Two-component methods of relativistic quantum chemistry: from the Douglas–Kroll approximation to the exact two-component formalism

Abstract

The two-component methods of relativistic quantum chemistry based on the Foldy–Wouthuysen (FW) transformations of the Dirac hamiltonian are reviewed. Following the strategy designed by Douglas and Kroll, the FW transformation is carried out in two steps. The first amounts to performing the exact free-particle FW transformation. At variance with other approaches, the second step is written in the form, which results in a nonlinear operator equation. This equation can be solved iteratively, leading to two-component hamiltonians of arbitrarily high accuracy in even powers of the fine structure constant. All these hamiltonians can be classified according to their completeness with respect to the leading order in the fine structure constant. On passing to the basis set representation one obtains the usual Douglas–Kroll hamiltonian and all possible higher-order approximations. By a simple modification of the operator equation which determines the block-diagonalizing transformation one can obtain numerical infinite-order solutions, i.e. one can obtain the exact numerical solution for the separation of the pure electronic part of the Dirac spectrum. This gives the exact two-component method for the use in relativistic quantum chemistry. The computational aspects of this approach are discussed as well. The transition from the Dirac formalism to any two-component approximation is accompanied by the change of all operators, including those which correspond to external perturbations and lead to properties of different orders. This so-called change of picture problem is given particular attention and its importance for certain operators is identified.