Black-Box Hartree–Fock Solver by Tensor Numerical Methods

Abstract. The Hartree–Fock eigenvalue problem governed by the 3D integro-differential operator is the basic model in ab initio electronic structure calculations. Several years ago the idea to solve the Hartree–Fock equation by a fully 3D grid based numerical approach seemed to be a fantasy, and the tensor-structured methods did not exist. In fact, these methods evolved during the work on this challenging problem. In this paper, our recent results on the topic are outlined and the black-box Hartree–Fock solver by tensor numerical methods is presented. The approach is based on the rank-structured calculation of the core hamiltonian and of the two-electron integrals tensor using the problem adapted basis functions discretized on n×n×n$n\times n\times n$ 3D Cartesian grids. The arising 3D convolution transforms with the Newton kernel are replaced by a combination of 1D convolutions and 1D Hadamard and scalar products. The approach allows huge spatial grids, with n 3 ≃10 15 $n^3\simeq 10^{15}$ , yielding high resolution at low cost. The two-electron integrals are computed via multiple factorizations. The Laplacian Galerkin matrix can be computed “on-the-fly” using the quantized tensor approximation of O(logn)$O(\log n)$ complexity. The performance of the black-box solver in Matlab implementation is compatible with the benchmark packages based on the analytical (pre)evaluation of the multidimensional convolution integrals. We present ab initio Hartree–Fock calculations of the ground state energy for the amino acid molecules, and of the “energy bands” for the model examples of extended (quasi-periodic) systems.