The Douglas-Kroll Transformation

If the expansion in powers of p/mc produces operators that can only be used in perturbation theory, is there an expansion that will produce operators that can be used variationally Such a transformation would have to provide an expansion in powers of the potential energy rather than the momentum. The Douglas-Kroll transformation 

The first step is the application of the free-particle Foldy-Wouthuysen transformation to the Dirac operator. As in the previous section, we use the Dirac Hamiltonian without subtracting the rest mass, 

Applying this to the Dirac Hamiltonian, and comparing to the free-particle transformed Hamiltonian of (16.20) above, we see that the only additional terms are those involving 

We can split the 1 2 2 term into scalar and spin-orbit operators, which makes it possible to define a spin-free one-electron Hamiltonian. The two-component Hamiltonian for the positive-energy states can then be written 

The Hamiltonian 1-C is bounded from below, and can therefore be used variationally. 

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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 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... 

U. Kaldor and B. A. Hess, Ghent. Phys. Lett., 230, 1 (1994). Relativistic All-Electron Coupled-Cluster Calculations on the Gold Atom and Gold Hydride in the Framework of the Douglas-Kroll Transformation. 

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 complicated nature of the operator VVi raises the question of whether a simpler alternative can be found. Barysz, Sadlej, and Snijders (1997) devised an approach which starts from the free-particle Foldy-Wouthuysen transformation, just as the Douglas-Kroll transformation does. Whereas the Douglas-Kroll approach seeks to eliminate the lowest-order odd term from the transformed Hamiltonian, their approach seeks to be correct to a particular order in 1 /c, and it provides a ready means for defining a sequence of approximations of increasing order in 1/c. It is important to note that, while the expansion in powers of 1 /c that resulted from the Foldy-Wouthuysen transformation in section 16.2 generates highly singular operators, this is not true per se of expansions in 1/c. What multiplies 1/c is all-important. In the case of the Foldy-Wouthuysen transformation it is and due to the fact that p can become... 

Equation (16.69) can now solved by an iterative procedure, making use of the orders of the various terms in powers of 1/c to define a procedure that should be convergent. Since (Hi)22 is of order and ( ii)2i is of order the maximum order of Xi is c (as was found for >Vi in the Douglas-Kroll transformation). The three terms on the right-hand side of (16.69) are then of order c, c, and c, respectively. The iteration is started by neglecting the second and third terms of (16.69). As H )22 is of order c, this gives a transformation that is correct to order c ... 

This is the equivalent of the second-order Douglas-KroU operator, but it only involves operators that have been defined in the free-particle Foldy-Wouthuysen transformation. As for the Douglas-Kroll transformed Hamiltonian, spin separation may be achieved with the use of the Dirac relation to define a spin-fi ee relativistic Hamiltonian, and an approximation in which the transformation of the two-electron integrals is neglected, as in the Douglas-Kroll-Hess method, may also be defined. Implementation of this approximation can be carried out in the same way as for the Douglas-Kroll approximation both approximations involve the evaluation of kinematic factors, which may be done by matrix methods. 

In this section, we use W for the electric perturbation. It should not be confused with the operators Wn used in the Douglas-Kroll transformation. 

The other transformations that we have considered, the Douglas-Kroll transformation and the Barysz-Sadlej-Snijders transformation, start with a free-particle Foldy-Wouthuysen transformation. This transformation is independent of V and it is a simple matter to separate the perturbation. The transformed perturbation operator can be written down directly from (16.44)... 

To illustrate, we use the second-order Barysz-Sadlej-Snijders transformation, which is more transparent than the Douglas-Kroll transformation. Introducing a perturbation parameter X, the Hamiltonian including the electric perturbation is... 

We could of course proceed as we did for the Douglas-Kroll transformation and use a transformation that depends only on the nuclear potential. This would remove the awkwardness of having 1 /r,y in the denominator, but we still have the product of c f(2mc — Vi) with 1 /r,-y to deal with. If we are only interested in spin-free relativistic effects, we could neglect the transformation of the electron-electron interaction, as we did in the Douglas-Kroll-Hess approximation. This approximation yields the Hamiltonian... 

Hess, B. A. (1989). Revision of the Douglas-Kroll transformation. Physical Review A, 39, 6016-6017. 

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