# Natural orbital population analysis in Gaussian

Within (restricted) density functional theory and Hartree-Fock, respectively the Kohn-Sham orbitals and spin orbitals are the natural orbitals, with occupation numbers either 2 or 0.

As I understood, this also applies to spin-unrestricted calculations, is this correct?

Now, when doing UDFT or UHF with Gaussian and specifying pop=NaturalOrbitals, the natural orbitals are different from the KS/canonical orbitals and some occupations numbers are also different from 2 and 0. Unfortunately Gaussian manual provides no reference to how the natural orbitals analysis is done exactly and the literature is fairly confusing (spin-projection vs non-spin projection etc.).

Could anybody give me an overview or reference regarding what Gaussian exactly does when this pop=NaturalOrbitals keyword is turned on?

• I really cannot find any documentation of what Gaussian actually does to produce the orbitals. You might have more luck asking that question on the CCL, as this mailing list is closer to the people who develop the code. However, judging from the definition in the Gold Book, natural orbitals, it seems to be a reasonably simple transformation. – Martin - マーチン Oct 25 '19 at 13:56
• Maybe this will help: doi.org/10.1103/PhysRev.97.1474 – Martin - マーチン Oct 25 '19 at 14:26
• Thanks @Martin-マーチン, those references are very useful. As much as I have since understood, the KS/canonical orbitals of a broken-symmetry UHF solution are the "natural spin orbitals" BUT if one constructs the spinless first order density matrix and diagonalizes that, then the natural orbitals will be different from the KS orbitals. – eimrek Oct 25 '19 at 14:41