Size-extensivity and connectedness of first-order exchange energy in symmetry-adapted perturbation theory: a graph-based approach

Abstract Given size-extensive monomer wave functions, the exchange (Pauli repulsion) energy defined within symmetry-adapted perturbation theory (SAPT) is a size-extensive quantity. Consequently, exchange energy expressions must be connected at every order of the intermolecular perturbation expansion. However, explicitly proving connectedness of approximate SAPT formulations is a challenging task. Here, we analyze the connectedness of the first-order exchange energy in two special cases: the second-quantized formulation of many-body SAPT and the multiconfigurational SAPT formulation based on antisymmetrized product of strongly orthogonal geminals. We present a graph-based proof in which connectedness is established via derivatives of chromatic polynomials, $$\pi '_G(x)$$ π G ′ ( x ) , evaluated at $$x=0$$ x = 0 . The proof holds to all orders of the intermolecular-overlap expansion.