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MiS Preprint
4/2012
Fast and Accurate 3D Tensor Calculation of the Fock Operator in a General Basis
Venera Khoromskaia, Dirk Andrae and Boris N. Khoromskij
Abstract
The present paper contributes to the construction of "black-box" solver for the Hartree-Fock equation by the grid-based tensor-structured methods. It focuses on the calculation of the Galerkin matrices for the Laplace and the nuclear potential operators by tensor operations using the generic set of basis functions with low separation rank, discretized on fine $N \times N\times N $ Cartesian grid.
We prove the $C h^2$ error estimate in terms of mesh parameter, $h=O(1/N)$, that allows to gain a guaranteed accuracy of the core Hamiltonian part in the Fock operator as $h\to 0$. However, the commonly used problem adapted basis functions have low regularity yielding the considerable increase of a constant $C$, hence, demanding rather large grid-size $N$ of about several tens of thousands to ensure the high resolution. Tensor-formatted arithmetics of complexity $O(N)$, or even $O(\log N)$, practically relaxes the limitations on the grid-size.
Our tensor-based approach allows to improve significantly the standard basis sets in quantum chemistry by including simple combinations of Slater-type, local finite element and other basis functions. Numerical experiments for moderate size organic molecules show efficiency of the accurate grid-based calculations to the core Hamiltonian in the range of grid parameter $N^3\approx 10^{15}$.
