Approximation method, first zeroth

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. 

The second approximation often used to truncate Cl wavefunctions is to neglect configurations formed by more than two-electron excitations. It is easily seen that they cannot mix into the Cl wavefunction in first order. The third method comes from perturbation theory. If one takes the Hartree-Fock energy as the zeroth-order approximation to the energy... 

In his first publications Huckel has already pointed out the possibility of the calculation of the transition frequencies from the calculated energy levels. An objection to this was, however, that in the so-called zeroth approximation (p. 279) the value of the parameters both in the valence bond and in the molecular orbital method had to be chosen quite different in order to calculate either the resonance energy or the transition frequencies in agreement with observation (Sklar, Forster)26. 

In this expression, Hg represents a purely electronic interaction between an undistorted system and the surface, and are electronic operators determining the interaction between a vibration (irrep F, component y) and the surface. Uiese latter operators must therefore have transformation properties dictated by the symmetry of both the adsorbed molecule and the surface. The form of (2) is suggestive of the standard method by which JT theory is developed, and this may be a desirable approach for future work. As a first approximation, however, we ignore the additional complication of surface-induced distortion and concentrate on the zeroth order term . 

Experience in a variety of applications of the C ASSCF method has shown it to be a valuable tool for obtaining good zeroth-order approximations to the wavefunctions. Attempts have been made to extend the treatment to include also the most important dynamical correlation effects. While this can be quite successful in some specific cases (see below for some examples), it is in general an impossible route. Dynamical correlation effects should preferably be included via multireference Cl calculations. It is then rarely necessary to perform very large CASSCF calculations. Degeneracy effects are most often described by a rather small set of active orbitals. On the other hand experience has also shown that it is important to use large basis sets including polarization functions in order to obtain reliable results. The CASSCF calculations will in such studies be dominated by the transformation step rather than by the Cl calculation. A mixture of first- and second-order procedures, as advocated above, is then probably the most economic alternative. 

Before discussing this new method it is useful to recall briefly the methods which we have already discussed. Note, first of all that calculations of the dielectric tensor must be based, as is known, upon a microscopic theory Such a theory for ionic crystals was first developed by Born and Ewald (2) for the infrared spectral region. The application of this approach for the region of exciton resonances has also been demonstrated in (3). In an approach identical to that of Born and Ewald (2) the mechanical excitons (see Section 2.2) are taken as states of zeroth-approximation. In the calculation of these states the Coulomb interaction between charges has to be taken into consideration without the contribution of the long-range macroscopic part of the longitudinal electric field. If this procedure can be carried out, then the Maxwell total macroscopic fields E and H can be taken as perturbations. In the first order of perturbation theory, we find... 

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]. 

Note that in this case the spin-orbit coupling is already included in zeroth order. Including the first-order term from an expansion of K defines the Eirst-Order Regular Approximation (FORA) method. A disadvantage of these methods is that they are not... 

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