Molecular Electronic-Structure Theory

by Helgaker, Trygve; Jorgensen, Poul; Olsen, Jeppe

Edition: 1st

ISBN13: 9780471967552

ISBN10: 0471967556

Format: Hardcover

Pub. Date: 2000-10-26

Publisher(s): WILEY

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Summary

Ab initio quantum chemistry has emerged as an important tool in chemical research and is appliced to a wide variety of problems in chemistry and molecular physics. Recent developments of computational methods have enabled previously intractable chemical problems to be solved using rigorous quantum-mechanical methods. This is the first comprehensive, up-to-date and technical work to cover all the important aspects of modern molecular electronic-structure theory. Topics covered in the book include: * Second quantization with spin adaptation * Gaussian basis sets and molecular-integral evaluation * Hartree-Fock theory * Configuration-interaction and multi-configurational self-consistent theory * Coupled-cluster theory for ground and excited states * Perturbation theory for single- and multi-configurational states * Linear-scaling techniques and the fast multipole method * Explicity correlated wave functions * Basis-set convergence and extrapolation * Calibration and benchmarking of computational methods, with applications to moelcular equilibrium structure, atomization energies and reaction enthalpies. Molecular Electronic-Structure Theory makes extensive use of numerical examples, designed to illustrate the strengths and weaknesses of each method treated. In addition, statements about the usefulness and deficiencies of the various methods are supported by actual examples, not just model calculations. Problems and exercises are provided at the end of each chapter, complete with hints and solutions. This book is a must for researchers in the field of quantum chemistry as well as for nonspecialists who wish to acquire a thorough understanding of ab initio molecular electronic-structure theory and its applications to problems in chemistry and physics. It is also highly recommended for the teaching of graduates and advanced undergraduates.

Author Biography

Trygve Helgaker Department of Chemistry, University of Oslo, Norway Poul Jorgensen and Jeppe Olsen Department of Chemistry, University of Aarhus, Denmark

Preface

xxi

Overview

xxv

Programs used in the preparation of this book

xxix

Second Quantization

(33)

The Fock space

(1)

Creation and annihilation operators

(4)

Creation operators

(1)

Annihilation operators

(1)

Anticommuntation relations

(1)

Number-conserving operators

(3)

Occupation-number operators

(1)

The number operator

(1)

Excitation operators

(2)

The representation of one- and two-electron operators

(5)

One-electron operators

(2)

Two-electron operators

(2)

The molecular electronic Hamiltonian

(1)

Products of operators in second quantization

(4)

Operator products

(3)

The canonical commutators

(1)

First- and second-quantization operators compared

(1)

Density matrices

(6)

The one-electron density matrix

(1)

The two-electron density matrix

(2)

Density matrices in spin-orbital and coordinate representations

(2)

Commutators and anticommutators

(2)

Nonorthogonal spin orbitals

(7)

Creation and annihilation operators

(2)

One- and two-electron operators

(1)

Biorthogonal operators

(1)

References

(1)

Further reading

(1)

Exercises

(1)

Solutions

(2)

Spin in Second Quantization

(46)

Spin functions

(1)

Operators in the orbital basis

(6)

Spin-free operators

(2)

Spin operators

(2)

Mixed operators

(1)

Spin tensor operators

(5)

Spin tensor operators

(2)

Creation and annihilation operators

(1)

Two-body creation operators

(1)

Excitation operators

(2)

Singlet excitation operators

(1)

Spin properties of determinants

(5)

General considerations

(1)

Spin projection of determinants

(1)

Total spin of determinants

(2)

Configuration state functions

(2)

The genealogical coupling scheme

(8)

Representations of determinants and CSFs

(1)

Genealogical coupling

(1)

Coupling coefficients

(1)

An example: three electrons in three orbitals

(1)

Completeness and orthonormality

(1)

Transformations between determinant and CSF bases

(1)

Genealogical coupling of operators

(1)

Density matrices

(19)

Orbital-density matrices

(2)

Spin-density matrices

(1)

Density functions

(2)

References

(1)

Further reading

(1)

Exercises

(4)

Solutions

(10)

Orbital Rotations

(27)

Unitary transformations and matrix exponentials

(6)

Matrix exponentials

(1)

Exponential representations of unitary matrices

(1)

Special unitary matrices

(1)

Orthogonal matrices

(1)

Evaluation of matrix exponentials

(1)

Nonunitary transformations

(2)

Unitary spin-orbital transformations

(3)

Unitary matrix expansions of creation and annihilation operators

(1)

Exponential unitary transformations of the elementary operators

(1)

Exponential unitary transformations of states in Fock space

(1)

Symmetry-restricted unitary transformations

(4)

The need for symmetry restrictions

(1)

Symmetry restrictions in the spin-orbital basis

(1)

Symmetry restrictions in the orbital basis

(2)

The logarithmic matrix function

(14)

Definition of the logarithmic matrix function

(1)

Expansion of the logarithmic matrix function

(1)

Properties of the logarithmic matrix function

(1)

References

(1)

Further reading

(1)

Exercises

(4)

Solutions

(8)

Exact and Approximate Wave Functions

107

(35)

Characteristics of the exact wave function

107

(4)

The variation principle

111

(15)

The variation principle

111

(1)

The variation method

112

(1)

Linear expansions and eigenvalue equations

113

(2)

Upper bounds and the Hylleraas--Undheim theorem

115

(2)

Nonlinear expansions

117

(2)

The Hellmann--Feynman theorem

119

(2)

The molecular electronic virial theorem

121

(2)

Variational reformulation of nonvariational energies

123

(3)

The variation principle summarized

126

(1)

Size-extensivity

126

(9)

Size-extensivity of exact wave functions

126

(3)

Size-extensivity of linear variational wave functions

129

(2)

Matrix representation of the noninteracting eigenvalue problem

131

(1)

Size-extensivity of exponential wave functions

132

(3)

Symmetry constraints

135

(7)

References

137

(1)

Further reading

137

(1)

Exercises

137

(2)

Solutions

139

(3)

The Standard Models

142

(59)

One- and N-electron expansions

143

(3)

A model system: the hydrogen molecule in a minimal basis

146

(16)

One-electron basis

146

(2)

N-electron basis

148

(1)

Density matrices and molecular integrals

148

(2)

Bonding and antibonding configurations

150

(2)

Superposition of configurations

152

(2)

Covalent and ionic states

154

(2)

Open-shell states

156

(2)

Electron correlation

158

(1)

The dissociation limit

159

(3)

Static and dynamical correlation

162

(1)

Exact wave functions in Fock space

162

(5)

Full configuration-interaction wave functions

162

(1)

The electronic ground state of the hydrogen molecule

163

(2)

The electronic ground state of the water molecule

165

(2)

The Hartree--Fock approximation

167

(9)

The Hartree--Fock model

167

(2)

The Fock operator and the canonical representation

169

(1)

Restricted and unrestricted Hartree--Fock theory

170

(1)

The correlation energy

170

(1)

The ground state of the hydrogen molecule

171

(1)

The bonded hydrogen molecule

172

(1)

The RHF dissociation of the hydrogen molecule

172

(1)

The UHF dissociation of the hydrogen molecule

173

(1)

The ground state of the water molecule

174

(1)

The dissociation of the water molecule

175

(1)

Final comments

176

(1)

Multiconfigurational self-consistent field theory

176

(5)

The multiconfigurational self-consistent field model

176

(1)

The ground state of the hydrogen molecule

177

(1)

The selection of MCSCF configuration spaces

177

(1)

The ground state of the water molecule

178

(2)

Final comments

180

(1)

Configuration-interaction theory

181

(5)

The configuration-interaction model

181

(1)

Single-reference CI wave functions

182

(1)

Multireference CI wave functions

183

(3)

Final comments

186

(1)

Coupled-cluster theory

186

(6)

The coupled-cluster model

186

(1)

The exponential ansatz of coupled-cluster theory

187

(2)

The ground state of the water molecule

189

(2)

The unrestricted coupled-cluster model

191

(1)

Approximate treatments of triple excitations

191

(1)

Final comments

191

(1)

Perturbation theory

192

(9)

Møller--Plesset perturbation theory

192

(1)

The ground state of the water molecule

193

(1)

Convergence of the Moslash;ller--Plesset perturbation series

193

(1)

The ground state of the hydrogen molecule

194

(2)

Final comments

196

(1)

References

196

(1)

Further reading

196

(1)

Exercises

196

(2)

Solutions

198

(3)

Atomic Basis Functions

201

(55)

Requirements on one-electron basis functions

201

(2)

One- and many-centre expansions

203

(1)

The one-electron central-field system

204

(3)

The angular basis

207

(11)

The spherical harmonics

207

(2)

The solid harmonics

209

(1)

Explicit Cartesian expressions for the complex solid harmonics

210

(4)

Explicit Cartesian expressions for the real solid harmonics

214

(1)

Recurrence relations for the real solid harmonics

215

(3)

Exponential radial functions

218

(11)

The Laguerre polynomials

219

(2)

The hydrogenic functions

221

(1)

The Laguerre functions

222

(1)

The carbon orbitals expanded in Laguerre functions

223

(2)

The nodeless Slater-type orbitals

225

(1)

STOs with variable exponents

226

(1)

STO basis sets

227

(2)

Gaussian radial functions

229

(27)

The harmonic-oscilltor functions in polar coordinates

230

(1)

The carbon orbitals expanded in HO functions

231

(1)

The nodeless Gaussian-type orbitals

232

(1)

The GTOs with variable exponents

233

(2)

The carbon orbitals expanded in GTOs

235

(1)

The HO functions in Cartesian coordinates

236

(1)

The Cartesian GTOs

237

(1)

References

238

(1)

Further reading

239

(1)

Exercises

239

(6)

Solutions

245

(11)

Short-Range Interactions and Orbital Expansions

256

(31)

The Coulomb hole

256

(3)

The Coulomb cusp

259

(3)

Approximate treatments of the ground-state helium atom

262

(5)

Configuration-interaction expansions

262

(2)

Correlating functions and explicity correlated methods

264

(2)

The Hylleraas function

266

(1)

Conclusions

267

(1)

The partial-wave expansion of the ground-state helium atom

267

(6)

Partial-wave expansion of the interelectronic distance

267

(1)

Partial-wave expansion of the wave function

268

(2)

The asymptotic convergence of the partial-wave expansion

270

(2)

The truncation error of the partial-wave expansion

272

(1)

The principal expansion of the ground-state helium atom

273

(5)

The principal expansion and its asymptotic convergence

273

(2)

Comparison of the partial-wave and principal expansions

275

(1)

The Coulomb hole in the principal expansion

276

(1)

Conclusions

276

(2)

Electron-correlation effects sumarized

278

(9)

References

278

(1)

Further reading

279

(1)

Exercises

279

(3)

Solutions

282

(5)

Gaussian Basis Sets

287

(49)

Gaussian basis functions

287

(1)

Gaussian basis sets for Hartree--Fock calculations

288

(12)

STO-kG basis sets

288

(3)

Primitive expansions of Hartree--Fock orbitals

291

(1)

Even-tempered basis sets

292

(2)

Contracted Gaussians

294

(1)

Segmented contractions

295

(2)

Simultaneous optimization of exponents and coefficients

297

(2)

Polarization functions

299

(1)

Gaussian basis sets for correlated calculations

300

(15)

Core and valence correlation energies

301

(3)

Atomic natural orbitals

304

(3)

Correlation-consistent basis sets

307

(5)

Extended correlation-consistent basis sets

312

(3)

Basis-set convergence

315

(12)

Basis-set convergence of the Hartree--Fock model

315

(2)

Basis-set convergence of correlated models

317

(5)

The asymptotic convergence of the correlation energy

322

(2)

Basis-set convergence of the binding energy

324

(3)

Basis-set superposition error

327

(9)

Basis-set superposition error and the counterpoise correction

327

(1)

BSSE in the neon dimer

328

(3)

BSSE in the water dimer

331

(2)

BSSE in the BH molecule

333

(1)

Summary

334

(1)

References

335

(1)

Further reading

335

(1)

Molecular Integral Evaluation

336

(97)

Contracted spherical-harmonic Gaussians

336

(3)

Primitive Cartesian GTOs

336

(1)

Spherical-harmonic GTOs

337

(1)

Contracted GTOs

338

(1)

Computational considerations

338

(1)

Cartesian Gaussians

339

(5)

Cartesian Gaussians

339

(1)

Recurrence relations for Cartesian Gaussians

340

(1)

The Gaussian product rule

341

(1)

Gaussian overlap distributions

341

(2)

Properties of Gaussian overlap distributions

343

(1)

Integrals over spherical overlap distributions

344

(1)

The Obara--Saika scheme for simple integrals

344

(5)

Overlap integrals

345

(1)

Multipole-moment integrals

346

(1)

Integrals over differential operators

347

(1)

Momentum and kinetic-energy integrals

348

(1)

Hermite Gaussians

349

(3)

Hermite Gaussians

349

(1)

Derivative and recurrence relations for Hermite Gaussians

350

(1)

Integrals over Hermite Gaussians

351

(1)

Hermite Gaussians and HO functions compared

352

(1)

The McMurchie--Davidson scheme for simple integrals

352

(5)

Overlap distributions expanded in Hermite Gaussians

353

(2)

Overlap distributions from Hermite Gaussians by recursion

355

(1)

The McMurchie--Davidson scheme for multipole-moment integrals

356

(1)

Gaussian quadrature for simple integrals

357

(4)

Orthogonal polynomials

358

(1)

Gaussian quadrature

359

(1)

Proof of the Gaussian-quadrature formula

360

(1)

Gauss--Hermite quadrature for simple integrals

361

(1)

Coulomb integrals over spherical Gaussians

361

(4)

Spherical Gaussian charge distributions

361

(1)

The potential from a spherical Gaussian charge distribution

362

(1)

The repulsion between spherical Gaussian charge distributions

363

(1)

The electrostatics of spherical Gaussian distributions

364

(1)

The Boys function

365

(7)

The Boys function

365

(1)

Evaluation of the Boys function

366

(2)

The incomplete gamma function

368

(1)

The error function

369

(1)

The complementary error function

370

(1)

The confluent hypergeometric function

371

(1)

The McMurchie--Davidson scheme for Coulomb integrals

372

(9)

Hermite Coulomb integrals

373

(1)

The evaluation of Hermite Coulomb integrals

374

(1)

Cartesian Coulomb integrals by Hermite expansion

375

(2)

Cartesian Coulomb integrals by Hermite recursion

377

(1)

Computational considerations for the one-electron integrals

377

(2)

Computational considerations for the two-electron integrals

379

(2)

The Obara--Saika scheme for Coulomb integrals

381

(6)

The Obara--Saika scheme for one-electron Coulomb integrals

382

(1)

The Obara--Saika scheme for two-electron Coulomb integrals

383

(2)

The electron-transfer and horizontal recurrence relations

385

(1)

Computational considerations for the two-electron integrals

386

(1)

Rys quadrature for Coulomb integrals

387

(11)

Motivation for the Gaussian-quadrature scheme

388

(1)

Gaussian quadrature for even polynomials and weight functions

388

(2)

Rys polynomials and Gauss--Rys quadrature

390

(2)

The Rys scheme for Hermite Coulomb integrals

392

(2)

The Rys scheme for Cartesian Coulomb integrals

394

(1)

Obara--Saika recursion for the two-dimensional Rys integrals

395

(2)

Computational considerations for the two-electron integrals

397

(1)

Scaling properties of the molecular integrals

398

(7)

Linear scaling of the overlap and kinetic-energy integrals

398

(2)

Quadratic scaling of the Coulomb integrals

400

(1)

Linear scaling of the nonclassical Coulomb integrals

401

(2)

The Schwarz inequality

403

(2)

The multipole method for Coulomb integrals

405

(12)

The multipole method for primitive two-electron integrals

405

(4)

Convergence of the multipole expansion

409

(1)

The multipole method for contracted two-electron integrals

409

(1)

Translation of multipole moments

410

(2)

Real multipole moments

412

(1)

The real translation matrix

413

(1)

The real interaction matrix

414

(1)

Evaluation of the scaled solid harmonics

415

(2)

The multipole method for large systems

417

(16)

The naive multipole method

417

(3)

The two-level multipole method

420

(1)

The fast multipole method

421

(2)

The continuous fast multipole method

423

(2)

References

425

(1)

Further reading

426

(1)

Exercises

426

(2)

Solutions

428

(5)

Hartree--Fock Theory

433

(90)

Parameterization of the wave function and the energy

433

(5)

Singlet and triplet CSFs

434

(1)

Orbital rotations

435

(2)

Expansion of the energy

437

(1)

The Hartree--Fock wave function

438

(5)

The Hartree--Fock wave function

438

(2)

Redundant parameters

440

(1)

The Brillouin theorem

441

(1)

Size-extensivity

442

(1)

Canonical Hartree--Fock theory

443

(7)

The Fock operator

444

(1)

Identification of the elements of the Fock operator

445

(2)

The Fock matrix

447

(1)

The self-consistent field method

448

(1)

The variational and canonical conditions compared

449

(1)

The RHF total energy and orbital energies

450

(4)

The Hamiltonian and the Fock operator

450

(1)

The canonical representation and orbital energies

450

(2)

The Hartree--Fock energy

452

(1)

Hund's rule for singlet and triplet states

452

(1)

The fluctuation potential

453

(1)

Koopmans' theorem

454

(4)

Koopmans' theorem for ionization potentials

454

(1)

Koopmans' theorem for electron affinities

455

(1)

Ionization potentials of H2O and N2

456

(2)

The Roothaan--Hall self-consistent field equations

458

(7)

The Roothaan--Hall equations

458

(2)

DIIS convergence acceleration

460

(3)

Integral-direct Hartree--Fock theory

463

(2)

Density-based Hartree--Fock theory

465

(13)

Density-matrix formulation of Hartree--Fock theory

465

(1)

Properties of the MO density matrix

466

(1)

Properties of the AO density matrix

467

(1)

Exponential parametrization of the AO density matrix

468

(1)

The redundancy of the exponential parametrization

469

(1)

Purification of the density matrix

470

(1)

Convergence of the purification scheme

471

(2)

The Hartree--Fock energy and the variational conditions

473

(2)

The density-based SCF method

475

(2)

Optimization of the SCF orbital-energy function

477

(1)

Linear scaling of the density-based SCF scheme

477

(1)

Second-order optimization

478

(12)

Newton's method

478

(2)

Density-based formulation of Newton's method

480

(1)

The electronic gradient in orbital-based Hartree--Fock theory

481

(1)

The inactive and active Fock matrices

482

(2)

Computational cost for the calculation of the Fock matrix

484

(1)

The electronic Hessian in orbital-based Hartree--Fock theory

485

(3)

Linear transformations in the MO basis

488

(1)

Linear transformations in the AO basis

489

(1)

The SCF method as an approximate second-order method

490

(6)

The GBT vector

491

(1)

The Fock operator

491

(2)

Identification from the gradient

493

(1)

Identification from the Hessian

494

(1)

Convergence rates

494

(2)

The SCF and Newton methods compared

496

(1)

Singlet and triplet instabilities in RHF theory

496

(8)

Orbital-rotation operators in RHF and UHF theories

497

(1)

RHF instabilities for nondegenerate electronic states

498

(1)

RHF energies of degenerate electronic states

499

(1)

Triplet instabilities in H2

500

(1)

Triplet instabilities in H2O

500

(2)

Singlet instabilities in the allyl radical

502

(2)

Multiple solutions in Hartree--Fock theory

504

(19)

References

506

(1)

Further reading

506

(1)

Exercises

506

(7)

Solutions

513

(10)

Configuration-Interaction Theory

523

(75)

The CI model

523

(4)

The CI model

524

(1)

Full CI wave functions

524

(2)

Truncated CI wave functions: CAS and RAS expansions

526

(1)

Size-extensivity and the CI model

527

(8)

FCI wave functions

528

(1)

Truncated CI wave functions

529

(1)

The Davidson correction

530

(1)

A numerical study of size-extensivity

531

(4)

A CI model system for noninteracting hydrogen molecules

535

(5)

The CID wave function and energy

535

(2)

The Davidson correction

537

(1)

The CID one-electron density matrix

537

(1)

The FCI distribution of excitation levels

538

(2)

Parametrization of the CI model

540

(3)

The CI expansion

540

(2)

The CI energy

542

(1)

Optimization of the CI wave function

543

(7)

The Newton step

544

(1)

Convergence rate of Newton's method for the CI energy

545

(2)

Approximate Newton schemes

547

(1)

Convergence rate of quasi-Newton schemes for the CI energy

548

(2)

Slater determinants as products of alpha and beta strings

550

(2)

The determinantal representation of the Hamiltonian operator

552

(2)

Direct CI methods

554

(15)

General considerations

554

(1)

Ordering and addressing of spin strings

555

(3)

The N-resolution method

558

(2)

The minimal operation-count method

560

(4)

Direct CI algorithms for RAS calculations

564

(3)

Simplifications for wave functions of zero projected spin

567

(1)

Density matrices

568

(1)

CI orbital transformations

569

(4)

Symmetry-broken CI solutions

573

(25)

References

574

(1)

Further reading

575

(1)

Exercises

575

(8)

Solutions

583

(15)

Multiconfigurational Self-Consistent Field Theory

598

(50)

The MCSCF model

598

(2)

The MCSCF energy and wave function

600

(10)

The parametrization of the MCSCF state

600

(1)

The Taylor expansion of the MCSCF energy

601

(2)

The MCSCF electronic gradient and Hessian

603

(1)

Invariance of the second-order MCSCF energy

604

(1)

Rank-1 contributions to the MCSCF electronic Hessian

604

(1)

Redundant orbital rotations

605

(3)

The MCSCF electronic gradient at stationary points

608

(1)

The MCSCF electronic Hessian at stationary points

609

(1)

The MCSCF Newton trust-region method

610

(6)

The Newton step

610

(1)

The level-shifted Newton step

611

(1)

The level-shift parameter

612

(2)

Step control for ground states

614

(1)

Step control for excited states

614

(1)

Trust-radius update schemes

615

(1)

The Newton eigenvector method

616

(5)

The MCSCF eigenvalue problem

616

(1)

The Newton eigenvector method

617

(2)

Norm-extended optimization

619

(1)

The augmented-Hessian method

620

(1)

Computational considerations

621

(9)

The MCSCF electronic gradient

622

(1)

MCSCF Hessian transformations

623

(2)

Inner and outer iterations

625

(1)

The structure of the MCSCF electronic Hessian

626

(2)

Examples of MCSCF optimizations

628

(2)

Exponential parametrization of the configuration space

630

(7)

General exponential parametrization of the configuration space

630

(1)

Exponential parametrization for a single reference state

631

(2)

A basis for the orthogonal complement to the reference state

633

(1)

Exponential parametrization for several reference states

634

(3)

MCSCF theory for several electronic states

637

(3)

Separate optimization of the individual states

637

(1)

State-averaged MCSCF theory

638

(2)

Removal of RHF instabilities in MCSCF theory

640

(8)

Bond breaking in H2O

640

(1)

The ground state of the allyl radical

641

(2)

References

643

(1)

Further reading

643

(1)

Exercises

643

(2)

Solutions

645

(3)

Coupled-Cluster Theory

648

(76)

The coupled-cluster model

648

(6)

Pair clusters

649

(1)

The coupled-cluster wave function

650

(1)

Connected and disconnected clusters

650

(1)

The coupled-cluster Schrodinger equation

651

(3)

The coupled-cluster exponential ansatz

654

(11)

The exponential ansatz

654

(1)

The coupled-cluster hierarchy of excitation levels

654

(3)

The projected coupled-cluster equations

657

(3)

The coupled-cluster energy

660

(1)

The coupled-cluster amplitude equations

660

(2)

Coupled-cluster theory in the canonical representation

662

(1)

Comparison of the CI and coupled-cluster hierarchies

662

(1)

Cluster-commutation conditions and operator ranks

663

(2)

Size-extensivity in coupled-cluster theory

665

(5)

Size-extensivity in linked coupled-cluster theory

665

(2)

Termwise size-extensivity

667

(1)

Size-extensivity in unlinked coupled-cluster theory

668

(1)

A numerical study of size-extensivity

669

(1)

Coupled-cluster optimization techniques

670

(4)

Newton's method

671

(1)

The perturbation-based quasi-Newton method

672

(1)

DIIS acceleration of the quasi-Newton method

672

(1)

Examples of coupled-cluster optimizations

673

(1)

The coupled-cluster variational Lagrangian

674

(3)

The coupled-cluster Lagrangian

674

(1)

The Hellmann--Feynman theorem

675

(1)

Lagrangian density matrices

676

(1)

The equation-of-motion coupled-cluster method

677

(8)

The equation-of-motion coupled-cluster model

677

(2)

The EOM-CC eigenvalue problem

679

(1)

The similarity-transformed Hamiltonian and the Jacobian

680

(1)

Solution of the EOM-CC eigenvalue problem

681

(2)

Size-extensivity of the EOM-CC energies

683

(1)

Final comments

684

(1)

The closed-shell CCSD model

685

(13)

Parametrization of the CCSD cluster operator

685

(1)

The CCSD energy expression

686

(1)

The T1-transformed Hamiltonian

687

(3)

The T1-transformed integrals

690

(1)

Representation of the CCSD projection manifold

691

(1)

The norm of the CCSD wave function

692

(1)

The CCSD singles projection

693

(2)

The CCSD doubles projection

695

(2)

Computational considerations

697

(1)

Special treatments of coupled-cluster theory

698

(6)

Orbital-optimized and Brueckner coupled-cluster theories

698

(4)

Quadratic configuration-interaction theory

702

(2)

High-spin open-shell coupled-cluster theory

704

(20)

Spin-restricted coupled-cluster theory

704

(3)

Total spin of the spin-restricted coupled-cluster wave function

707

(1)

The projection manifold in spin-restricted theory

708

(1)

Spin-adapted CCSD theory

709

(2)

References

711

(1)

Further reading

712

(1)

Exercises

712

(5)

Solutions

717

(7)

Perturbation Theory

724

(93)

Rayleigh-Schrodinger perturbation theory

725

(14)

RSPT energies and wave functions

726

(2)

Wigner's 2n + 1 rule

728

(6)

The Hylleraas functional

734

(2)

Size-extensivity in RSPT

736

(3)

Møller-Plesset perturbation theory

739

(10)

The zero-order MPPT system

740

(1)

The MP1 wave function

741

(1)

The MP2 wave function

742

(3)

The Moller--Plesset energies

745

(1)

Explicit expressions for MPPT wave functions and energies

746

(1)

Size-extensivity in Moller--Plesset theory

747

(2)

Coupled-cluster perturbation theory

749

(10)

The similarity-transformed exponential ansatz of CCPT

749

(2)

The CCPT amplitude equations

751

(1)

The CCPT wave functions

752

(1)

The CCPT energies

753

(1)

Size-extensivity in CCPT

754

(1)

The CCPT Lagrangian

755

(1)

The CCPT variational equations

756

(2)

CCPT energies that obey the 2n + 1 rule

758

(1)

Size-extensivity of the CCPT Lagrangian

759

(1)

Møller--Plesset theory for closed-shell systems

759

(10)

The closed-shell zero-order system

760

(1)

The closed-shell variational Lagrangian

761

(2)

The closed-shell wave-function corrections

763

(3)

The closed-shell energy corrections

766

(3)

Convergence in perturbation theory

769

(14)

A two-state model

770

(1)

Conditions for convergence

771

(1)

Intruders in the general two-state model

772

(4)

Prototypical intruders

776

(2)

Convergence of the Moller--Plesset series

778

(4)

Analytic continuation

782

(1)

Perturbative treatments of coupled-cluster wave functions

783

(13)

Perturbation analysis of the coupled-cluster hierarchy

784

(5)

Iterative hybrid methods

789

(4)

Noniterative hybrid methods: the CCSD(T) model

793

(2)

Hybrid and nonhybrid methods compared

795

(1)

Multiconfigurational perturbation theory

796

(21)

The zero-order CASPT Hamiltonian

796

(2)

Size-extensivity in CASPT

798

(2)

The CASPT wave function and energy

800

(1)

Sample CASPT calculations

801

(2)

References

803

(1)

Further reading

804

(1)

Exercises

804

(5)

Solutions

809

(8)

Calibration of the Electronic-Structure Models

817

(68)

The sample molecules

817

(2)

Errors in quantum-chemical calculations

819

(2)

Apparent and intrinsic errors

819

(2)

Statistical measures of errors

821

(1)

Molecular equilibrium structures: bond distances

821

(11)

Experimental bond distances

822

(1)

Mean errors and standard deviations

822

(3)

Normal distributions

825

(1)

Mean absolute deviations

825

(2)

Maximum errors

827

(1)

The CCSDT and CCSD(T) models

827

(1)

The effect of core correlation on bond distances

828

(1)

Trends in the convergence towards experiment

829

(2)

Summary

831

(1)

Molecular equilibrium structures: bond angles

832

(4)

Experimental bond angles

832

(1)

Calculated bond angles

833

(1)

Summary

834

(2)

Molecular dipole moments

836

(4)

Experimental dipole moments

836

(1)

Calculated dipole moments

837

(1)

Predicted dipole moments

838

(1)

Analysis of the calculated dipole moments

839

(1)

Summary

840

(1)

Molecular and atomic energies

840

(14)

The total electronic energy

841

(1)

Contributions to the total electronic energy

842

(2)

Basis-set convergence

844

(2)

CCSDT corrections

846

(1)

Molecular vibrational corrections

847

(2)

Relativistic corrections

849

(4)

Summary

853

(1)

Atomization energies

854

(11)

Experimental atomization energies

854

(1)

Statistical analysis of atomization energies

855

(3)

Extrapolation of atomization energies

858

(3)

Core contributions to atomization energies

861

(2)

CCSDT corrections

863

(1)

Summary

864

(1)

Reaction enthalpies

865

(9)

Experimental reaction enthalpies

865

(2)

Statistical analysis of reaction enthalpies

867

(3)

Extrapolation and covergence to the basis-set limit

870

(1)

Core contributions to reaction enthalpies

871

(2)

Summary

873

(1)

Conformational barriers

874

(5)

The barrier to linearity of water

875

(1)

The inversion barrier of ammonia

876

(1)

The torsional barrier of ethane

877

(2)

Summary

879

(1)

Conclusions

879

(6)

References

882

(1)

Further reading

882

(1)

Exercises

882

(1)

Solutions

883

(2)

List of Acronyms

885

(2)

Index

887

Molecular Electronic-Structure Theory

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