Transforming high-dimensional potential energy surfaces into a canonical polyadic decomposition using Monte Carlo methods.

A Monte Carlo method is proposed for transforming high-dimensional potential energy surfaces evaluated on discrete grid points into a sum-of-products form, more precisely into a Canonical Polyadic Decomposition form. To this end, a modified existing ansatz based on the alternating least squares method is used, in which numerically exact integrals are replaced with Monte Carlo integrals. This largely reduces the numerical cost by avoiding the evaluation of the potential on all grid points and allows the treatment of surfaces with many degrees of freedom. Calculations on the 15D potential of the protonated water dimer (Zundel cation) in a sum-of-products form are presented and compared to the results obtained in a previous work [M. Schröder and H.-D. Meyer, J. Chem. Phys. 147, 064105 (2017)], where a sum-of-products form of the potential was obtained in the Tucker format.