Efficient Semi-Numerical Implementation of Relativistic Exact Exchange within the Infinite-Order Two-Component Method Using a Modified Chain-of-Spheres Method

We present an efficient implementation of relativistic exact exchange within the infinite-order two-component method (IOTC) by employing a state-of-the-art seminumerical integration technique. For accurate consideration of the picture change, inherent to two-component methods, we propose a new scheme based on a relativistic or picture-change transformation of the density matrix, which provides a simple and efficient formulation of relativistically transformed quantities such as the electron density or exact exchange and thus avoids expensive integral transformations. We show that the new scheme does not introduce additional numerical or theoretical errors beyond the approximations of the IOTC method. For the efficient implementation of exact-exchange integrals, we build upon a modified version of the chain-of-spheres exact-exchange (COSX) method. In addition to the conventional overlap and density matrix screening by S- and P-junctions, respectively, we introduce a new simple screening technique in the sense of the original COSX method by additionally considering the asymptotic decay of the integrals over the Coulomb operator within the new F-junctions. Together with the picture-change transformation of the density matrix, this modified COSX method is shown to provide superior efficiency for the calculation of relativistic exact exchange compared to a conventional analytical direct self-consistent-field implementation of exact exchange.