GENERALIZATION OF THE BLANCHARD RULE

The old Kramers' rule is a useful recurrence relation for the calculat ion of diagonal [n, l\r(k)\n, l] matrix elements between hydrogenic wa ve functions. An improvement to such a relationship, which considers t he most general case of nondiagonal [n, l\r(k)\n', l'] matrix elements , is called Blanchard's rule. Both formulas were obtained by means of a method that uses the Schrodinger equation multiplied by an appropria te function, integral, and differential operators and boundary conditi ons. In the present work, using an alternative approach based on Hamil tonian identities, a general recurrence relation for the calculation o f nondiagonal multipolar matrix elements for any arbitrary central pot ential wave functions is presented. As expected, Kramers' and Blanchar d's equations are obtained as a particular case of the proposed formul a for hydrogenic potential wave functions. As a useful application of the improved Blanchard relationship, also presented are the generalize d quantum virial theorem and the generalization of the Pasternack-Ster nheimer selection rule that consider any central potential which is en ergy-dependent on the angular momentum. Likewise, the equivalent of Bl anchard's rule when applied to the harmonic oscillator and to the Krat zer potential wave functions, respectively, was found. These new recur rence 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 me thod proposed can also be extended to consider f(r) not equal r(k) mat rix elements for any potential wave functions as well as two-center in tegrals. (C) 1997 John Wiley & Sons, Inc.