I feel like this is a really stupid question, and I've been ashamed to ask for a while...

When I'm about to run some electronic structure calculation on a molecule, how do I know if I need to use an open-shell method (e.g. ROHF, UHF) as the first step in a correlated method like coupled cluster?

I understand that a single atom with an odd number of electrons will obviously have an unpaired electron. And, free radicals would be an obvious generalization of that.

I also understand that a molecule with (nearly-)degenerate frontier orbitals will need the multi-reference aspect of an open-shell method.

But how do I know about that degeneracy beforehand? Is it symmetry?

When I look at a molecule, what do I do to decide that RHF is not applicable?


1 Answer 1


When you're doing an electronic structure calculation, the input will include the following things:

  • Geometry
  • Basis set
  • Number of electrons
  • Spin multiplicity

If the spin multiplicity is not 1 (i.e. you are not dealing with a singlet state), then you're certainly dealing with an open-shell system.

The only case where this becomes highly non-trivial is when you're dealing with an open-shell singlet, meaning that there's a bunch of unpaired electrons, but overall the entire system still has a net spin multiplicity of 1.

Open shell singlets can be notoriously complicated. They arise (for example) in studies of one of the most difficult problems in all of quantum chemistry, which is to accurately study the electronic structure of FeMoco (arguably the most complicated enzyme known) as we discussed open-shell singlets in this paper. The answers to the question "Why is "open-shell singlet" not an oxymoron" are a good starting point to understand why such states arise.

So how do you know whether or not the system that you're studying needs to be modeled as an open-shell singlet?

Sometimes you know based on spectroscopic experiments what the occupation numbers of the valence orbitals should be (see the "configuration" column in my answer to: How to determine occupied and closed orbitals for a Molpro CASSCF calculation?). For example, you might already know by looking through the experimental literature, that the ground electronic state of the molecule in which you're interested has an equal (and non-zero) number of unpaired spin-up versus unpaired spin-down electrons.

What if there's not enough experimental data?

Then often the computational scientist's task is to do calculations for various values of the spin multiplicity (try making the molecule a singlet, triplet, etc.) and various electronic configurations to see which one gives the lowest energy. For very simple molecules you can infer what the ground state electronic configuration will be, but my answer to Total spin and/or multiplicity for transition metal ions? shows that even for a molecule with only two atoms (like the diatomic iron dimer) we don't know experimentally nor theoretically what the spin multiplicity of the ground electronic state is! My answer to the above-linked CASSCF question also shows examples where the ground state electronic configuration is still unknown. Another related thread is here: How to find out the multiplicities of molecules containing d and f block species?. So often we do not know in advance whether the system's ground state electronic configuration corresponds to an open-shell system or a closed-shell system, in fact we often don't even know the spin multiplicity so we frequently find ourselves doing calculations for many different configurations and attempting to determine which one has the lowest energy so that scientists in the future don't have to. In this regard, a literature search can be helpful: see if others have studied the same molecule and what they said about the open-shell vs closed-shell character of the ground electronic state.

Apart from what I've discussed above for ground electronic states, many people are also just interested in excited state chemistry, meaning that they're doing electronic structure calculations for many different states (including open-shell singlets, for example). In this case the decision is a lot easier :)

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    $\begingroup$ INMHO every single time one should calculate all the possible (realistic) spinstates. Even if the experimental value or other calculations are known, the ordering of spin-state for a given method can easily be different if there are other low energy spin states. $\endgroup$
    – Greg
    Sep 9, 2022 at 3:26
  • $\begingroup$ @Greg I totally agree (when resources allow for it). My answer does suggest to do calculations for various spin states. You also raise a good point that different methods can have different energetic orderings of spin-states. I totally agree with that! $\endgroup$ Sep 9, 2022 at 3:59
  • $\begingroup$ To make sure I understand: If the molecule has an odd number of electrons, then you know it can't be a singlet: odd # of electrons => open shell, right? Then, if there's an even number of electrons, that's where it's either "closed shell" or "open shell singlet", right? $\endgroup$ Sep 9, 2022 at 17:18
  • $\begingroup$ Regarding comparing the results for various spin states: how does that work if you're planning on using an expensive level of theory? Would you first run various spin states at a lower level, maybe with a smaller basis set, to determine the multiplicity to use for the more-expensive run? $\endgroup$ Sep 9, 2022 at 17:22
  • $\begingroup$ @ZaneBeckwith I find it interesting that you've cast 0 votes on this site (and only 1 on Physics)! Your first comment about odd and even numbers of electrons is correct. You also raise a good point about testing several states being an expensive process if you're using an expensive method. Yes, this would be done at whatever basis-set + theory-level would be feasible and practically effective, just like all of quantum chemistry. For large molecules it's not possible. I hope this answers your question! If it doesn't and you're still wondering about how to test various spin states, ask another! $\endgroup$ Sep 9, 2022 at 18:20

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