# Is it right to neglect very small imaginary frequencies?

I'm running a DFT optimization (B3LYP/def2-TZVP) and frequencies in ORCA for a molecule. And I get one or two very small (1-6 cm^-1) imaginary frequencies which corresponds to a slight bend of the molecule.

I assume that the frequencies arise from numerical integration errors as they decrease when I run the same calculation with better grid and tighter SCF.

So is it appropriate to just leave it as is if I plan to run electronic properties calculation on this geometry with more high-level functional and basis (polarizability, hyperpolarizability, and maybe TD-DFT)?

• +1 but what molecule is it? The smallest it ever got (with best grid and tightest SCF) was 1cm-1? – Nike Dattani Dec 15 '20 at 15:36
• Are these two imaginary frequencies on top of the three translation and three rotational zero frequencies? If not, it may be that imposing translation and/or rotational symmetry will help. – ProfM Dec 15 '20 at 17:41
• Orca gives 6 zero frequencies (as I suppose translation and rotational zero frequencies) and then 2 imaginary. – roma ichenko Dec 16 '20 at 1:00
• Yes, it does. Maybe I'm doing something wrong. Would it be a good idea to run IRC to follow this imaginary eigenvalue? – roma ichenko Dec 18 '20 at 2:34
• @romaichenko This looks like a numerical issue. Have you checked your numerical grids? They are sometimes an issue. In my experience, this may be particularly important if you're using a hybrid DFT with RIJCOSX, etc., or you have a reasonably flat PES (e.g., van der Waals complexes, etc.). By the way, this might be useful in general. – schneiderfelipe Dec 18 '20 at 2:48

## tldr: This is something of an eternal debate.

IMHO very small imaginary frequencies can be okay, but it depends on your system and needs.

As you might see from the various comments above, there are often different opinions on whether very small imaginary frequencies matter. The truth is, that it depends a bit on the size of the molecule and what you plan to do.

In principal, if you've reached a true local / global minima of the potential energy surface, there should never be imaginary frequencies, because if the second derivative is negative, you're not at a minima with respect to that normal mode.

In practice, for medium to large molecules it can be very hard to completely minimize and find a true minimum. Most geometry optimization methods include various routines to only stop when the forces are very small, the change in energy is very small, etc. But these don't guarantee a true minima, which is why you should calculate frequencies and check (which you mention in the question).

Remember that the potential energy surface has $$3N-6$$ dimensions for non-linear molecules. For 10 atoms, that's already 24 degrees of freedom. In many molecules, these may be correlated (e.g., think about twisting a dihedral angle in a protein - you might smash into another atom). Many times the potential energy surface can be close to flat, so finding the exact minima is time-consuming.

1. You should check for numerical noise, try to push for better optimization tolerances, integration grids, convergence, etc.

(It seems like you've done a lot of this based on your comments.)

1. It depends on your properties / needs. If you need rigorous thermochemistry, then there may be some energy error between your current geometry and a true minima. In that case, do what you can to remove the minima.

In your case, in my experience, the property calculations you plan are relatively insensitive to the small energy / geometry difference. Consider if the atomic positions move $$0.001Å$$ will the polarizability change much? Probably not. Similar story with TDDFT, which usually has a ~0.1-0.2eV error bar.

• Given your question and topic, if you were one of my students, I think you've done more than enough to get a true minima and I'd move on. If you were doing some sort of thermochemistry (e.g., reaction mechanism), I might suggest calculating an exact Hessian frequently (e.g. sites.google.com/site/orcainputlibrary/geometry-optimizations) – Geoff Hutchison Jan 1 at 19:38
• +10 as I'm really glad this unanswered question which as the subject of a lot of discussion in the comments, got answered, and it's very illuminating to learn from you about how this is something of an eternal debate! – Nike Dattani Jan 1 at 19:39
• I would expect some debate, because in principal if you have imaginary frequencies, it's not a true minima. In my experience, beyond very simple, rigid molecules it can be very hard to eliminate small (<$100 cm^{-1}$) imaginary frequencies and it often doesn't matter for the property calculation. If the dipole moment changes 0.001D, does that matter? Probably not. If the energy changes 1 kJ/mol, does that matter? Maybe yes. – Geoff Hutchison Jan 1 at 19:42
• The fact that it doesn't matter for property calculations is I think useful insight for the OP and for future members of this community, and it's one of those things that often comes with experience rather than from what we learn in textbooks. – Nike Dattani Jan 1 at 19:45