Elementary Methods of Molecular Quantum Mechanics

by Magnasco

• ISBN13: 9780444527783
• ISBN10: 0444527788
• Format: Hardcover
• Copyright: 2006-12-20
• Publisher: Elsevier Science

Summary

This is a teaching book for graduate University students of Chemistry and Physics, where emphasis is placed on methods of Quantum Mechanics from an elementary point of view as applied to the theoretical study of the electronic structure of atoms and molecules and their interactions. The book is not intended as a treatise covering all aspects of applications to Chemistry and to many problems in Molecular Physics, rather to present in the simplest possible way the mathematical machinery needed for a critical understanding of what is implied in the black boxes of the software which is today currently in use by chemists in this area. Many examples and mathematical points are given as Problems at the end of each Chapter, with a hint for their solution. An extensive set of alphabetically ordered references, the Author index and the Subject index complete the book. Book jacket.

Table of Contents

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