More detailed explanation of guess_mix

Hello everyone,
I was wondering a bit on how exactly the guess_mix option for UHF/UKS works and what the effect of spatial symmetry breaking is. Is this important for geometry optimizations only? Or does the spatial symmetry refer to energy level splitting?

I don’t quite understand the question, I suspect because we don’t have the same idea of what “symmetry breaking” is. I’ll give a very broad overview. Please let me know where I’m falling short.

Often, a molecule will have a point group symmetry. For example, water has a reflection plane and a rotation axis, so we say it has C2v symmetry. Given any orbital, we can apply all of the symmetry operations to it to generate more orbitals. For example, if you say there is a 1s orbital on one hydrogen and then rotate the molecule, you have produced a 1s orbital on the other hydrogen. By taking the appropriate linear combinations of orbitals, we can produce (potentially delocalized) orbitals such that every symmetry operation turns it into +/- itself. (There are more possibilities for non-abelian groups, but Psi only uses abelian groups.) We call this +/- behavior under symmetry operations the spatial symmetry.

The advantage of all this comes from the physical fact that rotating the molecule doesn’t change the energy means that every eigenstate of the Hamiltonian has a definite spatial symmetry. When our MOs are built from orbitals with spatial symmetry, our MO diagrams / determinants inherit it, and we can only consider the combinations with the same spatial symmetry as our target wavefunction.

Unfortunately, some electronic structure methods have solutions that are not eigenstates of some matrix. It is therefore theoretically possible for the solution to not have well-defined spatial symmetry. You can then either try to find all solutions, or solutions with the exact symmetry behavior (but probably also higher energy). This is called Lowdin’s symmetry dilemma. It’s rare, but it’s real.

This most frequently occurs in the Hartree-Fock method, where it means that the lowest-energy orbitals do not have a well-defined spatial symmetry. We call this symmetry breaking. Because this symmetry-breaking solution is an entirely different solution than the lowest-energy-keeping-symmetry, the orbital energy levels will naturally be different as well.

With all that background out of the way:
all that `guess_mix` does is rotate the HOMO and LUMO, so the new guess LUMO is 75% old LUMO and 25% old HOMO. The use case that the original developers had in mind was a case that the HOMO and LUMO had different spatial symmetry. Then, the new orbitals will not have a well-defined spatial symmetry. There’s nothing stopping you from using this keyword even when HOMO and LUMO have the same symmetry.

Thank you for the detailed background and explanation. Now I understand what spatial symmetry means in this context!

Thanks @jmisiewicz for the excellent primer on spatial symmetry breaking! Based on @sim_in_progress’s phrasing, though, I feel like I should add a little context to why people typically use it.

For systems like diradical molecules, there are two electrons which can arrange themselves in two nearly-degenerate orbitals to make six different configurations, which then combine to make six different electronic states. Of these six states, only the two high-spin components of the triplet are single-configurational, meaning that only one configuration contributes to the total wavefunction. To compute the energies of the other four – two 2-configurational, closed-shell singlets, one open-shell singlet, and the Ms = 0 component of the triplet – that means that you can’t use a single-configurational theory like Hartree-Fock of KS-DFT.

But if you don’t want to do MCSCF, what can you do?

That’s where symmetry breaking comes in. To compute a really dirty (but usually qualitatively) correct energy (or orbitals) for the open-shell singlet, we can break the spatial symmetry of the Hartree–Fock (or KS-DFT) orbitals by using the orbital rotations Jonathan discussed to mix the HOMO and LUMO. The reason that works is kinda messy, but suffice it to say that it does a decent job with MUCH less effort than doing MCSCF. And, especially if you do a broken-symmetry DFT computation, a much BETTER job than only MCSCF because of the dynamical correlation in whichever functional you choose (most folks do this with UB3LYP).

Hope that helps!

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Hey @dasirianni,
thank you for the further elaboration. I saw that guess_mix true gives me more accurate results than without it, but is still not complete as compared to CASSCF (which it can’t as DFT is single-determinant). May I ask what you mean by dynamical correlation? (I assume that is a property of hybrid functionals, as you mention B3LYP)?

Do you know why a mixing of 1:3 ratio was used for mixing? Is this to improve convergence by only adding a little bit of the other frontier orbital?

The exact number was arbitrary. The idea is that the HOMO-LUMO mixing should be small, but if we make it too small, we may land back at the symmetry-keeping solution.

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