Sturmians orthonormality relations

It is interesting to examine the momentum-space orthonormality relations of the Fourier transforms of hydrogenlike Sturmian basis sets. If we let... 

The momentum-space orthonormality relation for hydrogenlike Sturmian basis sets, equation) 17), can be shown to be closely related to the orthonormality relation for hyperspherical harmonics in a 4-dimensional space. This relationship follows from the results of Fock [5], who was able to solve the Schrodinger equation for the hydrogen atom in reciprocal space by projecting 3-dimensional p-space onto the surface of a 4-dimensional hypersphere with the mapping ... 

Goscinski s treatment of the orthonormality relations of Sturmian basis sets [3] is easy to generalize and we can see by an argument analogous to equations (9) and (10) that when P ,... 

Sturmian basis set obeys a potential-weighted orthogonality relationship analogous to equation (10). This still does not tell us how to normalize the functions, and in fact the choice is arbitrary. However, it will be convenient to choose the normalization in such a way that in momentum space the orthonormality relations become ... 

The reader may verify that these become the familiar hydrogenlike orbitals if k is replaced by Z n, where Z is the nuclear charge and n is the principal quantum number. It can be shown [19] that the Coulomb Sturmians obey a set of potential-weighted orthonormality relations of the form ... 

Ek is the same for all the members of a generalized Sturmian basis set, while the weighting factor jSv is chosen in such a way as to make all the members of the basis set isoenergetic. Generalized Sturmian basis sets can be shown to obey the following orthonormality relation [22, 23] ... 

The orthonormality relations for generalized Sturmian basis functions in direct space and reciprocal space can be written in the form [22-24]... 

This is, of course, also consistent with the potential-weighted orthonormality relation of the Coulomb Sturmian basis function, (7), as can be seen by making use of (132) for the special case where Xfl- = X = 0 and making the substitution k = ZJn. Looking at Table 6, we can see that for the special case where 5 = 0, the diagonal elements of T< T are equal to 1, while the off-diagonal elements vanish, as is required by the orthonormality relations (94). The momentum-space orthonormality relations for Coulomb Sturmians can be used to make a weakly... 

In the present paper, generalized Sturmians are introduced in Section 2. Their potential-weighted orthonormality relations (Section 3) permit us to write down a peculiar secular equation (Section 4). Examples of its solution for atomic problems are given in Section 5. In these calculations, generalized... 

Like the familiar one-particle Sturmians of Shull and Lowdin, generalized Sturmians obey potential-weighted orthonormality relations. To see this, we move the term in Vq to the right-hand side of equation (3), multiply by a conjugate function in the basis set, and integrate over the coordinates. This gives us the relation... 

We must remember that the subscript v represents a set of indices, and the constants jS may be independent of some of them. Orthogonality with respect to these minor indices must be established or constructed in some other way. Assuming that this has been done, we next need to normalize the generalized Sturmian basis set. It turns out that the most natural and convenient choice of normalization is that which yields the potential-weighted orthonormality relations in the form... 

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