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Quantum Chemical Methods

I. Hartree-Fock theory

D.R. Hartree, The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods, Proc. Cambridge Phil. Soc. 24, 89 (1928)
V. Fock, Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems, Z. Phys. 61, 126 (1930)

Roothaan-Hall equations
C.C.J. Roothaan, New Developments in Molecular Orbital Theory, Rev. Mod. Phys. 23, 69 (1951)
G.G. Hall, The molecular orbital theory of chemical valency VIII. A method of calculating ionization potentials, Proc. Roy. Soc. A205, 541 (1951)

unrestricted HF (UHF)
J.A. Pople and R.K. Nesbet, Self‐Consistent Orbitals for Radicals, J. Chem. Phys. 22, 571 (1954)

restricted open-shell HF (ROHF)
C.C.J. Roothaan, Self-Consistent Field Theory for Open Shells of Electronic Systems, Rev. Mod. Phys. 32, 179 (1960)

two-configurational SCF (TCSCF)
A.C. Wahl and G. Das, The Method of Optimized Valence Configurations: A Reasonable Application of the Multiconfiguration Self-Consistent-Field Technique to the Quantitative Description of Chemical Bonding, Adv. Quant. Chem. 5, 261 (1970)
F.W. Bobrowicz and W.A. Goddard, The Self-Consistent Field Equations for Generalized Valence Bond and Open-Shell Hartree—Fock Wave Functions, in Modern Theoretical Chemistry, Ed.: H.F. Schaefer III (Plenum, New York, 1977) Vol. 3, p. 79

complete-active space SCF (CASSCF)
H.J.Aa. Jensen and P. Jørgensen, A direct approach to second‐order MCSCF calculations using a norm extended optimization scheme, J. Chem. Phys. 80, 1204 (1980)
F. Lipparini and J. Gauss, Cost-Effective Treatment of Scalar Relativistic Effects for Multireference Systems: A CASSCF Implementation Based on the Spin-free Dirac–Coulomb Hamiltonian, J. Chem. Theor. Comp. 12, 4284 (2016); implementation in CFour

convergence acceleration in SCF (DIIS)
P. Pulay, Improved SCF convergence acceleration, J. Comp. Chem. 3, 556 (1982)

II. Many-body-perturbation theory (Møller-Plesset perturbation theory)

review articles
R.J. Bartlett, Many-Body Perturbation Theory and Coupled Cluster Theory for Electron Correlation in Molecules, Ann. Rev. Phys. Chem.32, 359 (1981)
D. Cremer, Møller–Plesset Perturbation Theory, in Encyclopedia of Computational Chemistry, Eds.: P.v.R. Schleyer et al., (Wiley, 1998), p. 1706

Møller-Plesset Hamiltonian
C. Møller and M.S. Plesset, Note on an Approximation Treatment for Many-Electron Systems, Phys. Rev. 46, 618 (1934)

many-body perturbation theory (also known as Møller-Plesset perturbation theory)
R.J. Bartlett and D.M. Silver, Pair-correlation energies in sodium hydride with many-body perturbation theory, Phys. Rev. A10, 1927 (1974)
R.J. Bartlett and D.M. Silver, Many‐body perturbation theory applied to electron pair correlation energies. I. Closed‐shell first‐row diatomic hydrides, J. Chem. Phys. 62, 3258 (1975)
R.J. Bartlett and D.M. Silver, Many‐body perturbation theory applied to electron pair correlation energies. II. Closed‐shell second‐row diatomic hydrides, J. Chem. Phys. 64, 4578 (1976)
R.J. Bartlett and I. Shavitt, Comparison of high-order many-body perturbation theory and configuration interaction for H2O, Chem. Phys. Lett. 50, 190 (1977)
J.A. Pople, J.S. Binkley, and R. Seeger, Theoretical models incorporating electron correlation, Int. J. Quant. Chem. Symp. 10, 1 (1976)
R. Krishnan and J.A. Pople, Approximate fourth‐order perturbation theory of the electron correlation energy, Int. J. Quant. Chem. 14, 91 (1978) (MP4(SDQ)
R. Krishnan, M.J. Frisch, and J.A. Pople, Contribution of triple substitutions to the electron correlation energy in fourth order perturbation theory, J. Chem. Phys. 72, 4244 (1980)

ROHF-MBPT/ROHF-MP
W.J. Lauderdale, J.F. Stanton, J. Gauss, J.D. Watts, and R.J. Bartlett, Many-body perturbation theory with a restricted open-shell Hartree—Fock reference, Chem. Phys. Lett. 187, 21 (1991)
W.J. Lauderdale, J.F. Stanton, J. Gauss, J.D. Watts, and R.J. Bartlett, Restricted open‐shell Hartree–Fock‐based many‐body perturbation theory: Theory and application of energy and gradient calculations, J. Chem. Phys. 97, 6606 (1992)

see also
P.J. Knowles, J.S. Andrews, R.D. Amos, N.C. Handy, and J.A. Pople, Restricted Møller—Plesset theory for open-shell molecules, Chem. Phys. Lett. 186, 130 (1991)

III. Coupled-Cluster Theory

book
I. Shavitt and R.J. Bartlett, Many-Body Methods in Chemistry and Physics, (Cambridge University Press, Cambridge, 2009)

review articles
R.J. Bartlett, Coupled-cluster approach to molecular structure and spectra: a step toward predictive quantum chemistry, J. Phys. Chem. 93, 1697 (1989)
R.J. Bartlett and J.F. Stanton, Applications of Post‐Hartree—Fock Methods: A Tutorial, Rev. Comp. Chem. 5, 65 (1994)
T.J. Lee and G.E. Scuseria, in Quantum Mechanical Electronic Structure Calculations, Ed.: S.R. Langhoff, (Kluwer, Dordrecht, 1995), p. 47
R.J. Bartlett, in Modern Electronic Structure Theory, Ed.: D.R. Yarkony (World Scientific, Singapore, 1995), p. 1047
J. Gauss, Coupled‐cluster Theory, in Encyclopedia of Computational Chemistry, Eds.: P.v.R. Schleyer et al. (Wiley, New York, 1998)), p. 615
T.D. Crawford and H.F. Schaefer III, An Introduction to Coupled Cluster Theory for Computational Chemists, Rev. Comp. Chem. 14, 33 (2000)
R.J. Bartlett and M. Musiał, Coupled-cluster theory in quantum chemistry, Rev. Mod. Phys. 79, 291 (2007)

original formulation of CC theory by Čížek
J. Čížek, On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell‐Type Expansion Using Quantum‐Field Theoretical Methods, J. Chem. Phys. 45, 4256 (1966)
J. Čížek, On the Use of the Cluster Expansion and the Technique of Diagrams in Calculations of Correlation Effects in Atoms and Molecules, Adv. Chem. Phys. 14, 35 (1966)

actual implementation and CC approximations:

CCD
R.J. Bartlett and G.D. Purvis III, Many‐body perturbation theory, coupled‐pair many‐electron theory, and the importance of quadruple excitations for the correlation problem, Int. J. Quantum Chem. 14, 561 (1978)
J.A. Pople, R. Krishnan, H.B. Schlegel, and J.S. Binkley, Electron correlation theories and their application to the study of simple reaction potential surfaces, Int. J. Quantum Chem.14, 545 (1978)

CCSD
G.D.Purvis III and R.J.Bartlett, A full coupled‐cluster singles and doubles model: The inclusion of disconnected triples, J. Chem. Phys. 76, 1910 (1982)

for more recent implementations, see
G.E. Scuseria, A.C. Scheiner, T.J. Lee, J.E. Rice, and H.F. Schaefer III, The closed‐shell coupled cluster single and double excitation (CCSD) model for the description of electron correlation. A comparison with configuration interaction (CISD) results, J. Chem. Phys. 86, 2881 (1987)
J.F. Stanton, J. Gauss, J.D. Watts, and R.J. Bartlett, A direct product decomposition approach for symmetry exploitation in many‐body methods. I. Energy calculations, J. Chem. Phys. 94, 4334 (1991); implementation in CFour
C. Hampel, K.A. Peterson, and H.-J. Werner, A comparison of the efficiency and accuracy of the quadratic configuration interaction (QCISD), coupled cluster (CCSD), and Brueckner coupled cluster (BCCD) methods, Chem. Phys. Letters 190, 1 (1992)

CCSDT-1
Y.S. Lee, S.A. Kucharski, and R.J. Bartlett, A coupled cluster approach with triple excitations, J. Chem. Phys. 81, 5906 (1984)

CCSDT-2 and CCSDT-3
J. Noga, R.J. Bartlett, and M. Urban, Towards a full CCSDT model for electron correlation. CCSDT-n models, Chem. Phys. Letters 134, 126 (1987)

CCSD+T(CCSD)
M. Urban, J. Noga, S.J. Cole, and R.J. Bartlett, Towards a full CCSDT model for electron correlation, J. Chem. Phys. 83, 4041 (1985)

CCSD(T)
K. Raghavachari, G.W. Trucks, J.A. Pople and M. Head-Gordon, A fifth-order perturbation comparison of electron correlation theories, Chem. Phys. Lett. 157, 479 (1989)
R.J. Bartlett, J.D. Watts, S.A. Kucharski, and J. Noga, Non-iterative fifth-order triple and quadruple excitation energy corrections in correlated methods, Chem. Phys. Lett. 165, 513 (1990)
J.F. Stanton, Why CCSD(T) works: a different perspective, Chem. Phys. Letters 281, 130 (1997); a posteriori rationalization of CCSD(T)

CCSD(T)_Lambda
T.D. Crawford and J.F. Stanton, Investigation of an asymmetric triple‐excitation correction for coupled‐cluster energies, Int. J. Quant. Chem. 70, 601 (1998)
S.A. Kucharski and R.J. Bartlett, Noniterative energy corrections through fifth-order to the coupled cluster singles and doubles method, J. Chem. Phys. 108, 5243 (1998)

CCSD(T-n)
J.J. Eriksen, K. Kristensen, T. Kjæegaard, P. Jørgensen, and J. Gauss, A Lagrangian framework for deriving triples and quadruples corrections to the CCSD energy, J. Chem. Phys. 140, 064108 (2014); theory
J.J. Eriksen, P. Jørgensen, and J. Gauss, On the convergence of perturbative coupled cluster triples expansions: Error cancellations in the CCSD(T) model and the importance of amplitude relaxation, J. Chem. Phys. 142, 014102 (2015); implementation for closed-shell systems up to n=4

CCn hierarchy

CC2
O. Christiansen, H. Koch, and P. Jørgensen, The second-order approximate coupled cluster singles and doubles model CC2, Chem. Phys. Letters 243, 409 (1995)

CC3
H. Koch, O. Christiansen, P. Jørgensen, A.M. Sanchez de Meras, T. Helgaker, The CC3 model: An iterative coupled cluster approach including connected triples, J. Chem. Phys. 106, 1808 (1997)

CCSDT
J. Noga and R.J. Bartlett, The full CCSDT model for molecular electronic structure, J. Chem. Phys. 86, 7041 (1987), Erratum J. Chem. Phys. 89, 3401 (1988)
G.E. Scuseria and H.F. Schaefer III, A new implementation of the full CCSDT model for molecular electronic structure, Chem. Phys. Letters 152, 382 (1988)
J.D. Watts and R.J. Bartlett, The coupled‐cluster single, double, and triple excitation model for open‐shell single reference functions, J. Chem. Phys. 93, 6104 (1990); UHF implementation

for the parallel CCSDT-1, CCSDT-2, CCSDT-3, CCSDT-4, CC3, and CCSDT implementation in CFOUR , see:
E. Prochnow, M.E. Harding and J. Gauss, Parallel Calculation of CCSDT and Mk-MRCCSDT Energies, J. Chem. Theor. Comp. 6, 2339 (2010)

CCSDTQ
S.A. Kucharski and R.J. Bartlett, Recursive intermediate factorization and complete computational linearization of the coupled-cluster single, double, triple, and quadruple excitation equations, Theor. Chim. Acta 80, 387 (1991)
N. Oliphant and L. Adamowicz, Coupled‐cluster method truncated at quadruples, J. Chem. Phys. 95, 6645 (1991)
S.A. Kucharski and R.J. Bartlett, The coupled‐cluster single, double, triple, and quadruple excitation method, J. Chem. Phys. 97, 4282 (1992)
D.A. Matthews and J.F. Stanton, Non-orthogonal spin-adaptation of coupled cluster methods: A new implementation of methods including quadruple excitations, J. Chem. Phys. 142, 064108 (2015)

CCSDT[Q]
S.A. Kucharski and R.J. Bartlett, Coupled-cluster methods that include connected quadruple excitations, T4: CCSDTQ-1 and Q(CCSDT), Chem. Phys. Letters 158, 550 (1989)

CCSDT(Q)
Y.J. Bomble, J.F. Stanton, M. Kállay, and J. Gauss, Coupled-cluster methods including noniterative corrections for quadruple excitations, J. Chem. Phys. 123, 054101 (2005)
M. Kállay and J. Gauss, Approximate treatment of higher excitations in coupled-cluster theory, J. Chem. Phys. 123, 214105 (2005)
M. Kállay and J. Gauss, Approximate treatment of higher excitations in coupled-cluster theory. II. Extension to general single-determinant reference functions and improved approaches for the canonical Hartree–Fock case, J. Chem. Phys. 129, 144101 (2008); CCSDT(Q) for ROHF, CCSDT(Q) variants A and B

CCSDT(Q-n) methods
J.J. Eriksen, K. Kristensen, T. Kjaeregaard, P. Jørgensen, and J. Gauss, A Lagrangian framework for deriving triples and quadruples corrections to the CCSD energy, J. Chem. Phys. 140, 064108 (2014); theory
J.J. Eriksen, D.A. Mattehws, P. Jørgensen, and J. Gauss, The performance of non-iterative coupled cluster quadruples models, J. Chem. Phys. 143, 041101 (2015); implementation for closed-shell systems up to n=4

general CC
M. Kállay and P.R. Surján, Higher excitations in coupled-cluster theory, J. Chem. Phys. 115, 2945 (2001)
J. Olsen, The initial implementation and applications of a general active space coupled cluster method, J. Chem. Phys. 113, 7140 (2000)
S. Hirata, Tensor Contraction Engine:  Abstraction and Automated Parallel Implementation of Configuration-Interaction, Coupled-Cluster, and Many-Body Perturbation Theories, J. Phys. Chem. A 107, 9887 (2003)

ROHF and QRHF-CC methods

CCSD
M. Rittby and R.J. Bartlett, An open-shell spin-restricted coupled cluster method: application to ionization potentials in nitrogen, J. Phys. Chem. 92, 3033 (1988)

CCSD(T)
J. Gauss, W.J. Lauderdale, J.F. Stanton, J.D. Watts, R.J. Bartlett, Analytic energy gradients for open-shell coupled-cluster singles and doubles (CCSD) calculations using restricted open-shell Hartree—Fock (ROHF) reference functions, Chem. Phys. Letters 182, 207 (1991)
J.D. Watts, J. Gauss, and R.J. Bartlett, Coupled‐cluster methods with noniterative triple excitations for restricted open‐shell Hartree–Fock and other general single determinant reference functions. Energies and analytical gradients, J. Chem. Phys. 98, 8718 (1993)

partially-spin-adapted CC methods
P.J. Knowles, C.Hampel, and H.-J. Werner, Coupled cluster theory for high spin, open shell reference wave functions, J. Chem. Phys. 99, 5219 (1993); Erratum J. Chem. Phys. 112, 3106 (2000)
P. Neogrády, M. Urban, and I. Hubač, Spin adapted restricted Hartree–Fock reference coupled cluster theory for open shell systems, J. Chem. Phys. 100, 3706 (1994)
P.G. Szalay and J. Gauss, Spin-restricted open-shell coupled-cluster theory, J. Chem. Phys. 107, 9028 (1997)

spin-restricted CC methods
P.G. Szalay and J. Gauss, Spin-restricted open-shell coupled-cluster theory, J. Chem. Phys. 107, 9028 (1997)
P.G. Szalay and J. Gauss, Spin-restricted open-shell coupled-cluster theory for excited states, J. Chem. Phys. 112, 4027 (1999)
I. Berente, P.G. Szalay, and J. Gauss, Spin-restricted coupled-cluster theory with triple excitations, J. Chem. Phys. 117, 7872 (2003)

spin-adapted CC methods
M. Heckert, O. Heun, J. Gauss, and P.G. Szalay, Towards a spin-adapted coupled-cluster theory for high-spin open-shell states, J. Chem. Phys. 124, 124105 (2006)

Brueckner CC methods
R.A. Chiles and C.E. Dykstra, An electron pair operator approach to coupled cluster wave functions. Application to He2, Be2, and Mg2 and comparison with CEPA methods, J. Chem. Phys. 74, 4544 (1981)
J.F. Stanton, J. Gauss, and R.J. Bartlett, On the choice of orbitals for symmetry breaking problems with application to NO3, J. Chem. Phys. 97, 5554 (1992)

orbital-optimized CC
G.E. Scuseria and H.F. Schaefer III, The optimization of molecular orbitals for coupled cluster wavefunctions, Chem. Phys. Lett. 142, 354 (1987)

QCISD and QCISD(T)
J.A. Pople, M. Head-Gordon, and K. Raghavachari, Quadratic configuration interaction. A general technique for determining electron correlation energies, J. Chem. Phys. 87, 5968 (1987)

unitary CC (UCC) methods
R.J. Bartlett, S.A. Kucharski, and J. Noga, Alternative coupled-cluster ansätze II. The unitary coupled-cluster method, Chem. Phys. Lett. 155, 133 (1989)
J.D. Watts, G.W. Trucks, and R.J. Bartlett, The unitary coupled-cluster approach and molecular properties. Applications of the UCC(4) method, Chem. Phys. Lett. 157, 359 (1989)

Mukherjee's Multireference CC method (Mk-MRCC)

U.S. Mahapatra, B. Datta, and D. Mukherjee, A size-consistent state-specific multireference coupled cluster theory: Formal developments and molecular applications, J. Chem. Phys. 101, 6171 (1999)
F.A. Evangelista, W.D. Allen, and H.F. Schaefer III, Coupling term derivation and general implementation of state-specific multireference coupled cluster theories, J. Chem. Phys. 127, 024102 (2007)
F.A. Evangelista, A.D. Simmonett, W.D. Allen, H.F. Schaefer III, and J. Gauss, Triple excitations in state-specific multireference coupled cluster theory: Application of Mk-MRCCSDT and Mk-MRCCSDT-n methods to model systems, J. Chem. Phys. 128, 124104 (2008)
F.A. Evangelista, E. Prochnow, J. Gauss, and H.F. Schaefer III, Perturbative triples corrections in state-specific multireference coupled cluster theory, J. Chem. Phys. 132, 074107 (2010)

for the parallel Mk-MRCCSDT implementation in CFOUR , see:
E. Prochnow, M.E. Harding, and J. Gauss, Parallel Calculation of CCSDT and Mk-MRCCSDT Energies, J. Chem. Theor. Comp. 6, 2339 (2010)

two-component CCSD and CCSD(T) approaches with spin-orbit coupling

F. Wang, J. Gauss, and C. van Wüllen, Closed-shell coupled-cluster theory with spin-orbit coupling, J. Chem. Phys. 129, 064113 (2008)

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CFOUR is partially supported by the U.S. National Science Foundation.