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Error estimates for the Coupled Cluster method∗

Published online by Cambridge University Press:  26 August 2013

Thorsten Rohwedder  and
Reinhold Schneider
Thorsten Rohwedder
Affiliation:
Sekretariat MA 5-3, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.. [email protected]
Reinhold Schneider
Affiliation:
Sekretariat MA 5-3, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.. [email protected]
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Abstract

The Coupled Cluster (CC) method is a widely used and highly successful high precisionmethod for the solution of the stationary electronic Schrödingerequation, with its practical convergence properties being similar to that of acorresponding Galerkin (CI) scheme. This behaviour has for the discrete CC method beenanalyzed with respect to the discrete Galerkin solution (the “full-CI-limit”) in[Schneider, 2009]. Recently, we globalized the CC formulation to the full continuousspace, giving a root equation for an infinite dimensional, nonlinear Coupled Clusteroperator that is equivalent the full electronic Schrödinger equation [Rohwedder, 2011]. Inthis paper, we combine both approaches to prove existence and uniqueness results,quasi-optimality estimates and energy estimates for the CC method with respect to thesolution of the full, original Schrödinger equation. The main property used is a localstrong monotonicity result for the Coupled Cluster function, and we give twocharacterizations for situations in which this property holds.

Keywords

Quantum chemistryelectronic Schrödinger equationcoupled Cluster methodnumerical analysisnonlinear operator equationquasi-optimalityerror estimators
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 6 , November 2013 , pp. 1553 - 1582
Copyright
© EDP Sciences, SMAI, 2013

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References

Anantharaman, A. and Cancès, E., Existence of minimizers for Kohn − Sham models in quantum chemistry. Ann. Institut Henri Poincaré, Non Linear Anal. 26 (2009) 2425. Google Scholar
S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations. Princeton University press, Princeton (1982).
H.W. Alt, Lineare Funktionalanalysis, Auflage. Springer, Berlin 5 (2006).
Arponen, J., Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems. Ann. Phys. 151 (1983) 311. Google Scholar
Auer, A.A. and Baumgärtner, G., Automatic Code Generation for Many-Body Electronic Structure Methods: The tensor contraction engine. Molecul. Phys. 104 (2006) 211. Google Scholar
Babuska, I. and Osborn, J.E., Finite Element-Galerkin Approximation of the Eigenvalues and Eigenvectors of Selfadjoint Problems. Math. Comput. 52 (1989) 275–297. Google Scholar
Bach, V., Lieb, E.H., Loss, M. and Solovej, J.P., There are no unfilled shells in unrestricted Hartree–Fock theory. Phys. Rev. Lett. 72 (1994) 2981. Google ScholarPubMed
Balabanova, N.B. and Peterson, K.A., 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. Google Scholar
W. Bangerth and R. Rannacher, Adaptive finite element methods for differential equations. Birkhäuser (2003).
Bartlett, R.J., Many-body perturbation theory and coupled cluster theory for electronic correlation in molecules. Ann. Rev. Phys. Chem. 32 (1981) 359. Google Scholar
Bartlett, R.J. and Musial, M., Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79 (2007) 291. Google Scholar
Bartlett, R.J. and Purvis, G.D., 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. Google Scholar
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.
U. Benedikt, M. Espig, W. Hackbusch and A.A. Auer, A new Approach for Tensor Decomposition in Electronic Structure Theory (submitted).
Bernholdt, D.E. and Bartlett, R.J., 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. Google Scholar
Bishop, R.F., An overview of coupled cluster theory and its applications in physics. Theor. Chim. Acta 80 (1991) 95. Google Scholar
Born, M. and Oppenheimer, R., Zur Quantentheorie der Molekeln. Ann. Phys. 389 (1927) 457. Google Scholar
Cancès, E., Chakir, R. and Maday, Y., Numerical Analysis of Nonlinear Eigenvalue Problems J. Scientific Comput. 45 (2010) 90. DOI: 10.1007/s10915-010-9358-1. Google Scholar
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).
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).
Christiansen, O., Coupled cluster theory with emphasis on selected new developments. Theor. Chem. Acc. 116 (2006) 106. Google Scholar
P.G. Ciarlet and J.L. Lions, Handbook of Numerical Analysis, Volume II: Finite Element Methods (Part I). Elsevier (1991).
P.G. Ciarlet and C. Lebris, Handbook of Numerical Analysis, Volume X: Special Volume. Computational Chemistry. Elsevier (2003).
Čížek, J., Origins of coupled cluster technique for atoms and molecules. Theor. Chim. Acta 80 (1991) 91. Google Scholar
Coerster, F., Bound states of a many-particle system. Nucl. Phys. 7 (1958) 421. Google Scholar
Coerster, F. and Kümmel, H., Short range correlations in nuclear wave functions. Nucl. Phys. 17 (1960) 477. Google Scholar
Computational Chemistry Comparison and Benchmark Data Base, National Institute of Standards and Technology. Available on www.cccbdb.nist.gov.
Crawford, T.D. and Schaeffer, H.F. III, An introduction to coupled cluster theory for computational chemists. Rev. Comput. Chem. 14 (2000) 33. Google Scholar
Dalgaard, and Monkhorst, H.J., Some aspects of the time-dependent coupled-cluster approach to dynamic response functions. Phys. Rev. A 28 (1983) 1217. Google Scholar
J.E. Dennis Jr. and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia (1996).
P.A.M. Dirac, Quantum Mechanics of Many-Electron Systems. Proc. of Royal Soc. London, Series A CXXIII (1929) 714.
R.M. Dreizler and E.K.U. Gross, Density functional theory. Springer (1990).
E. Emmrich, Gewöhnliche und Operator-Differentialgleichungen, Vieweg (2004).
Flad, H.J., Schneider, R. and Rohwedder, T., Adaptive methods in Quantum Chemistry. Zeitsch. f. Phys. Chem. 224 (2010) 651–670. Google Scholar
Fock, V., Konfigurationsraum und zweite Quantelung. Z. Phys. 75 (1932) 622. Google Scholar
Friesecke, G. and Goddard, B.D., 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. Google Scholar
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie Verlag (1974).
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.
Gwaltney, S.R., Sherrill, C.D., Head-Gordon, M. and Krylov, A.I., 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. Google Scholar
W. Hackbusch, Elliptic Differential Equations, vol. 18. of SSCM. Springer (1992),
Hampel, C. and Werner, H.-J., Local treatment of electron correlation in coupled cluster theory. J. Chem. Phys. 104 (1996) 6286. Google Scholar
Helgaker, T. and Jørgensen, P., Configuration-interaction energy derivatives in a fully variational formulation. Theor. Chim. Acta 75 (1989) 111127. Google Scholar
T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. John Wiley & Sons (2000).
Helgaker, T., Klopper, W. and Tew, D.P., Quantitative quantum chemistry. Mol. Phys. 106 (2008) 2107. Google Scholar
Hirata, S., 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. Google Scholar
Hunziker, W. and Sigal, I.M., The quantum N-body problem. J. Math. Phys. 41 (2000) 6. Google Scholar
Klopper, W., Manby, F.R., Ten-no, S. and Vallev, E.F., R12 methods in explicitly correlated molecular structure theory. Int. Rev. Phys. Chem. 25 (2006) 427. Google Scholar
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.
Kucharsky, S.A. and Bartlett, R.J., Fifth-order many-body perturbation theory and its relationship to various coupled-cluster approaches. Adv. Quantum Chem. 18 (1986) 281. Google Scholar
Kutzelnigg, W., Error analysis and improvement of coupled cluster theory, Theoretica Chimica Acta 80 (1991) 349. Google Scholar
Kümmel, H., Compound pair states in imperfect Fermi gases. Nucl. Phys. 22 (1961) 177. Google Scholar
Kümmel, H., Lührmann, K.H. and Zabolitzky, J.G., Many-fermion theory in expS- (or coupled cluster) form. Phys. Reports 36 (1978) 1. Google Scholar
S. Kvaal, Ab initio quantum dynamics using coupled-cluster, to appear in J. Chem. Phys. (2012).
Lee, T.J., 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. Google Scholar
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.
Lee, T.J. and Taylor, P.R., A diagnostic for determining the quality of single-reference electron correlation methods. Int. J. Quantum Chem. Symp. 23 (1989) 199–207. Google Scholar
X. Li and J. Paldus, Dissociation of N2 triple bond: a reduced multireference CCSD study. Chem. Phys. Lett. 286 12 (1998) 145–154.
Lieb, E.H. and Simon, B., The Hartree − Fock Theory for Coulomb Systems. Commun. Math. Phys. 53 (1977) 185. Google Scholar
Lieb, E.H., Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A 29 (1984) 3018. Google Scholar
I. Lindgren and J. Morrison, Atomic Many-body Theory. Springer (1986).
Lions, P.L., Solution of the Hartree Fock equation for Coulomb Systems. Commun. Math. Phys. 109 (1987) 33. Google Scholar
C. Lubich, From Quantum to Classical Molecular Dynamics: Reduced methods and Numerical Analysis. Zürich Lect. Adv. Math. EMS (2008).
Lyakh, D.I., Ivanov, V.V. and Adamowicz, L., 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. Google Scholar
Lyakh, D.I. and Bartlett, R.J., An adaptive coupled-cluster theory: @CC approach. J. Chem. Phys. 133 (2010) 244112. Google ScholarPubMed
Neese, F., Hansen, A. and Liakos, D.G., 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. Google ScholarPubMed
Mahapatra, U.S., Datta, B. and Mukherjee, D., A size-consistent state-specific multireference coupled cluster theory: Formal developments and molecular applications. J. Chem. Phys. 110 (1999) 6171–6188. Google Scholar
Nooijen, M., Shamasundar, K.R. and Mukherjee, D., Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Molecular Phys. 103 (2005) 2277. Google Scholar
J. Paldus, Coupled Cluster Theory, in Methods Comput. Molec. Phys., edited by S. Wilson and G.F.H. Diercksen. Plenum. New York (1992) 99.
Paldus, J., Takahashi, M. and Cho, B.W.H., Degeneracy and coupled-cluster Approaches 26 (1984) 237–244.
R.G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules. Oxford University Press (1994).
Persson, A., Bounds for the discrete part of the spectrum of a semibounded Schrödinger operator. Math. Scand. 8 (1960) 143. Google Scholar
Piecuch, P., Oliphant, N. and Adamowicz, L., A state-selective multireference coupled-cluster theory employing the single-reference formalism. J. Chem. Phys. 99 (1993) 1875. Google Scholar
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.
Pousin, J. and Rapaz, J., Consistenct, stability, a priori and a posteriori estimates for Petrov-Galerkin methods applied to nonlinear problems. Num. Math. 69 (1994) 213–231. Google Scholar
Raghavachari, K., Trucks, G.W., Pople, J.A. and Head–Gordon, M., A fifth-order perturbation comparison of electronic correlation theories. Chem. Phys. Lett. 157 (1989) 479. Google Scholar
Reiher, M., A Theoretical Challenge: Transition-Metal Compounds, Chimia 63 (2009) 140–145. CrossRefGoogle Scholar
M. Reed and B. Simon, Methods of Modern Mathematical Physics IV - Analysis of operators. Academic Press (1978).
T. Rohwedder, An analysis for some methods and algorithms of Quantum Chemistry, Ph.D. thesis, TU Berlin, available at http://opus.kobv.de/tuberlin/volltexte/2010/2852/ (2010).
T. Rohwedder, The continuous Coupled Cluster formulation for the electronic Schrödinger equation, submitted to M2AN.
W. Rudin, Functional Analysis. Tat McGraw & Hill Publishing Company, New Delhi (1979).
Saad, Y., Chelikowsky, J.R. and Shontz, S.M., Numerical Methods for Electronic Structure Calculations of Materials. SIAM Rev. 52 (2010) 1. Google Scholar
Schneider, R., Analysis of the projected Coupled Cluster method in electronic structure calculation. Num. Math. 113 (2009) 433. Google Scholar
Schütz, M. and Werner, H.-J., Low-order scaling local correlation methods. IV. Linear scaling coupled cluster (LCCSD). J. Chem. Phys. 114 (2000) 661. Google Scholar
Simon, B., Schrödinger operators in the 20th century. J. Math. Phys. 41 (2000) 3523. Google Scholar
A. Szabo and N.S. Ostlund, Modern Quantum Chemistry. Dover Publications Inc. (1992).
Thouless, D.J., Stability conditions and nuclear rotations in the Hartree − Fock theory. Nuclear Phys. 21 (1960) 225. Google Scholar
J. Wloka, Partial differential equations. Cambridge University Press, reprint (1992).
H. Yserentant, Regularity and Approximability of Electronic Wave Functions, in vol. 2000 of Lect. Notes Math. Ser. Springer-Verlag (2010).
E. Zeidler, Nonlinear Functional Analysis and Its Applications, Part II B: Nonlinear Monotone Operators. Springer (1990).
Zhislin, G.M., Discussion of the spectrum of Schrödinger operator for systems of many particles. Trudy Mosov. Mat. Obshch. 9 (1960) 81–128. Google Scholar