Large-scale Dirac–Fock–Breit method using density fitting and 2-spinor basis functions

Abstract

We present an efficient theory and algorithm for computing four-component relativistic Dirac-Fock wave functions using the Coulomb, Gaunt, and full Breit interactions. Our implementation is based on density fitting, and is routinely applicable to systems with 100 atoms and a few heavy elements. The small components are expanded using 2-spinor basis functions. We show that the factorization of 3-index half-transformed integrals before building Coulomb and exchange matrices is essential for efficient evaluation of the Fock matrix. With the Coulomb interaction, the computational cost for evaluating the Fock operator has been found to be only 70-90 times that in the non-relativistic density-fitted Hartree-Fock method. The prefactors have been 170 and 350-450 for the Gaunt and Breit interactions, respectively. The largest molecule to which we have applied the Dirac-Fock-Coulomb method is an Ac(III) motexafin complex (130 atoms, 556 electrons, 1289 basis functions), for which one self-consistent iteration takes around 1100 s using 1024 CPU cores. In addition, we have found that, while the standard fitting basis sets are accurate for Dirac-Fock-Coulomb calculations, their accuracy is very poor for Dirac-Fock-Gaunt and Breit calculations. We report a prototype of accurate fitting basis sets for these cases.

Fully relativistic complete active space self-consistent field for large molecules: Quasi-second-order minimax optimization

“…36 for an efficient algorithm), and (ps|tu) are the electron repulsion integrals over molecular spinors in chemists' notation. F I and F A are the closed (inactive) and active contributions to the Fock matrix F, respectively.F is the rectangular active space Fock matrix, which has column indices restricted in the active space while rows run over all MOs.…”

“…Slater determinants |I⟩ are constructed from orthonormal 4-spinor MOs, which are expanded using atomic, scalar basis sets, and restricted kinetic balance to generate the small component basis functions. 36 The MO coefficients C during an optimization are obtained from a previous coefficient C ′ as…”

“…A quasi-second-order algorithm 39 is used for orbital optimization together with density fitting, 40,41 as pioneered by Aquilante et al in the nonrelativistic framework. 38 The computational bottleneck in this algorithm with a moderate active space is computation of the so-called closed Fock operator, which implies that, whenever the standard Dirac-Hartree-Fock calculations are possible (Dirac-Hartree-Fock calculations on large molecules have been presented by us 36,37 and by others 42,43 ), the corresponding Dirac CASSCF is tractable. Our work will significantly extend its applicability from small molecules (often a few atoms) previously reported by others 44,45 to large molecules of materials interest and will be the basis of large-scale multireference electron correlation methods in the Dirac formalism.…”

“…In this work, we address this challenge by developing an efficient algorithm for the complete active space self-consistent field (CASSCF) method based on the Dirac equation (DiracCoulomb CASSCF), combining recent advances in DiracHartree-Fock theory for large molecules 36,37 and modern CASSCF algorithms using density fitting. 38 Our method has been implemented in an efficient program that utilizes largescale parallel computer hardware.…”

Communication: An efficient algorithm for evaluating the Breit and spin–spin coupling integrals

“…An efficient, parallel Dirac-Fock method with the full Breit interaction will be reported elsewhere. 28 All the computer codes have been implemented in the BAGEL package 27 under GNU Public License, and will be released to public in the near future.…”