First-Order Regular Approximation method

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. 

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]. 

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. 

C. V. Wiillen, Molecular density functional calculations in the regular relativistic approximation Method, application to coinage metal diatomics, hydrides, fluorides and chlorides, and comparison with first-order relativistic calculations, J. Chem. Phys. 109, 392-399 1998. 

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