On the spin separation of algebraic two-component relativistic Hamiltonians

The separation of the spin-free and spin-dependent terms of a given relativistic Hamiltonian is usually facilitated by the Dirac identity. However, this is no longer possible for the recently developed exact two-component relativistic Hamiltonians derived from the matrix representation of the Dirac equation in a kinetically balanced basis. This stems from the fact that the decoupling matrix does not have an explicit form. To resolve this formal difficulty, we first define the spin-dependent term as the difference between a two-component Hamiltonian corresponding to the full Dirac equation and its one-component counterpart corresponding to the spin-free Dirac equation. The series expansion of the spin-dependent term is then developed in two different ways. One is in the spirit of the Douglas-Kroll-Hess (DKH) transformation and the other is based on the perturbative expansion of a two-component Hamiltonian of fixed structure, either the two-step Barysz-Sadlej-Snijders (BSS) or the one-step exact two-component (X2C) form. The algorithms for constructing arbitrary order terms are proposed for both schemes and their convergence patterns are assessed numerically. Truncating the expansions to finite orders leads naturally to a sequence of novel spin-dependent Hamiltonians. In particular, the order-by-order distinctions among the DKH, BSS, and X2C approaches can nicely be revealed. The well-known Pauli, zeroth-order regular approximation, and DKH1 spin-dependent Hamiltonians can also be recovered naturally by appropriately approximating the decoupling and renormalization matrices. On the practical side, the sf-X2C+so-DKH3 Hamiltonian, together with appropriately constructed generally contracted basis sets, is most promising for accounting for relativistic effects in two steps, first spin-free and then spin-dependent, with the latter applied either perturbatively or variationally.