Explicit Formulas for the N th-Order Wavefunction and Energy in Nondegenerate Rayleigh–Schrödinger Perturbation Theory

The wavefunction and energy in nth order of Rayleigh–Schrödinger perturbation theory are shown to be given by En = ∑ σ1,σ2···,σn; (σ1+2σ2+···+nσn=n;σi≥0,i=1,2,···,n) (σ1!σ2!···σn!)−1(d / dE0)Σσi−1〈V〉σ1〈Va−1V〉2σ··· × 〈V(a−1V)n−1〉σn, χn = (a−1V)n| 0 〉 + ∑ j=1n−1 ∑ (σ1+2σ2+···=n−j;σi≥0,i=1,2,···) (σ1!σ2···)−1(d / dE0)Σσi−1(V〉1σ〈Va−1V〉2σ·mc· × 〈V(a−1V)n−j−1〉σn−1(d / dE0) (a−1V)i| 0 〉. Here |0〉 is the unperturbed eigenfunction of H0 with energy E0, V is the perturbation, 〈V···V〉 denotes 〈0|V···V|0〉, and a−1 is (1 − |0 〉〈0 |) (E0 − H0)−1(1 − |0〉〈0|). The wavefunction is given in the socalled “intermediate normalization.” Partial summations of these formulas give exactly the wavefunction and energy in Brillouin–Wigner perturbation theory.