Efficient Four-Component Dirac-Coulomb-Gaunt Hartree-Fock in the Pauli Spinor Representation

Four-component Dirac-Hartree-Fock is an accurate mean-field method for treating molecular systems where relativistic effects are important. However, the computational cost and complexity of the two-electron interaction make this method less common, even though we can consider the Dirac-Hartree-Fock Hamiltonian the "ground truth" of the electronic structure, barring explicit quantum electrodynamical effects. Being able to calculate these effects is then vital to the design of lower scaling methods for accurate predictions in computational spectroscopy and properties of heavy element complexes that must include relativistic effects for even qualitative accuracy. In this work, we present a Pauli quaternion formalism of maximal component and spin separation for computing the Dirac-Coulomb-Gaunt Hartree-Fock ground state, with a minimal floating point operation count algorithm. This approach also allows one to explicitly separate different spin physics from the two-body interactions, such as spin-free, spin-orbit, and spin-spin contributions. Additionally, we use this formalism to examine relativistic trends in the periodic table and analyze the basis set dependence of atomic gold and gold dimer systems.