Douglas-Kroll higher-order

Apart from primary structural and energetic data, which can be extracted directly from four-component calculations, molecular properties, which connect measured and calculated quantities, are sought and obtained from response theory. In a pilot study, Visscher et al. (1997) used the four-component random-phase approximation for the calculation of frequency-dependent dipole polarizabilities for water, tin tetrahydride and the mercury atom. They demonstrated that for the mercury atom the frequency-dependent polarizability (in contrast with the static polarizability) cannot be well described by methods which treat relativistic effects as a perturbation. Thus, the varia-tionally stable one-component Douglas-Kroll-Hess method (Hess 1986) works better than perturbation theory, but differences to the four-component approach appear close to spin-forbidden transitions, where spin-orbit coupling, which the four-component approach implicitly takes care of, becomes important. Obviously, the random-phase approximation suffers from the lack of higher-order electron correlation. 

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. 

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

One-component calculations or two-component calculations including also spin-orbit coupling effects provide a firm basis for the calculations of higher-order relativistic corrections by means of perturbation theory. Several quasi-relativistic approximations have been proposed. The most successful approaches are the Douglas-Kroll-Hess method (DKH) [1-7], the relativistic direct perturbation theory (DPT) [8-24], the zeroth-order regular approximation (ZORA) [25-48], and the normalized elimination of small components methods (NESC) [49-53]. Related quasi-relativistic schemes based on the elimination of the small components (RESC) and other similar nonsingular quasi-relativistic Hamiltonians have also been proposed [54-61]. 

We have developed two quasi-relativistic approaches. One is the RESC method [147-149], and the other is the higher order Douglas-Kroll (DK) method [150-152]. In the following sections we will introduce RESC and higher order DK methods briefly. 

The Douglas-Kroll (DK) approach [153] can decouple the large and small components of the Dirac spinors in the presence of an external potential by repeating several unitary transformations. The DK transformation is a variant of the FW transformation [141] and adopts the external potential Vg t an expansion parameter instead of the speed of light, c, in the FW transformation. The DK transformation correct to second order in the external potential (DK2) has been extensively studied by Hess and co-workers [154], and has become one of the most familiar quasi-relativistic approaches. Recently, we have proposed the higher order DK method and applied the third-order DK (DK3) method to several systems containing heavy elements. 

Notice that the additional term only involves VVi. This is an instance of the familiar 2n- -1 rule of perturbation theory. Here, the operators up to VV are all that are needed to determine the Hamiltonian of order 2n - -1. Higher-order transformations have also been derived and examined by Wolf et al. (2002), to which the reader is referred for details. The Douglas-Kroll Hamiltonian of order n is often written as Hdkii or... 

This is a scalar term the spin-dependent terms that are second order in the potential have a higher leading power of l(mc). The Douglas-Kroll correction including the nuclear potential term can be derived from the lowest-order part of the term,... 

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