Generalized Foldy–Wouthuysen transformation for relativistic two‐component methods: Systematic analysis of two‐component Hamiltonians

Abstract

AbstractThe generalized Foldy–Wouthuysen (GFW) transformation was proposed as a generic form that unifies four types of transformations in relativistic two‐component methods: unnormalized GFW(UN), and normalized form 1, form 2, and form 3 (GFW(N1), GFW(N2), and GFW(N3)). The GFW transformation covers a wide range of transformations beyond the simple unitary transformation of the Dirac Hamiltonian, allowing for the systematic classification of all existing two‐component methods. New two‐component methods were also systematically derived based on the GFW transformation. These various two‐component methods were applied to hydrogen‐like and helium‐like ions. Numerical errors in energy were evaluated and classified into four types: the one‐electron Hamiltonian approximation, the two‐electron operator approximation, the newly defined “picture difference error (PDE),” and the error in determining the transformation, and errors in multi‐electron systems were discussed based on this classification.