Potential-weighted orthonormality relations

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 

Subtracting the complex conjugate of equation (10) from equation (9), and making use of the Hermiticity of the operator on the left, we obtain 

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 

In the case of atomic calculations, this choice of normalization does not need to be imposed. It results spontaneously from the form of the generahzed Sturmian basis set. 

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Making use of the potential-weighted orthonormality relation (58) and the subsidiary conditions (62) and (63), we obtain ... 

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 ... 

From the potential-weighted orthonormality relation (14), it follows that... 

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... 

Finally, multiplying from the left by a conjugate function in the basis set, integrating over space and spin coordinates, and making use of the potential-weighted orthonormality relations (13), we obtain the set of secular equations ... 

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 ... 

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