Representation of the Exchange Terms in Fock's Equations by a Quasi-Potential

Fock's equations for the ($\mathrm{nl}$) wave functions of an atom can be expressed in the form of Schrödinger's equation for a single particle by including in the potential energy a term $\eta (nl; r)$ expressing the "exchange" terms in the equation. As a further approximation, this term can be replaced by some average over the various ($\mathrm{nl}$) wave functions. This "exchange" contribution to the potential is evaluated for the ($\mathrm{nl}$) wave functions of ${\mathrm{Cu}}^{+}$, and an appropriate mean is compared with the "averaged exchange potential," based on the statistical model, suggested by Slater and used by Pratt in calculations for ${\mathrm{Cu}}^{+}$. It is found that the range of variation of $\eta (nl; r)$ with ($\mathrm{nl}$) is of the order of a factor 2, so that its replacement by an average over ($\mathrm{nl}$) is only a rough way of treating exchange effects.