The continuous Coupled Cluster formulation for the electronic Schrödinger equation

Thorsten Rohwedder

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 2, page 421-447
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Rohwedder, Thorsten. "The continuous Coupled Cluster formulation for the electronic Schrödinger equation." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.2 (2013): 421-447. <http://eudml.org/doc/273292>.

@article{Rohwedder2013,
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.},
author = {Rohwedder, Thorsten},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {quantum chemistry; electronic Schrödinger equation; coupled cluster method; numerical analysis; nonlinear operator equation},
language = {eng},
number = {2},
pages = {421-447},
publisher = {EDP-Sciences},
title = {The continuous Coupled Cluster formulation for the electronic Schrödinger equation},
url = {http://eudml.org/doc/273292},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Rohwedder, Thorsten
TI - The continuous Coupled Cluster formulation for the electronic Schrödinger equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 2
SP - 421
EP - 447
AB - 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.
LA - eng
KW - quantum chemistry; electronic Schrödinger equation; coupled cluster method; numerical analysis; nonlinear operator equation
UR - http://eudml.org/doc/273292
ER -

References

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  1. [1] A.A. Auer and M. Nooijen, Dynamically screened local correlation method using enveloping localized orbitals. J. Chem. Phys. 125 (2006) 24104. 
  2. [2] R.J. Bartlett, Many-body perturbation theory and coupled cluster theory for electronic correlation in molecules. Ann. Rev. Phys. Chem. 32 (1981) 359. 
  3. [3] R.J. Bartlett and M. Musial, Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79 (2007) 291. 
  4. [4] 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. 
  5. [5] U. Benedikt, M. Espig, W. Hackbusch and A.A. Auer, Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2. J. Chem. Phys. 134 (2011) 054118. 
  6. [6] F.A. Berezin, The Method of Second Quantization. Academic Press (1966). Zbl0151.44001MR208930
  7. [7] R.F. Bishop, An overview of coupled cluster theory and its applications in physics. Theor. Chim. Acta 80 (1991) 95. 
  8. [8] S.F. Boys, Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another. Rev. Mod. Phys. 32 (1960) 296. MR122174
  9. [9] A. Chamorro, Method for construction of operators in Fock space. Pramana 10 (1978) 83. 
  10. [10] O. Christiansen, Coupled cluster theory with emphasis on selected new developments. Theor. Chem. Acc. 116 (2006) 106. 
  11. [11] P.G. Ciarlet (Ed.) and C. Lebris (Guest Ed.), Handbook of Numerical Analysis X : Special Volume. Comput. Chem. Elsevier (2003). Zbl1052.81001
  12. [12] J. Čížek, Origins of coupled cluster technique for atoms and molecules. Theor. Chim. Acta 80 (1991) 91. 
  13. [13] F. Coerster, Bound states of a many-particle system. Nucl. Phys. 7 (1958) 421. 
  14. [14] F. Coerster and H. Kümmel, Short range correlations in nuclear wave functions. Nucl. Phys. 17 (1960) 477. Zbl0094.43903
  15. [15] Computational Chemistry Comparison and Benchmark Data Base. National Institute of Standards and Technology, available on http://cccbdb.nist.gov/ 
  16. [16] T.D. Crawford and H.F. Schaeffer III, An introduction to coupled cluster theory for computational chemists. Rev. Comput. Chem. 14 (2000) 33. 
  17. [17] 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). Zbl0619.47005MR883643
  18. [18] V. Fock, Konfigurationsraum und zweite Quantelung. Z. Phys. 75 (1932) 622. Zbl0004.28003
  19. [19] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Østergaard Sørensen, Sharp regularity results for Coulombic many-electron wave functions. Commun. Math. Phys. 255 (2005) 183. Zbl1075.35063MR2123381
  20. [20] C. Hampel and H.-J. Werner, Local treatment of electron correlation in coupled cluster theory. J. Chem. Phys. 104 (1996) 6286. 
  21. [21] T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. John Wiley & Sons (2000). 
  22. [22] P.D. Hislop and I.M. Sigal, Introduction to spectral theory with application to Schrödinger operators. Appl. Math. Sci. 113 Springer (1996). Zbl0855.47002MR1361167
  23. [23] W. Hunziker and I.M. Sigal, The quantum N-body problem. J. Math. Phys. 41 (2000) 6. Zbl0981.81026MR1768629
  24. [24] T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. X (1957) 151. Zbl0077.20904MR88318
  25. [25] 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. 
  26. [26] W. Kutzelnigg, Error analysis and improvement of coupled cluster theory. Theor. Chim. Acta 80 (1991) 349. 
  27. [27] 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. 
  28. [28] H. Kümmel, Compound pair states in imperfect Fermi gases. Nucl. Phys. 22 (1961) 177. Zbl0094.43902MR129368
  29. [29] H. Kümmel, K.H. Lührmann and J.G. Zabolitzky, Many-fermion theory in expS- (or coupled cluster) form. Phys. Rep. 36 (1978) 1. 
  30. [30] 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. 
  31. [31] 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. 
  32. [32] M. Nooijen, K.R. Shamasundar and D. Mukherjee, Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Mol. Phys. 103 (2005) 2277. 
  33. [33] J. Pipek and P.G. Mazay, A fast intrinsic localization procedure for ab initio and semiempirical linear combination of atomic orbital wave functions. J. Chem. Phys. 90 (1989) 4919. 
  34. [34] 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. 
  35. [35] M. Reed and B. Simon, Methods of Modern Mathematical Physics IV – Analysis of operators. Academic Press (1978). Zbl0242.46001MR493421
  36. [36] 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/. 
  37. [37] T. Rohwedder and R. Schneider, An analysis for the DIIS acceleration method used in quantum chemistry calculations. J. Math. Chem.49 (2011) 1889–1914. Zbl1252.81135MR2833010
  38. [38] 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. Zbl1297.65139MR3110488
  39. [39] W. Rudin, Functional Analysis. Tat McGraw & Hill Publishing Company, New Delhi (1979). Zbl0253.46001MR1157815
  40. [40] R. Schneider, Analysis of the projected Coupled Cluster method in electronic structure calculation, Numer. Math. 113 (2009) 433. Zbl1170.81043MR2534132
  41. [41] M. Schütz and H.-J. Werner, Low-order scaling local correlation methods. IV. Linear scaling coupled cluster (LCCSD). J. Chem. Phys. 114 (2000) 661. 
  42. [42] B. Simon, Schrödinger operators in the 20th century. J. Math. Phys. 41 (2000) 3523. Zbl0981.81025MR1768631
  43. [43] A. Szabo and N.S. Ostlund, Modern Quantum Chemistry. Dover Publications Inc. (1992). 
  44. [44] G. Teschl, Mathematical methods in quantum mechanics with applications to Schrödinger operators. AMS Graduate Stud. Math. 99 (2009). Zbl1166.81004MR2499016
  45. [45] D.J. Thouless, Stability conditions and nuclear rotations in the Hartree-Fock theory. Nucl. Phys. 21 (1960) 225. Zbl0097.43602MR144694
  46. [46] J. Weidmann, Lineare Operatoren in Hilberträumen, Teil I : Grundlagen, Vieweg u. Teubner (2000). Zbl0344.47001MR1887367
  47. [47] J. Weidmann, Lineare Operatoren in Hilberträumen, Teil II : Anwendungen, Vieweg u. Teubner (2003). Zbl0344.47001MR2382320
  48. [48] H. Yserentant, Regularity and Approximability of Electronic Wave Functions. Springer-Verlag. Lect. Notes Math. Ser. 53 (2010). Zbl1204.35003MR2656512

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