Unitary series expansion

Evidently, this fomuila is not exact if fand vdo not connnute. However for short times it is a good approximation, as can be verified by comparing temis in Taylor series expansions of the middle and right-hand expressions in (A3,11,125). This approximation is intrinsically unitary, which means that scattering infomiation obtained from this calculation automatically conserves flux. 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order... 

The most general ansatz to construct a unitary transformation U = fiW) as an analytical function of an antihermitean operator W is a power series expansion,... 

By using the general power series expansion for U all the infinitely many parametrisations of a unitary transformation are treated on equal footing. However, the question about the equivalence of these parametrisations for application in the Douglas-Kroll method, which represents a crucial point, is more subtle and will be analysed in the next section. It is especially not clear a priori, if the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behaviour as a correct power in the external potential, have to be checked for every single transformation Ui of Eq. (73). 

Table 11.1 Coefficients of the power series expansion of the unitary transformation U for five different parametrizations [606,611]. The first two coefficients have been fixed to be o = = 1- Note that all coefficients are only given with an accuracy of three digits after the...

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. 

For truncated expansions of U the optimal parametrization behaves significantly better than all other choices for the coefficients flj., as is clearly demonstrated by the deviations of U from unitarity presented in Table 11.2. Truncation of any power series applied in decoupling transformations necessarily requires that the operator norm of W is smaller than unity to guarantee convergence, since otherwise the higher-order terms in W would dominate the expansion. This implies that an even better performance of the optimum unitary parametrization as compared to all other choices for the coefficients is gained than the data in Table 11.2 suggest. 

Consequently, a high- or even infinite-order DKH Hamiltonian requires a predefinition of the order in V which determines the Taylor expansion length of all individual unitary matrices in the sequence. Before we come back to this issue, the convergence of the DKH series needs to be discussed. 

This energy is to be expanded as a perturbation series in 1/c and the appropriate functionals varied subject to the unitary normalization conditions to arrive at stationarity conditions. The stationarity conditions naturally lead to the DHF equations, so we expect the variation of the perturbation functionals to involve an expansion of the DHF operators. 

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