Expansion of the Transformation DKH

The DKH approach requires the free-particle FW transformation as the initial transformation to obtain the transformed Hamiltonian matrix Ho [610], which is composed of even terms (i.e., diagonal terms, denoted E in basis-set representation) and odd terms (i.e., off-diagonal terms, denoted O in basis representation) 

The basis-set representation of the antihermitean matrix operators Wjt is denoted as Wfc. To keep the convention for the antihermitean matrix consistent with chapter 12 and with the research literature, the letter W is used although it could be confused with the relativistic potential-energy integral defined in Eq. (14.17). 

For the different parametrization schemes of the HJfc [611], the exponential parametrization can be chosen as it requires the lowest number of matrix multiplications [646]. The number of matrix multiplications can be further reduced by two additional considerations elaborated in Ref. [647]. First, the intermediate operator products which do not contribute to the final DKH Hamiltonian can be neglected. For example, in the fcth step the matrix is multiplied to an intermediate M of order I in the potential. If fc - - 21 n k + l and M is even, the multiplication with Wfc can be skipped, because the intermediate term, which is the product of and M/, is odd and then does not contribute to the nth order DKH Hamiltonian. The further multiplication to yields an even matrix but goes beyond nth order. Second, the DKH Hamiltonian matrix is taken from the upper part of the transformed four-component Hamiltonian matrix while the lower part is not required. For instance, if fc + Z = n and M/ is odd, the product with the odd operator W/t contributes to the final DKH Hamiltonian, but the matrix multiplications to obtain the lower part result can be neglected. The symmetry of the matrices can also be exploited [647]. Noting that the odd matrices O are hermitean, = O, and the matrices Wjt are antihermitean, = —W/t- The algorithm requires the evaluation of their commutator. 

The resulting numbers of matrix multiplications required for the construction of the DKH Hamiltonian of orders 2 to 14 are listed in Table 14.1 [647]. 

The number of multiplications for low-order DKH is surprisingly small. For example, DKH2 requires only one and DKH3 requires three more. The explicit formulae for the relevant even terms 2 and 3 in DKH2 and DKH3 are 

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