Coupled-cluster theory revisited. II: Analysis of the single-reference coupled-cluster equations. (English) Zbl 1515.81231
Summary: In a series of two articles, we propose a comprehensive mathematical framework for Coupled-Cluster-type methods. In this second part, we analyze the nonlinear equations of the single-reference Coupled-Cluster method using topological degree theory. We establish existence results and qualitative information about the solutions of these equations that also sheds light of the numerically observed behavior. In particular, we compute the topological index of the zeros of the single-reference Coupled-Cluster mapping. For the truncated Coupled-Cluster method, we derive an energy error bound for approximate eigenstates of the Schrödinger equation.
For Part I, see [the authors, ibid. 57, No. 2, 645–670 (2023; doi:10.1051/m2an/2022094)].
For Part I, see [the authors, ibid. 57, No. 2, 645–670 (2023; doi:10.1051/m2an/2022094)].
MSC:
| 81V55 | Molecular physics |
| 81-08 | Computational methods for problems pertaining to quantum theory |
| 81-10 | Mathematical modeling or simulation for problems pertaining to quantum theory |
| 47H11 | Degree theory for nonlinear operators |
Keywords:
quantum mechanics; many-body problem; quantum chemistry; electronic structure; coupled-cluster theory; nonlinear analysis; topological degree; Brouwer degreeSoftware:
HOMPACKReferences:
| [1] | J.S. Arponen, Variational principles and linked-cluster exp S expansions for static and dynamic many-body problems. Ann. Phys. 151 (1983) 311-382. · doi:10.1016/0003-4916(83)90284-1 |
| [2] | V. Bach, E.H. Lieb, M. Loss and J.P. Solovej, There are no unfilled shells in unrestricted Hartree-Fock theory, in The Stability of Matter: From Atoms to Stars. Springer (1997) 309-311. |
| [3] | R.J. Bartlett, S.A. Kucharski and J. Noga, Alternative coupled-cluster ansätze II. The unitary coupled-cluster method. Chem. Phys. Lett. 155 (1989) 133-140. · doi:10.1016/S0009-2614(89)87372-5 |
| [4] | N. Benedikter, Hartree-Fock theory. Online lecture notes (2017). |
| [5] | A. Buffa, Remarks on the discretization of some noncoercive operator with applications to heterogeneous Maxwell equations. SIAM J. Numer. Anal. 43 (2005) 1-18. · Zbl 1128.78010 |
| [6] | A. Buffa, R. Hiptmair, T. von Petersdorff and C. Schwab, Boundary element methods for Maxwell transmission problems in lipschitz domains. Numer. Math. 95 (2003) 459-485. · Zbl 1071.65160 |
| [7] | E. Cances, M. Defranceschi, W. Kutzelnigg, C. Le Bris and Y. Maday, Computational quantum chemistry: a primer. Handb. Numer. Anal. 10 (2003) 3-270. · Zbl 1070.81534 |
| [8] | J. Cronin, Fixed Points and Topological Degree in Nonlinear Analysis. Vol. 11. American Mathematical Society (1995). · doi:10.1090/surv/011 |
| [9] | G. Dinca and J. Mawhin, Brouwer degree and applications. Preprint (2009). |
| [10] | G. Dinca and J. Mawhin, Brouwer Degree (The Core of Nonlinear Analysis). Birkhäuser Basel (2021). · Zbl 07301627 · doi:10.1007/978-3-030-63230-4 |
| [11] | P. Drábek and J. Milota, Methods of Nonlinear Analysis: Applications to Differential Equations. Springer Science & Business Media (2007). · Zbl 1176.35002 |
| [12] | F.M. Faulstich, A. Laestadius, O. Legeza, R. Schneider and S. Kvaal, Analysis of the tailored coupled-cluster method in quantum chemistry. SIAM J. Numer. Anal. 57 (2019) 2579-2607. · Zbl 1435.81069 |
| [13] | F.M. Faulstich, M. Máté, A. Laestadius, M.A. Csirik, L. Veis, A. Antalik, J. Brabec, R. Schneider, J. Pittner, S. Kvaal and O. Legeza, Numerical and theoretical aspects of the DMRG-TCC method exemplified by the nitrogen dimer. J. Chem. Theory Comput. 15 (2019) 2206-2220. · doi:10.1021/acs.jctc.8b00960 |
| [14] | G. Friesecke, The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions. Arch. Ration. Mech. Anal. 169 (2003) 35-71. · Zbl 1035.81069 · doi:10.1007/s00205-003-0252-y |
| [15] | J. Geertsen, M. Rittby and R.J. Bartlett, The equation-of-motion coupled-cluster method: excitation energies of Be and CO. Chem. Phys. Lett. 164 (1989) 57-62. · doi:10.1016/0009-2614(89)85202-9 |
| [16] | T. Helgaker, P. Jorgensen and J. Olsen, Molecular Electronic-Structure Theory. John Wiley & Sons (2014). |
| [17] | K. Jankowski and K. Kowalski, Physical and mathematical content of coupled-cluster equations. II. On the origin of irregular solutions and their elimination via symmetry adaptation. J. Chem. Phys. 110 (1999) 9345-9352. · doi:10.1063/1.478900 |
| [18] | K. Jankowski and K. Kowalski, Physical and mathematical content of coupled-cluster equations. IV. Impact of approximations to the cluster operator on the structure of solutions. J. Chem. Phys. 111 (1999) 2952-2959. · doi:10.1063/1.479576 |
| [19] | K. Jankowski, K. Kowalski and P. Jankowski, Multiple solutions of the single-reference coupled-cluster equations. I. H4 model revisited. Int. J. Quantum Chem. 50 (1994) 353-367. · doi:10.1002/qua.560500504 |
| [20] | K. Jankowski, K. Kowalski, I. Grabowski and H. Monkhorst, Correspondence between physical states and solutions to the coupled-cluster equations. Int. J. Quantum Chem. 75 (1999) 483-496. · doi:10.1002/(SICI)1097-461X(1999)75:4/5<483::AID-QUA14>3.0.CO;2-M |
| [21] | F. Kossoski, A. Marie, A. Scemama, M. Caffarel and P.-F. Loos, Excited states from state specific orbital optimized pair coupled cluster. J. Chem. Theory Comput. 17 (2021) 4756-4768. · doi:10.1021/acs.jctc.1c00348 |
| [22] | K. Kowalski and K. Jankowski, Full solution to the coupled-cluster equations: the H4 model. Chem. Phys. Lett. 290 (1998) 180-188. · doi:10.1016/S0009-2614(98)00464-3 |
| [23] | A. Laestadius and F.M. Faulstich, The coupled-cluster formalism – a mathematical perspective. Mol. Phys. 117 (2019) 2362-2373. |
| [24] | A. Laestadius and S. Kvaal, Analysis of the extended coupled-cluster method in quantum chemistry. SIAM J. Numer. Anal. 56 (2018) 660-683. · Zbl 1397.81009 |
| [25] | M. Lewin, Geometric methods for nonlinear many-body quantum systems. J. Funct. Anal. 260 (2011) 3535-3595. · Zbl 1216.81180 · doi:10.1016/j.jfa.2010.11.017 |
| [26] | M. Lewin, Existence of Hartree-Fock excited states for atoms and molecules. Lett. Math. Phys. 108 (2018) 985-1006. · Zbl 1384.81148 |
| [27] | E.H. Lieb and B. Simon, On solutions to the Hartree-Fock problem for atoms and molecules. Journal Chem. Phys. 61 (1974) 735-736. |
| [28] | P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems. Commun. Math. Phys. 109 (1987) 33-97. · Zbl 0618.35111 · doi:10.1007/BF01205672 |
| [29] | D. O’Regan, Y.J. Cho and Y.-Q. Chen, Topological Degree Theory and Applications. CRC Press (2006). · Zbl 1095.47001 |
| [30] | J. Paldus, M. Takahashi and B. Cho, Degeneracy and coupled-cluster approaches. Int. J. Quantum Chem. 26 (1984) 237-244. · doi:10.1002/qua.560260824 |
| [31] | W.V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations. Vol. 171. CRC Press (1992). |
| [32] | P. Piecuch and K. Kowalski, In search of the relationship between multiple solutions characterizing coupled-cluster theories, in Computational Chemistry: Reviews of Current Trends. World Scientific (2000) 1-104. |
| [33] | P. Piecuch, S. Zarrabian, J. Paldus and J. Čžek, Coupled-cluster approaches with an approximate account of triexcitations and the optimized-inner-projection technique. II. Coupled-cluster results for cyclic-polyene model systems. Phys. Rev. B 42 (1990) 3351. |
| [34] | V.V. Prasolov, Problems and Theorems in Linear Algebra. Vol. 134. American Mathematical Society (1994). · Zbl 0803.15001 |
| [35] | T. Rohwedder, The continuous Coupled Cluster formulation for the electronic Schrödinger equation. ESAIM: Math. Modell. Numer. Anal.-Modél. Math. Anal. Numér. 47 (2013) 421-447. · Zbl 1269.82032 · doi:10.1051/m2an/2012035 |
| [36] | T. Rohwedder and R. Schneider, Error estimates for the coupled cluster method. ESAIM: Math. Modell. Numer. Anal.-Modél. Math. Anal. Numér. 47 (2013) 1553-1582. · Zbl 1297.65139 · doi:10.1051/m2an/2013075 |
| [37] | R. Schneider, Analysis of the projected coupled cluster method in electronic structure calculation. Numer. Math. 113 (2009) 433-471. · Zbl 1170.81043 |
| [38] | I.V. Skrypnik, Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. Vol. 139. American Mathematical Society (1994). · Zbl 0822.35001 |
| [39] | J.P. Solovej, Many body quantum mechanics. Lecture Notes (2007). |
| [40] | L.T. Watson, S.C. Billups and A.P. Morgan, Algorithm 652: Hompack: a suite of codes for globally convergent homotopy algorithms. ACM Trans. Math. Softw. (TOMS) 13 (1987) 281-310. · Zbl 0626.65049 · doi:10.1145/29380.214343 |
| [41] | H. Yserentant, Regularity and Approximability of Electronic Wave Functions. Springer (2010). · Zbl 1204.35003 |
| [42] | P. Zabrejko, Rotation of vector fields: definition, basic properties, and calculation, in Topological Nonlinear Analysis II. Springer (1997) 445-601. · Zbl 0887.47042 |
| [43] | E. Zeidler and P.R. Wadsack, Nonlinear Functional Analysis and its Applications: Fixed-Point Theorems/Transl. by Peter R. Wadsack. Springer-Verlag (1993). · Zbl 0794.47033 |
| [44] | T.P. Živković, Existence and reality of solutions of the coupled-cluster equations. Int. J. Quantum Chem. 12 (1977) 413-420. |
| [45] | T.P. Živković and H.J. Monkhorst, Analytic connection between configuration-interaction and coupled-cluster solutions. J. Math. Phys. 19 (1978) 1007-1022. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
