Exact Relations Related to OPM

Before investigating the correlation component of in niore detail it seems worthwhile to list a few exact relations which emerge from an analysis of the OPM integral equation. One first recognizes that the OPM equation determines xc only up to an additive constant. In fact, as norm-conservation requires that 

It has already been mentioned that for physical reasons the exact exchange potential of finite systems must as3miptoticaIIy behave as 

It is thus very pleasing that one finds exactly this behavior for the solution of (2.27) Equation (2.49) can be verified by an examination of the x-only OPM integral equation for large r, requiring the standard normalization limj. .cx) x( ) = 0, i.e. the validity of (2.48) [3]. Equation (2.49) provides an alternative to (2.48) for the normahzation of v - 

It seems worthwhile to point out that the behavior (2.49) can not be obtained by differentiation of the asymptotic form of the exact exchange energy density Cx. The asymptotic form of Cx follows from (2.19) [46], 

One can also establish a necessary condition for the eigenvalue-dependence of i xc [47]. In fact, direct integration over (2.27) yields 

---