Atoms in molecules

C O N T E N T S......Page ix
List of Symbols......Page xiv
1.1 Atoms & the molecular structure hypothesis......Page p1
1.2 Necessary criteria for a theory of atoms in molecules......Page p2
1.3 The role of the charge density......Page p5
E1.1 Density matrices......Page p11
2.1 Introduction......Page p13
2.2.1 The dominant form in the charge density......Page p14
2.2.2 Critical points & their classification......Page p16
2.2.3 Critical points of molecular charge distributions......Page p19
2.3.1 Trajectories of gradient vector field......Page p22
2.3.2 Phase portraits of the gradient vector Held......Page p25
2.4.1 Equivalence of topological & quantum definitions of atom......Page p28
2.4.2 Chemical bonds & molecular graphs......Page p32
2.4.3 Rings and cages......Page p35
E2.1.1 Local maxima in p......Page p40
E2.1.2 Theorems on properties of p......Page p45
E2.2.1 Eigenvalues & eigenvectors of Hessian of p......Page p47
E2.2.2 Analytical expr., trajectory near critical point......Page p49
3.1 The notion of structure in chemistry......Page p53
3.2.1 Equivalence relation, qualitative discussion......Page p54
3.2.2 Changes in structure......Page p58
3.2.3 Structure diagrams......Page p64
3.2.4 Bonds and structure......Page p70
3.3.1 A coming together of mathematics & chemistry......Page p87
3.3.2 The equivalence relation & structural stability......Page p88
3.3.3 Stable & unstable intersec. of lines & surfaces in R3......Page p91
3.3.4 Criteria for structural stability......Page p94
3.4.1 Topological vs. energetic stabilities of molecular structures......Page p95
3.4.2 Structural homeomorphism, p & nuclear potential......Page p97
3.4.3 Mechanics of p in 1-electron case......Page p101
3.5 Summary......Page p102
E3.2 Natural coordinate system for an atomic basin......Page p103
4.1 Introduction......Page p110
4.2.1 Isolating the unstable piece of a critical point......Page p111
4.2.2 Elementary catastrophes......Page p112
4.3.1 Opening a ring structure—the fold catastrophe......Page p113
4.3.2 General analysis of 3-centre systems......Page p115
4.3.3 Formation of a cage structure......Page p121
5.1.1 From Lewis to quantum mechanical models......Page p130
5.1.2 Role of charge density in defining structure......Page p132
5.2.1 Observational basis for a quantum atom......Page p133
5.2.2 Observables & properties for total system......Page p137
5.3 Need for a subsystem variation principle......Page p146
5.3.1 Schroedinger’s derivation of wave mechanics......Page p150
5.3.2 Variational definition of subsystem, its properties......Page p154
E5.1 The variation of Hamiltonian—based functionals......Page p161
E5.2 Vectors, tensors, and dyadics......Page p164
E5.3 Divergence of a vector & Gauss’s theorem......Page p166
6.1.1 The charge & current densities......Page p169
6.1.2 Variational derivation of the atomic force law......Page p172
6.1.3 The atomic virial theorem......Page p175
6.1.4 Correspondence between local & subsystem mechanics......Page p178
6.2.1 Single—particle basis for atomic properties......Page p179
6.2.2 Definition of atomic properties......Page p180
6.3.2 The virial & partitioning of energy of interaction......Page p185
6.3.3 The electronic energy of an atom......Page p189
6.3.4 Potential energy contributions to an atomic energy......Page p191
6.4 Properties of atoms in molecules......Page p195
6.4.1 Properties determined by electronic charge density......Page p196
6.4.2 Transferability of atomic properties......Page p209
6.4.3 Origin of strain energy in cyclic hydrocarbon molecules......Page p219
6.4.4 Origin of rotation & inversion barriers......Page p221
6.4.5 Perfect transferability—an unattainable limit......Page p235
6.5 Hypervirial theorem & definition of bond properties......Page p238
6.5.1 An atomic property as sum of bond contributions......Page p239
6.5.2 Surface integrals proportional to the energy......Page p240
7.1.1 The electron pair at any cost......Page p248
7.1.2 Are electrons localized in pairs?......Page p249
7.1.3 Properties of the Laplacian of charge density......Page p252
7.1.4 The valence shell charge concentration (VSCC)......Page p258
7.1.5 Bonded & non-bonded charge concentrations......Page p260
7.2.1 Résumé of the VSEPR model......Page p265
7.2.2 Valence-shell charge conc. with four maxima......Page p266
7.2.3 Valence-shell charge conc. with 5 and 6 maxima......Page p268
7.2.4 Effects of charge transfer on properties of VSCC......Page p274
7.3.1 The Laplacian & the local energy of p......Page p275
7.3.2 Lewis acid—base reactions......Page p277
7.4.1 Definition of atomic interactions......Page p288
7.4.2 Classification of atomic interactions......Page p290
7.4.3 Hydrogen bonds & van der Waals molecules......Page p302
7.4.4 Intermediate interactions......Page p307
7.5.1 The electrostatic theorem & chemical binding......Page p315
7.5.2 The virial theorem & chemical bonding......Page p322
7.5.3 A summary......Page p331
E7.1 The pair density & the localization of electrons......Page p332
E7.2 Local charge conc. & partial localization of Fermi hole......Page p343
8.1.1 Introduction......Page p352
8.1.2 State vectors, state functions, & transformation functions......Page p353
8.1.3 Unitary transformations......Page p359
8.1.4 Canonical transformations & classical mechanics......Page p361
8.2.1 The principle of stationary action......Page p369
8.2.2 Applying the principle of stationary action......Page p371
8.3.1 Properties of the quantum mechanical Lagrangian......Page p376
8.3.2 Atomic action & Lagrangian integrals......Page p378
8.3.3 Variation of the atomic action integral......Page p380
8.3.4 Pr. of stationary action in Schroedinger representation......Page p382
8.3.5 Atomic statement of the principle of stationary action......Page p384
8.3.6 Examples of unitary transformations......Page p391
8.4.1 Variational derivation of the atomic force law......Page p393
8.4.2 Differential force law......Page p396
8.4.3 Variational derivation of the atomic virial theorem......Page p398
8.4.4 Local virial relationship......Page p401
8.4.5 Summary of the atomic variation principle......Page p402
8.5.1 The Lagrangian and Hamiltonian......Page p403
8.5.2 Definition of an atom in an external field......Page p406
8.5.3 Atomic force & virial theorems w. external fields......Page p410
8.5.4 Atomic contrib. to molecular polarizability & susceptibility......Page p416
8.6 The action principle—-in the past & future......Page p422
__ and of the Laplacian distribution function......Page p427
__  bonded & non bonded radii, ground-state diatomic molecules......Page p428
Table A4. Atomic charges, moments, volumes for diatomic molecules......Page p430
Table A5. Radii, spherical atoms and ions......Page p432
INDEX......Page p433