Dirac-Fock-Slater potential

Figure 8 compares the Dirac-Fock-Slater potential for Si-Si collisions with the corresponding Moliere potential. While for low intemuclear separations the two potentials are almost indistinguishable from each other, significant differences occur for medium and large intemuclear separations, where both potentials are weak. Nevertheless, trajectories calculated with these two potentials differ significantly at large impact parameters, where the DFS potential becomes attractive (Fig. 9). 

In Fig. 7 the purely repulsive Moliere potential is compared with the (partly) attractive Morse potential. It is to be noted from this comparison that the Morse (like the Lennard-Jones) potential, while providing a realistic description of the attractive part of the interaction, becomes insufficient at low intemuclear separations where the purely repulsive potentials are more adequate. While more sophisticated potentials calculated, for example, by employing the Dirac-Fock-Slater (DFS) method (Eckstein etai, 1992), have recently become available, such potentials are generally more complicated and are available only in numerical form, and are thus not very handy for the calculations of interest here. For Si-Si collisions, the interaction potential, being repulsive for small and attractive for... 

Fig. 8. A comparison of the Si-Si screening function calculated within the Dirac-Fock-Slater model (Eckstein et al., 1992) and for the Molifere potential.

W.-D. Sepp, D. Kolb, W. Sengjer, H. Har-tung, B. Fricke. Relativistic Dirac-Fock-Slater program to calculate potential-energy curves for diatomic molecules. Phys. Rev. A, 33(6) (1986) 3679-3687. 

The ionization potentials and low excitation energies calculated for El 22 are shown in Table 2.7. More values may be found in [60]. Intermediate Hamiltonian values for E122 and its monocation were calculated by the Dirac-Coulomb and Dirac-Coulomb-Breit schemes, to obtain the effect of the Breit interaction (2.2). The Breit term contribution is small (0.01-0.04 eV) for transitions not involving/ electrons but increases to 0.07-0.1 eV when/ orbital occupancies are affected, as observed above (Section 2.3.1). The ground state is predicted to be 8s 8p7d, in agreement with early Dirac-Fock(-Slater) calculations [55-57], and not the 8s 8p configuration obtained by density functional theory [58]. The separation of the... 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

The potential surrouding each atom in a molecule is not the same as that for the free atom, because electron transfer occurs between atoms in the molecule. This means that atomic orbitals in the molecule are distinct from those in the free atom. Accordingly, it is necessary to use atomic orbitals optimized for each atomic potential in the molecule, as basis functions. In the present methods, the molecular wave functions were expressed as linear combinations of atomic orbitals obtained by numerically solving the Dirac-Slater or Hartree-Fock-Slater equations in the atomic-like potential derived from the spherical average of the molecular charge density around the nuclei [15]. Thus the atomic orbitals used as basis functions were automatically optimized for the molecule and thus the minimum size of the present basis set has enough flexibility to form accurate molecular orbitals. 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

There are several theoretical studies on LaO and related lanthanide oxides. We have already mentioned the ligand-field theory model calculations of Field (1982) as well as Carette and Hocquet (1988). More recently, Kotzian et al. (1991a,b) have applied the INDO technique (Pople et al. (1967) extended to include spin-orbit coupling [see also Kotzian et al. (1989a, b)] to lanthanide oxides (LaO, CeO, GdO and LuO). The authors call it INDO/S-CI method. The INDO parameters were derived from atomic spectra, model Dirac Fock calculations on lanthanide atoms and ions to derive ionization potentials, Slater-Condon factors and basis sets. The spin-orbit parameter is derived from atomic spectra in this method. 

In short Cl expansions, one may set up explicitly those CSFs which allow us to assign a correlating radial function to a given radial function of the Dirac-Hartree-Fock Slater determinant. This correlating function has got some well-defined properties for instance, the virtual radial function and the one to be correlated should live in the same spatial region. However, this can create additional technical difficulties for the guess of those radial functions which have been introduced to account for the correlation of a particular shell in the Cl expansions with more than one CSF. The above-mentioned model potentials are not the best choice, and additional adjustments to them are necessary so that it can be certain that the shells to be correlated live in the same radial space. 

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