Perturbation versus variation treatment of regular relativistic Hamiltonians

A two-parameter form of the Dirac Hamiltonian, which follows from our earlier papers and can be treated in the framework of double perturbation theory, is derived. The usual 1c2 perturbation expansion is partitioned into a partial expansion in terms of 1c2 and additional expansion into a series of the metric perturbation parameter λ2. The block diagonalization of the two-parameter Dirac Hamiltonian corresponds to infinite summation with respect to the first of these parameters and leads to what is known as the 2-component Hamiltonian of the zeroth-order regular approximation (ZORA, CPD). When corrected through first order in λ2 the ZORA Hamiltonian reproduces exactly all terms of the Pauli approximation. The block diagonalized Hamiltonian can be further modified to a form which has the non-relativistic energy operator as the leading term. However, this form has been found to possess several inconvenient features which are remedied by initializing the perturbation expansion with the ZORA Hamiltonian. This stresses the importance of the variation solution of the ZORA equation as the lowest order approximation for the expansion in the metric perturbation parameter λ2.