Effective interaction in the Rayleigh–Schrödinger perturbation theory

We present a unified description of the effective interaction vv in the Rayleigh–Schrödinger perturbation theory. First, we generalize the well-known bracketing expression for the energy shift ΔE in a one-dimensional model space to express the effective interaction vv in a multi-dimensional model space. Second, we show that the generalized bracketing representation has a natural graphic expression in terms of folded diagrams. The present work thus gives a unified understanding of the effective interaction (i) in one- and multi-dimensional model spaces and (ii) in algebraic (bracketing) and graphic (folded diagram) representations.