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

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Abstract
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.

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