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

Venera Khoromskaia

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Submission date: 23. Jun. 2009 (revised version: March 2010)
Pages: 20
published in: Computational methods in applied mathematics, 10 (2010) 2, p. 204-218 
DOI number (of the published article): 10.2478/cmam-2010-0012
Bibtex
MSC-Numbers: 65F30, 65F50, 65N35
Keywords and phrases: Hartree-Fock operator, exchange matrix, discrete tensor operations
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Abstract:
We introduce the novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is the (nonlocal) integral operator in . Usually, this challenging computational problem is solved by laborious analytical evaluation of the two-electron integrals using ``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 core of our method is the low-rank tensor representation of arising functions and operators on Cartesian grid, and 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 computation of the exchange operator and a recent tensor method of the Coulomb matrix evaluation are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide the new tool for algebraic optimization of the Galerkin basis in the case of large molecules.