Error estimates for the Coupled Cluster method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 6, p. 1553-1582
The Coupled Cluster (CC) method is a widely used and highly successful high precision method for the solution of the stationary electronic Schrödinger equation, with its practical convergence properties being similar to that of a corresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method been analyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in [Schneider, 2009]. Recently, we globalized the CC formulation to the full continuous space, giving a root equation for an infinite dimensional, nonlinear Coupled Cluster operator that is equivalent the full electronic Schrödinger equation [Rohwedder, 2011]. In this paper, we combine both approaches to prove existence and uniqueness results, quasi-optimality estimates and energy estimates for the CC method with respect to the solution of the full, original Schrödinger equation. The main property used is a local strong monotonicity result for the Coupled Cluster function, and we give two characterizations for situations in which this property holds.
Classification:  65Z05,  81-08,  70-08
Keywords: quantum chemistry, electronic Schrödinger equation, coupled cluster method, numerical analysis, nonlinear operator equation, quasi-optimality, error estimators
     author = {Rohwedder, Thorsten and Schneider, Reinhold},
     title = {Error estimates for the Coupled Cluster method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {6},
     year = {2013},
     pages = {1553-1582},
     doi = {10.1051/m2an/2013075},
     zbl = {1297.65139},
     mrnumber = {3110488},
     language = {en},
     url = {}
Rohwedder, Thorsten; Schneider, Reinhold. Error estimates for the Coupled Cluster method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 6, pp. 1553-1582. doi : 10.1051/m2an/2013075.

[1] A. Anantharaman and E. Cancès, Existence of minimizers for Kohn − Sham models in quantum chemistry. Ann. Institut Henri Poincaré, Non Linear Anal. 26 (2009) 2425. | Numdam | MR 2569902 | Zbl 1186.81138

[2] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations. Princeton University press, Princeton (1982). | MR 745286 | Zbl 0503.35001

[3] H.W. Alt, Lineare Funktionalanalysis, Auflage. Springer, Berlin 5 (2006). | Zbl 1099.46001

[4] J. Arponen, Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems. Ann. Phys. 151 (1983) 311.

[5] A.A. Auer and G. Baumgärtner, Automatic Code Generation for Many-Body Electronic Structure Methods: The tensor contraction engine. Molecul. Phys. 104 (2006) 211.

[6] I. Babuska and J.E. Osborn, Finite Element-Galerkin Approximation of the Eigenvalues and Eigenvectors of Selfadjoint Problems. Math. Comput. 52 (1989) 275-297. | MR 962210 | Zbl 0675.65108

[7] V. Bach, E.H. Lieb, M. Loss and J.P. Solovej, There are no unfilled shells in unrestricted Hartree-Fock theory. Phys. Rev. Lett. 72 (1994) 2981.

[8] N.B. Balabanova and K.A. Peterson, Basis set limit electronic excitation energies, ionization potentials, and electron affinities for the 3d transition metal atoms: Coupled cluster and multireference methods. J. Chem. Phys. 125 (2006) 074110.

[9] W. Bangerth and R. Rannacher, Adaptive finite element methods for differential equations. Birkhäuser (2003). | MR 1960405 | Zbl 1020.65058

[10] R.J. Bartlett, Many-body perturbation theory and coupled cluster theory for electronic correlation in molecules. Ann. Rev. Phys. Chem. 32 (1981) 359.

[11] R.J. Bartlett and M. Musial, Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79 (2007) 291.

[12] R.J. Bartlett and G.D. Purvis, Many-body perturbation theory, coupled-pair many-electron theory, and the importance of quadruple excitations for the correlation problem. Int. J. Quantum Chem. 14 (1978) 561.

[13] R. Becker and R. Rannacher, An optimal control approach to error estimation and mesh adaptation in finite element methods. Acta Numerica 2000. Edited by A. Iserles. Cambridge University Press (2001) 1. | MR 2009692 | Zbl 1105.65349

[14] U. Benedikt, M. Espig, W. Hackbusch and A.A. Auer, A new Approach for Tensor Decomposition in Electronic Structure Theory (submitted).

[15] D.E. Bernholdt and R.J. Bartlett, A Critical Assessment of Multireference-Fock Space CCSD and Perturbative Third-Order Triples Approximations for Photoelectron Spectra and Quasidegenerate Potential Energy Surfaces. Adv. Quantum Chemist. 34 (1999) 261.

[16] R.F. Bishop, An overview of coupled cluster theory and its applications in physics. Theor. Chim. Acta 80 (1991) 95.

[17] M. Born and R. Oppenheimer, Zur Quantentheorie der Molekeln. Ann. Phys. 389 (1927) 457. | JFM 53.0845.04

[18] E. Cancès, R. Chakir and Y. Maday, Numerical Analysis of Nonlinear Eigenvalue Problems J. Scientific Comput. 45 (2010) 90. DOI: 10.1007/s10915-010-9358-1. | MR 2679792 | Zbl 1203.65237

[19] P. Cársky, J. Paldus and J. Pittner, Recent Progress in Coupled Cluster Methods, Theory and Applications. In vol. 44 of series: Challenges Adv. Comput. Chem. Phys. Springer (2010).

[20] T. Chan, W.J. Cook, E. Hairer, J. Hastad, A. Iserles, H.P. Langtangen, C. Le Bris, P.L. Lions, C. Lubich, A.J. Majda, J. Mclaughlin, R.M. Nieminen, J.T. Oden, P. Souganidis and A. Tveito, Encyclopedia Appl. Comput. Math. Springer. To appear (2013).

[21] O. Christiansen, Coupled cluster theory with emphasis on selected new developments. Theor. Chem. Acc. 116 (2006) 106.

[22] P.G. Ciarlet and J.L. Lions, Handbook of Numerical Analysis, Volume II: Finite Element Methods (Part I). Elsevier (1991). | MR 1115235 | Zbl 0905.00032

[23] P.G. Ciarlet and C. Lebris, Handbook of Numerical Analysis, Volume X: Special Volume. Computational Chemistry. Elsevier (2003). | Zbl 1052.81001

[24] J. Čížek, Origins of coupled cluster technique for atoms and molecules. Theor. Chim. Acta 80 (1991) 91.

[25] F. Coerster, Bound states of a many-particle system. Nucl. Phys. 7 (1958) 421.

[26] F. Coerster and H. Kümmel, Short range correlations in nuclear wave functions. Nucl. Phys. 17 (1960) 477. | MR 127273 | Zbl 0094.43903

[27] Computational Chemistry Comparison and Benchmark Data Base, National Institute of Standards and Technology. Available on

[28] T.D. Crawford and H.F. Schaeffer Iii, An introduction to coupled cluster theory for computational chemists. Rev. Comput. Chem. 14 (2000) 33.

[29] Dalgaard and H.J. Monkhorst, Some aspects of the time-dependent coupled-cluster approach to dynamic response functions. Phys. Rev. A 28 (1983) 1217.

[30] J.E. Dennis Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996). | MR 1376139 | Zbl 0847.65038

[31] P.A.M. Dirac, Quantum Mechanics of Many-Electron Systems. Proc. of Royal Soc. London, Series A CXXIII (1929) 714. | JFM 55.0528.04

[32] R.M. Dreizler and E.K.U. Gross, Density functional theory. Springer (1990). | Zbl 0723.70002

[33] E. Emmrich, Gewöhnliche und Operator-Differentialgleichungen, Vieweg (2004).

[34] H.J. Flad, R. Schneider and T. Rohwedder, Adaptive methods in Quantum Chemistry. Zeitsch. f. Phys. Chem. 224 (2010) 651-670.

[35] V. Fock, Konfigurationsraum und zweite Quantelung. Z. Phys. 75 (1932) 622. | Zbl 0004.28003

[36] G. Friesecke and B.D. Goddard, Explicit large nuclear charge limit of electronic ground states for Li, Be, B, C, N, O, F, Ne and basic aspects of the periodic table. SIAM J. Math. Anal. 41 (2009) 631-664. | MR 2507464 | Zbl 1187.81103

[37] H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie Verlag (1974). | MR 636412 | Zbl 0289.47029

[38] S.R. Gwaltney and M. Head-Gordon, A second-order correction to singles and doubles coupled-cluster methods based on a perturbative expansion of a similarity-transformed Hamiltonian 323 (2000) 2128.

[39] S.R. Gwaltney, C.D. Sherrill, M. Head-Gordon and A.I. Krylov, Second-order perturbation corrections to singles and doubles coupled-cluster methods: General theory and application to the valence optimized doubles model. J. Chem. Phys. 113 (2000) 3548-3560.

[40] W. Hackbusch, Elliptic Differential Equations, vol. 18. of SSCM. Springer (1992), | MR 1197118 | Zbl 0755.35021

[41] C. Hampel and H.-J. Werner, Local treatment of electron correlation in coupled cluster theory. J. Chem. Phys. 104 (1996) 6286.

[42] T. Helgaker and P. Jørgensen, Configuration-interaction energy derivatives in a fully variational formulation. Theor. Chim. Acta 75 (1989) 111127.

[43] T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. John Wiley & Sons (2000).

[44] T. Helgaker, W. Klopper and D.P. Tew, Quantitative quantum chemistry. Mol. Phys. 106 (2008) 2107.

[45] S. Hirata, Tensor contraction engine: Abstraction and automated parallel implementation of Configuration-Interaction, Coupled-Cluster, and Many-Body perturbation theories. J. Phys. Chem. A 46 (2003) 9887.

[46] W. Hunziker and I.M. Sigal, The quantum N-body problem. J. Math. Phys. 41 (2000) 6. | MR 1768629 | Zbl 0981.81026

[47] W. Klopper, F.R. Manby, S. Ten-no and E.F. Vallev, R12 methods in explicitly correlated molecular structure theory. Int. Rev. Phys. Chem. 25 (2006) 427.

[48] P. Knowles, M. Schütz and H.-J. Werner, Ab Initio Methods for Electron Correlation in Molecules, Modern Methods and Algorithms of Quantum Chemistry, vol. 3 of Proceedings, Second Edition, edited by J. Grotendorst. John von Neumann Institute for Computing, Jülich, NIC Series, ISBN 3-00-005834-6 (2000) 97-179.

[49] S.A. Kucharsky and R.J. Bartlett, Fifth-order many-body perturbation theory and its relationship to various coupled-cluster approaches. Adv. Quantum Chem. 18 (1986) 281.

[50] W. Kutzelnigg, Error analysis and improvement of coupled cluster theory, Theoretica Chimica Acta 80 (1991) 349.

[51] H. Kümmel, Compound pair states in imperfect Fermi gases. Nucl. Phys. 22 (1961) 177. | MR 129368 | Zbl 0094.43902

[52] H. Kümmel, K.H. Lührmann and J.G. Zabolitzky, Many-fermion theory in expS- (or coupled cluster) form. Phys. Reports 36 (1978) 1.

[53] S. Kvaal, Ab initio quantum dynamics using coupled-cluster, to appear in J. Chem. Phys. (2012).

[54] T.J. Lee, Comparison of the T1 and D1 diagnostics for electronic structure theory: a new definition for the open-shell D1 diagnostic. Chem. Phys. Lett. 372 (2003) 362-367.

[55] T.J. Lee and G.E. Scuseria, Achieving chemical accuracy with Coupled Cluster methods, in Quantum Mechanical Electronic Structure Calculations with Chemical Accuracy, edited by S.R. Langhof. Kluwer Academic Publishers, Dordrecht (1995) 47.

[56] T.J. Lee and P.R. Taylor, A diagnostic for determining the quality of single-reference electron correlation methods. Int. J. Quantum Chem. Symp. 23 (1989) 199-207.

[57] X. Li and J. Paldus, Dissociation of N2 triple bond: a reduced multireference CCSD study. Chem. Phys. Lett. 286 12 (1998) 145-154.

[58] E.H. Lieb and B. Simon, The Hartree − Fock Theory for Coulomb Systems. Commun. Math. Phys. 53 (1977) 185. | MR 452286

[59] E.H. Lieb, Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A 29 (1984) 3018.

[60] I. Lindgren and J. Morrison, Atomic Many-body Theory. Springer (1986).

[61] P.L. Lions, Solution of the Hartree Fock equation for Coulomb Systems. Commun. Math. Phys. 109 (1987) 33. | MR 879032 | Zbl 0618.35111

[62] C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced methods and Numerical Analysis. Zürich Lect. Adv. Math. EMS (2008). | MR 2474331 | Zbl 1160.81001

[63] D.I. Lyakh, V.V. Ivanov and L. Adamowicz, State-specific multireference complete-active-space coupled-cluster approach versus other quantum chemical methods: dissociation of the N2 molecule. Mol. Phys. 105 (2007) 1335-1357.

[64] D.I. Lyakh and R.J. Bartlett, An adaptive coupled-cluster theory: @CC approach. J. Chem. Phys. 133 (2010) 244112.

[65] F. Neese, A. Hansen and D.G. Liakos, Efficient and accurate approximations to the local coupled cluster singles doubles method using a truncated pair natural orbital basis. J. Chem. Phys. 131 (2009) 064103.

[66] 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. 110 (1999) 6171-6188.

[67] M. Nooijen, K.R. Shamasundar and D. Mukherjee, Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Molecular Phys. 103 (2005) 2277.

[68] J. Paldus, Coupled Cluster Theory, in Methods Comput. Molec. Phys., edited by S. Wilson and G.F.H. Diercksen. Plenum. New York (1992) 99.

[69] J. Paldus, M. Takahashi and B.W.H. Cho, Degeneracy and coupled-cluster Approaches 26 (1984) 237-244.

[70] R.G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules. Oxford University Press (1994).

[71] A. Persson, Bounds for the discrete part of the spectrum of a semibounded Schrödinger operator. Math. Scand. 8 (1960) 143. | MR 133586 | Zbl 0145.14901

[72] P. Piecuch, N. Oliphant and L. Adamowicz, A state-selective multireference coupled-cluster theory employing the single-reference formalism. J. Chem. Phys. 99 (1993) 1875.

[73] P. Piecuch, K. Kowalski, P.-D. Fan and I.S.O. Pimienta, New alternatives for electronic structure calculations: Renormalized, extended, and generalized coupled-cluster theories, in vol. 12 of Progr. Theoret. Chemist. Phys., edited by J. Maruani, R. Lefebvre, E. Brändas. Kluwer, Dordrecht (2003) 119-206.

[74] J. Pousin and J. Rapaz, Consistenct, stability, a priori and a posteriori estimates for Petrov-Galerkin methods applied to nonlinear problems. Num. Math. 69 (1994) 213-231. | MR 1310318 | Zbl 0822.65034

[75] K. Raghavachari, G.W. Trucks, J.A. Pople and M. Head-Gordon, A fifth-order perturbation comparison of electronic correlation theories. Chem. Phys. Lett. 157 (1989) 479.

[76] M. Reiher, A Theoretical Challenge: Transition-Metal Compounds, Chimia 63 (2009) 140-145.

[77] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV - Analysis of operators. Academic Press (1978). | MR 493421 | Zbl 0242.46001

[78] T. Rohwedder, An analysis for some methods and algorithms of Quantum Chemistry, Ph.D. thesis, TU Berlin, available at (2010).

[79] T. Rohwedder, The continuous Coupled Cluster formulation for the electronic Schrödinger equation, submitted to M2AN. | Numdam | Zbl 1269.82032

[80] W. Rudin, Functional Analysis. Tat McGraw & Hill Publishing Company, New Delhi (1979). | MR 1157815 | Zbl 0253.46001

[81] Y. Saad, J.R. Chelikowsky and S.M. Shontz, Numerical Methods for Electronic Structure Calculations of Materials. SIAM Rev. 52 (2010) 1. | MR 2639608 | Zbl 1185.82004

[82] R. Schneider, Analysis of the projected Coupled Cluster method in electronic structure calculation. Num. Math. 113 (2009) 433. | MR 2534132 | Zbl 1170.81043

[83] M. Schütz and H.-J. Werner, Low-order scaling local correlation methods. IV. Linear scaling coupled cluster (LCCSD). J. Chem. Phys. 114 (2000) 661.

[84] B. Simon, Schrödinger operators in the 20th century. J. Math. Phys. 41 (2000) 3523. | MR 1768631 | Zbl 0981.81025

[85] A. Szabo and N.S. Ostlund, Modern Quantum Chemistry. Dover Publications Inc. (1992).

[86] D.J. Thouless, Stability conditions and nuclear rotations in the Hartree − Fock theory. Nuclear Phys. 21 (1960) 225. | MR 144694 | Zbl 0097.43602

[87] J. Wloka, Partial differential equations. Cambridge University Press, reprint (1992). | MR 895589 | Zbl 0623.35006

[88] H. Yserentant, Regularity and Approximability of Electronic Wave Functions, in vol. 2000 of Lect. Notes Math. Ser. Springer-Verlag (2010). | MR 2656512 | Zbl 1204.35003

[89] E. Zeidler, Nonlinear Functional Analysis and Its Applications, Part II B: Nonlinear Monotone Operators. Springer (1990). | MR 1033498 | Zbl 0684.47029

[90] G.M. Zhislin, Discussion of the spectrum of Schrödinger operator for systems of many particles. Trudy Mosov. Mat. Obshch. 9 (1960) 81-128. | Zbl 0121.10004