Elementary Methods of Molecular Quantum Mechanics Book

Preface     xv
Basic Principles of Quantum Mechanics     1
The Orbital Model     1
The Fundamental Postulates of Quantum Mechanics     2
Correspondence Between Observables and Operators     3
State Function and Average Values of Observables     7
Time Evolution of State Function     8
The Physical Principles of Quantum Mechanics     9
Wave-Particle Dualism     9
Atomicity of Matter     12
Schroedinger Wave Equation     15
Born Interpretation     16
Measure of Observables     17
The Mathematics of Quantum Mechanics     19
Dirac Notation and Sets of Normalizable Functions     20
Linear Operators     22
Hermitian Operators     23
Expansion Theorem: From Operators to Matrices     25
[Characters not reproducible] Vector Operator and Its Properties     27
Systems of Orthogonal Coordinates     28
Generalized Coordinates     29
Fundamental Physical Constants and Atomic Units     34
Problems 1     35
Solved Problems     41
Elementary Matrix Methods     57
Introduction     57
Elements of Matrix Algebra     58
Definitions     58
Properties of Matrices     58
Properties of Determinants     59
Special Matrices 1     61
Special Matrices 2     63
Matrix Eigenvalue Problem     64
Systems of Linear Equations     64
Eigenvalue Equation     65
Pseudoeigenvalue Equation     68
Functions of Hermitian Matrices     69
Analytic Functions     69
Canonical Form     70
Lagrange Interpolation Formula     70
Cayley-Hamilton Theorem     71
Problems 2     72
Solved Problems     78
The Particle in the Box     103
Introduction     103
The Free Particle in One Dimension     103
The 3-dimensional Box of Sides a, b, c     105
Particle in a 1-dimensional Box with Impenetrable Walls     106
Particle in a 1-dimensional Box of Finite Height     108
Problems 3     112
Solved Problems     113
The Hydrogen-Like System     117
Introduction     117
Separation of the Motion of the Centre-of-Mass     119
Separation of the Radial Equation in Spherical Coordinates     121
Solution of the Radial Equation     123
Solution of the Angular Equation     128
Solution of the [Phi]-Equation     128
Solution of the [Theta]-Equation     129
Hydrogen-Like Orbitals, Eigenvalues and Quantum Numbers     132
Properties of Ground and Excited States     138
1s Ground State     138
Excited 2p State     140
Expectation Values for Ground and First Excited States     141
Slater and Gaussian Atomic Orbitals     142
Slater Orbitals (STOs)     142
Gaussian Orbitals (GTOs)     144
Problems 4     147
Solved Problems     150
The Variation Method     163
Introduction     163
The Variation Method     164
Variational Principles     164
Properties of the Variational Solutions     165
Variational Approximations     166
Basis Functions and Variational Parameters     167
Non-Linear Parameters     168
The 1s Ground State of the Hydrogenic System     168
The First 2s, 2p Excited States of the Hydrogenic System     171
The 1s[superscript 2] Ground State of the He-Like System     174
Linear Parameters and the Ritz Method     178
Orthonormal Basis     178
Non-Orthogonal Basis     182
Atomic Applications of the Ritz Method     183
The First 1s2s Excited State of the He-Like System     183
The First 1s2p Excited State of the He-Like System     185
Results for Hydrogenic AOs     188
Molecular Applications of the Ritz Method     188
The Ground and First Excited State of the H[superscript +][subscript 2] Molecular Ion     189
The Interaction Energy and Its Components     192
The Wentzel-Kramers-Brillouin (WKB) Method     197
Problems 5     200
Solved Problems     203
The Electron Spin     215
Introduction     215
Electron Spin According to Pauli and the Zeeman Effect     216
Theory of 1-Electron Spin     220
Matrix Representation of Spin Operators     225
Theory of 2-Electron Spin     227
Theory of N-Electron Spin     229
The Kotani Synthetic Method     233
Lowdin Spin Projection Operators     234
Problems 6     236
Solved Problems     239
Many-Electron Wavefunctions: Slater, Hartree-Fock and Related Methods     255
Introduction     256
Antisymmetry of the Electronic Wavefunction and the Pauli Principle     256
Two-Electron Wavefunctions     256
Three-Electron Wavefunctions     257
Many-Electron Wavefunctions and the Slater Method     258
Electron Distribution Functions     263
1-Electron Distribution Functions: General Definitions     263
Electron Density and Spin Density     264
2-Electron Distribution Functions: General Definitions     268
Spinless Pair Functions and the Correlation Problem     268
Average Values of 1- and 2-Electron Operators     272
Symmetrical Sums of 1-Electron Operators     272
Symmetrical Sums of 2-Electron Operators     273
Average Value of the Electronic Energy     274
The Slater Rules     275
Pople's Two-Dimensional Chart of Quantum Chemistry     276
Hartree-Fock Theory for Closed Shells     279
Basic Theory and Properties of the Fundamental Invariant p     279
Electronic Energy for the HF Wavefunction     280
Roothaan Variational Derivation of the HF Equations     282
Hall-Roothaan Formulation of the LCAO-MO-SCF Equations      285
Mulliken Population Analysis     288
Atomic Bases in Quantum Chemical Calculations     291
Localization of Molecular Orbitals     296
Huckel theory     298
Recurrence Relation for the Linear Chain     300
General Solution for the Linear Chain     300
General Solution for the Closed Chain     302
Alternant Hydrocarbons     304
An Introduction to Band Theory of Solids     309
Semiempirical MO Methods     311
Extended Huckel Theory (EHT)     311
CNDO Method     312
INDO Method     316
ZINDO Method     316
Post-Hartree-Fock Methods     317
Configuration Interaction (CI)     317
Multiconfiguration SCF (MC-SCF)     318
Explicitly Correlated Non-Orbital Methods     320
Second-Order Moller-Plesset (MP2) Theory     322
MP2-R12 Method     324
CC-R12 Method     325
A Short Outline of Second Quantization     327
Density Functional Theory (DFT)     328
Problems 7     332
Solved Problems     339
Molecular Symmetry and Group Theoretical Methods     363
Introduction      363
Symmetry and Quantum Mechanics     364
Molecular Symmetry     365
Symmetry Operations as Transformation of Coordinate Axes     368
Passive and Active Representations of Symmetry Operations     368
Symmetry Transformations in Coordinate Space     370
Symmetry Operators and Transformations in Function Space     372
Matrix Representatives of Symmetry Operators     376
Similarity Transformations     377
Group Theoretical Methods     378
Axioms of Group Theory     378
Examples of Groups     379
Isomorphism     382
Conjugation and Classes     383
Representations and Characters     384
Irreducible Representations     387
Construction of Symmetry-Adapted Functions     390
The Wigner Method     392
Subgroups and Direct-Product Groups     394
Applications     395
The Fundamental Theorem of Symmetry     395
Selection Rules     395
Ground State Electron Configuration of Polyatomic Molecules     396
An Outline of Continuous and Permutation Groups     398
Continuous Groups     398
Continuous Lie Groups      398
Transformation Properties of Spherical Harmonics     399
Rotation Groups     400
Permutation Group     403
Problems 8     404
Solved Problems     413
Angular Momentum Methods for Atoms     439
Introduction     439
The Vector Model     440
Coupling of Angular Momenta     440
LS Coupling and Multiplet Structure     443
Construction of States of Definite Angular Momentum     448
The Matrix Method     448
The Projection Operator Method     454
An Outline of Advanced Methods for Coupling Angular Momenta     454
Clebsch-Gordan Coefficients and Wigner 3-j and 9-j Symbols     455
Gaunt Coefficients and Coupling Rules     456
Problems 9     458
Solved Problems     462
Valence Bond Theory and the Chemical Bond     473
Introduction     473
The Born-Oppenheimer Approximation     475
The Chemical Bond in H[subscript 2]     477
Failure of the MO Theory for Ground State H[subscript 2]     478
The Heitler-London Theory for H[subscript 2]     484
Equivalence Between MO-CI and Full VB for Ground State H[subscript 2] and Improvements in the Wavefunction     488
The Orthogonality Catastrophe in the Covalent VB Theory for Ground State H[subscript 2]     494
Elementary Valence Bond Methods     502
General Formulation of VB Theory     502
Construction of VB Structures for Multiple Bonds     506
The Allyl Radical (N = 3)     507
Cyclobutadiene (N = 4)     509
VB Description of Simple Molecules     511
Pauling VB Theory for Conjugated and Aromatic Hydrocarbons     522
Pauling Formula for the Matrix Elements of Singlet Covalent VB Structures     523
Cyclobutadiene     525
Butadiene     526
Allyl Radical     528
Benzene     529
Naphthalene     538
Derivation of the Pauling Formula for H[subscript 2] and Cyclobutadiene     541
Hybridization and Directed Valency in Polyatomic Molecules     545
sp[superscript 2] Hybridization in H[subscript 2]O     545
VB Description of H[subscript 2]O     547
Properties of Hybridization     549
The Principle of Maximum Overlap in VB Theory     552
An Outline of Recent Advances in VB Theory     554
Modern VB Theories     554
The Spin-Coupled VB Theory      557
Problems 10     560
Solved Problems     563
Rayleigh-Schroedinger Perturbation Methods for Stationary States     577
Introduction     577
RS Perturbation Theory for Stationary States     578
RS Perturbation Equations and Energy Corrections     578
The Orthogonality Conditions     580
First-Order Perturbation Theory for Degenerate Eigenvalues     581
Properties of the Perturbation Solutions     582
Expansion in Eigenstates     584
Unsold Approximation     585
Variational Approximations for the Second-Order Energy     586
Variation-Perturbation Method     586
Kirkwood Approximation     587
The Ritz Method for [Characters not reproducible]: Expansion in Pseudostates     588
Static Multipole Polarizabilities for H(1s)     590
Dipole Polarizability     590
Exact Solution of the General First-Order RS Differential Equation for H(1s) in a Uniform Electric Field     593
Variational Approximations     597
Electric Properties of Molecules     600
Problems 11     604
Solved Problems     606
Atomic and Molecular Interactions     617
Introduction      617
Interatomic Interactions     618
RS Perturbation Theory of the H-H[superscript +]Interaction     618
Non-Expanded Interaction Energies up to Second Order     618
Expanded Interaction Energies up to Second Order     622
RS Perturbation Theory of the H-H Interaction     623
Non-Expanded Interaction Energies up to Second Order     623
Expanded Interaction Energies up to Second Order     626
HL Theory as a First-Order Perturbation Theory Including Exchange     629
Accurate Theoretical Results for Simple Molecular Systems     633
An Outline of a Perturbation Theory for Molecular Interactions     635
MS-MA Perturbation Theory of Molecular Interactions     635
First-Order Exchange-Overlap Energy     638
Non-Expanded RS Intermolecular Energies     640
Expanded Dispersion Interactions Between Molecules     644
Angle-Dependent C[subscript 6] Dispersion Coefficients for Simple Molecular Systems     645
Isotropic C[subscript 6] Dispersion Coefficients from Dipole Polarizability Pseudospectra     648
The Van der Waals Bond     650
Problems 12     655
Solved Problems     656
Evaluation of Molecular Integrals over STOs     663
Introduction      663
The Basic Integrals     664
The Indefinite Integral     664
Definite Integrals and Auxiliary Functions     665
1-Centre Integrals     667
1-Electron Integrals     667
2-Election Integrals     670
Evaluation of the Electrostatic Potential J[subscript 1s]     670
Spherical Coordinates     670
Spheroidal Coordinates     673
The (1s[superscript 2]=970 1s[superscript 2]) Electron Repulsion Integral     674
Same Orbital Exponent     674
Different Orbital Exponents     674
General Formula for 1-Centre 2-Electron Integrals     675
2-Centre Integrals over 1s STOs     676
1-Electron Integrals     677
2-Electron Integrals     679
Limiting Values of 2-Centre Integrals     686
On the General Formulae for 2-Centre Integrals     690
Spheroidal Coordinates     690
Spherical Coordinates     692
A Short Note on Multicentre Integrals     693
3-Centre 1-Electron Integral over 1s STOs     693
4-Centre 2-Electron Integral over 1s STOs     694
Problems 13     696
Solved Problems     697
References      709
Author Index     717
Subject Index     723