Four-Component Electronic Structure Methods for Molecules

Abstract

In this chapter we consider the extension of 4-component relativistic methods from atomic to molecular systems, in particular the challenges arising from the introduction of the algebraic approximation. In order to analyze the variational stability of the relativistic many-electron Hamiltonian we derive a variational theory of QED in the semiclassical limit using the second quantization formalism and exponential parametrization. In QED the negative-energy orbitals are filled leading to a true minimization principle for the electronic ground state, whereas in the standard 4-component approach these orbitals are empty and treated as an orthogonal complement, thus leading to a minimax principle. We emphasize the non-uniqueness of the resulting no-pair Hamiltonian of the standard approach. 4-component methods allow the continuous update of the Hamiltonian and thereby complete relaxation of the electronic wave function. We also discuss more practical aspects of the implementation of 4-component relativistic methods. We carefully analyze their computational cost and conclude that the difference with respect to non-relativistic methods constitute a prefactor and not a difference in order. We furthermore discuss how computational cost may be reduced while staying at the 4-component level, e.g. by exploiting the atomic nature of the small component density.