Elements of Quantum Chemistry

460 Pages·1981·12.554 MB·English

by  Rudolf Zahradník, Rudolf Polák (auth.)

ELEMENTS
OF QUANTUM
CHEMISTRY
ELEMENTS
OF QUANTUM
CHEMISTRY
by
R UDOLF ZAHRADNIK
J. Heyrovsky Institute of Physical Chemistry and Electrochemistry,
Czechoslovak Academy of Sciences, Prague
RUDOLF POLAK
1. Heyrovsky Institute of Physical Chemistry and Electrochemistry,
Czechoslovak Academy of Sciences, Prague
PLENUM PRESS. NEW YORK AND LONDON
SNTL • PUBLISHERS OF TECHNICAL LITERATURE, PRAGUE
Distributed throughout the world with the exception of the Socialist countries by
Plenum Press,
a Division of Plenum Publishing Corporation,
221 West 11th Str«t, New York 10011
ISBN-13: 918-94-009-6261-5 c-JSBN-J 3: 918-1-4613-3921-2
001: 10.1001/918-1-4613-3921-2
Library of Congress Catalog Card Number 11-85610
© 1980 Rudolf Zahradnik, RudoU Polak
Soficover reprint of the hardcover 15 t edition 1980
Translated by Jill Horky
English edition first published in 1980 simultaneously by Plenum Publishing Corporation and
SNTL- Publishers of Technical Literature, Prague
No part of this book may be reproduced, stored in a retrieval system, or transmitted
in any form or by any means, electronic, mechanical photocopying, microfilming, recording,
or otherwise, without written permission from the Publisher
CONTENTS
1. Introduction 9
2. A brief comment on the development of the theory of the chemical
bond 11
3. The time-independent Schrodinger equation 13
3.1 Introduction of the equation 13
3.2 Formulation of the SchrOdinger equation for simple systems 18
3.2.1 A particle in a one-dimensional potential box 18
3.2.2 The harmonic oscillator 18
3.2.3 The hydrogen atom 19
3.2.4 The hydrogen molecular ion, H; 19
3.3 Examples of the solution of the SchrOdinger equation 20
3.3.1 The free particle 20
3.3.2 A particle in a potential box; the solution and its consequences 22
3.3.3 The harmonic oscillator 27
3.3.4 The rigid rotator 33
3.3.5 The hydrogen atom 35
References 46
4. Mathematics and logic of quantum mechanics 47
4.1 Linear operators and their properties 47
4.2 Axiomatic foundation of quantum mechanics 49
4.3 Consequences of the axiomatic system 51
4.4 Constants of motion. The Pauli principle 55
4.5 Matrix representation of operators and operations with matrices 65
4.6 Approximate solution of the Schrodinger equation: variation and perturbation
methods 70
References 79
5. Basic approximations in the theory of the chemical bond 80
5.1 Introductory comments 80
5.2 Neglecting of non-electrostatic interactions 81
5.3 The Born-Oppenheimer and adiabatic approximations 82
5.4 The method of configuration interaction 87
5.5 The independent electron model (one-electron approximation) 92
5.6 The method of molecular orbitals as linear combinations of atomic orbitals 100
References 102
6
6. Symmetry in quantum chemistry 103
6.1 Introduction 103
6.2 Symmetry transformations of the Hamiltonian 106
6.3 The principal symmetry groups and their notation 110
6.4 Matrix representation of symmetry groups 114
6.5 Selection rules for matrix elements 125
6.6 Symmetry and hybrid orbitals 128
6.7 Spin and spatial symmetry of many-electron systems 137
6.8 Perturbation treatment for symmetrical systems 147
References 148
7. Atomic orbitals (AO) and molecular orbitals (MO) 150
7.1 The significance of hydrogen type orbitals; atomic orbitals 150
7.2 Hybridization 151
7.3 Molecular orbitals 154
References 160
8. M any-elect ron atoms 161
8.1 The one-electron approximation and the periodic system of the elements 161
8.2 The total angular momentum 165
References 169
9. Diatomic molecules 170
9.1 Introductory comments; the hydrogen molecular ion, H; 170
9.2 The H2 molecule 174
9.3 Calculation of the molecular integrals 179
9.4 General diatomic molecules and correlation diagrams 184
References 189
10. Calculation methods in the theory of the chemical bond 190
10.1 Introductory remarks 190
10.2 All-valence electron MO-LCAO methods 195
10.2.1 Methods explicitly considering electron repulsion 195
10.2.2 Methods using an effective Hamiltonian 207
10.3 It-Electron theory 209
10.3.1 It -a-Electron separation 209
10.3.2 The Pople version of the SCF method for It-electron systems 211
10.3.3 The Pariser-Parr method of limited configuration interaction 214
10.3.4 A survey of semiempirical It-electron methods 216
10.3.5 Very simple It-electron version of the MO method 222
10.3.6 Perturbation methods within the framework of the simple MO method 233
10.4 The FE-MO method 238
10.5 Valence bond theory (VB method) 239
10.6 The crystal field and ligand field theories 249
10.6.1 Introductory comments 249
10.6.2 The electrostatic model (crystal field) 251
10.6.3 Ligand field theory 259
References 260
7
11. Use of the solution to the Schrodinger equation 263
11.1 Quantities related to the molecular energy (the total electron energy, ionization
potential, electron affinity, excitation energy) 263
11.2 'Quantities derived from the wave function 272
11.2.1 Introductory comments 272
11.2.2 Density matrix 273
11.2.3 Localized orbitals 278
11.2.4 Electron distribution in molecules 282
11.2.5 Dipole moment 285
11.2.6 Nodal planes of molecular orbitals: the Woodward-HolTmann rules 289
References 293
12. Examples of the study of polyatomic molecules 295
12.1 Introductory comments 295
12.2 Inorganic compounds 295
12.3 Organic compounds 304
12.4 Examples of systems studied in biochemistry 308
References 311
13. Molecular spectroscopy 312
13.1 Phenomenological description 312
13.1.1 Introductory comments 312
13.1.2 Units and the spectral regions 313
13.1.3 Absorption and emission spectra, the population of excited states 316
13.2 Excitation within a single electronic level 321
13.2.1 Introductory comments on radiofrequency spectroscopy 321
13.2.2 Nuclear quadrupole resonance (NQR) 323
13.2.3 The elementary theory of magnetic resonance 324
13.2.4 Nuclear magnetic resonance (NMR) 326
13.2.5 Electron spin resonance (ESR) 333
13.2.6 Pure rotational spectra 340
13.2.7 Vibrational spectroscopy 341
13.2.8 Raman spectroscopy 345
13.3 Excitation within the framework of several electronic levels 347
13.3.1 Absorption spectra in the ultraviolet and visible regions 347
\3.3.2 Luminescence phenomena (fluorescence, phosphorescence) 366
13.3.3 Photochemistry 369
References 376
14. Magnetic properties of molecules 377
References 382
15. Thermochemical properties and molecular stability 383
15.1 Heats of formation and atomization 383
15.2 Delocalization energies of conjugated compounds 385
15.3 Stabilization of coordination compounds 387
Reference 390
8
16. Chemical reactivity 391
16.1 Introductory comments 391
16.2 Empirical approach 393
16.3 Theoretical approach 396
16.3.1 Qualitative considerations 396
16.3.2 Quantitative considerations. Calculations of absolute values of equilibrium
and rate constants 410
16.4 Calculations of relative equilibrium and rate constants 417
16.5 Compromise approach: the quantum chemical treatment 419
16.5.1 Reactions of conjugated compounds 419
16.5.2 Substitution reactions of complexes of the transition elements 434
References 436
17. Weak interactions 438
17.1 Introduction 438
17.2 van der Waals interaction between a pair of linear oscillators 439
17.3 Various means of calculating intermolecular interaction energies 442
17.4 Application of weak interactions from the point of view of physical chemistry 446
References 449
Index 451
1. INTRODUCTION
The post-war generation of chemists learned to handle a blow pipe at
the university as thoroughly as modern chemistry students learn to write
computer programmes. Even after World War II the rule of three was
considered to be sufficient mathematical knowledge for chemists and the
short course of "higher mathematics" at technical universities was the
test most feared by chemistry students. However, even then some en
visaged the theoretical derivation of information on the properties of
molecules from knowledge of the bonding of the component atoms.
During the last quarter of this century, amazing changes have
occurred in chemistry, some of them almost incredible. Dirac's famous
clairvoyant statement* has been partially realized. Incorporation of
quantum mechanics into chemistry encountered numerous difficulties.
After all, the reserve of experimental chemists is not surprising. For
decades the hydrogen and helium atoms and the hydrogen molecule
belonged among the systems most frequently investigated by theoreti
cians. Later these systems were supplemented by ethylene and benzene.
The authors of this book can therefore recall with understanding the words
of the late Professor Lukes: "Well, when they succeed in computing a
molecule of some alkaloid by those methods of yours ..." . Unfortunately,
the calculations on calycanin were not completed before his death.
Now there is no need to convince even the members of the older
generation of the usefulness of quantum chemistry for chemists. Even
the most conservative were convinced after the introduction of the W ood
ward-Hoffmann rules.
* "The underlying physical laws necessary for the mathematical theory of a large part
of physics and the whole of chemistry are thus completely known, and the difficulty is only
that the exact application of these laws leads to equations much too complicated to be
soluble. It therefore becomes desirable that approximate practical methods of applying
quantum mechanics should be developed, which can lead to an explanation of the main
features of complex atomic systems without too much computation". [Po A. M. Dirac:
Proc. Roy. Soc. (London) 123, 714 (1929).]
10
This book is concerned, on the one hand, with an introduction
to the theory of the chemical bond to a degree necessary for active
understanding of quantum chemical semi-empirical methods (Chapter 10,
which completes the methodical part) and, on the other hand, with the
study of the relationships between the structures of molecules and their
properties. Among these properties, both the static characteristics (thermo
chemical, electric, magnetic, optical) and the dynamic characteristics,
chemical reactivity characterized by the equilibrium and velocity con
stants, will be discussed. It is necessary to define the meaning of the term
"structure" more precisely. In a narrow sense structure means the way
in which atoms are bonded in molecules or the arrangement of molecules
in a crystal lattice. In recent years, structure in this sense has often been
determined directly using X-ray analysis. Here, as a rule, for the probable
structure of a compound the theoretical characteristics will be determined
by computation and afterwards will be compared with the experimental
results.
An attempt will be made to acquaint the reader with these com
parisons in such a way as to enable him not only to perform similar
comparisons himself but also to open new possibilities. In comparing
theoretical and experimental quantities, both a more profound qualitative
explanation of the studied properties and processes and quantitative
interpretation of experimental data will be necessary. This approach will
help in generalizing the knowledge obtained and in condensing large
groups of experimental data into empirical formulae, in which, of course,
quantities appear resulting from quantum-chemical calculations. These
relationships will be used as interpolation formulae and will permit
estimation of the values of experimental characteristics in substances not
yet prepared, whose properties are of interest. Moreover, there is also
the very attractive possibility of using the quantum theory of the chemical
bond not only for the interpretation, but also for the prediction of
properties.
2. A BRIEF COMMENT
ON THE DEVELOPMENT
OF THE THEORY
OF THE CHEMICAL BOND
It is admirable that, as early as in the nineteenth century, chemists
succeeded in defining concepts of the structure of substances that are
in remarkable agreement with modern knowledge of the quantum theory
of the chemical bond and with direct structural determinations using
electron or neutron diffraction and X -ray analysis. Only in the theory
published in 1916 by Kossel and Lewis did electrons assume a decisive
role in concepts of the origin of the chemical bond. (The electron was
discovered by Thomson only 19 years earlier, and 5 years earlier
Rutherford proposed the planetary model of the atom.) The basic
concepts of this very successful and innovative theory are based on the
ideas of electrovalency and covalency, which are still accepted at the
present time. This theory of the chemical bond forms a basis for the
theory of mesomeric and inductive effects which contributed consider
ably to the rationalization of organic and inorganic chemistry (Robinson,
Ingold, Arndt, Eistert). The work carried out by their predecessors
(Kekule, Cooper, Butlerov, Werner, and in spatial structure Le Bel and
van't Hoff) is of essential importance.
The difficulties encountered in classical mechanics will be men
tioned in another context. Here, however, it should be noted that
classical Newtonian mechanics is useful for the description and predic
tion of phenomena in the middle and macro cosmos. The growing need
to describe the motion of particles forming molecules and atoms led to
the establishment of a new mechanics, quantum mechanics, in the
twenties of this century.
The fundamental equation of this new mechanics, the Schrodinger
equation, can be obtained in two ways. The method given by SchrOdinger
is apparently less complicated, proceeding from the concept that electron
motion can be described in terms used for the description of wave
motion, leading to the term "wave mechanics".
Quite independently, the same result was achieved by Heisenberg,
who made use of matrix properties. Although the two approaches are