Douglas-Kroll-transformed Hamiltonian

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. 

Here we see a modified kinetic energy term that is cut off near the nuclei, a spin-free relativistic correction, and a spin-orbit term, both of which are regularized and behave as 1/r for small r. We may compare this with the regularization of the free-particle Foldy-Wouthuysen or Douglas-Kroll transformed Hamiltonian of section 16.3. The regularization clearly corrects the overestimation of relativistic effects that plagues the Pauli Hamiltonian. There is another consequence of the small r behavior. Since the relativistic correction operator behaves like 1 jr, it ought to be possible to use T zoRA variationally—and in fact we may demonstrate that there is a variational lower bound. 

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... 

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 ... 

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. 

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. 

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]. 

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]. 

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... 

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... 

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. 

Operators that result from a DK transformation are directly given in the momentum representation. Hess et al. [29,31] developed a very efficient strategy to evaluate the corresponding matrix elements in a basis set representation it employs the eigenvectors of the operator as approximate momentum representation [29,31]. In practice, the two-component DK Hamiltonian is built of matrix representations of the three operators p, V, pVp + id(pV x p). This Douglas-Kroll-Hess (DKH) approach became one of the most successful two-component tools of relativistic computational chemistry [16,74]. In particular, many applications showed that the second-order operator 2 Is variationally stable [10,13,14,31,75,76,87]. 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

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