Systematic treatment of relativistic effects accurate through arbitrarily high order in α2

A systematic method for the generation of two-component relativistic Hamiltonians for use in relativistic quantum chemistry is presented and discussed. The free particle Foldy–Wouthuysen transformation of the Dirac Hamiltonian is performed prior to the determination of the block-diagonalizing unitary transformation. The latter can be determined iteratively through arbitrarily high leading order with respect to α (fine structure constant). Certain freedom in the initialization of the iterative solution leads to the whole class of two-component Hamiltonians h2k which are exact through the order of α2k and differ in contributions of all higher orders in α2. The efficiency of different iterative schemes is analyzed. Also the relation between the present method and the Douglas–Kroll approximation is investigated. The performance of two-component Hamiltonians for k=2, 3, and 4 is studied numerically in calculations of energies of the 1s1/2 level in heavy hydrogen-like ions. Their performance in calculations of the valence-determined atomic and molecular properties is investigated by computing the ionization potential of Au and spectroscopic constants of the AuH molecule. The total energy of these systems strongly depends on the level of exactness with respect to α2. However, for moderately relativistic systems the α4-class Hamiltonian derived in this paper is found to be sufficient for accurate calculations of valence-determined properties.