Ideas of Quantum Chemistry (eBook)

Front cover 1
Ideas of Quantum Chemistry 4
Copyright page 5
Contents 8
Introduction 22
Chapter 1. The Magic of Quantum Mechanics 36
1.1 History of a revolution 39
1.2 Postulates 50
1.3 The Heisenberg uncertainty principle 69
1.4 The Copenhagen interpretation 72
1.5 How to disprove the Heisenberg principle? The Einstein-Podolsky-Rosen recipe 73
1.6 Is the world real? 75
1.7 The Bell inequality will decide 78
1.8 Intriguing results of experiments with photons 81
1.9 Teleportation 82
1.10 Quantum computing 84
Chapter 2. The Schrödinger Equation 90
2.1 Symmetry of the Hamiltonian and its consequences 92
2.2 Schrödinger equation for stationary states 105
2.3 The time-dependent Schrödinger equation 111
2.4 Evolution after switching a perturbation 114
Chapter 3. Beyond the Schrödinger Equation 125
3.1 A glimpse of classical relativity theory 128
3.2 Reconciling relativity and quantum mechanics 144
3.3 The Dirac equation 146
3.4 The hydrogen-like atom in Dirac theory 158
3.5 Larger systems 164
3.6 Beyond the Dirac equation… 165
Chapter 4. Exact Solutions - Our Beacons 177
4.1 Free particle 179
4.2 Particle in a box 180
4.3 Tunnelling effect 188
4.4 The harmonic oscillator 199
4.5 Morse oscillator 204
4.6 Rigid rotator 211
4.7 Hydrogen-like atom 213
4.8 Harmonic helium atom (harmonium) 219
4.9 What do all these solutions have in common? 223
4.10 Beacons and pearls of physics 224
Chapter 5. Two Fundamental Approximate Methods 230
5.1 Variational method 231
5.2 Perturbational method 238
Chapter 6. Separation of Electronic and Nuclear Motions 252
6.1 Separation of the centre-of-mass motion 256
6.2 Exact (non-adiabatic) theory 259
6.3 Adiabatic approximation 262
6.4 Born-Oppenheimer approximation 264
6.5 Oscillations of a rotating molecule 264
6.6 Basic principles of electronic, vibrational and rotational spectroscopy 270
6.7 Approximate separation of rotations and vibrations 273
6.8 Polyatomic molecule 276
6.9 Non-bound states 282
6.10 Adiabatic, diabatic and non-adiabatic approaches 287
6.11 Crossing of potential energy curves for diatomics 290
6.12 Polyatomic molecules and conical intersection 295
6.13 Beyond the adiabatic approximation… 303
Chapter 7. Motion of Nuclei 310
7.1 Rovibrational spectra - an example of accurate calculations: atom - diatomic molecule 313
7.2 Force fields (FF) 319
7.3 Local Molecular Mechanics (MM) 325
7.4 Global molecular mechanics 327
7.5 Small amplitude harmonic motion - normal modes 329
7.6 Molecular Dynamics (MD) 339
7.7 Simulated annealing 344
7.8 Langevin Dynamics 345
7.9 Monte Carlo Dynamics 346
7.10 Car-Parrinello dynamics 349
7.11 Cellular automata 352
Chapter 8. Electronic Motion in the Mean Field: Atoms and Molecules 359
8.1 Hartree-Fock method - a bird's eye view 364
8.2 The Fock equation for optimal spinorbitals 369
8.3 Total energy in the Hartree-Fock method 386
8.4 Computational technique: atomic orbitals as building blocks of the molecular wave function 389
8.5 Back to foundations… 404
Results of the hartree–fock method 414
8.6 Mendeleev Periodic Table of Chemical Elements 414
8.7 The nature of the chemical bond 418
8.8 Excitation energy, ionization potential, and electron affinity (RHF approach) 424
8.9 Localization of molecular orbitals within the RHF method 431
8.10 A minimal model of a molecule 452
Chapter 9. Electronic Motion in the Mean Field: Periodic Systems 463
9.1 Primitive lattice 466
9.2 Wave vector 468
9.3 Inverse lattice 471
9.4 First Brillouin Zone (FBZ) 473
9.5 Properties of the FBZ 473
9.6 A few words on Bloch functions 474
9.7 The infinite crystal as a limit of a cyclic system 480
9.8 A triple role of the wave vector 483
9.9 Band structure 484
9.10 Solid state quantum chemistry 495
9.11 The Hartree-Fock method for crystals 503
9.12 Long-range interaction problem 510
9.13 Back to the exchange term 520
9.14 Choice of unit cell 523
Chapter 10. Correlation of the Electronic Motions 533
Variational methods using explicitly correlated wave function 537
10.1 Correlation cusp condition 538
10.2 The Hylleraas function 541
10.3 The Hylleraas CI method 541
10.4 The harmonic helium atom 542
10.5 James-Coolidge and Kolos-Wolniewicz functions 543
10.6 Method of exponentially correlated Gaussian functions 548
10.7 Coulomb hole ("correlation hole") 548
10.8 Exchange hole ("Fermi hole") 551
Variational methods with slater determinants 555
10.9 Valence bond (VB) method 555
10.10 Configuration interaction (CI) method 560
10.11 Direct CI method 568
10.12 Multireference CI method 568
10.13 Multiconfigurational Self-Consistent Field method (MC SCF) 570
Non-variational methods with slater determinants 574
10.14 Coupled cluster (CC) method 574
10.15 Equation-of-motion method (EOM-CC) 583
10.16 Many body perturbation theory (MBPT) 586
10.17 Moller-Plesset version of Rayleigh-Schrödinger perturbation theory 593
Chapter 11. Electronic Motion: Density Functional Theory (DFT) 602
11.1 Electronic density - the superstar 604
11.2 Bader analysis 606
11.3 Two important Hohenberg-Kohn theorems 614
11.4 The Kohn-Sham equations 619
11.5 What to take as the DFT exchange-correlation energy Exc? 625
11.6 On the physical justification for the exchange correlation energy 627
11.7 Reflections on the DFT success 637
Chapter 12. The Molecule in an Electric or Magnetic Field 650
12.1 Hellmann-Feynman theorem 653
Electric phenomena 655
12.2 The molecule immobilized in an electric field 655
12.3 How to calculate the dipole moment 668
12.4 How to calculate the dipole polarizability 670
12.5 A molecule in an oscillating electric field 680
Magnetic phenomena 682
12.6 Magnetic dipole moments of elementary particles 683
12.7 Transitions between the nuclear spin quantum states - NMR technique 687
12.8 Hamiltonian of the system in the electromagnetic field 688
12.9 Effective NMR Hamiltonian 693
12.10 The Ramsey theory of the NMR chemical shift 701
12.11 The Ramsey theory of NMR spin-spin coupling constants 703
12.12 Gauge invariant atomic orbitals (GIAO) 708
Chapter 13. Intermolecular Interactions 716
Theory of intermolecular interactions 719
13.1 Interaction energy concept 719
13.2 Binding energy 722
13.3 Dissociation energy 722
13.4 Dissociation barrier 722
13.5 Supermolecular approach 724
13.6 Perturbational approach 727
13.7 Symmetry adapted perturbation theories (SAPT) 745
13.8 Convergence problems 756
13.9 Non-additivity of intermolecular interactions 761
Engineering of intermolecular interactions 776
13.10 Noble gas interaction 776
13.11 Van der Waals surface and radii 777
13.12 Synthons and supramolecular chemistry 779
Chapter 14. Intermolecular Motion of Electrons and Nuclei: Chemical Reactions 797
14.1 Hypersurface of the potential energy for nuclear motion 801
14.2 Accurate solutions for the reaction hypersurface (three atoms) 810
14.3 Intrinsic reaction coordinate (IRC) or statics 816
14.4 Reaction path Hamiltonian method 818
14.5 Acceptor-donor (AD) theory of chemical reactions 833
14.6 Barrier for the electron-transfer reaction 863
Chapter 15. Information Processing - the Mission of Chemistry 883
15.1 Complex systems 887
15.2 Self-organizing complex systems 888
15.3 Cooperative interactions 889
15.4 Sensitivity analysis 890
15.5 Combinatorial chemistry - molecular libraries 890
15.6 Non-linearity 892
15.7 Attractors 893
15.8 Limit cycles 894
15.9 Bifurcations and chaos 895
15.10 Catastrophes 897
15.11 Collective phenomena 898
15.12 Chemical feedback - non-linear chemical dynamics 901
15.13 Functions and their space-time organization 910
15.14 The Measure of information 910
15.15 The mission of chemistry 912
15.16 Molecular computers based on synthon interactions 913
Appendices 922
A. A Reminder: Matrices and determinants 924
1. Matrices 924
2. Determinants 927
B. A few words on spaces, vectors and functions 930
1. Vector space 930
2. Euclidean space 931
3. Unitary space 932
4. Hilbert space 933
5. Eigenvalue equation 935
C. Group theory in spectroscopy 938
1. Group 938
2. Representations 948
3. Group theory and quantum mechanics 959
4. Integrals important in spectroscopy 964
D. A two-state model 983
E. Dirac delta function 986
1. Approximations to delta(x) 986
2. Properties of delta(x) 988
3. An application of the Dirac delta function 988
F. Translation vs momentum and rotation vs angular momentum 990
1. The form of the U operator 990
2. The Hamiltonian commutes with the total momentum operator 992
3. The Hamiltonian, J2 and Jz do commute 993
4. Rotation and translation operators do not commute 995
5. Conclusion 995
G. Vector and scalar potentials 997
H. Optimal wave function for a hydrogen-like atom 1004
I. Space- and body-fixed coordinate systems 1006
J. Orthogonalization 1012
1. Schmidt orthogonalization 1012
2. Löwdin symmetric orthogonalization 1013
K. Diagonalization of a matrix 1017
L. Secular equation (h-epsilons)c=0 1019
M. Slater-condon rules 1021
N. Lagrange multipliers method 1032
O. Penalty function method 1036
P. Molecular integrals with gaussian type orbitals 1s 1039
Q. Singlet and triplet states for two electrons 1041
R. The hydrogen molecular ion in the simplest atomic basis set 1044
S. Population analysis 1050
T. The dipole moment of a lone electron pair 1055
U. Second quantization 1058
V. The hydrogen atom in the electric field - variational approach 1064
W. NMR shielding and coupling constants - derivation 1067
1. Shielding constants 1067
2. Coupling constants 1070
X. Multipole expansion 1073
Y. Pauli deformation 1085
Z. Acceptor-donor structure contributions in the mo configuration 1093
Name Index 1100
Subject Index 1112