Relativistic two-component

In this Chapter, we will show how a whole family of one- and two-component quasi-relativistic Hamiltonians can conveniently be derived. The operator difference between the quasi-relativistic Hamiltonians and the Dirac equation can be explicitly identified and used in perturbation expansions. Expressions are derived for a direct perturbation theory scheme based on quasi-relativistic two-component Hamiltonians. The remaining difference between the variational energy obtained using quasi-relativistic Hamiltonians and the energy of the Dirac equation is estimated numerically by applying the direct perturbation theory ap-... 

With equation (20) as starting point, the first approximation one can make in order to derive quasi-relativistic two-component equations is to assume that the upper (0 ) and the lower (0y) components are identical. Note that the ansatz... 

Let us first discuss the usual spin-orbit pseudopotentials, which can be defined in a general way via relativistic two-component pseudopotentials Uf r). They originate fi om the definition of a Schrodinger-like valence model Hamiltonian in a two-component form shown here for an atom... 

Several excellent reviews of the relativistic two-component methods have recently appeared [12-15]. This review is not aimed at the completeness of the... 

We could continue now with the Dirac equation and derive expressions for the molecular properties using standard perturbation theory. However, as stated earlier, the exposition in these notes is restricted basically to a non-relativistic treatment with the exception that we want to include also interactions with the spin of the electrons. The appropriate operator can be found by reduction of the Dirac equation to a non-relativistic two-component form, which can be achieved by several approaches. Here, we want to discuss only the simplest approach, the so-called elimination of the small component. 

In the case of relativistic two-component PPs, the inclusion of SO effects requires some modification of the analytical PP form. At the Dirac-Hartree-Fock (DHF) level, the degeneracy of the orbitals is reduced and depends in addition to n and / also on the total angular momentum quantum number j implying a semilocal PP with a //-dependency [33]... 

Malkin I, Malkina OL, Malkin VG and Kaupp M 2005 Relativistic two-component calculations of electronic g-tensors that include spin polarization. J. Chem. Phys. 123(24), 244103-16. 

Hess, B.A. (1986) Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators. Physical Review A, 33, 3742-3748. 

Barysz, M. and Sadlej, A.J. (2002) Infinite-order two-component theory for relativistic quantum chemistry. Journal of Chemical Physics, 116, 2696-2704. 

Ilias, M. and Saue, T. (2007) An infinite-order two-component relativistic Hamiltonian by a simple one-step transformation. Journal of Chemical Physics, 126, 064102-1-064102-9. 

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

At this point, it is appropriate to present a brief discussion on the origin of the FC operator (d function) in the two-component form (Pauli form) of the molecular relativistic Hamiltonian. Many textbooks adopt the point of view that the FC is a relativistic effect, which must be derived from the Dirac equation [50,51]. In other textbooks or review articles it is stressed that the FC is not a relativistic effect and that it can be derived from classical electrodynamics [52,53] disregarding the origin of the gyromagnetic factor g—2. In some textbooks both derivations are presented [54]. The relativistic derivations suffer from the inherent drawbacks in the Pauli expansion, in particular that the Pauli Hamiltonian can only be used in the context of the first-order perturbation theory. Moreover, the origin of the FC term appears to be different depending on whether one uses the ESC method or FW transformation. 

Earlier we mentioned briefly that the electron spin is perfectly consistent with the non-relativistic four-component Levy-Leblond theory [44,45]. The EC type interaction does not manifest in Dirac or Levy-Leblond theory. We shall show that on reducing the four-component Levy-Leblond equation into a two-component form the EC contribution arises naturally. A non-relativistic electron in an electromagnetic radiation field is described by the Levy-Leblond equation given by... 

The only calculation we found for CdH is the work of Balasubramanian [68], using Cl with relativistic effective core potentials. The coupled-cluster results are presened in Table 6. Calculated values for R , cOg and Dg agree very well with experiment. Relativity contracts the bond by 0.04 and reduces the binding energy by 0.16 eV. The one- and two-component DK method reproduce the relativistic effects closely. Similar trends are observed for the excited states (Tables 7-9). Comparison with experiment is difficult for these states, since many of the experimental values are based on incomplete or uncertain data [65]. Calculated results for the CdH anion are shown in Table 10. The... 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

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