Kroll Transformation

The singularity that may arise when using the Breit-Pauli Hamiltonian can cause instability in variational calculations and it is desirable that such singularity be avoided formally. Douglas and KrolE showed that the singularity 

here s and O are even and odd operators, respectively, as we have seen in the Foldy-Wouthuysen transformation. The next step is to choose other unitary operators and apply them successively such that 

In most current applications of the Douglas-Kroll transformation, the Hamiltonian is truncated at second order in the successive unitary transformation, that is, Ui, and the resulting Hamiltonian can be written in 

The spin-dependent part of Eq. [83] is the one-electron and two-electron spin-orbit operators. 

The spin-orbit operators in the Douglas-Kroll Hamiltonian still contain the 1/r term however, that term is offset by the l/( , -b term, where as 

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Since working with the full four-component wave function is so demanding, different approximate methods have been developed where the small component of the wave function is eliminated to a certain order in 1/c or approximated (like the Foldy-Wouthuyserd or Douglas-Kroll transformations thereby reducing the four-component wave function to only two components. A description of such methods is outside the scope of this book. 

Hess, B.A. and Kaldor, U. (2000) Relativistic all-electron coupled-cluster calculations on Au2 in the framework of the Douglas—Kroll transformation. Journal of Chemical Physics, 112, 1809-1813. 

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]. 

The second major method leading to two-component regular Hamiltonians is based on the Douglas-Kroll transformation (Douglas and Kroll 1974 Hess 1986 Jansen and Hess 1989). The classical derivation makes use of two successive unitary transformations... 

It is clear from Ho that the Douglas-Kroll transformation makes use of a model space of relativistic free-particle spinors, and that it is defined by a perturbative expansion with the external potential as perturbation. Indeed, using the formulas given above, we get the familiar expressions for the second-order Douglas-Kroll-transformed Dirac operator, which is often dubbed Douglas-Kroll-Hess (DKH) operator... 

If a multiparticle system is considered and the election interaction is introduced, we may use the Dirac-Coulomb-Breit (DCB) Hamiltonian which is given by a sum of one-particle Dirac operators coupled by the Coulomb interaction 1 /r,7 and the Breit interaction Bij. Applying the Douglas-Kroll transformation to the DCB Hamiltonian, we arrive at the following operator (Hess 1997 Samzow and Hess 1991 Samzow et al. 1992), where an obvious shorthand notation for the indices pi has been used ... 

The Douglas-Kroll transformation can be carried out to higher orders, if desired (Barysz et al. 1997). In this way, arbitrary accuracy with respect to the eigenvalues of D can be achieved. 

Hamiltonian resulting from the Douglas-Kroll transformation is particularly indicated for heavy elements and in variational calculations, because it is bounded from below (Samzow et al. 1992). The raw integrals are by now combined with AO and MO information from a variety of standard program packages (Molecule-Sweden, Columbus, Turbomole) and SOMF integrals are provided for BnSoc, Columbus, Molcas and LuciaRel. 

Since even terms are always block-diagonal, they can, for later convenience at the discussion of the Douglas-Kroll transformation, always be decomposed into their diagonal (2 x 2)-blocks,... 

The decoupling of the Dirac Hamiltonian Hd in the framework of the Doug-las-Kroll transformation is achieved by a systematic expansion of the Hamiltonian in ascending powers of the external potential V, whereby odd terms are systematically removed step by step. This procedure imposes the restriction that this expansion shall always be possible on a suitably chosen domain of the... 

DK approximation and it will be shown that the result is independent of the chosen parametrisation. This approach has not been investigated in the literature so far. We will denote the resulting operator equations as the generalised Douglas-Kroll transformation. We conclude this section by a presentation of some technical aspects of the implementation of the DK Hamiltonian into existing quantum chemical computer codes. 

Douglas-Kroll transformation to fifth and higher order in the external potential is subject to further investigations. 

The Douglas-Kroll transformation [40] of the Dirac-Coulomb Hamiltonian in its implementation by HeB [41-45] leads to one of the currently most successful and popular forms of a relativistic no-pair Hamiltonian. The one-electron terms of the Douglas-Kroll-HeB (DKH) Hamiltonian have the form... 

Extended Douglas—Kroll transformations applied to the relativistic many-electron Hamiltonian... 

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]. 

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. 

Before leaving the theoretical formalism section, it is important to note that perturbation theory for relativistic effects can also be done at the fo n-con onent level, i.e. before elimination of the small component by a Foldy-Wouthitysen (FW) or Douglas-Kroll transformation. This is best done with direct perturbation theory (DPT) [71]. DPT involves a change of metric in the Dirac equation and an expansion of this modified Dirac eqtiation in powers of c . The four-component Levy-Leblond equation is the appropriate nonrelativistic limit. Kutzelnigg [72] has recently worked out in detail the simultaneous DPT for relativistic effects and magnetic fields (both external and... 

DKee Douglas-Kroll (transformation) including electron-electron... 

The reduction of the relativistic many-electron hamiltonian by expansion in powers of the external field is the second-order Douglas-Kroll transformation [29], and has been used with success by Hess and co-workers [30]. The operators which result from this transformation are non-singular, but the integrals over the resulting operators are complicated and have to be approximated, even for finite basis set expansions. The reduction of the Dirac-Coulomb-Breit equation to two-component form using direct perturbation theory has been described by Kutzelnigg and coworkers [26, 27, 31], Rutkowski [32], and van Lenthe et al. [33]. 

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