Accelerating Fock Build via Hybrid Analytical-Numerical Integration
Yong Zhang, Rongding Lei, Bingbing Suo, Wenjian Liu · The Journal of Physical Chemistry A · 2025
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 D R = D – D ⊕ A . While J [ D ⊕ A ] is evaluated analytically, J [ D R ] is evaluated fully numerically [with the multipole expansion of the Coulomb potential (MECP)] during the SCF iterations. Upon convergence, D R is further split into those of near ( D RC ) and distant ( D RL ) atomic orbital (AO) pairs, such that J [ D RC ] and J [ D RL ] 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 [ D R ] 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 R ] 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 R is further split into those of near ( D RC ) and distant ( D RL ) AO pairs followed by a rescaling, leading to D ~ R C and D ~ R L, respectively. K [ D ~ R C ] and K [ D ~ R L ] 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 ( < μ E h / 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.