## Grid-based tensor numerical methods in electronic structure calculations

Venera Khoromskaia

Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Allemagne

Lundi 26 Novembre 2012, 14^{h}00

*vekh@mis.mpg.de*

bibliothèque LCT, tour 12 - 13, 4e étage

The novel grid-based tensor-structured methods for the numerical solution of the Hartree-Fock equation
[3] are presented, which can be used as well in other quantum chemistry simulations. The efficient algorithms
for the separable representation and calculation of the discretized functions and integral operators
in R^{3} are based on the canonical, Tucker and mixed tensor formats [1]. The core of our “black-box”
solver is the rank-structured computation of the Laplace, nuclear potential, nonlinear Hartree and the
(nonlocal) exchange parts of the Fock operator, using a general basis, discretized on a sequence of
*n* × *n* × *n* Cartesian grids [1,2,5]. The arising 3D and 6D convolution integrals are replaced by 1D
algebraic operations in *O*(*n* log *n*) complexity, yielding high resolution at low cost. In our multilevel
scheme the nonlinear parts of the Fock operator are calculated in DIIS iterations “on the fly”, thus eliminating
storage demands.

We demonstrate that the tensor methods are as well advantageous in a “black-box” calculation of the
two-electron integrals (TEI), giving a choice to use generic basis sets including, for example, combinations
of plane waves, Gaussians, Slater-type and local hat-functions [4]. Our numerical algorithm to
compute the Cholesky decomposition of TEI matrix is based on multiple factorizations, which yield an
almost unreducible number of product basis functions building the TEI tensor, depending on a threshold
*ε* > 0. The factorized TEI matrix can be applied in tensor calculations of MP2 energy correction
[6]. Numerical tests for compact (3D) molecules demonstrate efficiency of the grid-based tensor methods
in electronic structure calculations. High resolution is achieved by employing large spatial grids up
to the size *n*^{3}
≈ 10^{14} in Matlab simulations.

http://personal-homepages.mis.mpg.de/vekh

[1] B. N. Khoromskij and V. Khoromskaia, SIAM J. Sci. Comput. 2009, 31, 3002.

[2] V. Khoromskaia, Comput. Methods Appl. Math. 2010, 10, 204.

[3] B. N. Khoromskij, V. Khoromskaia and H.-J. Flad, SIAM J. Sci. Comput. 2011, 33, 45.

[4] V. Khoromskaia, B. N. Khoromskij and R. Schneider, Preprint 29/2012 MIS MPG, Leipzig, 2012.

[5] V. Khoromskaia, D. Andrae and B. N. Khoromskij, Comput. Phys. Comm. 2012, 183, 2392.

[6] V. Khoromskaia and B. N. Khoromskij, Preprint MIS MPG, Leipzig, 2012, in progress.