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lectronic tructure

and the Properties of Solids

THE PHYSICS OF THE CHEMICAL BOND

Walter A. Harrison

STANFORD UNIVERSITY

DOVER PUBLICATIONS, INC., New York

Copyright © 1980, 1989 by Walter A. Harrison.

All rights reserved under Pan American and International Copyright

Conventions.

Published in Canada by General Publishing Company, Ltd., 30 Lesmill Road,

Don Mills, Toronto, Ontario.

This Dover edition, first published in 1989, is an unabridged, corrected

republication of the work first published by W. H. Freeman and Company, San

Francisco, 1980. The author has written a new Preface for the Dover edition. The

"Solid State Table of the Elements," a foldout in the original edition, is herein

reprinted as a double-page spread.

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

Library of Congress Cataloging-in-Publication Data

Harrison, Walter A. (Walter Ashley), 1930-

Electronic structure and the properties of solids: the physics of the chemical

bond / by Walter A. Harrison.

p. cm.

"An unabridged, corrected republication of the work first published by

W. H. Freeman and Company, San Francisco, 1980"-T.p. verso.

Bibliography: p.

Includes index.

ISBN 0-486-66021-4

I. Electronic structure. 2. Chemical bonds. 3. Solid state physics. 4. Solid

state chemistry. 1. Title.

QC 176.8.E4H37 1989

530.4' II-dc20 89-34153

CIP

To my wife, Lucky,

and to my sons, Rick, John, Bill, and Bob

Preface to the Dover Edition

Recent Developments

IT IS WITH GREAT PLEASURE that I greet the Dover edition of this book,

which joins my Solid State Theory as affordable physics. It comes with some

minor corrections to the last printing by W. H. Freeman and Company.

This text appeared in 1980, very early in the development of the simplified

methods for calculating properties in the context of tight-binding theory. As

mentioned in the original preface, the derivation of the basic formulae for

interatomic couplings only arose during the production of the first edition.

Fortunately, all the essentials of the theory were complete enough to be

included. There have been a number of developments since the appearance

of the book which both simplify the theory and make it more accurate. It has

not been possible to incorporate these in this edition but it may be helpful to

give references to the principal ones.

Perhaps the most significant was a redetermination of the parameters

giving the coupling between atomic orbitals on neighboring atoms.1 By

incorporating an additional atomic orbital in pelturbation theory, as done for

other reasons by Louie,2 it was possible to fit a larger set of energy-band

values and the fitting was more stable. The resulting couplings were rather

different (y]SS(T = -1.32, Y]spu = 1.42, Y]ppu = 2.22, and y]PP7T= -0.63, rather than

the adjusted values given in Table 2-1). The additional atomic orbital could

then be discarded and with the new parameters it became possible to abandon

the distinction between two types of covalent energies (V2 and V2h) and the

viii Preface to the Dover Edition

corresponding two types of polar energies (V3 and V3h); one could use those

based upon hybrids for dielectric as well as bonding properties. This was a

very considerable simplification with no appreciable loss of accuracy. Since

we were changing the couplings, we also changed over to the use of Hartree￾Fock term values, from page 534, instead of the Herman-Skillman term

values from the Solid State Table. The latter were appropriate when most of

our comparisons were with band calculations which utilized similar

approximations to those used in the Herman-Skillman tables. We tend now to

compare more with experiment and the Hartree-Fock tables are closer to the

experimental term values.

A second simplification was the introduction of overlap repulsions

between atoms in covalent solids as a power-law variation, 7]oV22/IEhl, with

the coefficient 7]0 adjusted to give the correct lattice spacing.3 A similar

form varying as the inverse eighth power of spacing was introduced for ionic

solids. 4 This is not quite as accurate nor general as exponential forms but by

using the algebraic form with the leading factor fit to obtain the known

equilibrium spacing, it was possible to write all terms in the energy in terms

of the parameters of the theory (V 1, V 2, and V3) and thus to obtain

elementary formulae for properties such as the bulk modulus. This does

produce appreciable errors, however, and more accurate procedures have

been developed by van Schilfgaarde and Sher.5

Extended bond orbitals were introduced on page 83 of the text, but few of

the corresponding corrections to the properties were calculated. Since

publication corrections have been made to the total energy of semiconductors

to obtain cohesion,3 heats of solution,6 and corrections to the dielectric

properties.7 There have also been studies of Coulomb effects8 in

semiconductors and insulators, including self-consistency and the "many￾body" enhancement of the gap, in the same spirit as the analyses in this text.

We completed the evaluation of the effective interaction between ions in

metals introduced on page 3 87, using the Fermi -Thomas dielectric function

from page 378. This led to the remarkably simple form V(d) = Z2e2e - Kd X

cosh2Krcld and a good description of the bonding properties of simple

metals. 9 We also followed up the analysis of transition-metals given in

Chapter 20 in a series of studies,10 and on the analysis of transition-metal

compounds11 given in Chapter 19. As might be expected, we also made

application of the elementary theory of electronic structure to the newly

discovered high-temperature superconductors.12

Recent studies by Zaanen, Sawatzky, and Allen13 have made it clear

that the origin of the metal-insulator transition in transition-metal

compounds, discussed in Section 19-B, is not associated with the s- to d-state

promotion to which we attributed it and nothing from that section should be

used without considering these more recent and complete studies.

T~ere have been very dramatic developments in the understanding of

semIconductor surface reconstructions discussed in Section lO-B. A number

of theoretical studies showed that Coulomb effects will prevent the Jahn￾Preface to the Dover Edition

Teller ~istortion proposed by Haneman and discussed in Section lO-B.

Pandey prop.os~d that the observed two-by-one reconstruction of the silicon

(111) surface IS mstead due to a 1T-bonded chain configuration, which is now

generally ac~epted. Th~ two-by-one reconstruction on the silicon (lOa)

surface., WhICh we attnbute? to a "ridge" structure, is now generally

r~coglllzed to be the Schher-Farnsworth dimer formation, which we

d~scussed but thought an unlikely structure. It is also established that the

dimers are canted as proposed by Chadi. 15 The adatom model of the seven￾by-s~ven reconstruction on silicon (111) surfaces, which we proposed in

S~ctIon 10-D, w.as .spectacularly confirmed using the scanning tunneling

mIcroscope b'y Bmlllg, Rohr~r, Gerber and Weibel,16 with almost exactly the

Lander-Mo.rnson. p~ttern whIch ~e suggested. However, further studies by

!ak.ayanagI, TalllshIro, TakahashI, and Takahashi17 indicated a much more

mtrIcate structure including also stacking faults and dimers; that model is

generally .accept~d. Finally the natural semiconductor band line-ups

proposed m SectIOn 10-F were brought into question by Tersoff ,18 who

su.gges~e? that there were "neutral points" in the energy bands which would

a~me.' fIxmg the band ?ff-sets at heterojunctions. In the context of the tight￾bIlldI~g ~heory of thIS 1text thes~ neutral .points are the average hybrid

energieS ~n each crystal.. 9 A.ny dI~ference m the average hybrid energy on

the two SIdes of a heteroJunctIOn WIll be reduced by a factor of the dielectric

constant of the systems. The reason the natural band line-ups of Section lO-F

worked as well as .they d.id is that ~he average hybrid energies are frequently

the sa~e so no dielectnc screenmg is necessary. The theory based upon

matchmg average hybrids19 is just as simple and more general and accurate

than that given here.

These n:ore rece~t de.velopments have strengthened and supported the

me~hods. dI.scussed III thIS text. Except for the new choice of parameters,

WhICh elImmated the awkward u.se of two ~ets of covalent and polar energies,

the.se developments do not .m?dIfY.the baSIC theory described, but simply add

to It .. 1 hope that the descnptron gIven here can continue to be useful to the

matenals scientist and physicist.

References:

Walter A. Harrison

April 1988

lW. A. ~anison, New tight-binding parameters for covalent solids obtained using Louie

Penpheral States, Phys. Rev. B24, 5835 (1981).·

2S. Louie, New localiz~d-orbital method/or calculating the electronic structure of

molecules and sohds: covalent semIconductors, Phys. Rev. B22, 1933 (1980).

IX

x Preface to the Dover Edition

3W. A. Harrison, Theory of the two-center bond, Phys. Rev. B27, 3592 (1983).

4W. A. Harrison, Overlap interaction and bonding in ionic solids, Phys. Rev. B34, 2787

(1986).

SM. van Schilfgaarde and A. Sher, Tight-binding theory and elastic constants, Phys. Rev.

B36, 4375 (1987).

6E. A. Kraut and W. A. Harrison, Heats of solution and substitution in semiconductors, J.

Vac. Sci. and Techno!. B2, 409 (1984), Lattice distortion and energies of atomic

substitution, ibid B3, 1231 (1985), and W. A. Harrison and E. A. Kraut, Energies of

substitution and solution in semiconductors, Phys. Rev., in press.

7W. A. Harrison, The dielectric properties of semiconductors, Microscience 4, 121

(1983).

8W. A. Harrison, Coulomb interactions in semiconductors and insulators, Phys. Rev.

B31, 2121 (1985).

9W. A. Harrison and J. M. Wills, Interionic interactions in simple metals, Phys. Rev.

B25, 5007 (1982), and J. M. Wills and W. A. Harrison, Further studies on interionic

interactions in simple metals and transition metals, Phys. Rev. B29, 5486 (1984).

lOS. Froyen, Addendum to "Universal LCAO parameters for d-state solids", Phys. Rev.

B22, 3119 (1980); W. A. Harrison, Electronic structure off-shell metals, Phys. Rev ..

B28, 550 (1983), J. M. Wills and W. A. Harrison, Interionic interactions in transition

metals, Phys. Rev. B28, 4363 (1983); W. A. Harrison, Localization inf-shell metals,

Phys. Rev. B29, 2917 (1984); G. K. Straub and W. A. Harrison, Analytic methods

for calculation of the electronic structure of solids, Phys. Rev. B31, 7668 (1985).

llW. A. Harrison and G. K. Straub, Electronic structure and bonding in d- andJ-metal AB

compounds, Phys. Rev. B35, 2695 (1987).

12W. A. Harrison, Elementary theory of the properties of the cup rates , in Novel

Superconductivity, edited by Stuart A. Wolf and Vladimir Z. Kresin, Plenum Press,

(New York, 1987), p. 507; W. A. Harrison, Superconductivity on an YBa2Cu307

lattice, Phys. Rev. B, in press.

13J. Zaanen, G. A. Sawatzky, and J. W. Allen, Band gaps and electronic structure of

transition-metal compounds, Phys. Rev. Letters 55, 418 (1985).

14K. C. Pandey, New lr-bonded chain model for Sir 111 )-(2x1) surface, Phys. Rev.

Letters 47,1913 (1981).

1SD. J. Chadi, Atomic and electronic stuctures of reconstructed Si (100) surfaces, Phys.

Rev. Letters 43, 43 (1979).

16G. Binnig, H. Rohrer, Ch. Gerber, and E. Weibel, 7x7 reconstruction on Si (111)

resolved in real space, Phys. Rev. Letters, 50, 120 (1983).

17Takayanagi, Y. Tanishiro, M. Takahashi, and S. Takahashi, Structural analysis of Si

(11 I )-7x7 by UHV-transmission electron diffraction and microscopy, J. Vac. Sci. and

Te~hno!. A3, 1502 (1985).

Preface to the Dover Edition Xl

18J. Tersoff, Theory of semiconductor heterojunctions: the role of quantum dipoles, Phys.

Rev. B30, 4874 (1984).

19W. A. Harrison and J. Tersoff, Tight-binding theory of heterojunction band lineups and

interface dipoles, J. Vac. Sci. and Techno!. B4, 1068 (1986).

Preface to the First Edition

IN THE PAST FEW YEARS the understanding of the electronic structure of solids has

become sufficient that it can now be used as the basis for direct prediction of the

entire range of dielectric and bonding properties, that is, for the prediction of

properties of solids in terms of their chemical composition. Before that, good

theories of generic properties had been available (for example, the free-electron

theory of metals), but these theories required adjustment of parameters for each

material. It had also been possible to interpolate properties among similar

materials (as with ionicity theory) or to make detailed prediction of isolated

properties (such as the energy bands for perfect crystals). The newer predictions

have ranged from Augmented Plane Wave (APW) or multiple-scattering tech￾niques for calculating total energies in perfect crystals, possible with full-scale

computers, to elementary calculations of defect structures, which can be done with

linear combinations of atomic orbitals (LCAO theory) or pseudopotentials on

hand-held calculators. The latter, simpler category is of such importance in the

design of materials and in the interpretation of experiments that there is need for a

comprehensive text on these methods. This book has been written to meet that

need.

The Solid State Table of the Elements, folded into the book near the back

cover, exemplifies the unified view of electronic structure which is sought, and its

relation to the properties of solids. The table contains the parameters needed to

calculate nearly any property of any solid, using a hand-held calculator; these

are parameters such as the LCAO matrix elements and pseudopotential core radii,

in terms of which elementary descriptions of the electronic structure can be given.

The approach used throughout this book has been to simplify the description of

xiv Preface to the First Edition

the electronic structure of solids enough that not only electronic states but also

the entire range of properties of those solids can be calculated. This is always

possible; the only questions are: how difficult is the calculation, and how accurate

are the results? For determining the energy bands of the perfect crystal, the

simplified approach does not offer a competitive alternative .to m~re tradi~ional

techniques; therefore, accurate band calculations are used as mput mformatlOn￾just as experimental results are used-in establishing understa~ding, tests,. and

parameters. It is only with great difficulty that these band-calculatIOnal technIques

can be extended beyond the energy bands of the perfect crystal. On the other

hand, the simplified approaches explained in this book, though they give only

tolerable descriptions of the bands, can easily be applied to the entire range of

dielectric, transport, and bonding properties of imperfect as well as perfect solids. In

most cases, they give analytic forms for the results which are easily evaluated

with a hand-held electronic calculator.

Linear combinations of atomic orbitals are used as a basis for studying covalent

and ionic solids; for metals the basis consists of plane waves. Both bases are

related, however, and the relations between the parameters of the two systems are

identified in the text. The essential point is not which basis is used for expansion:

either basis can give an arbitrarily accurate description if carried far enough. T~e

point is that isolating the essential aspects within either fr~mework, and. t~~n dis￾carding (or correcting for) the less essential aspects, provIdes the p~sslblhty for

making simple numerical estimates. It is also at the root of what IS meant by

"learning the physics of the system" (or" learning the chemistry of the system,"

if one is of that background.)

Use of LCAO and plane wave bases does not necessarily make the parts of the

text where they are used independent, since we continually draw on insight from

both outlooks. The most striking case of this is an analysis in Chapter 2 in which

the requirement that energy bands be consistent for both bases provides formulae

for the interatomic matrix elements used in the LCAO studies of sp-bonded solids.

This remarkable result was obtained only in late 1978 by Sverre Froyen and me,

and it provided a theoretical basis for what had been empirical formulae when

the text was first drafted. The development came in time to be included as a

fundamental part of the exposition; it followed on the heels of the much more

intricate formulation of the corresponding LCAO matrix elements in transition

metals and transition metal compounds, which is described in Chapter 20.

Neither of these developments has yet appeared in the physics journals. Indeed,

because the theoretical approaches have been developing so rapidly, several studies

contained here are original with this book. The analysis of angular forces in ionic

crystals-the chemical grip-is one such case, and there are a n.umber of others.

I think of the subject as new; the text could not have been WrItten a few years

ago and certainly some changes would be made if it were to be writt~n a few years

from now. However, I believe that the main features of the theory will not change,

as the general theory of pseudo potentials has not changed fundamentally since the

writing of Pseudopotentials in the Theory of Metals at the very inception of that

field. In any case, the subject is much too important to wait for exposition until

every avenue has been explored.

Preface to the First Edition

The text itself is designed for a senior or first-year graduate course. It grew out

of a one-quarter course in solid state chemistry offered as a sequel to a one-quarter

solid state physics course taught at the level of Kittel's Introduction to Solid State

Physics. A single quarter is a very short time for either course. The two courses,

though separate, were complementary, and were appropriate for students of

physics, applied physics, chemistry, chemical engineering, materials science, and

electrical engineering.

Serving so broad an audience has dictated a simplified analysis that depends on

three approximations: a one-electron framework, simple approximate interatomic

matrix elements, and empty-core pseudopotentials. Refinement of these methods

is not difficult, and is in fact carried out in a series of appendixes. The text begins

with an introduction to the quantum mechanics needed in the text. An introductory

course in quantum mechanics can be considered a prerequisite. What is reviewed

here will not be adequate for a reader with no background in quantum theory,

but should aid readers with limited background.

The problems at the ends of chapters are an important aspect of the book. They

clearly show that the calculations for systems and properties of genuine and current

interest are actually quite elementary. A set of problem solutions, and comments on

teaching the material, are contained in a teacher's guide that can be obtained from

the publisher.

I anticipate that some users will object that much of the material covered in this

book is so recent it is not possible to feel as comfortable in teaching it as in

teaching a more settled field such as solid state physics. I believe, however, that the

subject dealt with here is so important, particularly now that techniques such as

molecular beam epitaxy enable one to produce almost any material one designs,

that no modern solid state scientist should be trained without a working knowledge

of the kind of solid state chemistry described in this text.

Walter A. Harrison

June 1979

xv

Contents

PART I ELECTRON STATES

1 The Quantum-Mechanical Basis

A. Quantum Mechanics

B. Electronic Structure of Atoms

C. Electronic Structure of Small Molecules

D. The Simple Polar Bond

E. Diatomic Molecules

2 Electronic Structure of Solids

A. Energy Bands

B. Electron Dynamics

C. Characteristic Solid Types

D. Solid State Matrix Elements

E. Calculation of Spectra

PART II COVALENT SOLIDS

3 Electronic Structure of Simple Tetrahedral Solids

A. Crystal Structures

B. Bond Orbitals

C. The LCAO Bands

D. The Bond Orbital Approximation and Extended Bond Orbitals

E. Metallicity

F. Planar and Filamentary Structures

2

3

8

16

20

22

31

32

36

38

46

55

59

61

62

64

71

80

88

90

xviii Contents

Contents XIX

11 Mixed Tetrahedral Solids 257 4 Optical Spectra 96

A. Tetrahedral Complexes 258 A. Dielectric Susceptibility 97 B. The Crystal Structure and the Simple Molecular Lattice 261 B. Optical Properties and Oscillator Strengths 100 C. The Bonding Unit 263 C. Features of the Absorption Spectrum 105 D. Bands and Electronic Spectra 267 D. X 1 and the Dielectric Constant 110 E. Mechanical Properties 275

F. Vibrational Spectra 277 5 Other Dielectric Properties 118 G. Coupling of Vibrations to the Infrared 282

A. Bond Dipoles and Higher-Order Susceptibilities . 118

B. Effective Atomic Charge 124

C. Dielectric Screening 127 PART III CLOSED-SHELL SYSTEMS 289 D. Ternary Compounds 129

E. Magnetic Susceptibility 131 12 Inert-Gas Solids 291

A. Interatomic Interactions 292 6 The Energy Bands 137 B. Electronic Properties 295

A. Accurate Band Structures 138

B. LCAO Interpretation of the Bands 142 13 Ionic Compounds 299

C. The Conduction Bands 151 A. The Crystal Structure 299 D. Effective Masses 155

B. Electrostatic Energy and the Madelung Potential 303 E. Impurity States and Excitons 163

C. Ion-Ion Interactions 307

D. Cohesion and Mechanical Properties 309 7 The Total Energy 167

E. Structure Determination and Ionic Radii 314

A. The Overlap Interaction 168

B. Bond Length, Cohesive Energy, and the Bulk Modulus 171 14 Dielectric Properties of Ionic Crystals 318 C. Cohesion in Polar Covalent Solids 173

A. Electronic Structure and Spectra 319

8 B. Dielectric Susceptibility 326 Elasticity 180

C. Effective Charges and Ion Softening 331 A. Total Energy Calculations 181 D. Surfaces and Molten Ionic Compounds 336 B. Rigid Hybrids 185

C. Rehybridization 191

D. The Valence Force Field 193

E. Internal Displacements, and Prediction of C44 197 PART IV OPEN-SHELL SYSTEMS 339

15 Simple Metals 341 9 Lattice Vibrations 203 A. History of the Theory 342

A. The Vibration Spectrum 204 B. The Free-Electron Theory of Metals 345

B. Long Range Forces 210 C. Electrostatic Energy 349

C. Phonons and the Specific Heat 215 D. The Empty-Core Pseudopotential 350

D. The Transverse Charge 218 E. Free-Electron Energy 353

E. Piezoelectricity 224 F. Density, Bulk Modulus, and Cohesion 354

F. The Electron-Phonon Interaction 225

16 Electronic Structure of Metals 359 10 Surfaces and Defects 229 A. Pseudopotential Perturbation Theory 360

A. Surface Energy and Crystal Shapes 230 B. Pseudopotentials in the Perfect Lattice 364

B. Surface Reconstruction 233 C. Electron Diffraction by Pseudopotentials 367

C. The Elimination of Surface States, and Fermi Level Pinning 243 D. Nearly-Free-Electron Bands and Fermi Surfaces 369

D. Adsorption of Atoms and the 7 x 7 Reconstruction Pattern 247 E. Scattering by Defects 373

E. Defects and Amorphous Semiconductors 249 F. Screening 376

F. Photothresholds and Heterojunctions 252

xx Contents

17 Mechanical Properties of Metals 383

A. The Band-Structure Energy 384

B. The Effective Interaction Between Ions, and Higher-Order Terms 386

C. The Phonon Spectrum 390

D. The Electron-Phonon Interaction and the Electron-Phonon

Coupling Constant

E. Surfaces and Liquids

18 Pseudopotential Theory of Covalent Bonding

A. The Prediction of Interatomic Matrix Elements

B. The Jones Zone Gap

C. Covalent and Polar Contributions

D. Susceptibility

E. Bonding Properties

F. Ionic Bonding

G. Interfaces and Heterojunctions

19 Transition-Metal Compounds

A. d States in Solids

B. Monoxides: Multiplet d States

C. Perovskite Structures; d Bands

D. Other Compounds

E. The Perovskite Ghost

F. The Chemical Grip

G. The Electrostatic Stability of Perovskites

H. The Electron-Phonon Interaction

396

399

407

408

410

416

419

421

424

425

430

431

433

438

452

455

459

468

471

20 Transition Metals 476

A. The Bands 477

B. The Electronic Properties and Density of States 488

C. Cohesion, Bond Length, and Compressibility 494

D. Muffin-Tin Orbitals and the Atomic Sphere Approximation 500

E. d Resonances and Transition-Metal Pseudopotentials 508

F. Local Moments and Magnetism 520

APPENDIXES

A. The One-Electron Approximation

B. Nonorthogonality of Basis States

C. The Overlap Interaction

D. Quantum-Mechanical Formulation of Pseudopotentials

E. Orbital Corrections

Solid State Table of the Elements

Bibliography and Author Index

Subject Index

531

536

539

543

546

552

555

571

Electronic Structure and the Properties of Solids

PART I

ELECTRON

STATES

IN THIS PART of the book, we shall attempt to describe solids in the simplest

meaningful framework. Chapter 1 contains a simple, brief statement of the

quantum-mechanical framework needed for all subsequent discussions. Prior

knowledge of quantum mechanics is desirable. However, for review, the premises

upon which we will proceed are outlined here. This is followed by a brief descrip￾tion of electronic structure and bonding in atoms and small molecules, which

includes only those aspects that will be directly relevant to discussions of solids.

Chapter 2 treats the electronic structure of solids by extending the framework

established in Chapter 1. At the end of Chapter 2, values for the interatomic

matrix elements and term values are introduced. These appear also in a Solid

State Table of the Elements at the back of the book. These will be used extensively

to calculate properties of covalent and ionic solids.

The summaries at the beginnings of all chapters are intended to give readers

a concise overview of the topics dealt with in each chapter. The summaries will

also enable readers to select between familiar and unfamiliar material.

CHAPTER 1

The Quantum￾Mechanical Basis

SUMMARY

This chapter introduces the quantum mechanics required for the analyses in this text. The

state of an electron is represented by a wave function I/J. Each observable is represented

by an operator 0. Quantum theory asserts that the average of many measurements of an

observable on electrons in a certain state is given in terms of these by J ljJ*OI/Jd3 r. The

quantization of energy follows, as does the determination of states from a Hamiltonian

matrix and the perturbative solution. The Pauli principle and the time-dependence of the

state are given as separate assertions.

In the one-electron approximation, electron orbitals in atoms may be classified accord￾ing to angular momentum. Orbitals with zero, one, two, and three units of angular momen￾tum are called s, p, d, andf orbitals, respectively. Electrons in the last unfilled shell of sand p

electron orbitals are called valence electrons. The principal periods of the periodic table

contain atoms with differing numbers of valence electrons in the same shell, and the

properties of the atom depend mainly upon its valence, equal to the number of valence

electrons. Transition elements, having different numbers of d orbitals orf orbitals filled, are

found between the principal periods.

When atoms are brought together to form molecules, the atomic states become

combined (that is, mathematically, they are represented by linear combinations of atomic

orbitals, or LeAD's) and their energies are shifted. The combinations of valence atomic

orbitals with lowered energy are called bond orbitals, and their occupation by electrons

bonds the molecules together. Bond orbitals are symmetric or nonpolar when identical

atoms bond but become asymmetric or polar if the atoms are different. Simple calculations

of the energy levels are made for a series of nonpolar diatomic molecules.

I-A Quantum Mechanics 3

I-A Quantum Mechanics

For the purpose of our discussion, let us assume that only electrons have impor￾tant quantum-mechanical behavior. Five assertions about quantum mechanics

will enable us to discuss properties of electrons. Along with these assertions, we

shall make one or two clarifying remarks and state a few consequences.

Our first assertion is that

(a) Each electron is represented by a wave function, designated as tjJ(r). A wave

function can have both real and imaginary parts. A parallel statement for light

would be that each photon can be represented by an electric field 8(r, t). To say

that an electron is represented by a wave function means that specification of the

wave function gives all the information that can exist for that electron except

information about the electron spin, which will be explained later, before assertion

(d). In a mathematical sense, representation of each electron in terms of its own

wave function is called a one-electron approximation.

(b) Physical observables are represented by lineal' operators on the wavefunction.

The operators corresponding to the two fundamental observables, position and

momentum, are

position +--+ r,

h

momentum +--+ p = --;- V,

I

(1-1)

where h is Planck's constant. An analogous representation in the physics of light is

of the observable, frequency of light; the operator representing the observable is

proportional to the derivative (operating on the electric field) with respect to time,

a/at. The operator r in Eq. (1-1) means simply multiplication (of the wave func￾tion) by position r. Operators for other observables can be obtained from

Eq. (1-1) by substituting these operators in the classical expressions for other

observables. For example, potential energy is represented by a multiplication by

V(r). Kinetic energy is represented by p2/2m = - (h2/2m)"I/ 2 . A particularly impor￾tant observable is electron energy, which can be represented by a Hamiltonian

operator:

_h2

H = 2m V2 + V(r). (1-2)

The way we use a wave function of an electron and the operator representing an

observable is stated in a third assertion:

(c) The average value qf measurements qf an observable 0, for an electron with

wave function tjJ, is

(1-3 )

4 The Quantum-Mechanical Basis

(If IjJ depends on time, then so also will (0).) Even though the wave function

describes an electron fully, different values can be obtained from a particular

measurement of some observable. The average value of many measurements of the

observable 0 for the same IjJ is written in Eq. (1-3) as (0). The integral in the

numerator on the right side of the equation is a special case of a matrix elemellt; in

general the wave function appearing to the left of the operator may be different

from the wave function to the right of it. In such a case, the Dirac notation for the

matrix element is

(1-4 )

In a similar way the denominator on the right side of Eq. (1-3) can be shortened to

(1jJ IIjJ ). The angular brackets are also used separately. The bra (11 or (1jJ 1 I means

1jJ1(r)*; the ket 12) or 11jJ2) means 1jJ2(r). (These terms come from splitting the

word" bracket.") When they are combined face to face, as in Eq. (1-4), an integra￾tion should be performed.

Eq. (1-3) is the principal assertion of the quantum mechanics needed in this

book. Assertions (a) and (b) simply define wave functions and operators, but

assertion (c) makes a connection with experiment. It follows from Eq. (1-3), for

example, that the probability of finding an electron in a small region of space, d3 r,

is 1jJ*(r)ljJ(r)d3r. Thus 1jJ*1jJ is the probability density for the electron.

It follows also from Eq. (1-3) that there exist electron states having discrete or

definite values for energy (or, states with discrete values for any other observable).

This can be proved by construction. Since any measured quantity must be real,

Eq. (1-3) suggests that the operator 0 is Hermitian. We know from mathematics

that it is possible to construct eige1lstates of any Hermitian operator. However, for

the Hamiltonian operator, which is a Hermitian operator, eigenstates are ob￾tained as solutions of a differential equation, the time-indepellde1lt Schroedi1lger

equatioll ,

HIjJ(r) = EIjJ(r), (1-5)

where E is the eigenvalue. It is known also that the existence of boundary condi￾tions (such as the condition that the wave functions vanish outside a given region

of space) will restrict the solutions to a discrete set of eigenvalues E, and that these

different eigenstates can be taken to be orthogonal to each other. It is important to

recognize that eigenstates are wave functions which an electron mayor may not

have. If an electron has a certain eigenstate, it is said that the corresponding state

is occupied by the electron. However, the various states exist whether or not they

are occupied.

We see immediately that a measurement of the energy of an electron repre￾sented by an eigenstate will always give the value E for that eigenstate, since the

I-A Quantum Mechanics 5

average value of the mean-squared deviation from that value is zero:

(1-6)

We have used the eigenvalue equation, Eq. (1-5), to write H 11/1) = E 11jJ). The

electron energy eigenstates, or e1lergy levels, will be fundamental in many of the

discussions in the book. In most cases we shall discuss that state of some entire

system which is of minimum energy, that is, the ground state, in which, therefore,

each electron is represented by an energy eigenstate corresponding to the lowest

available energy level.

In solving problems in this book, we shall not obtain wave functions by solving

differential equations such as Eq. (1-5), but shall instead assume that the wave

functions that interest us can be written in terms of a small number of known

functions. For example, to obtain the wave function 1/1 for one electron in a

diatomic molecule, we can make a linear combination of wave functions IjJ 1 and

1jJ2, where 1 and 2 designate energy eigenstates for electrons in the separate atoms

that make up the molecule. Thus,

(1-7)

where U1 and U2 are constants. The average energy, or e1lergy expectation value, for

such an electron is given by

UtU1(1jJ1I H I1jJ1) + UtU2(1jJ1I H I1jJ2) + U~U1(1jJ2IHI1jJ1) + U~U2(1jJ2IHI1jJ2)

UtU 1(1jJ1 I 1jJ1) + utu/1jJ111jJ2) + U~U1(1jJ21I/1J + U!U2(1jJ211jJ2)

(1-8)

The states compnsmg the set (here, represented by I!/J 1) and IIjJ 2») in

which the wave function is expanded are called basis states. It is customary to

choose the scale of the basis states such that they are Ilormalized; that is,

(1jJ 1 IIjJ 1) = (1jJ 211jJ 2) = l. Moreover, we shall assume that the basis states are

orthogonal: (1jJ111jJ2) = O. This may in fact not be true, and in Appendix B we

carry out a derivation of the energy expectation value while retaining overlaps in

(1jJ 1 11/12)' It will be seen in Appendix B that the corrections can largely be ab￾sorbed in the parameters of the theory. In the interests of conceptual simplicity,

overlaps are omitted in the main text, though their effect is indicated at the few

places where they are of consequence.

6 The Quantum-Mechanical Basis

We can use the notation Hij = < 1/1 d H 11/1); then Eq. (1-8) becomes

<I/IIHII/I) uiu1H 11 + UiU2H12 + U!UI H21 + u!u2H22

(1/111/1) = UtUl+ U!U2 (1-9)

(Actually, by Hermiticity, H21 = Ht2, but that fact is not needed here.)

Eq. (1-7) describes only an approximate energy eigenstate, since the two terms

on the right side are ordinarily not adequate for exact description. However,

within this approximation, the best estimate of the lowest energy eigenvalue can

be obtained by minimizing the entire expression (which we call E) on the right in

Eq. (1-9) with respect to U1 and U2' In particular, setting the partial derivatives of

that expression, with respect to ut and u! ,equal to zero leads to the two equations

H 11 u1 + H12 U2 = EUl; t

H21 U1 + H22 U2 = EU2'(

(1-10)

(In taking these partial derivatives we have treated Ub ui, U2, and u! as indepen￾dent. It can be shown that this is valid, but the proof will not be given here.)

Solving Eqs. (1-10) gives two values of E. The lower value is the energy expectation

value of the lowest energy state, called the bonding state. It is

E - H11 + H22 _ J(H11 - H22)2 + H H b - 2 2 12 21' (1-11 )

An electron in a bonding state has energy lowered by the proximity of the two

atoms of a diatomic molecule; the lowered energy helps hold the atoms together

in a bond. The second solution to Eqs. (1-10) gives the energy of another state, also

in the form of Eq. (1-7) but with different U1 and U2' This second state is called the

antibondillg state. Its wave function is orthogonal to that of the \'londing state; its

energy is given by

E=H11+H22+J(H11-H22)2+H H a 2 2 12 21' (1-12 )

We may substitute either of these energies, Eb or E., back into Eqs. (1-10) to

obtain values for U1 and U2 for each of the two states, and therefore, also the form

of the wave function for an electron in either state.

A particularly significant, simple approximation can be made in Eqs. (1-11) or

(1-12) when the matrix element H12 is much smaller than the magnitude of the

difference IH11 - Hd. Then, Eq. (1-11) or Eq. (1-12) can be expanded in the

perturbation H 12 (and H 2 d to obtain

(1-13 )

i-A Quantum Mechanics 7

for the energy of a state near H 11; a similar expression may be obtained for an

energy near H 22' These results are part of perturbatioll theory. The corresponding

result when many terms, rather than only two, are required in the expansion of the

wave function is

(1-14)

Similarly, for the state with energy near H 11' the coefficient U2 obtained by

solving Eq. (1-10) is

(1-15)

The last step uses Eq. (1-13). When H21 is small, U2 is small, and the term U2 1/12(1')

in Eq. (1-7) is the correction to the unperturbed state, 1/1 1 (r), obtained by perturba￾tion theory. The wave function can be written to first order in the perturbation,

divided by H 11 - H 22, and generalized to a coupling with many terms as

(1-16)

The perturbation-theoretic expressions for the electron energy, Eq. (1-14), and

wave function, Eq. (1-16), will be useful at many places in this text.

All of the discussion to this point has concerned the spatial wave function 1/1(1')

of an electron. An electron also has spin. For any 1/1(1') there are two possible spin

states. Thus, assertion (a) set forth earlier should be amended to say that an

electron is described by its spatial wave function and its spin state. The term

"state" is commonly used to refer to only the spatial wave function, when electron

spin is not of interest. It is also frequently used to encompass both wave function

and electron spin.

In almost all systems discussed in this book, there will be more than one

electron. The individual electron states in the systems and the occupation of those

states by electrons will be treated separately. The two aspects cannot be entirely

separated because the electrons interact with each other. At various points we

shall need to discuss the effects of these interactions.

In discussing electron occupation of states we shall require an additional

assertion-the Pauli principle:

(d) Only two electrons can occupy a single spatial state; these electrons must be of

opposite spin. Because of the discreteness of the energy eigenstates discussed

above, we can use the Pauli principle to specify how states are filled with electrons

to attain a system of lowest energy.

Because we shall discuss states of minimum energy, we shall not ordinarily be

interested in how the wave function changes with time. For the few cases in which

that information is wanted, a fifth assertion applies: