Efficient HF exchange evaluation through Fourier convolution in Cartesian grid for orbital-dependent density 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 et al., J. Chem. Theor. Comput. 13, 2571]. 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 πelectron 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.

A Simple Effective SCF Method for Computing Optical Gaps in Organic Chromophores

“…The triplet calculation is restricted open‐shell (RO) so that all orbitals in the configuration are well defined. The K if integral is then numerically evaluated accurately, using the Cartesian grid DFT program, developed in our laboratory, [48,50,54–59] with orbitals from GAMESS. Over the years, this has been applied to a number of problems quite successfully.…”

Excitation energies through Becke's exciton model within a Cartesian-grid KS DFT

“…These are computed using our in-house pseudopotential KS-DFT program in CCG [48]. An initial version was first developed by the corresponding author in 2008, on the basis of works [33][34][35][36][37], which has been later extended by his group [38][39][40][41]. Since the details have been published in the above references, here we give only some essential aspects.…”

“…where v hxc refers to the classical (Hartree) and non-classical (XC) potential combined. The construction of HF exchange in KS-Fock matrix has been documented in our earlier work [41]; hence not repeated here.…”

“…In the present work, we have adopted the above approach for lowest single-particle excitation energy corresponding to singlet excited state in molecules. The required computations are performed following a pseudopotential KS-DFT implemented in Cartesian coordinate grid (CCG), as developed in our laboratory [33][34][35][36][37][38][39][40][41]. The pertinent two-electron integral (from appropriate orbitals obtained after solving the lowest triplet excited state) is carried out numerically using a recently designed algorithm [41].…”

“…The required computations are performed following a pseudopotential KS-DFT implemented in Cartesian coordinate grid (CCG), as developed in our laboratory [33][34][35][36][37][38][39][40][41]. The pertinent two-electron integral (from appropriate orbitals obtained after solving the lowest triplet excited state) is carried out numerically using a recently designed algorithm [41]. This employs the Fourier convolution theorem in conjunction with a RS Coulomb interaction kernel.…”