Molecular Electronic-Structure Theory

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

• ISBN13: 9780471967552
• ISBN10: 0471967556
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-10-26
• Publisher: WILEY

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

<B>Trygve Helgaker </B> Department of Chemistry, University of Oslo, Norway <BR> <B>Poul Jorgensen</B> and <B>Jeppe Olsen</B> Department of Chemistry, University of Aarhus, Denmark

Table of Contents

Preface	xxi
Overview	xxv
Programs used in the preparation of this book	xxix
Second Quantization	1	(33)
The Fock space	1	(1)
Creation and annihilation operators	2	(4)
Creation operators	3	(1)
Annihilation operators	4	(1)
Anticommuntation relations	5	(1)
Number-conserving operators	6	(3)
Occupation-number operators	6	(1)
The number operator	7	(1)
Excitation operators	7	(2)
The representation of one- and two-electron operators	9	(5)
One-electron operators	9	(2)
Two-electron operators	11	(2)
The molecular electronic Hamiltonian	13	(1)
Products of operators in second quantization	14	(4)
Operator products	14	(3)
The canonical commutators	17	(1)
First- and second-quantization operators compared	18	(1)
Density matrices	19	(6)
The one-electron density matrix	20	(1)
The two-electron density matrix	21	(2)
Density matrices in spin-orbital and coordinate representations	23	(2)
Commutators and anticommutators	25	(2)
Nonorthogonal spin orbitals	27	(7)
Creation and annihilation operators	27	(2)
One- and two-electron operators	29	(1)
Biorthogonal operators	30	(1)
References	31	(1)
Further reading	31	(1)
Exercises	31	(1)
Solutions	32	(2)
Spin in Second Quantization	34	(46)
Spin functions	34	(1)
Operators in the orbital basis	35	(6)
Spin-free operators	36	(2)
Spin operators	38	(2)
Mixed operators	40	(1)
Spin tensor operators	41	(5)
Spin tensor operators	41	(2)
Creation and annihilation operators	43	(1)
Two-body creation operators	43	(1)
Excitation operators	44	(2)
Singlet excitation operators	46	(1)
Spin properties of determinants	46	(5)
General considerations	47	(1)
Spin projection of determinants	48	(1)
Total spin of determinants	49	(2)
Configuration state functions	51	(2)
The genealogical coupling scheme	53	(8)
Representations of determinants and CSFs	54	(1)
Genealogical coupling	55	(1)
Coupling coefficients	56	(1)
An example: three electrons in three orbitals	57	(1)
Completeness and orthonormality	58	(1)
Transformations between determinant and CSF bases	59	(1)
Genealogical coupling of operators	60	(1)
Density matrices	61	(19)
Orbital-density matrices	61	(2)
Spin-density matrices	63	(1)
Density functions	64	(2)
References	66	(1)
Further reading	66	(1)
Exercises	66	(4)
Solutions	70	(10)
Orbital Rotations	80	(27)
Unitary transformations and matrix exponentials	80	(6)
Matrix exponentials	81	(1)
Exponential representations of unitary matrices	81	(1)
Special unitary matrices	82	(1)
Orthogonal matrices	83	(1)
Evaluation of matrix exponentials	83	(1)
Nonunitary transformations	84	(2)
Unitary spin-orbital transformations	86	(3)
Unitary matrix expansions of creation and annihilation operators	87	(1)
Exponential unitary transformations of the elementary operators	88	(1)
Exponential unitary transformations of states in Fock space	89	(1)
Symmetry-restricted unitary transformations	89	(4)
The need for symmetry restrictions	89	(1)
Symmetry restrictions in the spin-orbital basis	90	(1)
Symmetry restrictions in the orbital basis	91	(2)
The logarithmic matrix function	93	(14)
Definition of the logarithmic matrix function	93	(1)
Expansion of the logarithmic matrix function	94	(1)
Properties of the logarithmic matrix function	95	(1)
References	95	(1)
Further reading	95	(1)
Exercises	95	(4)
Solutions	99	(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