Is it reasonable to use natural orbitals of an unrestricted HF calculation as a basis to start a CASSCF calculation for a radical

Question

Let's say we want to run a CASSCF calculation of a radical, for example the cation of a neutral closed-shell molecule (therefore, we calculate an open-shell radical cation). MOLCAS, as well as probably every other program package out there, needs starting orbitals when performing a CASSCF calculation. For a neutral species, these are normally assumed to be the regular old Hartree-Fock orbitals. Usually, when dealing with a cation, it seems that the Hartree-Fock-SCF calculation is performed for the neutral species or the di-cationic species to yield the necessary HF starting-orbitals. They are in these cases the result of closed-shell calculations. MOLCAS also permits the use of a UHF calculation as the basis for a subsequent CASSCF calculation. Then, the natural orbitals of the unrestricted calculation are used.

Is the usage of the natural orbitals of an unrestricted HF calculation reasonable as a basis for a subsequent CASSCF calculation? If not, why? When is it a good idea / when is it a bad idea? What are the differences in using natural UHF orbitals in contrast to the SCF orbitals of the neutral or di-cationic species? What are the pitfalls?

Edit: I am aware (or at least it is my understanding) that conceptually it is no problem to perform the CASSCF using the natural UHF orbitals, as it could for example also be performed using only the alpha orbitals of the UHF calculation. My question is therefore aimed at finding out in which cases it is generally a good or bad idea to use the natural UHF orbitals.

Answer 1

The best starting orbitals for a CASSCF calculation are optimised orbitals of another CASSCF (or RASSCF) calculation. That sounds a bit ridiculous, but this is probably the best way to figure out the active space.

At first you are probably choosing something quite small, like CAS(2,2) to CAS(4,4). For these calculations it barely matters what kind of starting orbitals you are choosing. The change will be negligible and probably depending heavily on the system you are using.

The advantage of calculating the cations by dismissing the radicals is often a good choice since you can run a very fast RHF calculation and the orbitals are symmetric, which is what you need for CAS.
Natural orbitals of an UHF calculation are also a quite sensible choice, because they already allow for fractional occupation and should also be symmetric. The only downside I can think of is, that this kind of start orbital generation takes a lot longer.
Similarly you can choose to run a ROHF calculation, if you are uncomfortable with too much cationic charge.
As you have said you can choose also an UHF calculation by throwing away one set. I would advise caution here. Sometimes - probably especially when CAS is necessary - spin contamination in UHF is significant. Therefore one set might not give you anything close and you could end up choosing the wrong CAS.

I am not aware of any major pitfalls except for the just mentioned choosing of a completely wrong active space.

In any case it is a good idea to give the CASSCF calculation a few cycles and see how the active space is developing. Sometimes you need to adjust the active space long before the calculation converges and then you will have already a much better guess than with your initial starting orbitals.

I personally use the following routine

R(O)HF [x+] -> CAS(x, ⌈½x⌉ +1) 
            -> RAS(x+2n, ⌈½x⌉ +n +1) 
            -> CAS(x+2n, ⌈½x⌉ +n +1) 
            -> RAS(...) 
            -> CAS(...) -> ...

to narrow down the active space and include all orbitals I want and need. In most cases I am not converging the intermediate calculations, except for the minimal CAS.
As you can see the procedure that follows after your start orbitals is much™ more time consuming, so it really does not matter.

Answer 2

Hartree-Fock Orbitals are Always the Wrong Choice for CAS Calculations

In Hartree-Fock (HF) theory, only the occupied orbitals contribute to the electronic energy. Due to the presence of Fock exchange, the Fock operator is different for occupied and virtual orbitals. For occupied orbitals, the Fock exchange exactly cancels the Coulomb interaction of the electron with itself. Hence, electrons in occupied orbitals experience the mean-field potential of the N-1 remaining electrons. Since virtual orbitals do not contribute to the potential, there exists no exchange term to cancel part of the Coulomb potential for these orbitals. Therefore, virtual orbitals experience the full potential due to the N electrons in the occupied orbital space. Compared to the occupied orbitals, these orbitals are relatively diffuse and high in energy. They rather correspond to anionic states and do not represent a good estimate to describe charge-neutral excited states. Consequently, HF virtual orbitals are not a reasonable starting point for CAS calculations, since their character can change dramatically once they become partially occupied.

Though it may seem counterintuitive in an ab-initio framework, Kohn-Sham (KS) DFT provides a much better starting point. Due to the lack of orbital-dependent Fock exchange in (semi-)local DFT, the occupied and virtual KS orbitals are generated in the same mean-field Coulomb and exchange potential. Thus, occupied and virtual orbitals are described on equal footing, i.e., both experience a mean-field potential from approximately N-1 electrons. Therefore, virtual KS orbitals represent a much better guess for charge-neutral excited states (cf. the reasonable performance of TD-DFT compared to CIS for vertical excitation energies). A comprehensive discussion of the band gap in Hartree-Fock theory, as well as in "exact" and approximate KS-DFT, is given in the following paper:

Baerends & Gritsenko & van Meer: The Kohn-Sham gap, the fundamental gap and the optical gap: the physical meaning of occupied and virtual Kohn-Sham orbital energies

There are alternative options, if one wants to avoid (semi-)local KS-DFT orbitals, which are inevitably affected by the so-called self-interaction error. For example, fractional occupation number (FON)-HF orbitals from an electronic temperature treatment serve as a good starting point, since the virtual orbitals become partially occupied. Such a treatment comes at no extra computational cost and is used, e.g., in FOMO-CASCI. Using natural transition orbitals (NTOs) form a CIS calculation is more physically motivated. While it is somewhat more elaborate, the additional computational cost is negligible compared to the subsequent CAS treatment. The higher quality for the initial set of orbitals will more than pay back through a faster and more robust convergence of the CAS procedure.

Answer 3

I can't tell you when a UHF reference wave-function would be a good idea (maybe if you don't have ROHF), but I think it might be a BAD idea if UHF generates a significantly spin-contaminated wave-function (UHF wave-functions are not eigenfunctions of S^2). Orbitals in your CASSCF active space will obviously be optimized, but the non-active occupied and external spaces will forever stay, in your case, UHF natural orbitals.

Two configuration MCSCF (TCSCF) is a tiny bit more expensive than UHF but results in S^2 eigenfunctions. Also, I have had luck with MP2 natural orbitals (RHF/ROHF reference).