Results for the exact exchange

The x-only ground-state energies of noble gas atoms obtained by solution of Equations (4.38)-(4.41) for different forms of the electron-electron interaction are listed in Table 4.1. 

In the Coulomb limit a direct comparison with fully numerical RHF calculations is possible. Due to the multiplicative nature of u PM, the OPM energies are higher than the RHF data. The actual differences, however, are extremely small. As a consequence, basis-set limitations easily dominate over these conceptual differences, as can be seen from the Coulomb-Breit energies in Table 4.1. Finally, the comparison of the Coulomb-Breit values with those found by inclusion of the complete Ej, demonstrates the size of the retardation corrections to the Breit interaction. It is obvious that these corrections are only relevant for truly heavy atoms. 

The importance of a self-consistent treatment of the transverse interaction is examined in Table 4.2. 

The fully self-consistent handling is compared with a perturbative evaluation of only the beyond-Breit terms and a perturbative treatment of the complete Ej. Even for the heaviest atoms the perturbative evaluation of the retardation corrections to the Breit term seems to be sufficient. On the other hand, use of first-order perturbation theory for the complete Ej leads to errors of the order of 1 eV for heavy atoms. An accurate description of inner shell transitions in these systems requires the inclusion of second-order Breit corrections. 

In principle, not only low-order perturbative Ec can be obtained in this way, but also resummed forms like the RPA (Engel and Facco Bonetti 2000). In practice, however, the resulting functionals are computationally much more demanding than the exact Ex, so that until now only the lowest-order contribution has been applied. Within the no-pair approximation and neglecting the transverse interaction, this second-order term reads 

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Eqs. (42) and (43) embody the Holas-March results for the exact exchange-correlation potential in terms of low-order density matrices both fully interacting and non-interacting matrices built from SKS one-electron wave functions being involved. 

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