Zeroth-order regular approximation ZORA

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. 

Autschbach and Ziegler performed the relativistic calculations of spin-spin coupling constants (isotropic part) and anisotropies in heavy atom compounds with the two-component zeroth-order regular approximation (ZORA) method. The experimentally determined reduced spin-spin coupling tensor elements Kjk A,B) between two magnetically active nuclei, A and... 

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]. 

Another method to avoid the singularities in the vicinity of the nuclei was proposed by van Lenthe et al. [26,27,35]. They suggested an method that also includes the interaction potential, V (r), in the denominator of the ansatz for the small component. This ansatz was used in the derivation of the so called zeroth-order regular approximation (ZORA) Hamiltonian. The ZORA ansatz can thus be the written as... 

Figure 1. The Exponential Regular Approximation (ERA with y= 1), the Zeroth-Order-Regular Approximation (ZORA), and the Kinetic-Energy Balance Condition (KEBC) ansatz functions for uranium (Z = 92). The distance R is given in bohrs.

The zeroth-order regular approximation (ZORA) Hamiltonian can be derived from the upper part of the transformed Dirac equation (20). By using the ZORA ansatz for the small component (5) and assuming that the upper and the lower components are equal, the final ZORA equation for the upper component becomes... 

One of the shortcomings of the BP approach is that the expansion in (p/mc) is not justified in the case where the electronic momentum is too large, e.g. for a Coulomb-like potential. The zeroth-order regular approximation (ZORA) [142,143] can avoid this disadvantage by expanding in E/ 2mc — P) up to the first order. The ZORA Hamiltonian is variationally stable. However, the Hamiltonian obtained by a higher order expansion has to be treated perturbatively, similarly to the BP Hamiltonian. Other quasi-relativistic methods have been proposed by Kutzelnigg [144,145] and DyaU [146]. 

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". ... 

This enables us to avoid the divergence of the expansion due to Ef(2mc —V) < 1. Moreover, it is usually appropriate to approximate K as unity. This approximation is called the zeroth-order regular approximation (ZORA). 

VMC method [14, 15] by deriving the relativistic local energy of the scalar version of the zeroth-order regular approximation (ZORA) Hamiltonian [16-19] as the first attempt to develop the relativistic QMC method. 

Coulomb-like potential. The zeroth-order regular approximation (ZORA) avoids this disadvantage by expanding in /(2c — V) up to the first order so that the ZORA Hamiltonian is variationally stable. The ZORA Hamiltonian was first derived by Chang et al. in 1986 [16], and later rediscovered as an approximation to the FW transformation by van Lenthe et al. [ 17-19]. The ZORA Hamiltonian of one electron in the external potential V is given by... 

Amsterdam Density Functional (ADF) is an accurate, parallelized, and powerful computational chemistiy program used to understand and predict chemical stmcture and reactivity with DFT [60], It is a popular tool to predict and imderstand magnetic, electric, optical, and vibrational spectra [61]. Heavy elements and transition metals can be modeled with ADF s relativistic zeroth order regular approximation (ZORA) approach and all-electron basis sets for the whole periodic table. It can be nsed to compnte IR freqnencies and intensities, vibrational circular dichroism (VCD), mobile block Hes-... 

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. 

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