Generalized Douglas-Kroll Transformation

A. Wolf, M. Reiher, B. Hess, Two-component methods and the generalized Douglas-Kroll transformation, in P. Schwerdtfeger (Ed.), Relativistic Electronic Structure Theory, Part 1, Fundamentals, Elsevier, Netherlands, 2002, pp. 627-668. 

Reiher, M. and Wolf A. (2004) Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas—Kroll—Hess transformation up to arbitrary order. Journal of Chemical Physics, 121, 10945-10956. 

There are many problems in e.g. catalysis in which relativity may play a deciding role in the chemical reactivity. These problems generally involve large organic molecules which cannot be handled within the Dirac Fock framework. It is therefore necessary to reduce the work by making additional approximations. Generally used approaches are based on the Pauli expansion or on the Douglas Kroll transformation [3]. 

General two-component methods have been discussed in various chapters of the first part of this book, for instance in chapter 11 on Two-Component Methods and the Generalised Douglas-Kroll Transformation by Wolf, Reiher and Hess [165], in chapter 12 by Kutzelnigg on Perturbation Theory of Relativistic Effects [166] and in chapter 13 by Sundholm on Perturbation Theory Based on Quasi-Relativistic Hamiltonians [167]. 

Molecules are more difficult to treat accurately than atoms, because of the reduced symmetry. An additional complication arises in relativistic calculations the Dirac-Fock-(-Breit) orbitals will in general be complex. One way to circumvent this difficulty is by the Douglas-Kroll-Hess transformation [57], which yields a one-component function with computational effort essentially equal to that of a nonrelativistic calculation. Spin-orbit interaction may then be added as a perturbation, implementation to AuH and Au2 has been reported [58]. Progress has also been made in the four-component formulation [59], and the MOLFDIR package [60] has been extended to include the CC method. Application to SnH4 has been described [61] here we present a recent calculation of several states of CdH and its ions [62], with one-, two-, and four-component methods. 

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). 

Of course, what has just been stated for the one-electron Dirac Hamiltonian is also valid for the general one-electron operator in Eq. (11.1). However, the coupling of upper and lower components of the spinor is solely brought about by the off-diagonal ctr p operators of the free-partide Dirac one-electron Hamiltonian and kinetic energy operator, respectively. We shall later see that the occurrence of any sort of potential V will pose some difficulties when it comes to the determination of an explicit form of the unitary transformation U. A universal solution to this problem will be provided in chapter 12 in form of Douglas-Kroll-Hess theory. 

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