Accelerating Fock Build via Hybrid Analytical-Numerical Integration

A hybrid analytical-numerical integration scheme is introduced
to accelerate the Fock build in self-consistent field (SCF) and time-dependent
density functional theory (TDDFT) calculations. To evaluate the Coulomb
matrix J[D], the density matrix D is first decomposed into two parts, the superposition of atomic
density matrices D⊕A and the rest DR = D–D⊕A. While J[D⊕A] is evaluated analytically, J[DR] is evaluated fully numerically [with the
multipole expansion of the Coulomb potential (MECP)] during the SCF
iterations. Upon convergence, DR is further split into those of near (DRC) and distant (DRL) atomic orbital (AO) pairs, such that J[DRC] and J[DRL] are evaluated seminumerically
and fully numerically (with MECP). Such a hybrid J-build
is dubbed “analytic-MECP” (aMECP). Likewise, the analytic
evaluation of K[D⊕A] and seminumerical
evaluation of K[DR] are also invoked for the construction of the exchange matrix K[D] during the SCF iterations. The chain-of-spheres
(COSX) algorithm [Chem. Phys. 356, 98 (2009]) is employed for K[DR] but with a
revised construction of the S-junctions for overlap AO pairs. To distinguish
from the original COSX algorithm (which does not involve the partition
of the density matrix D), we denote the presently revised
variant as COSx. Upon convergence, DR is further split into those of near (DRC) and distant (DRL) AO pairs followed by a rescaling, leading to D̃RC and D̃RL, respectively. K[D̃RC] and K[D̃RL] are then evaluated analytically and seminumerically
(with COSx), respectively. Such a hybrid K-build is dubbed
“analytic-COSx” (aCOSx). Extensive numerical experimentations
reveal that the combination of aMECP and aCOSx is highly accurate
for ground state SCF calculations (<μEh/atom error in energy) and is particularly efficient
for calculations of large molecules with extended basis sets. As for
TDDFT excitation energies, a medium grid for MECP and a coarse grid
for COSx are already sufficient.