Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods

Abstract

-We propose a novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is a (nonlocal) integral operator in R 3 × R 3 . Usually, this challenging computational problem is solved by analytical evaluation of two-electron integrals using the "analytically separable" Galerkin basis functions, like Gaussians. Instead, we employ the agglomerated "grey-box" numerical computation of the corresponding six-dimensional integrals in the tensor-structured format which does not require analytical separability of the basis set. The point of our method is a low-rank tensor representation of arising functions and operators on an n × n × n Cartesian grid and the implementation of the corresponding multi-linear algebraic operations in the tensor product format. Linear scaling of the tensor operations, including the 3D convolution product, with respect to the one-dimension grid size n enables computations on huge 3D Cartesian grids thus providing the required high accuracy. The presented algorithm for evaluation of the exchange operator and a recent tensor method for the computation of the Coulomb matrix are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide a new tool for algebraic optimization of the Galerkin basis in the case of large molecules.2000 Mathematics Subject Classification: 65F30, 65F50, 65N35, 65F10.

Black-Box Hartree–Fock Solver by Tensor Numerical Methods

“…In particular, the opportunity to verify the results of the 3D grid-based tensor approach by comparison with the 3D calculations by analytical methods of high accuracy using a standard package [44] helped to convince the scientific community [23,21].…”

“…Thus, it became possible to avoid the cubic scaling of computational work in the numerical treatment of functions and operators in 3D problems. The tensor-structured numerical methods for the 3D grid-based calculation of the Coulomb and exchange integral operators have been introduced in [23,21,22], when it was shown that the 3D convolution integration can be reduced to tensor calculus based on combinations of 1D convolutions, and 1D Hadamard and scalar products.…”

Tensors-structured numerical methods in scientific computing: Survey on recent advances

“…Tensor numerical methods are proved to be efficient for data-sparse representation of functions and operators in the Hartree-Fock and Kohn-Sham models in electronic structure calculations [7,39,59,57,23,76,47]. Fully tensorized numerical approach for solving the Hartree-Fock equation, discretized over N × N × N grid, by tensor truncated iteration of complexity O(N log N ), was recently presented in [58].…”

QTT Representation of the Hartree and Exchange Operators in Electronic Structure Calculations

“…The exchange matrix K(C) is computed in three steps, see [23,24]. For a = 1, ..., N orb , and ν = 1, ..., N b , compute the convolution integrals,…”

“…• tensor calculation of the Coulomb and exchange integrals in R 3 by means of a combination of 1D Hadamard and scalar products and 1D convolutions [30,23,24].…”