Monotonicity in Coupled-Cluster Methods
Andre Laestadius1, Fabian M. Faulstich1, Simen Kvaal1, Örs Legeza2 and Reinhold Schneider3
1Hylleraas Centre for Quantum Molecular Sciences, Department of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway
2Strongly Correlated Systems ”Lendület” Research Group, Wigner Research Center for Physics, H-1525, Budapest, Hungary
3Modeling, Simulation and Optimization in Science, Department of Mathematics, Technische Universität Berlin, Sekretariat MA 5-3, Straße des 17. Juni 136, 10623 Berlin, Germany
The coupled-cluster (CC) method – that makes use of an exponential ansatz for the solution of the Schrödinger equation – is a highly successful approach to treat electron correlation. The CC method and its variations have been analyzed [1–4] within the ERC project BIVAQUM [5]. This project studies a generalized variational principle, the so-called bivariational principle, where the bra and ket wave functions of the Rayleigh–Ritz quotient are treated as truly independent variables. This brief presentation aims at explaining the basic mathematical concepts used to prove a locally unique solution of different CC methods. This includes notions such as local strong monotonicity and Lipschitz continuity. Their connections to a HOMO-LUMO gap and the fluctuation potential, defined as the difference between the system’s Hamiltonian and the Fock operator, are here further elaborated.
Keywords: coupled-cluster method; tailored coupled-cluster method, extended coupled-cluster method; bivariational principle; uniqueness and existence; local strong monotonicity
References
(1) A. Laestadius and S. Kvaal, Analysis of the Extended Coupled-Cluster Method in Quantum Chemistry, SIAM J. Numer. Anal. 56, 660, (2018).
(2) F. M. Faulstich, A. Laestadius, S. Kvaal, Ö. Legeza, and R. Schneider, Analysis of the Coupled-Cluster Method Tailored by Tensor-Network States in Quantum Chemistry, arXiv:1802.05699, (2018).
(3) A. Laestadius and F. M. Faulstich, The Coupled-Cluster Formalism – A Mathematical Perspective, to appear in Mol. Phys., (2019).
(4) F. M. Faulstich, M. Máté, A. Laestadius, M. A. Csirik, L. Veis, A. Antalik, J. Brabec, R. Schneider, J. Pittner, S. Kvaal, and Ö. Legeza, Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer, arXiv:1809.07732, (2018).
(5) S. Kvaal, http://www.bivaqum.no
CONTACT Email: andre.laestadius@kjemi.uio.no