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Using a Non-direct Product Single Particle Function Basis within the Multi-Configurational Time-Dependent Hartree (MCTDH) Approach

Banff International Research Station for Mathematical Innovation and Discovery

The theoretical investigation of the vibrational quantum dynamics of molecular systems has been a central task in molecular physics and physical chemistry since the foundation of quantum mechanics. However, solving the corresponding Schroedinger equation (both time-dependent and time-independent) still nowadays is in general a formidable task for molecules with more than five or six atoms. Exact numerical methods like the multi-configurational time-dependent Hartree (MCTDH) approach have been developed to cope with the problem of efficiently solving the time-dependent Schroedinger equation. The key idea, in contrast to standard variational approaches, is to use a time-dependent basis set (single-particle functions) which optimally represents the wave function at each instant of time. This significantly reduces the basis set size and hence the numerical cost. However, the (original) MCTDH approach, like standard methods, suffers from what mathematicians call "the curse of dimensionality”, that is, an exponential scaling of the numerical effort with the number of degrees of freedom of the molecular system under study. This scaling is due to the fact that although the 1-D basis functions are optimized, a direct product (or tensor product) basis set is used. Significantly better numerical scaling is achieved by employing a non-direct product base. In this case, only a fraction of the full direct product space is used by choosing product basis functions according to some physically reasonable criterion. Here, we show that is is possible to combine the idea of using a non-direct basis set with the MCTDH method in an efficient way. Assuming a sum-of-products Hamiltonian, the crucial step in the algorithm is the evaluation of matrix-vector products by doing restricted sums sequentially. We illustrate the performance of this approach by calculating the first 69 vibrational eigenstates of acetonitrile, a 12-dimensional problem. Yielding accurate results, the numerical effort is decreased by about one order of magnitude compared to MCTDH calculations employing direct product bases.

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

video/mp4

Author affiliation: Queen's University

2016-08-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58627

Unreviewed

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