Improving Density Functional Theory: General Computation of Exact Exchange Holes and A New Algorithm for Fast Gaussian Transform

Abstract

Exchange and correlation holes are unique quantum concepts for understanding the nature of electron interactions based on quantum conditional probabilities. Among those, the exact exchange hole is of special interest since it is derived rigorously from first principles without approximations and is often modeled by approximate exchange expressions of density functional theory. In this work, the algorithm for the computation of the exact exchange hole for a given reference point is developed and implemented for molecular orbitals in Gaussian basis functions. The formulas include the spherical surface integral of a Gauss function. A computationally concise algorithm is proposed for obtaining the expansion coefficients of polynomial terms when the coordinate system is transformed from Cartesian to spherical. The result expression includes a number of cases of elementary integrals, the most difficult integral can be formed by linearly combining modified Bessel functions of the first kind. Direct applications of the standard approach using Mathematica and GSL are found to be inefficient and limited in the range of the parameters for the Bessel function. We propose an asymptotic function for the Bessel function. The relative error of the asymptotic function is in the order of 1e-16 with the first five terms of the asymptotic expansion. This new capability is used to explore the extent to which current popular model exchange holes resemble or differ from the exact exchange hole. Point-wise accuracy of the exchange holes for isolated atoms is important in local hybrid schemes, real-space models of static correlation and other. We find in this vein that among the models tested here only the BR89 exchange hole seems more or less suitable for that purpose, while better approximations are still very much on demand. Analyzing the deviations of model exchange holes from the exact exchange hole in molecules like H2 and Cr2 upon bond stretching reveals new aspects of the left-right static correlation. To speed up the computation of exchange-correlation of density functional theory, we derive a new algorithm for calculating the electron density in gaussian functions with polynomial factors and variable scales in an efficient way. Such a problem belongs to the general of computing Gaussian functions of distributed source points on distributed target points. The current best algorithm is the so-called the Fast Generalized Gauss Transform. We have improved upon it on several fronts. The Cardinal B-spline interpolation is applied, reducing the computational cost by almost half. This improvement enables the treatment of Gaussians with variable scales by segmenting the range of the scales. The application of this improved Fast Gauss Transform method is also applied to Gaussian functions with arbitrary polynomial factors. A scheme is also designed to determine all the parameters of the algorithm based on one input parameter. Benchmark calculations show that the new algorithm can be 180 times faster than the direct calculation. And the speed of the new algorithm can be 9 times faster than Strain’s variable-scale algorithm. The Cardinal B-spline method can be 60 times faster than using the plane-wave directly with a single scale and can be even faster for variable scales. The memory cost is also reduced in one dimension in our new algorithm.