CCL:G: is CAS-SCF first-order density matrix idempotent

Dear colleagues,

We recently ran some CAS-SCF calculations in Gaussian 16. When visualizing the orbitals in GaussView 5 or outputting the natural orbitals to a WFX file (e.g., pop=NO   output=wfx), the natural orbitals are listed as fully occupied or fully empty. (In other words, the CAS-SCF first-order density matrix is claimed to be idempotent.) This is different than the behavior that occurs with Gaussian's MP2, coupled-cluster, SAC-CI, or CISD  calculations, which print fractional orbital occupancies, and generally speaking have non-idempotent first-order density matrices. The systems we are studying are some small diatomic molecules (e.g., Be2, B2, etc.)

I am confused about whether the claimed whole number occupancies of the CAS-SCF natural orbitals is a real effect or whether it is a bug in the Gaussian software? According to literature references, an N-representable first-order density matrix can in general have fractional natural orbital occupancies (i.e., it can be non-idempotent). But also, there is not a one-to-one map between the first-order density matrix and physical observables: two calculations with identical electron density distributions can have different first-order density matrices. Consequently, I have not yet been able to rule out whether this is just a representation issue.

Can anyone offer specific insights into the following questions:

(1) Has anyone observed a non-idempotent first-order density matrix (i.e., fractional natural orbital occupancies) when performing CAS-SCF calculations using any software program?

(2) Can anyone offer a concrete answer as to whether the CAS-SCF first-order density matrix should or should not in general be idempotent? (Please omit discussions of trivial one or two-electron systems, since they are too simple.) Is there theory or literature references to back this up?

(3) If the CAS-SCF first-order density matrix is not supposed to be idempotent, is there a reasonably easy and fast way to extract the CAS-SCF natural orbital fractional occupancies from Gaussian 16 calculations? (Caution: This needs to be based on actual experience, rather than conjectures.) 

By the way, this issue is not new to Gaussian 16; we observed the same behavior for Gaussian 09. At this point, I do not understand whether this behavior is "real" or a "bug". Can anyone provide clarification?


Tom Manz
associate professor
Chemical & Materials Engineering
New Mexico State University