Handbook of Computational Quantum Chemistry. DAVID B. COOK The Department of Chemistry, University of Sheffield

Transcription

1 Handbook of Computational Quantum Chemistry DAVID B. COOK The Department of Chemistry, University of Sheffield Oxford New York Tokyo OXFORD UNIVERSITY PRESS 1998

2 CONTENTS 1 Mechanics and molecules Introduction Time-independent Schrödinger equation The Born-Oppenheimer model The Pauli principle The orbital model The determinantal method Physical interpretation Non-determinantal forms The Variation principle 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

3 xii CONTENTS 2 The Hartree-Fock Method Introduction The variational method The differential Hartree-Fock equation Canonical form Orbital energies Physical Interpretation Direct parametric minimisation 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 Introduction Notation The expansion The energy expression The numerator: Hamiltonian mean value The denominator: normalisation condition The Hartree-Fock equation "Normalisation": the Lagrangian Preliminary summary Some technical manipulations Canonical orbitals The total energy Summary 90

4 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 D An artifact of expansion D.4 Lowest State of a given symmetry E Single determinant: choice of orbitals E.5 Orthogonal invariance E.6 Koopmans' theorem E.7 Localised orbitals E.8 "Zeroth-order" perturbed orbitals A special case: closed Shells Introduction Notation for the closed-shell case Closed-shell expansion The closed-shell "HF" equation Closed-shell summary Implementation of the closed-shell case Preview Vectors, matrices and arrays The implementation: getting started The implementation: repulsion integral access Building a testbench: conventional SCF Another testbench: direct SCF Summary What next? 162

5 xiv CONTENTS 5.A Jacobi diagonalisation A.1 Introduction A.2 The problem A.3 The Solution A.4 Implementation A.5 Other diagonalisation methods B Orthogonalisation B.6 Introduction B.7 Functions of a matrix B.8 Implementation C getint and data for H^O \11 5.D Coding the Standard index loops Improvements: tools and methods Introduction Versions: conditional compilation Improved diagonalisation Simple interpolation Improving the formation of G(R) Summary Molecular integrals: an introduction Introduction Basis functions AOs and atom-centred-functions Multi-dimensional integral evaiuation Molecular integrals over STOs Basis functions of convenience Gaussian basis functions The contraction technique 234

6 CONTENTS xv 8 Molecular integrals: implementation Introduction Data structures Normalisation Overview; the general structure Complex code management: the WEB System AworkingWEB Some comments on the WEB The füll integral codes A Source for the WEB of fmch Repulsion integral storage Introduction A storage algorithm Implementation: putint A partner for putint; getint Conclusion "Virtual Orbitals" Introduction Virtual orbitals in practice The Virtual space in LCAO Conclusions A Perturbation theory A.1 Introduction A.2 Perturbation theory A.3 Perturbation theory for matrix equations 301

7 xvi CONTENTS 11 Choice of tools Existing Software Why ratfor? The Revision Control System: RCS A RCS: version control 310 ll.a.l Motivation A.2 Introduction A.3 Getting started with RCS Open Shells: implementing UHF Introduction Choice of constraints Organising the basis Integrals over the spin-basis Implementation J and K for GUHF The GUHF testbench Interpreting the MO coefficients DODS or GUHF? Version 1 of the SCF code WEB Output for function scf Comments A WEB Source for the scf code B Blocking the Hartree-Fock matrix B.1 The block form of the HF matrix B.2 Implementation 352

8 CONTENTS xvii 12.C The Aufbau principle C.3 Introduction C.4 The second Variation C.5 Special case: a single excitation Population analysis Introduction Densities and spin-densities Basis representations: charges Basis-function analysis A cautionary note Multi-determinant forms Implementation The general MO functional A generalisation Shells of orbitals The variational method A single "Hartree-Fock" Operator Non-orthogonal basis Choice of the arbitrary matrices Implementation: Stacks of matrices A Projection Operators and SCF A.1 Introduction: Optimum single determinant A.2 Alternative SCF conditions A.3 R matrices as projection Operators 403

9 xviii CONTENTS 15 Spin-restricted open shell Introduction The ROHF model Implementation A WEB for spin-restricted open shell Banana skins: unexpected disasters Symmetry restrictions Anions Aufbau exceptions Summary Molecular symmetry Introduction Symmetry and the HF method Permutational symmetry of the basis Implementation Permutation symmetry: summary Symmetry orbital transformations Introduction Symmetry-adapted basis Generation of symmetry Orbitals Conclusions A symmetry-adapted SCF method Introduction Permutations only Füll implementation; linear combinations Summary 494

10 CONTENTS xix 19.A Kronecker product notation A.1 Basis transformations A.2 Basis-product transformations A.3 Density matrix transformations A.4 Transformations in the HF matrix A.5 Practice Linear multi-determinant methods Correlation and the Hartree-Fock model The configuration interaction method The valence bond method Restricted Cl Symmetry-restricted Cl More general Cl Nesbet's method for large matrices "Direct" Cl Conclusions A The "orthogonal VB" model B DCI matrix elements The valence bond model Non-orthogonality in expansions Spins and spin functions Spin eigenfunctions and permutations Spin-free VB theory Summary 544

11 xx CONTENTS 22 Doubly-occupied MCSCF Introduction: natural orbitals Paired-excitation MCSCF Implementation Partial Paired-Excitations; GVB Details of GVB Implementation Interpreting the McWeenyan Introduction Stationary points Many shells Summary Core potentials Introduction Simple orthogonalization Transforming the Hartree-Fock equation The pseudopotential Arbitrariness in the pseudo-orbital Modelling atomic pseudopotentials Modelling atomic core potentials Several valence electrons Atomic cores in molecules Summary 589

12 CONTENTS xxi 25 Practical core potentials Introduction Forms for the core potentials Core potential integrals Implementation SCF perturbation theory Introduction Two forms for the HF equations Self-consistent perturbation theory The method Conciusions Time-dependent perturbations: RPA Introduction Time-dependent Hartree-Fock theory Oscillatory time-dependent perturbations Seif consistency Implementation A "Random phase approximation" B Time-dependent Variation principle Transitions and stability Introduction Transitions The transition frequencies Finite perturbations; oscillations Stability; the time-independent case Implementation 639

13 xxii CONTENTS 29 Two-electron transformations Orbital transformations Strategy Transformation without sorting Transformations with sorting Summary AA bitof fun: MP A.1 Derivation A.2 Implementation Geometry optimisation: derivatives Introduction Derivatives and perturbation theory Derivatives of variational Solutions Parameter-dependent basis functions The derivative of the SCF energy Derivatives of molecular integrals Derivatives of non-variational energies Higher derivatives Summary The Semi-empirical approach Introduction Use of Coulomb's law Atomic data Simulation or calibration? General conclusions 691

14 CONTENTS xxüi 32 Density functional theory Introduction Hohenberg and Kohn's proofs Kohn-Sham equations: introduction Kohn-Sham equations Non-Iocal Operators in orbital theories Implementing the Kohn-Sham equations A precursor: The Hartree-Fock-Slater model Implementation of the Kohn-Sham method The kinetic energy density Gradients in the exchange-correlation energy Numerical integration of densities Summary Semi-numerical methods Non-variational expansions The pseudospectral method The discrete variational method Additional reading and other material Additional reading Additional material by ftp 734