Handbook of Computational Quantum Chemistry

Transcription

1 Handbook of Computational Quantum Chemistry David B. Cook Dept. of Chemistry University of Sheffield DOVER PUBLICATIONS, INC. Mineola, New York

2 F Contents 1 Mechanics and molecules Introduction..... Time-independent Schrodinger equation. The Born-Oppenheimer model. The Pauli principle The orbital model. The determinantal method Physical interpretation Non-determinantal forms The variation principle Summary l.a Atomic units 24 l.b Standard Notation for Quantum Chemistry l.b.1 Introduction l.b.2 The Hamiltonian 28 l.b.3 Many-electron wavefunctions. 29

3 Xll l.b.4 l.b.5 CONTENTS 29 Spin-orbitals Linear expansions for the spatial orbitals. l.b.6 Primitive Gaussians... l.b.7 Single determinant energy expression l.b.8 Notation for repulsion integrals l.b.9 Spatial orbital repulsion integrals. l.b.lo Basis function repulsion integrals cc 2 The Hartree-Fock Method 2.1 Introduction The variational method. 2.3 The differential Hartree-Fock equation 2.4 Canonical form 2.5 Orbital energies. 2.6 Physical interpretation 2.7 Direct parametric minimisation. 2.8 Summary Single-determinant energy expression A.l Introduction A.2 The normalisation integral 61 2.A.3 One-electron terms 65 2.A.4 Two-electron terms. 2.A.5 Summary

4 >NTENTS contents The matrix SCF equations ntrod uction Notation The expansion The energy expression 80 Xlll 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 A Atomic orbitals B Charge density C Properties of the J and K matrices C.1 Mathematical properties C.2 Physical interpretation C.3 Supermatrices D An artifact of expansion D.l Lowest state of a given symmetry 108

5 xiv CONTENTS 3.E Single determinant: choice of orbitals llo 3.E.1 Orthogonal invariance E.2 Koopmans' theorem E.3 Localised orbitals E.4 "Zeroth-order" perturbed orbita Is ll3 c 4 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? A Jacobi diagonalisation A.1 Introduction A.2 The problem A.3 The solution A.4 Implementation A.5 Other diagonalisation methods. 182

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7 xvi 8 Molecular integrals: implementation 8.1 Introduction Data structures 8.3 Normalisation. CONTENTs Overview; the general structure Complex code management: the WEB system A working WEB Some comments on the WEB The full integral codes A Source for the WEB of fmch 9 Repulsion integral storage 9.1 Introduction A storage algorithm. 9.3 Implementation: putint 9.4 A partner for putint; getint 9.5 Conclusion "Virtual orbitals" 10.1 Introduction Virtual orbitals in practice 10.3 The virtual space in LCAO 10.4 Conclusions lo.a Perturbation theory 10.A.1 Introduction.. 10.A.2 Perturbation theory. 10.A.3 Perturbation theory for matrix equations

8 NTS contents Choice of tools Existing software Why ratfor? The Revision Control System: RCS 269 n.a RCS: version control xvii n.a.l Motivation n.a.2 Introduction. 288 n.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 o o 0 o 0 o I o 4 :I 12.A WEB Source for the scf code

9 xviii CONTENTs 12.B Blocking the Hartree-Fock matrix 38~ 12.B.1 The block form of the HF matrix 382 col 14, 12.B.2 Implementation C The Aufbau principle C.3 Introduction ! 12.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 422

10 NTENTs 382 contents 14.A Projection operators and SCF A.l Introduction: optimum single determinant A.2 Alternative SCF conditions R matrices as projection operators A Spin-restricted open shell ntrod uction The ROHF model Implementation A WEB for spin-restricted open shell 445 XIX Banana skins: unexpected disasters Symmetry restrictions Anions Aufbau exceptions Summary Molecular symmetry Introduction ~09 9 Q 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

11 XX 19 A symmetry-adapted SCF method 19.1 Introduction Permutations only 19.3 Full implementation; linear combinations 19.4 Summary.... CONTENTS col A Kronecker product notation 19.A.1 Basis transformations. 19.A.2 Basis-product transformations 19.A.3 Density matrix transformations. 19.A.4 Transformations in the HF matrix 19.A.5 Practice ' 20 Linear multi-determinant methods 20.1 Correlation and the Hartree-Fock model The configuration interaction method 20.3 The valence bond method 20.4 Restricted Cl Symmetry-restricted Cl 20.6 More general Cl Nesbet's method for large matrices 20.8 "Direct" Cl 20.9 Conclusions A The "orthogonal VB" model B DCI matrix elements 568

12 JTS )65 i66 68 contents The valence bond model 21 Non-orthogonality in expansions Spins and spin functions 0 0 o Spin eigenfunctions and permutations 2 4 Spin-free VB theory o Summary 0 0 o Doubly-occupied MCSCF 22 1 Introduction: natural orbitals Paired-excitation MCSCF Implementation... o Partial Paired-Excitations; GVB Details of GVB 22.6 Implementation 23 Interpreting the McWeenyan 2301 Introduction Stationary points 23.3 Many shells 23.4 Summary 24 Core potentials 24.1 Introduction Simple orthogonalization 2403 Transforming the Hartree-Fock equation 2404 The pseudopotential Arbitrariness in the pseudo-orbital 24.6 Modelling atomic pseudopotentials Modelling atomic core potentials Several valence electrons Atomic cores in molecules Summary xxi

13 xxii CONTENTS 25 Practical core potentials Introduction.... * Forms for the core potentials Core potential integrals Implementation c 26 SCF perturbation theory Introduction Two forms for the HF equations Self-consistent perturbation theory The method Conclusions Time-dependent perturbations: RPA Introduction Time-dependent Hartree-Fock theory Oscillatory time-dependent perturbations Self 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

14 NTENrs contents 63( Two-e I ec tron transformations Orbital transformations Strategy Transformation without sorting XXlll Transformations with sorting Summary A A bit of fun : MP A.l 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. 741 i Atomic data Simulation or calibration? General conclusions

15 xxiv 32.1 Introduction Hohenberg and Kohn's proofs 32.3 Kohn-Sham equations: introduction Kohn-Sham equations 32.5 Non-local operators in orbital theories CONTENI~ o Implementing the Kohn-Sham equations 33.1 A precursor: The Hartree-Fock-Siater model 33.2 Implementation of the Kohn-Sham method 33.3 The kinetic energy density Gradients in the exchange-correlation energy 33.5 Numerical integration of densities 33.6 Summary Semi-numerical methods 34.1 Non-variational expansions 34.2 The pseudospectral method 34.3 The discrete variational method Additional reading and other material 35.1 Additional reading Additional material by ftp