# How to properly denote imaginary magnitudes of vibrational frequencies

Vibrational frequency computations provide the magnitudes of vibrational frequencies for each vibrational mode of a molecule ($3N-6$ degrees of freedom for non-linear molecules and $3N-5$ for linear molecules). Sometimes these magnitudes can be non-real (i.e. imaginary). This is common when characterizing a transition state which contain one, and exactly one, imaginary mode of vibration.

Some popular software packages will print these magnitudes as a negative number. This syntax has unfortunately propagated itself into common, everyday practice even if technically incorrect.

My question is, what is the best way to denote these imaginary magnitudes, especially in the context of a peer-reviewed journal? Should we stick to using the minus sign convention (e.g. -15) with a note explicitly defining the meaning of the $-$ sign, or should they simply be listed with an $i$ (e.g. 15$i$)?

(I've just been told to NEVER use the minus sign convention as this will incur the wrath of the reviewers. Please feel free to verify.)

• I would love to know why you would include the magnitude of an imaginary vibration. Since this "vibration" may not be observed, it has also no physical meaning. BTW: The minus results from harmonic approximations used in quantum chemical packages as it is defined as $i^2=-1$. Apr 14, 2014 at 10:35
• When you characterize small molecular clusters, people are interested in knowing the nature of the stationary point. The magnitude of the harmonic vibrational frequencies can become important if the magnitudes are small (in this context, close to zero). Small magnitudes can arise from numerical noise or perhaps an insufficient level of theory. Say you used 3 different levels of theory (LOT) to characterize molecule X. X is a minimum at LOT1 and LOT2 but is a TS at LOT3. If the magnitude was only a few wave numbers at LOT3, one may expect it to be a minimum. That is why we report them. Apr 14, 2014 at 11:49
• Okay. I did not understand that you meant a value for a wave number. I thought you were referring to the amplitude. However, the obtained value has no physical meaning whatsoever. Any mentioning this in the paper (not for recomputing purpose in the supporting information) should be avoided. If you are talking about a "real" TS, it should be one at all LOT and you should confirm it via IRC calculations. Apr 14, 2014 at 12:02
• Well, IRCs are only really necessary if you are trying to connect stationary points on a PES. If you're just anchoring the PES of some small molecular cluster, the connection between the various stationary points may not be necessary (lies beyond the scope of the study). Mentioning the Hessian index in a paper is indeed important if you are trying to illustrate the deficiency of a particular LOT (such as DFT) in characterizing stationary points. We do high-level CCSD(T) type calculations in my lab. These types of things are DEFINITELY worth mentioning. Apr 14, 2014 at 12:27
• Agreed, I did not know you were doing benchmark calculations. Anything recent worth sharing, I'd love to have a good read :) Apr 14, 2014 at 12:42

The suffix i to the positive wavenumber to denote an imaginary frequency is seemingly accepted by reviewers, see Photochemical transformations of 5-methyltetrazole. Matrix isolation FTIR and DFT studies as one example I could find in a very quick search.

• Impact factor of 2.4? I can roll with that. Feb 17, 2014 at 20:45
• @LordStrykerI am not familiar with that phrase. Feb 17, 2014 at 21:43
• The journal you linked has an impact factor of 2.4 which means that the journal has 'enough clout' for me to assume that the convention that is accepted by the journal is going to be generally satisfactory elsewhere. Feb 17, 2014 at 21:48
• Thanks for the explanation! I just looked there because J. Photochem. Photobiol. A isn't behind a paywall for me. Feb 17, 2014 at 21:57

I'm completely out of my depth here, but I would, since the minus sign convention seems to be frowned upon, look for other symbols:

• As mentioned, with an $$i$$ denoting the imaginarity: $$15i$$

This is maybe not so beautiful if we're actually talking about the absolute value of the imaginary magnitude, because in mathematics the $$i$$ clearly denotes the imaginary part of an imaginary number.

• If not yet taken, I would give my vote for the asterisk (*): $$15^*$$

• Since we're talking about transition state magnitudes, maybe the double dagger ($$\ddagger$$): $$15^\ddagger$$

Of course this notation would have to be defined somewhere towards the beginning of your article, stating that this symbol denotes imaginary (transition state) magnitudes.

Using this method you could even have the option of distinguishing between normal state and transition state imaginary magnitudes: Asterisk for the former, double dagger for the latter.

• I should have mentioned that this does indeed refer to the absolute value of the magnitudes. Feb 17, 2014 at 20:20
• Well, then I personally think you should find some other way of marking it as based on an imaginary magnitude. The $i$ is not technically correct ;-) In any way, the reviewers will let you know whether they approve of your newly conceived notation or not... Feb 17, 2014 at 20:22
• I must know what is technically correct. Don't leave me on a cliffhanger! Feb 17, 2014 at 20:29