Robust and efficient variational fitting of Fock exchange

Abstract

We propose a new variational fitting approach for Fock exchange that requires only the calculation of analytical three-center electron repulsion integrals. It relies on localized molecular orbitals and Hermite Gaussian auxiliary functions. The working equations along with a detailed description of the implementation are presented. The computational performance of the new algorithm is analyzed by benchmark calculations on systems with different dimensionality. Comparison with standard four-center and three-center electron repulsion integral Hartree-Fock calculations shows an excellent accuracy-performance relation.

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm

“…These approaches may be broadly classified into two categories: local fit metrics [43][44][45][46] and local fit domains. 42,[47][48][49][50] In the local fit metric approach, the fit coefficients are determined by minimizing the self-repulsion between the fitted and exact densities as measured by a rapidly decaying local analog of the Coulomb operator, rather than the more typical full Coulomb metric. Local fit domain approximations are applied by expanding a given primary basis product only with the auxiliary basis functions within a predefined spatial domain; they differ from local metrics in that locality is achieved explicitly through a constraint rather than implicitly through the decay of the metric.…”

Hybrid Functionals with Variationally Fitted Exact Exchange

“…(35) is added to the selfinteraction including Hartree energy with Coulomb fitting, the variationally fitted self-interaction-free Hartree-Fock energy expression is obtained. 30 On the other hand, if the fitted exact exchange energy is combined with exchange-correlation functionals, hybrid ADFT energy expressions without four-center ERIs are obtained. In particular, we find the following ADFT or DF-DFT energy expressions for the here discussed B3LYP (Eq.…”

“…Moreover, in the case of Hartree-Fock, the approximated energy expression is self-interaction-free. 30 After expanding the orbital products in Eq. (29), we find:…”

“…24 For this reason, a new local density-fitting (LDF) approach for the exact exchange (EXX) energy was recently developed in our group. 30 Following the variational fitting of the Coulomb potential, the LDF-EXX approach finds the variationally best fit for the orbital products…”

“…We then continue to discuss the LDF exact exchange algorithm recently developed in our group. 30 Finally, we show how the combination of these two methods performs for three popular hybrid DFAs: B3LYP, 10 PBE0, 34,35 and M06-2X. 36 …”