Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-16T14:05:07.272Z Has data issue: false hasContentIssue false
  • English
  • Français

The continuous Coupled Cluster formulation for the electronic Schrödinger equation∗

Published online by Cambridge University Press:  11 January 2013

Thorsten Rohwedder
Thorsten Rohwedder*
Affiliation:
Sekretariat MA 5-3, Institut für Mathematik, TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany.. rohwedde@math.tu-berlin.de
Get access

Abstract

Nowadays, the Coupled Cluster (CC) method is the probably most widely used high precision method for the solution of the main equation of electronic structure calculation, the stationary electronic Schrödinger equation. Traditionally, the equations of CC are formulated as a nonlinear approximation of a Galerkin solution of the electronic Schrödinger equation, i.e. within a given discrete subspace. Unfortunately, this concept prohibits the direct application of concepts of nonlinear numerical analysis to obtain e.g. existence and uniqueness results or estimates on the convergence of discrete solutions to the full solution. Here, this shortcoming is approached by showing that based on the choice of an N-dimensional reference subspace R of H1(ℝ3 × {± 1/2}), the original, continuous electronic Schrödinger equation can be reformulated equivalently as a root equation for an infinite-dimensional nonlinear Coupled Cluster operator. The canonical projected CC equations may then be understood as discretizations of this operator. As the main step, continuity properties of the cluster operator S and its adjoint S† as mappings on the antisymmetric energy space H1 are established.

Keywords

Quantum chemistryelectronic Schrödinger equationcoupled cluster methodnumerical analysisnonlinear operator equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 2 , March 2013 , pp. 421 - 447
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Auer, A.A. and Nooijen, M., Dynamically screened local correlation method using enveloping localized orbitals. J. Chem. Phys. 125 (2006) 24104. Google Scholar
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
Benedikt, U., Espig, M., Hackbusch, W. and Auer, A.A., Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2. J. Chem. Phys. 134 (2011) 054118. Google ScholarPubMed
F.A. Berezin, The Method of Second Quantization. Academic Press (1966).
Bishop, R.F., An overview of coupled cluster theory and its applications in physics. Theor. Chim. Acta 80 (1991) 95. Google Scholar
Boys, S.F., Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another. Rev. Mod. Phys. 32 (1960) 296. Google Scholar
Chamorro, A., Method for construction of operators in Fock space. Pramana 10 (1978) 83. Google Scholar
Christiansen, O., Coupled cluster theory with emphasis on selected new developments. Theor. Chem. Acc. 116 (2006) 106. Google Scholar
P.G. Ciarlet (Ed.) and C. Lebris (Guest Ed.), Handbook of Numerical Analysis X : Special Volume. Comput. Chem. 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 http://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
H.L. Cycon, R.G. Froese, W. Kirsch and B. Simon, Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Series Theor. Math. Phys. Springer (1987).
Fock, V., Konfigurationsraum und zweite Quantelung. Z. Phys. 75 (1932) 622. Google Scholar
Fournais, S., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T. and Østergaard Sørensen, T., Sharp regularity results for Coulombic many-electron wave functions. Commun. Math. Phys. 255 (2005) 183. Google Scholar
Hampel, C. and Werner, H.-J., Local treatment of electron correlation in coupled cluster theory. J. Chem. Phys. 104 (1996) 6286. Google Scholar
T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. John Wiley & Sons (2000).
P.D. Hislop and I.M. Sigal, Introduction to spectral theory with application to Schrödinger operators. Appl. Math. Sci. 113 Springer (1996).
Hunziker, W. and Sigal, I.M., The quantum N-body problem. J. Math. Phys. 41 (2000) 6. Google Scholar
Kato, T., On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. X (1957) 151. 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
Kutzelnigg, W., Error analysis and improvement of coupled cluster theory. Theor. Chim. Acta 80 (1991) 349. Google Scholar
W. Kutzelnigg, Unconventional aspects of Coupled Cluster theory, in Recent Progress in Coupled Cluster Methods, Theory and Applications, Series : Challenges and Advances in Computational Chemistry and Physics 11 (2010). To appear.
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. Rep. 36 (1978) 1. 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.
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
Nooijen, M., Shamasundar, K.R. and Mukherjee, D., Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Mol. Phys. 103 (2005) 2277. Google Scholar
Pipek, J. and Mazay, P.G., A fast intrinsic localization procedure for ab initio and semiempirical linear combination of atomic orbital wave functions. J. Chem. Phys. 90 (1989) 4919. 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
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, TU Berlin, Ph.D. thesis (2010). Available on http://opus.kobv.de/tuberlin/volltexte/2010/2852/.
Rohwedder, T. and Schneider, R., An analysis for the DIIS acceleration method used in quantum chemistry calculations. J. Math. Chem. 49 (2011) 1889–1914. Google Scholar
T. Rohwedder and R. Schneider, Error estimates for the Coupled Cluster method. on Preprint submitted to ESAIM : M2AN (2011). Available on http://www.dfg-spp1324.de/download/preprints/preprint098.pdf.
W. Rudin, Functional Analysis. Tat McGraw & Hill Publishing Company, New Delhi (1979).
Schneider, R., Analysis of the projected Coupled Cluster method in electronic structure calculation, Numer. 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).
Teschl, G., Mathematical methods in quantum mechanics with applications to Schrödinger operators. AMS Graduate Stud. Math. 99 (2009). Google Scholar
Thouless, D.J., Stability conditions and nuclear rotations in the Hartree-Fock theory. Nucl. Phys. 21 (1960) 225. Google Scholar
J. Weidmann, Lineare Operatoren in Hilberträumen, Teil I : Grundlagen, Vieweg u. Teubner (2000).
J. Weidmann, Lineare Operatoren in Hilberträumen, Teil II : Anwendungen, Vieweg u. Teubner (2003).
Yserentant, H., Regularity and Approximability of Electronic Wave Functions. Springer-Verlag. Lect. Notes Math. Ser. 53 (2010). Google Scholar