Toward Well-Defined Analytic Block-Diagonal Hamiltonians

This Hamiltonian can then be used variationally in quantum chemical calculations, since because of its derivation no negative energy states can occur. It should be anticipated that this Hamiltonian is conceptually equivalent to the infinite-order Douglas-Kroll-Hess Hamiltonian to be discussed in section 12.3, because both schemes do not apply any expansion in 1/c. Also the expressions for Ep and Ap are strictly evaluated in closed form within both approaches. However, whereas Douglas-Kroll-Hess theory yields analytic exressions for each order in V, the infinite-order two-component method summarizes all powers of V in the final matrix representation of/+. 

Toward Well-Defined Analytic Block-Diagonal Hamiltonians 

however, an elegant and legitimate expansion of the decoupled Hamiltonian similar to Eq. (11.93) is to be preserved for both analytical and numerical investigations, one has to classify each term of this expansion according to a new order parameter. This is necessarily the scalar potential V (in the absence of any vector potentials A), which is the only remaining possibility since all other parameters like me and p are already all associated with 1 /c. 

An expansion in terms of V, i.e., the Douglas-Kroll-Hess expansion, is the only valid analytic expansion technique for the Dirac Hamiltonian, where the final block-diagonal Hamiltonian is represented as a series of regular even terms of well-defined order in V, which are all given in closed form. For the derivation, the initial transformation step has necessarily to be chosen as the closed-form, analytical free-particle Foldy-Wouthuysen transformation defined by Eq. (11.35) in order to provide an odd term depending on the external potential that can then be diminished. We now address these issues in the next chapter. 

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Toward Well-Defined Analytic Block-Diagonal Hamiltonians 465 —, j Ep - mc + ApVAp + Ap(r-PpV(r-PpAp (1 + Pp Q)... 

Toward Well-Defined Analytic Block-Diagonal Hamiltonians 467... 

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