Linear scaling computation of the Fock matrix. VIII. Periodic boundaries for exact exchange at the $\Gamma$-point

TL;DR

This paper introduces a translationally invariant formulation for periodic Hartree-Fock calculations at the $b5$-point, enabling linear scaling computation of the exchange matrix using the MIC and ONX algorithms, with demonstrated accuracy on materials like diamond and ice.

Contribution

It presents a novel periodic HF $b5$-point method using MIC and ONX for linear scaling exchange matrix computation, bridging real-space and reciprocal-space approaches.

Findings

Convergence to the k-space limit demonstrated for MgO, ice, and diamond.

Accurate lattice constant for diamond obtained with HF-MIC and PBE0.

Linear scaling exchange matrix computation shown for condensed-phase materials.

Abstract

A translationally invariant formulation of the Hartree-Fock (HF) $\Gamma$-point approximation is presented. This formulation is achieved through introduction of the Minimum Image Convention (MIC) at the level of primitive two-electron integrals, and implemented in a periodic version of the ONX algorithm [J. Chem. Phys, {\bf 106} 9708 (1997)] for linear scaling computation of the exchange matrix. Convergence of the HF-MIC $\Gamma$-point model to the HF ${\bf k}$-space limit is demonstrated for fully periodic magnesium oxide, ice and diamond. Computation of the diamond lattice constant using the HF-MIC model together with the hybrid PBE0 density functional [Theochem, {\bf 493} 145 (1999)] yields $a_0=3.569$\AA with the 6-21G* basis set and a $3\times3\times3$ supercell. Linear scaling computation of the HF-MIC exchange matrix is demonstrated for diamond and ice in the condensed phase