Sequential Unitary Decoupling Transformations

Since the energy-independent operator X is not known in closed form, a decomposition of the overall unitary transformation U of Eq. (11.15) into a sequence of tailored unitary transformations, as in Eq. (11.91), 

The original idea of this procedure dates back to 1974 and is due to Douglas and Kroll [238] who mention it in the appendix of their work on the lowest state of He, which they study with the Bethe-Salpeter equation. More than a decade later the paper by Douglas and Kroll was rediscovered by Hess [624] who at first had to struggle with the huge problem to transform the idea into a method that allows actual calculations on molecular systems. He invoked the idea of producing a basis set that diagonalizes the matrix representation 

Relativistic Quantum Chemistry. Markus Reiher and Alexander Wolf 

In the Douglas-Kroll-Hess expansion, the block-diagonal Hamiltonian of Eq. (11.15) can be formally expressed as a series of even terms of well-defined order in the external scalar potential V, 

It should be recalled that, because of the presence of the external potential and the nonlocal form of Ep given by Eq. (11.11), all operators resulting from these unitary transformations are well defined only in momentum space (compare the discussion of the square-root operator in the context of the Klein-Gordon equation in chapter 5 and the momentum-space formulation of the Dirac equation in section 6.10). Whereas So acts as a simple multiplicative operator, all higher-order terms containing the potential V are integral operators and completely described by specifying their kernel. For example, the 

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As before, we stick to the one-electron case and leave the generalization to N electrons to the reader. The proper choice of the DKH expansion parameter, V or V(A), for the DKH transformation is a decisive question in the sequential unitary transformation scheme as it produces a series expansion of the block-diagonal operator with each term to be classified according to a well-defined order in the expansion parameter. In the case of a one-step decoupling scheme (for instance, in a purely numerical fashion as suggested by Barysz and Sadlej for the Hamiltonian see section 11.6) all derivations of this section are also valid. 

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