Wulfman integrals

In (38), hi(uj) is an harmonic polynomial of order / in ui, U2 and M3, while hi sj) is the same harmonic polynomial with uj replaced by Sj. The Shibuya-Wulfman integrals can then be calculated by resolving the product of hyperspherical harmonics in (37) into terms of the form u hi(uj). To illustrate this second method for the evaluation of... 

As has been pointed out by Shibuya, Wulfman [7] and Aquilanti [18], the expansion coefficients in equation (A4) are just the integrals shown in equation (37), i.e., the Shibuya-Wulfman integrals. To see this, we can take the Fourier transform of (A4), which gives us the relationship... 

The functions Fkj s) are illustrated in Table 6, while Shibuya-Wulfman integrals derived from equation (73)-(75) are shown in Table 7 for the first few... 

Just as in the case of the Shibuya-Wulfman integrals, we can make use of equation (73), from which we obtain ... 

Shibuya-Wulfman Integrals and Sturmian Overlap Integrals Evaluated... 

Using this result, together with Fock s solution (95), we can rewrite the Shibuya-Wulfman integrals of (114) in the form... 

But in discussing the direct-space formulation of the problem, we gave a different formula for the Shibuya-Wulfman integrals... 

To see that these two definitions of the Shibuya-Wulfman integrals are really the same, we insert the expression for yT (x) in terms of its Fourier transform into the direct-space definition of T>>T. Thus we obtain the relation... 

From (130), it can also be seen that the Shibyya-Wulfman integrals can be interpreted as a species of nuclear attraction integrals, as Koga has pointed out [33, 34]. To see this, we note that ydx) obeys the Schrodinger equation... 

As Vincenzo Aquilanti has shown, the Shibuya-Wulfman integrals can be related to the effect of a translation on Coulomb Sturmians. Combining (75) and (137), we obtain... 

In other words, if a Coulomb Sturmian located on one center is expanded in terms of Coulomb Sturmians located on another center, the expansion coefficients are Shibuya-Wulfman integrals. It should be noted, however, that this expansion is... 

Thus the Shibuya-Wulfman integrals form a representation of the group of translations. 

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