First-Order Regular Approximation

Note that in this case the spin-orbit coupling is included already in zero order. Including the first-order term from an expansion of K defines the First-Order Regular Approximation (FORA) method. 

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... 

A truncation of this expansion for u) defines the zeroth- and first-order regular approximation abbreviated as ZORA and FORA, respectively [702]. The ZORA equation is then obtained from... 

FORA stands for first-order regular approximation, and is the first-order correction to the ZORA Hamiltonian. 

CPD=Chang - Pelissier- Durand DCB = Dirac - Coulomb -Breit DHF = Dirac-Hartree-Fock DK = Douglas-Kroll FORA = first-order regular approximation MVD = mass-velocity-Darwin term QED = quantum electrodynamics ZORA = zero-order regular approximation. 

First-Order Regular Approximation (FORA) method, 209... 

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 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]. 

These problems are avoided if one uses regular Hamiltonians which are bounded from below. Many applications are based on the so-called zero order regular approximation (ZORA), which has been extensively investigated by the Amsterdam group [46-50]. It can be viewed as the first term in a clever expansion of the elimination of the small component, an expansion which already covers, at zeroth order, a substantial part of the relativistic effects. In fact ZORA is a rediscovery of the so-called CPD Hamiltonian (named after the authors. Ref. [60]). 

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-... 

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... 

In the development of the Pauli Hamiltonian in section 17.1, truncation of the power series expansion of the inverse operator after the first term yielded the nonrelativistic Hamiltonian. In (18.1), the zeroth-order term is the Hamiltonian first developed by Chang, Pelissier, and Durand (1986), often referred to as the CPD Hamiltonian. The name given by van Lenthe et al. is the zeroth-order regular approximation, ZORA, which we will adopt here. The zeroth-order Hamiltonian is... 

The operator in the second bracket is the ZORA Hamiltonian, and it is sandwiched by normalization operators. If we expand these operators as we did above, we get the FORA Hamiltonian as the first term. The higher terms differ, however, because the final energy in the previous series must be the Dirac energy, whereas here it is the energy for the approximate Hamiltonian. Inclusion of the normalization terms corresponds to a resummation of certain parts of the ZORA perturbation series to infinite order, and the name coined by Dyall and van Lenthe (1999) is lORA—infinite-order regular approximation. 

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