Douglas-Kroll-Hess

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. 

Accounting for Relativistic Effects within the Douglas-Kroll-Hess Theory... 

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. 

Bioinorganic systems often contain heavy elements that need to be treated with an explicit relativistic method. It is now possible to carry out an explicit relativistic electronic structure calculation on the fly (152). The scalar-relativistic Douglas - Kroll - Hess method was implemented by us recently in the BOMD simulation framework (152). To use the relativistic densities in a non-relativistic gradient calculations turned out to be a valid approximation of relativistic gradients. An excellent agreement between optimized structures and geometries obtained from numerical gradients was observed with an error smaller than 0.02 pm. 

DKH2 Second order Douglas-Kroll-Hess transformation... 

In a related study Ilias, Furdik and Urban have calculated FCu, FAg and FAu using the CCSD(T) method and considering relativistic effects by the nopair one-component Douglas-Kroll-Hess approximation. These are stable diatomic molecules in the S ground state with the bonding primarily arising from a s orbital formed by the 2p valence orbital of F and the ns valence orbital of the metal. 

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. 

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. 

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... 

Mayer, M. (1999) A parallel density functional method implementation of the two-component Douglas-Kroll-Hess method and applications to relativistic effects in heavy-element chemistry. PhD thesis, Technical University of Munich. 

Rosch, N., Kriiger, S., Mayer, M. andNasluzov.V. A. (1996) The Douglas-Kroll-Hess approach to relativistic density functional theory Methodological aspects and applications to metal complexes and clusters. In Recent Developments and Applications of Modem Density Functional Theory (ed. J. M. Seminario), pp. 497-566. Elsevier. 

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). 

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 formalism described here to derive energy-consistent pseudopotentials can be used for one-, two- and also four-component pseudopotentials at any desired level of relativity (nonrelativistic Schrbdinger, or relativistic Wood-Boring, Douglas-Kroll-Hess, Dirac-Coulomb or Dirac-Coulomb-Breit Hamiltonian implicit or explicit treatment of relativity in the valence shell) and electron correlation (single- or multi-configurational wavefunctions. The freedom... 

---