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Sparse Grids and the Hierarchical Expansion of the Kinetic Energy Operator for the Anharmonic Vibrational Problem

Banff International Research Station for Mathematical Innovation and Discovery

The determination of the fully anharmonic vibrational spectrum and the corresponding eigenstates is one of the prototypical quantum dynamical problems. Even though many approximate methods have been developed in the past and normal modes are computed routinely nowadays, the solution of the full problem still poses computational challenges for medium (i.e. dozens of degrees of freedom) to large sized systems. A powerful and systematically improvable approach employs the vibrational self-consistent field (VSCF) method and its correlated extensions, like the vibrational configuration interaction (VCI) or the vibrational coupled cluster (VCC) expansions. It has long been known that the rate of convergence of these expansions depends critically on the choice of basis states and their correlations which may be reduced by a skilled choice of the underlying coordinate system. For molecular systems the topology of the Born--Oppenheimer potential energy surface (PES) in the vicinity of the minima is frequently best described in terms of a set of curvilinear internal coordinates, which has been shown by Wilson and coworkers more than 60 years ago. We have recently developed a hierarchical expansion of the resulting curvilinear kinetic energy operator which allows for systematic approximations that are well suited for the VSCF and VCI methods [1]. The combination with sparse grid approximations for the many-body sub-operators yields a method which is adaptive on two levels [2]. Based on some heuristic criteria derived from perturbation theory an efficient fully self-adaptive method has been devised which generates an approximate vibrational Hamiltonian for a given system (PES and underlying coordinate system) and a prescribed tolerance for some part of the vibrational spectrum without further input. [1] Daniel Strobusch and Christoph Scheurer, A General Nuclear Motion Hamiltonian and Non-Internal Curvilinear Coordinates. J. Chem. Phys. {\bf 138(9)} 094107 (2013). [2] Daniel Strobusch and Christoph Scheurer, Adaptive Sparse Grid Expansions of the Vibrational Hamiltonian. J. Chem. Phys. {\bf 140(7)} 074111 (2014).

Mathematics; Quantum theory; Partial differential equations; Applied computer science

video/mp4

Author affiliation: Technische Universität München

2016-07-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58562

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/