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 chemistry, electronic Schrödinger equation, coupled cluster method, numerical analysis, nonlinear operator equation
@article{M2AN_2013__47_2_421_0, author = {Rohwedder, Thorsten}, title = {The continuous {Coupled} {Cluster} formulation for the electronic {Schr\"odinger} equation}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {421--447}, publisher = {EDP-Sciences}, volume = {47}, number = {2}, year = {2013}, doi = {10.1051/m2an/2012035}, mrnumber = {3021693}, zbl = {1269.82032}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012035/} }
TY - JOUR AU - Rohwedder, Thorsten TI - The continuous Coupled Cluster formulation for the electronic Schrödinger equation JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 421 EP - 447 VL - 47 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012035/ DO - 10.1051/m2an/2012035 LA - en ID - M2AN_2013__47_2_421_0 ER -
%0 Journal Article %A Rohwedder, Thorsten %T The continuous Coupled Cluster formulation for the electronic Schrödinger equation %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 421-447 %V 47 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012035/ %R 10.1051/m2an/2012035 %G en %F M2AN_2013__47_2_421_0
Rohwedder, Thorsten. The continuous Coupled Cluster formulation for the electronic Schrödinger equation. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 (2013) no. 2, pp. 421-447. doi : 10.1051/m2an/2012035. http://www.numdam.org/articles/10.1051/m2an/2012035/
[1] Dynamically screened local correlation method using enveloping localized orbitals. J. Chem. Phys. 125 (2006) 24104.
and ,[2] Many-body perturbation theory and coupled cluster theory for electronic correlation in molecules. Ann. Rev. Phys. Chem. 32 (1981) 359.
,[3] Coupled-cluster theory in quantum chemistry. Rev. Mod. Phys. 79 (2007) 291.
and ,[4] 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.
and ,[5] Tensor decomposition in post-Hartree-Fock methods. I. Two-electron integrals and MP2. J. Chem. Phys. 134 (2011) 054118.
, , and ,[6] The Method of Second Quantization. Academic Press (1966). | MR | Zbl
,[7] An overview of coupled cluster theory and its applications in physics. Theor. Chim. Acta 80 (1991) 95.
,[8] Construction of some molecular orbitals to be approximately invariant for changes from one molecule to another. Rev. Mod. Phys. 32 (1960) 296. | MR
,[9] Method for construction of operators in Fock space. Pramana 10 (1978) 83.
,[10] Coupled cluster theory with emphasis on selected new developments. Theor. Chem. Acc. 116 (2006) 106.
,[11] P.G. Ciarlet (Ed.) and C. Lebris (Guest Ed.), Handbook of Numerical Analysis X : Special Volume. Comput. Chem. Elsevier (2003).
[12] Origins of coupled cluster technique for atoms and molecules. Theor. Chim. Acta 80 (1991) 91.
,[13] Bound states of a many-particle system. Nucl. Phys. 7 (1958) 421.
,[14] Short range correlations in nuclear wave functions. Nucl. Phys. 17 (1960) 477. | Zbl
and ,[15] Computational Chemistry Comparison and Benchmark Data Base. National Institute of Standards and Technology, available on http://cccbdb.nist.gov/
[16] An introduction to coupled cluster theory for computational chemists. Rev. Comput. Chem. 14 (2000) 33.
and ,[17] Schrödinger Operators with Applications to Quantum Mechanics and Global Geometry, Series Theor. Math. Phys. Springer (1987). | MR | Zbl
, , and ,[18] Konfigurationsraum und zweite Quantelung. Z. Phys. 75 (1932) 622. | Zbl
,[19] Sharp regularity results for Coulombic many-electron wave functions. Commun. Math. Phys. 255 (2005) 183. | MR | Zbl
, , and ,[20] Local treatment of electron correlation in coupled cluster theory. J. Chem. Phys. 104 (1996) 6286.
and ,[21] Molecular Electronic-Structure Theory. John Wiley & Sons (2000).
, and ,[22] Introduction to spectral theory with application to Schrödinger operators. Appl. Math. Sci. 113 Springer (1996). | MR | Zbl
and ,[23] The quantum N-body problem. J. Math. Phys. 41 (2000) 6. | MR | Zbl
and ,[24] On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. X (1957) 151. | MR | Zbl
,[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] Error analysis and improvement of coupled cluster theory. Theor. Chim. Acta 80 (1991) 349.
,[27] 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] Compound pair states in imperfect Fermi gases. Nucl. Phys. 22 (1961) 177. | MR | Zbl
,[29] Many-fermion theory in expS- (or coupled cluster) form. Phys. Rep. 36 (1978) 1.
, and ,[30] 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.
and ,[31] 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.
, and ,[32] Reflections on size-extensivity, size-consistency and generalized extensivity in many-body theory. Mol. Phys. 103 (2005) 2277.
, and ,[33] A fast intrinsic localization procedure for ab initio and semiempirical linear combination of atomic orbital wave functions. J. Chem. Phys. 90 (1989) 4919.
and ,[34] A fifth-order perturbation comparison of electronic correlation theories. Chem. Phys. Lett. 157 (1989) 479.
, , and ,[35] Methods of Modern Mathematical Physics IV - Analysis of operators. Academic Press (1978). | MR | Zbl
and ,[36] 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] An analysis for the DIIS acceleration method used in quantum chemistry calculations. J. Math. Chem. 49 (2011) 1889-1914. | MR | Zbl
and ,[38] 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. | Numdam | MR | Zbl
and ,[39] Functional Analysis. Tat McGraw & Hill Publishing Company, New Delhi (1979). | MR | Zbl
,[40] Analysis of the projected Coupled Cluster method in electronic structure calculation, Numer. Math. 113 (2009) 433. | MR | Zbl
,[41] Low-order scaling local correlation methods. IV. Linear scaling coupled cluster (LCCSD). J. Chem. Phys. 114 (2000) 661.
and ,[42] Schrödinger operators in the 20th century. J. Math. Phys. 41 (2000) 3523. | MR | Zbl
,[43] Modern Quantum Chemistry. Dover Publications Inc. (1992).
and ,[44] Mathematical methods in quantum mechanics with applications to Schrödinger operators. AMS Graduate Stud. Math. 99 (2009). | MR | Zbl
,[45] Stability conditions and nuclear rotations in the Hartree-Fock theory. Nucl. Phys. 21 (1960) 225. | MR | Zbl
,[46] Lineare Operatoren in Hilberträumen, Teil I : Grundlagen, Vieweg u. Teubner (2000). | MR | Zbl
,[47] Lineare Operatoren in Hilberträumen, Teil II : Anwendungen, Vieweg u. Teubner (2003). | MR | Zbl
,[48] Regularity and Approximability of Electronic Wave Functions. Springer-Verlag. Lect. Notes Math. Ser. 53 (2010). | MR | Zbl
,Cited by Sources: