The old Kramers' rule is a useful recurrence relation for the calculation ² < k < : of diagonal n, l r n, l matrix elements between hydrogenic wave functions. An improvement to such a relationship, which considers the most general case of nondiagonal ² < k < : n, l r nЈ, lЈ matrix elements, is called Blanchard's rule. Both formulas were obtained by means of a method that uses the Schrodinger equation multiplied by an appropriate function, integral, and differential operators and boundary conditions. In the present work, using an alternative approach based on Hamiltonian identities, a general recurrence relation for the calculation of nondiagonal multipolar matrix elements for any arbitrary central potential wave functions is presented. As expected, Kramers' and Blanchard's equations are obtained as a particular case of the proposed formula for hydrogenic potential wave functions. As a useful application of the improved Blanchard relationship, also presented are the generalized quantum virial theorem and the generalization of the Pasternack᎐Sternheimer selection rule that consider any central potential which is energy-dependent on the angular momentum. Likewise, the equivalent of Blanchard's rule when applied to the harmonic oscillator and to the Kratzer potential wave functions, respectively, was found. These new recurrence relations reduce to particular cases which are in good agreement with published results which were derived using well-known approaches such as the hypervirial commutator algebra procedure. Finally, the method Ž .
k proposed can also be extended to consider f r / r matrix elements for any potential wave functions as well as two-center integrals.