Linear scaling computation of the Fock matrix VII. Periodic Density Functional Theory at the $\Gamma$-point

TL;DR

This paper extends linear scaling quantum chemical methods for Density Functional Theory to periodic systems at the Gamma-point, enabling efficient calculations of large condensed-phase systems with controlled accuracy.

Contribution

It introduces a periodic version of hierarchical cubature and combines it with tree-code algorithms for Coulomb summation, achieving linear scaling in periodic DFT calculations.

Findings

Demonstrated convergence to k-space integration limit for MgO and NaCl.

Achieved linear scaling construction of the Fock matrix for diamond with up to 512 atoms.

Controlled error in Fock matrix construction for large systems.

Abstract

Linear scaling quantum chemical methods for Density Functional Theory are extended to the condensed phase at the $\Gamma$-point. For the two-electron Coulomb matrix, this is achieved with a tree-code algorithm for fast Coulomb summation [J. Chem. Phys. {\bf 106}, 5526 (1997)], together with multipole representation of the crystal field [J. Chem. Phys. {\bf 107}, 10131 (1997)]. A periodic version of the hierarchical cubature algorithm [J. Chem. Phys. {\bf 113}, 10037 (2000)], which builds a telescoping adaptive grid for numerical integration of the exchange-correlation matrix, is shown to be efficient when the problem is posed as integration over the unit cell. Commonalities between the Coulomb and exchange-correlation algorithms are discussed, with an emphasis on achieving linear scaling through the use of modern data structures. With these developments, convergence of the $\Gamma$-point supercell approximation to the ${\bf k}$-space integration limit is demonstrated for MgO and NaCl. Linear scaling construction of the Fockian and control of error is demonstrated for RBLYP/6-21G* diamond up to 512 atoms.