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_⊕ and the rest D = D-D_⊕. While J[D_⊕] is evaluated analytically, J[D] is evaluated fully numerically [with the multipole expansion of the Coulomb potential (MECP)] during the SCF iterations. Upon convergence, D is further split into those of near (D) and distant (D) atomic orbital (AO) pairs, such that J[D] and J[D] 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_⊕] and seminumerical evaluation of K[D] 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[D] 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, D is further split into those of near (D) and distant (D) AO pairs followed by a rescaling, leading to $$ and $$, respectively. $$ and $$ 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 ($$ 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.