Geometric Approximation in Perturbation Theory

Even if the terms in a perturbation expansion show geometric behavior only in exceptional cases, e.g., the Hartree-Fock hydrogen atom, the geometrical sum rule leads to remarkable numerical accuracy in a number of cases beyond the apparent prerequisites for its applicability. The rule is here derived by variational perturbation theory and it is seen that it holds, whenever the norm of the first-order wave function is negligible with respect to unity. In other cases it holds in a modified form.