# 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? Dec 15, 2020 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. Dec 15, 2020 at 17:41
• Orca gives 6 zero frequencies (as I suppose translation and rotational zero frequencies) and then 2 imaginary. Dec 16, 2020 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? Dec 18, 2020 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. Dec 18, 2020 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 principle, 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 minimum 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 minimum, 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 $$\sim 0.1$$$$0.2$$ eV 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) Jan 1, 2021 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! Jan 1, 2021 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. Jan 1, 2021 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. Jan 1, 2021 at 19:45
• As a somewhat random aside, I have had situations where the imaginary frequencies refuse to go away even with extremely tight tolerances on what I knew was a very flat part of the PES. It turns out that many optimization algorithms adaptively shrink the optimization step-size which, for a flat enough potential can cause the optimization to stop early even when using very tight thresholds. The solution turned out to be to use a substantially larger initial step size than one would normally use. I only say this to point out that sometimes truly reaching a minimum is very difficult. Feb 15 at 21:38

One may be tempted to think that a $$<\pu{10 cm-1}$$ imaginary frequency introduces a negligible error to the energy or the property of interest, and this is usually true. But one frequently overlooks the fact that, even an infinitesimal imaginary frequency has a finite impact on the Gibbs free energy, and that finite impact is usually non-negligible.

To see this, consider a molecule whose lowest frequency is only slightly above zero. In the RRHO approximation (as used by programs like Gaussian), this vibrational mode gives a very negative contribution to the Gibbs free energy, which diverges logarithmically to negative infinity as the frequency approaches zero:

\begin{align} G(\nu) &= \frac{1}{2}h\nu + kT\ln \left( 1 - \exp\left\{-\frac{h\nu}{kT}\right\} \right)\\ &= \frac{1}{2}h\nu + kT\ln \left( \frac{h\nu}{kT} + O(\nu^2) \right). \tag{1} \end{align}

This is a result of the logarithmic divergence of the entropy:

$$\begin{gather} S(\nu) = k\left[ \frac{h\nu}{kT\left( \exp\left\{\frac{h\nu}{kT}\right\} - 1 \right)} - \ln\left( 1 - \exp\left\{-\frac{h\nu}{kT}\right\} \right) \right]. \tag{2} \end{gather}$$

But when $$\nu$$ is imaginary, it contributes zero to the Gibbs free energy (and the entropy), because only real frequencies enter the partition function. Therefore, if a small positive frequency is erroneously calculated as imaginary, there is a positive error in the Gibbs free energy, and that error can be arbitrarily large even if the error in the frequency is infinitesimal! (The good news is, because the divergence of the RRHO Gibbs free energy is only logarithmic, the probability of having a certain magnitude of error decreases exponentially w.r.t. the error magnitude.)

Things improve somewhat with programs like ORCA or Turbomole, which use the quasi-RRHO (or QRRHO) method[1] instead. In the QRRHO method, the limit $$G(\nu=0)$$ is regularized to a finite value, which numerically equals to $$\pu{-2.68 kcal/mol}$$. So with an infinitesimal error in the frequency you can make an error of up to $$\pu{+2.68 kcal/mol}$$ in the Gibbs free energy, which is still quite sizeable for most studies! Even worse you may have more than one false imaginary frequency per molecule, and as the errors are always positive, they add up.

The lesson is that, numerical error-induced imaginary frequencies are much bigger problems for Gibbs free energies than for most other properties, and a small (even infinitesimal) imaginary frequency does not guarantee a small error in the Gibbs free energy. This is due to the fact that the Gibbs free energy is not a continuous function w.r.t. the frequency when the latter changes from real to imaginary.

### References:

1. Grimme, S. Supramolecular Binding Thermodynamics by Dispersion-Corrected Density Functional Theory. Chem. Eur. J. 2012, 18 (32), 9955–9964. DOI: 10.1002/chem.201200497.
• +1 Great point. I was approaching from the context of the optimization and "general properties." If you want accurate thermochemistry, you absolutely need to clean up imaginary frequencies. Feb 12 at 22:53