnik et alU5 of the highpressure limiting rate constants k recoo were derived at an a/P ratio of 0.46. The error bounds were given by the values of the rate constants calculated with parameter a corresponding to the lower and upper limit of the recommended ratio of ot/p = 0.46 ± 0.09, i.e. for a = 0.37(3 and for a = 0.55p\ The temperature dependence of the calculated high-pressure limiting rate constants derived in the range of 200 - 500 K can be expressed by

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

181

kioH,~ = (4.4 + 2.4)xlO- n x(T/300)°'

cm 3 molecule~V

kipo.- = (2.0 + 1.2)xlO- u x(T/300) a:

cm molecule-1 s_1

kiao,- = (1.8 ± l.l)xlO" u x(T/300) o:

crr^molecule'V 1

kiBK),^ = (1.4 + 0.8)xl0" u x(T/300) 0:

cm3molecule_1s_1

k2H,„ = (3.0 ± 0.7)xl0" lo x(T/300) o:

cm 3 molecule~V

k2Fi«, = (3.4 ± l.l)xl0- n x(T/300) 0 :

cm3molecule~'s~'

k2Ci,~ = (3.6 ± 0.9)xlO- H x(T/300) o:

cm3molecule"1s_1

W - = ( 2 - 6 ± 0.7)xl0- n x(T/300) 0:

cm 3 molecule~V

where k]0x,oo and k2x,» denote the rate constants for the CF3 + OX and CF 3 0 + X reactions, respectively. The calculated values of the rate constants at 300 K are relatively high and cover the range of 1.4 x l O 1 1 to 3.0 x l O 1 0 cm3molecule"1s"1. The fastest is the most exothermic reaction, CF 3 0 + H. The results of theoretical calculations cannot be verified directly because of the lack of experimental data. Experiments by Zhang et al.}45 for the CF 3 0 + F reaction lead to a value of k2F,„o of (5.8 ± 0.8) x 10"n cm3molecule"1s"1 at 300 K, which is 70% higher than that predicted theoretically.135 However, the experimental estimate is uncertain because it was derived from a single measurement carried out twenty years ago.145 The high-pressure limiting rate constants for the respective CF3OX dissociation reactions obtained by Brudnik et a/.135 via the theoretical equilibrium constant can be expressed as k-lOH,"

(4.3 ± 2.4)xl016x(T/300)025xexp(-58600/T)

s"1

k-lFO,=

(2.1 ± 1.2)xl017x(T/300)057xexp(-47600/T)

s"1

k-icio,=

(2.1 ± 1.2)xl017x(T/300)"068xexp(-46600/T)

s"1

k-lBrO,=

(1.7 ± 0.9)xl0 17 x(T/300) 067 xexp(-48200/T)

s"1

k-2H,o

(1.9 ± 0.5)xl0 l5 x(T/300) 098 xexp(-59900/T)

s_1

k-2F,o

(4.4 ± 1.5)xl015x(T/300)0O2xexp(-23500/T)

s_1

k-2ci,=

(4.1 ± l.l)xl0 15 x(T/300)" 027 xexp(-27700/T)

s_1

k-2Br,o

(3.0 ± 0.8)xl0 15 x(T/300) 041 xexp(-26600/T)

s"1

and

182

J. T. Jodkowski

A comparison of theoretical and experimental results is possible only for the dissociation reaction CF3OF —> CF 3 0 + F, which was studied experimentally in the temperature range of 476 - 507 K. The calculated value of k_2F,«, of 1.7 x 10"5 s"1 (at 500 K) is slightly lower than the 2.4 x 10~5 s"1 estimated by Cobos147 and Czarnowski et a/.148'149 and is in excellent agreement with the value of 1.7 x 10"5 s"1 obtained by Kennedy and Levi.144 The analytical expressions derived by Brudnik et a/.135 allow a description of the reaction kinetics. This has significant importance for chemical modeling due to the lack of experimental measurements. However, direct experimental investigations on the kinetics of this class of reactions are necessary to verify their importance for atmospheric chemistry.

4. Reactions of formaldehyde and acetaldehyde with hydroxyl radicals Aldehydes are among the most abundant atmospheric carbonyls, which are emitted into the atmosphere by a wide variety of anthropogenic and natural processes. Microbiological processes, vegetation emissions, and biomass fires are major biogenic sources of alhehydes.150 Large concentrations of aldehydes are also formed by the partial oxidation of hydrocarbon fuels.8 Reactions with OH, H0 2 , and NO3 as well as photolysis are the possible fates of aldehydes in the atmosphere. The kinetics of OH reactions with a variety of aldehydes has been extensively studied in a wide temperature range. The negative temperature dependence of the bimolecular rate constants for these reactions is well established, except for the reactions of formaldehyde and acetaldehyde, for which the dependence is more complex.151152

4.1. The reaction HCHO + OH The photochemical oxidation of methane is the most important source of formaldehyde. Atmospheric formaldehyde is also produced by the photochemical oxidation of non-methane hydrocarbons. The kinetics of the reaction HCHO + OH has been studied both experimentally and theoretically.151"162 Kinetic isotopic effects for some deuterated formaldehyde isotopomers have been reported.153"155 Results of experimental and theoretical studies151"162 indicate a complex reaction mechanism consisting of three competitive reaction channels

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

HCHO + OH ->HCO + H 2 0

183

(1)

-> H + CO + H 2 0

(2)

-» H + HCOOH

(3)

Under atmospheric conditions, H and HCO react with 0 2 , yielding CO and H 2 0. Thus, pathways (1) and (2) lead to the same final products. Calculations of the potential energy surface of the reaction system provide more detailed information on the reaction mechanism. In Fig. 22 is shown the profile of the potential energy surface calculated at the CCSD(T)/aug-cc-pVDZ//MP2/augcc-pVDZ level by d'Anna et al.156 There are three possible reaction pathways, all consisting of three elementary steps. Two routes are related to H-abstraction, while the another proceeds by OH-addition. All the reaction pathways are exothermic, but the formation of HCO and H 2 0 is favored energetically. The total energy of all the molecular structures taking part in the reaction mechanism of the Habstraction channels is lower than the reactants' energy. In contrast, the first elementary step of the OH-addition channel is related to a sizable energy barrier.

50 TS OHadd •

HCHO + OH ^/

""

1 >-

T S H

Pre-Reaction Adduct

-50

\

•

T S

\ \

UJ

•

el

_ ^ _ \

\\

TSHei

/

•

•

\ ^H+CO+H 2 0

O DC HI

z

H

\ \ \ \

\ -100

-150

WcOOH

HCOOH+OH ' HCO+H20

Post-Reaction Adduct

•

Fig. 22. Energy levels of the HCHO + OH reaction system obtained at the CCSD(T)/aug-cc-pVDZ//MP2/aug-cc-pVDZ level.156

184 J. T. Jodkowski The temperature dependence of the reaction kinetics was analyzed in terms of the method developed by Jodkowski et al.29'30 This approach allows a description of the kinetics of the bimolecular reaction, which proceeds with the formation of intermediate complexes. The rate constant k*, related to the Habstraction channel, describes the rate of formation of the post-reaction adduct (AD2) under the assumption that the pre-reaction adduct (ADl) is not stabilized by collisions, but at once undergoes possible forward or backward processes. If the pressure is high enough to enable an efficient collisional stabilization of ADl, the kinetics of its formation needs more advanced treatment. The temperature dependence of the rate constant calculated by different methods by d'Anna et al.156 is shown in Fig. 23. The calculated highpressure limiting value k ^ i ^ for the formation of ADl at room temperature is almost one order of magnitude higher than that estimated experimentally. Therefore, the formation of ADl occurs in the low-pressure regime. The strong-collision values sc k ADlj0 for the formation of ADl calculated at a helium pressure of 1 bar are depicted by a dashed line in Fig. 23. The upper limit of the low-pressure limiting rate constant k.ADi,o is given by its strongcollision value, but under experimental conditions k AD i 0 is expected to be 1 - 2

,_ T-

tt

a> 3 O

5 •

4

a> o

E 3 E

CO

o

^

o

2 -

•

^ ™

z o 1 o+ X o X X n

'•

•

0.001

0.002

0.003 T /K-1

• 0.004

0.005

1

Fig. 23. The observed150'151 and calculated156 rate constants for the reaction HCHO + OH. The solid line denotes the calculated rate constant k*, whereas the dashed line corresponds to the values of the strongcollision, low-pressure limiting rate constant' 'kADi.o calculated for the formation of the pre-reaction adduct.

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

185

orders of magnitude lower then ICADI.O- The rate constant for the AD1 adduct formation increases when the temperature decreases. This may significantly contribute to the overall rate constant at low temperatures, i.e. below 250 K, and pressure dependence of the reaction kinetics may occur. Unfortunately, there are no credible kinetic measurements in this temperature range.151'152 At temperatures higher than ambient, the reaction kinetics is described by the rate constant k*. The derived temperature dependence of k* is realistic.156 The values of k* calculated by d'Anna et a/.156 reproduce available experimental data well, and are comparable to those recommended by recent kinetic data evaluations in a wide temperature range. Kinetic isotope effect (KIE) investigations provide additional information of essential importance for the credibility of the theoretical analysis. The values of KIE calculated by d'Anna et al.156 are in excellent agreement with those measured for the studied isotopomers.153'156'162 This supports the interpretation of the reaction mechanism proposed by d'Anna et al.156

4.2. The reaction CH3CHO + OH Acetaldehyde is emitted into the atmosphere from anthropogenic and biogenic sources or can be produced in the photooxidation of other organic pollutants. The oxidation of acetaldehyde in the troposphere is an efficient source of peroxyacetylnitrate (PAN), which is an NOx reservoir and, ultimately, a source of OH radicals in the dark.8 It is believed that the degradation of acetaldehyde in the atmosphere is initiated mainly by reaction with hydroxyl radicals. The negative temperature dependence of the reaction rate constant observed in the range 200 - 550 K suggests that this reaction proceeds with the formation of the intermediate complexes.151152'156163 The following five pathways are ascribed to the reaction mechanism. CH3CHO + OH ->CH 3 C0 + H 2 0

(1)

-» CH3 + CO + H 2 0

(2)

-> CH2CHO + H 2 0

(3)

-> CH3 + HCOOH

(4)

-» CH3COOH + H

(5)

More detailed information on the possible reaction pathways is derived from calculations of the potential energy surface. Fig. 24 shows the profile of the potential energy surface obtained by d'Anna et al}56 at the CCSD(T)/aug-cc-

186

J. T. Jodkowski

50 '

TS °Hiaa

/ /

0

\

CH3CHO+OH/ \

1

"methyl absll

-

/j TS \\11

TS

1 "aldabst \\

Pre-Reaction Adducts

o E

3 > o

TS

1 11 \ 11

H

ellm

rn

I TS

"

C H 3 elinl 1

50

1 j

DC LU

\|CH,COOH + H • \ CH2CHO+H20

z

--|\CH3+CO+H20

UJ

CH3CH(0)OH H20-CH2CHO

-100 1

HCOOH + CHj

CH3CO+H20 •

• Pos t-Reaction Adduct

•

-150 Fig. 24. Energy levels of the CH3CHO + OH reaction system obtained at the CCSD(T)/aug-cc-pVDZ//MP2/aug-cc-pVDZ level.156

pVDZ//MP2/aug-cc-pVDZ level. The OH-addition pathway leads to the formation of CH3COOH. The other routes are related to H-abstraction reactions, which yield different products due to the presence of non-equivalent hydrogens in the CH3 and CHO groups of acetalaldehyde. All routes were found to be exothermic and each consists of three elementary steps. HaWabstraction should be the dominant reaction pathway because it proceeds without an energy barrier. The energy barriers related to Hmethyl-abstraction and OH-addition were found to be 12 kJmol 1 and 31 kJmoi 1 , respectively. The addition pathway is therefore of less importance under atmospheric conditions. Results of the ab initio calculations by d'Anna et al.156 show a similarity in the molecular and thermochemical properties of the structures important for die

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

187

kinetics of Hald-abstraction in the CH3CHO/HCHO + OH reaction systems. The mechanism of H-abstraction for CH3CHO + OH seems to be analogous to that found for HCHO + OH because of a predicted distinct dominance of the HaW-abstraction route. The rate constant measured for the CH3CHO + OH reaction shows some peculiarities. The overall rate constant for CH3CHO + OH is expected to be two times lower than that for the HCHO + OH reaction due to the presence of only one Haid atom in the CH3CHO molecule. However, the rate constant for CH3CHO + OH measured at low temperatures is significantly higher than that for HCHO + OH.156 In addition, the experimental rate constant for CH3CHO + OH shows a distinct negative dependence on temperature over a wide range of temperatures. This suggests a more complex, and pressure- and temperature-dependent, mechanism of H-abstraction for CH3CHO + OH. Results of ab initio calculations by d'Anna et al.156 show the existence of two non-equivalent pre-reaction adducts, with the oxygen atom of the attacking OH radical oriented towards the hydrogen of the aldehydic or methyl groups of CH3CHO. The pre-reaction adducts formed may produce the postreaction adduct in the second elementary step. They can also isomerize one into the other, because the energy barrier calculated for this process is less than 2 kJmol"1. In this way, the initial attack of OH on the methyl group of CH3CHO may also result in the formation of the HaW-abstraction product. The formation of the pre-reactive adducts may contribute a dominant share to the overall rate constant for the decay of reactants at low temperatures. A pressure dependence of the reaction kinetics should occur due to the collisional stabilization of the energized molecules of the pre-reactive adducts. The results of the rate constant calculations by d'Anna et al}56 seem to confirm this reaction mechanism. In Fig. 25 is shown the temperature dependence of the observed and calculated rate constants. The rate constant k* describes the rate of formation of the post-reaction adduct under the assumption that the pre-reactive adducts are not stabilized by collisions, whereas kadd describes the kinetics of formation of the stable pre-reactive complexes at a total pressure of 1 bar. Thus the overall rate constant for the decay of reactants (denoted in the figure by a solid line) is given by the sum k* + kadd. The values of k* predicted by d'Anna et al}56 distinctly underestimate the reaction rate at low temperatures, but they approach the results of measurements at temperatures above 700 K. The limiting rate constants kadd)00 and kadd,o for the addition channels were analyzed in terms of statistical unimolecular rate theory. Results of the calculations show a fall-off behavior of the reaction kinetics under typical atmospheric conditions corresponding to a total pressure of 1 bar. Therefore, all kadd values were derived from the

188 J. T. Jodkowski

0.001

0.002

0.003 1

T- /K"

0.004

1

Fig. 25. Observed 151152155 and calculated156 rate constants of the CH 3 CHO + OH reaction products. Long-dashed curve denotes the rate constant k*, the short-dashed curve the rate constant kadd, and the solid curve the overall rate constant k* + kadd.

theoretical fall-off curves at the given temperature and at a helium pressure of 1 bar. A strong negative dependence on temperature of the calculated kadd is responsible for the distinct increase in the overall rate constant with decreasing temperature observed below 300 K. The sum of k* + kadd allows a theoretical description of the experimental findings in a wide temperature range. The agreement between experiment and the theoretical results of d'Anna et a/.156 is very good, in spite of possible uncertainties related to computational simplifications (rigid rotor/harmonic oscillator) and additional complications of the reaction mechanism, e.g. the equilibrium between pre-reactive adducts. The calculated values of the KIE are also in excellent agreement with measurements available for some isotopomers.156 However, further experiments at different total pressures are necessary to verify the proposed reaction mechanism.

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

189

5. Reactions of vinoxy and acetonyl radicals with nitric oxide The vinoxy CH2CHO and 1-methylvinoxy (acetonyl) radicals are key intermediates in the mechanisms of many reactions of importance for atmospheric and combustion chemistry. The formation of vinoxy radicals has been observed in several chemical processes. They may be formed in reactions of OH radicals with ethylene oxide (C 2 H 4 0) and with acetylene (C2H2) in the presence of 0 2 . 8 They are also produced in reactions of 0( 3 P) atoms with alkenes and in the reactions of reactive atoms such as F or 0( 3 P) with acetaldehyde.164165 The 1-methylvinoxy (acetonyl) radical CH 2 C(CH 3 )0 is considered an important intermediate in the atmospheric oxidation of acetone initiated by the OH radical.166"171 Spectroscopic studies by Washida et al.x(A and Williams et a/.171 allow estimation of the rate constant for the reaction of acetonyl with 0 2 . The vinoxy radical was an object of extensive investigation using various methods: spectroscopic,168172173 theoretical,174"178 and a few experimental kinetic studies " mostly limited to room temperature. Three kinetics groups studied the reaction of CH2CHO with 0 2 ; however, their results were strongly contradictory.179183184 Only two studies concerning the reaction of vinoxy radical with NO have been published.179'182 The kinetics of the reaction with NO was studied experimentally by Gutman and Nelson.179 Delbos et al.1 studied the kinetics of the reaction of vinoxy and 1-methylvinoxy radicals with nitric oxide using both experimental and theoretical methods. There is no information in the literature on the kinetics of 2-methylvinoxy with NO.

5.1. The reaction of vinoxy radical with NO The electronic structure of the vinoxy radical is usually considered a resonance structure between the two localized electronic states CH2=CHO° and °CH2CHO. Both quantum mechanical ab initio calculations174"178182 and microwave spectroscopy172'173 indicate a dominant contribution of the °CH2-CHO form. In consequence, the nitroso-compound ON-CH2CHO is expected to be a major product of the addition reaction of nitric oxide to the vinoxy radical. Delbos et a/.182 carried out measurements of the rate constant for the reaction CH2CHO + NO (+He) -» Products

(1)

in a wide range of temperatures and pressures of helium bath-gas. The discharge flow reactor, in the low-pressure range (below 10 mbar), and the laser photolysis technique for measurements at higher pressures were used in

190 J. T. Jodkowski

Fig. 26. Profile of the potential energy surface for the CH2CHO + NO reaction system obtained at the G2 level.182

their experiments. The experiments were supported by theoretical investigations. The results of ab initio calculations provide structural and energetic information and an insight into the reaction mechanism. The potential energy surface was explored for the possible existence of minimum energy structures as well as transition states in the CH2CHO + NO reaction system. The molecular properties of the stationary points of the potential energy surface enable one to describe the kinetics of the other reaction pathways. The profile of the potential energy surface is shown in Fig. 26. Both adducts, 2-nitrosoethanal (ON-CH2CHO) and ethenenitrite (CH2=CHONO), are expected as the major products of NO-addition to the vinoxy radical: CH2CHO + NO -> ON-CH2CHO

(la)

CH2CHO + NO -> CH2=CHONO

(lb)

The nitroso-adduct ON-CH2CHO is formed by the attack of nitric oxide on the alkyl carbon atom of the radical. Addition of NO to the oxygen of the vinoxy radical CH2=CHO leads to the formation of CH2=CHONO. Results of ab initio calculations by Delbos et al.m show the occurrence of structural isomers of ON-CH2CHO and CH2=CHONO. Three structural stereoisomers of ONCH2CHO have been found to correspond to the different orientations of the N-

Theoretical Description

of the Kinetics of Gas-Phase Reactions

191

O group in the molecule. The internal rotation around the C-0 and O-N axes of CH2=CHONO leads to its four isomers. The relative differences in total energy of the respective stereoisomers are small. Therefore, only the most stable structures of the adducts were taken into account in describing the reaction kinetics. The energy levels in the figure correspond to those structures only. The potential energy profile shows that the nitroso-adduct may also be formed by the isomerization reaction CH2=CHONO -» ON-CH2CHO

(lc)

The total energy of the transition state, which describes this reaction path, was found to be lower than the reactants' energy. The formed ethenenitrite molecule possesses an excess energy to pass the barrier. If the pressure is not high enough for a fast collisional stabilization of CH2=CHONO, it may be entirely converted to ON-CH2CHO. The other products, such as nitroethene (CH 2 =CHN0 2 ) or ketene (CH2CO), may also be produced in the reactions CH2=CHONO -» CH 2 =CHN0 2

(1 d)

CH2CHO + NO -> CH2CO + HNO

(1 e)

However, the reaction products CH 2 =CHN0 2 and CH2CO are formed with negligible efficiency, due to the high energy barriers. The final reaction products are formed in a parallel reaction channel. The competitive addition reactions are barrierless, but the formation of the nitrosoadduct is thermodynamically favored. However, the results of a theoretical analysis by Delbos et al.iS2 suggest that practically all the CH2=CHONO is converted by isomerization to ON-CH2CHO. The kinetics of the CH2CHO + NO reaction is then described by the rate of formation of ON-CH2CHO. The decay of vinoxy radicals and the formation of ON-CH 2 CHO are considered as proceeding at the same rate. The fall-off curve parameters, i.e. the limiting rate constants and the broadening factor, were derived within the Troe theory.17"23 The best fit to experiments by Delbos et al.1S2 was obtained with the set of fitting parameters *i,-

(2.9 ± 0.4) x 1 0 " x (T/300) 030

cm 3 molecule"V

*i,o/[He]

(5.8 ±0.9) x 10"29 x (T/300) -60

cn/molecule'V 1

^cent,He(T)

0.460 x (T/300)"07

The theoretical fall-off curves by Delbos et al.182 shown in Fig. 27 reproduce the results of their measurements very well. The high-pressure value

192

J. T. Jodkowski

-10.0

M=He

M=N2

•

O 295K

298K

'_ O 351 K -10.5 • • 373K II

\ • 413K II •5 -11.0 o _g> o E E .0

.*"

-11.5 -

D) O

-12.0

-12.5

;

/

4*V^£A/^

m 1111 a 0

1

1

.

1 2 logpM/mbar

1

.

3

4

Fig. 27. Fall-off curves and results of experiments179182 for the CH2CHO + NO (+ M) reaction.

of 2.9 x 10"11 cm3molecule"1s"1 at room temperature is in line with the (2.5±0.6)xlfJ n cnrWlecule'V1 derived by Gutman and Nelson.179 The limiting low-pressure rate constant kuo depends very strongly and negatively on temperature. The results of the theoretical analysis were verified by refitting the experimental results of Gutman and Nelson, obtained for M = N2 at 295 K. The fall-off curve determined by the derived values of A:li0/[N2] = (7.0 ± 0.7) x 10"29 cn/molecule'Y1 and Fcent = 0.485 very well reproduce the measurements of Gutman and Nelson.179 The excellent agreement between theoretical and experimental results allows one to expect that the kinetic parameters derived by Delbos et a/.182 credibly describe the kinetics of CH2CHO + NO in a wider range of temperatures.

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

193

5.2. The reaction ofacetonyl radical with NO The kinetics of the reaction of acetonyl (1-methylvinoxy) radical with nitric oxide was studied both experimentally and theoretically by Delbos et al.m Theoretical investigations were based on ab initio calculations at the G2 level of the potential energy surface of the reaction system. The profile of the potential energy surface is shown in Fig. 28. The geometrical parameters of the molecular structures taking part in the reaction mechanism are similar to the corresponding structures found for the CH 2 C(H)0 + NO reaction system. The skeleton geometry of the structure changes only slightly when an H atom is replaced by a CH3 group. The nitroso-adduct ON-CH 2 C(CH 3 )0 is the most stable product in the CH 2 C(H)0 + NO reaction system. Three and four stereoisomers have been found for the nitroso- and nitrite-adduct, respectively. These structural isomers are related to another by internal rotation of the respective fragments. Such transformations are related to a small energy barrier. Therefore, only the most stable stereoisomers presented in the figure were taken into account in the kinetic analysis. The adducts ON-CH 2 C(CH 3 )0 and CH2=C(CH3)ONO are formed by direct addition as CH 2 C(CH 3 )0 + NO -> ON-CH 2 C(CH 3 )0

(la)

CH 2 C(CH 3 )0 + NO -> CH 2 =C(CH 3 )0N0

(lb)

200 TS2

150 7

~ 100

TS3

I 50 >C5 CC UJ

ui

71 CKCHO+NO

0

-. 0

.' -18 , TS1 '-53 -50 CH.CO+HNO

-100

-150

-79 CH,=CHONO

CH2=CHNO.

-125 ON-CH2CHO

Fig. 28. Profile of the potential energy surface for the CH2C(CH3)0 + NO reaction system obtained at the G2 level.182

194

J. T. Jodkowski

but formation of the nitroso-adduct can also be additionally supported by isomerization of the nitrite-structure CH2=C(CH3)ONO -> ON-CH 2 C(CH 3 )0

(1 c)

The possible reaction pathway leading to the formation of the nitro-compound CH2=C(CH3)ONO -> CH 2 =C(CH 3 )N0 2

(1 d)

is only of minor importance, in accordance with the very high barrier for this process.

-10.0

-11.5

1 2 logpM/mbar

Fig. 29. Fall-off curves and results of measurements for the CH2C(CH3)0 + NO (+ He) reaction system in the range 298 - 447 K.182

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

195

The mechanism of the reaction CH 2 C(CH 3 )0 + NO is very similar to that discussed for CH2CHO + NO. The formation of ON-CH 2 C(CH 3 )0 is the dominant reaction pathway. The fall-off behavior of the reaction system was also analyzed within the Troe formalism.17"23 The fall-off curves of Delbos et a/.182 were constructed on the basis of their experimental results. The best fit leads to results corresponding to the fall-off parameters (3.1 ± 0.4) x 10 11 x (T/300) 016

crn'moleculeV 1

27 63 *i.o/[He] = (1.6 ±0.5) x 10" x (T/300)"

cn/molecuIeV 1

*i.« =

^cent,He(T) =

0.477 x (T/300) 18

The kinetic parameters given above lead to the fall-off curves shown in Fig. 29. The experimental points are in good agreement with values predicted theoretically, especially at room temperature. The low-pressure limiting value of ki?0/[He] for the CH 2 C(CH 3 )0 + NO (+ He) reaction system are almost 30 times higher than those obtained for CH2CHO + NO (+ He). The strong negative temperature dependence of the low-pressure limiting rate constant is similar to that observed for the addition reactions of some radicals with molecular oxygen or nitrogen dioxide.183"185

6. The reactions of alkyl radicals Alkyl radicals are common intermediate species formed during the atmospheric degradation reactions of most organic compounds. The Habstraction reactions from alkanes by halogen atoms and OH and N 0 3 radicals are efficient sources of atmospheric alkyl radicals.186 The most significant fate of alkyl radicals in the atmosphere is related to reactions with molecular oxygen yielding alkylperoxy radicals R0 2 . 8 These 0 2 -addition reactions proceed very rapidly under atmospheric conditions due to the high concentration of molecular oxygen. Reactions of alkyl radicals with OH, NO, and N 0 2 are also of great interest for modeling processes important in atmospheric chemistry and combustion.

6.1. The reaction CH3 + OH Methyl and hydroxyl radicals are formed at an early stage in hydrocarbon combustion processes. The kinetics of the recombination reaction CH3 + OH (+ M) -> CH3OH (+ M)

(1)

196

J. T. Jodkowski

has been studied both experimentally187"192 and theoretically.188'191"195 Various experimental techniques have been applied. The results of kinetic measurements by Fagerstrom et a/.191,192 were also supported by theoretical analysis. The free radicals CH3 and OH can be produced by different methods. A wide group of radicals is formed in the hydrogen abstraction reactions of hydrogen-containing molecules with fluorine atoms. Fluorine atoms can be produced by pulse radiolysis of the three-component mixtures S,¥(JCWJY\TP in the reactions SF6 + 800 keV e" -» SF6* + e"

(2a)

SF6* -> SF4 + 2F

(2b)

-> SF5 + F

(2c)

Under the experimental conditions of Fagerstrom et a/.191'192 fluorine atoms were mainly produced by reaction (2b), with a great excess of SF6. The rest of the SF6 was also used as a third body. The fluorine atoms formed are quantitatively converted into CH3 and OH radicals in the reactions F + CH4 -> HF + CH3

(3)

F + H 2 0 -> HF + OH

(4)

The title reaction proceeds in competition with the self-reactions of methyl and hydroxyl radicals CH3 + CH3 (+ M) -> C2H6 (+ M)

(5)

OH + OH (+ M) -» H 2 0 2 (+ M)

(6)

OH + OH -> H 2 0 + O

(7)

Reaction (1) is the dominant reaction channel at room temperature.187191"195 Shock tube experiments196 carried out at 1200 K and 1 bar pressure of Ar have also shown that the CH3 and OH radicals decay mainly in the recombination reaction yielding CH3OH (75%). However, the importance of the other reaction channels has recently been emphasized, especially at high 1 8 7 1Q7

temperatures. ' The kinetics of the title reaction was studied experimentally by Fagerstrom et a/.191'192 in the temperature range of 283 — 373 K and at total SF6 pressures between 85 - 1 0 0 0 mbar. Under experimental conditions, at ambient temperature and 1 bar pressure of the bath-gas, the kinetics of recombination

Theoretical Description

• 283 K

-9.9

of the Kinetics

of Gas-Phase Reactions

197

C H 3 + OH

• 298 K • 331 K

I -10.0 o

-10.1 2.0 2.5 logpM/mbar

Fig. 30. Fall-off curve and experimental results for the recombination of CH3 + OH (+ SF6) at 283, 298, and 341 K and for the recombination OH + OH (+ SF6) at 298 and 361 K (insert).192

CH 3 + OH is distinctly pressure dependent. A computer simulation of the reaction kinetics, taking into account the competitive reactions OH + CH4 -» CH3 + H 2 0 OH + O -> H + O z CH3 + O -> HCHO + H CH3 + H (+ M) -> CH4 (+ M)

(8) (9) (10)

CH3 + H 2 0 -> CH4 + OH

(11) (12)

CH3 + H 2 0 2 -> H 0 2 + H2Q

(13)

shows that the time profiles of the CH 3 and OH concentrations are very sensitive to the values of the rate constant of the OH radical recombination reaction (6). Knowledge of the rate constant k6 under the experimental

198

J. T. Jodkowski

conditions is of significant importance for a kinetic description of the reaction system. Therefore, the kinetics of this reaction, which is pressure and temperature dependent, was independently studied by Fagerstrom et al.192 The analysis of the pressure dependence of the rate constants for the CH3 + OH and OH + OH recombination reactions was based on the construction of the theoretical fall-off curve. The fall-off curves obtained theoretically by Fagerstrom et al.192 are shown in Fig. 30. The temperature dependence of the limiting rate constants for both the CH3 + OH (+ SF6) and OH + OH (+ SF6) recombination reactions can be expressed in the form k,,. = (1.4 ± 0.1) x 10 10 x (T/300) 0 '

cm3molecule"V1

ki,o/[SF6] = (2.5 ±0.7) x 10"27 x (T/300)"38

cm 6 moleculeV

ke,- = (3.0 ± 0.5) x 10"11 x (T/300) 037

cm 3 molecule"V

k6,o/[SF6] = (1.1 ± 0.2) x 10"30 x (T/300)'1-4

cm 6 molecule"V

and

The derived by Fagerstrom et al.192 high- and low-pressure limiting rate constants well describe the kinetic behavior of the reaction system in a wide range of temperatures and SF6 pressures. They are also in agreement with results of recent studies on the kinetics of the CH3 + OH and OH + OH recombination reactions.6'198"202

6.2. The reaction C2H5 + OH Both ethyl and hydroxyl radicals are formed at an early stage in several types of reactions of the hydrocarbon combustion processes. The kinetics of the title reaction C2H5 + OH (+M) -> C2H5OH (+ M)

(1)

was studied by Fagerstrom et al.203 This reaction takes place in competition with the other radical processes C2H5 + OH -> C2H4 + H 2 0

(2)

C2H5 + OH -> CH3 + H + HCHO

(3)

C2H5 + OH -» CH3 + CH2OH

(4)

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

199

The radicals were produced by pulse radiolysis of the gas mixture C2H6/H20/SF6 with a great excess of SF6. The self-reactions of the formed C2H5 and OH radicals were also observed C2H5 + C2H5 (+ M) -> C4H10 (+ M) OH + OH (+ M) -> H 2 0 2 (+ M)

(5) (6)

The relative importance of the above reaction depends on the composition of the reaction mixture. At low ratios of [CH3]/[OH], almost all methyl radicals are consumed by reaction (1), whereas the ethyl self-reaction becomes dominant at high ratios. The experimental results of Fagerstrom et al.203 were analyzed in terms of a computer simulation of the reaction kinetics. The following reactions were also taken into account in describing the behavior of the reaction system F + H 2 0 -> HF + O H

(7)

C 2 H 5 + C 2 H 5 -> C2H4 + C 2 H 6

(8)

O H + O H -> H 2 0 + O

(9)

O H + C 2 H 6 -> H 2 0 + C 2 H 5

(10)

CH 3 + CH 3 (+ M ) -> C 2 H 6 (+ M ) CH 3 + C 2 H 5 (+ M ) H> C 3 H 8 (+ M )

(11) (12)

H + C 2 H 5 (+ M ) -> C 2 H 6 (+ M )

(13)

H + O H (+ M ) -» H 2 0 (+ M )

(14)

H + H (+ M ) -> H 2 (+ M )

(15)

Experiments performed at room temperature and at SF6 pressures of 250-1000 mbar show that the kinetics of decay of the C2F5 radicals does not depend on the total pressure. The reaction of the oxygen atoms produced in the hydroxyl radical self-reaction with the ethyl and hydroxyl radicals is not important in this pressure range and has been omitted in the reaction scheme. The value of k1+2 = (1.1 ± 0.2) x 10"10 cm3molecule"1 s"1 estimated by Fagerstrom et al.203 can be considered as an upper limit of the high-pressure limiting rate constant k loo for reaction (1). One version of the statistical adiabatic channel models, i.e. the maximum free energy method,20 was applied in the theoretical analysis of Fagerstrom et al.203 The calculated high-pressure limiting rate constant ki,«, = ( 1 . 3 ± 0 . 2 ) x l 0 " 1 0

cm 3 moleculeV

200

J. T. Jodkowski

at 298 K is very close to the value of k1+2 measured experimentally. In the temperature range of 200 - 400 K, the high-pressure limiting rate constant k)i0o can be considered as temperature independent. The value of the low-pressure limiting rate constant can only be preliminarily estimated theoretically because of the lack of experimental evidence for a pressure dependence of the reaction kinetics. The calculations of Fagerstrom et al.203 show that the magnitude of the low-pressure limiting rate constant k]>0/[SF6] at room temperature is 10 25 cm6molecule 2s_1. This implies a narrow fall-off range. A pressure dependence of the rate constant for reaction (1) is expected at SF6 pressures lower than 0.1 mbar. Therefore, no pressure dependence of the reaction kinetics was observed by Fagerstrom et al. 3

6.3. The reaction CH3 + NO Reactions of nitric oxide with alkyl radicals are recognized as key reactions in the polluted atmosphere.8 They are also important in combustion processes, where the fate of NO is of great interest in the context of air pollution and possible De-NOx, processes. The reaction of nitric oxide with methyl radicals CH3 + NO (+M) -> CH3NO (+M)

(1)

has been investigated experimentally and theoretically.204"214 The bimolecular rate constant was found to be strongly pressure dependent and the reaction is still in the fall-off region even at a total pressure of 1 atm. The nitrosomethane molecule formed can also initiate other processes, and hydrogen cyanide has been postulated as one of the possible end-products.215"218 The kinetics of the reaction of NO with methyl radicals in the presence of acetone as the bath-gas was studied by Jodkowski et al.212 using time-resolved infrared absorption spectroscopy. The kinetics of CH3 + NO was studied under pseudo-first-order conditions, i.e. with [NO] » [CH3], while the contribution from the self-reaction of methyl radicals CH3 + CH3 (+ M) -> C2H6 (+ M)

(2)

was negligible. The kinetic measurements were analyzed in terms of Troe models. 1723 The obtained kinetic parameters were compared with those derived by different research groups using various methodology and experimental techniques.205 212

Theoretical Description of the Kinetics of Gas-Phase Reactions 201

logpM/mbar

Fig. 31. A comparison of the theoretical fall-off curves of Jodkowski et al. 90S 919 and results of experiments205 "212 for CH3 + NO (+ M) obtained at and results of experiments " for room temperature for other bath-gases

Theoretical fall-off curves derived from the calculated limiting rate constants, k loo and k 10 /[SF 6 ] were refined by fitting them to the experimental findings using the least-squares minimization procedure. The best fit was reached for values of the internal parameters corresponding to the following values of the limiting rate constants k,,.

=(l.l±0.2)xlO"nx(T/300)°

cm3molecule"1s"1

and kli0/[ CH3COCH3] = (1.2 + 0.1) x l(T y x (T/300),-3.5

cm6molecule~2s_1

Low-pressure limiting rate constants for other bath-gases were also derived by Jodkowski et al.212 by fitting the theoretical fall-off curve to available

202

J. T. Jodkowski

• 298K, (CH3)2CO • 296 K, Ar A 407 K, Ar -11

• 509 K, Ar

-12 O

-13 logpM/mbar

Fig. 32. A comparison of the theoretical fall-off curves of Jodkowski et al2n and results of experiments for CH3 + NO obtained at temperatures in the range of 296 - 509 K with M = Ar.210

experimental data at a fixed value of the limiting high-pressure rate constant, ki,„o. The calculated fall-off curves for different diluent gases are compared with available experimental data in Fig. 31, whereas Fig. 32 shows the temperature dependence of the rate constants derived by Jodkowski et al.2i2 by refitting the experimental data of Davies et al.210 The figures illustrate the high quality of the theoretical description. The predicted fall-off curves very well reproduce experiments performed at various temperatures and for different bath-gases. The calculated values of k 10 /[M] increase as He < Ar < N 2 < SF6 < CH3COCH3 < C3H8, which is in agreement with the relative efficiencies of the bath-gases reported for other termolecular reactions.6

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

203

6.4. The reaction CF3 + N02 Trifluoromethyl radicals, CF3, are released in the atmospheric decomposition of many hydrofluorocarbons (HFCs) containing -CF3 group. It has been established that the kinetics of the reactions of CF 3 radicals with 0 3 , H0 2 , NO, and N 0 2 is of special importance for the complex chemistry of the ozone layer.8 Several products, e.g. CF 3 0, CF 3 0 2 , and CF 2 0, have been identified. The reaction of CF3 radicals with nitrogen dioxide has been the subject of experimental and theoretical studies. " Reported values of the overall rate constant for the decay of CF 3 radicals are in the range of (0.3 - 3.2) x 1 0 " cm3molecule ' s 1 . Both CF 2 0 and FNO are found as the major products of the reaction CF3 + N0 2 . However, the reaction mechanism and the branching ratios are still uncertain. Valuable information on the kinetics and mechanism of CF3 + N 0 2 comes from the experimental and theoretical study by Pagsberg et al.226 The experiments, performed using time-resolved infrared diode laser spectroscopy, were supported by ab initio calculations to provide insight into the potential energy surface of the reaction system. The reaction was initiated by pulse radiolysis of Ar/SF6/CF3I mixtures. The abstraction reaction of the iodine atom from CF3I by fluorine atoms leads to the formation of CF 3 radicals. The reactions CF 3 + N 0 2 -^ CF 2 0 + FNO

(1 a)

CF3 + N 0 2 -> CF 3 0 + NO

(lb)

CF3 + N 0 2 -^ CF 3 0 + F + NO

(lc)

CF3 + N 0 2 (+ M) -> CF3NOz (+ M)

(Id)

CF3 + N 0 2 (+ M) -^ CF3ONO (+ M)

(1 e)

CF3 + CF3 (+ M) -* C2F6 (+ M)

(If)

were taken into account in describing the reaction kinetics. At relatively high concentrations of [N0 2 ], e.g. for [N0 2 ] » [CF3], the decay of trifluoromethyl radicals is described by the overall rate constant k! = k ]a + ki b + k lc + k ld + k le . Under experimental conditions, i.e. at SF6 pressures between 4 - 2 2 mbar, the rate constant ^ was not sensitive to changes in the total pressure.226 The value of k, = (1.5 ± 0.2) x 10"11 cnrWleculeY 1 at 298 K derived by Pagsberg et al.226 is in line with other estimates, e.g. 1.0 x 10"" by Bevilacqua et al.222, 1.6 x 10"11 and 2.1 x 10 11 by Sehested et al.223, 2.4 x 10~u by Oum et al.225, 2.5 x 10"" by Sugawara et al.220 and Francisco et al.221, and 3.2 x 10"11 cnVmoleculeV 1 obtained by Vakhtin.224

204 J. T. Jodkowski

In contrast to ki, the rate constants which describe the formation of the major reaction products, CF 2 0 and FNO, depend distinctly on total pressure. The values of the rate constant k(CF 2 0) estimated by Pagsberg et al.226 for the formation of CF 2 0 were found to be in the range of (0.5 - 1.4) x 10 " cm3molecule"1s"1 at pressures of SF6 of 11 - 88 mbar. A similar dependence on total pressure is observed in the formation of FNO, with the rate constant k(FNO) between (1.7 - 9.2) x 10"12 cm3molecule"1s"1 at total pressures between 5.5 and 44.0 mbar. Both rate constants, k(CF 2 0) and k(FNO), were found to increase with increasing pressures of SF6. The difference in the pressure dependence of the rate constants for CF 3 decay and products formation can be explained in terms of quantum chemistry. Calculations of the potential energy surface show that the reaction mechanism is more complex than was previously considered. The profile of the potential energy surface derived at the G2 level by Pagsberg et al}26 is shown in Fig. 33. The energy diagram shows that CF3 radicals are mainly consumed by two competitive reaction channels, resulting in formation of CF3ONO and

CF3+N02 0 -50 -100

1 -150 LU

-200 -250 -300

Fig. 33. The energy diagram for the CF3 + N0 2 reaction system calculated at G2 level.226

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

205

CF 3 N0 2 . The formation of CF3ONO is favored as the more exothermic process. No evidence for the formation of CF 3 N0 2 was observed by Pagsberg et al.226 at total pressures in the range of 4 - 88 mbar at room temperature. However, the branching ratio of reaction (Id) may increase with increasing bath-gas pressure. It is worth noting that the total energy of the transition state, obtained at the G2 level, is 200 kJ mol 1 lower than the reactants' energy. The formed (CF3ONO)* molecule has an excess energy high enough to pass the energy barrier. The product molecules are formed in vibrationally excited states as (CF 2 0)* and (FNO)*. Their collisional deactivation leads to the final reaction products according to the reaction mechanism proposed by Pagsberg .

1 226

et al. CF3 + N 0 2 t ; (CF3ONO)* t > (CF20)* + (FNO)*

i(+ M) I (+ M) CF 2 0

FNO

The observed pressure dependence of the formation kinetics of CF 2 0 and FNO, ascribed to the relaxation of the vibrationally excited product molecules (CF 2 0)* and (FNO)*, have also been detected by infrared chemilumine„„.

221,223,225

scence. Values of the high-pressure limiting rate constants for the competing recombination reactions (Id) and (le) were calculated by Pagsberg et al.226 using the statistical adiabatic channel model.20 Values of krecoo of 9.1 x 10~12 cm3molecule"1s"1 and 1.4x10"" cm3molecule"1s"1 were obtained at 298 K for the formation of CF 3 N0 2 (Id) and CF3ONO (le), respectively. The value of kleoo is very close to the value of k2 = (1.5 ± 0.2) x 10"11 cir^moleculeV 1 at room temperature measured by Pagsberg et al.226 On the other hand, the sum of the calculated high-pressure limiting rate constants for the competing recombination channels, kidoo + k1<Si„ of 2.3 x 10"11 cm3molecule"1s"1 at 298 K, is in line with available results of the most credible measurements6 of the overall rate constant for the decay of CF3 radicals, i.e. ki = (2.1 - 2.5) x 10"" cm3molecuIe"'s"1. This provides strong support for the reaction mechanism proposed by Pagsberg et al.226 Very recent measurements of k] also seem to give evidence for their interpretation of experimental findings.6'227

7. Reactions of amidogen radicals The major source of atmospheric amidogen radicals is ammonia, which plays an important role in neutralizing acids in the atmosphere.8 Ammonia is produced by biological processes and hence exists in both clean and polluted

206

J. T. Jodkowski

atmosphere. It is also produced by livestock wastes and fertilizer. Part of the atmospheric ammonia is generated by industrial activities and natural processes in the soil. Ammonia can react with hydroxyl radicals, forming an amidogen radical NH3 + OH -> NH2 + H 2 0 This reaction proceeds relatively slowly. The rate constant of 1.6 xlO"13 cm3molecule"'s"1 at room temperature8 corresponds to the lifetime for NH3 of 72 days with respect to its reaction with OH radicals at their typical average daytime atmospheric concentration of 1 x 106 radicals cm"3. The fate of NH2 radicals formed is uncertain. It was suggested, on thermochemical grounds, that the major removal processes for NH2 radicals in the troposphere are reactions with 0 2 , NO, and N0 2 . 8 The recombination reactions NH2 + NH2, NH 2 + OH, and NH2 + CH 3 discussed further can also play an important role in the chemistry of the polluted atmosphere. The kinetic analysis of these reaction systems is based on the papers by Fagerstrom et al.228 and Jodkowski et al.229

7.1. The reaction NH2 + NH2 The kinetics of the self-reaction of amidogen radicals, NH2 + NH2, has been studied extensively using different techniques and bath-gases.228"239 The apparent bimolecular rate constant was found to be strongly pressure dependent. The amidogen radicals can recombine or disproportionate: NH2 + NH2 (+ M) -> N2H4 (+ M) NH2 + NH2 -> NH + NH3

(1) (2)

Both reaction channels are exothermic; however, reaction (2) proceeds slowly. The upper limit of the reported values of k2 does not exceed 3.3 x 10"15 cm3molecule Y 1 at room temperature,237 so the recombination reaction (1) is the main reaction channel. The experimental investigations by Fagerstrom et al.22S have shown that at a total pressure of 1000 mbar of SF6, the reaction is still in the fall-off regime. The fluorine atoms' abstraction reactions, produced by radiolysis of the gas mixture containing SF6 in great excess, can initiate the H-abstraction reactions

Theoretical Description of the Kinetics of Gas-Phase Reactions 207

F + NH3 -» HF + NH2 F + NH2 -» HF + NH

(3) (4)

which must be taken into account in describing the reaction kinetics. A separate set of experiments by Fagerstrom et al.22S was carried out in order to estimate the rate constants for the above reactions. The derived values of k3 = 1.2 x 10"10 cm3molecule"V1 and k4 = 1.2 x 10"10 cnrWlecule's"1 were then used in the kinetic modeling of the reaction system. The experimental results were then analyzed in terms of the theory developed by Troe et al.17'23 Theoretical analysis of the results of Fagerstrom et al. leads to expressions for the limiting rate constants as kj,.

(1.2±0.3)xl0 10 x(T/300) 0 - 27

kiytSFd = (2.0 ± 0.4) x 10-29 x (T/300)-

cm3molecule_1s"' cm6molecule~2s_1

1 2 logpH/mbar

Fig. 34. Theoretical fall-off curves228 and experimental228'231,233'235 results for the recombination NH2 + NH2 obtained at room temperature for other bath-gases.

208

J. T. Jodkowski

The derived value of ki,«, was also used in calculations of ki,o/[M] for other bath-gases M by fitting to available experimental data.231'233'235 Results of these calculations are shown in Fig. 34. The theoretical fall-off curves presented in the figure well reproduce the results of experiments and allow a credible description of the reaction kinetics in a wide pressure range for different diluent gases.

7.2. The reaction NH2 + OH The reaction between amidogen and hydroxyl radicals was found to be a key reaction in the modeling of thermal De-NOx reaction kinetics.8 However, there were no direct measurements of the kinetics of the reaction NH2 + OH until the experiments performed by Fagerstrom et al. 8 The following parallel reaction channels were taken into account in describing the reaction kinetics NH2 + OH (+ M) -> NH2OH (+ M)

(1)

NH2 + OH -> NH + H 2 0

(2)

NH2 + OH -> O + NH3

(3)

All the above reactions are exothermic, but the addition channel seems to be the main route dominant at room temperature and atmospheric pressure. The NH2 and OH radicals were produced by pulse radiolysis of the gas mixture SF6/NH3/H20, with SF6 always in great excess. The fluorine atoms formed initiate the hydrogen atom abstraction reactions F + NH3 -> HF + NH2

(4)

F + H 2 0 -» HF + OH

(5)

F + NH2 -> HF + NH

(6)

The reaction between NH2 and OH radicals proceeds in competition with other reactions OH + OH (+ M) -> H 2 0 2 (+ M) OH + OH -> H 2 0 + O

(7) (8)

which should also be considered in the kinetic description of the reaction system. The potential importance of the other reactions occurring in the reaction system, e.g. OH + NH3, OH + O, and OH + H 2 0 2 , was found to be

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

209

negligible.226 A value of the rate constant ki = 9.1 x 10"n cm3molecule"'s"1 at 298 K has been obtained by Fagerstrom et al?n No pressure dependence of kj was observed in the range of 500 - 1000 mbar of SF6. The theoretical analysis based on the maximum free energy method20 predicts a weak temperature dependence of the high-pressure limiting rate constant, ki,„ as ki,„ = (9.3 ± 1.7) x 10"1' x (T/300)02

cm3molecule"'s"'

This theoretical value is very close to that obtained experimentally. The lack of a pressure effect of the reaction kinetics suggests that the pressure dependence of the rate constant k ; occurs at pressures lower than those used in the experiments of Fagerstrom et al.22S The center of the fall-off curve predicted theoretically is located below 100 mbar of SF6.

7.3. The reaction NH2 + CH3 The combination reaction CH3 + NH2 may be of practical importance for modeling combustion processes due to the role of amidogen radicals in the conversion of fuel-bound nitrogen into N2 or NO.240 However, our knowledge of the reaction kinetics is very limited. All kinetic information is derived from direct measurements by Jodkowski et al.229 They suggest, on thermochemical grounds, that only exothermic reaction channels are possible, resulting in the stabilization of the excited adduct molecule [CH3NH2]*, and two others involving H2-elimination, i.e. yielding HCNH2 + H2 and H2CNH + H2 as final products. However, signals corresponding to HCNH2 and H2CHN were not observed in the reaction products.229 H2-elimination is associated with a sizable energy barrier, so these reactions may be omitted in the reaction mechanism at ambient temperature. The addition reaction NH2 + CH3 (+ M) -> CH3NH2 (+ M)

(1)

proceeds in competition with the other reactions of amidogen and methyl radicals NH2 + NH2 (+ M) -> N2H4 (+ M) NH2 + NH2 -> N2H2 + H2 CH3 + CH3 (+ M) -> C2H6 (+ M)

(2) (3) (4)

210 J. T. Jodkowski The amidogen and methyl radicals were produced by hydrogen atom abstraction from NH3 and CH4 by F atoms, according to F + NH3 -> HF + NH2

(5)

F + CH4 -> HF + CH3

(6)

The decay of fluorine atoms in the other reactions was estimated to be less than 1%.229'241 Pulse radiolysis combined with UV absorption was employed by Jodkowski et al.229 at pressures of 500 - 1000 mbar (M = SF6), while a fastflow system with a quadruple mass spectrometer was applied at pressures in the range of 0.7 - 5.1 mbar (M = He). The influence of other homogeneous reactions on the reaction kinetics was also analyzed by computer simulation and found to be negligible. The rate constant ki = (1.3 + 0.3) x 10"10 cm3molecule"1s"1 at 298 K has been estimated at pressures of 500 - 1000 mbar of SF5.229 The theoretical analysis of the reaction kinetics was based on the approach developed by Troe et al.11'23 The fall-off curves for the reaction CH3 + NH2 are shown in Fig. 35. The predicted fall-off curves are in good

-11.0 0

1 2 logpM/mbar

3

Fig. 35. Fall-off curves and experimental results of Jodkowski et al.2 'for the reaction CH3 + NH2 obtained at room temperature for M ; SF6 and M = He.

Theoretical Description

of the Kinetics

of Gas-Phase Reactions

211

agreement with the results of two sets of experiments performed in different pressure regions and for other bath-gases.229 The best fit of the fall-off curves to experimental results corresponds to the high- and low-pressure limiting rate constants, expressed as ki„ =(1.3±0.3)xl010x(T/300)042

cm 3 moleculeV

k!,o/[He] = (1.8 ±0.5) x 10"27 x (T/300) 35

cm 6 moleculeV

OH, NH2, and CH3 form a series of isoelectronic radicals of importance for atmospheric chemistry. The high-pressure value of 1.3 x 10"10 cm3molecule"'s"1 for the reaction CH3 + NH2 at 298 K229 is one of the highest in the group of isoelectronic radical-radical reactions: 1.4 xlO" 10 (for CH3 + OH)192, 1.2 x 1(T10 (NH2 + NH2)226, 9.3 x 1 0 " (NH2 + OH)226, 5.8 x 10"11 (CH3 + CH3)242'243, and 3.0 x 10 10 cm3molecule"1s"1 (OH + OH)192.

8. Reactions of fluorine atoms Fluorine chemistry in the stratosphere has also been considered and attention has been drawn to the atmospheric chemistry of the FO x radicals. The compounds with O-F bonds have gained interest in connection with the ozone depletion problem. It has been suggested that FO and F0 2 radicals formed in the atmospheric degradation of hydrofluorocarbons (HFCs) could destroy ozone in chain reaction processes. Experimental studies of this hypothesis led to the conclusion that catalytic cycles involving F, FO, and F0 2 are irrelevant with respect to the chlorine cycle.8 However, kinetic investigations of the reactions of fluorine atoms with 0 2 and NOx provide useful information on the fluorine chemistry in the polluted atmosphere.

8.1. The reaction F + NO The role of FNO in atmospheric chemistry is related to the destruction of stratospheric ozone.244"246 The reaction F + NO has been extensively studied using different experimental techniques, including mass spectrometry, chemiluminescence, ESR, IR, and UV spectroscopy.241'247"251 The pressure dependence of the kinetics of FNO formation was recently studied by Pagsberg et al.251 in order to obtain the fall-off curve and the high- and lowpressure limiting rate constants. The reaction was initiated by pulse radiolysis of a SF6/NO gas mixture. In the presence of NO, the decay of the formed

212

J. T. Jodkowski

fluorine atoms takes place by pressure-dependent association reactions in both competing reaction channels F + NO (+ M) -> FNO (+ M)

(1)

F + F (+ M) -> F2 (+ M)

(2)

The experiments of Pagsberg et al.251 were performed at bath-gas pressures (M = SF6) of 10 - 1000 mbar. A strong pressure dependence of the apparent bimolecular rate constants was found in this pressure range. The results obtained by IR and UV spectroscopy were in good agreement and have been included in the theoretical analysis.251 The spectroscopic and molecular properties of F, NO, and FNO are well known. The kinetic analysis of the reaction system was based on the construction of the theoretical fall-off curve. The best fit of the fall-off curve to experiments at 298 K was reached by

0

2 logpM/mbar

Fig. 36. Theoretical fall-off curves and experimental results for the reaction F + NO (+ SF6) -> FNO (+ SF6) obtained at 298 K and 341 K. 2 4 8 ' 2 5 1

Theoretical Description

-

2

-

of the Kinetics

1 0 1 logpM/mbar

of Gas-Phase Reactions

213

2

Fig. 37. Theoretical fall-off curves251 and experimental247'248'251 results for the reaction F + NO (+ M) —> FNO (+ M) obtained at room temperature for other bath-gases.

Pagsberg et al.251 at values of the limiting rate constants of k1)00 = 4.2 x 10"n cm3molecule"1s"1 and kij0/[SF6] = 4.1 x 10~31 cm6molecule~2s \ The pressure dependence of the rate constant for various bath-gases is shown in Figs. 36 and 37. The results of the measurements were obtained at low total pressures, in an almost linear region of the fall-off curve.251 The curvature of the fall-off curve in this pressure range is too small to verify the high-pressure value of ki,„ predicted theoretically. The calculated value of ki,„o may be uncertain by as much as 60%. A more realistic estimation of k ^ needs experimental investigation at total pressures above 10 bar. No dependence on temperature of ki,oo in the range 200 - 400 K is predicted by the maximum free energy method.20 In the temperature range 200 - 400 K the limiting rate constants can be expressed in the form

214

J. T. Jodkowski

k,,.= (4.2!o 5 8 )xlO" n

cm'moleculeV 31

1

ki,o/[SF6] = (4.1 ± 0.8) x 10" x (T/300) °

cm 6 moleculeV

The temperature dependence of kij0/[SF6] derived from experiments at 298 K and 341 K is relatively weak and close to that observed experimentally for CI + NO -> C1NO (ocT1-4)4'5 and I + NO -> INO (°cT 10 ). 17 The values of ki,o/[M] for other bath-gases, obtained by refit to available experimental data, fulfil the inequality Ne < He < Ar < N 2 < C2F6 < C0 2 < SF6 < CF4, which describes the relative efficiencies of different bath-gases observed for termolecular reactions.252

8.2. The reaction F + N02 Results of early experiments on the kinetics of the reaction F + NO were analyzed under the assumption that FN0 2 is the sole reaction product.247 A possible occurrence of FONO isomeric reaction products was discussed in more recent studies.253'254 In a recent investigation by Pagsberg et al.254 the kinetics of the reaction F + N 0 2 was studied with SF6 pressures of 5 - 1000 mbar. No evidence of the formation of the isomeric species FONO was found. The fluorine atoms (produced by pulse radiolysis of a SF 6 /N0 2 mixture) are consumed by the competing reaction channels F + NO z (+ M) -> FN0 2 (+ M)

(1)

F + F (+ M) -> F 2 (+ M)

(2)

The rate constants of the above reactions strongly depend on the bath-gas pressure. Time-resolved IR absorption spectroscopy was used by Pagsberg et al.254 to study the formation kinetics of FN0 2 , whereas, for comparison, the decay of N 0 2 was monitored by UV spectroscopy at 298 K and 341 K. The molecular properties of the reactants and product are known very well. The kinetic analysis was based on the concepts introduced by Troe et al.17'23 and the theoretical fall-off curve was constructed. The pressure dependence of the reaction kinetics corresponding to the fall-off curve shown in Fig. 38 is given by the following values of the limiting high- and lowpressure constants ki,. = (3.5 ± 1.7) x 10"11 x (T/300)0-15

cm3molecule V

k!,o/[SF6] = (1.1 ± 0.5) x 10"31 x (T/300)"24

cm 6 moleculeV

Theoretical Description

0

1

of the Kinetics

2 logpM/mbar

of Gas-Phase Reactions

215

3

Fig. 38. Fall-off curves and experimental results for the recombination F + N0 2 for the other bath-gases obtained at room temperature and at 341 K (insert).

The theoretical fall-off curve well describes the results of measurements. The values ki,o/[He]= (7.2 ±2.2) x 1011 cnAnoleculeV and klj0/[N2] = (9.9 ± 5.0) x 10~31 cm6molecule~V at 298 K were derived by Pagsberg et al.254 from experimental data of Zetzsch,247 and Fasano and Nogar. 5 The calculated high-pressure value k1>M = 3.5 x 10"11 cir^molecule'V1 at 298 K254 can be compared with the reported rate constants for similar addition reactions, e.g. 1.0 x 10"10 for CI + NO z -» C1N02,5 2.7 x 10"" for Br + N 0 2 -» BrN0 2 , 5 and6.6x 10 H cm 3 molecule"V for 1 + N 0 2 - > I N 0 2 . 5

8.3. The reaction F + 02 The kinetics and equilibrium of the addition reaction F + 0 2 t ^ F 0 2 has been the subject of several measurements using various techniques and with

216

J. T. Jodkowski

0A7 OS^ Oft")

different bath-gases and their total pressures. ' " A weak negative temperature dependence of the rate constant for the forward reaction, F + 0 2 was observed at temperatures below 360 K. 2 4 7 ' 2 5 7 ' 2 5 8 Indirect kinetic measurements for the reverse reaction, i.e. dissociation of F0 2 , show a typical positive temperature dependence of the rate constant with an activation energy of 4 6 - 6 7 kJ mol1.263"266 The observed F 0 2 radical decay kinetics may also be used to determine the equilibrium constant. The first direct kinetic measurements of the equilibrium constants for the reaction F + 0 2 i ^ F 0 2 were carried out by Pagsberg et al?62 in the temperature range of 295 - 359 K. The following reactions F + 0 2 (+ M) ±^ F0 2 (+ M)

(1,-1)

F + F 0 2 -> F 2 0 2

(2a)

F + F02 -»F2 + 02

(2b)

F0 2 + F 0 2 (+ M) -» F 2 0 4 (+ M) F 0 2 + F 0 2 -> F 2 + 0 2 + 0 2 F + F (+ M) -> F 2 (+ M)

(3a) (3b) (4)

were taken into account in the computer simulation of the reaction kinetics. The entropic contributions were calculated on the basis of statistical thermodynamics. A Third Law analysis leads to a value of the enthalpy of formation of F 0 2 at room temperature of 26 ± 2 kJmoi"1.262 This corresponds to the F-0 bond strength of 53 kJ mol"1 at 298 K, which is greater than the ClO bond strength of 32 ± 5 kJ mol"1 in C102.267 The thermodynamic properties of F 0 2 obtained in recent investigations by Troe et al.26S are in very good agreement with those estimated by Pagsberg et al.262 Analysis of the experimental data of Pagsberg et al.262 shows that the experiments were made at total pressures, where the kinetics of the association reaction corresponds to the third-order process F + 0 2 (+ M) —> F 0 2 (+ M), with M = Ar. The low-pressure limiting rate constant derived by Pagsberg et al.262

klj0/[Ar] = (4.5 ±0.3) x 10"33 x (T/300) 16

an 6 moleculeV

is in good agreement with the reported values of (3.6 ± 2.5) x 10~33 at 295 K,260-261 4.4 xlO' 3 3 at 300 K,268 (5.8 ± 1.7) x 10 33 at 300 K,258 and (6.1 ± 1.9) x 10~33 cn/molecule'V 1 at 295 K,255 but almost two times lower

Theoretical Description n

1

of the Kinetics

>

1

>

of Gas-Phase Reactions

217

r

-10 ' a> 3 O

-11

a> o

E E

eo

o -12

.* O) o -13

0

1

2

3

4

logpAr/bar Fig. 39. Fall-off curve and results of experimental measurements for the recombination reaction F + 0 2 (+ Ar) at 300 K.268

than the (8.3 ± 1.1) x 10"33 cm6molecule V obtained at 293 K by Chen et al.256 and Chegodaev et al.257 The curvature of the predicted fall-off curve is very weak. Therefore, a realistic estimation of the high-pressure limiting rate constant needs experiments at very high total pressures. Recent kinetic studies268"270 performed at high pressures enable one to estimate a highpressure value of ki,«,= 1.2 x 10"10

cm 3 molecule~y

which may be considered temperature independent in the temperature range 100 - 373 K. The results of kinetic measurements by Troe et al.26s carried out in a wide range of pressures, are shown in Fig. 39. The high-pressure limit was not quite reached, even at the highest pressures of that study.268 The rate constant of 1.9 X 1 0 " cm3molecule"1s"1 measured at Ar pressure of 850 bar is still 6 times lower than the predicted high-pressure limit.

218

J. T. Jodkowski

Acknowledgments I would like to express my great appreciation to Dr. Katarzyna Brudnik, who has been my closest collaborator for many years, for her constructive suggestions and valuable advice concerning the manuscript. A great part of my theoretical investigations discussed in this review could not have been performed without the generous allotment of computer time by The Wroclaw Center of Networking and Supercomputing.

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164. N. Washida, S. Inomata, M. Furubayashi, J. Phys. Chem. A, 102, 7924 (1998). 165. R. Quandt, Z. Min, X. Wang, R. Bersohn, J .Phys. Chem. A, 102, 60 (1998). 166. V. Schmidt, G.Y. Zhu, K.H. Becker, E.H. Fink, Ber. Bunsenges. Phys. Chem. 89, 321 (1985). 167. J. Sehested, L.K. Christensen, O.J. Nielsen, T.J. Nielsen, Intern. J. Chem. Kinet. 30, 921 (1998). 168. G. Inoue, H. Akimoto, J. Chem. Phys. 74, 425 (1981) 169. G. Vasvary, I. Szilagyi, A. Bencsura, S. Dobe, T. Berces, E. Henon, S. Canneaux, F. Bohr, Phys. Chem. Chem. Phys. 3, 551 (2001). 170. M. Wollenhaupt, S.A. Carl, A. Horowitz, J.N. Crawley, J. Phys. Chem. A, 104, 2695 (2000). 171. S. Williams, E. Zingher, J.C. Weisshaar, J. Phys. Chem. A, 102, 2297 (1998). 172. Y. Endo, S. Saito, E. Hirota, J. Chem. Phys. 83, 2026 (1985). 173. H.E. Hunziker, H. Kneppe, H.R. Wendt, J. Photochem. 17, 377 (1981). 174. N.C Baird, R.R. Gupta, K.F. Taylor, J. Am. Chem. Soc. 101, 4531 (1979). 175. M. Dupuis, J.J. Wendoloski, W.A. Lester Jr., J. Chem. Phys. 76, 488 (1982). 176. E.S. Huyser, D. Feller, W.T. Bordon, E.R. Davidson, /. Am. Chem. Soc. 104, 2957 (1982). 177. M. Yamaguchi, T. Momose, T. Shida, J. Chem Phys. 93, 4211 (1990). 178. M. Yamaguchi, Chem. Phys. Lett. 221, 531 (1994). 179. D. Gutman, H.H. Nelson, J. Phys. Chem. 87, 3902 (1983). 180. M. Lorenz, D. Rhase, R. Zellner, B. Fritz, Ber. Bunsenges. Phys. Chem. 89, 341 (1985). 181. L. Zhu, G. Johnson, J. Phys. Chem. 99, 15114 (1995). 182. E. Delbos, P. Devolder, L. ElMaimouni, C. Fittschen, K. Brudnik, J.T. Jodkowski, E. Ratajczak, Phys. Chem. Chem. Phys. 4, 2941-2149 (2002); Erratum: Phys. Chem. Chem. Phys. 4, 3931 (2002). 183. F. Danis, F. Caralp, M.T. Rayez, R. Lesclaux, J. Phys. Chem. 95, 7300 (1991). 184. K. Luther, K. Oum, J. Troe, J. Phys. Chem. A, 105, 5535 (2001). 185. I. Bridier, F. Caralp, H. Loirat, R. Lescaux, B. Veyret, K.H. Becker, A. Reimer, F. Zabel, J. Phys. Chem. 95, 3594 (1991). 186. R. Atkinson, Atmos. Envir. A, 24, 1 (1990) 187. H. Oser, N.D. Stothard, R. Humpfer, H.H. Grotheer, J. Phys. Chem. 96, 5359 (1992). 188. K.J. Hughes, A.R. Pereira, M.J. Pilling, Ber. Bunsenges. Phys. Chem. 96, 1352 (1992). 189. T.J. Sworski, C.J. Hochanadel, P. Orgen, J. Phys. Chem. 84, 129 (1980). 190. C. Anastasi, S. Beverton, T. Ellermann, P. Pagsberg, J. Chem. Soc. Faraday Trans. 87, 2325 (1991). 191. K. Fagerstrom, A. Lund, G. Mahmoud, J.T. Jodkowski, E. Ratajczak, Chem. Phys. Lett. 204, 226 (1993). 192. K. Fagerstrom, A. Lund, G. Mahmoud, J.T. Jodkowski, E. Ratajczak, Chem. Phys. Lett. 22A, 43 (1994).

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W. Tsang, R.F. Hampson, J. Phys. Chem. Ref. Data, 15, 1087 (1986). A.M. Dean, P.R. Westmoreland, Intern. J. Chem. Kinetics, 19, 207 (1987). M.J.T. Jordan, S.C. Smith, R.G. Gilbert, J. Phys. Chem. 95, 8685 (1991). J.F. Bott, N. Cohen, Intern. J. Chem. Kinetics, 23, 1017 (1991). Ch. Dombrowsky, A. Hoffmann, M. Klatt, H.Gg. Wagner, Ber. Bunsenges. Phys. Chem. 95, 1685 (1991). H. Oser, N.D. Stothard, R. Humpfer, H.H. Grotheer, Th. Just, Symp. Int. Combust. Proc. 24, 591 (1992). R. Humpfer, H. Oser, H.-H. Grotheer, Th. Just, Symp. Int. Combust. Proc. 25, 721 (1994). R. Humpfer, H. Oser, H.-H. Grotheer, Intern. J. Chem. Kinet. 27, 577 (1995). R. Forster, M. Frost, D. FuUe, H.F. Hamann, H. Hippler, A. Schlepegrell, J. Troe, J. Chem. Phys. 103, 2949 (1995). D. Fulle, H.F. Hamann, H. Hippler, J. Troe, J. Chem. Phys. 105, 1001 (1996). K. Fagerstrbm, A. Lund, G. Mahmoud, J.T. Jodkowski, E. Ratajczak, Chem. Phys. Lett. 208, 321 (1993). N. Basco, D.G.L. James, R.D. Suart, Intern. J. Chem. Kinetics, 2, 215 (1970). H.E. Van den Bergh, A.B. Callear, Trans. Faraday Soc. 67, 2017 (1971). M.J. Pilling, J.A. Robertson, G.J. Rogers, Intern. J. Chem. Kinetics, 8, 883, (1976). J. Pratt, I. Veltman, J. Chem. Soc. Faraday Trans. I, 72, 2477 (1976). N. Washida, J. Chem. Phys. 73, 1665 (1980). A.B. Vakhtin, A.K. Petrov, Chem. Phys. 149, 427 (1991). J.W. Davies, N.J.B. Green, M.J. Pilling, /. Chem. Soc. Faraday Trans. 87, 2317 (1991). T.J. Wallington, M.M. Maricq, T. Ellermann, O.J. Nielsen, J. Phys. Chem. 96, 982 (1992). J.T. Jodkowski, E. Ratajczak, A. Sillesen, P. Pagsberg, Chem. Phys. Lett. 203, 490 (1993). E.W. Kaiser, J. Phys. Chem. 91, 11681 (1993). M.T. Nguyen, D. Sengupta, L.G. Vanquickenborne, J. Phys. Chem. 100, 10956 (1996). C.P. Fenimore, J.W. Jones, J. Phys. Chem. 65, 1532 (1961). J.C. Guibet, A. VanTiggelen, Rev. Inst. Fr. Petrol, 12,1284 (1963). C. Morley, Combustion Flame, 27, 189 (1976). B.S. Haynes, Combustion Flame, 28, 81 (1977). M.J. Rossi, J.R. Barker, D.M. Golden, J. Chem. Phys. 71, 65 (1979). K. Sugawara, T. Nakanaga, H. Takeo, C. Matsumura, J. Phys. Chem. 93, 1894 (1989). J.S. Francisco, Z. Li, Chem. Phys. Lett. 162, 528 (1989). T.J. Bevilacqua, D.R. Hanson, C.J. Howard, J. Phys. Chem. 97, 3750 (1993). J. Sehested, O.J. Nielsen, C.A. Rinaldi, S.I. Lane, J.C. Ferrero, Intern. J. Chem. Kinet. 28, 579 (1996).

226 J. T. Jodkowski 224. A.B. Vakhtin, Intern. J. Chem. Kinetics, 29, 203 (1997). 225. K.W. Oum, G. Hancock, J. Phys. Chem. 101, 2634 (1997). 226. P. Pagsberg, J.T. Jodkowski, E. Ratajczak, A. Sillesen, Chem. Phys. Lett. 286, 138(1998). 227. C. Breheny, G. Hancock, C. Morrell, Phys. Chem. Chem. Phys. 2, 5105 (2000). 228. K. Fagerstrom, J.T. Jodkowski, A. Lund, E. Ratajczak, Chem. Phys. Lett. 236, 102(1995). 229. J.T. Jodkowski, E. Ratajczak, K. Fagerstrom, A. Lund, N.D. Stothard, R. Humpfer, H.H. Grotheer, Chem. Phys. Lett. 240, 63 (1995). 230. M. Gehring, K. Hoyermann, H.Gg. Wagner, J. Wolfram, Ber. Bunsenges. Phys. Chem. 75, 1287(1971). 231. S. Gordon, W. Mulac, P. Nagia, J. Phys. Chem. 75, 2087 (1971). 232. R.A. Back, T. Yokota, Intern. J. Chem. Kinetics, 5, 1039 (1973). 233. P.V. Khe, J.C. Soulignac, R. Lesclaux, J. Phys. Chem. 81, 210 (1977). 234. P.B. Pagsberg, J. Eriksen, H.C. Christensen, J. Phys. Chem. 83, 582 (1979). 235. V.A. Lozovskii, V.A. Nadtochenko, O.M. Sarkisow, S.G. Cheskis, Kinetics Katal.20, 111(1979). 236. O.M. Sarkisow, S.G. Cheskis, V.A. Nadtochenko, E.A. Sviridenkov, V.I. Vedeneev, Arch. Combustion, 4, 111 (1984). 237. P. Dransfeld, W. Hack, H. Kurzke, F. Temps, H.Gg. Wagner, Symp. Intern. Combustion Proc. 20, 655 (1985). 238. R. Patrick, D.M. Golden, J. Phys. Chem. 88, 491 (1984). 239. S.A Mulenko, Rev. Roum. Phys. 32, 173 (1987). 240. J.A. Miller, C.T. Bowman, Prog. Energy Combustion Sci. 15, 287 (1989). 241. P. Pagsberg, B. Sztuba, E. Ratajczak, A. Sillesen, Acta Chem. Scand., 45, 329 (1991). 242. H. Hippler, K. Luther, A.R. Ravishankara, J. Troe, Z Physik. Chem. 142, 1 (1984). 243. D. Walter, H.H. Grotheer, J.W. Davies, M.J. Pilling, A. Wagner, 23th Symp. (Intern.) on Combustion, 107-114, The Combustion Inst. (1990). 244. T.S. Dibble, J.S. Francisco, J. Phys. Chem. 98, 5010 (1994). 245. T.J. Wallmgton, W.F. Schneider, O.J. Nielsen, J. Sehested, J. Phys. Chem. 98, 10373 (1994). 246. T.S. Dibble, J.S. Francisco, J. Phys. Chem. 98, 10374 (1994). 247. C. Zetzsch, European Symposium on Combustion, Ed. F.S. Weinberg, Academic Press, New York, 1973, p. 35. 248. E.D. Skolnik, S.W. Veysey, M.G. Ahmed, W.E. Jones, Can. J. Chem. 53, 3188 (1975). 249. P. Kim, D.I. MacLean, W.G. Valance, J. Phys. Chem. 84, 1806 (1980). 250. T.J. Wallington, T. Ellermann, O.J. Nielsen, J. Sehested, J. Phys. Chem. 98, 2346 (1994). 251. P. Pagsberg, A. Sillesen, J.T. Jodkowski, E. Ratajczak, Chem. Phys. Lett. 249, 358 (1996).

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Chapter 5: Predicting and Understanding the Signs of One- and Two-Bond Spin-Spin Coupling Constants across X-H-Y Hydrogen Bonds

Janet E. Del Bene" and Jose Elguerob "Department of Chemistry, Youngstown State University Youngstown, Ohio 44555 USA Instituto de Quimica Medica, CSIC, Juan de la Cierva 3, 28006 Madrid, Spain

Abstract One-bond X-H Ch-H) and H-Y ( % - Y ) and two-bond X-Y (2hJX-Y) spinspin coupling constants across X-H-Y hydrogen bonds have been calculated for a variety of hydrogen-bonded complexes formed from the second period elements 13C, 15N, 17 0, and 19F. These calculations, carried out using the ab initio equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method, have led to generalizations concerning the signs of the one- and two-bond reduced coupling constants ^KX-H, lhKH-Y, and 2hKx.Y)- These generalizations make possible the prediction of the signs of the corresponding spin-spin coupling constants (J) measured experimentally, taking into account the magnetogyric ratios of the pair of coupled atoms. Insight into the signs of the reduced coupling constants can be gained through the newlyproposed Nuclear Magnetic Resonance Triplet Wavefunction Model (NMRTWM).

229

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J. E. Del Bene and J. Elguero

1. Introduction There are three spin-spin couplings that can arise between the atoms which form an X-H-Y hydrogen bond. Two are one-bond couplings, the first being coupling between the covalently-bonded X and H atoms ('JX-H). and the second between the atoms H and Y which form the hydrogen bond (lhJH-Y). The third coupling is a two-bond coupling between the hydrogen-bonded X and Y atoms (2hJx-y)- Coupling constants have signs associated with them, which have been determined experimentally for many molecules, and for a few hydrogen-bonded complexes. Until recently, the only model used to predict these signs was the Dirac Vector Model.1 However, as will be illustrated below, this model often fails, even for simple molecules. Our interest in the properties of hydrogen-bonded complexes has led us to investigate by means of ab initio calculations both one- and twobond spin-spin couplings involving hydrogen-bonded atoms. We have also proposed a new model, the Nuclear Magnetic Resonance Triplet Wavefunction Model (NMRTWM),2 to give insight into the signs of these coupling constants. In this chapter we present some of the computed results, make generalizations about the signs of reduced oneand two-bond coupling constants (K),3"5 demonstrate that knowledge of the reduced coupling constants can lead to reliable predictions of the signs of J, and provide insights into these signs using NMRTWM. The first experimental measurements of two-bond spin-spin couplings across hydrogen bonds were carried out in the late 1990s by Dingley and Grzesiak.6 Their landmark paper reported experimental values for N-N coupling constants across the N-H...N hydrogen bonds in the AU and GC base pairs. This paper generated a great deal of interest and excitement in the chemical and biochemical communities, since such measurements were viewed as having the potential for providing structural information about hydrogen-bonded complexes, particularly in solution. Since then, other experimental studies of spin-spin coupling constants across hydrogen bonds have been reported, and these have been summarized in books and review articles.7"14 In particular, the work of Limbach and his associates has provided a rich source of experimental data, particularly for systems with F-H-N hydrogen bonds.15"18 There have also been a number of theoretical studies of coupling constants across hydrogen bonds, carried out using ab initio or density

Predicting and Understanding

the Signs across X-H- Y Hydrogen Bonds

231

functional methods. These studies have also been summarized in various books and review articles.7"14 It is not the purpose of this chapter to review this work, but rather to focus on the results of ab initio EOMCCSD calculations carried out in this laboratory to determine one- and two-bond spin-spin coupling constants for a large number of hydrogenbonded complexes. The EOM-CCSD method has been shown to yield coupling constants in agreement with experimental data, reproducing both signs and magnitudes of coupling constants in isolated A. J 0 0 J

1 0 00~

.—

molecules ' " and hydrogen-bonded complexes. " Despite the fact that it is a computationally demanding method, its reliability makes it the theoretical method of choice. Using this method in systematic studies of hydrogen-bonded complexes has enabled us to formulate generalizations about the signs of one- and two-bond coupling constants for hydrogenbonded complexes, and these generalizations are the primary emphasis of this chapter. The signs of coupling constants will be interpreted in terms of the Nuclear Magnetic Resonance Triplet Wavefunction Model (NMRTWM).2

2. Method of Calculation The structures of monomers and hydrogen-bonded complexes reported in this chapter were obtained at second-order M0ller-Plesset perturbation theory (MP2)28"31 with the 6-31+G(d,p) basis set.3235 All of the monomers and most of the hydrogen-bonded complexes are fully optimized structures with no imaginary frequencies. In addition, some complexes were optimized under certain constraints, such as those with short, symmetric hydrogen bonds, which may be transition structures for proton transfer. The X-Y, X-H, and Y-H distances for all structures are reported along with the corresponding coupling constants. Since complexes with symmetric hydrogen bonds have short X-X distances, including these permits evaluation of coupling constants for a wider range of intermolecular distances. Spin-spin coupling constants were computed for monomers and complexes using the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method in the CI(configuration interaction)-like approximation,3 ~39 with the Ahlrichs40 qzp basis set on C, N, O, and F, qz2p on the hydrogen-bonded H atom, and Dunning's cc-pVDZ basis41'42 on all other H atoms. In the

232

J. E. Del Bene and J, Elguero

nonrelativistic approximation, the total spin-spin coupling constant is the sum of four components: the paramagnetic spin-orbit (PSO), diamagnetic spin-orbit (DSO), Fermi-contact (FC), and spin-dipole (SD) terms. Since spin-spin coupling constants are second-order properties, they are dependent on excited-state wavefunctions. The PSO and DSO terms arise from coupling between the ground state and excited singlet states through the appropriate coupling operators; the FC and SD terms arise from coupling of the ground state to excited triplet states. All four terms were evaluated for the great majority of complexes. However, for certain complexes full calculations are not feasible so only the FC term was evaluated. Approximating the total coupling constant (J) by the FC term was done only when justified by the results of full calculations on similar complexes. Geometry optimizations were carried out using the Gaussian 98 suite of programs,43 and coupling constants were obtained using ACES II.44 All calculations were performed on the Cray SV1 or the Itanium Cluster at the Ohio Supercomputer Center.

3. Results and Discussions 3.1 Two-bond X-Y Coupling Constants JX-Y Our initial studies of spin-spin coupling constants across hydrogenbonded complexes focused exclusively on two-bond couplings (2hJx-v) across X-H-Y hydrogen bonds, where "2h" indicates that the coupling is across two bonds, one of which is a hydrogen bond, J is the coupling constant, and "X-Y" identifies the coupled atoms. Given that the first two-bond coupling constants were measured for N-N coupling in the AU and GC pairs, we computed N-N coupling constants across the N-H...N hydrogen bond in the complex formed between hydrogen isocyanide and hydrogen cyanide (CNH...NCH). To our surprise at the time, the computed N-N coupling constant of 7.2 Hz at the approximate N-N distance in the AU and GC pairs (2.90 A) was in very good agreement with the experimentally-measured coupling constant of about 7 Hz.6 Since the CNH:NCH complex is not one that would be chosen as a model for base pairs, further investigation was warranted. A variety of hydrogen-bonded complexes with sp, sp2, and sp3 N atoms as proton

Predicting and Understanding the Signs across X-H- Y Hydrogen Bonds

233

donors and acceptors were selected for systematic investigation.45 Some of the complexes included are neutral complexes, others are charged; some complexes are stabilized by traditional hydrogen bonds, others by proton-shared hydrogen bonds. (The characteristics of traditional and symmetric or quasi-symmetric proton-shared hydrogen bonds, and the relationship between hydrogen bond type and two-bond spin-spin coupling constants has been discussed previously.46) The coupling constants for a subset of these complexes are reported in Table 1. It is important to note that for complexes stabilized by N-H-N hydrogen bonds, JN.N is dominated by the Fermi-contact term. This dominance is not a result of the cancellation of other terms, but arises because the FC term is more than an order of magnitude greater than any other term. It is possible to correlate the computed coupling constants 2hJN-N to N-N distances, as illustrated in Fig. 1. Each point on this graph represents the value of the N-N coupling constant at the optimized N-N distance found in a particular complex. Does this correlation imply that JN-N is independent of the hybridization of the hydrogen-bonded N atoms? This question cannot be answered by a simple yes or no. Obviously, a single curve correlates coupling constants for two N atoms irrespective of their hybridization, which suggests that 2HJN-N is independent of hybridization. However, hybridization is an important factor in determining the intermolecular distance, and this distance does correlate with 21IJN-N- Hence, there is a correlation between hybridization and 21IJN-N5 but it is an indirect one. The value of 2IIJN-N obtained from the curve shown in Fig. 1 at a distance of 2.90 A is 7.5 Hz, in agreement with the value of 7.2 Hz obtained for the CNH:NCH complex, and with the experimental value of about 7 Hz for the base pairs. Thus, this curve should be useful for extracting N-N distances from experimentally measured N-N coupling constants across N-H-N hydrogen bonds. A second feature of Fig. 1 that is of particular importance in the context of this chapter is that all of the N-N coupling constants are positive. What are the signs of coupling constants between other pairs of hydrogen-bonded second-period atoms? Table 1 provides coupling constant data for a subset of complexes stabilized by C-H-N hydrogen bonds.47 All of the complexes investigated in ref. 47 are stabilized by traditional C-H...N hydrogen bonds except for two cationic complexes, pyridinium:CNH and NH4+:CNH, which are stabilized by traditional N-H+...C hydrogen bonds. From Table 1 it can be seen that the FC term

234

J. E. Del Bene and J. Elguero

2h

T dominates and determines JC-NSince the FC term is distance 2h Z dependent, "JCN also depends on the C-N distance. The FC terms and 2h JC-N for a set of twenty-three complexes are plotted in Fig. 2 as a function of the C-N distance. The best-fit second order curves relating the FC terms and 2hJC-N to the C-N distance are superimposable. Moreover, it can be seen from Fig. 2 that all 2hJc-N have negative signs.

Table 1. Computed Fermi contact terms, total coupling constants 2hJX-Y (Hz), reduced coupling constants [2hKX-y (x 1019) N A"2 m"3], and X-Y distances (A) for X-H-Y hydrogen bonds with X, Y = 13C, ,5N, 170, and 19Fa. FC C-H-N hydrogen bonds NCH:NCH NCH:pyridine NCH:NH3 HNCH+:NCH NCH:NC

2hj

2h

\c J^X-Y

R(X-Y)

-7.3 -12.7 -11.0 -40.0 -24.7

-7.3 -12.7" -11.0 -40.1 -24.8

+23.8 +41.5 +35.9 + 130.9 +81.0

3.316 3.163 3.204 2.832 2.940

N2H5" Pyridinium:NCH (HCN)2H+ CNH:NCH CNH:pyridine CNH:NH3

12.9 17.2 10.4 8.2 32.5 5.6 10.7 8.7

12.9 17.2 10.3 8.2" 32.6 5.6 10.7" 8.7"

+ 104.5 +139.4 +83.5 +66.4 +264.1 +45.4 +86.7 +70.5

2.705 2.597 2.608 2.872 2.521 2.996 2.793 2.846

O-H-N hydrogen bonds HCNH+:OC HOH:NCH HOH:NC H2OH+:NCH

11.3 1.1 6.6 34.1

11.3 1.1 6.6 34.1

+68.4 +6.7 +40.0 +206.5

2.785 3.126 2.820 2.471

F-H-N hydrogen bonds FH:NHF2 FH:NCLi FH:pyridine Pyridinium.FH NH4+:FH HCNH+:FH

-16.1 -47.5 -57.0 -26.9 -28.7 -94.5

-17.9 -47.8 -57.0" -26.9" -28.4 -94.3

+15.6 +41.7 +49.7 +23.5 +24.8 +82.3

2.865 2.660 2.611 2.882 2.813 2.603

N-H-N hydrogen bonds N2H7+ (C3v) (DM)

Predicting and Understanding the Signs across X-H- Y Hydrogen Bonds 235 Table 1 (continued).

FC

2hT Jx-Y

K-X-Y

R(X-Y)

C-H-O hydrogen bonds F2HCH:OCH2 F2HCH:OH2 HNCH+:OC NCH:OC

-1.6 -5.4 -13.9 -2.7

-1.6 -5.5 -14.0 -2.7

+3.9 +13.4 +34.2 +6.6

3.142 3.338 3.079 3.478

O-H-O hydrogen bonds 02H5+ (CO)2H+ 02H3 (H 2 0) 2 (H2CO)2H+

39.9 42.9 18.0 1.5 21.0

39.5 42.8 16.3 1.3 21.0 b

+178.8 +193.7 +73.8 +5.9 +95.1

2.385 2.394 2.443 2.914 2.409

F-H-O hydrogen bonds FH:OC FH:OH2 COH+:FH FH:OCH2 HFH+:OC H2COH+:FH H2OH+:FH

0.9 -22.1 -176.0 -13.5 -176.1 -35.0 -73.6

1.3 -18.4 -173.7 -12.2 -173.8 -33.3 -71.0

-0.8C +12.0 +113.3 +8.0 +113.4 +21.7 +46.3

3.051 2.661 2.362 2.670 2.362 2.574 2.523

C-H-F hydrogen bonds NCH:FH HNCH+:FH OCH+:FH FH:CO

31.9 140.5 225.7 31.8

32.0 141.5 228.3 33.7

+11.3 +49.8 +80.3 +11.9

3.216 2.862 2.713 3.054

a) b) c)

(HF)2 is not included in this table. The value of 2hJx-y has been estimated from the FC term. The signs of the FC term and 2 h J F 0 become positive at shorter F-O distances.

236

J. E. Del Bene and J. Elguero

14 •

10

2.6

2.8

3.0

2.9

3.1

3.2

N-N (A)

Fig. 1. JN.N versus the N-N distance for neutral and cationic complexes with N-H-N hydrogen bonds. -45 -40 -35 -30 -25 -20 -15 -10 -

-5 0 2.8

2.9

3.0

3.1

3.2

3.3

3.4

3.5

C-N (A)

Fig. 2. JC„N versus the C-N distance for neutral and cationic complexes with CH-N hydrogen bonds.

Predicting and Understanding the Signs across X-H- Y Hydrogen Bonds

237

-120

-40

2.5

2.8

N-F (A)

Fig. 3. 2hJF_N versus the F-N distance for neutral and cationic complexes stabilized by F-H-N hydrogen bonds. Squares: cations; Triangles: neutrals. Correlating 2HJF-N to F-N distances for complexes stabilized by F-HN hydrogen bonds requires that the complexes be subdivided into cationic complexes with N-H+-F hydrogen bonds, and neutral complexes with F-H-N hydrogen bonds.48'49 Table 1 lists the Fermi-contact terms and JF-N for a subset of these complexes. Figure 3 shows the correlation between the computed F-N coupling constants and F-N distances for the neutral and the cationic complexes. As evident from Fig. 3, there is more scatter in these data than observed for the curves relating N-N and C-N coupling constants to N-N and C-N distances, respectively. Nevertheless, an important feature of these curves is that all F-N coupling constants are negative. It is also interesting to note the significant overlap in the range of F-N distances for neutral complexes with F-H...N hydrogen bonds and cationic complexes with N-FT...F hydrogen bonds. At the same F-N distance, 2hJF-N is significantly greater for a cationic complex compared to a neutral one, since the degree of proton-shared character is greater in the cationic N-H + ...F hydrogen bond.48'49 Table 1 also contains X-Y distances, the FC terms, and 2hJx-Y for selected complexes stabilized by traditional or proton-shared O-H-N, C-H-O, O-H-O, F-H-O, and C-H-F hydrogen bonds. The signs of the FC

238

J. E. Del Bene and J. Elguero

terms and total coupling constants may be positive or negative, but are always the same for complexes with the same X and Y. How can the signs of these coupling constants be understood? It is well-known that the proper way for comparing coupling constants involving different atoms is to use reduced coupling constants (2hKx_Y).50 2h Kx_Y is proportional to 2hJx_Y, 2h

Kx_Y «

2h

JX-Y/(yx)(YY)

(1)

where, y x and yy are the magnetogyric ratios of X and Y. The signs of y x are positive for 13C and 19F, and negative for 15N and 1 7 0. When the dependence of 2hJX-Y on the signs of y x and yY is removed, all reduced two-bond coupling constants 2hKx.y (as well as reduced FC terms) for X-Y coupling across an X-H-Y hydrogen bond are positive,3 as evident from Table 1. This is a very important generalization, for in practice it permits the prediction of the signs of experimentally measured coupling constants, 2hJX-Y- 2hJx-Y will be positive if neither or both X and Y have negative magnetogyric ratios; 2hJX-y W1H D e negative if either X or Y has a negative magnetogyric ratio. Notably absent from Table 1 are F-H-F hydrogen bonds. Both 2hJF.F and 2hKF.F for the equilibrium structure of (HF)2 are negative, and are thus exceptions to the generalization above. However, both the distance and angular dependence of 2hJF.F are unique, and the FC term is not a good approximation to the total coupling constant. A detailed discussion of two-bond coupling constants for this unusual complex can be found in a previous paper.51 The reduced FC terms and reduced coupling constants for F-F coupling in both FHF" and F2H3+ are large and positive, but accurate estimates of 2hJF.F require that other terms contributing to the total coupling constant be evaluated. Hence, their exclusion from this analysis. It should also be noted that the FH:OC complex is an apparent exception to the generalization that 2hKx.Y is positive. Howver, the value of the FC term for F-0 coupling is positive (the reduced FC term is negative) but less than 1 Hz at equilibrium, becomes zero when the equilibrium F-0 distance of 3.05 A decreases to 2.95 A, and then becomes negative at shorter distances, with a value of -10.3 Hz at a distance of 2.65 A. The distance dependence of the FC term for this complex is similar to that observed for (HF)2, which has a negative FC term at equilibrium but positive FC terms at shorter F-F distances.

Predicting and Understanding the Signs across X-H- Y Hydrogen Bonds

239

How well does the prediction that all reduced two-bond spin-spin coupling constants across hydrogen bonds are positive compare with experimental data? Unfortunately, the number of experimental measurements of such couplings in hydrogen-bonded complexes is limited. However, those that are available are consistent with this prediction. The N-N coupling constants for the AU and GC pairs6 and molecules with intramolecular N-H-N hydrogen bonds are positive.52"55 F-N coupling constants have been measured in various systems as a function of temperature for complexes with F-H-N hydrogen bonds by Limbach et al.15'18 2HJF-N is negative, which means that 2hKF.N is positive, in agreement with the above generalization. It is hoped that additional experimental data on the signs of two-bond coupling constants will become available soon, so that this prediction can be further tested. 3.2 One-bond X-H Coupling Constants JXH Although our initial work with spin-spin coupling constants in hydrogenbonded complexes focused on two-bond couplings, the two one-bond couplings between X and H (!JX-H) and H and Y (1HJH-Y) have also been investigated.4'5 To evaluate the reliability of the EOM-CCSD method for these studies, one-bond X-H coupling constants for a set of monomers that are potential C-H, N-H, O-H, or F-H proton donors in hydrogenbonded complexes were computed and compared with experimental values.56"59 The experimental values of 'JX-H for donors are plotted against the computed values in Fig. 4. The equation of the line is 'JX-H (exptl) = (1.02 ±0.02) lJx_n (calc)

(2)

with a correlation coefficient of 0.997. These data illustrate that the agreement between theory and experiment is really quite good, particularly since the coupling constants extend over a range of approximately 700 Hz. It is significant that data points for 15N-!H and n O- 1 H coupling appear in the lower-left quadrant of Fig. 4, indicating that the computed and experimental signs of 'JN-H and 'J 0 -H are negative. Points for 13C-1H and 19F-!H coupling appear in the upper right quadrant of Fig. 4, corresponding to positive values of both computed and experimental coupling constants. Since the magnetogyric ratios of 13C, 19 F, and 'H are positive and those of 15N and 17 0 are negative, 'KX-H is positive for C-H, N-H, O-H, and F-H. Does ^X-H remain positive when

240

J- E. Del Bene and J. Elguero

X-H is a proton donor, particularly if the X-H bond is significantly stretched in a complex with a proton-shared hydrogen bond? 600

-200 -200

1

-100

J 0

100

200 1

300

400

500

600

J X 4 ( (calc) (Hz)

Fig. 4. Experimental versus computed values of the one-bond X-H spin-spin coupling constants 'JX-HTable 2. Computed X-H distances (A), X-'H Fermi contact terms and total coupling constants 'J X .H (HZ), and reduced coupling constants ['^X-H ( X 1019) N A"2 rn 3 ] for X-H-Y hydrogen bonds with C = 13C, 15N, n O , and 19F. R(X-H) Complexes with C-H donors NCH 1.067 NCH:OC 1.067 NCH:FH 1.069 NCH:NCH 1.072 NCH:NH3 1.081 NCH:pyridine 1.082 NCH:NC" 1.110 NCH:CN-(C„V) 1.114 NCH:CN (D^) 1.391

FC

'JX-H

'KX.H

251.4 252.2 253.8 254.4 252.7 251.9 248.6 248.0 99.9

251.7 252.5 254.1 254.7 253.0 251.93 249.3 248.7 101.3

83.3 83.6 84.1 84.3 83.7 83.4 82.5 82.3 33.5

Predicting and Understanding

the Signs across X-H- Y Hydrogen Bonds

Table 2 (continued). R(X-H)

FC

F2HCH:OCH2 F2HCH:OH2

1.086 1.083 1.084

HNCH+ HNCH+:OC HNCH+:FH HNCH+:CO HNCH+:NCH HNCH+:CNH (C w ) HNCH+:CNH (DroA)

Jx-H

KX-H

171.3 178.5 180.8

172.0 178.5a 181.5

56.9 59.1 60.1

1.079 1.086 1.091 1.100 1.129 1.146 1.376

332.5 327.4 324.8 321.0 299.2 288.6 130.8

332.5 327.7 325.3 321.6 300.1 289.9 132.7

110.1 108.5 107.7 106.5 99.3 96.0 43.9

OCH+ OCH+:OC OCH+:CO (C w ) OCH+:FH

1.091 1.109 1.160 1.121

345.0 330.3 292.6 319.5

345.6 331.4 294.3 320.7

114.4 109.7 97.4 106.2

Complexes with N-H CNH CNH:NCH CNH:NH3 CNH:pyridine CNH:NC" (Co„v) CNH:NC (Dmh)

1.000 1.012 1.035 1.040 1.141 1.268

-114.4 -114.9 -109.2 -107.4 -79.2 -43.1

-115.8 -115.7 -109.2a -107.4a -79.5 -43.4

95.1 95.0 89.7 88.2 65.3 35.6

Pyrrole Pyrrole:NCH Pyrrole:NH3

1.007 1.011 1.021

-90.8 -93.6 -93.7

-93.4 -93.6 -93.7a

76.7 76.9 77.0

NH3 NHJiNHz^C,) NH3:NH2-(C2)

1.011 1.052 1.304

-58.4 -63.2 -22.0

-61.6 -64.4 -22.0

50.6 52.9 18.1

HCNH+ HCNH + :OC HCNH+:FH (HCN)2H+ (£>„„)

1.017 1.033 1.046 1.261

-149.0 -142.0 -136.7 -56.6

-149.3 -142.2 -136.9 -56.9

122.6 116.8 112.4 46.7

CF2H2

241

242

J. E. Del Bene and J. Elguero

Table 2 (continued). R(X-H)

FC

Pyridinium Pyridinium:FH Pyridinium:NCH Pyridium:CNH

1.017 1.021 1.035 1.042

-91.6 -93.2 -92.0 -90.7

-91.6a -93.2a -92.0a -90.7 a

75.2 76.5 75.6 74.5

NH/ NH4+:OC NH4+:FH NH4+:CO NH4+:NCH NH4+:CNH NH4+:NH3 (C3v) NH4+:NH3 (D3d)

1.022 1.027 1.029 1.035 1.049 1.057 1.113 1.299

-73.4 -74.4 -74.5 -73.7 -72.7 -71.2 -60.5 -26.4

-75.0 -75.8 -75.9 -75.0 -73.7 -72.1 -61.2 -26.5

61.6 62.3 62.3 61.6 60.5 59.2 50.3 21.8

Complexes with O-H donors H2Q HOH:NCH HOH:OH2 HOH:NC" HOH:OH(C ; ) HOH:OH(C 2 )

0.963 0.967 0.970 0.992 1.096 1.222

-65.9 -70.2 -71.6 -74.2 -49.5 -19.9

-77.9 -80.3 -81.0 -80.5 -51.6 -20.1

47.8 49.3 49.7 49.4 31.7 12.3

COH+ (CO)2H+ (D^)

0.997 1.197

-180.6 -68.0

-183.7 -68.2

112.8 41.9

H2COH+ H2C02H+:FH (H2CO)2H+(C2/l)

0.985 1.006 1.205

-86.2 -85.1 -27.7

-90.3 -87.7 -27.8

55.4 53.8 17.1

HsQ!

H2OH+:FH H2OH+:CO H2OH+:NCH H2OH+:OH2 (C2)

0.980 1.011 1.039 1.134 1.194

-115.1 -106.8 -97.2 -64.7 -47.5

-120.7 -110.1 -99.6 -65.1 -47.5

74.1 67.6 61.1 40.0 29.2

Complexes with F-H donors FH FH.CO FH:NCH FH:OC

0.926 0.922 0.927 0.928

309.3 353.4 370.2 323.9

495.4 518.6 516.7 497.6

43.8 45.9 45.7 44.0

J

X-H

Kx-H

Predicting and Understanding the Signs across X-H-Y Hydrogen Bonds

243

Table 2 (continued). R(X-H)

FC

Jx-H

Kx-H

FH.FH FH:OCH2 FH:OH2 FH:NCLi FH:NH3 FHF"

0.932 0.943 0.943 0.955 0.963 1.150

338.1 341.0 349.0 356.4 325.5 82.1

497.8 476.8 482.8 470.1 431.5 101.2

44.0 42.2 42.7 41.6 38.2 9.0

FHj4 HFH+:OC HFH+:FH (C2h)

0.969 1.093 1.151

556.6 299.4 186.3

631.4 315.8 195.3

55.8 27.9 17.3

a) Estimated from the Fermi-contact term.

Table 2 presents computed X-H distances, FC terms, 'JX-H, and 'K X -H for complexes in which the X-H donors are C-H, N-H, O-H, and F-H. The data of Table 2 indicate that the FC term is an excellent approximation to 'JX-H for complexes in which X-H is either C-H or N-H. This approximation is not as good for complexes with O-H as the donor, especially when the donor molecule is H 2 0. The FC term is a poor approximation to ^F-H, especially for complexes in which the neutral HF molecule is the proton donor. Thus, it appears that the dominance of the FC term decreases as the number of lone pairs of electrons on X increases. This observation is consistent with that made by Pecul et al,60 who noted that the performance of DFT calculations of coupling constants deteriorates as the number of electron pairs on the coupled atoms increases. Values of the reduced one-bond X-H coupling constants for monomers and complexes are plotted against X-H distances in the scattergram shown in Fig. 5. There are two very striking observations that can be made from this figure. The first is that there appears to be little if any correlation between reduced X-H coupling constants and X-H distances. The second is that all reduced one-bond X-H coupling constants ('K X -H) are positive, irrespective of the nature of the hydrogen bond or the charge on the complex.

244

J. E. Del Bene and J. Elguero

Q

a 4

Q

0

a o Ri 4

$>o

<£k

^ P =ta a

d°*aa

o 41

a

<> , , , , ft. , 0.90

0.95

1.00

1.05

1.10 R(X-H)(A)

1.15

1.20

1.25

1.30

Fig. 5. Scattergram for 'K X -H versus the X-H distance for monomers and complexes. Circles: C-H; Squares: N-H; Triangles: O-H; Diamonds: F-H.

1.04

1.05

1.06

1.07

1.08

1.09

1.1

1.11

1.12

R(N-H) (A)

Fig. 6. One-bond N-H coupling constants ' j N . H versus the N-H distance for the stretched N-H bond in NH4+ and the hydrogen-bonded N-H bond in complexes with NH4+. Diamonds: NH4+; Squares: Complexes.

Predicting and Understanding

the Signs across X-H- Y Hydrogen Bonds

245

In a recent paper, Cremer noted that there have been many attempts to correlate KX-H to monomer X-H distances, but these have been unsuccessful.61 Our attempt to find such a correlation in hydrogenbonded complexes began with the recognition that 'JX-H in complexes is strongly dependent on the hydrogen bond itself and hydrogen bond type. This is illustrated in Fig. 6, in which 'JN.H for a stretched N-H bond in an isolated NH4+ cation is compared to 'JN.H for stretched N-H bonds in complexes with NH4+ as the proton donor. From Fig. 6 it can be seen that 'JN-H in the isolated cation is nearly constant over the relatively short range of N-H distances considered, decreasing only slightly in absolute value as the N-H distance increases from 1.022 to 1.113 A. The near constancy of 'JN.H reflects the absence of an explicit distance-dependent term in the Fermi-contact operator.62 In contrast, 'JN.H for the complexes decreases dramatically as the N-H distance increases from 1.027 A in NH4+:OC to 1.113 A in N2H7+. Why are the N-H coupling constants in the monomer and complexes different when the two N-H distances are the same? It is well-known that hydrogen bond formation changes ground-state electron densities on hydrogen-bonded atoms, increasing these densities on X and Y, and decreasing the density on the hydrogenbonded proton. It can therefore be inferred that the lower ground-state electron densities at the hydrogen-bonded proton are responsible in part for the decrease in 'JN.H in hydrogen-bonded complexes. Moreover, the electron density at the hydrogen-bonded proton decreases as the N-H distance and the degree of proton-shared character of the hydrogen bond increase, as evident from chemical shift data.4'15'16'18 In an attempt to reduce the dependence of 'KX-H on the particular monomer, plots of the ratio Kc/Km versus Rc/Rm were constructed, where Kc and Km are the reduced X-H coupling constants in the complex and corresponding monomer, respectively, and Re and Rm are the X-H distances in the complex and monomer, respectively. If these plots are constructed separately for complexes with C-H, N-H, O-H, and F-H donors, then the slopes and intercepts of the resulting first-order curves are found to increase as the electronegativity of X increases. Plots of the slopes and intercepts of these lines versus the Pauling electronegativity are shown in Fig. 7. As a next step, the monomer dependence of coupling constants in complexes was further reduced by subtracting from the reduced coupling constants and distances of complexes the coupling constants and distances, respectively, of the corresponding monomers.

246

J. E. Del Bene and J.

Elguero

When the slopes of the four curves that relate [(Rc - R J / R J to [(Kc Km)/Km] for complexes with C-H, N-H, O-H, and F-H donors are plotted against the square of the Pauling electronegativity, a linear relationship is found with a correlation coefficient of 0.999.4 Thus, the very wellknown relationship between coupling constants and electronegativity is apparent.61'6368 5.5-,

5

4.5

4

|

3.5

3 3

2.5

2

1.5

2.25

2.45

2.65

2.85

3.05 3.25 3.45 Pauling electronegativity

3.65

3.85

4.05

4.25

Fig. 7. Slopes and intercepts of straight lines that relate reduced X-H coupling constants to X-H distances plotted against the Pauling electronegativity of X for X-H = C-H, N-H, O-H, and F-H. Diamonds: Slopes; Squares: Intercepts. Figure 8 presents a plot of [(Jc - JJ/J m ] versus [(Rc - Rm)/Rm] x P2, where P2 is the square of the Pauling electronegativity, for 16 different X-H proton donors and 64 hydrogen-bonded complexes with these donors. There are two remarkable features of this plot. The first is that a linear relationship exists between these variables, with a correlation coefficient of 0.97. This relationship is dramatic, given that the raw data

Predicting and Understanding the Signs across X-H- Y Hydrogen Bonds

247

exhibit such scatter in Fig. 5. The second feature is the replacement of the reduced coupling constant ('K X -H) by the coupling constant which would be measured experimentally ('JX-H)- This replacement is possible because it is the dimensionless ratio [(Kc - K^/K^] which is plotted against [(Rc - Rm)/RnJ x P2, and the quantity [(Kc - K^/KjJ is equal to [(Jc - Jm)/JnJ- That is, plotting the ratio removes the dependence of the coupling constant on the magnetogyric ratios of X and H. Thus, for the first time, the one-bond coupling constants 'JX-H for different X in complexes with C-H, N-H, O-H, and F-H proton donors have been directed related.

f{Rcomp-Rmon)/Rmon]*(P"2)

Fig. 8. ['JX-H (complex) - 'JX-H (monomer)]/1 JX-H (monomer) versus {[RX-H (complex) - RX-H (monomer)]/RX-H (monomer)} x P2, where P2 is the square of the Pauling electronegativity.

248

J. E. Del Bene and J. Elguero

3.3 One-bond H-Y Coupling Constants

JY-H

3.3.1. Complexes with traditional hydrogen bonds Table 3 presents X-Y distances, H-Y distances, FC terms, lhJn-Y and lh KH-Y for complexes with N, O, or F as the proton-acceptor atom, and C-H, N-H, O-H, and F-H as the proton donors. The 19 complexes with traditional N-H...N hydrogen bonds comprise by far the most extensive set. The proton acceptors include nitrogen atoms that are sp (HCN), sp2 (pyridine and pyrazine), and sp3 (NH3) hybridized. The FC term approximates the H-N coupling constant (lhJH-N ) quite well for these complexes, with the largest difference being 0.5 Hz for FH:NCH. However, the values of lhJH-N are relatively small, varying between 1 and 5 Hz. These small coupling constants are not surprising in view of the long H-N distances. However, the most interesting feature of these coupling constants is that all lhJH-N values are positive. Since the magnetogyric ratio of 'H is positive and that of 15N is negative, all reduced H-N coupling constants (lhKH-N) across X-H-N hydrogen bonds are negative, as seen in Table 3. The proton acceptors in complexes with X-H...O hydrogen bonds are CO, H2CO, and H 2 0, which have O atoms that are sp, sp2, and sp3 hybridized, respectively. The 6 complexes with CO as the proton acceptor have linear X-H...0 hydrogen bonds, while those with H2CO and H 2 0 have hydrogen bonds that deviate from linearity, but by less than 10°. The FC term is a good estimate of lhJH-o, although the values of JH-o are relatively small. Nevertheless, all one-bond H-O coupling constants are positive. Since 17 0 has a negative magnetogyric ratio, the reduced coupling constants lhKH-o are again negative, as seen in Table 3. Complexes stabilized by X-H...F hydrogen bonds have FH as the proton acceptor molecule. Once again, the FC term is a good approximation to lhJH-F- The H-F coupling constants for the 7 complexes in this group vary from -10 to -71 Hz. Since *H and 19F have positive magnetogyric ratios, both lhJH-F and lhKH.F are negative. Thus, all reduced one-bond H-Y coupling constants (lhKH-Y) for complexes with traditional hydrogen bonds are negative.5

Predicting and Understanding

the Signs across X-H- Y Hydrogen Bonds

249

Table 3. Computed X-Y and Y-H distances (A), Fermi-contact terms and coupling constants lhJH-Y (Hz), and reduced coupling constants [lhKH.Y (x 1019) N A"2 m"3] for complexes with traditional X-H-Y hydrogen bonds with Y = 15N, l7 0, and 19F. lh

R(X-Y)

R(H-Y)

FC

%_ Y

KH-Y

X-H-N hydrogen bonds NCH NCH:NCH 3.316 F3CH:NCH 3.456 HNCH+:NCH 2.832 CNH:NCH 2.996 Pyrrole:NCH 3.160 Pyridinium:NCH 2.872 Pyrazinium:NCH 2.833 H3NH+:NCH 2.830 HOH:NCH 3.126 FH:NCH 2.817

2.244 2.373 1.704 1.984 2.149 1.836 1.792 1.781 2.159 1.879

2.1 1.3 3.2 3.3 2.5 5.1 5.0 4.9 2.7 4.2

1.8 1.3' 3.5 2.9 2.5' 5.1' 5.0' 4.6 2.4 3.7

-1.5 -1.1 -2.9 -2.4 -2.1 -4.2 -4.1 -3.8 -2.0 -3.0

Azines NCH:pyridine CNH:pyridine FH: pyridine FH:pyrazine

3.163 2.793 2.611 2.638

2.080 1.753 1.644 1.678

3.4 2.8 2.1 2.9

3.4' 2.8' 2.1' 2.9'

-2.8 -2.3 -1.7 -2.4

NHs NCH:NH3 F3CH:NH3 CNH:NH3 Pyrrole:NH3 FH:NH3

3.204 3.341 2.846 3.036 2.637

2.123 2.255 1.811 2.015 1.673

2.8 1.9 3.1 3.1 2.9

2.7 1.9' 3.1' 3.1' 2.6

-2.2 -1.6 -2.5 -2.5 -2.1

X-H—O hydrogen bonds CO NCH:OC 3.478 OCH+:OC 2.921 HNCH+:OC 3.079 HCNH+:OC 2.785 H3NH+:OC 3.035 FH:OC 3.051

2.411 1.812 1.993 1.752 2.008 2.124

0.7 6.0 4.1 4.5 3.8 2.5

0.5 5.9 3.9 4.3 3.5 2.1

-0.3 -3.6 -2.4 -2.6 -2.1 -1.3

250

J. E. Del Bene and J. Elguero

Table 3 (continued). ih T J H-Y

R(X-Y)

R(H-Y)

FC

H,CO FH:OCH2

2.670

1.743

5.6

5.0

-3.1

H,0 HOH:OH2 FH:OH2

2.914 2.661

1.946 1.718

4.9 7.3

4.7 7.2

-2.9 -4.4

X-H--F hydrogen bonds HF NCH:FH 3.216 OCH+:FH 2.713 HNCH+:FH 2.862 HCNH+:FH 2.603 H3NH+:FH 2.813 H2OH+:FH 2.523 FH:FH 2.777

2.148 1.601 1.778 1.561 1.808 1.514 1.856

-9.9 -68.0 -47.0 -50.3 -42.3 -55.1 -28.7

-10.0 -70.8 ^7.6 -52.3 ^11.6 -55.6 -26.9

-0.9 -6.3 -4.2 -A.6 -3.7 -4.9 -2.4

r>-H-Y

a) Estimated from the Fermi-contact term.

3.3.2. JH-Y for complexes with proton-shared hydrogen bonds For complexes with traditional hydrogen bonds, 'KX-H is positive and lh KH-Y is negative. Therefore, there must be a change in the signs of the X-H and H-Y coupling constants as the proton is transferred from X to Y and the hydrogen-bonded ion-pair X"...+H-Y is formed. Such sign changes have been observed experimentally by Limbach and his coworkers in studies of the temperature dependence of F-H and H-N coupling constants for the FHxollidine complex.151618 A previous EOM-CCSD investigation of F-H and H-N coupling constants in model systems FH:NH3 and FH:pyridine supported the experimental findings that these coupling constants change sign as the proton is transferred from F to N.27 Sign changes for one-bond coupling constants that occur as a proton is transferred from X to Y can be most easily illustrated for a cationic complex in which the two molecules that compete for the proton are identical, using N2H7+ (H3Na...H+...NbH3) as an example. Although the equilibrium structure of N2H7+ has C3v symmetry, there is a transition

Predicting and Understanding the Signs across X-H-Y Hydrogen Bonds

251

structure of D3d symmetry that has a symmetric Na...H...Nb hydrogen bond. The two N-H distances in the D3d structure are 1.299 A , and the two N-H coupling constants ('JNB-H) and (lhJH-Nt>) have equal values of -26.5 Hz, indicating that both 'KN^H and lhKH-Nb are positive. How these coupling constants vary as the proton is transferred from one nitrogen (Na) to the other (Nb) is illustrated in Fig. 9. At an Na-H distance of 1.00 A , "JNa-n i s -73.2 Hz and lhJH-Nb is +3.3 Hz. As the proton moves from Na to Nb, lhJH-Nb becomes 0.0 Hz before becoming negative and equal to 'jNa-H when the hydrogen bond is symmetric. As the proton continues to move toward Nb, the Nb-H coupling constant acquires a larger negative value and the Na-H coupling constant eventually becomes positive as Nb-H becomes the proton donor and Na the proton acceptor. It is interesting that lhJH-Nb changes sign near the equilibrium Na-H distance of 1.113 A. This explains the very small value of -0.3 Hz for lh JH-Nb for the equilibrium C3v structure of N2H7+. It is also consistent with the difficulties of measuring H-Y coupling constants experimentally. Table 4 provides data illustrating that 'KX-H and lhKH-Y are both positive for complexes stabilized by symmetric or quasisymmetric proton-shared hydrogen bonds.

Na-H (A)

Fig. 9. The variation of the Fermi-contact term for Na-H and H-Nb coupling in N2H7+ (H3Na-H...NbH3)+ as a function of the Na-H distance. Circles: FC term for Na-H; Squares: FC term for H-Nb.

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J. E. Del Bene and J. Elguero

Table 4. Computed X-Y and Y-H distances (A), Fermi-contact terms and coupling constants lhJH-Y (Hz), and reduced coupling constants [lhKH-Y ( x 1019) N A~2 m~3] for X-H-Y hydrogen bonds with significant proton-shared character. R(X-Y)

R(H-Y)

Symmetric hydrogen bonds N-H-N (HCN)2H+ Dxh 2.521 2.598 (NH3)2H+ D3d

1.260 1.299

O-H-O (H20)2H+ C2 (CO)2H+ Dmh

2.386 2.394

F-H-F (HF)2H+ C2h

2.302

FC

HIT J

lhTT

H-Y

K-H-Y

-56.3 -26.4

-56.6 -26.5

46.5 21.8

1.193 1.197

^7.5 -68.0

-47.5 -68.2

29.2 41.9

1.151

186.3

195.3

17.3

1.591 1.093 1.271 1.337 1.134

-0.1 298.9 -39.3 -33.9 -65.1

-0.3 315.3 -39.7 -34.3 -65.4

0.2 27.9 24.4 28.2 40.1

Quasi-symmetric hydrogen bonds H3NH+:NH3 C3v COH+:FH HFH+:OC H2OH+:NCH HCNH+:OH2

2.704 2.362 2.362 2.471 2.471

Since both reduced one-bond spin-spin coupling constants in hydrogen-bonded complexes with proton-shared hydrogen bonds are positive, experimental determination of the signs of these coupling constants could be used as a means to detect the presence of a protonshared (or low-barrier) hydrogen bond. If the signs of 'KX-H and IhKH-Y are the same, then the hydrogen bond is proton-shared, or has moved far enough along the proton transfer coordinate to acquire sufficient protonshared character. This prediction is in agreement with the signs of the two one-bond spin-spin coupling constants reported by Limbach and coworkers for the proton-shared complex of HF with collidine.15'1618 These investigators reported values of +30 and -50 Hz for 'JF-H and lh JH-N> respectively, for an FHrcollidine complex with a proton-shared

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253

hydrogen bond. Thus, both 'KF-H and lhKH-N are large and positive for this complex. 3.4. Models for the Signs of Reduced Coupling Constants 3.4.1. The Dirac Vector Model Now that generalizations have been made concerning the signs of reduced one- and two-bond coupling constants 2hKx.Y, 'KX-H and lhKH-Y for X-H-Y hydrogen bonds, it is a natural next step to ask what determines these signs. In the early 1950s, the Dirac Vector Model was proposed and used to predict the signs of coupling constants in molecules. According to this model for a general A-B-C-D system, if the nuclear magnetic moment of A is "up", it polarizes the electron pair that forms the A-B bond such that the electron nearer A has spin "down", and the other has spin "up". This leads to the magnetic moment of nucleus B being "down", which then leads to an "up", "down" alignment of the pair of electrons that form the B-C bond. This induces the nuclear magnetic moment of C to be "up", as illustrated in Fig. 10. If two nuclei have an antiparallel alignment of nuclear magnetic moments, then the reduced spin-spin coupling constant between that pair is positive.69 From the Dirac Vector Model comes the well-known rule that reduced one-bond coupling constants are positive, two-bond negative, three-bond positive, and so on.1 Dirac Vector Model A

t

B

it

i

C

n

t

D

it

i

Fig. 10. The Dirac Vector Model showing the orientation of nuclear and electron spins. How well does this Model predict the signs of reduced coupling constants? Unfortunately, it is not possible to answer this question based on experimental data for hydrogen-bonded complexes, since such data are scarce. However, this question can be addressed by comparing the

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signs of experimental one- and two-bond coupling constants in small molecules to the predictions of this model, and to computed EOM-CCSD coupling constants. Table 5 presents coupling constants for one-bond C-C and N-N couplings in C2H6 and N2H4, respectively, as well as twobond H-H couplings in H 2 C=0 and H2C=C=0. Since the two coupled atoms in these molecules are identical, both the coupling constant (J) and the reduced coupling constant (K) have the same sign. As evident from this table, both experiment70 and theory indicate that the one-bond C-C coupling constant in C2H6 is positive, in agreement with the Dirac Vector Model. However, both experiment71 and theory show that the one-bond N-N coupling constant is negative in the nitrogen analogue, N2H4, in contradiction to the Model. A similar situation arises for the two-bond H-H couplings in ketene (H2C=C=0) and formaldehyde (H2C=0). According to the Dirac Vector Model, these H-H couplings should be negative, as is the case for ketene. However, the two-bond H-H coupling constant for formaldehyde is positive.72 In addition, the Dirac Vector Model would predict that the signs of the reduced two-bond X-Y coupling constants across X-H-Y hydrogen bonds are negative, which is not the case. Neither can this model explain the sign change of the onebond coupling constants along the proton transfer coordinate. Table 5. Computed and experimental one- and two-bond Fermi-contact terms and spin-spin coupling constants nJx-x (Hz) for small molecules.

Molecule

n, X-X

FC

H3C-CH3 H2N-NH2

1,C-C 1,N-N

H2C=C=0 H 2 C=0

2, H-H 2, H-H

Jx-x

Exptl.

+34.6 -8.0

+35.9 -6.2

+34.6a -6.7"

-14.0 +33.2

-14.2 +33.5

-15.6C +41°

a) Ref. 70. b) Ref. 71. c) Ref. 72.

3.4.2. NMRTWM The initial impetus which led to the formulation of the Nuclear Magnetic Resonance Triplet Wavefunction Model (NMRTWM)2 came from the

Predicting and Understanding the Signs across X-H- Y Hydrogen Bonds

255

puzzling orientation and distance dependence of the sign of F-F coupling constants in (HF)2 clusters,51 and the failure of the Dirac Vector Model to provide any insight into such sign changes. It should be noted that the Dirac Vector Model is based on a description of bonding in ground electronic states. However, spin-spin coupling constants are second-order properties, which depend on excited state wavefunctions. Any model for the signs of these coupling constants must include some dependence on electronic excited states. As noted above, there are four terms which contribute to the total coupling constant, namely, the PSO, DSO, FC, and SD terms. In principle, these terms can be computed from the familiar sum-over-states expression for J.62'73 (This approach is not computationally feasible in practice.) However, from the sum-over-states expression it can be seen that the sign and magnitude of J for a pair of atoms is the result of a sum of individual contributions that arise from coupling between the groundstate wavefunction and excited-state wavefunctions of the correct symmetry through the appropriate nuclear spin operators on the two coupled atoms. The PSO and DSO terms couple excited singlet states to the ground state; the FC and SD terms arise from coupling between the ground state and excited triplet states. Since for hydrogen-bonded complexes the FC term generally dominates J, NMRTWM focuses on that term. There is an alternate, also well-known expression for computing J, namely 2 JAB= 5 E/duAd|aB (3) where UA and \iR are the magnetic moments of nuclei A and B. The algorithm used in practical calculations of J is based on this second derivative expression. Since this expression indicates that the energy of the system is dependent on the magnetic moments of the coupled nuclei, then the wavefunctions must also be dependent on these moments. The hypothesis that forms the basis of NMRTWM is that if the wavefunctions are dependent on the magnetic moments, then the orientations of magnetic moments may depend on the phases or nodal properties of the excited-state wavefunctions. How this model applies to hydrogen-bonded complexes will now be illustrated, focusing on the Fermi-contact term and the nodal properties of wavefunctions for lowerenergy excited triplet states.

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Figure 11 shows the phases and nodal patterns of a-type wavefunctions for lower-energy excited triplet states for a complex with a traditional X-H...Y hydrogen bond, illustrating only those wavefunctions that do not accidentally have a node passing through one of the atoms. The wavefunctions can only be of o-type, since they must have "s" electron density at the nuclei. NMRTWM assumes that the orientation of the nuclear magnetic moment vector responds to the phases of excited triplet state wavefunctions, and arbitrarily assigns nuclear spin "up" (T) when the wavefunction is positive at an atom, and "down" (1) when it is negative. When the magnetic moments of two nuclei have a parallel alignment, then the contribution to the reduced coupling constant from the triplet state with that alignment is negative; when the alignment is antiparallel, the contribution is positive.69 Thus, according to NMRTWM, the sign of the FC term is determined by a competition between positive and negative contributions made from otype excited states that make up the triplet manifold and couple to the ground state.

t

t

t

t

t

i

Fig. 11. Nodal patterns and nuclear magnetic moment alignments for o-type lower-energy triplet states for a traditional X-H-Y hydrogen bond. Four different nodal patterns are illustrated in Fig. 11 for a complex with a traditional X-H...Y hydrogen bond. How would wavefunctions with such patterns contribute to the reduced two-bond spin-spin

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257

coupling constant KX-Y? Wavefunction "a" in Fig. 11 has no nodes intersecting the X-Y axis, which means that the magnetic moments of nuclei X and Y have a parallel alignment, and the contribution to the reduced FC term is negative from such a state. The number of state wavefunctions with no nodes would be expected to be small, since the states involved are excited triplet states. The reduced FC term for X-Y coupling is also negative for a type "d" function, which has two nodes intersecting the X-Y axis between X and Y. However, there are also two types of wavefunctions that have only one node intersecting the X-Y axis, and these are illustrated as types "b" and "c". In type "b", the node intersects the X-H bond, while in "c" it intersects the H...Y hydrogen bond. In both cases, the nuclear magnetic moments of X and Y have an antiparallel alignment, and the contributions to the reduced FC term from both types of states are positive. Since 2hKX-y is positive, it is type "b" and "c" excited states that must dominate, although it is not possible to determine which is more important without a complete sum-over-states calculation, which is not feasible.74

t

t

t

1

I

t

I

t

Fig. 12. Nodal patterns and nuclear magnetic moment alignments for o-type lower-energy triplet states for a symmetric X.. .H.. .X hydrogen bond. How would this picture change when the hydrogen bond is a symmetric, proton-shared hydrogen bond? Figure 12 illustrates three types of lower-energy excited-state triplet wavefunctions. Types "a" and

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J. E. Del Bene and J. Elguero

"c" have no nodes and two nodes, respectively, intersecting the X-X axis. States such as these have a parallel alignment of the magnetic moments of the two X atoms, thereby making negative contributions to the reduced FC term. However, state "b" has only one node intersecting the X-X axis, which by symmetry must pass through the H atom. The nuclear magnetic moments of the two X atoms are antiparallel, and the sign of the reduced FC term is positive. Thus, NMRTWM indicates that it is states of this type with one node (or an odd number of nodes) intersecting the X-X axis that dominate and determine the sign of the reduced FC term and 2hKX-x for complexes with symmetric hydrogen bonds. Figures 11 and 12 may also be used to gain insight into the signs of the two reduced one-bond spin-spin coupling constants. All 'KX-H for both traditional and symmetric hydrogen bonds are positive, which means that the dominant triplet states must be those that have one node (or an odd number of nodes) intersecting the X-H bond. For traditional hydrogen bonds, the states that make positive contributions to the reduced X-H FC term and must therefore dominate are those labeled "b" and "d" in Fig. 11. While it is not necessary that the same states dominate for different coupling constants, it is interesting to note that contributions from type "b" states are positive for both 2hKX-y and 'KX-HHowever, when the hydrogen bond is symmetric, the "b" states that determine 2hKX-x make no contribution to ! K X -H, since these states have a node passing through the H atom. It must be type "c" states with an odd number of nodes intersecting the two X-H bonds that dominate, making the reduced FC terms and ^X-H positive for the two equivalent X-H bonds. Consideration of the signs of lhKY-H requires that traditional and proton-shared hydrogen bonds be treated separately, since as noted above, lhKH-y changes sign along the proton transfer coordinate. For a traditional hydrogen bond, both the FC term and lhKH-y can have relatively small values, but both are negative. A negative sign is consistent with the dominance of states with wavefunctions labeled "a" and "b" in Fig. 11, both of which have no nodes intersecting the X-Y axis between H and Y. However, for complexes with a symmetric hydrogen bond, then states with wavefunctions labeled "c" must dominate, since both reduced X-H coupling constants are equal and positive.

Predicting and Understanding the Signs across X-H- Y Hydrogen Bonds

259

Analysis of the signs of 2hKX-Y, 'KX-H, and lhKH.Y using NMRTWM indicates that the signs of the reduced FC terms are determined as a result of competing positive and negative contributions from excited atype triplet states that couple to the ground state. Although it is not necessary that the same triplet states dominate each of the three different coupling constants, for complexes with traditional hydrogen bonds, this could be the case. Thus, excited triplet states with type "b" wavefunctions which have one node (or an odd number of nodes) intersecting the X-H bond and no nodes (or an even number of nodes) intersecting the H...Y hydrogen bond make positive contributions to the reduced FC term for X-Y and X-H coupling, and negative contributions to the reduced H-Y FC term. The signs of these contributions are consistent with the signs of the corresponding reduced coupling constants. However, as the proton moves from X to Y, the nature of the excited-state wavefunctions and those states that dominate must change as a proton-shared hydrogen bond is formed. The transition among dominant wavefunctions as the proton moves must be smooth and continuous. If along the proton-transfer coordinate a symmetric X...H...X hydrogen bond is formed, then the states that dominate and make 2hKx_x positive make no contribution to 'KX-H and lhKH-x since the wavefunctions for these states have a node through the H atom. Both 'KX-H and lhKH-x must be dominated by states that have an odd number of nodes intersecting the two X-H bonds. If the proton-shared hydrogen bond formed during the proton transfer process is quasi-symmetric, similar arguments hold for sign changes of 'KX-H and lhKH-Y- The signs of 2hKX-Y, 'K X -H, and lhKH-Y for traditional and proton-shared hydrogen bonds are summarized in Fig. 13.

+ rv

<

11

-KX-H-

+

+

X-Y ~~~

1

|

n

~K

J

J

T

—lhKH.YJ

,v

L1KX.H+

__-V

^KH-YJ

+

Fig. 13. The signs of 2hKx_Y , 'KX-H, and lhKH.Y for complexes with traditional and proton-shared hydrogen bonds.

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J. E. Del Bene and J. Elguero

4. Concluding Remarks Prior to the mid-1990s, there had been no experimental or ab initio computed spin-spin coupling constants reported for both one- and twobond couplings across X-H-Y hydrogen bonds. Thus, in about a decade this field has grown remarkably, as indicated by the number of papers, review articles, and books that have been published on this subject. Nevertheless, this field is still in its infancy. For further progress to occur, theorists need to devise ways of applying reliable theoretical methods to the larger hydrogen-bonded complexes that can be studied experimentally; experimentalists need to provide coupling constant data for the smaller complexes that are amenable to high-level theoretical studies. NMRTWM needs to be further developed to become a more quantitative model. The pursuit of these goals will lead to the bridging of gaps that exist between theory and experiment, a better understanding of the physical meaning of coupling constants and their implications for hydrogen-bonded complexes, and the routine extraction of reliable structural data for hydrogen-bonded complexes from NMR measurements. There is no doubt that this area of research will continue to be a very active one, filled with new and surprising findings.

Acknowledgments This work was supported by a grant from the U.S. National Science Foundation (NSF CHE-9873815) and by the Spanish DGI/MCyT (project no. BQU-2003-01251). The authors gratefully acknowledge this support, as well as the computational support provided by the Ohio Supercomputer Center. The authors would also like to thank the American Chemical Society and John Wiley and Sons, Inc. for permission to reprint figures and tables.

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70. J. B. Stothers (1972) Carbon-13 NMR Spectroscopy, Academic Press, New York, p 371. 71. S. Berger, S. Braun, and H.-O. Kalinowski (1997) NMR Spectroscopy of the Non-Metallic Elements, John Wiley & Sons, Chichester, p. 274. 72. J. W. Emsley, J. Feeney, and L. H. Sutcliffe (1967) High Resolution Nuclear Magnetic Resonance Spectroscopy, Pergammon Press, Oxford. 73. H. Sekino and R. J. Bartlett (1994) Nonlinear Optical Materials, S. Kama, ed. American Chemical Society, Washington, D. C. 74. J. Oddershedde, P. Jorgensen, and N. H. Beebe (1997) Chem. Phys. 25, 451.

Chapter 6: Towards the Elucidation of the Activation of Cisplatin in Anticancer Treatment

Jaroslav V. Burda", Jifi Sponerbc and Jerzy Leszczynskid "Department of Chemical Physics and Optics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague 2, Czech Republic Institute of Biophysics, Academy of Sciences of the Czech Republic, Krdlovopolskd 135, 612 65 Brno, Czech Republic c

Institute of Organic Chemistry and Biochemistry, Academy of Sciences of the Czech Republic, Flemingovo ndmest(2, 16610 Prague 6, Czech Republic d

Computational Center for Molecular Structure and Interactions, Department of Chemistry, Jackson State University, Jackson, Mississippi 39217, USA

Abstract The first step in biological activity of cisplatin, one of the most efficient anticancer drugs, involves the hydration process. This chapter reviews the results of theoretical studies concerning the activation of cisplatin and its analogs. The complexity of the studied phenomena requires application of different theoretical tools. The energetic description of the extended hydration scheme was performed for the square-planar Pt(II) and Pd(II) complexes. High-accuracy methodology was adopted for the predictions of their heats of reactions. Molecular mechanism of the ligand replacement by

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water molecules was investigated using the supermolecular approach. In this model the reactant metal complex was examined together with an additional water molecule. Based on such approach, not only thermodynamical but also kinetical parameters can be determined. The CCSD(T) computational level was used for the determination of the reaction energy profiles. Despite the high accuracy, it was found that the obtained results do not sufficiently match the experimental data. This causes the necessity of the improvement of the gas-phase calculations using the implicit solvent in the polarizable continuum model. The COSMO method was applied on the B3LYPreoptimized geometries and was augmented by the CCSD(T) single point calculations. These final results are in excellent accord with both the experimental data as well as other calculated values.

1. The Effect of Cisplatin in the Cell Cisplatin (cis-dichlorodiammineplatinum cisDDP) belongs to one of the most frequently explored platinum compounds since Rosenberg's discovery of its anticancer activities.1 It was found that cisplatin makes a bridge connection between two (usually) adjacent guanine bases in DNA predominately forming 1,2-intrastrand GpG complexes (about 65% of the occurrences). Some other intrastrand coordinations occur less frequently, namely with the l,2-d(ApG) and the l,3-d(GpXpG) base sequences. However, coordination to the l,2-d(ApA) or the l,2-d(GpA) fragment was observed only very rarely. The high mobility group (HMG) proteins can recognize the bend on the platinated DNA a-helix and bind to it.2 Structures of such Pt-DNA-HMG complexes were solved recently based on crystallographic data.3"5 Cisplatin can also interact with other molecules in a cellular environment, like peptide structures6"10 or RNA's. 11 The main cisplatin targets, before nucleus is reached, are side chains of amino acids on the surface of proteins or peptides like glutathione.12'13 A considerable amount of effort has been devoted to the study of these interactions. Sulphurcontaining amino acids like methionine and cysteine were frequently considered.14"16 Some attention was also paid to interactions of cisplatin with N-side chains of amino acids like histidine,17"20 arginine or amides of aspartic21 and glutamic acids22 as well as other amino acids.23'24

Towards the Elucidation of the Activation of Cisplatin 267

Figure 1. The X-ray structure of the cisplatin adduct to dodecamer sequence of DNA d(CCTCTG*G*TCTCC)/d(GGAGACCAGAGG) (pdb structure la84). 136

2. Experimental Data on Cisplatin Hydration Before all of the above mentioned interactions can occur, cisplatin undergoes the hydration process. Many studies have been devoted to the

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thermodynamics and kinetics of cisplatin hydration. As to the experimental works, the thermodynamic data for the replacement of the chloride ligand by water molecules were reported by Hindmarch et al.25 for both the hydration steps: cis-[Pt(NH2)2Cl2]

+ H20^=±^Cr+cis-[Pt(NH^2Cl(H20)]+

(i)

IS

cis-[Pt(NH,)2Cl(H20)]+

+ H20+==±±Cr

+cis-[Pt{NH3)2(H20)2]1+

The values of equilibrium constants logKi = -2.19 and logK2 = -3.53 in 1.0 M NaC104 solution were found. Arpalahti et al.26'27 determined the equilibrium constants for analogous processes of transplatin. Slightly smaller hydration constants were reported: logKi = -2.92 and logK2 = -4.41 in more diluted 0.1 M NaC104 solution. Due to their dependence on the pH value of the environment, the hydrated complexes can be subsequently stabilized by deprotonation of the aqua ligands.28 This means that in the basic solutions (with pH > 7) the activity of the platinum complexes will be reduced: K

4.

7

cis-[Pt(NHl)iCl(H20)]<=r-a^->H++cis-[Pt(NHi)2Cl(OH)] K

1

cis - [Pt(NH3)7 (H,0)2 ]2+ ^=*±3H

+

+ cis - [ Ft (Mf,)a {H20){OH)X

K

•> ds-[R(m3)3{H1C)(OH)]+e=^H*+ciB-[Pt(iM3)3(OH)i]

(2)

The values of these pKa were determined for both cisplatin and transplatin in many laboratories.29"32 Approximate values of individual pKa based on several measurements can be estimated as pKal = 5.5, pKa2 = 7.3, and pKa3 = 6.6 for cisplatin. The estimations for transplatin are pKal = 4.4, pKa2 - 7.3, and pKa3 = 5.6. It can be noticed that the hydrated transplatin deprotonates in the first step more readily (or at lower pH) than cisplatin. The same is also true for the corresponding singly-hydrated platinum complexes (pKa3). The kinetic parameters of the hydration processes were also studied. It is widely accepted that the substitution reactions occur predominately by an associative mechanism in the case of Pt(II) and transition metals universally.33 Despite the fact that large bulky ligands can slow down the reaction course substantially, switching to the alternative dissociative

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mechanism is not likely in these cases. Because of presence of the water molecule in reactants of eq. 1, the forward reaction can be regarded as a kinetic reaction of a pseudo-first order. Nevertheless, the backward reaction follows the second order kinetic formalism. Experimentally determined rate constants for dechlorination were published by Arpalahti et al.26'27 for the first and second steps and both cis- and trans-conformers at T = 318.2 K and pH 2.8-3.4 and in a 0.1 M NaC104 solution. Similarly, the rate constants for cisplatin interactions with guanine were estimated. In these studies either water replacement by the base {kP1 a kP2) or hydration process of the already coordinated cisplatin with guanine (k3) were considered: cis-[Pt(NH3)2ClN7(G)]+ +H20—^Cr

+cis-[Pt{NH3)2{H20)Nl{G)T

cis-[Pt{NH,)2O(H20)T+G-^^H20+cis-[Pt(NH3)2aNl{G)]+ cis-[Pt(NH3)2(H20)2f+ +G-^^H20+cis-[Pt{NHl)2{H20)Nl{G)T

(3)

Based on several studies,29'35"41 it is also possible to determine the rate constants of the reactions in eq. 3: k3 = 7.6 x 10"5 [s 1 ], kP1 = 1.5 x 10"4, and kP2 = 92 [ M ' S ' 1 ] (averaging the cisplatin concentration by about 5 |XMand the guanine concentration by about 0.8 mM for bom subsequent reactions). Note that the value of the k3 rate constant is comparable with values kj and k2 of the reactions from eq. 1 and Table 7 (cf. below). Also, based on the values of kP1 and kP2 , it is obvious that guanine interacts substantially faster with the diaqua Pt-complex (by about 5 orders of magnitude). The experimental data unambiguously reveal the important difference between the activation processes in the first and second step. Despite the question whether the equilibrium for the hydration process can occur, there is a factor which facilitates the reaction course, namely the decreased concentration of CI" anions during cisplatin passage from blood (with [CI] about 100 mM) to a cellular environment (with [CI] about 10-80 mM). The chloride concentration further drops down in the cellular nucleus to a value about 4 mM. This forces higher yields of the hydration products, according to LeChatelier-Braun-van Hoff's principle.

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3. Quantum Chemical Calculations Related to Cisplatin Studies One of the first researchers who applied the theoretical approaches to study cisplatin was Lipinski.42 In his pioneering work, many important aspects of cisplatin behavior were investigated using a semiempirical approach. This includes hydration, cisplatin dimerization, and bonding to phosphate and guanine. The first ab initio calculations of cisplatin complexes with DNA bases were performed by Basch et al.43 who also introduced the relativistic pseudopotentials for the platinum atom. The calculated coordination energies were already very accurate (e.g., bond dissociation energy (BDE) of the PtN7(guanine) dative bond was estimated to be approximately 114 kcal/mol). In 1990 Kozelka and coworkers carried out classical molecular mechanics calculations on the cisplatin-DNA oligomeric sequence.44"46 Further, they recently reported explicit-solvent nanosecond-scale molecular dynamics simulation of a platinated DNA duplex.47 Since the empirical parameters for platinum were not available, they fitted the results of ab initio Hartree-Fock obtained in a minimal basis set. Despite the very approximate estimation of the empirical force field parameters, they were able to predict the experimental measurements with very good accuracy. Individual conformers of cis/transDDP were investigated by Pavankumar et al.48 combining the the second order M0ller-Plesset perturbation theory (MP2) approximation with several basis sets for the determination of the bonding properties, vibrational frequencies, and charge densities. Similar work was performed by Carloni and coworkers49'50 using DFT techniques for the gas-phase and solid state calculations. They also performed a comparative study of the properties of cisplatin and carboplatin.51 Different functionals were utilized for the determination of the molecular structure and vibrational spectra of carboplatin and cisplatin,52 and the calculated spectroscopic data were compared for both platinum complexes with experimental measurements. The transition states for the chloride ligand replacement by water and guanine were studied by Chval et al.53 The complex of cisplatin with l,2-d(GpG) bases was examined by Carloni and coworkers.54 In this study some hydration aspects of cisplatin were predicted using Car-Parrinello MD simulations.

Towards the Elucidation of the Activation of Cisplatin 271

H3N

CI

H3NV

C1

Hj/

Tt H

3N

Cisplatin

/O

f

Oo \)

/

Carboplatin ^ Q

Competitive effects for the interactions of cisplatin with various active sites in the cellular environment are discussed in papers ofDeubel. 5556 Inthe earlier paper, energetic and structural data of complexes with the different substituted ligands were explored. The more recent work deals with kinetic factors in the relation to the transition state (TS) for water replacement of the semihydrated cisplatin complex (cis-[Pt(NH3)2(H20)Cl]+ ) with either an N- or S-containing ligand (thiopheneimidazol, dimethyl sulphide, or methanethiolate which serve as amino acid models). Deubel concluded the kinetic preference of N-sites over S-nucleophiles where the important role is played by the electrostatic terms. In addition, the aliphatic/aromatic character of the substituent as well as the influence of different dielectric constants of the environment are very important. A more realistic model for the aqualigand replacement with adenine and guanine was studied in works of Chval et al.53'57 and Eriksson and coworkers.58 They performed independently the estimation of the thermodynamic and kinetic parameters of this process. The coordination of cisplatin to DNA purine bases represents a frequent topic of many recent papers. A study on the comparison of the cisplatin and cispalladium interaction with guanine was published by the Leszczynski group.59 In addition, a possible chelation of cisplatin to the 0 6 and N7 guanine sites was explored. They showed that the possible deformations of Pt-adducts with a GC base pair can be stabilized in the case of an isolated pair resulting in the alteration of the (non-planar) H-bonding pattern. The Car-Parrinello/Molecular Dynamics (CPMD) study of cisplatin bound to DNA oligomer models was performed by Spiegel et al.60 where besides the structural and energetical parameters, NMR shifts of the 195Pt nucleus were also observed. Large fluctuation of these shifts during the dynamical simulations was reported. Comprehensive computational studies of cisplatin compounds were performed by the Lippard group. In one of their works the strength of the Nglycosyl bonds after platination was investigated. It was shown that, while

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protonation of the N7 position of guanine accelerates the depurination reaction, the cisplatin adduct does not change the glycosyl bond substantially.61 They also explored the character of Pt-X bonds (X=C1, O, N, and C) for various ligands: CI, H 2 0, NH3, CO, and guanine.62 The decreased strength of the C-O bond (in carbon-oxide containing ligands) was demonstrated with a decreasing total charge of platinum complexes. The question of guanine preference over adenine was examined in another work.61 The same study also addressed the existence of the TS structure and the role of kinetic aspects on the preference of guanine adducts. The conclusion about the estimation of activation barriers are in very good accord with previously mentioned studies of Chval et al.57 and Eriksson and coworkers.58 The thermodynamic results can be compared with similar results from our group63'64 where cisplatin bridges were examined between purine bases together with the influence of the sugar-phosphate backbone on the Pt-N7 bonds. Mispairs in DNA caused by different stabilizations of rare tautomers after platination of the N4-cytosine and N6-adenine sites were explored by Sponer et al.,65 and a similar study was conducted for the N7guanine and adenine sites by Burda et al.66'67 Possible point mutations could occur in the case of adenine and cytosine where the imino-tautomer is more preferable over regular N6 and N4 amino forms under amino-platination. Interestingly, it was argued that the amino-platination promotes the tautomerism via substantial changes of the electronic structure of bases; therefore, the effect remains fully expressed under polar solvent conditions. A similar effect was also revealed for mercury and may be a common source of mutations in DNA upon metal binding. In contrast, the tautomers obtained by the N7 metalation are stabilized essentially by the electrostatic effects, and the formation of these tautomers is thus expected to be inhibited to a large extent by polar solvents. A similar conclusion was derived for the adenine protonation upon platinum binding to ring nitrogen atoms.68 While the gas phase calculations show that the adenine proton affinity is dominated by the formal charge of the metal adduct, the corresponding solution pKa values are independent of the charge of the metal in a wide range of - 1 to +3. The platination nevertheless shifts the adenine pKa due to nonelectrostatic effects. These nonelectrostatic effects are also seen in the gas phase QM calculations, albeit being much weaker compared to the electrostatics that dominate in the gas phase. Another recent study (combination of experimental and computational approaches) revealed that platination can cause cytosine deamination which also is a possible source of

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mutations in DNA under metal binding. The effect of platination on Watson-Crick base pair interactions was explored,67 and the study clearly shows that the G-C base pair strength is increased via polarization effects.

4. The Topics of the Review There are several questions which will be addressed within this review. • The thermodynamic surface for the hydration of square-planar platinum(II) will be explored, and general conclusions on the solvation will be drawn. • A comparison of the thermodynamical hydration surfaces of platinum and palladium complexes will be done showing that the SAR (structure and activity relationship) is not valid in relation to the cisplatin anticancer activity. • Closer insight to the detailed molecular mechanism of the cisplatin ligand replacement can be used for the determination of both the thermodynamic and kinetic parameters from the Eyring TS theory. • The influence of the surrounding environment will be examined in the last part revealing the key role of the proper screening of electrostatic interactions for accurate prediction of physico-chemical behavior of the systems under study.

5. Calculations of Platinum(II) Hydration Surface 5.1 Computational Model A very accurate computational methodology based on MP4 single-point energy calculations was chosen for the evaluation of the selected hydration schemes. The methodology is based on the Gaussian theory ideas70 that were modified for the determination of the total energies of all involved platinum complexes. The necessary modifications for the inclusion of the pseudopotentials were indicated so that the following "recipe" could be suggested: • The most stable rotamers for all investigated species were found at the MP2/6-31G(d) level using the frozen-core approximation. The MWB-60

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energy-averaged pseudopotentials from the Stuttgart/Dresden laboratory71 were used for a description of the Pt atom. The 5 s and 5p electrons of Pt were also kept in the frozen core at this step. The suggested basis set was extended with one set of f functions. Their exponent (af = 0.9804) was obtained by minimizing the ground state energy of the Pt atom at the CCSD level. The global minima of the considered complexes were further reoptimized at the MP2(full) level. • Zero-point vibration energies (ZPVE) were determined at the HF level with the same basis set after another reoptimization at this level. • Spin-orbit (SO) coupling corrections were calculated for the Pt atom since the relativistic effects are essential for species containing heavy elements. Other scalar relativistic corrections like the Darwin and massvelocity terms are supposed to be implicitly included in (quasi)relativistic pseudopotentials because they mostly affect the core region of the considered heavy element. Their secondary influence can be seen in the contraction of the outer s-orbitals and the expansion of the d-orbitals. This is considered in the construction of the pseudoorbitals. The effective SO operator can be written within pseudopotential (PS) treatment in the form71"75 U 2AV^SAUr = Y, °

V

so(0

a

)P X)l s P X) i ri i '

where/,• and s, are corresponding

momentum operators of the i'th electron, and X labels the atomic core described with the pseudopotentials. Pt( ' is the projection operator: P U)

i

=

X l ^ X 1 ^ I •T h e

radial

Parts

of the

pseudopotentials are fitted

m=-l

in terms of Gaussian functions: A V ^ ' (ra) = £ AB^

e x p ( - ^ A ) rfx).

k

In our previous work76 the Pitzer's version77"79 of the COLUMBUS program80 was used. Here, an implementation of the SO coupling in the MOLPRO program suite81 was employed. For the SO interaction energies, averaged CAS wave functions were used with 6 singlet and 10 triplet reference states. With these numbers, the order of atomic spectrum lines (the 3 D, 3 F, 'D and : S states) was successfully reproduced. The three highest occupied and three lowest unoccupied orbitals were chosen as the active space. Some preliminary calculations demonstrated that the results are not too sensitive to the size of the active space. A larger space (e.g., 4+4 or more) would be substantially more time demanding with practically the same

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effect on the SO coupling energies. To the contrary, it was of crucial importance to keep the number of states treated in the averaged CAS calculation and in the following CI-SO procedure as large as possible. A smaller number of states (5 triplet and 6 singlet states) did not reproduce the calculated spectrum of the Pt atom even qualitatively.82 • The inclusion of higher components of the correlation energy (not included in up to the fourth order of perturbation theory, MP4) was performed with the CCSD(T) method. This method is known to be very reliable for systems containing transition metals, while some similar approaches like QCISD(T) failed.83'84 • The role of increased size of basis sets was examined in three sets of calculations: a) with another set of diffuse functions (the MP4/6-31+G(d)) calculations, where the original set of the Pt atom pseudo-orbitals was also augmented with s-, p-, and d-functions (as = 0.0075, ap = 0.013, and da = 0.025) for the sake of consistency. b) with extension by polarization functions (computed at the MP4/631G(2df,p) level); for the Pt atom a set of the f-functions was replaced by a new set of 2fg AOs (af = 1.4193, 0.4662, and ag = 1.2077) optimized for the Pt ground state with the CCSD method. c) the last basis set extension resulted in calculations at the MP2(full)/aug-cc-pvtz level. A new contraction scheme was used for the platinum atom (8s,7p,6d)/[6s,5p,4d] where the three outermost d-primitive gaussian functions were excluded from contraction. This basis set was augmented with the diffuse functions mentioned above and with a set of 3f, 2g polarization functions (af = 1.8466, 0.8065, 0.2937, ag = 2.0049, 0.7040) (optimized for the Pt ground state). • Utilizing the PS treatment, some portion of the correlation energy corresponding to inclusion of the inner Pt electrons was lost in the MP2(full) calculations. Therefore, at least part of this lost energy can be estimated in the so-called core-polarization potentials (CPP) which were used for the estimation of the core-valence electron interactions as originally suggested by Preuss85 and Miiller.86 The effective core-polarization potentials can be written in the form VCPP —

^T,ac-fc-fc^

where/ c is the electric field

vector of electrons and the other core shells which act on a given core shell

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J. V. Burda, J. Sponer and J.

Leszczynski

c: fc = ^5j.C(ric,pc)-^-^Zc, i

ic

, with ric= r,-r c ; and/? c c = RC~RC.. c'*c

c'c

The expression C(ric ,pc) represents a so-called cutoff function. This function was introduced mainly as a short-range correction of the core effects. 86 In this form of CPP, there are two explicit variables, a core polarizability ac and a parameter pc in the cutoff function of the potential. The core polarizability was estimated from a calculation on the Pt18+ ion at the non-relativistic HF level. For this purpose the Huzinaga well-tempered uncontracted basis set87'88 was used. The set of 5g functions with exponents, taken from the most diffuse f-functions, was added for better flexibility. The polarizability was calculated using analytical derivatives with the Gaussian 98 program package. The value obtained is a(Pt18+) = 0.1297 A.3 The polarizability obtained in this way can serve as a rough estimate only. The other variable parameter p in the cutoff function for the short range potential C(ric, pc) - (l-e{riJPc]') was in the original work86 adjusted to the ionization potential (IP). However in the case of small-core pseudopotentials, the IPs are not so sensitive to the change of the cutoff parameter. Thus, another quantity had to be investigated for this purpose. We have found that the eigenvalue of the electrons in the 5 s AO could be considered, and the cutoff parameter was adjusted so that the orbital energy of the 5s fits the corresponding value obtained from the Dirac-Fock equations published by Desclaux.89 In this way the p = 1.245 value was achieved. The final AE0(G3) energies at OK can be evaluated according to: A£ 0 (G3) = AE(combined) + E(ZPVE) with AE(combined) defined as: AE(combined) = E[MP4 / 6 - 3 lG(d)] + AE(+) + AE(2df, p) + AE(CC) + AE(CPP) + AE(G3large) + AE(SO) where individual terms come from the following equations: AE(CC) = E[CCSD(T)/6-3lG(d)]-E[MP4/6-31G(d)] A£(+) = E[MP4/6 - 31 + G(d)] E[MP4/6-3lG(d)] AE(2df) = E[MP4/6-3lG(2df,p)]-E[MP4/6-3lG(d)] AE(G3large) = E[MP2(full) I aug -cc- pvtz] - E[MP2/6-3lG(2df, p)] - E[MP216-31 +G{d)] + E[MP216-31G( d)] AE(CPP) = E[MP41 CPP + 6 - 3 IG(d)] - E[MP416 - 3 lG(d)]

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All studied complexes were considered in a square-planar arrangement. The electronic ground state of such platinum complexes is close-shell singlet. The main focus of the investigations is a comparison of cis- and transplatin. Nevertheless the charged complexes Pt(NH3)42+ and PtCl42" were also considered under the reaction (hydration):

Y

X | Pt

x + Solv. *-

V

Y

Pt

Solv.

(eq. 4)

•V

Z

Z

All of the reactions where chloride or the NH3 group was replaced with either H 2 0 and OH" were examined. This should, to a certain amount, reflect processes with both a constant and a varied pH of the solution. 5.2 Optimized Structures of Square-Planar Platinum Complexes The most stable rotamers were found for the considered complexes. The individual Pt-L bond distances (L = heavy atom of the ligand) are collected in Table 1. In accordance with Pavankumar et al.,48 it was found the C2v conformer is the global minimum for the cisplatin complex. For the transDDP complex, the expected candidate for the global minimum was rotamer with C2h point group symmetry (with axes perpendicular to ligand plane, cf. Figure 2). However, another structure with C2v symmetry (axes colinear with Cl-Pt-Cl bonds) was found to be slightly lower in energy (0.5 kJ/mol). This conclusion was also verified with the additional set of optimizations where larger basis set was employed - at the MP2(FC)/aug-cc-pvtz level. This interesting finding should not be overemphasized since internal molecular movement can easily cover such a difference at higher temperature. ci

CI .H

H CI

C

2v

CI

-2h

Figure 2. A comparison of the C2h and C2v geometries of the trans-DDP complex.

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Table 1. Pt-ligand distances (in A) obtained at the MP2/6-31G(d) optimization level. Geometry [Pt(NH3)4]2+ [Pt(NH 3 ) 3 (H 2 0)] 2+ [Pt(NH3)3(OH)]+ cis-Pt(NH3)2Cl2 cis-[PtCl(NH 3 ) 2 (H 2 0)] + cis-PtCl(NH3)2(OH) cis-[Pt(NH 3 ) 2 (H 2 0) 2 ] 2+ cis-[Pt(NH 3 ) 2 (H 2 0)(OH)] + cis-Pt(NH3)2(OH)2 cis-PtCl2(NH3)(H20) cis-[PtCl2(NH3)(OH)]" cis-PtCl 2 (H 2 0) 2 cis-[PtCl2(H20)(OH)]cis-[PtCl2(OH)2]2cis-PtCl(NH3)(H20)(OH) cis-[PtCl(NH3)(OH)2 ]• trans-Pt(NH3)2Cl2 trans-[PtCl(NH 3 ) 2 (H 2 0)] + trans-PtCl(NH3)2(OH) trans-[Pt(NH3)2(H20)2]2+ trans- [Pt(NH 3 ) 2 (H 2 0)(OH)] + trans-Pt(NH3)2(OH)2 trans-PtCl2(NH3)(H20) trans-[PtCl2(NH3)(OH)]trans-PtCl 2 (H 2 0) 2 trans~[PtCl2(H20)(OH)]trans-[PtCl2(OH)2]2trans-[PtCl(NH3)(OH)2][PtCl3(H20)]" [PtCl 3 (OH)f

[Ptcuf

Pt-N 2.055 2.045/2.057/2.012 2.035/2.098/2.035 2.056 2.068/2.002 2.040/2.081 2.004 2.092/1.996 2.130 2.044 2.016 2.067 2.047 2.025 2.030/2.045 2.019 2.049 2.031/2.038 2.014 1.984 2.069 2.003 -

Pt-Cl 2.275 2.254 2.283 2.239/2.273 2.351/2.301 2.238 2.324/2.256 2.372/2.406 2.248 2.358 2.293 2.228 2.320 2.290 2.315/2.312 2.289 2.301/2.326 2.338/2.337 2.310 2.248/2.306/2.306 2.346/2.385/2.320 2.329

Pt-Oa 2.077 1.950 2.078 1.953 2.081 2.074/1.944 1.967 2.123 1.977 2.109 2.112/1.989 1.997/1.980 2.106/1.955 1.971/1.973 2.120 1.979 2.033 2.148/1.927 1.999 2.070 1.960 2.024 2.177/1.935 2.026/2.026 2.010/2.007 2.117 1.987 -

'in mixed complexes the Pt-0 bonds longer than 2.0 A belong to H 2 0, others to OH".

In addition, the application of different computational techniques could possibly change the preference. The schematic pictures of the global minima

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279

for the most important compounds of the reaction path are demonstrated in Figure 3. Following the suggested Gaussian theory, all systems were reoptimized at the MP2 level without any frozen core orbitals. The shortest Pt-bonding distances were found for negatively charged OH" ligands (on average about 1.96 A). Ammonia and water coordinations are similar; the average Pt-N distance is about 2.04 A and Pt-0 about 2.10 A.

>*++ + ds-DDP

d.-PKNH.MOH),

trans-DDP

,rans.P,
cls-PI(NHJ,(H,0)1

^.p.c.^oH

trans-Pt(NH,),(H,0),

,r.ns.P«C^H,OH

Figure 3. Schematic pictures of some of the important global minima structures on the hydration energy surface. The longest bonds are the Pt-Cl bonds. All the Pt-ligand distances exhibit relatively large fluctuation, for example, in the Pt-Cl bond they vary from 2.23 A (in positively charged particle [trans-Pt(NH3)2Cl(H20)]+ ) to 2.35 A (in negatively charged [cis-Pt(NH3)Cl2(OH)]" species). The total charge of the complex influences the bond length; the longer bonds in negatively charged complexes reflect the larger contribution of the electron repulsion. All Pt complexes basically preserve the square planar arrangement, one of the largest deviations of the dihedral angle from the plane being 8° in the case of the trans-Pt (NH3)2(OH)2 complex. Bond angles also remain close to 90°; the smallest bond angle is Z N-Pt-0 = 82° in the negatively charged cisPt(NH3)Cl2(OH) complex.

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J. V. Burda, J. Sponer and J.

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5.3 The Analysis of the Terms in the AEQ (G3) Evaluation In the applied methodology, various corrections are added to the MP2(full)/6-31G(d) level, assuming that these contributions are basically additive. Table 2 contains the individual contributions of the examined corrections. It can be seen that an increase of the order of PT leads to lower total energies by about 30 kcal/mol (column labeled AE(MP4)). The column AE(CC) sums all the contributions from the single and double excitations together with the non-iterative triple contributions, according to the CCSD(T) theory. These contributions are very close to the MP4-SDTQ results; the differences are not larger than a few kcal/mol. Comparing the MP4-SDQ correlation energy with the corresponding CCSD energy, the values obtained within PT are generally smaller for positively charged complexes and larger for neutral and negatively charges species. Energy contributions of triples are systematically lower in the case of the CCSD(T) method (up to 2 kcal/mol). Hence, for positively charged systems, these two trends (larger triples contributions and smaller SDQ values) compensate at the MP4 level. In the case of 2+ charged complexes, the final CCSD(T) energies are smaller than the corresponding MP4 energies. The next columns of Table 2 display the role of larger basis sets augmented by the polarization and diffuse functions. Here it can be noticed that the polarization functions are more important than the inclusion of the diffuse functions. The only exception can be noticed in the presence of the OH" ion which seems to be fairly sensitive to the additional diffuse functions. In these cases the additional set of diffuse functions gives improvement of about 12kcal/mol in comparison, for example, with complexes with water. Such a trend is easy to understand because diffuse functions are more important for the description of the loosely bonded electrons in negatively charged systems. The column (AE(MP2/G3)) contains the differences between the MP2(full) energies with a basis set of augmented-triple-zeta quality and the 6-31G(d) basis set. The size of these numbers clearly demonstrates the fact that the split-valence double-zeta basis sets are far from being complete. The ZPVE contributions closely reflect the number of bonded hydrogens because the vibrations of the light elements can more easily absorb energy and thus represent the most important contribution to the final ZPVE.

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Very important also is the spin-orbit coupling. From Table 2, it can be seen that the heavier the complex, the larger the SO splitting. The negatively charged complexes also exhibit larger SO energies. Nevertheless these trends are only approximate since other effects are also important. From the calculated set, it is difficult to decide whether the cis- or trans-isomers could be more stabilized by the SO coupling. 5.4 Thermodynamical Surface for the Hydration ofPt(II) Complexes The energetics for the designed hydration processes was determined using the AE0 (G3) values. Figure 4 displays the reaction energies schematically. In the fraction numbers, the numerator represents the energy of the cisconfiguration and the denominator value of the trans-conformer. In the left part of Figure 4 where the charged [Pt(NH3)4]2+ complexes are displayed, the exchange of the first ammonia molecule with water proceeds with an endofhermic step (AE = 20 kcal/mol) if the total charge of the complex is preserved. This finding is in good accord with the generally known fact that transition metals prefer coordination with soft atoms like nitrogen or sulfur rather than the coordination with oxygen. Further hydration with the second water molecule requires a similar amount of energy. In the case of hydration with the OH" ion, the positive charge of the complex is reduced, and a large amount of energy is released (more than 230 kcal/mol). Replacement with the second OH" ion leads to a neutral complex releasing about 130 kcal/mol. Hence the final and most stable product of hydration of positively charged [Pt(NH3)4]2+ is the neutral Pt(NH3)2(OH)2 complex. This process is associated with a decrease in pH because two protons are the product of the hydration if the total charge is preserved. However, considerations of the role of pH is not simple since the (NH4)+ cations can be partially formed, reducing the concentration of the released protons. In the right part of the scheme, the negatively charged [PtCl4]2" complex is considered. Here, the substitution of the first chloride with water is about twice more exothermic than the substitution with the hydroxyl anion. Nevertheless the energies of the two reactions are relatively close. Water substitution becomes energetically more profitable only when diffuse functions are included in the basis set.

284

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Leszczynski

PtCl(NH 3 ) 2 (OH)

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Towards the Elucidation

of the Activation

of Cisplatin

285

The reason for the larger solvation energy can be seen in the higher stability of the singly charged [PtCl3(H20)]~ complex over the [PtCl3(OH)]2~ complex with two excess electrons. The second hydration step also favors a singly charged mixed water-hydroxyl complex ([PtCl2(H20)(OH)]~). This species represents the most stable complex on the reaction surface of [PtCl4]2". Replacing the second chloride with a water molecule is a fairly endothermic step, and such complexes (neutral cis/trans-PtCl2(H20)2) lie remarkably higher than the charged [PtCl2(OH)2]2" species. Assuming that the [PtCl 2 (H 2 0)(OH)r system is the final product of the [PtCU]2" solvation, the hydration process leads to a decrease in the pH since one proton from the dissociated water is released (and only partially compensated by HC1 formation). In comparison with [Pt(NH3)4]2+, the hydration process of [PtCl4]2" is substantially less exothermic. Such a difference can be understood from the point of view of IPs and electron affinities (EA). Generally speaking, IP energies are much larger than EA energies for neutral molecules. The hydration process of cis- and trans-DDP represents the most important case. The first step was found exothermic for substitution of both CI" and NH3 ligands with the hydroxyl anion. A larger amount of energy is released when the ammine ligand is replaced, forming the negative [PtCl2(NH3)(OH)]" species. Subsequent deammination process of this complex is endothermic considering both water and the hydroxyl ion. Nevertheless dechlorination of this complex and replacement with a hydroxo-ligand leads to the release of more than 20 kcal/mol of energy. This mixed monoamine-monochloro-dihydroxoplatinum anion represents the global minimum of the whole hydration scheme. When the hydroxyl group is considered in the second solvation step of the PtCl(NH3)2(OH) complex, the exothermic behavior is obtained for both dechlorination and deammination reactions. A smaller release of energy (slightly less than 30 kcal/mol) is connected with dechlorination, while more than 40 kcal/mol is gained in the NH3 replacement reaction resulting in the global minimum structure. Thus, the interaction of cis- and transplatin with the water solvent provides the [PtCl(NH3)(OH)2]" anion as their most stable products followed by the neutral Pt(NH3)2(OH)2 complex. These compounds represent the lowest lying complexes on the reaction surface. This conclusion is in partial agreement with experimental findings that cisplatin usually interacts with DNA bases in the hydrated form where two ammonia ligands are present.

286

J. V. Burda, J. Sponer and J.

Leszczynski

Notice that within the suggested reactions, two protons dissociate, decreasing the pH of the solution. The hydration of charged [Pt(NH3)4]2+ leads to the Pt(NH3)2(OH)2 complex. Neutral cis-/trans- DDP would thermodynamicaily prefer the [PtCl(NH3)(OH)2]" complex, which is slightly more stable. The cis-isomer can be considered as the global minimum on the generalized hydration surface. The position of a complex on this surface is determined as a sum of the total energies of the given complex and its ligand supplement. The ligand supplement for the complex contains the complete set of ligands (4x Cl\ 4x NH3, 2x H 2 0, and 2x OH"), omitting the ligands that are present in the given complex. This hydration energy surface for complexes from Figure 4 is drawn in Figure 5. As the reference energy, the value for the [PtCU]2" complex was set to zero, and the relative energies of the other complexes were determined. Figure 5 shows the positions of the complexes which have relative hydration energies lower than 50 kcal/mol. The most stable structures are the [PtCl(NH3)(OH)2]" anions with the trans isomer being approximately 2 kcal/mol above the cis isomer. The Pt(NH3)2(OH)2 complexes are slightly more than 20 kcal/mol higher. In this case, the cisPt(NH3)2(OH)2 isomer is about 5 kcal/mol above the trans isomer. Besides these very stable structures, other complexes were also found in their proximity: [PtCl2(NH3)(OH)]", cis-/trans-[PtCl2(H20)(OH)]" which is slightly higher than the products of [PtCl4]2" hydration, and the PtCl(NH3)2(OH) complexes. Interestingly, the mixed PtCl(NH3)(H20)(OH) complexes lie lower than PtCl2(NH3)2. It can be seen that among these complexes, there are three neutral structures and two negatively charged (1-) species for both cis and trans isomers. In these particular cases one or two ammine-ligands with a soft nitrogen (according to the HSAB principle90) are replaced with a harder oxygen, which has smaller affinity to platinum (and to transition metals in general). If such species are to be stable, additional forces must strengthen the dative bonds, which could be done by the additional electrostatic interactions Pt54 - O8" involving the negatively charged oxygen from the hydroxyl groups.

Towards the Elucidation

of the Activation

of Cisplatin

287

Pt(NH3)2(H20)(OH)+ cis/trans 40

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288

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Leszczynski

6. Comparison of the Hydration Energy Surfaces of Pt(II) and Pd(II) Complexes Palladium compounds have long been explored in relation to the possible interactions with DNA bases. In addition to numerous experimental work91"93 a lot of theoretical papers have appeared due to the recent progress in computational quantum chemistry. Some of these studies provide the comparison of both structural and thermodynamical properties of palladium and platinum complexes. Generally, structural similarities are usually demonstrated.910 As to other, for example, catalytic properties, the behavior of these two metals is not so similar. One of the reasons can be seen in the different electronic structure of the atoms. While Pd in the ground state favors state 'S with a complete d-shell, the ground state of atomic Pt is 3D where the 6s orbital is also partially occupied.76 The DFT methods used for Pd(II) halide systems94"96 show fairly good reliability. The BP86 functional (Becke's 88 exchange functional) was applied to the square planar palladium compound [PdCU]2"- Recently an investigation of the solvation of square planar palladium complexes was published by Swang.97 There, infrared and NMR spectroscopy results were compared with ab initio calculations at the Hartree-Fock level using MIDI basis set demonstrating relatively good agreement between the experimental and computational techniques. It is well established that contrary to cisplatin, the analogous palladium compound is inactive as an anticancer drug.98 It was observed experimentally that numerous analogous species with Pt(II) and Pd(II) have very similar structural parameters, but their kinetics are substantially different (ca. 104 times faster in the case of Pd(II) compounds).910 The difference in the kinetics of the chemical reactions is important in understanding the various catalytic and biochemical roles of palladium- and platinum-based compounds. For this purpose, the detailed energetics of the hydration processes for the selected square palladium complexes were examined using high-level quantum chemical calculations in order to compare the results with the corresponding energies of the platinum systems. The chosen methodology is based on the known role of individual contributions from Gaussian theory discussed above. The single-point calculations utilizing MP4(SDTQ) with the standard 6-31+G(d) Pople's basis set are sufficiently accurate. Analogous MWB-28 pseudopotentials were used for the description of Pd valence electrons, augmenting the suggested set of pseudo-orbitals with a set of

Towards the Elucidation

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289

diffuse (a s = 0.008, a p = 0.012, and ccd = 0.03) and polarization (otf = 1.47) functions for the sake of basis consistency. 6.1 Structures of the Palladium

Complexes

The most important structural data are summarized in Table 3. The largest bond lengths are observed in the case of Pd-Cl bonds (about 2.31 A). The Pd-O distances for aqua-ligands are about 2.10 A, and the Pd-N bond lengths about 2.04 A (note the analogous Pt-distances!). Some interesting exceptions can be seen, for example, in the [Pd(NH3)3(OH)]+ complex where NH 3 in the trans position to the OH group has a relatively long Pd-N distance of 2.11 A. The reason for such an extension stems from an induction effect which causes a shift in electron density towards the negatively charged OH group, resulting in a mild weakening of the Pd-N bond. The same effect is responsible for the prolongation of Pd-Cl bonds (up to 2.42 A) in [PdCly(OH)x]2" complexes. In the case of Pd-O bond lengths, the shortest bond lengths occur for the hydroxyl groups. Here, the influence of Coulombic forces leads to decreased Pd-O bond distances. Another interesting point is that the Pd-N distances are generally shorter than the PdO distances. This is the same feature as revealed for Pull) complexes, and generally it is not in accord with the covalent and/or atomic radii of nitrogen and oxygen. Both covalent and atomic radii are a little larger in the case of nitrogen. The observation can be explained by considering that the nitrogen atom belongs to the so-called soft (or borderline) atoms (ligands)99100 and that the oxygen atom belongs to hard atoms.90 A similar comparison of the interactions between magnesium and zincgroup cations with the nitrogen sites of DNA bases and water oxygen atoms can be found in one of our recent studies.101 While in the case of the magnesium ion, the ratio of the Mg-O/Mg-N distances roughly follows the ratio of the O/N atomic radii. Zinc clearly favors interactions with nitrogen over oxygen; Zn-N distances are shorter than Zn-O.

290

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Leszczynski

Table 3. Bond lengths of individual Pd complexes (in A) obtained at the MP2/631G(d) level of calculation. Complex [Pd(NH 3 ) 4 ] /+ [Pd(NH 3 ) 3 (H 2 0)] 2+ [Pd(NH3)3(OH)]+ cis-DDPd cis-[PdCl(NH 3 ) 2 (H 2 0)] + cis-PdCl(NH3)2(OH) cis-[Pd(NH 3 ) 2 (H 2 0) 2 ] 2+ cis-[Pd(NH 3 ) 2 (H 2 0)(OH)] + cis-Pd(NH3)2(OH)2 cis-PdCl 2 (NH 3 )(H 2 0) cis-[PdCl2(NH3)(OH)]" cis-PdCl 2 (H 2 0) 2 cis-[PdCl2(H20)(OH)]cis-[PdCl 2 (OH) 2 f cis-[PdCl(NH 3 )(H 2 0) 2 ] + cis-PdCl(NH3)(H20)(OH) cis-[PdCl(NH3)(OH)2]" trans-DDPd trans-[PdCl(NH 3 ) 2 (H 2 0)] + trans-PdCl(NH3)2(OH) trans-[Pd(NH 3 ) 2 (H 2 0) 2 ] 2+ trans-[Pd(NH 3 ) 2 (H 2 0)(OH)] + trans-Pd(NH3)2(OH)2 trans-PdCl2(NH3)(H20) trans-[PdCl2(NH3)(OH)]" trans-PdCl 2 (H 2 0) 2 trans-[PdCl2(H20)(OH)]trans-[PdCl2(OH)2]2trans-[PdCl(NH 3 )(H 2 0) 2 ] + trans-PdCl(NH3)(H20)(OH) trans-[PdCl(NH3)(OH)2][PdCl2(HCl)(H20)] [PdCl3(H20)][PdCl3(OH)]2[PdCU]2"

Pd-N 2.054 2.042/2.016/2.054 2.044/2.110/2.044 2.070 2.008/2.076 2.099/2.060 2.004 2.012/2.104 2.080 2.056 2.031 1.997 2.085 2.064 2.025 2.032/2.048 2.023 2.041 2.049/2.038 2.020/2.021 1.989 2.086 2.067 1.985 2.019 -

Pd-Cl 2.254 2.229 2.271 2.250/2.223 2.345/2.293 2.223 2.257/2.322 2.385/2.429 2.204 2.242 2.358 2.275/2.276 2.209 2.302 2.271/2.271 2.312/2.322 2.267/2.267 2.301/2.336 2.360 2.226 2.307 2.310 2.215/2.213/2.433 2.297/2.235/2.297 2.366/2.389/2.332 2.326

Pd-0 2.083 1.925 2.082 1.925 2.083 1.917/2.078 1.946/1.946 2.133 1.952 2.115 1.953/2.127 1.985/1.961 2.086/2.120 1.922/2.121 1.955/1.953 2.129 1.956 2.047 1.906/2.155 1.976/1.976 2.069 1.933 2.032 1.909/2.188 2.001 2.045 1.942/2.087 1.984 2.150 2.121 1.959 -

The obtained geometric parameters are in a good accord with the bond distances published in another study97 where the Pd-0 distances are 2.12 A

Towards the Elucidation

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291

for formic acid ligands and 2.01 A for formate ligands. (These results were obtained at the Hartree-Fock level with a slightly smaller basis set.) Recently a study of the excited states of cis/trans dichlorodiammineplatinum (DDP) and the palladium analogue has appeared.102 The optimized geometries at the B3LYP/6-311G(3d,3p) level with Hay&Wadt ECP103 (Pd-N and Pd-Cl distances are 2.20/2.10 A and 2.31/2.36 A for the cis/trans conformers, respectively) are in agreement with our results. All the bond lengths follow the trends found in experimental measurements: d(Pd-N) = 2.13/2.05 and d(Pd-Cl) = 2.26/2.33 A (cis/trans).104 Comparing the corresponding platinum and palladium structures from Tables 1 and 3, it can be noticed the metal-ammine distances are very slightly (but still significantly) shorter in the case of Pd. On the contrary metal-chloride and more pronouncedly metal-oxygen (especially from hydroxo ligand) exhibit elongation in Pd-containing complexes. The average elongation is about 0.023 A for Pd-OH bonds. This clearly points to some small differences in softness of both considered metals.105"108 6.2 Comparison ofPd(II) and Pt(II) Hydration

Surfaces

The MP4 single-point calculations were performed for the optimized structures, and the hydration surface was evaluated according to equation 4. The individual reaction energies were used to compose an extended reaction scheme analogous to Figure 4. The discussion of the energies of individual reactions can follow the pattern of the corresponding Pt(II) complexes and, therefore, will be omitted. The computational model used for the hydration of chosen platinum and palladium complexes allows a straightforward comparison of both hydration surfaces. Figure 6 depicts a scheme with the differences between corresponding Pd and Pt reaction energies. For the sake of simplicity, a label Px was used for naming the metal atom in the complexes. Figure 6 underlines the striking extent of the similarity between the energetics of the both metals (in square planar coordination), at least for the hydration reactions presently investigated. The largest difference of the reaction energies is about 10 kcal/mol. This process is however irrelevant for metal hydration since it describes replacement of a chloride particle with an ammine-ligand in the formation of the cis-[Px(NH3)2(H20)(OH)]+ complexes. Actually there are

292

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PxCKNHjMOH)

/

\ PxCUNH.VOW

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PxCl 2 (NH 3 )(H 2 0)

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1.0/2.6

-1.5/-0.5

PxCl ; nMH,«OH)

Figure 6. A differential hydration surface for the Pd(II) and Pt(II) complexes. The values are taken as a difference of the reaction energies E(Pd) - E(Pt) in kcal/mol.

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only five reactions where the energy difference is larger than 4.5 kcal, and three of them contain this complex. The other two are related to a similar product of cis-Px(NH3)2(OH)2 . This is also in agreement with the fact that hydrated Pd(NH3)2(OH)2 is slightly less stable. And consequently, the energy difference between the cis and trans conformer is also slightly increased. In the other cases, there is not only qualitative but also quantitative agreement for all of the considered reactions. Regarding the markedly different biochemical roles of Pt- and Pd-based compounds, the very high degree of similarity between the calculated hydration surfaces of the Pd and Pt compounds does not correspond with the QSAR principle. The differences are considerably smaller compared to those we have recently noticed between Zn2+ and Mg2+ cations.100'109 This clearly demonstrates that an understanding of the anticancer activity of the cisplatin complex is related not only to the thermodynamical description, but also some additional factors have to be included. This will be discussed in the next section.

7. Detailed Molecular Mechanism of the Hydration Processes of Platinum and Palladium Complexes Deeper insight into the reaction mechanism of the hydration of cisplatin, transplatin, and their Pd-analogues is examined in this part. For such studies, the supermolecular approach was chosen where both reactants (metal complex with water) or both product molecules (metal complex with remoted ammonia or chloride particle) were treated as one system. This system was considered electroneutral and optimized using MP2 in singlet electronic groundstate. The most stable conformers of the reactants and products were selected for further analyses. For the determination of the transition state (TS) structures, an associative reaction mechanism was considered in accord with the studies of other authors.33'34'53"55,60,110"113 For some difficult-to-optimize transition states, the QST3 algorithm of Schlegel was found to be very efficient.114 Since frequency analysis was necessary for obtaining vibrational degrees of freedom, geometry re-optimizations of the global minima structures were carried out at the Hartree-Fock (HF) level followed by vibrational modes determinations. The 6-31+G(d) basis set was chosen for the first row elements. The chlorine, palladium, and platinum atoms were described by Stuttgart's relativistic energy-averaged pseudopotentials: CI (MWB-10),115 Pd (MWB-28),116 and Pt (MWB-60)71

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with consistent extensions of the polarization and diffuse functions, analogous to the methodology from the previous section. Lastly, single-point calculations were performed at the CCSD(T)/6-31++G(d,p) level. The main aim of this section is to provide the detailed mechanism of the exchange reaction.

Reactant

Transition State

Product

Figure 7. The supermolecular description of the reaction coordinate. In the case of the second hydration step of cis-Px-X2YW complexes, there are two possibilities how to replace one of the X ligands, yielding both cis-/trans-Px-XYWW' conformers (X, Y = CI or NH3 and W, W = H 2 0 or OH). The solvent molecule can be placed in cis- or trans-coordination relative to the first solvent-ligand. Therefore three reactions are considered for these two cis-reactants. The association energies between the two parts of the supermolecule were determined considering the basis set superposition error (BSSE)" of the Boys-Bernardi"8 counterpoise corrections. Based on the known vibrational modes, as well as the energy and geometry characteristics of the studied reactions, the rate constants can be estimated according to Eyring's Transition State Theory (TST). The canonical rate constant for a bimolecular reaction at a given temperature proceeds according to the following equation: TC7-

kT

F 15

h

FMF

-.exp(EJkT)

(5)

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where F are molecular partition functions per unit volume for TS, Mcomplex, and water; Ea is the activation energy. The rate constants were determined with the program DOIT (Dynamics Of Instanton Tunneling) obtained from Smerdachina et al.119 7.1 The Supermolecules

and TS Structures

The reactants and products of hydration reactions are square planar complexes with one additional remote particle (water for reactants, and ammonium or chloride for products) in the vicinity of the Pt(If)-complex. Several positions of these molecules were examined for every system. Optimized interatomic distances between Pt and other heavy-atoms for the global minima are present in Table 4, and all examined reactions are drawn in Figure 8 where the used notation is also shown. The remote molecule is usually associated through two H-bonds, and the specific location of this molecule is controlled by the common rules valid for H-bonding. A relatively short Pt...O distance for the remote water (< 3.2 A) in the cisDDP+H 2 0 supermolecule was found. Here, besides the two H 2 N-H...O H-bonds, the additional weak interaction of HO-H.. .Pt can be noticed, which holds the water molecule above the Pt-complex plane. Comparing the bond lengths of individual coordinated ligands, the remarkable influence of the trans-effect can be noticed. The ligand-strength diminishes as follows: hydroxyl group > chloride » ammonium > water. This relationship can be obtained from the comparison of many factors. For example, there are longer Pt-N and shorter Pt-Cl bonds in the cisplatin complex than in trans-DDP (where both chlorines compete with each other) or in cis- and trans-PtCl2(H20)2 complexes where a more pronounced reduction of the Pt-Cl bond distances occurs in the cis-conformer. The stronger trans effect of the hydroxyl group is evident from the previous sections where negatively charged complexes were also treated and a comparison of the [PtCU]2" and [PtCl3(OH)]2" anions clearly supports the order.

296 J. V. Burda, J. Sponer and J. Leszczynski Table 4, The bond distances of Px-X where Px=Pd or Pt, and X=N, O or CI. Pt-N 2.051 2.035 2.035 2.014 2.016 2.023 1.983 1.996 2.022 2.020 2.033 2.559 3.537 2.000 2.001 2.032 2.037 1.992 2.473 3.676 1.986 2.042 2.055 2.049 2.031 2.026 2.011 2.005 2.021 1.972 1.984 TS14 2.021 P14 2.021

ROl TSOl POl TS02 P02 R03 TS03 P03 TS04 P04 R05 TS05 P05 TS06 P06 TS07 P07 R08 TS08 P08 TS09 RIO TS10 P10 a TS11 Pll TS12 P12 R13 TS13 pl3a

Pt-N 2.052 2.619 3.582 2.047 2.046 2.034 2.533 3.770 2.029 2.050

2.056 2.587 3.482 2.661 3.613 2.068 2.064 2.027 2.508 3.488 2.023 2.034

Pt-Cl 2.277 2.243 2.253 2.284 2.282 2.312 2.301 2.306 2.260 2.259 2.247 2.246 2.246 2.252 2.254 2.286 2.283 2.284 2.299 2.304 2.246 2.282 2.257 2.257 2.301 2.292 2.688 3.996 2.298 2.324 2.302 2.814 3.939

Pt-Cl 2.277 2.295 2.286 2.693 3.885 2.284 2.301 2.288 2.796 3.921 2.290 2.240 2.253 2.677 3.725 2.654 3.850 2.312 2.229 2.286 2.892

Pt-0 3.183 2.412 2.087 2.344 2.027 3.802 2.295 2.047 2.344 2.063 2.100 2.107 2.076 2.100 2.069 2.099 1.986 2.057 2.061 2.003 2.074 3.609 2.362 2.079 2.480 2.101 2.339 2.040 3.632 2.423 2.062 2.415 2.080

Pt-0 Pd-N 2.059 2.041 2.045 2.044 2.024 2.020 2.003 1.998 2.012 2.014 3.600 2.042 2.424 2.518 2.102 3.521 2.399 2.021 2.039 2.004 2.277 2.026 2.068 2.036 3.702 1.995 2.258 2.464 2.043 3.746 2.366 1.979 2.011 2.063 1.992 2.078 2.009 2.080 1.985 2.060 1.992 2.056 1.984 2.065 1.983 1.996 2.023 2.010 2.005 2.012 2.022 2.004 1.976 2.012 1.959 2.030

Pd-N 2.059 2.575 3.580 2.045 2.046 2.033 2.503 3.827 2.026 2.044

2.090 2.587 3.481 2.642 3.633 2.093 2.020 2.035 2.493 3.493 2.025 2.028

Pd-Cl 2.260 2.243 2.237 2.258 2.264 2.301 2.286 2.294 2.257 2.244 2.231 2.239 2.234 2.228 2.238 2.256 2.263 2.269 2.277 2.289 2.243 2.279 2.276 2.258 2.303 2.292 2.676 3.957 2.259 2.320 2.287 2.716 3.927

Pd-Cl 2.260 2.275 2.268 2.664 3.791 2.272 2.285 2.276 2.806 3.903 2.274 2.228 2.237 2.643 3.646 2.636 3.729 2.302 2.277 2.270 2.892

Pd-0 3.138 2.358 2.096 2.309 2.022 3.776 2.251 2.044 2.260 2.066 2.110 2.114 2.083 2.100 2.072 2.142 1.990 2.056 2.097 2.007 2.067 3.580 2.270 2.106 2.481 2.138 2.280 2.061 3.614 2.385 2.035 2.349 2.032

Pd-0

3.576 2.382 2.110 2.357 2.034 2.241 2.078 3.728 2.199 2.049 2.288 1.927 1.965 1.917 1.896 1.896 1.922 2.045 2.111 1.930 2.005 2.064 1.992

"water and chloride are in the cis-conformation in the product complex of the reaction plO while they are in the trans position in product p l 3 .

Towards the Elucidation of the Activation of Cisplatin 297

,

j£

TSUI

Tsnj TS02

V

P<

© **

, WPiH

0

QP05

«J

T0PO'

R05

^*

TSOT

Figure 8. Optimized structures of reactants, transition states, and products in a supermolecular description of hydration reactions with the used reactions and supermolecules labeling.

298 J. V. Burda, J. Sponer and J. Leszczynski

"-

*»

"X

v

•rsi2

2v„.

W

^

PIJ

TS13

Figure 8. (cont.)

Table 4 collects the Pt-X distances of the transition structures. All of them have a shape of deformed trigonal-bipyramide. The bond distances between Pt and the entering and leaving ligands are substantially longer than the corresponding bond lengths in the reactant or product complexes. As discussed below, all of the hydration reactions are endothermic and in accord with the Hammond postulate, the TS structures resemble the products of the

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reactions. This can be demonstrated for example by the fact that the bond distance for the arriving ligand (Pt-O) is always shorter than Pt-N for the departing ammine ligand despite the shorter Pt-N bonds in the stable complexes. The Pt(II) structural data can be compared with the analogous data for the Pd(II) complexes which are summarized in the second part of Table 4. The most stable conformers of the reactants and products for Pd-containing supermolecules possesses a remote water, ammonia, or chloride oriented in the same manner as in the corresponding platinum complexes. Also, the relationship of the Pd-X bond lengtihs between metal and individual ligands are very close to the Pt analogues. Basically all of the structural similarities already discussed in the previous section as well as in other computational102'120'121 and experimental9'91'122'123 studies are also reflected in the Pd compounds. 7.2 Gibbs Energies of the Ligand Replacement

and

Association

Energies The reaction energies and Gibbs energies for the processes displayed in Figure 8 are collected in Table 5. In the upper part, the values for the platinum complexes are displayed. From the reaction energies and the Gibbs enthalpies, it can be seen that all of the hydration reactions are slightly endothermic. The hydration reaction requires energy of 8 - 21 kcal/mol. The estimated values for the Gibbs enthalpies approximately follow the reaction energies. In the case of chloride replacement, usually larger Gibbs enthalpies reflect the loss of energy from three rotational degrees of freedom of the chloride anion in comparison with a water molecule in reactants. To the contrary, the higher contents of vibronic energy of ammonium stabilize the products in the case of NH3 detachment reactions. The role of a higher level of correlation contributions is obvious from a comparison of the MP2 and CCSD(T) columns of Table 5. While the dechlorination process does not exhibit any substantial deviations (maximum difference is 0.2 kcal/mol), the average decrease of the NH3 detachment energy is about 1.6 kcal/mol. Nevertheless, even these changes are relatively very small. The differences in the MP2 and CCSD activation energies in the AEa(Pt) and AEa(Pd) columns in Table 5 have two sources: a) the corrections on the proper configuration of the reactant states (discussed below) and b) the differences

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J. V. Burda, J. Sponer and J.

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in the correlation energies at the both levels. The correlation energies at the CCSD(T) level are systematically lower by about 1.5-5.3 kcal/mol. In all cases where the cis- and trans- diammine- or dichloro- conformers appear, the cis-structures are less stable due to the higher repulsion of both ligands in the cis-arrangement; thus, lower endothermicity of the reaction involving the cis-conformers can be noticed. For example, comparing the cisDDP+H 2 0 and transDDP+H 2 0 complexes (reactants in the first hydration step), the trans-conformer is about 12 kcal/mol more stable. The relative stability of the supermolecules can be obtained from the generalized hydration hypersurface defined above. The reference zero energy was set in this case for the most stable system: trans-Pt(NH3)2Cl(OH)+H20. This second step reactant is derived from the trans-[Pt(NH3)2Cl(H20)]+ product complex replacing water with an OH" group in order to keep the supermolecules neutral in this reaction branch. The higher stabilization of this reactant complex is due to the stronger electrostatic interaction of the Pt atom with OH" ligand. Such a replacement can be considered to be a reaction caused by the increased pH of the environment.124 The generalized hydration surface is drawn in Figure 9. The ordering of the cis/trans conformers is apparent, as discussed above. From Figure 9, it can be seen that the most stable (final) product of hydration is the mixed cisPt(NH3)Cl(H20)(OH)+NH3 (rl3) complex. The trans-Pt(NH3)Cl(H20)(OH) +NH3 supermolecule lies about 4 kcal/mol higher followed by the transPt(NH3)2(H20)(OH)+Cl (about 5 kcal/mol) and the cisPt(NH3)2(H20)(OH)+Cl (8 kcal/mol) complexes. Hence, the mixed monoammine-monochloro-diaqua-platinated complexes and the diamminediaqua-complexes are lying in near proximity on the potential energy surface as can be seen in the above sections for both of the isolated Pt(II) and Pd(II) complexes. It was established that the most stable complex on the hydration surface of both metals is cis-[Px(NH3)Cl(OH)2]\ In the present study, the most stable hydration product is cis-Px(NH3)Cl(H20)(OH). It is evident that the character of the mixed monoammine-monochloro-complex is preserved. The difference is caused by the fact that only neutral supermolecules were investigated in this section. Thus, no extra electrostatic work, which is necessary for the separation of the two (charged) particles to infinity, appears in the current scheme. This is also the reason why all of the reaction energies can be found in a relatively narrow range (within 12 kcal/mol), while differences up to 200 kcal/mol were observed in the previous

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(thermodynamical) schemes which include also both cationic (2+, 1+) and anionic (2-, 1-) complexes. Table 5. Reaction energies for the hydration of platinum complexes; subscript r denotes a reaction, and a denotes activation energies or Gibbs enthalpies. Index c means activation energies corrected on the proper reactant conformer. Gas-phase COSMO MP2

CCSD(T)

CCSD(T)

AEr(Pt) AGr(Pt) AEa(Pt) AGa(Pt) AGr(Pt) AEa(Pt) AGr(Pt) AEa(Pt) rOl r02 r03 r04 r05 r06 r07 r08 r06 rlO rll rl2 rl3 rl4

10.1 9.5 32.8 30.2 8.7 28.9 10.5 11.4 29.0 29.7 10.2 26.3' 15.0 14.6 36.2 33.1 13.8 33.6 17.9 19.3 31.2 32.2 17.8 29.4 17.0 15.7 39.2 35.9 15.2 30.2' 9.5 10.0 30.5 30.6 9.7 28.4 19.2 19.0 37.6 37.7 18.6 29.4' 21.8 20.5 45.1 41.9 19.9 37.4' 16.8 17.6 36.2 36.8 16.9 34.0 12.8 12.4 34.3 32.7 11.5 30.6 13.8 13.0 39.7 35.9 12.4 30.4' 16.2 16.5 31.4 31.8 16.5 26.0' 17.3 17.2 42.2 40.1 15.2 38.2 20.7 20.7 36.2 35.7 20.6 29.3' AE r (Pd) AG r (Pd) AE a (Pd) AG a (Pd) AG r (Pd) AE a (Pd) rOl 6.1 6.3 21.5 20.1 5.2 18.3 r02 9.5 11.0 20.3 21.3 9.8 18.2C r03 12.1 11.9 26.6 24.6 11.0 23.7 r04 18.0 19.3 23.4 24.5 17.7 21.8 r05 18.9 18.7 27.4 27.4 r06 9.4 10.0 22.6 22.9 r07 13.5 12.1 28.2 25.1 r08 18.0 16.9 31.9 29.4 r06 16.6 17.4 29.3 29.8 rlO 9.5 9.0 22.1 20.7 rll 9.9 9.0 27.6 24.8 rl2 13.6 16.3 22.1 22.3 rl3 16.4 13.9 33.4 27.7 rl4 17.9 16.3 30.0 27.1 x rXY labels are taken from Figure 8.

12.8 6.7 11.5 9.4 10.1 7.0 12.6 12.9 11.4 10.9 9.4 8.4 12.2 10.7

32.3 22.7' 31.1 21.5 30.9' 24.7 25.4' 32.6' 26.5 35.8 33.6' 24.5C 35.1 24.6'

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J. V. Burda, J. Sponer and J.

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This conclusion matches well with the experimental data. In the work of Jestin et al.,39 the cis-diammine-chloro-aqua-complex was considered for direct coordination with the N7-site of the DNA base. The replacement of the aqua-ligand with guanine was found to be substantially faster than the formation of the diaqua-Pt(II) complex. The obtained hydration energies can be compared with the values of experimentally determined equilibrium constants published in studies for both cisplatin26 and transplatin.27 These constants were determined at 318.2 K in a 0.1 M NaC104 solution in pH 2.8-3.4, as mentioned in the Introduction section, and correspond to AG = 1 . 5 and 2.1 kcal/mol for cisplatin and transplatin, respectively. Clearly, the present values are larger (cf. Table 5). Nevertheless, relative agreement should be noticed. The equilibrium constants for the second hydration step of cis-/trans-[Pt(NH3)2(H20)2]2+ cannot be compared directly since we did not consider the charged complexes. The published experimental equilibria26'27 correspond to AG = 2.3 and 3.0 kcal/mol. Despite the differing total charge, the qualitative accordance with hydration of the [Pt(NH3)2(H20)(OH)]+ complexes is preserved. Nevertheless, the mixed Pt(NH3)Cl(H20)(OH) structures were found to be thermodynamically more stable. This is caused by the omitted environmental effects in the model used. The results for the improved models are presented in the next section. The reaction energies and the Gibbs enthalpies for hydration of the Pd(II) complexes are summarized in the lower part of Table 5. In this case, only the first hydration step was evaluated at the CCSD(T) level due to the large amount of computational time necessary for such simulations and only minor differences in the reaction energies in comparison with the MP2 results. All of the reaction energies of the palladium complexes show endothermic character similar to the cis-/transplatin complexes, and even the absolute values strongly resemble the solvation of these complexes. The largest difference is about 3.9 kcal/mol and an average difference of approximately 1.8 kcal/mol (at the MP2 level). The numbers are really convincing concerning the extended similarity of the reaction surface or landscape (in AG). This is also in agreement with the previous parts and 1 ni i on lot

related studies. ' ' Practically, the whole previous energy discussion in this section concerning the Pt(II) hydration applies also to the Pd(II) reactions. If the generalized hydration surface is drawn in the same manner into Figure 9, "Pd/Pt degeneracy" of the reactants and products energies

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would be clearly seen, with the exception of the energies of the TS structures.

o

I 40 | bo

0

Reactants

TS

Products

first hydration step

Reactants

TS

Products

second hydration step

Figure 9. The generalized hydration energy surface for Pt(II) complexes at the CCSD(T) level. Two different parts in the 2nd hydration step are connected with two different types of reactants (with water and with the hydroxyl group): • and A belong to trans-reactants; •,•, and * to cis-reactants; solid and half-filled symbols connected with solid lines label dechlorination; open or open with cross symbols and dotted lines designate ammonia detachment. The strength of the association interactions of the two parts of a supermolecule has been examined and is summarized in the first (gas-phase) column of Table 6 where three sections related to different remote ligands (water, ammonia, and chloride) are present. Surprisingly, the water associates are not usually as strongly bonded as the ammonia complexes in the product states. An explanation of the lower water association energies can be derived from the fact that polarized water, as a coordinated ligand,

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makes stronger H-bonds with remote molecules than Pt-polarized NH3 ligands. Such a conclusion is also confirmed by the fact that, for instance, leaving ammonia in a product state of trans-Pt(NH3)Cl2(H20) forms H-bonds more preferably (by 5 kcal/mol) with H 2 0 and CI ligands than with NH3 and CI. In the reactant part of Table 6, higher association energies can be observed in the case of the second hydration step where one ligand of the solvent (H 2 0 or OH") is already coordinated to Pt. There is no substantial difference in the association energies when a remote water interacts with a neutral water-ligand or hydroxyl anion as follows from the data of the R05 and R08 rows or of the RIO and R13 rows of Table 6. Similarly, the association energies of NH3 are not influenced in such cases, too. The relatively large association energies for the remote chloride species are a consequence of the separation of charged particles (Cl~ and 1+ charged Ptcomplex), that is, the result of electrostatic work. This situation resembles closely the determination of the hydration energies in the sections with isolated Pt(II) and Pd(II) complexes above where the charged molecules and particles were considered. The electrostatic work would be substantially diminished in any polarized solvent, especially in a water environment where efficient screening is present.126 7.3 Activation Energies and Estimation of Rate Constants All of the Pt transition states adopt five-coordinated trigonal-bipyramidal structures, which lie more than 25 kcal/mol above the energy of the reactant supermolecules on the potential energy surface. In Table 5, the last two columns in the MP2 part and the last column of the gas-phase CCSD(T) sections collect the activation energies of TS for the studied hydrations. The lowest activation barrier was predicted for the cisplatin hydration when the first and then second chloride anions are replaced (about 26 kcal/mol). Thus, these dechlorinations represent the fastest reactions considered here. Moreover, the first step is one of the less endothermic processes (ca. 9 kcal/mol) in the whole hydration scheme. The highest barriers (more than 35 kcal/mol) were calculated for the formation of the trans-PtCl2(H20)2+NH3 and trans-Pt(NH 3 )Cl(H 2 0)(0H)+NH 3 complexes. Based on the MP2 activation energies of the Pt complexes, the replacement of the ammonium ligand should always be slower than the dechlorination process. Thus, a kinetic factor would prevail, and the chloride anions would be preferentially replaced. However, this hypothesis was not completely and unambiguously confirmed by the CCSD(T) calculations. The activation

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energies even at the CCSD(T) level are still always higher for the ammonium detachment, but the differences sometimes markedly diminish. Table 6. The association energies between both parts of the supermolecule (in kcal/mol); an asterisk (*) labels the two reactions where the deammination product supermolecule contains a neutral NH3 molecule and an NH/ cation in the COSMO calculations; NH3 was obtained in all corresponding gas-phase products; all energies are BSSE-corrected with inclusion of deformation contributions. Gas-phase COSMO H 2 0+ reactant -10.4 ROl -5.8 -9.2 R03 -2.9 -13.8 R05 -7.4 R08 -12.7 -7.4 RIO -12.7 -8.3 R13 -13.5 -8.8 NH4 + product -12.7 POl -15.6 -16.3* P03 -14.7 -11.0 P05 -15.0 -16.7 P08 -16.5 -15.7 PIO -15.8 -13.6* Pll -11.6 CI" + product P02 -112.5 -8.3 P04 -116.9 -5.4 P06 -121.0 -9.0 P07 -117.0 -4.8 P12 -109.5 -2.4 P14 -111.5 -1.9 An estimation of the TST rate constants was performed from the determined structures, energies, and vibrational modes of all of the reaction components. These constants are collected for both directions of the reactions (k1; k_i, k2, and k_2) at temperature 310 K in Table 7. Comparing the calculated constants for the dechlorination process in the 1st hydration step with the available experimental values,25,39124 a qualitative agreement has been reached. It is clear that differences of several orders of magnitude in the values of the rate constants represent a variation of only a few kcal/mol on the hydration surface. Hence, the requirement for accuracy for such kinetic quantity is at the limit of the current computational techniques for the

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examined systems. From a comparison of forward and backward rate constants, it follows that neither the height of the activation barrier nor the geometry of TS (its location on the energy surface) are sufficiently accurate for quantitative assessments. In the lower section of Table 5, the energy characteristics of the palladium transition states are collected. One can notice that the Pd activation barriers are substantially lower than the corresponding barriers for hydration of the Pt(II) complexes. The average difference between the corresponding barriers is about 10 kcal/mol. The smallest difference of 6.2 kcal/mol occurs in the second chloride replacement of the transPt(NH3)2Cl(OH) structure. It should be mentioned that the global minima on the reactant side do not always coincide with the conformation from which the hydration process starts. For example, for chloride replacement of cisplatin, the global minimum structure must be transferred to the conformation where water is H-bonded simultaneously to ammonium and chloride (about 0.5 kcal/mol higher). In this case, it is expected that the relatively fast movement of the (remote) water molecule over the metal complex can occur, and the required conformation is temporarily available. Such "water transfer" does not represent a reaction bottleneck and will not slowdown the whole hydration process. This assumption is based on the fact that the corresponding minimum (the proper conformation of the given reactant) is energetically close to the global minimum (less than 5 kcal/mol in all considered cases). Since the water transfer is connected only with H-bond reformation, which is weak in comparison with the dative bond rearrangements, the energy of TS between such H-bonded minima is not high. Moreover, in reality, usually more water molecules can be expected in the vicinity of the platinum complex. Nevertheless, these considerations are very important for a correct estimation of the rate constants which are very sensitive (exponentially) to any energy change. Consequently, if the activation energies are only slightly lower (up to 4 kcal/mol), the rate constants are reduced by nearly two orders of magnitude. A complete set of rate constants for hydration of cis/transplatin together with the results for the first solvation step of the analogous Pd complexes and some experimentally known values are summarized in Table 7. The average difference corresponds to 106 times shorter reaction time for the hydration of palladium complexes (Table 7). This conclusion is in good accord with the experimental findings for the

Towards the Elucidation of the Activation of Cisplatin 307

reactions with DNA bases where palladium complexes are expected to interact roughly 104 times faster than platinum(II) complexes.9 Table 7. Forward (k b k2) and backward (k_1( k_2) rate constants estimated from TST atT = 310K. Pt(MP2) r01a r02 exp.a exp.c exp.d r03 r04 exp.b r05 r06 r07 r08 r09 rlO rll rl2 exp.a exp.e exp.f rl3 rl4 exp.b

Gas- phase Pt(CCSD) ki

k.,

ki

ki

1.9e-02 8.3e-01

2.0e-06 5.1e-07

1.1 17.7

1.9 84. 6.0e-02 7.6e-03

3.2e-ll 6.1e-01 Lie-10 4.7e+03

2.0e-09 2.1e-09

5.9 7.5e+04

1.7e-09 1.3e-04 1.9e-04 5.2e-05 l.le-04 l.le-08 5.4e-04 l.le-03

1.7e-01 3.1e+02 2.2

k2 3.5e-13 1.7e-09 1.7e-14 1.9e-17 9.2e-14 5.5e-ll 3.4e-13 2.4e-10

k.2 4.1e-02 2.0e-02 4.1e-01 5.5e-03 1.1 3.1e-02 5.0e-04 l.le+02

k2 7.1e-07 4.9e-08 1.0e-08 5.7e-12 3.4e-12 2.1e-08 1.2e-06 1.6e-06

k.2 1.4 7.7e-01 3.4e+01 2.4e-01 4.9e+01 5.3e-03 2.6e-01 2.2e+03

k2

t2

3.9e-16 4.5e-13

5.0e-04 1.7e+02

2.7e-13 3.2e-08

2.9 4.7e+03

3.6e-09 7.0e-09

Pd(MP2) ki

r01a r02 r03 r04

COSMO Pt(CCSD)

k.,

4.6e-02 1.3e+03 5.3e-03 3.6e+05 2.9e-05 6.7e+03 3.1e-05 1.4e+09

Pd(CCSD) k, k.! 8.2 5.2e+04 1.6e-01 1.6e+07 3.1e-03 1.2e+05 4.4e-04 1.0e+10

'ref.26; bref.27; cref.25; dref.39; eref.134; and fref.135.

2.7e-10 5.5e-01 2.1e-04 8.8e-01 5.6e-07 5.4e+01 9.5e-09 2.2e+00 3.5e-04 7.7e+02 4.5e-10 6.3e-03 2.6e-09 1.7e-01 4.4e-06 1.4e+01 2.3e-04 9.9e-01 8.0e-5 2.0e-5 3.8e-ll 2.4e-04 4.0e-06

3.1e-01 1.0e+02 2.0e-04

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8. Changes in the Thermodynamic and Kinetic Parameters under the Effects of Solvation In the last section, the influence of the adjacent molecules is examined. The applied computational model is based on the same supermolecular approach as above considering the molecular complexes to be neutral and in the singlet electronic ground state. Since the dominant effect of the solute-solvent interaction (at least in the case of water solution) is of electrostatic origin, one possible approach used to study such phenomena is to replace the explicit solvent molecules by a dielectric continuum. This continuum is represented by polarization charges appearing on finite-sized elements on the boundary surface between the solute and the continuum.127 This is the common description for the whole class of the so-called "Apparent Surface Charge" methods (ASC). The COSMO method belongs to this ASC group. The advantage of COSMO is its better description of systems in solvent with high permitivity128 so that it is suitable for the simulation of a water environment. The "United Atomic Topological Model" (UATM) was used for building the cavity. In UATM, hydrogen atoms are not surrounded by their own cavity sphere but are considered to be a subpart of the cavity sphere of the hydrogen-bonded atom. This strategy reduces the number of spheres and still provides high enough accuracy, as pointed by Tomasi ' and Amovilli. The second advantage of this choice is that the method is faster and has better convergence. The utilization of the supermolecular model for the hydration reaction of cisplatin within the UATM molecular cavity is schematically depicted in Figure 7. The designed molecular complexes of the reactants, products, and transition states were optimized using the Becke3LYP functional of the DFT technique and the COSMO method. The used basis set is the same as in the previous in vacuo model. The single point energy determination was performed with the CCSD(T) method and the 6-31++G(d,p) basis set within the COSMO formalism. The active space contained all of the orbitals except those belonging to frozen core electrons (Is of the O and N atoms; inner electrons of Pt and CI were covered within the ECP approach). 8.1 Structures of Supermolecules

in the COSMO Model

The Pt-L distances in structures within the PCM approximation are summarized in Table 8. From the table, two basic features can be noticed.

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First, all distances are increased under the solvent effects, which truly reflects the fact that some portion of electron density of the Pt-complexes is involved in the interactions with the surrounding solvent molecules in more realistic models. Comparing the gas-phase and COSMO models, the average difference in the Pt-L dative bonds is about 0.04 A. Secondly, the largest differences in comparison with the gas-phase structures occur for the Pt-L distances between metal and the weakly bonded remote particles. This can be explained by the fact that these remote ligands are also attracted by the electrostatic charge induced on the cavity surface (simulating the interactions with solution). The largest difference in the molecular parameters occurs in the cisplatin+water complex. The main change can be seen in the shift of the H-bonded water from its position above the Pt-complex plane in the gasphase case towards the plane in the COSMO case (cf. Figure 10a). Usually, weakly bonded particles (water, ammonium, or chloride) were stabilized in the complex plane in both the isolated and the COSMO structures. Arrangement with remote particles above the complex plane is very rare since sufficient additional stabilization is required in this case. Such a stabilization can be found in the R01 reactant in the gas-phase as mentioned above. There, the weak bonding interaction between one of the water protons and the platinum cation stabilizes the water molecule above the complex plane. Passing from the gas phase to the condensed phase model, this interaction was replaced by interaction of the water molecule with induced charges on the cavity surface.126 The additional electrostatic forces diminish this Pt.. .H interaction so that the water molecule could not be kept above the Pt-plane and is transferred towards the complex plane. It can be concluded that in analogy with the gas-phase results, the TS structures more closely resemble the products than the reactants which is in good accord with the Hammond postulate. The other related difference between the results of the gas phase and the COSMO approach reflects a partial change of the reaction coordinate in most of the cases of the deammination reactions (cf. Figure 10b). The ion-pair structures in the frame of the supermolecule are formed as the products in both hydration steps. The NH4+ ion is released from the platinum complex instead of the ammonia molecule. The extra proton is taken from the ligated water. The preference for the N H / cation is about 2.6 kcal/mol. Consequently, the difference between the gas-phase product conformers P10 and P13 disappears. The formation of the ion-pair structure is in good accordance with the quantum chemical calculations of the amino acids. There, the formation of the NH2-

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COOH structure represents the most stable arrangement in the gas-phase calculations, while the NH3+ - COO" zwitterions become the global minima in studies employing PCM techniques. This was for the first time mentioned by Tomasi131 for glycine and later also confirmed by others.I32,133 Except the different character of the product, the transfer from NH3 to N H / does not substantially change the position of the remote ligand. As is demonstrated in Table 8 and Figure 10b, the remote ligand remains in a very similar location in both the gas-phase and the COSMO models.

Figure 10. Superimposed optimized geometries from the gas-phase and COSMO calculations for: (a) the reactant supermolecule of the hydration process rOl: cisPt(NH3)2Cl2+H20; arrow points to the change of water position passing from the gas-phase (thin sticks) to the PCM structure (balls&sticks), (b) the product supermolecule plO: neutral (gas phase - thin sticks) and ion pair structures (PCM ball&sticks). Arrow shows the proton transfer from an aqua-ligand to a released NH4+ particle. 8.2 Thermodynamics and Activation Energies The complete reaction Gibbs free energies of the hydration processes and their activation barriers calculated at the CCSD(T) level are summarized in the COSMO section of Table 5. At first glance, one can notice that practically all of the hydration energies are reduced by several kcal/mol in comparison with the gas-phase results. The reaction energies are usually slightly less endothermic for the chloride replacement. The lowest AG hydration energies are predicted in both steps of cisplatin dechlorination: 6.7 and 8.4 kcal/mol. On the contrary, the deammination of cisplatin is

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accompanied by the largest endothermicity increase of approximately 12.8 kcal/mol in the first step and 12.9 kcal/mol in the second step. Such trends were not observed in the gas-phase calculations. Moreover, the lowest Gibbs energy in the gas-phase is connected with the deammination processes rOl (8.7 kcal/mol) and rlO (8.1 kcal/mol). The predicted energies of the first hydration-dechlorination step (r02 and r04) can be compared with the value of AG = 3.1 kcal/mol for cisplatin and AG = 4.1 for transplatin obtained from experimental studies26'27 and evaluated from the equilibrium constants determined as pK] =2.19 and 2.92 for both cis- and transplatin, respectively. It can be seen that the inclusion of environmental effects substantially improves the predicted energies. The deviation of about 3 - 5 kcal/mol can be considered as an error bar of the model used for the calculations, and some uncertainty should be connected with the determination of the experimental values of pK. A similar comparison can be done for the second step, hydration processes rl2 and rl4, combining reactions Ka3-K2-Kai from eqs. 1 and 2. The evaluation of the corresponding Gibbs energies gives AG2 = AG2 + AGai - AGa3 (assuming the same temperature T = 310 K in all three measurements). In this way, the Gibbs energies AG2 = 3.4 and 4.6 kcal/mol can be predicted for cis- and transplatin, respectively. Here however, the uncertainty in the "experimental" AG2* values is substantially higher. One can conclude that, despite the fact that the calculated energies do not match exactly the experimental data, the appropriate trend in going from the gas-phase to the COSMO values is obtained. Comparing the above results with the analogous data of Zhang,110 very good agreement can be observed. However, since the latter study assumed a slightly different charged reactant (cis-[Pt(NH3)2Cl(H20)]++H20) for the second hydration step, a somewhat higher reaction energy of 14.1 kcal/mol is obtained compared to our value of 7.8 kcal/mol (or AG = 8.4 kcal/mol) for the neutral complex of cis-Pt(NH3)2Cl(OH)+H20. This is in very good accord with the lower stabilization of cisplatin in the diaqua than the aquahydroxo-form as demonstrated already in previous sections. The last column of Table 5 contains information about the reaction barrier heights. The differences between the energies of the optimized TS structures and their reactant conformers, which were obtained going downhill from these TS (using the Intrinsic Reaction Coordinate (IRC) algorithm), were used for the determination of the activation energies at the CCSD(T) level of calculation. It is obvious (the last column of Table 5) that

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Table 8. The optimized Pt-L distances between Pt and the N, O, or CI atom (in A) in the COSMO model and a comparison with gas-phase results from Table 4. Structures P06 = P09, P10 = P13. COSMO ROl TSOl POl TS02 P02 R03 TS03 P03 TS04 P04 R05 TS05 P05 TS06 P06 TS07 P07 R08 TS08 P08 TS09 RIO TS10 P10 TS11 Pll TS12 P12 R13 TS13 TS14 P14

Pt-N 2.065 2.052 2.052 2.058 2.070 2.063 2.030 2.036 2.055 2.067 2.056 2.603 3.809 2.035 2.025 2.049 2.065 2.028 2.364 3.855 2.015 2.068 2.077 2.058 2.056 2.057 2.057 2.087 2.066 2.025 2.056 2.061

Pt-N 2.065 2.617 3.735 2.045 2.034 2.064 2.592 3.898 2.058 2.062 2.074 2.658 3.519 2.673 3.926 2.094 2.026 2.060 2.717 2.057 2.060

Pt-Cl 2.361 2.362 2.369 2.354 2.350 2.369 2.357 2.357 2.344 2.321 2.363 2.332 2.317 2.340 2.312 2.352 2.345 2.357 2.350 2.357 2.336 2.360 2.342 2.369 2.363 2.358 3.009 4.104 2.371 2.390 2.873 4.093

Pt-Cl 2.361 2.339 2.367 2.798 4.083 2.348 2.361 2.352 2.799 4.092 2.322 2.311 2.362 2.890 3.938 2.741 4.031 2.338 2.352 2.357 2.822 -

differences between PCM and gas-phase Pt-0 3.862 2.365 2.040 2.467 2.082 3.961 2.366 2.069 2.421 2.096 2.099 2.108 2.106 2.106 2.098 2.118 2.036 2.094 2.130 2.116 2.096 3.738 2.351 2.038 2.463 2.100 2.405 2.086 3.740 2.249 2.553 2.119

Pt-O Pt-N Pt-N Pt-Cl 0.015 0.015 0.084 0.017 -0.003 0.119 0.017 0.153 0.116 0.044 -0.002 0.070 0.054 -0.012 0.067 0.040 0.030 0.057 0.047 0.059 0.056 0.040 0.128 0.051 0.033 0.029 0.084 0.047 0.012 0.062 3.798 0.023 0.116 2.358 0.044 0.086 2.028 0.272 0.071 2.385 0.035 0.088 2.089 0.024 0.058 2.369 0.017 0.066 2.065 0.027 0.062 3.757 0.036 0.073 2.487 -0.110 0.051 1.9863 0.179 0.053 2.417 0.029 0.090 2.021 0.026 0.018 0.078 2.008 0.022 0.071 0.085 2.044 0.009 0.037 0.112 2.000 0.025 0.012 0.062 1.988 0.031 0.313 0.066 2.003 0.046 0.026 0.321 2.005 0.082 -0.038 0.107 2.032 0.045 0.033 0.073 2.017 0.053 0.209 0.066 1.998 0.034 0.035 0.059 1.986 0.039 0.026 0.154

Pt-Cl Pt-0 Pt-O 0.084 0.679 0.044 -0.047 0.081 -0.047 0.105 0.123 0.198 0.055 0.064 0.159 0.060 0.071 0.064 0.022 0.003 0.077 0.171 0.033 0.032 0.000 0.198 0.071 0.001 -0.066 0.109 0.030 -0.074 0.213 0.006 -0.014 0.213 0.029 0.050 0.087 0.019 0.093 0.180 0.050 -0.003 0.026 0.037 0.055 0.123 0.069 0.230 0.071 0.113 -0.057 -0.070 0.022 0.051 0.129 0.010 -0.011 0.016 -0.041 0.035 -0.017 0.015 -0.001 -0.004 0.066 0.020 0.046 0.022 0.108 -0.174 0.138 0.039

0.009 0.012 0.023 0.027

all of the dechlorination reactions have substantially lower reaction barriers than the corresponding deammination processes. This trend is also partially notable from the gas-phase results; however, in solvent, it is much clearer

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and unambiguous. The relatively large difference between dechlorination and deammination occurs in the first hydration step, about 10 kcal/mol for both cis-/transplatin complexes. In the second step, the difference is slightly smaller but still remains over 6 kcal/mol. These differences distinctly speak out for the kinetic preference of the dechlorination process (cf. discussion below). For the deammination processes (with exclusion of the r04 and r l l reactions) the ligand exchange and the proton transfer from water to ammonia occur simultaneously within the reaction course. However, the heights of the activation barriers are not affected by this fact, since the proton transfer starts on the reaction coordinate a little later after passing the TS structure, going already downhill towards the products on the energy surface. The association energies of platinum complexes with water, ammonia, or chloride species were determined at the CCSD(T)/6-31++G(d.p) level. The obtained results are collected in the second (COSMO) column of Table 6. The obtained values can be compared with the gas-phase calculations. It is apparent that the H-bond interactions of a water molecule with the Ptcomplex are decreased in the COSMO model, on average by about 5 kcal/mol. In the case of the NH3/NH/ detachment, the changes are practically negligible (4 kcal/mol at most). The products of the dechlorination processes represent the most striking cases since the remarkable reduction of the association energies was achieved. The explanation can be deduced from the efficient screening of the electrostatic interaction between the positively charged platinum complex and the chloride ligand. However, this fact does not mean that the electrostatic forces were diminished since other Coulombic interactions based on the interaction between the monomers and the induced charges on the cavity boundaries were introduced. These induced charges realistically simulate the effect of the solvent environment. Therefore it can be assumed that the bulk water environment efficiently compensates the electrostatic work, which is necessary for the separation of the two charge carriers (the CI" anion and [Ptcomplex]+ as well as the NH4+ cation and [Pt-complex]"). 8.3 Evaluation of the Rate Constants The TST rate constants, determined according to eq. 5, are collected in the last two columns of Table 7. The rate constants for the hydrationdechlorination processes in the first step are generally about four orders of

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magnitude higher than the corresponding values for the deammination reaction. For the replacement of the second water molecule, the difference decreases but still remains about three orders of magnitude higher. It means that dechlorination reactions are kinetically preferred over deammination reactions. The detection of the process, where ammonium is released, would be very difficult in real experimental conditions since very low concentration of cisplatin (usually about 5 (J.M) is used. For a comparison with experimental measurements, there are several results available.2527'39 From Table 7, it follows that a very good agreement with the experimental data was achieved. The differences are within an order of magnitude for the forward reaction and slightly worse for reverse processes. For the second dechlorination step, reactions rl2 and rl4, agreement with the published measurements is also fairly good, but here slightly different reactions are considered. Unfortunately there are no data for the evaluation of kinetic parameters such as AG2 in the thermodynamic part. Nevertheless, the rate constants for the second step are not too far from the experimental data, and substantial improvement of the results obtained with the COSMO model in comparison with the in vacuo calculation was achieved. A comparison with other theoretical studies devoted to similar systems110"112 can be performed, at least for dechlorination reactions. Besides cisplatin, the ethylenediammine-dichloro-Pt(II) complexes were also studied. Zhang et al.110 have estimated the energetic and kinetic parameters from a similar model, employing several PCM models and the DFT calculation with a double-zeta quality basis set and with the pseudopotential approach. Their results are in a very good accord with the relevant part of the present data. Similarly, de Costa et al.111,112 made estimations of rate constants for Pt(en)Cl2 dechlorination reactions combining the general Born-Onsager SCRF or PCM models with the DFT or MP2 level of theory. They reported activation barriers of 32 and 23 kcal/mol for gas-phase and PCM, respectively. Also, their rate constants are fairly close to our results (k!=1.9e-ll for gas-phase and k!=4.8e-5 for PCM). This similarity originate, to a large extent, from the chemical propinquity (or relations) of those Pt(II) complexes

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9. Conclusions Recent progress in computational methodologies and improvements in computational facilities allow new insight into many biological processes to be obtained by the accurate, ab initio calculations. Among such processes is the biological activity of cisplatin reviewed in this chapter. The essential results could be summarized as follows: • There are no substantial differences in the thermodynamic behavior of cis and trans isomers during the hydration process. • The most stable products of hydration reactions are [PtCl(NH3)(OH)2]" and Pt(NH3)2(OH)2. These results are in partial agreement with the experimental findings that the hydrated products of cis-DDP contain two ammonia (and two solvent molecules) coordinated to the Pt atom. • The significant similarity of the thermodynamic parameters of the square-planar Pt(II) and Pd(II) complexes was predicted. Since there is considerable difference in the biological activity it is concluded the Pd and Pt compounds do not fulfill the QSAR principle. • Reaction barriers were determined from the TS structures. In agreement with many experimental findings, the activation energies are distinctly lower for palladium complexes than for the corresponding Pt structures; the average difference is 9.7 kcal/mol, which indicates that the hydration processes are about 106 times faster in the palladium case. This is 1-2 orders of magnitude more than the values obtained from experimental data. • Combining the PCM approach with the gas phase methodology is the most promising way to understand and model the effect of a water solution. The COSMO model was used for the determination of the interaction energies and the activation barriers at the CCSD(T)/6-31++G(d,p) level. • The endothermicity of the hydration reactions was confirmed. The Gibbs energies amount to AG = 6.7 and 12.8 kcal/mol for the chloride and ammonium replacements of the water. Analogous values for transplatin are 9.4 and 11.5 kcal/mol. These dechlorination results match the experimental data fairly accurately. • The activation of cisplatin can be regarded purely based on thermodynamic properties. Since the stabilization energies are lowered within each hydration step by about 10 kcal/mol the hydrated complexes become less stable, it means more reactive.

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• The determined rate constants of the dechlorination process are by several (3-4) orders of magnitude larger than those for the deammination process. Very good agreement with the available experimental data was achieved. • The thermodynamically most stable products from the first and second sections (the [PtCl(NH3)(OH)2]_ [PdCl(NH3)(OH)2]" complexes) will be formed with very low probability due to the discriminating kinetic factors for the deammination process.

Acknowledgment The study was supported by NSF-MSMT grant No. 1P05ME784(JVB), grant GAUK No. 2003/181/B_CH(JVB), project MSM 0021620835 (JVB), NIH grant S06 GM008047(JL), NSF-CREST grant HRD-0318519(JL), ONR grant N00034-03-l-0116(JL), grant 203/05/0009, the Grant Agency of the Czech Republic (JS) and AVOZ5 0040507, MSMT CR (JS). The authors thank the Meta-Centers in Prague (Charles University and Czech Technical University), Brno (Masaryk University) and Pilsen (University of West Bohemia) for the generous support of the computational resources.

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INDEX ab initio 121-123, 131, 132, 135, 229-231, 260 activation energies 301, 304, 306, 310, 311, 315 anisotropy Hamiltonian 90, 98 anticancer drug 265, 288 association energies 294, 299, 303-305, 313 atmospheric chemistry 139, 149, 172,175,176,179,182,195, 211 basis set, atomic gaussian type of orbitals (GTO) 37,49-50, 63-64 lobe functions 49 Slater type of orbitals (STO) 50 Born-closure 133 BSSE 294,305 catalysis 121 CCSD(T) 266, 275, 280, 282, 294, 299-305, 308, 310, 311, 315 chemical shift 245 chemisorption 121 combustion 140, 147, 148, 150, 162,189, 195, 198, 200, 209 computer simulations 1-2 Car-Parrinello 72 multi-level or multi-scale 16-18, 49, 54, 60-61, 73 Quantum Monte Carlo 76 conductor-like model of the solvent (COSMO) 68 conjointness of exchange- and kinetic energy functionals 44 constrained search (Levy) 6, 9-10, 15, 24 COSMO 266,301,305,307-315 coupled cluster 229,231 coupling constant 229-234, 237-240, 243-260 operator 232 dative bond 270, 286, 306, 309 Density Functional Theory 84, 91, 92, 115 DFT 243,270,288,308,314 dipole moment 124,133 Dirac Vector Model 230, 253-255 discrete momentum representation (DMR) 123, 125, 129, 131, 132 dissociation reaction 143,181,182 DNA 266, 267, 270-273, 285, 288, 289, 302, 307 damage 122 effective potential in Kohn-Sham equations 7-8, 11-12, 17, 46, 68 KSCED 11-12, 17-18, 48, 63-67, 70-72

323

324

Index

electron density 245, 256 N-representability 6, 8, 14-15 overlap 21, 26-27, 32, 34, 36-37, 57-58, 60, 65 partitioning 3,21 spill-out 19 v-representability (noninteracting pure-state) 7-8, 12-13, 15, 17, 21, 23-24 electron energy loss spectroscopy (EELS) 121, 129, 132 electron scattering 121-123, 125, 131, 132, 135 electronegativity 245-247 electronic excitations 45,67-72 structure 90, 92, 98, 102, 106, 114 electrostatic charge 309 effect 272 embedding 18-19, 63, 65 forces 309,313 interaction 273,300,313 origin 308 term 271 work 300,304,313 energy surface 279, 286, 288, 300, 303, 304, 306, 313 EOM-CCSD 229, 231, 239, 250, 254 equation-of-motion 229,231 equilibrium constants 268,302,311 structure 238,250 Euler-Lagrange minimization 5,7, 11 exchange parameters 112-115 excited state 121, 122,125,126 excited-state 232, 255-257, 259 experimental 230, 231, 233, 239, 240, 250, 252-254, 260 Fermi-contact 232, 233, 237, 243, 245, 249-252, 254, 255 freeze-and-thaw cycle 16-17, 52-55, 60, 69-70 functional of exchange-correlation energy 7-9, 20, 37-38, 44, 63 enhancement factor 30-34, 40 first functional derivative (potential) 11,17, 23, 38, 41, 68-69, 75 Generalized Gradient Approximation (GGA) 31-32, 39-40, 42-45, 57, 75 second functional derivative (response kernel) 47, 69 kinetic energy 28, 42-44 Generalized Gradient Approximation (GGA) 24, 31-32, 44-45, 56-58, 63, 75 regular gradient expansion approximation (GEA) 24, 26-27, 30, 39-40 Thomas-Fermi approximation 23-24,31 von Weizsaecker term 5, 24, 26-28, 31 semi-local 37,57-58

Index 325 total energy embedding (Wesolowski and Warshel) 13-16, 20 Hohenberg-Kohn 4-5,20 Kohn-Sham 6-8,20,41 subsystem (Cortona) 8-11, 20, 36, 37, 42-45 Gaussian orbitals 94 generalized-gradient approximation Gordon-Kim model 10-11,38 Green function 125,126 grid adaptive 51-53 supermolecular 50-53 ground state 232, 255, 256, 259

91

Hammond postulate 298, 309 harmonic approximation 126,132,135 Hartree-Fock 124,129 Heisenberg Hamiltonians 83, 86 Hohenberg-Kohn theorems 4-5,13 HSAB 286 hybrid methods 18,49 hybridization 233 hydration process 265, 267-269, 283, 285, 288, 293, 306, 310, 311, 315 hydrogen abstraction reaction 140, 147, 153, 156, 157, 176, 196 hydrogen bond 229-240, 243, 245, 248-254, 256-260 hydrogen bonding 26, 33-38, 58, 69 hyperfine tensor 63, 76 kinetic parameters 268, 271, 273, 308, 314 Kohn-Sham equations 7,13,24,46 Kohn-Sham Equations with Constrained Electron Density (KSCED) LCAO 110 Lieb-Oxford bound 40 ligand field 64-67 Lippmann-Schwinger equation 123, 125 Local Density Approximation 21, 23, 37-41, 67, 75-76 magnetic anisotropy barrier 95,103,109 moment 253,255-258 magnetogyric ratio 229, 238, 239, 247, 248 Mn4 dimer 106,107,112 monomer 90, 106-108, 112 Mn12-acetate 87, 89, 90, 98, 102-106, 111-114 molecular magnets 83, 84, 86, 89, 90, 111, 112, 115

3, 12, 28, 74

326

Index

M0ller-Plesset 125,131,231 MP2 270, 273-275, 277-282, 290, 293, 299, 302, 304, 307, 314 MP4 273, 275, 280-282, 288, 291 NMRTWM 229-231, 254-256, 258-260 non-additive kinetic energy bi-functional 9-12, 19-20, 24, 26-28, 32-34, 36, 40, 42-43, 49, 52 approximations to 23 GEA0 21,23-24,29,38-39,41 GEA2 24, 29-30, 34 GGA 24, 29, 33-34, 36, 39 definition 8 exact properties 21, 27-28, 37 for unrepresentable pairs of electron densities 22-23 in four-electron systems 21-22,27 nonrelativistic approximation 232 Nuclear Magnetic Resonance 229-231, 254, 261, 264 open-shell systems 63-67, 76 optimized effective potential 73 orbital-free (or KSCED) embedding potential orbitals 7-11,22-23 embedded 12-15, 20, 45, 64-67, 72 highest occupied (HOMO) 72 Kohn-Sham 22,45 localized for subsystems 50-51, 65 lowest unoccupied (LUMO) 72 orthogonality 7, 11-12, 15, 20

17-18, 63-65, 67, 69-72

paramagnetic 232 plane-wave 124-127, 132 Poisson's equation 94 polarizability (dynamic) 46, 48 polarizable continuum 68 polarization electronic 19, 43, 65-66, 69-70 polyatomic molecules 121-123, 128, 131, 132 polycyclic aromatic hydrocarbons 56-60 potential energy surface ab initio calculations 140, 141, 147, 165, 173, 193, 203 characteristic points activated complex 142,147 generalized transition state 142 intermediate complex 141, 146, 150, 153, 155, 162, 173, 176, 184, 185 saddle point 141,142 transition state 141, 147, 150, 152, 153, 155, 159, 190, 191, 205 profile energy barrier 140, 143, 148, 153, 155, 156, 159, 162, 163, 166, 167, 169, 183, 186, 187, 191, 193, 207, 209

Index 327 reaction coordinate 142 reaction path 141, 150, 152, 156, 159, 172, 183, 185, 186, 190, 191, 194, 195 proton transfer 18,60 proton-acceptor 248,251 pseudopotential 19, 20, 270, 273, 274, 276, 288, 293 pyrazine 248,249 pyridine 234, 240, 241, 248-250 QSAR

293,315

rate constant 269, 294, 295, 304-307, 313, 314, 316 high-pressure limiting 140, 144, 145, 180, 181, 199, 200, 205, 209, 211, 217 low-pressure limiting 144, 145, 184, 195, 200, 201, 211, 216 overall 143, 144, 146, 148, 149, 153, 154, 158, 159, 162, 165-168, 185, 187, U 203 rate theory of bimolecular reactions canonical variational transition state theory (CVT) 142 transition state theory (TST) 141, 15 3, 170, 174 tunneling correction 141,142 variational transition state theory (VTST) 142 of unimolecular reactions 145, 187 RRKM method 144, 146, 160, 170 statistical adiabatic channel model 145, 147, 180, 199, 205 fall-off range 144,145,200 reaction enthalpy 148, 156 field 68 kinetics 139, 140, 143, 146, 147, 150, 153, 154, 165, 169, 182, 184, 185, 187, 191, 197, 199, 200, 203, 207, 208-210, 214, 216 mechanism 140, 141, 146, 149, 150, 152, 153, 155, 159, 162, 169, 170, 173, 174, 176, 182, 183, 185, 187, 188, 190, 193, 203-205, 209 branching ratio 148-150, 152-154, 156, 161, 165-169, 203, 205 complex reaction 140,146,162,182 elementary reaction 140 multi-step reaction 140, 146, 176 reaction channel 148-150, 153, 155, 156, 159, 161-163, 165-171, 182, 191,196, 204, 206, 208, 209, 212, 214 reactions of acetaldehyde 182, 185, 189 atoms bromine 149, 158, 159, 161, 177-179 chlorine 148, 149, 154, 155, 159, 161, 162, 168, 211 fluorine 149, 150, 152, 155, 156, 162, 177, 179, 196, 203, 206, 208, 210-212,214 hydrogen 162, 165 formaldehyde 182

328

Index

free radicals 1-methylvinoxy (acetonyl) 189,193 amidogen 205, 206, 208-210 ethyl 198, 199 hydroxyl 148-150, 155, 156, 167, 168, 182, 185, 195, 196, 198, 199, 206, 208 hydroxymethyl 147, 148, 150, 154-157, 159, 161, 162, 165-171 methoxy 147-150, 152-157, 160, 161, 165-171 methyl 149, 165, 168, 195, 196, 199, 200, 209 trifluoromethoxy 172 trifluoromethyl 203 vinoxy 189-191 methanol 147-150, 152, 154-156, 158, 159, 161, 162, 165-168, 170 molecular oxygen 195, 215-217 nitric oxide 172, 189, 190, 193, 200 nitrogen dioxide 195, 203 trifluoromethanol 171, 172, 176-179 trifluoromethyl hypohalites 178,179 recombination reaction 143, 144, 178-180, 195-198, 205, 206, 217 reduced coupling constant 230,234, 238-240, 243, 245-249, 252-254, 256-259 resonant tunneling 84, 88-90, 95, 112 scattering angle 130, 131, 133 Schrodinger equation 124 solvatation 18,71-72 solvatochromism 18,68,71 solvent disorder 106 effects 309 spin contamination 111, 114 excitations 86,103,104,112,115 spin-dipole 232 spin-orbit coupling 84,95-97,103 spin-spin coupling 229-233, 239, 240, 252-255, 258, 260 static-exchange approximation 123, 124, 127, 131-133, 135 sum-over-states 255,257 Thomas-Fermi model 5 time-dependent density functional theory (TDDFT) 45 linear response (LR-TDDFT) 45-48, 68, 71-72 transition operator 125 structure 231 transverse anisotropy 105 triplet 229-232,254-259 TST 273, 294, 305, 307, 313

Index 329 unimolecular decomposition

121, 222

van der Waals complexes 10, 37-44, 55-59 variational integration mesh 92 vibrational 121-123, 125, 126, 128, 129, 132-134 wave-function based methods 2, 38, 57 configuration interactions 68 coupled cluster 38 Hartree-Fock 10,38 MP2 57 multireference 68 zeolite 61-62 ZPVE 274,280-282

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CONTENT INDEX Volume 1 Relativistic Many-Body Calculations on Atoms and Molecules (Y. Ishikawa & U. Kaldor) Modern Developments in Hartree-Fock Theory: Fast Methods for Computing the Coulomb Matrix (M. Challacombe, E. Schwegler & J. Almlof) Local Shape Analysis of Macromolecular Electron Densities (P. G. Mezey) Liquid-State Quantum Chemistry: Computational Applications of the Polarizable Continuum Models (J.-L. Rivail & D. Rinaldi) Elemental Boron Route to Stuffed Fullerenes (E. D. Jemmis & B. Kiran) Interactions of DNA Bases and the Structure of DNA. A Nonempirical Ab Initio Study with Inclusion of Electron Correlation (/. Sponer, P. Hobza & J. Leszczynski) Computational Approaches to the Design of Safer Drugs and Their Molecular Properties (TV. Bodor & M. -J. Huang)

Volume 2 The Electron Propagator Picture of Molecular Electronic Structure (J. V. Ortiz) SAC-CI Method: Theoretical Aspects and Some Recent Topics (H. Nakatsuji) Quantum Monte Carlo and Electronic Structure (R. N. Barnett & W. A. Lester, Jr.) Molecular Structure and Infrared Spectra of the DNA Bases and Their Derivatives: Theory and Experiment (M. /. Nowak, L. Lapinski, J. S. Kwiatkowski & J. Leszczynski) Derivation and Assessment of a New Set of Ab Initio Potentials and Its Application to Molecular Dynamics Simulations of Biological Molecules in Vacuo, in Crystal and in Aqueous Solution (M. Aida) Practical Exercises in Ab Initio Quantum Chemistry — The World Wide Web as a Teaching Environment (H. P. Liithi, G. Vacek, A. Hilger & W. Klopper) Volume 3 Multireference Brillouin-Wigner Coupled-Cluster Theory (7. Hubac, J. Masik, P. Mach, J. Urban & P. Babinec) Model Core Potentials: Theory and Applications (M. Klobukowski, S. Huzinaga & Y. Sakai) Stratospheric Bromine Chemistry: Insights from Computational Studies (S. Guha & J. S. Francisco) 331

332

Content

Index

Bonding in Gas-Phase Sulfur Radicals (D. C. Young & M. L. McKee) Chemistry of the Liquid State: Current Trends in Quantum-Chemical Modeling (L. Gorb & J. Leszczynski) Chemistry and the Internet (K. Flurchick, M. Hurley, J. K. Labanowski, G. H. Lushington & T. L. Windus) Volume 4 Topography of Atomic and Molecular Scalar Fields (S. R. Gadre) The Ab Initio Model Potential Method: A Common Strategy for Effective Core Potential and Embedded Cluster Calculations (L. Seijo & Z. Barandiaran) Continuum Models of Macromolecular Association in Aqueous Solution (M. A. Olson) Interactions of Nucleic Acid Bases: The Role of Solvent (M. Orozco, E. Cubero, B. Hernandez, J. M. Lopez & F. J. Luque) Recent Advances in Multireference M0ller-Plesset Method (K. Hirao, K. Nakayama, T. Nakajima & H. Nakano) Detonation Initiation and Sensitivity in Energetic Compounds: Some Computational Treatments (P. Politzer & H. E. Alper) Volume 5 In Search of the Relationship between Multiple Solutions Characterizing CoupledCluster Theories (P. Piecuch & K. Kowalski) Computational Time-Dependent Two-Electron Theory and Long-Time Propagators (C. A. Weatherford) Self-Consistent Field Theory of Weakly Bonded Systems (E. Gianinetti, I. Vandoni, A. Famulari & M. Raimondi) Aromatic DNA Base Stacking and H-Bonding (J. Sponer, P. Hobza & J. Leszczynski) Direct Ab Initio Dynamics Methodology for Modeling Kinetics of Biological Systems (T. N. Truong & D. K. Maity) Molecular Structure and Vibrational IR Spectra of Fluoro, Chloro and Bromosubstituted Methanes, Silanes and Germanes: An Ab Anitio Approach (J. S. Kwiatkowski & J. Leszczynski) Volume 6 Relativistic Multireference M0ller-Plesset Perturbation Theory (Y. Ishikawa & M. J. Vilkas)

Content Index

333

15 Years of Car-Parrinello Simulations in Physics, Chemistry and Biology (U. Rothlisberger) Methods of Combined Quantum/Classical (QM/MM) Modeling for Organometallic and Metallobiochemical Systems (7. B. Bersuker)

Large

A Review of Ab Initio Calculations on Proton Transfer in Zeolites (M. Allavena & D. White) Ionic Clusters with Weakly Interacting Components-Magic Numbers Rationalized by the Shell Structure (S. Roszak & J. Leszczynski) Turning Point Quantization and Scalet-Wavelet Analysis (C. R. Handy) Volume 7 Molecules as Components in Electronic Devices: A First-Principles Study (M. Di Ventra) Tackling DNA with Density Functional Theory: Development and Application of Parallel and Order-N DFT Methods (C. F. Guerra, F. M. Bickelhaupt, E. J. Baerends & J. G. Snijders) Low-Scaling Methods for Electron Correlation (S. Saeb0) Iterative and Non-Iterative Inclusion of Connected Triple Excitations in CoupledCluster Methods: Theory and Numerical Comparisons for Some Difficult Examples (J. D. Watts) Explicitly Correlated Coupled Cluster R12 Calculations (J. Noga & P. Valiron) Ab Initio Direct Molecular Dynamics Studies of Atmospheric Reactions: Interconversion of Nitronium Ions and Nitric Acid in Small Clusters (Y. Ishikawa & R. C. Binning, Jr.) Volume 8 Computational Modelling of the Solvent Effects on Molecular Properties: An Overview of the Polarizable Continuum Model Approach (R. Cammi, B. Mennucci & J. Tomasi) Electronic and Nonlinear Optical Properties of 2-Mefhyl-4-Nitroaniline Clusters {M. Guillaume, B. Champagne, F. Castet & L. Ducasse) Ab Initio Calculations of the Intermolecular Nuclear Spin-Spin Coupling Constants (M. Pecul & J. Sadlej) Base Polyad Motifs in Nucleic Acids - Biological Significance, Occurrence in Three-Dimensional Experimental Structures and Computational Studies (M. Meyer & J. Siihnel)

334

Content

Index

Model Calculations of Radiation Induced Damage in DNA Constituents Using Density Functional Theory (D. M. Close) Excited States of Nucleic Acid Bases (M. K. Shukla & J. Leszczynski)

Volume 9 Molecular Electronics with Gaussian98/03 (J. J. Palacios, A. J. E. Louis, E. SanFabidn, J. A. Verges & Y. Garcia)

Perez-Jimenez,

Molecular Dynamics Simulations of Single Molecule Atomic Force Microscope Experiments (W. Nowak & P. E. Marszalek) Molecular Dynamics Simulations of a Molecular Electronics Device: The NanoCell (J. Seminario, P. Derosa, L. Cordova & B. Bozard) Computation of Excited State Potential Energy Surfaces via Linear Response Theories Based on State Specific Multi-Reference Coupled Electron-Pair Approximation Like Methods (S. Chattopadhyay, D. Pahari, U. Mahapatra & D. Mukherjee) Modelling of Anisotropic Exchange Coupling in Rare-Earth-Transition-Metal Pairs: Applications to Yb 3+ -Mn 2+ and Yb 3+ -Cr 3+ Halide Clusters and Implications to the Light Up-Conversion (M. Atanasov, C. Daul & H. U. Giidel) Is a Dihydrogen Bond a Unique Phenomenon? (S. J. Grabowski & J. Leszczynski)

Computational There have been important developments in the last decade: computers are faster and more powerful, code features are enhanced and more efficient, and larger molecules can be studied — not only in vacuum but also in a solvent or in crystal. Researchers are using new techniques to study larger systems and obtain more accurate results. This is impetus for the development of more efficient methods based on the first-principle multi-level f simulations appropriate for complex species. f Among the cutting-edge methods and studies reviewed in this decennial volume of the series are the Density Functional Theory (DFT) method, vibrational electron energy loss spectroscopy (EELS), computational models of the reaction rate theory, the nuclear magnetic resonance triplet wavefunction model (NMRTWM) and biological reactions that benefit from computational studies.

^ f

"... constitutes convincing proof of steady progress in developing computational chemistry methodology. ... excellent reading material for a wide range of readers." Professor W. A. Sokalskt Wroclaw University of Technology, Poland "A good balance between chapters that focus on theory and chapters that focut on applications. The theoretical developments discussed here are o\'general Interest to practitioners of computational chemistiy."

Professor Kang iiway Chuan National University of Singapore, Singapore

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