Evaluation of the uncorrelated energy with the Dirac–Fock method

Deslattes, R. D.; Kessler, E. G. Jr; Indelicato, P.; Lindroth, E.

doi:10.1107/97809553602060000592

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, p. 205

Section 4.2.2.9. Evaluation of the uncorrelated energy with the Dirac–Fock method

R. D. Deslattes,^c E. G. Kessler Jr,^f P. Indelicato^e and E. Lindroth^g

4.2.2.9. Evaluation of the uncorrelated energy with the Dirac–Fock method

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The first step in the calculation, following Indelicato and collaborators (Indelicato & Desclaux, 1990; Indelicato & Lindroth, 1992; Mooney et al., 1992; Lindroth & Indelicato, 1993; Indelicato & Lindroth, 1996) consists in evaluating the best possible energy with relativistic corrections, within the independent electron approximation, for each hole state (here $[1s_{{1\over2}}]$ , $[2p_{{1\over2}}]$ , $[2p_{{3\over2}}]$ , $[3p_{{1\over2}}]$ , $[3p_{{3\over2}}]$ for K, L_II, L_III, M_II, M_III, respectively). Such a calculation must provide a suitable starting point for adding all many-body and QED contributions. We have thus chosen the Dirac–Fock method in the implementation of Desclaux (1975, 1993). This method, based on the Dirac equation, allows treatment of arbitrary atoms with arbitrary structure and has been widely used for this kind of calculation. We have used it with full exchange and relaxation (to account for inactive orbital rearrangement due to the hole presence). The electron–electron interaction used in this program contains all magnetic and retardation effects, which is very important to have good results at large Z. The magnetic interaction is treated on an equal footing with the Coulomb interaction, to account for higher-order effects in the wavefunction (which are also useful for evaluating radiative corrections to the electron–electron interaction). All these calculations must be done with proper nuclear charge models to account for finite-nuclear-size corrections to all contributions. For heavy nuclei, nuclear deformations must be accounted for (Blundell, Johnson & Sapirstein, 1990; Indelicato, 1990). For all elements for which experiments have been performed, we used experimental nuclear charge radii. For the others we used a formula from Johnson & Soff (1985), corrected for nuclear deformations for Z $[\gt]$ 90. Contribution of deformation to the r.m.s. radius (the only parameter of importance to the atomic calculation) is roughly constant (0.11 fm) for Z $[\gt]$ 90. There is an unknown region, between Bi and Th (83 $[\lt]$ Z $[\lt]$ 90), where deformation effects start to be important, but for which they are not known. When experiments are done for a particular isotope, we calculated separately the energies for each isotope.

As mentioned in the introduction, there are special difficulties involved when dealing with atoms with open outer shells (obviously this is the most common case). Computing all energies $[E_{{J}}]$ for total angular momentum J would be both impossible and useless. The Dirac–Fock method circumvents this difficulty. One can evaluate directly an average energy that corresponds to the barycentre of all $[E_{{J}}]$ with weight ( [2{J}+1] ). There are still a few cases for which the average calculation cannot converge (when the open shells have identical symmetry). In that case, the outer electrons have been rearranged in an identical fashion for all hole states of the atom, to minimize possible shifts due to this procedure.

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, p. 205