Unitary Transformations of the Dirac Hamiltonian

Abstract

The separation of the spin-dependent terms in the Dirac Hamiltonian enables us to make an approximation in which the spin-free terms are included in the orbital optimization and the spin-dependent terms may be treated later as a perturbation. In this process, the parameter space required to treat the large and small components has not changed. Even with the extraction of (σ ·p) from the small component, we still have to calculate integrals involving pfL, which essentially regenerates the original small component space and so the integral work has not really changed. What has been achieved is the ability to use the machinery of spin algebra from nonrelativistic theory, but we are left with a large and a small component. The obvious next step is to separate the large and small components, or the positiveand negative-energy states. The small component can be eliminated from the Dirac equation by algebraic manipulation, but this leaves the energy in the denominator. It would be preferable to obtain an energy-independent Hamiltonian that acted only on positive-energy states and that could therefore be represented as two-component spinors. If, following this separation, it were possible to separate out the spin-free and spin-dependent terms, we would have a spin-free Hamiltonian that would operate on a one-component wave function, and we would then be able to use all the machinery of nonrelativistic quantum chemistry but with modified one- and two-electron integrals. The matrix form of the Dirac Hamiltonian suggests that we should seek a unitary transformation that will make it diagonal with respect to the large- and small-component spinor spaces. Such a transformation is called a Foldy–Wouthuysen transformation (Foldy and Wouthuysen 1950). Although in their original paper only the free-particle transformation was derived, together with an iterative decoupling procedure that will be described later in this chapter, the term Foldy–Wouthuysen transformation has come to mean any unitary transformation that decouples the large and small components, either exactly or approximately, and we will use it in this sense.