Many-Electron Atoms

Abstract

A system consisting of a nucleus of charge +Ze and n electrons (1, 2,…, i, j,…,n) will be chosen as a model. Two types of interactions occur: mutual repulsion of the electrons and attraction between the nucleus and the electrons. The expression for the potential energy of the i-th electron assumes the form

$$ {{V}_{i}}({{r}_{1}},{{r}_{2}},...,{{r}_{n}}) = - \frac{{Z{{e}^{2}}}}{{4\pi {{\varepsilon }_{0}}\left| {{{r}_{i}}} \right|}} + \sum\limits_{{j( \ne i)}}^{n} {\frac{{{{e}^{2}}}}{{4\pi {{\varepsilon }_{0}}\left| {{{r}_{i}} - {{r}_{j}}} \right|}}} , $$

(8-1)

where r _i is the position vector of the i-th electron. It is assumed that the immobile atomic nucleus is located at the origin of the coordinate system. The two-electron part of the operator can be simplified (cf. Section 5.5) if the i-th electron is assumed to be exposed to the effect of an averaged potential of all the remaining electrons: V _i then depends only on r _i :

$$ {{V}_{i}}({{r}_{i}}) \approx - \frac{{Z{{e}^{2}}}}{{4\pi {{\varepsilon }_{0}}\left| {{{r}_{i}}} \right|}} + \sum\limits_{{j( \ne i)}}^{n} {\frac{{{{e}^{2}}}}{{4\pi {{\varepsilon }_{0}}\left| {{{r}_{i}} - {{r}_{j}}} \right|}}} $$

(8-2)