Is it right to neglect very small imaginary frequencies?

Question

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)?

Answer 1

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.)

2. 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.

Answer 2

Just want to add a small comment upon Geoff's excellent answer.

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.