where [e.sub.n](x) = [square root of (2/[pi])] sin(nx) is a

complete orthonormal set of eigenvectors of A.Then, the following properties hold:

This ensures that the orbitals form a complete orthonormal set, so that other quantities can be expanded in this basis.

Likewise, the so-called natural orbitals [16] that diagonalize the one-particle reduced density matrix and thereby enable its most efficient representation cannot be employed; although the natural orbitals constitute a complete orthonormal set, they do not, in general, correspond to the eigenfunctions of a single-particle Hamiltonian with known eigenvalues, and for a collinear magnetic system they are also spin dependent.

(iii) There exists a

complete orthonormal set [{[[PHI].sub.i]}.sup.[infinity].sub.i=1], of eigenvectors of A, such that

Later in this section (see Theorem 4.10), we prove that [E.sub.n] is, in fact, a complete orthonormal set in [H.sub.n] for each n [member of] N.

Since the set E = {[e.sub.m] | m [member of] [N.sub.0]} of eigenfunctions of A is a complete orthonormal set in [L.sup.2] [a, b], we see that

As remarked in Section 3, E is a complete orthonormal set in [L.sup.2] [a, b]; consequently, (4.19) implies that f = 0 in [L.sup.2] [a, b].

Guseinov, "New complete orthonormal sets of exponential-type orbitals in standard convention and their origin," Bulletin of the Chemical Society of Japan, vol.

Guseinov, "One-range addition theorems for noninteger n slater functions using complete orthonormal sets of exponential type orbitals in standard convention," Few-Body Systems, vol.

Guseinov, "Unified treatment of one-range addition theorems for complete orthonormal sets of generalized exponentialtype orbitals and noninteger N slater functions," Bulletin of the Chemical Society of Japan, vol.

By considering this deficiency, Guseinov had suggested a new centrally symmetric potential of the SF field and on the basis of this idea Guseinov proposed the new complete orthonormal sets of [[psi].sup.([alpha])]-ETOs.

To obtain the expansion of NISTO's about a new center by the use of complete orthonormal sets of [[PSI].sup.([alpha])]-ETOs we can use the method for the expansion of ISTOs.