Radial integrals, Hartree-Fock calculations

Fig. 15.9 Relativistic Hartree-Fock calculations of some actinide radial integrals (a) f (//) (i>) (c) G (/

Table 1. Stevens multiplicative factors associated with equivalent operators for the ground states of rare earth ions and the calculated Hartree-Fock radial integrals (r > in atomic units of length...

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. 

Electron densities calculated from Hartree-Fock wave functions are quite accurate. Figure 11.1 compares the radial distribution function of argon (found by integrating the electron density over the angles d and and multiplying the result by r ) calculated by the Hartree-Fock method with the experimental radial distribution func-... 

The numerical integration also can be used to calculate the matrix elements of the exchange-correlation potential. For the numerical integration, the atomic partition method proposed by Savin [392] and Becke [393] has been adopted and combined with Gauss-Legendre (radial) and Lebedev (angular) quadratures [394]. The Kohn-Sham LCAO periodic method based on numerical integration at each cycle of the self-consistent-field process is computationally more expensive than the periodic LCAO Hartree-Fock method that is almost fully analytical. 

As to spherical parameters, at least the Slater integrals and the spin-orbit coupling constant, it should be possible to evaluate them with precision from the radial part of the f-electron wave function obtained by solving the Hartree or Hartree-Fock equations. Unfortunately, the F calculated this way are 30-50% greater than the fitted ones. It seems that the lanthanides are almost alone (even if there are fourteen elements ) in the Periodic Table in displaying such pronounced misbehaviour. Elements to help understand this may be found in a paper by Rajnak and Wyboume (1963) who showed that the effects of configuration interactions in configurations may be represented by... 

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