Parallel and Low-Order Scaling Implementation of Hartree–Fock Exchange Using Local Density Fitting

Abstract

Calculations using modern linear-scaling electron-correlation methods are often much faster than the necessary reference Hartree-Fock (HF) calculations. We report a newly implemented HF program that speeds up the most time-consuming step, namely, the evaluation of the exchange contributions to the Fock matrix. Using localized orbitals and their sparsity, local density fitting (LDF), and atomic orbital domains, we demonstrate that the calculation of the exchange matrix scales asymptotically linearly with molecular size. The remaining parts of the HF calculation scale cubically but become dominant only for very large molecular sizes or with many processing cores. The method is well parallelized, and the speedup scales well with up to about 100 CPU cores on multiple compute nodes. The effect of the local approximations on the accuracy of computed HF and local second-order Møller-Plesset perturbation theory energies is systematically investigated, and default values are established for the parameters that determine the domain sizes. Using these values, calculations for molecules with hundreds of atoms in combination with triple-ζ basis sets can be carried out in less than 1 h, with just a few compute nodes. The method can also be used to speed up density functional theory calculations with hybrid functionals that contain HF exchange.

Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2)

“…A more detailed investigation of such approximation and procedures is currently under way in our laboratory. Additionally, the computation of the Fock matrix in CASSCF calculations can be speeded up using local density fitting approximations, similarly to the Hartree-Fock method 161,162 , and we are currently working on a new MCSCF method that is suitable to treat large molecules.…”

Efficient evaluation of the geometrical first derivatives of three-center Coulomb integrals

“…(49) to produce all the λ, β, ζ-scaled classes of these ERIs that are going to be required by Eqs. (50) and (51), perform the primitive contraction on these scaled classes, and then apply Eqs. (50) and (51) to build up l a and l b , respectively.…”

“…(50) and (51), perform the primitive contraction on these scaled classes, and then apply Eqs. (50) and (51) to build up l a and l b , respectively. We refrain from presenting the necessary index ranges for the intermediates here since the presence of the scaling indices makes them quite complicated.…”

“…At the contracted level, we can work in two directions: in GHP Cart,1 , we build up l a and l b by using Eqs. (50) and 51, respectively, while the GHP Cart,2 scheme uses Eq. 50to construct the appropriate…”

“…Utilizing the J-engine scheme and related ideas, 36,44 it is possible to only partially calculate the integral-derivatives and, by reverse operations, transform the density matrix into a form appropriate for the contraction with intermediate differentiated ERIs rather than with those over atomic orbitals (AOs). The approach is especially advantageous for Kohn-Sham SCF gradients, but it can also be beneficial for DF Hartree-Fock or hybrid density functional calculations if the J-engine is applied along with reduced-cost schemes [45][46][47][48][49][50][51] for the computation of the Fock-exchange contribution to the gradient. In the latter cases, however, three-center ERI derivatives still have to be computed.…”