Recurrence relations for the real solid harmonics

In Section 6.4.4, we derived an explicit expression for the real solid harmonics. Often, it is more convenient to calculate the solid harmonics by recursion, in particular when the full set of solid harmonics up to a given angular momentum / is needed. In the present subsection, we shall derive a set of recurrence relations for the real solid harmonics of the form 

The factor has here been introduced since, in our discussion of multipole expansions of the Coulomb integrals in Section 9.13, we shall work with different sets of scaled solid harmonics. Of course, when the solid harmonics are calculated from an explicit expression such as (6.4.47), any scaling factor is trivially incorporated and requires no special attention. 

To simplify matters, we shall consider the generation of the solid harmonics Sim in two steps. First, we evaluate all functions of the type 5 , / by diagonal recursion, where the quantum numbers I and m are changed simultaneously. Next, the remaining solid harmonics Sim are generated by vertical recursion, where m is kept fixed and I is incremented. While the diagonal recurrence relations are easily established, the derivation of the vertical recurrences requires more work. 

elementary algebra shows that these functions satisfy the diagonal recurrence relations 

To obtain the desired vertical recurrence relations for the solid harmonics, it is sufficient to determine a recurrence for the expansion coefficients of the form 

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We begin in Section 6.4.1 by reviewing, without proofs or derivations, the standard properties of the spherical harmonics and their relationship to the associated Legendre polynomials. The closely related solid harmonics are next introduced in Section 6.4.2. In Sections 6.4.3 and 6.4.4, we derive explicit Cartesian expressions for the complex and real solid harmonics, respectively. Finally, in Section 6.4.5, we derive a set of recurrence relations for the real solid harmonics. 

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