Natural orbitals is one solution. I'm not sure if that particular form is necessary.
One application is the evaluation of integrals of coordinate products of a particular order. These yield terms in the "primitive" or "traced" multipole expansion, which then can be turned into the traceless multipoles. These are useful for characterizing and distinguishing nearly iso-energetic electronic excited states, and for comparison with experimental results.
(background (Zangwill, Modern Electrodynamics)
(Sonia Coriani, 10.1063/1.1562198)
Two examples:
Orca CCSD(T) natural orbitals to molden fed to multiwfn can reproduce the multipoles from cfour exactly, and can give the hexadecapole moment that I want to distinguish two rydberg effective potentials.
The same can be done with cfour, but only if the fix I put in that forum is used, and only if g iar the highest functions.
The MRCC scf molden files are really excellent, They are spherical and well-formed through H functions.
Steve