# Efficient HF exchange evaluation through Fourier convolution in Cartesian grid for orbital-dependent density functionals

## TL;DR

This paper introduces a fast, purely numerical Cartesian grid method for calculating Hartree-Fock exchange in density functional theory, utilizing Fourier convolution and range-separated Coulomb kernels for improved efficiency and scalability.

## Contribution

It presents a novel Fourier convolution-based numerical approach for HF exchange evaluation that scales logarithmically with molecular size, extending to global hybrid functionals within pseudopotential Kohn-Sham theory.

## Key findings

- Excellent agreement with semi-numerical methods in total and orbital energies
- Logarithmic scaling with molecular size demonstrated
- Potential for developing advanced range-separated hybrid functionals

## Abstract

We present a purely numerical approach in Cartesian grid, for efficient computation of Hartree-Fock (HF) exchange contribution in the HF and density functional theory models. This takes inspiration from a recently developed algorithm [Liu \emph{et al.}, J.~Chem.~Theor.~Comput.~ \textbf{13}, 2571 (2017)]. A key component is the accurate evaluation of electrostatic potential integral, which is the rate-determining step. This introduces the Fourier convolution theorem in conjunction with a range-separated Coulomb interaction kernel. The latter is efficiently mapped into real grid through a simple optimization procedure, giving rise to a constraint in the range-separated parameter. The overall process offers logarithmic scaling with respect to molecular size. It is then extended towards global hybrid functionals such as B3LYP, PBE0 and BHLYP within pseudopotential Kohn-Sham theory, through an LCAO-MO ansatz in Cartesian grid, developed earlier in our laboratory. For sake of comparison, a parallel semi-numerical approach has also been worked out that exploits the familiar Obara-Saika recursion algorithm. An excellent agreement between these two routes is demonstrated through total energy and orbital energy in a series of atoms and molecules (including 10 $\pi$-electron molecules), employing an LANL2DZ-type basis function. A critical analysis of these two algorithms reveals that the proposed numerical scheme could lead to very attractive and competitive scaling. The success of our approach also enables us for further development of optimally tuned range-separated hybrid and hyper functionals.