Zeroth order regular approximation for relativistic effects

Within the DFT framework, we apply two different approaches to deal with relativistic effects, the so-called quasi-relativistic (QR) method (73) and the more modem "Zeroth Order Regular Approximation for Relativistic Effects" (ZORA) (14-16). The QR method is also known as the Pauli approach. 

Wolff, S. K., Ziegler, T., van Lenthe, E., Baerends, E. J., 1999, Density Functional Calculations of Nuclear Magnetic Shieldings Using the Zeroth-Order Regular Approximation (ZORA) for Relativistic Effects ZORA Nuclear Magnetic Resonance , J. Chem. Phys., 110, 7689. 

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. 

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]. 

Three quasi-relativistic approaches that are variationally stable are the Doug-lass-Kroll-Hess transformation of the no-pair Hamiltonian (for example, see Ref. 11, 20, 23-29), the zeroth order regular approximation, ZORA, (for example, see Ref. 30-34), and the approach of Barysz and Sadlej (for example, see Ref. 36). The results of the first two approaches differ considerably even when used by the same authors,which led them to try the third approach. A calibration study suggests that relativistic effects on heavy atom shieldings are significantly underestimated by ZORA in comparison to the four-component relativistic treatment, but that the neighboring proton chemical shifts are closer to experi-... 

For transition metals from the second and third rows (like Mo and W), the inner electrons start to move at speeds approaching that of light. Consequently, relativistic effects may become sizeable. There are many ways to treat these effects, but the two most common ones for enzyme models are effective core potentials (ECP) or the zeroth-order regular approximation (ZORA). ECP means that the inner electrons are replaced by special one-electron potentials, which actually make the calculations faster. ZORA calculations, on the other hand, require special basis sets and are more demanding than non-relativistic calculations. However, in our test calculations, the two approaches have given similar results, with differences of 0-5 kj mol". ... 

S. K. Wolff, T. Ziegler, E. van Lenthe, E. J. Baerends. Density functional calculations of nudear magnetic shiddings using the zeroth-order regular approximation (ZORA) for relativistic effects ZORA nuclear magnetic resonance. /. Chem. Phys., 110(1999) 7689-7698. 

A truncation of the expansion (3.5) defines the zero- and first-order regular approximation (ZORA, FORA) (van Lenthe et al. 1993). A particular noteworthy feature of ZORA is that even in the zeroth order there is an efficient relativistic correction for the region close to the nucleus, where the main relativistic effects come from. Excellent agreement of orbital energies and other valence-shell properties with the results from the Dirac equation is obtained in this zero-order approximation, in particular in the scaled ZORA variant (van Lenthe et al. 1994), which takes the renormalization to the transformed large component approximately into account, using... 

If we wish to incorporate some level of relativistic effects into the zeroth-order Hamiltonian, we cannot start from Pauli perturbation theory or direct perturbation theory. But can we find an alternative expansion that contains relativistic corrections and is valid for all r that is, can we derive a regular expansion that is convergent for all reasonable values of the parameters The expansion we consider in this chapter has roots in the work by Chang, Pelissier, and Durand (1986) and HeuUy et al. (1986), which was developed further by van Lenthe et al. (1993, 1994). These last authors coined the term regular approximation because of the properties of the expansion. 

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