# Why rotate orbitals in a CASSCF calculation?

I'm trying to get the rate constant for the reaction of propene to give propenyl (radical) and H.

Using Molpro, I calculated the energy for the optimized geometry of the reactant using CASSCF.

It is required at one point to rotate two orbitals (the one with energy just below the HOMO and one in the valence shell).

I don't understand the point in doing this; what's the purpose?

• Are you talking about orthogonalization of orbitals? Jun 7 '17 at 17:42
• No, it's the rotate command, it's used to swap two orbitals at different energy and, if you are considering them in the active space they have different population too. Jun 7 '17 at 18:11
• If I understand you correctly, you are swapping two occupied orbitals, is that right? What is the symmetry of the rotated orbitals? Is that relevant at all?
– user41033
Jun 8 '17 at 15:52
• I might guess that you are having issues during orbital optimization and the rotating of two orbitals is a convergence trick. Jun 12 '17 at 8:37

The reason for rotation of orbitals (really the reordering of orbitals) is that the starting orbitals are usually not in the correct order for any sort of MCSCF calculation (CASSCF, RASSCF, GASSCF, adding correlation, ...). Doing a single CASSCF calculation with no "preparation" beforehand should force most programs to start with Hartree-Fock-like orbitals; this is the reference wavefunction $\Psi_0$. Hartree-Fock includes almost no static correlation to speak of, which can manifest in an incorrect ordering of the orbitals when compared to MOs that do have static correlation accounted for.

From here:

Reorder your orbitals if necessary before running the MCSCF. Except for the most trivial cases, the orbitals that belong in the active space are not the orbitals that come straight out of your single-reference calculation. If they were, then you might not be running MCSCF to begin with.

Poor convergence of an MCSCF procedure is a good sign that the active space orbitals are not the correct ones. It is not a matter of the ordering within a space, since the energy is invariant to rotation/swapping within a space (inactive/frozen, active, or secondary/unoccupied). It is that an orbital Hartree-Fock predicts as a frozen core orbital should be a valence-type orbital in the active space, etc.

One of the reasons that MCSCF calculations are multi-step calculations and shouldn't be done in one shot is the generation of (hopefully) better orbitals than HF ones, inspection and any necessary rotation, then the MCSCF calculation. The best starting orbitals for MCSCF are other MCSCF orbitals, but you probably don't have those. Usual choices are MP2 or CI natural orbitals, which gives an indication of the total size of the active space and adds some dynamic correlation to "patch up" the lack of any real correlation in HF orbitals.

In my limited experience, HF orbitals are a fine starting point as long as the order is correct.

I'm aware of two reasons why this is done, and both are ways to go around a convention in the implementation of CASSCF that is shared by all the codes I know.

In general, the active space in CASSCF is comprised by a contiguous band of orbitals (in energy terms), which includes the $\mathrm{N}$ highest occupied orbitals and the $\mathrm{N'}$ lowest unoccupied orbitals. In ORCA (and, for what I could tell from the documentation, also in Molpro), there is no option to select which orbitals are included to generate the CAS - you can only select how many. In other words, if a relevant orbital is lower (if occupied) or higher (if empty) in energy than orbitals you are not interested in, you are stuck with either adding these uninteresting orbitals to the CAS as well (computationally expensive) or rotating orbitals so that you artificially swap their energies and "trick" the code into using them.

I've seen this used to build a symmetrically consistent CAS (if you are only interested in orbitals with a certain molecular symmetry) and in systems where small energy gaps mean that MO ordering can change throughout, say, a trajectory - to ensure that your CAS is always composed of the relevant orbitals. I can't offer deeper theoretical justification as I'm not a CASSCF expert.

Eventually I had an answer by the professor so, here's the full story:

As I said I had to study a reaction with formation of two radicals. Performing a first calculation with two active orbitals I obtain an estimation of the energy of the molecule and occupation of HOMO and LUMO.

Having done this, I proceed in elongating the bond I want to break, until an imaginary frequency is detected, indicating the presence of a saddle point in my PES. Each calculation performed will have an inactive variable (to avoid optimization of geometry) and selecting the same active space, it will converge to the previous solution found.

The purpose of swapping two occupied orbitals is to highlight a delocalization of the sigma orbital between C and H that's the one that will pass from a double occupation to a single occupation.

This sigma orbital usually has an energy much lower than the HOMO, so in the first CASSCF calculation is computed as a closed orbital.

Using the command rotate I perform again the calculation this time expecting that the sigma orbital will be the HOMO instead of the pi bond of propene.

This answer is not yet backed by theory, even if I guess pentavalentcarbon was pretty accurate in describing the process of multi-reference calculations.

Thank you all!