Size-Extensivity of First-order Exchange Energy in Symmetry-Adapted Perturbation Theory: a Graph-based Approach

We present a graph-based proof of the size-extensivity of the first-order exchange energy in symmetry-adapted perturbation theory (SAPT). The connectedness of the exchange energy expression can be established via derivatives of chromatic polynomials $\chi'_G(x)$ evaluated at $x=0$. The proof holds to all orders of the overlap expansion. As an extension, we demonstrate the linked character of the first-order exchange energy obtained from strongly orthogonal geminal wave functions. In the diagrammatic formulation, the proof ultimately relies on the vanishing property of $\chi'_G(0)$ for disjoint graphs