Handbook of Computational Quantum Chemistry

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ISBN: 9780486443072 | 0486443078
Cover: Paperback
Copyright: 8/2/2005

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This comprehensive text provides upper-level undergraduates and graduate students with an accessible introduction to the implementation of quantum ideas in molecular modeling, exploring practical applications alongside theoretical explanations. Topics include the Hartree-Fock method; matrix SCF equations; the implementation of the closed-shell case; and an introduction to molecular integrals that extends to their implementation. Other subjects include open shells; population analysis; molecular symmetry; linear multi-determinant methods; time-dependent perturbations; density functional theory; and implementing the Kohn-Sham equations. The text concludes with suggestions for additional reading. 1998 ed.

Mechanics and molecules

(36)

Introduction

(4)

Time-independent Schrodinger equation

(2)

The Born-Oppenheimer model

(2)

The Pauli principle

(2)

The orbital model

(3)

The determinantal method

(2)

Physical interpretation

(2)

Non-determinantal forms

(1)

The variation principle

(4)

Summary

(1)

Atomic units

(3)

Standard Notation for Quantum Chemistry

(10)

Introduction

(1)

The Hamiltonian

(1)

Many-electron wavefunctions

(1)

Spin-orbitals

(1)

Linear expansions for the spatial orbitals

(1)

Primitive Gaussians

(1)

Single determinant energy expression

(1)

Notation for repulsion integrals

(1)

Spatial orbital repulsion integrals

(1)

Basis function repulsion integrals

(2)

The Hartree-Fock Method

(38)

Introduction

(1)

The variational method

(2)

The differential Hartree-Fock equation

(7)

Canonical form

(2)

Orbital energies

(1)

Physical interpretation

(1)

Direct parametric minimisation

(2)

Summary

(1)

Single-determinant energy expression

(21)

Introduction

(2)

The normalisation integral

(5)

One-electron terms

(4)

Two-electron terms

(5)

Summary

(5)

The matrix SCF equations

(40)

Introduction

(2)

Notation

(1)

The expansion

(2)

The energy expression

(1)

The numerator: Hamiltonian mean value

(3)

The denominator: normalisation condition

(2)

The Hartree-Fock equation

(1)

``Normalisation'': the Lagrangian

(1)

Preliminary summary

(1)

Some technical manipulations

(3)

Canonical orbitals

(3)

The total energy

(1)

Summary

(1)

Atomic orbitals

(3)

Charge density

100

(3)

Properties of the J and K matrices

103

(5)

Mathematical properties

103

(3)

Physical interpretation

106

(1)

Supermatrices

107

(1)

An artifact of expansion

108

(2)

Lowest state of a given symmetry

108

(2)

Single determinant: choice of orbitals

110

(5)

Orthogonal invariance

110

(2)

Koopmans' theorem

112

(1)

Localised orbitals

113

(1)

``Zeroth-order'' perturbed orbitals

113

(2)

A special case: closed shells

115

(8)

Introduction

115

(1)

Notation for the closed-shell case

116

(1)

Closed-shell expansion

116

(1)

The closed-shell ``HF'' equation

117

(4)

Closed-shell summary

121

(2)

Implementation of the closed-shell case

123

(76)

Preview

124

(1)

Vectors, matrices and arrays

125

(6)

The implementation: getting started

131

(17)

The implementation: repulsion integral access

148

(10)

Building a testbench: conventional SCF

158

(7)

Another testbench: direct SCF

165

(8)

Summary

173

(1)

What next?

174

(1)

Jacobi diagonalisation

175

(9)

Introduction

175

(1)

The problem

176

(1)

The solution

177

(2)

Implementation

179

(3)

Other diagonalisation methods

182

(2)

Orthogonalisation

184

(6)

Introduction

184

(2)

Functions of a matrix

186

(1)

Implementation

187

(3)

getint and data for H2O

190

(4)

Coding the standard index loops

194

(5)

Improvements: tools and methods

199

(18)

Introduction

199

(1)

Versions: conditional compilation

200

(7)

Improved diagonalisation

207

(3)

Simple interpolation

210

(2)

Improving the formation of G(R)

212

(3)

Summary

215

(2)

Molecular integrals: an introduction

217

(38)

Introduction

218

(1)

Basis functions

219

(2)

AOs and atom-centred-functions

221

(1)

Multi-dimensional integral evaluation

222

(1)

Molecular integrals over STOs

223

(9)

Basis functions of convenience

232

(2)

Gaussian basis functions

234

(17)

The contraction technique

251

(4)

Molecular integrals: implementation

255

(42)

Introduction

256

(1)

Data structures

256

(4)

Normalisation

260

(3)

Overview; the general structure

263

(6)

Complex code management: the WEB system

269

(7)

A working WEB

276

(11)

Some comments on the WEB

287

(1)

The full integral codes

288

(1)

Source for the WEB of fmch

289

(8)

Repulsion integral storage

297

(14)

Introduction

297

(1)

A storage algorithm

298

(2)

Implementation: putint

300

(7)

A partner for putint; getint

307

(2)

Conclusion

309

(2)

``Virtual orbitals''

311

(20)

Introduction

311

(1)

Virtual orbitals in practice

312

(5)

The virtual space in LCAO

317

(5)

Conclusions

322

(1)

Perturbation theory

323

(8)

Introduction

323

(1)

Perturbation theory

324

(5)

Perturbation theory for matrix equations

329

(2)

Choice of tools

331

(12)

Existing software

331

(4)

Why ratfor?

335

(2)

The Revision Control System: RCS

337

(2)

RCS: version control

339

(4)

Motivation

339

(1)

Introduction

340

(1)

Getting started with RCS

341

(2)

Open shells: implementing UHF

343

(56)

Introduction

344

(1)

Choice of constraints

344

(3)

Organising the basis

347

(1)

Integrals over the spin-basis

348

(2)

Implementation

350

(1)

J and K for GUHF

351

(6)

The GUHF testbench

357

(3)

Interpreting the MO coefficients

360

(3)

DODS or GUHF?

363

(1)

Version 1 of the SCF code

363

(5)

WEB output for function scf

368

(8)

Comments

376

(1)

WEB Source for the scf code

377

(5)

Blocking the Hartree-Fock matrix

382

(12)

The block form of the HF matrix

382

(2)

Implementation

384

(10)

The Aufbau principle

394

(5)

Introduction

394

(1)

The second variation

395

(1)

Special case: a single excitation

396

(3)

Population analysis

399

(10)

Introduction

399

(1)

Densities and spin-densities

400

(1)

Basis representations: charges

401

(3)

Basis-function analysis

404

(2)

A cautionary note

406

(1)

Multi-determinant forms

407

(1)

Implementation

408

(1)

The general MO functional

409

(32)

A generalisation

409

(1)

Shells of orbitals

410

(2)

The variational method

412

(4)

A single ``Hartree-Fock'' operator

416

(3)

Non-orthogonal basis

419

(1)

Choice of the arbitrary matrices

420

(2)

Implementation: stacks of matrices

422

(11)

Projection operators and SCF

433

(8)

Introduction: optimum single determinant

433

(2)

Alternative SCF conditions

435

(1)

R matrices as projection operators

436

(5)

Spin-restricted open shell

441

(32)

Introduction

441

(1)

The ROHF model

442

(2)

Implementation

444

(1)

A WEB for spin-restricted open shell

445

(28)

Banana skins: unexpected disasters

473

(6)

Symmetry restrictions

474

(1)

Anions

475

(2)

Aufbau exceptions

477

(1)

Summary

478

(1)

Molecular symmetry

479

(26)

Introduction

479

(1)

Symmetry and the HF method

480

(3)

Permutational symmetry of the basis

483

(5)

Implementation

488

(16)

Permutation symmetry: summary

504

(1)

Symmetry orbital transformations

505

(10)

Introduction

505

(4)

Symmetry-adapted basis

509

(3)

Generation of symmetry orbitals

512

(2)

Conclusions

514

(1)

A symmetry-adapted SCF method

515

(26)

Introduction

515

(3)

Permutations only

518

(10)

Full implementation; linear combinations

528

(5)

Summary

533

(1)

Kronecker product notation

534

(7)

Basis transformations

534

(1)

Basis-product transformations

535

(1)

Density matrix transformations

536

(1)

Transformations in the HF matrix

537

(2)

Practice

539

(2)

Linear multi-determinant methods

541

(30)

Correlation and the Hartree-Fock model

541

(2)

The configuration interaction method

543

(1)

The valence bond method

544

(1)

Restricted CI

545

(5)

Symmetry-restricted CI

550

(3)

More general CI

553

(1)

Nesbet's method for large matrices

554

(6)

``Direct'' CI

560

(5)

Conclusions

565

(1)

The ``orthogonal VB'' model

566

(2)

DCI matrix elements

568

(3)

The valence bond model

571

(16)

Non-orthogonality in expansions

571

(2)

Spins and spin functions

573

(3)

Spin eigenfunctions and permutations

576

(5)

Spin-free VB theory

581

(4)

Summary

585

(2)

Doubly-occupied MCSCF

587

(18)

Introduction: natural orbitals

588

(2)

Paired-excitation MCSCF

590

(5)

Implementation

595

(1)

Partial Paired-Excitations; GVB

596

(3)

Details of GVB

599

(5)

Implementation

604

(1)

Interpreting the McWeenyan

605

(6)

Introduction

605

(2)

Stationary points

607

(1)

Many shells

608

(2)

Summary

610

(1)

Core potentials

611

(26)

Introduction

612

(2)

Simple orthogonalization

614

(1)

Transforming the Hartree-Fock equation

615

(3)

The pseudopotential

618

(3)

Arbitrariness in the pseudo-orbital

621

(3)

Modelling atomic pseudopotentials

624

(2)

Modelling atomic core potentials

626

(3)

Several valence electrons

629

(4)

Atomic cores in molecules

633

(1)

Summary

634

(3)

Practical core potentials

637

(16)

Introduction

637

(1)

Forms for the core potentials

638

(3)

Core potential integrals

641

(10)

Implementation

651

(2)

SCF perturbation theory

653

(16)

Introduction

654

(1)

Two forms for the HF equations

654

(3)

Self-consistent perturbation theory

657

(2)

The method

659

(8)

Conclusions

667

(2)

Time-dependent perturbations: RPA

669

(14)

Introduction

669

(1)

Time-dependent Hartree-Fock theory

670

(1)

Oscillatory time-dependent perturbations

671

(4)

Self consistency

675

(1)

Implementation

676

(2)

``Random phase approximation''

678

(2)

Time-dependent variation principle

680

(3)

Transitions and stability

683

(8)

Introduction

683

(1)

Transitions

684

(1)

The transition frequencies

685

(1)

Finite perturbations; oscillations

686

(2)

Stability; the time-independent case

688

(1)

Implementation

689

(2)

Two-electron transformations

691

(32)

Orbital transformations

691

(2)

Strategy

693

(2)

Transformation without sorting

695

(10)

Transformations with sorting

705

(3)

Summary

708

(1)

A bit of fun: MP2

709

(14)

Derivation

709

(3)

Implementation

712

(11)

Geometry optimisation: derivatives

723

(16)

Introduction

724

(1)

Derivatives and perturbation theory

725

(2)

Derivatives of variational solutions

727

(2)

Parameter-dependent basis functions

729

(1)

The derivative of the SCF energy

730

(4)

Derivatives of molecular integrals

734

(1)

Derivatives of non-variational energies

735

(2)

Higher derivatives

737

(1)

Summary

737

(2)

The Semi-empirical approach

739

(24)

Introduction

739

(2)

Use of Coulomb's law

741

(1)

Atomic data

742

(2)

Simulation or calibration?

744

(1)

General conclusions

745

(3)

Introduction

748

(2)

Hohenberg and Kohn's proofs

750

(5)

Kohn-Sham equations: introduction

755

(2)

Kohn-Sham equations

757

(3)

Non-local operators in orbital theories

760

(3)

Implementing the Kohn-Sham equations

763

(16)

A precursor: The Hartree-Fock-Slater model

764

(2)

Implementation of the Kohn-Sham method

766

(5)

The kinetic energy density

771

(1)

Gradients in the exchange-correlation energy

772

(1)

Numerical integration of densities

773

(3)

Summary

776

(3)

Semi-numerical methods

779

(12)

Non-variational expansions

779

(3)

The pseudospectral methods

782

(4)

The discrete variational method

786

(5)

Additional reading and other material

791

Additional reading

791

(3)

Additional material by ftp

794

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Handbook of Computational Quantum Chemistry

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