Exact decoupling of the Dirac Hamiltonian. I. General theory

Abstract

Exact decoupling of positive- and negative-energy states in relativistic quantum chemistry is discussed in the framework of unitary transformation techniques. The obscure situation that each scheme of decoupling transformations relies on different, but very special parametrizations of the employed unitary matrices is critically analyzed. By applying the most general power series ansatz for the parametrization of the unitary matrices it is shown that all transformation protocols for decoupling the Dirac Hamiltonian have necessarily to start with an initial free-particle Foldy–Wouthuysen step. The purely numerical iteration scheme applying X-operator techniques to the Barysz–Sadlej–Snijders (BSS) Hamiltonian is compared to the analytical schemes of the Foldy–Wouthuysen (FW) and Douglas–Kroll–Hess (DKH) approaches. Relying on an illegal 1/c expansion of the Dirac Hamiltonian around the nonrelativistic limit, any higher-order FW transformation is in principle ill defined and doomed to fail, irrespective of the specific features of the external potential. It is shown that the DKH method is the only valid analytic unitary transformation scheme for the Dirac Hamiltonian. Its exact infinite-order version can be realized purely numerically by the BSS scheme, which is only able to yield matrix representations of the decoupled Hamiltonian but no analytic expressions for this operator. It is explained why a straightforward numerical iterative extension of the DKH procedure to arbitrary order employing matrix representations is not feasible within standard one-component electronic structure programs. A more sophisticated ansatz based on a symbolical evaluation of the DKH operators via a suitable parser routine is needed instead and introduced in Part II of this work.