Douglas-Kroll transformations of higher order

More accurate two-component Hamiltonians can be obtained by successive application of higher-order DK transformations [12,49], 

The resulting expression for the third-order DK Hamiltonian is remarkably simple [17] 

Note that this third-order Hamiltonian can still be expressed via the first-order generator ITi which is of first order in the potential V. In general, DK Hamiltonians up to order 2 +l in the potential V can be expressed by the generators W i, -.W,. [17]. In this way, expressions of DK Hamiltonians of fourth and fifth order have been derived [17]. 

The choice of a particular parameterization for the unitary decoupling transformation U is only relevant for DK Hamiltonians of fifth and higher orders because the expansion of U is unambiguous up to second order in the generator W [12]. The various parameterizations of U, for instance t/ = (1 + W ) -1- W 

Wolf et al. recently proposed a generalized DK transformation as an alternative approach to higher-order DK Hamiltonians [89] they define the unitary transformation by a series in the generator W  

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