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Calculating Vibrational Spectra with Symmetrized Sum-of-Product Basis Functions

Banff International Research Station for Mathematical Innovation and Discovery

The memory cost of representing vibrational wavefunctions of polyatomic molecules with more than 6 atoms is huge if direct product basis sets are used. However the $n^D$ numbers which represent a wavefunction ($n$ being the number of 1-D basis functions for each of the $D$ coordinates) can often be deduced from a much smaller set of coefficients by using approximate, low-rank tensor decompositions. One possible choice is the CP format [1], i.e. a sum-of-products of 1-D functions whose memory cost scales as $\mathcal{O}(nD)$. In the reduced-rank block power method (RRBPM) introduced in [2], the rank of the SOP basis functions is reduced by iteratively optimizing the 1-D functions from which they are made. The optimized SOP basis functions span a subspace close to that of the desired eigenfunctions. Reduction is done with alternating least square algorithm of ref. [3]. Only 1-D operations are required and there is no need to store multidimensional vectors in memory. With this method one can compute vibrational eigenstates of systems up to $D=30$ using a few GB of memory. The RRBPM can be improved in several ways, one of which is the use of molecular symmetry [4]. Symmetry properties can be exploited in the RRBPM by using symmetry-constrained sum-of-products without jeopardizing the memory advantage of the algorithm. Since the Hamiltonian is totally symmetric, it is possible to compute states of a particular symmetry by using start vectors each of which has the same symmetry. This makes symmetry assignments easier and improves the accuracy in comparison with symmetry-free calculations. The ideas are illustrated by applying them to the acetonitrile molecule $CH_3CN$ (a 12D problem). [1] T. G. Kolda and B. W. Bader, SIAM Review {\bf 51} 455 (2009). [2] A. Leclerc and T. Carrington, J. Chem. Phys. {\bf 140} 174111 (2014). [3] G. Beylkin and M. J. Mohlenkamp, SIAM J. Sci. Comput. {\bf 26} 2133 (2005). [4] A. Leclerc and T. Carrington, submitted to Chem. Phys. Lett.

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

video/mp4

Author affiliation: Université de Lorraine

2016-07-29

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58588

Unreviewed

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