Direct DKH Transformation of First-Order Energy

Assume the unperturbed Dirac Hamiltonian given by Eq. (15.15) has been decoupled by the perturbation-independent unitary transformation U to yield the standard (i.e., perturbation-free) block-diagonal infinite-order DKH Hamiltonian with the unitary transformation U decomposed into a sequence of unitary transformations Lf, to be applied stepwise as discussed in depth in chapter 12. The transformed unperturbed wave function for positive-energy solutions is — according to Eq. (11.16) — given by 

the individual unitary transformations do not depend on the perturbation. Since it is the unperturbed wave function jpi that enters the linear-response energy correction of Eq. (15.7), the two-component DKH expression for is most easily derived by smart insertion of unperturbed identity operators (1 = U U) according to 

In contrast to the decoupled DKH Hamiltonian the transformed property operator X khoo = UXU is not block-diagonal, but its off-diagonal components do not affect because of the vanishing lower components of In practical calculations this is even the case for any finite-order DKH scheme because the lower two components are never calculated and are therefore set equal to zero. Since the first-order energy correction is given as the unperturbed expectation value of the property operator X, a compact yet slightly imprecise notation (X) has been introduced in Eq. (15.60) which suppresses the information on the state for which the expectation value is evaluated. It is crucial to realize that all terms contributing to the transformed property 

2 Explicit Expressions up to Third Order in the Unperturbed Potential 

For any arbitrary electromagnetic property operator of the form given by Eq. (15.25) the even components up to third order in V read [764] 

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