The dependence on and continuity of the energy and other molecular properties with respect to the number of electrons

It was recently shown that the size consistency of the energy implies that, for any system with a rational number of electrons, the energy is given by the weighted average of the two systems with the nearest integer numbers of electrons. Specifically, E[N+P/Q] =  (1−P/Q)E[N] + (P/Q)E[N+1]. This paper extends that analysis, showing that the same result holds for irrational numbers of electrons. This proves that the energy is a continuous …