The Douglas-Kroll-Hess transformation

The Douglas-Kroll-Hess (DKH) transformation [52, 53] is closely related to the FW transformation. In a non-expanded own the two transformations are identical, but while in the FW context one expands in powers of the natural perturbation parameter one now expands in powers of the external potential (or the nuclear charge). 

Although it is usually not done so, one can present the DKH transformation in the following way [51]. We proceed as in sec. 3.1, but we introduce the expansion parameter and expand in terms of this. We start with 

We count orders in by a superscript. If the potential is explicitly given as e.g. V = —Z/r, on may as well use Z as expansion parameter. 

Obviously is a free-particle Hamiltonian, since V enters earliest to first order in We have further  

In the context of the FW transformation (section 2.7), in order to evaluate the Xk, we had to invert commutators with / mc or equivalently anticommutators with mc, which is quite trivial. The inversion of the anticommutators with mc + in (201) or (202) is much harder. It can either be done exactly in a momentum representation, or approximately by diagonalizing the matrix representation of this operator in a given basis. 

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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. 

Since it is actually impossible to sum the infinite series, this summation is terminated at particular numbers of unitary transformations 2 in DK2, 3 in DK3 and so forth. This method is also called the Douglas-Kroll-Hess transformation, because it was revised by Hess and coworkers (Jansen and Hess 1989). 

Chapter 11 introduced the basic principles for elimination-of-the-small-component protocols and noted that the Foldy Wouthuysen scheme applied to one-electron operators including scalar potentials yield ill-defined 1 /c-expansions of the desired block-diagonal Hamiltonian. In contrast, the Douglas Kroll-Hess transformation represents a unique and valid decoupling protocol for such Hamiltonians and is therefore investigated in detail in this chapter. 

For the discussion of the Douglas-Kroll-Hess transformation in chapter 12 the Fourier transformation of a product of two functions, h x) = f(x)gix), has been employed. In one dimension it is given by the convolution integral of the Fourier transformations of / and g. 

For the discussion of the Douglas-Kroll-Hess transformation in chapter 12 it has been advantageous to consider the momentum-space representation of the Coulomb potential, which may be obtained via a Fourier transformation of V (r). It is given by... 

J. C. Boettger. Approximate two-electron spin-orbit coupling term for density-functional-theory DFT calculations using the Douglas-Kroll-Hess transformation. Phys. Rev. B, 62(12) (2000) 7809-7815. 

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. 

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]. 

An expansion in terms of V, i.e., the Douglas-Kroll-Hess expansion, is the only valid analytic expansion technique for the Dirac Hamiltonian, where the final block-diagonal Hamiltonian is represented as a series of regular even terms of well-defined order in V, which are all given in closed form. For the derivation, the initial transformation step has necessarily to be chosen as the closed-form, analytical free-particle Foldy-Wouthuysen transformation defined by Eq. (11.35) in order to provide an odd term depending on the external potential that can then be diminished. We now address these issues in the next chapter. 

R. Fukuda, M. Hada, H. Nakatsuji. Quasirelativistic theory for the magnetic shielding constant. I. Farmulation of Douglas-Kroll-Hess transformation for the magnetic field and its application to atomic systems. J. Chem. Phys., 118(3) (2003) 1015-1026. 

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. 

The transformed Hamiltonians that we have derived allow us to calculate intrinsic molecular properties, such as geometries and harmonic frequencies. We would like to be able to calculate response properties as well, with wave functions derived from the transformed Hamiltonian. If we used a method such as the Douglas-Kroll-Hess method, it would be tempting to simply evaluate the property using the nonrelativistic property operators and the transformed wave function. As we saw in section 15.3, the property operators can have a relativistic correction, and for properties sensitive to the environment close to the nuclei where the relativistic effects are strong, these corrections are likely to be significant. To ensure that we do not omit important effects, we must derive a transformed property operator, starting from the Dirac form of the property operator. 

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... 

The modified two-electron terms contain all the relativistic integrals, which means that the integral work is no different from that in the full solution of the Dirac-Hartree-Fock equations. It would save a lot of work if we could approximate the integrals, in the same way as we did for the Douglas-Kroll-Hess approximation. To do so, we must use the normalized Foldy-Wouthuysen transformation. The DKH approximation neglects the commutator of the transformation with the two-electron Coulomb operator, and in so doing removes all the spin-dependent terms. We must therefore also use a spin-free one-electron Hamiltonian. The approximate Hamiltonian (in terms of operators rather than matrices) is... 

A regular alternative to the Foldy-Wouthuysen transformation was given by Douglas and Kroll and later developed for its use in electronic structure calculations by Hess et al. The Douglas-Kroll (DK) transformation defines a transformation of the external-field Dirac Hamiltonian Hq of equation (11) to two-component form which leads, in contrast to the Foldy-Wouthuysen transformation, to operators which are bounded from below and can be used variationally, similarly to those of the regular approximations discussed above. As in the FW transformation, it is not possible in the DK formalism to give the transformation in closed form. Rather, it is... 

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. 

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... 

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