The Douglas-Kroll Method

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 

1-electron Hilbert space It requires the construction of a sequence of unitary transformations Ui, i = 1,2,3.) which eliminate the lowest-order odd term Oi in the ith step in order to arrive at the block-diagonal Hamiltonian H,  

The original idea of this procedure dates back to 1974 and is due to Douglas and ECroll [56]. In the following years it was brought to the attention of the community and developed to a powerful computational tool for relativistic quantum chemistry [57,58]. 

It should be mentioned that there are only a few restrictions on the choice of the matrices C/j. Firstly they have to be unitary and analytical (holomorphic) functions on a suitable domain of, and secondly they have to permit a decomposition of Hm in even terms of definite order in the external potential according to Eq. (73). It is thus possible to parametrise them without loss of generality by a power series expansion in an odd and antihermitean operator Wi of ith order in the external potential. In the following, the physical consequences of this freedom in the choice of the unitary transformations will be investigated. Therefore we shall start with a discussion of all possible parametrisations in terms of such power series expansions. Afterwards the most general parametrisation of Ui is applied to the Dirac Hamiltonian in order to derive the fourth-order 

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. 

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Malkin, L, Malkina, O.L. and Malkin, V.G. (2002) Relativistic calculations of electric field gradients using the Douglas—Kroll method. Chemical Physics Letters, 361, 231-236. 

By using the general power series expansion for U all the infinitely many parametrisations of a unitary transformation are treated on equal footing. However, the question about the equivalence of these parametrisations for application in the Douglas-Kroll method, which represents a crucial point, is more subtle and will be analysed in the next section. It is especially not clear a priori, if the antihermitean matrix W can always be chosen in the appropriate way the mandatory properties of W, i.e., its oddness, antihermiticity and behaviour as a correct power in the external potential, have to be checked for every single transformation Ui of Eq. (73). 

The calculations were performed with several different levels of correlation treatment Hartree-Fock (HF), configuration interaction with single and double excitations (SDCI), Multiconfiguration self consistent field (CAS), and multireference configuration interaction (MRCI). Relativistic efferts were accounted for using either the Douglas-Kroll method or a relativistic effective core potential approach (RECP). 

We extend the method over all three rows of TMs. No systematic study is available for the heavier atoms, where relativistic effects are more prominent. Here, we use the Douglas-Kroll-Hess (DKH) Hamiltonian [14,15] to account for scalar relativistic effects. No systematic study of spin-orbit coupling has been performed but we show in a few examples how it will affect the results. A new basis set is used in these studies, which has been devised to be used with the DKH Hamiltonian. 

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. 

Thakkar and Lupinetti5 have used the coupled-cluster method in conjunction with the Douglas-Kroll relativistic Hamiltonian to obtain a very accurate value for the static dipole polarizability of the sodium atom. Their revised value for a(Na) = 162.88 0.6 au resolves a previous discrepancy between theory and experiment and when combined with an essentially exact value for lithium, establishes the ratio a(Li)/a(Na) = 1.0071 0.0037, so that, because of the... 

Accounting for relativistic effects in computational organotin studies becomes complicated, because Hartree-Fock (HF), density functional theory (DFT), and post-HF methods such as n-th order Mpller-Plesset perturbation (MPn), coupled cluster (CC), and quadratic configuration interaction (QCI) methods are non-relativistic. Relativistic effects can be incorporated in quantum chemical methods with Dirac-Hartree-Fock theory, which is based on the four-component Dirac equation. " Unformnately the four-component Flamiltonian in the all-electron relativistic Dirac-Fock method makes calculations time consuming, with calculations becoming 100 times more expensive. The four-component Dirac equation can be approximated by a two-component form, as seen in the Douglas-Kroll (DK) Hamiltonian or by the zero-order regular approximation To address the electron cor-... 

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

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... 

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]. 

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. 

Scalar relativistic effects (e.g. mass-velocity and Darwin-type effects) can be incorporated into a calculation in two ways. One of these is simply to employ effective core potentials (ECPs), since the core potentials are obtained from calculations that include scalar relativistic terms [50]. This may not be adequate for the heavier elements. Scalar relativity can be variationally treated by the Douglas-Kroll (DK) [51] method, in which the full four-component relativistic ansatz is reduced to a single component equation. In gamess, the DK method is available through third order and may be used with any available type of wavefunction. 

Another group of methods successfully used for calculations of the electronic structures of the heaviest element molecules are effective core potentials (ECP) (see the Chapters of M. Dolg and Y.-S. Lee in these issues). The relativistic ECPs (RECP) were applied to calculations of the electronic structures of halides and oxyhalides of Rf and Sg and of some simple compounds (mostly hydrides and fluorides) of elements 113 through 118 [126-131]. Using energy-adjusted pseudo-potentials (PP) [132] electronic structures and properties, and the influence of relativistic effects were studied for a number of compounds of elements at the end of the 6d series (elements 111 and 112), as well as at the beginning of the 7p series (elements 113 and 114) (see Refs. 26 and 133 for reviews and references therein). Some other methods, like the Douglas-Kroll-Hess (DKH) [134], were also used for calculations of small heaviest-element species (e.g. IIIH [95]). 

These problems can be solved if one starts from the (untruncated ) Foldy-Wouthuysen transformation for a free particle, the only case for which the transformation is known anal3d ically, and incorporates the effects of the external potential on top. Along these lines, the so-called Douglas-Kroll-HeB (DKH) method [61-64] is constructed which is probably the most successful quasi-relativistic method in wave function based quantum chemistry. No details will be given here since this topic has been extensively discussed in volume 1 [34] of this series. Meanwhile several density functional implementations exist based on the Douglas-Kroll-HeJ3 approach [39-45]. In recent years. 

Quantum chemistry with the Douglas-Kroll-Hess approach to relativistic density functional theory Efficient methods for molecules and materials... 

We review the Douglas-Kroll-Hess (DKH) approach to relativistic density functional calculations for molecular systems, also in comparison with other two-component approaches and four-component relativistic quantum chemistry methods. The scalar relativistic variant of the DKH method of solving the Dirac-Kohn-Sham problem is an efficient procedure for treating compounds of heavy elements including such complex systems as transition metal clusters, adsorption complexes, and solvated actinide compounds. This method allows routine ad-electron density functional calculations on heavy-element compounds and provides a reliable alternative to the popular approximate strategy based on relativistic effective core potentials. We discuss recent method development aimed at an efficient treatment of spin-orbit interaction in the DKH approach as well as calculations of g tensors. Comparison with results of four-component methods for small molecules reveals that, for many application problems, a two-component treatment of spin-orbit interaction can be competitive with these more precise procedures. 

The Douglas-Kroll approach to relativistic electronic structure theory in the framework of density functional theory was reviewed focussing on recent method developments and illustrative applications which demonstrate the capabilities of this approach. Compared to other relativistic methods, which often are only applied to small molecules for demonstration purposes, the DK approach has been used in a variety of fields. Besides the very popular pseudopotential approach, which accounts for relativistic effects by means of a potential replacing the core electrons, until now the scalar relativistic variant of the second-... 

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). 

The lOTC method presented in this review is obviously related to earlier attempts to reduce the four-component Dirac formalism to computationally much simpler two-component schemes. Among the different to some extent competitive methods, the priority should be given to the Douglas-Kroll-Hess method [13,53,54]. It was Bernd Hess and his work which was our inspiration to search for the better solutions to the two-component methodology. 

The very large nonrelativistic electron correlation contribution at the MP2 level is rather disturbing. An interpretation of this pattern has been reported in ref. [15], The CCSD and CCSD(T) methods lead to more correct energy denominators and diminish the electron correlation correction. However, the correlation correction to Pzzz remains large. The magnitude, in absolute value, of the Douglas -Kroll (DK) [16, 17] relativistic correction increases with the atomic number of the metal for HgS is quite significant, it is comparable with the correlation contribution (Table 5.2). 

The scalar relativistic (SR) corrections were calculated by the second-order Douglas-Kroll-Hess (DKH2) method [53-57] at the (U/R)CCSD(T) or MRCI level of theory in conjunction with the all-electron aug-cc-pVQZ-DK2 basis sets that had been recently developed for iodine [58]. The SR contributions, as computed here, account for the scalar relativistic effects on carbon as well as corrections for the PP approximation for iodine. Note, however, that the Stuttgart-Koln PPs that are used in this work include Breit corrections that are absent in the Douglas-Kroll-Hess approach [58]. 

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