Laplace-transformed atomic orbital-based M{o}ller-Plesset perturbation theory for relativistic two-component Hamiltonians

We present a formulation of Laplace-transformed atomic orbital-based second-order M{\o}ller-Plesset perturbation theory (MP2) energies for two-component Hamiltonians in the Kramers-restricted formalism. This low-order scaling technique can be used to enable correlated relativistic calculations for large molecular systems. We show that the working equations to compute the relativistic MP2 energy differ by merely a change of algebra (quaternion instead of real) from their non-relativistic counterparts. With a proof-of-principle implementation we study the effect of the nuclear charge on the magnitude of half-transformed integrals and show that for light elements spin-free and spin-orbit MP2 energies are almost identical. Furthermore, we investigate the effect of separation of charge distributions on the Coulomb and exchange energy con- tributions, which show the same long-range decay with the inter-electronic / atomic distance as for non-relativistic MP2. A linearly scaling implementation is possible if the proper distance behavior is introduced to the quaternion Schwarz-type estimates as for non-relativistic MP2.