Perturbation theory by the moment method and point-group symmetry

We analyze earlier applications of perturbation theory by the moment method (also called inner product method) to anharmonic oscillators. For concreteness we focus on two-dimensional models with symmetry $C_{4v}$ and $C_{2v}$ and reveal the reason why some of those earlier treatments proved unsuitable for the calculation of the perturbation corrections for some excited states. Point-group symmetry enables one to predict which states require special treatment.