# Quantum Chemistry: Small imaginary frequencies

I am running a freq calculation on a geometry optimized structure using the ORCA quantum chemistry program. The output of the freq. calculation contains small neg. frequencies. Normally, these are considered to be saddle points, and thus would lead to the conclusion that the geometry does not represent a ground state.

freq.             IR Int.
-17.6900          0.000
-16.8900          0.000
-10.6600          0.000
6.4000            0.000
11.7200           0.000
... more small frequencies with IR Int. 0
30.3700           0.000
36.7000           19.3544


However, I am wondering if this is really the case, as the output also assigns an IR intensity of 0.000 to these modes, and states that "The first frequency considered to be a vibration is 14", which is the first positive mode. I am a little bit stuck, as obviously a negative frequency shouldn't appear in the IR spectrum, so does ORCA just removes those frequencies from the IR, even though they are there? Or does the IR-Intensity of 0 imply that the negative frequencies are actually just a precision error of the calculation, and I can still assume the structure to be a ground state.

Normally, the criterion is "no negative frequencies". On the other hand, not much energy should be stored in these modes, so I wonder how this will affect the computed energies.

• If my distant memory from quantum chem classes is correct, small imaginary frequencies occur all the time because I forgot. But we should only care about those imaginary frequencies above some arbitrary threshold I also forgot, because only those are able to show us saddle points. I distinctly remember at least one practice calculation we did where we were told to ignore the negative frequency. – Jan Oct 10 '17 at 17:19
• What method? If DFT, what functional, what grid, what kinds of RI, ...? – pentavalentcarbon Oct 10 '17 at 18:29
• @pentavalentcarbon Is that really important? I tried AM1 and PM3 both give small imaginary frequencies. Calculations should only be semiquantitatively. The aim is to calculate Proton affinities, and a publication by Dewar has shown AM1 to perform quite well for this kind of task. Systems are big (around ~140 atoms), so I only consider semiempirical methods. – logical x 2 Oct 10 '17 at 18:36
• Yes, it can be. You just gave me a crucial piece of information; big and/or floppy systems often require tighter optimization convergence criteria (and therefore more cycles) than smaller systems. This probably isn't noise, it's probably an unconverged geometry. – pentavalentcarbon Oct 10 '17 at 18:49
• Though I am a big fan of ORCA, I would not recommend using it for semi-empirical methods like AM1 or PM$x$. MOPAC does just fine for those. Further, simply ignoring these frequencies can lead to huge inaccuracies, because if one additional vibrational modes makes it to the positive (say, in a different conformation), you suddenly get more energy. The cleanest thing to do is "get only positive frequencies". Some investigators have also simply flipped the sign (yeah, I know). – TAR86 Oct 11 '17 at 4:58

It's probably okay, but an indication you don't have a true minimum in the geometry.

It's important to calculate frequencies to know if you have a true minimum of the potential energy surface. Presence of imaginary frequencies are indeed indications of saddle points.

That's simple theory. Imaginary frequencies indicate problems in the curvature with respect to that degree of freedom - it's simply not a minimum.

If you want a true minimum, you'll need to improve convergence and numerical accuracy.

Now the practical side...

I'm not familiar enough with ORCA to know if the translational and rotational degrees of freedom are removed from the frequency list. Some programs list these, and some do not. Obviously these include 5 or 6 degrees of freedom which are inherently low energy and due to numerical instability or incomplete geometric convergence can sometimes be imaginary frequencies. Your list includes three very small imaginary frequencies, so they could be translations or rotations.

(I also mention this, because ORCA indicates the first vibration is at ~14 $\mathrm{cm}^{-1}$ and the code is likely removing the lowest frequencies as translations and vibrations.)

For even moderate-sized molecules, and particularly for large molecules, it can be very, very hard to converge geometries to true minima.

Consider your list - with multiple vibrational frequencies of 14-30 $\mathrm{cm}^{-1}$. Those are incredibly low energy breathing modes. Gradients and energy changes along those dimensions will be small and a great number of optimization steps will be required.

This is one reason that geometry optimization methods "give up" if the gradients get very small and displacements get very small. It may not be a true minimum, but in all likelihood, it's very, very close and would require a large computational effort to get the true minimum.

Note: Don't read this and think "oh, I don't need to bother." Your needs may vary. If you're doing accurate work, you absolutely need to push hard on geometry convergence.