Hartree radial integrals

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. 

The quantities 6 are the Stevens factors (a, fir, y, for n = 2, 4 and 6, respectively), (rn) are Hartree-Fock radial integrals and A are crystal-field potentials. The quantities O are Stevens operators and Hm in the last term of eq. (11) represents the molecular-field acting on the rare earth moment. 

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

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... 

Notice that 1 haven t made any mention of the LCAO procedure Hartree produced numerical tables of radial functions. The atomic problem is quite different from the molecular one because of the high symmetry of atoms. The theory of atomic structure is simplified (or complicated, according to your viewpoint) by angular momentum considerations. The Hartree-Fock limit can be easily reached by numerical integration of the HF equations, and it is not necessary to invoke the LCAO method. 

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