Explicit diagrammatic solution of normalised, nondegenerate Rayleigh-Schr\"odinger perturbation theory

We solve the coupled recurrence relations for eigenenergies and -vectors in nondegenerate Rayleigh-Schr"odinger perturbation theory under the constraint that the approximate eigenvector be normalised to $1$ in every order. The series can be expressed in terms of diagrams that were first introduced by C. Bloch [Nucl. Phys. 6, 329 (1958)] for the degenerate, unnormalised case. Normalisation increases the number of terms and introduces a nontrivial dependence on the diagrams’ topology to the coefficients.