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Helgaker T., Jorgensen P., Olsen J. Molecular Electronic-Structure Theory. Part 2
Helgaker T., Jorgensen P., Olsen J.  Molecular Electronic-Structure Theory. Part 2









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: Molecular Electronic-Structure Theory. Part 2

: Helgaker T., Jorgensen P., Olsen J.

:

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.


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$\Delta SCF$ method      456
$\mathrm{H_2}$ in minimal basis      146
$\mathrm{H_2}$ in minimal basis, antibonding configuration      148 150
$\mathrm{H_2}$ in minimal basis, bonding configuration      148 150
$\mathrm{H_2}$ in minimal basis, configuration interaction      152
$\mathrm{H_2}$ in minimal basis, density functions      151 153 156 157
$\mathrm{H_2}$ in minimal basis, density matrices      148
$\mathrm{H_2}$ in minimal basis, dissociation limit      159
$\mathrm{H_2}$ in minimal basis, MOs      146
$\mathrm{H_2}$ in minimal basis, N-electron basis      148
$\mathrm{H_2}$ in minimal basis, one-electron basis      146
$\mathrm{H_2}$ in minimal basis, open-shell states      156
$\mathrm{H_2}$ in minimal basis, potential-energy curves (illustration)      161
3-21G basis set      297
6-311G basis set      300
6-31G basis set      297
6-31G basis set, 6-31G* basis set      300
6-31G basis set, 6-31G** basis set      300
6-31G basis set, carbon exponents and contraction coefficients      298
6-31G basis set, scaling of exponents      298
Active electrons      178
Active orbitals      178 434 526 599
Addition theorem for regular solid harmonics      407
Addition theorem for spherical harmonics      263 280
Addition theorem for spherical harmonics in Racahs normalization      406
Alpha strings      550 (see also Spin strings)
Analytic continuation      782
Angular-momentum integrals      348 (see also Molecular integral evaluation)
Angular-momentum operators, orbital angular momentum      110 205 240
Angular-momentum operators, spin angular momentum      38 45 47 65 109 704
Annihilation operators      4 8
Annihilation operators, nonorthogonal spin orbitals      28
Annihilation operators, spin properties      43
Annihilation operators, unitary transformations      86
ANO, atomic natural orbital      304
Anticommutation relations      5 8
Anticommutation relations in nonorthogonal spin-orbital basis      28
Anticommutation relations, annihilation operators      4
Anticommutation relations, creation and annihilation operators      5
Anticommutation relations, creation operators      4
Anticommutators      25 (see also Commutation relations)
Anticommutators, expansion rules      26
AO, atomic orbital      144
Apparent error      819 (see also Errors in calculations)
Asymptotic behaviour of electronic systems      110
Atomic natural orbitals (ANOs)      301 304
Atomic natural orbitals (ANOs), ANO basis sets      305
Atomic natural orbitals (ANOs), ANO energy contributions in carbon atom      306
Atomic natural orbitals (ANOs), ANO occupation numbers of carbon atom      305
Atomic natural orbitals (ANOs), ANO radial distribution functions      306
Atomic orbitals (AOs)      144 201 287 336
Atomization energies      854 (see also Calibration of methods)
Atomization energies, basis-set extrapolation      858
Atomization energies, calculated and experimental atomization energies (table)      865
Atomization energies, comparison of CCSD(T) and CCSDT      863
Atomization energies, core correlation      861
Atomization energies, experimental atomization energies      854
Atomization energies, experimental reference data (table)      855
Atomization energies, statistical analysis of errors in calculated atomization energies      855
aug-cc-pCVXZ, augmented cc-pCVXZ [basis]      312
aug-cc-pVXZ basis sets      312 314
aug-cc-pVXZ basis sets, dipole moments      837
aug-cc-pVXZ basis sets, weak interactions      328 331
aug-cc-pVXZ, augmented cc-pVXZ [basis]      312 314
Back-door intruders      772
Baker Campbell Hausdorff (BCH) expansion      81
Baker Campbell Hausdorff (BCH) expansion of annihilation and creation operators      88 102
Baker Campbell Hausdorff (BCH) expansion of CCPT energy function      753
Baker Campbell Hausdorff (BCH) expansion of CCPT similarity-transformed Fock operator      750 762
Baker Campbell Hausdorff (BCH) expansion of coupled-cluster similarity-transformed Hamiltonian      661 701 710
Baker Campbell Hausdorff (BCH) expansion of Hartree Fock energy function      437
Baker Campbell Hausdorff (BCH) expansion of state-averaged MCSCF energy function      639
Baker Campbell Hausdorff (BCH) expansion, asymmetric BCH expansion      85 469
Baker Campbell Hausdorff (BCH) expansion, standard BCH expansion      81
Balanced level of description      820 (see also Errors in calculations)
Basis functions (one-electron)      see also Basis sets
Basis functions (one-electron), algebraic approach      201
Basis functions (one-electron), angular basis      207
Basis functions (one-electron), atoms-in-molecule approach      204
Basis functions (one-electron), comparison of radial functions (Box. 6.1)      219
Basis functions (one-electron), Gaussian-type orbitals (GTOs)      232 287 336
Basis functions (one-electron), general considerations      201
Basis functions (one-electron), harmonic-oscillator (HO) functions      230 236
Basis functions (one-electron), hydrogenic functions      221
Basis functions (one-electron), Laguerre functions      222
Basis functions (one-electron), numerical orbitals      201 223 264 291 296 307 318 319
Basis functions (one-electron), separation in angular and radial parts      205
Basis functions (one-electron), Slater-type orbitals (STOs)      225
Basis functions (one-electron), universal HO basis      203
Basis sets, basis functions (one-electron), Gaussian basis sets      287
Basis sets, STO basis sets      227
Basis-set convergence      see also Basis-set extrapolation
Basis-set convergence, $\mathrm{C_2H_6}$ torsional barrier      877
Basis-set convergence, $\mathrm{H_2O}$ barrier to linearity      875
Basis-set convergence, $\mathrm{H_2O}$ dimer interaction energy      332
Basis-set convergence, $\mathrm{H_2O}$ total energy and geometry      320
Basis-set convergence, $\mathrm{NH_3}$ inversion barrier      876
Basis-set convergence, $\mathrm{N_2}$ total energy, binding energy and geometry      320 323
Basis-set convergence, atomization energies      854
Basis-set convergence, BH total energy and binding energy      326 333
Basis-set convergence, bond angles      832
Basis-set convergence, bond distances      821
Basis-set convergence, carbon total energy      311 314 315
Basis-set convergence, conformational energies      874
Basis-set convergence, convergence with respect to the cardinal number      322 842
Basis-set convergence, convergence with respect to the number of AOs      324
Basis-set convergence, correlated models      317 844
Basis-set convergence, correlation-consistent and R12 (MP2 and CCSD) energies      320 328 331 333
Basis-set convergence, dipole moments      836
Basis-set convergence, electronic energies of sample molecules      841 844
Basis-set convergence, Hartree Fock convergence (energy)      315 844
Basis-set convergence, Hartree Fock convergence (geometry)      317 821 832
Basis-set convergence, helium atom (cc-pVXZ)      318
Basis-set convergence, helium atom (partial-wave expansion)      271
Basis-set convergence, helium atom (principal expansion)      274 318
Basis-set convergence, neon dimer interaction energy      329
Basis-set convergence, reaction enthalpies      865
Basis-set convergence, relation to BSSE      329 333
Basis-set error      819 (see also Errors in calculations)
Basis-set extrapolation      322 (see also Basis-set convergence)
Basis-set extrapolation, $\mathrm{H_2O}$ dimer interaction energy      333
Basis-set extrapolation, atomization energies      858
Basis-set extrapolation, electronic energies of sample molecules      841
Basis-set extrapolation, helium atom      272 274
Basis-set extrapolation, Ne, $\mathrm{N_2}$ and $\mathrm{H_2O}$ correlation energies      320 322 323
Basis-set extrapolation, reaction enthalpies      870
Basis-set extrapolation, relation to principal expansion      322 324
Basis-set extrapolation, two-point extrapolation      323 842
Basis-set superposition error (BSSE)      327
Basis-set superposition error (BSSE), $\mathrm{H_2O}$ dimer interaction energy      331
Basis-set superposition error (BSSE), BH binding energy      333
Basis-set superposition error (BSSE), counterpoise correction      327
Basis-set superposition error (BSSE), ghost basis      327
Basis-set superposition error (BSSE), neon dimer interaction energy      328
BCC, Brueckner coupled-cluster [model]      701
BCH, Baker Campbell Hausdorff [expansion]      81
Beta strings      550 (see also Spin strings)
Binding energy      see Dissociation energy
Binomial probability distribution      539
Biorthogonal configuration basis in closed-shell CCPT      762
Biorthogonal configuration basis in closed-shell CCSD theory      691
Biorthogonal configuration basis in EOM-CC theory      679
Biorthogonal operators      30 570
Bipolar multipole expansion      407
Bond angles      832 (see also Calibration of methods)
Bond angles, calculated and experimental bond angles (table)      836
Bond angles, experimental bond angles      832
Bond angles, experimental reference data (table)      833
Bond angles, statistical analysis of errors in calculated bond angles      833
Bond distances      821 (see also Calibration of methods)
Bond distances, calculated and experimental bond distances (table)      831
Bond distances, comparison of CCSD(T) and CCSDT bond distances      827
Bond distances, core correlation      828
Bond distances, experimental bond distances      822
Bond distances, experimental reference data (table)      823
Bond distances, maximum errors of calculated distances      827
Bond distances, mean absolute errors of calculated distances      825
Bond distances, mean errors of calculated distances      822
Bond distances, normal distributions of errors in calculated distances      825
Bond distances, standard deviations of calculated distances      822
Bond distances, trends in convergence      829
Born-Oppenheimer approximation      111
Boxes, Box. 1.1: fundamentals of second quantization      8
Boxes, Box. 1.2: first and second quantizations compared      18
Boxes, Box. 13.1: excitation commutators      688
Boxes, Box. 13.2: excitation commutators applied to Hartree Fock state      688
Boxes, Box. 6.1: one-electron radial functions      219
Boys function      363 365
Boys function as special case of the confluent hypergeometric function      372
Boys function, evaluation of Boys function      366
Boys function, properties of Boys function      365
Boys function, recurrence relations for Boys function      367
Boys function, relation to error function      369
Boys function, relation to incomplete gamma function      368
Branching diagram      54
Brillouin theorem      441
Brillouin theorem in Hartree Fock theory      441
Brillouin theorem, GBT in MCSCF theory      608
Brillouin theorem, GBT in OCC theory      700
Brillouin theorem, GBT vector      491
Brillouin theorem, generalized Brillouin theorem (GBT)      442 491 608 700
Brillouin Wigner perturbation theory (BWPT)      728 804
Brueckner coupled-cluster (BCC) theory      701
BSSE, basis-set superposition error      327
BWPT, Brillouin Wigner perturbation theory      728 804
Calibration of methods      817
Calibration of methods, atomization energies      854
Calibration of methods, bond angles      832
Calibration of methods, bond distances      821
Calibration of methods, conformational energies      874
Calibration of methods, dipole moments      836
Calibration of methods, electronic energies of sample molecules      841
Calibration of methods, errors in calculations      819
Calibration of methods, experimental reference data reaction enthalpies      865
Calibration of methods, relativistic corrections      849
Calibration of methods, statistical measures of errors      821
Calibration of methods, vibrational energies      847
Canonical commutators      17 18
Canonical orbitals      169 444
Canonical orbitals in CASPT      797
Canonical orbitals in coupled-cluster theory      662 672
Canonical orbitals in MCSCF optimization (as initial orbitals)      628
Canonical orbitals in Muller Plesset theory      192 740 760
Canonical orbitals in Roothaan Hall SCF theory      459
Canonical orbitals, canonical conditions      446
Canonical orbitals, comparison of canonical and variational conditions      449
Canonical orbitals, Hartree Fock electronic Hessian in canonical representation      488 512
Canonical orbitals, Koopmans theorem      454
Canonical orbitals, orbital energies      450
Canonical orbitalssuitability as correlating orbitals      304
cardinal number      308 (see also Correlation-consistent basis sets)
Cardinal number, convergence with respect to cardinal number      322
Cardinal number, dependence of computational cost on cardinal number      310
Cardinal number, number of AOs as a function of cardinal number      310 312 314
Cartesian GTOs      237 336 339
Cartesian GTOs, derivatives      340
Cartesian GTOs, factorization in Cartesian directions      339
Cartesian GTOs, illustration of Cartesian GTOs      340
Cartesian GTOs, recurrence relations      340
Cartesian GTOs, self-overlap integral      340 351
Cartesian GTOs, transformation to spherical-harmonic GTOs      337
CAS      see Complete active space (CAS)
CAS, complete active space      178 526 599
CASPT      796 (see also Multiconfigurational perturbation theory)
CASPT, CAS perturbation theory      796
CASPTn, nth-order CASPT      800
CASSCF      178 599
CASSCF, CAS self-consistent field [model]      178 599
Cauchys interlace theorem      116
cc-pCVXZ basis sets      312 (see also Correlation-consistent basis sets)
cc-pCVXZ, correlation-consistent polarized core-valence X-tuple zeta [basis]      312
cc-pVXZ basis sets      308 (see also Correlation-consistent basis sets)
cc-pVXZ, correlation-consistent polarized valence X-tuple zeta [basis]      308
CCD, coupled-cluster doubles [model]      649
CCn, coupled-cluster [rank] n [hybrid model]      790
CCPT, coupled-cluster perturbation theory      724 750
CCS, coupled-cluster singles [model]      789
CCSD + T(CCSD) CCSD [model] with approximate triples correction      794
CCSD theory (closed-shell)      685
CCSD theory (closed-shell), AO-based (integral-direct) algorithm      697
CCSD theory (closed-shell), biorthogonal configuration basis      692
CCSD theory (closed-shell), computational considerations      697
CCSD theory (closed-shell), energy expression      686
CCSD theory (closed-shell), norm of wave function (singles and doubles part)      692
CCSD theory (closed-shell), numerical illustration of size-extensivity in CCSD theory      669
CCSD theory (closed-shell), projected equations      661
CCSD theory (closed-shell), projected equations (explicit doubles part)      695
CCSD theory (closed-shell), projected equations (explicit singles part)      693
CCSD theory (closed-shell), projection manifold      691
CCSD theory (closed-shell), singlet CCSD cluster operator      685
CCSD theory (closed-shell), T1 transformed integrals      690
CCSD theory (closed-shell), T1-transformed Hamiltonian      687
CCSD(T), CCSD [model] with approximate triples correction      191 793
CCSD, coupled-cluster singles-and-doubles [model]      188 656
CCSDT, coupled-cluster singles-doubles-and-triples [model]      188 656
CCSDTQ, coupled-cluster singles-doubles-triples-and-quadruples [model]      657
CCSD[T], CCSD [model] with approximate triples correction      794
Central-field systems (one-electron)      204
Centrifugal potential      206
CFMM, continuous fast multipole method      424
Chebyshev polynomials      358 (see also Orthogonal polynomials)
Chemical accuracy      267 858 870
CI, configuration-interaction [model]      181 523
CID, CI doubles [model]      529
CISD, CI singles-and-doubles [model]      182 526
CISDT, CI singles-doubles-and-triples [model]      183
CISDTQ, CI singles-doubles-triples-and-quadruples [model]      183
Clebsch Gordan coefficients      56
Cluster operator in CCPT      750 760
Cluster operator in iterative hybrid methods      789
Cluster operator in noniterative hybrid methods      793
Cluster operator in spin-adapted coupled-cluster theory      710
Cluster operator in spin-restricted coupled-cluster theory      706
Cluster operator, contribution to coupled-cluster energy      660 686
Cluster operator, relation to MP1 wave-function correction      742
Cluster operator, singlet doubles cluster operator      686
Cluster operator, singlet singles cluster operator      685
Cluster operator, spin-orbital cluster operator      188 654
Cluster-commutation condition      661 663
Commutation relations, canonical commutators      17
Commutation relations, cluster operators      655
Commutation relations, cluster-commutation relations      663
Commutation relations, coupled-cluster excitation operators      650
Commutation relations, elementary (annihilation and creation) operators      5 28
Commutation relations, excitation operators (orbital basis)      46
Commutation relations, excitation operators (spin-orbital basis)      26
Commutation relations, Fock operator      662 741 761
Commutation relations, ON operators      7
Commutation relations, orbital angular-momentum operators      240
Commutation relations, shift operators      39 41
Commutation relations, spin tensor operators      41
Commutators      25 (see also Commutation relations)
Commutators, double commutators      27
Commutators, expansion rules      26
Commutators, nested commutators      27
Commutators, S commutators      85 469
Complementary error function      370 401 402 424
Complete active space (CAS)      178 526
Complete active space (CAS), CASSCF      178 599
Complete active space (CAS), minimal active space      178
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