Slater radial integrals

Usually, for both theoretical and semi-empirical determination of energy spectra, radial integrals that do not depend on term energy of the configuration are used. More exact values of the energy levels are obtained while utilizing the radial wave functions, which depend on term. Therefore, there have been attempts to account for this dependence in semi-empirical calculations. Usually the Slater parameters are multiplied by the energy dependent coefficient... 

This basic parameterization scheme, used at the time of the last A.C.S. symposium on lanthanide and actinide chemistry ( 5), has been discussed in detail by Wybourne 06). In applying the scheme, the free-ion Hamiltonian was first diagonalized and then the crystal-field interaction was treated as a perturbation. This procedure yielded free-ion energy levels that frequently deviated by several hundred cm from the observed energy levels.a In addition, the derived parameters such as the Slater radial integral, f(2), and the spin-orbit radial integral did not follow an expected systematic pattern across the lanthanide or actinide series ( 7). ... 

Crosswhite (23) has used the correlated multiconfiguration Hartree-Fock scheme of Froese-Fisher and Saxena (24) with the approximate relativistic corrections of Cowan and Griffin (25) to calculate the Slater, spin-orbit, and Marvin radial integrals for all of the actinide ions. A comparison of the calculated and effective parameters is shown in Table II. The relatively large differences between calculation and experiment are due to the fact that configuration interaction effects have not been properly included in the calculation. In spite of this fact, the differences vary smoothly and often monotonically across the series. Because the Marvin radial integral M agrees with the experimental value, the calculated ratios M3(HRF)/M (HRF) =0.56 and M4 (HRF)/M° (HRF) =0.38 for all tripositive actinide ions, are used to fix M and M4 in the experimental scheme. 

The F- Indues are Slater radial integrals. Using these is easier than it appears, since Slater has tabulated all possible values for use in matrix diagonalization. 

Fk interaction with magnetic field Slater s radial integrals... 

Because of the greater extension of the 5f radial wavefunctions with respect to those of the shielding 7s and 7p shells, they are more sensitive to changes in the valence-electron situation than for the corresponding lanthanide cores. Nevertheless, their rigidity is remarkable as compared to that for the valence electrons themselves. This can be seen quantitatively in the plots for the Slater electrostatic interaction integrals (Sf, Sf) and spin-orbit radial integral C, which dominate the atomic Hamiltonian for all cases of interest to us here. 

First, we already mentioned that the Hartree-Fock Slater integrals F do not match the fitted ones the ab initio values are 1.1 to 1.5 timesthe phenomenological ones. This difference can be taken into account by an expansion of the rare earth radial wavefunction which, in counterpart, gives higher radial integrals (r ). If A is the mean ratio between the phenomenological F and the Hartree-Fock ones, then may replace the simple... 

The principal computational approaches to molecular electronic structme that developed from about 1950 onwards had molecular orbitals (MOs) as their basis. Since the molecular orbitals were expressed in terms of linear combinations of atomic orbitals (LCAOs), the orbital remained a feature of the quantum mechanical account of molecular structure. Initially at least it was not possible to realise fully the LCAO MO approach because, except in diatomic systems, the integrals over the orbitals, which were exponential in their radial parts (Slater orbitals) proved too difficult to evaluate quickly enough to make non-empirical calculation feasible. But even given these limitations, it was already clear that the role of the bond in the emerging discipline of computational quantum chemistry was going to be problematic... 

Here, k is an integer of values 2,4 and 6,/ are the coefficients representing the angular part of the wave function [29] and are the electrostatic Slater two-electron radial integrals given by Equation 1.17. 

The Slater—Condon integrals Ft(ff), Ft(fd), and Gj-(fd), which represent the static electron correlation within the 4f" and 4f 15d1 configurations. They are obtained from the radial wave functions R, of the 4f and 5d Kohn—Sham orbitals of the lanthanide ions.23,31... 

International Tables for Crystallography 1992). The function <]/> for Slater-type radial functions can be expressed in terms of a hypergeometric series (Stewart 1980), or in closed form (Avery and Watson 1977, Su and Coppens 1990). The latter are listed in appendix G. As an example, for a first-row atom quadrupolar function (/ = 2) with n, = 2, the integral over the nonnormalized Slater function is... 

---