HPC in Computational Chemistry: Bridging Quantum Mechanics ...
Handbook of Computational Quantum Chemistry · Handbook of Computational Quantum Chemistry DAVID B....
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Handbook of Computational Quantum Chemistry DAVID B. COOK The Department of Chemistry, University of Sheffield Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1998
Transcript of Handbook of Computational Quantum Chemistry · Handbook of Computational Quantum Chemistry DAVID B....
Handbook of Computational Quantum Chemistry
DAVID B. COOK The Department of Chemistry, University of Sheffield
Oxford New York Tokyo OXFORD UNIVERSITY PRESS
1998
CONTENTS
1 Mechanics and molecules 1
1.1 Introduction 1
1.2 Time-independent Schrödinger equation 4
1.3 The Born-Oppenheimer model 6
1.4 The Pauli principle 8
1.5 The orbital model 10
1.6 The determinantal method 13
1.7 Physical interpretation 15
1.8 Non-determinantal forms 17
1.9 The Variation principle 18
1.10 Summary 21
l . A Atomic units 23
l . B Standard Notation for Quantum Chemistry 25
l .B . l Introduction 25
1.B.2 The Hamiltonian 25
l.B.3 Many-electron wavefunctions 26
1.B.4 Spin-orbitals 27
l.B.5 Linear expansions for the spatial orbitals 27
1.B.6 Primitive Gaussians 28
l.B.7 Single determinant energy expression 29
1.B.8 Notation for repulsion integrals 31
l.B.9 Spatial orbital repulsion integrals 32
l.B.10 Basis function repulsion integrals 32
xii CONTENTS
2 The Hartree-Fock Method 34
2.1 Introduction 34
2.2 The variational method 35
2.3 The differential Hartree-Fock equation 36
2.4 Canonical form 44
2.5 Orbital energies 45
2.6 Physical Interpretation 47
2.7 Direct parametric minimisation 48
2.8 Summary 49
2.A Single-determinant energy expression 50
2.A.1 Introduction 50
2.A.2 The normalisation integral 52
2.A.3 One-electron terms 56
2.A.4 Two-electron terms 60
2.A.5 Summary 65
3 The matrix SCF equations 70
3.1 Introduction 70
3.2 Notation 72
3.3 The expansion 73
3.4 The energy expression 75
3.5 The numerator: Hamiltonian mean value 75
3.6 The denominator: normalisation condition 79
3.7 The Hartree-Fock equation 80
3.8 "Normalisation": the Lagrangian 81
3.9 Preliminary summary 82
3.10 Some technical manipulations 83
3.11 Canonical orbitals 87
3.12 The total energy 89
3.13 Summary 90
CONTENTS xiii
3.A Atomic orbitals 92
3.B Charge density 94
3.C Properties of the J and K matrices 97
3.C.1 Mathematical properties 97
3.C.2 Physical interpretation 99
3.C.3 Supermatrices 100
3.D An artifact of expansion 102
3.D.4 Lowest State of a given symmetry 102
3.E Single determinant: choice of orbitals 104
3.E.5 Orthogonal invariance 104
3.E.6 Koopmans' theorem 105
3.E.7 Localised orbitals 106
3.E.8 "Zeroth-order" perturbed orbitals 107
4 A special case: closed Shells 108
4.1 Introduction 108
4.2 Notation for the closed-shell case 109
4.3 Closed-shell expansion 109
4.4 The closed-shell "HF" equation 110
4.5 Closed-shell summary 113
5 Implementation of the closed-shell case 114
5.1 Preview 114
5.2 Vectors, matrices and arrays 115
5.3 The implementation: getting started 121
5.4 The implementation: repulsion integral access . . . . 137
5.5 Building a testbench: conventional SCF 147
5.6 Another testbench: direct SCF 154
5.7 Summary 162
5.8 What next? 162
xiv CONTENTS
5.A Jacobi diagonalisation 164
5.A.1 Introduction 164
5.A.2 The problem 165
5.A.3 The Solution 166
5.A.4 Implementation 167
5.A.5 Other diagonalisation methods 170
5.B Orthogonalisation 171
5.B.6 Introduction 171
5.B.7 Functions of a matrix 173
5.B.8 Implementation 174
5.C g e t i n t and data for H^O \11
5.D Coding the Standard index loops 181
6 Improvements: tools and methods 185
6.1 Introduction 185
6.2 Versions: conditional compilation 186
6.3 Improved diagonalisation 192
6.4 Simple interpolation 195
6.5 Improving the formation of G(R) 197
6.6 Summary 199
7 Molecular integrals: an introduction 201
7.1 Introduction 201
7.2 Basis functions 202
7.3 AOs and atom-centred-functions 203
7.4 Multi-dimensional integral evaiuation 205
7.5 Molecular integrals over STOs 206
7.6 Basis functions of convenience 215
7.7 Gaussian basis functions 216
7.8 The contraction technique 234
CONTENTS xv
8 Molecular integrals: implementation 236
8.1 Introduction 236
8.2 Data structures 237
8.3 Normalisation 240
8.4 Overview; the general structure 243
8.5 Complex code management: the WEB System . . . . 249
8.6 AworkingWEB 256
8.7 Some comments on the WEB 266
8.8 The füll integral codes 267
8.A Source for the WEB of fmch 268
9 Repulsion integral storage 274
9.1 Introduction 274
9.2 A storage algorithm 274
9.3 Implementation: p u t i n t 276
9.4 A partner for p u t i n t ; g e t i n t 282
9.5 Conclusion 284
10 "Virtual Orbitals" 285
10.1 Introduction 285
10.2 Virtual orbitals in practice 286
10.3 The Virtual space in LCAO 291
10.4 Conclusions 295
10.A Perturbation theory 296
10.A.1 Introduction 296
10.A.2 Perturbation theory 296
10.A.3 Perturbation theory for matrix equations 301
xvi CONTENTS
11 Choice of tools 303
11.1 Existing Software 303
11.2 Why ratfor? 306
11.3 The Revision Control System: RCS 308
1 1 . A RCS: version control 310
l l . A . l Motivation 310
11.A.2 Introduction 310
11.A.3 Getting started with RCS 311
12 Open Shells: implementing UHF 314
12.1 Introduction 314
12.2 Choice of constraints 315
12.3 Organising the basis 317
12.4 Integrals over the spin-basis 318
12.5 Implementation 320
12.6 J and K for GUHF 321
12.7 The GUHF testbench 326
12.8 Interpreting the MO coefficients 329
12.9 DODS or GUHF? 332
12.10 Version 1 of the SCF code 333
12.11 WEB Output for function scf 337
12.12 Comments 345
12.A WEB Source for the scf code 346
12.B Blocking the Hartree-Fock matrix 351
12.B.1 The block form of the HF matrix 351
12.B.2 Implementation 352
CONTENTS xvii
12.C The Aufbau principle 363
12.C.3 Introduction 363
12.C.4 The second Variation 363
12.C.5 Special case: a single excitation 365
13 Population analysis 367
13.1 Introduction 367
13.2 Densities and spin-densities 368
13.3 Basis representations: charges 369
13.4 Basis-function analysis 372
13.5 A cautionary note 374
13.6 Multi-determinant forms 375
13.7 Implementation 376
14 The general MO functional 377
14.1 A generalisation 377
14.2 Shells of orbitals 378
14.3 The variational method 380
14.4 A single "Hartree-Fock" Operator 383
14.5 Non-orthogonal basis 386
14.6 Choice of the arbitrary matrices 388
14.7 Implementation: Stacks of matrices 390
14.A Projection Operators and SCF 400
14.A.1 Introduction: Optimum single determinant 400
14.A.2 Alternative SCF conditions 402
14.A.3 R matrices as projection Operators 403
xviii CONTENTS
15 Spin-restricted open shell 406
15.1 Introduction 406
15.2 The ROHF model 407
15.3 Implementation 408
15.4 A WEB for spin-restricted open shell 409
16 Banana skins: unexpected disasters 436
16.1 Symmetry restrictions 437
16.2 Anions 438
16.3 Aufbau exceptions 439
16.4 Summary 441
17 Molecular symmetry 442
17.1 Introduction 442
17.2 Symmetry and the HF method 443
17.3 Permutational symmetry of the basis 445
17.4 Implementation 450
17.5 Permutation symmetry: summary 466
18 Symmetry orbital transformations 467
18.1 Introduction 467
18.2 Symmetry-adapted basis 470
18.3 Generation of symmetry Orbitals 473
18.4 Conclusions 476
19 A symmetry-adapted SCF method 477
19.1 Introduction 477
19.2 Permutations only 480
19.3 Füll implementation; linear combinations 489
19.4 Summary 494
CONTENTS xix
19.A Kronecker product notation 495
19.A.1 Basis transformations 495
19.A.2 Basis-product transformations 495
19.A.3 Density matrix transformations 497
19.A.4 Transformations in the HF matrix 498
19.A.5 Practice 500
20 Linear multi-determinant methods 501
20.1 Correlation and the Hartree-Fock model 501
20.2 The configuration interaction method 502
20.3 The valence bond method 503
20.4 Restricted Cl 504
20.5 Symmetry-restricted Cl 510
20.6 More general Cl 512
20.7 Nesbet's method for large matrices 513
20.8 "Direct" Cl 519
20.9 Conclusions 524
20.A The "orthogonal VB" model 525
20.B DCI matrix elements 527
21 The valence bond model 530
21.1 Non-orthogonality in expansions 530
21.2 Spins and spin functions 531
21.3 Spin eigenfunctions and permutations 535
21.4 Spin-free VB theory 539
21.5 Summary 544
xx CONTENTS
22 Doubly-occupied MCSCF 545
22.1 Introduction: natural orbitals 545
22.2 Paired-excitation MCSCF 548
22.3 Implementation 553
22.4 Partial Paired-Excitations; GVB 553
22.5 Details of GVB 556
22.6 Implementation 561
23 Interpreting the McWeenyan 562
23.1 Introduction 562
23.2 Stationary points 563
23.3 Many shells 565
23.4 Summary 566
24 Core potentials 567
24.1 Introduction 567
24.2 Simple orthogonalization 569
24.3 Transforming the Hartree-Fock equation 570
24.4 The pseudopotential 574
24.5 Arbitrariness in the pseudo-orbital 576
24.6 Modelling atomic pseudopotentials 579
24.7 Modelling atomic core potentials 581
24.8 Several valence electrons 584
24.9 Atomic cores in molecules 588
24.10 Summary 589
CONTENTS xxi
25 Practical core potentials 591
25.1 Introduction 591
25.2 Forms for the core potentials 591
25.3 Core potential integrals 595
25.4 Implementation 604
26 SCF perturbation theory 605
26.1 Introduction 605
26.2 Two forms for the HF equations 606
26.3 Self-consistent perturbation theory 609
26.4 The method 610
26.5 Conciusions 618
27 Time-dependent perturbations: RPA 621
27.1 Introduction 621
27.2 Time-dependent Hartree-Fock theory 621
27.3 Oscillatory time-dependent perturbations 623
27.4 Seif consistency 626
27.5 Implementation 627
27.A "Random phase approximation" 629
27.B Time-dependent Variation principle 631
28 Transitions and stability 633
28.1 Introduction 633
28.2 Transitions 634
28.3 The transition frequencies 635
28.4 Finite perturbations; oscillations 636
28.5 Stability; the time-independent case 638
28.6 Implementation 639
xxii CONTENTS
29 Two-electron transformations 640
29.1 Orbital transformations 640
29.2 Strategy 641
29.3 Transformation without sorting 643
29.4 Transformations with sorting 654
29.5 Summary 656
2 9 . A A b i to f fun: MP2 657
29.A.1 Derivation 657
29.A.2 Implementation 660
30 Geometry optimisation: derivatives 671
30.1 Introduction 671
30.2 Derivatives and perturbation theory 672
30.3 Derivatives of variational Solutions 674
30.4 Parameter-dependent basis functions 676
30.5 The derivative of the SCF energy 677
30.6 Derivatives of molecular integrals 681
30.7 Derivatives of non-variational energies 682
30.8 Higher derivatives 684
30.9 Summary 684
31 The Semi-empirical approach 686
31.1 Introduction 686
31.2 Use of Coulomb's law 687
31.3 Atomic data 689
31.4 Simulation or calibration? 690
31.5 General conclusions 691
CONTENTS xxüi
32 Density functional theory 693
32.1 Introduction 693
32.2 Hohenberg and Kohn's proofs 695
32.3 Kohn-Sham equations: introduction 700
32.4 Kohn-Sham equations 703
32.5 Non-Iocal Operators in orbital theories 705
33 Implementing the Kohn-Sham equations 708
33.1 A precursor: The Hartree-Fock-Slater model . . . . 708
33.2 Implementation of the Kohn-Sham method 710
33.3 The kinetic energy density 715
33.4 Gradients in the exchange-correlation energy . . . . 717
33.5 Numerical integration of densities 717
33.6 Summary 720
34 Semi-numerical methods 722
34.1 Non-variational expansions 722
34.2 The pseudospectral method 724
34.3 The discrete variational method 729
35 Additional reading and other material 732
35.1 Additional reading 732
35.2 Additional material by f t p 734
