0030-400X/03/9501- $24.00 © 2003 MAIK “Nauka/Interperiodica” 0006 Optics and Spectroscopy, Vol. 95, No. 1, 2003, pp. 6–13. Translated from Optika i Spektroskopiya, Vol. 95, No. 1, 2003, pp. 11–18. Original Russian Text Copyright © 2003 by Kozlov. INTRODUCTION Completely accounting for electron correlations is possible only for atoms with a very small number of electrons. In the general case, calculations of atomic structure are only possible in terms of some approxi- mate methods. Presently, there are several approaches to calculation of many-electron atoms (see, for exam- ple, [1, 2]). Generally, some variant of a mean field is used as an initial approximation. Most often, it is a Har- tree–Fock field for one of the electronic configurations. For atoms with one valence electron above occupied shells of the core, it is reasonable to use a Hartree–Fock core potential. In the case of two or more valence elec- trons, it seems reasonable to construct an initial approx- imation with complete or partial allowance made for the field of valence electrons. The many-body perturba- tion theory (MBPT), which is based on such an initial approximation, is more intricate, since it contains a large class of additional diagrams. It is customary to call them subtraction diagrams. The possibility of partially taking into account the field of valence electrons is provided for in the method of the effective Hamiltonian for valence electrons [3– 5]. For example, calculations for the Hg and Ba atoms [6] were carried out for the potential V (N) , where N is the total number of electrons in the atom, including the two valence electrons. In this case, the correlation correc- tions turn out to be significantly smaller than in the case when the initial approximation is constructed on the basis of the core potential V (N – 2) . Thus, one might expect to obtain higher accuracy in the calculations based on the V (N) approximation. In this study, we ana- lyze higher order corrections for the potentials V (N – 2) and V (N) on the example of a simple model. A completely satisfactory theory for taking into account electron correlations in atoms has not yet been developed, in particular, due to the absence of an ade- quate smallness parameter. Even when the correlation corrections are small, this parameter cannot be distin- guished explicitly. This makes it difficult to estimate ignored terms and the accuracy of calculations. There- fore, even rough estimates of the effective smallness parameter are of interest. We suggest a simple estimation of the accuracy of the method of the effective Hamiltonian for valence electrons [3–5]. In this method, the valence correlations are rigorously taken into account and the core–valence correlations are considered in the second order of the MBPT. The contribution of the latter correlations to the energy can be roughly estimated as (1) where V ' is the energy of residual interaction and ∆ cv is the excitation energy of the core. Let us introduce the parameter (2) which characterizes the magnitude of the residual inter- action, and express this parameter from (1): (3) The latter equation allows us to relate λ eff to well- known quantities. The results of calculations in the sec- ond order of the MBPT can be used to obtain a correla- tion correction to the energy. Then, the terms ignored are of the third order with respect to the residual inter- action and, consequently, (4) δ E c v V ' 〈 〉 2 ∆ c v ------------, = λ eff V ' 〈 〉 ∆ c v ----------, = λ eff δ E c v ∆ c v ----------- . = error V ' 〈 〉 3 ∆ c v 2 ------------ ∼ λ eff δ E c v . = ATOMIC SPECTROSCOPY Higher Orders in Residual Two-Electron Interaction M. G. Kozlov Institute of Nuclear Physics, Russian Academy of Sciences, Gatchina, 188300 Russia e-mail: mgk@MF1309.spb.edu Received October 3, 2002 Abstract—Precise methods of calculations of many-electron atoms are under intensive development. In such calculations, higher orders of the perturbation theory in regard to residual interaction of electrons should be taken into account. In this study, a model of an atom with two electrons in its core and two valence electrons is considered. In terms of this model, expansion with respect to residual interaction is analyzed for two initial approximations based on the potentials V (2) and V (4) . It turns out that the higher order corrections are nearly the same in both cases. At the same time, the potential V (2) has a number of advantages. © 2003 MAIK “Nauka/Interperiodica”.