Two-Component Methods

Abstract

The Douglas-Kroll and other recently developed two-component methods, based on similar ideas, are reviewed and discussed. A unified derivation of these methods through the solution of certain operator equations is presented. These solutions can be obtained by imposing additional conditions on the accuracy of the resulting two-component Hamiltonians. The traditional Douglas-Kroll Hamiltonian is shown to be the best two-component Hamiltonian which approximates the Dirac operator simultaneously through the fourth order in the fine structure constant and through the second order with respect to the external interaction potentials. Several higher order methods are presented as well. The main attention is focused on the infinite-order solution of the operator equation for the block-diagonalizing transformation of the Dirac Hamiltonian. This is accomplished by a further modification of this equation and leads to the infinite-order two-component theory ‘for electrons only’, i.e., the resulting two-component Hamiltonian gives the Dirac spectrum of electronic solution. This ‘exact’ two-component method is combined with the Hess technique to handle the momentum space integrals and its implementation is shown to give the electronic Dirac levels with arbitrarily high accuracy. The new infiniteorder method is also shown to save the supersymmetric features of the 4-component theory by reproducing the accidental degeneracies present in the Dirac spectrum. It is concluded that relativistic quantum chemistry ‘for electrons only’ can be done in exclusively two-component form. The same method produces another two-component Hamiltonian for the negative energy spectrum. This Hamiltonian can be used to build a finite basis set theory of quantum electrodynamic corrections. In addition to reviewing the latest developments in two-component relativistic methods rooted in the Douglas-Kroll approach also the problem of the change of picture for other than energy operators is surveyed. The same problem is shown to occur when the two-component methods are extended to deal with electron-electron interactions. Some recent ideas in this area are reviewed.