# Zero point correction and free energy calculation with imaginary modes?

I am a beginner in DFT. Recently I have been working on calculating the energy (with zero point correction), and may be free energy in the future, for a structure in both stable and unstable configurations.

Obviously, with the unstable structure, some of the phonon modes will be imaginary. I was wondering how to do zero-point correction in this case: do I just discard all imaginary modes or is there some sophisticated ways to do this properly?

• +1 and welcome to our new community! Thank you for contributing your question here and we hope to see much more of you in the future !!! I made a minor edit: let's stick to one question per post. If two questions are asked then people will hesitate to write an answer if they know the answer to one question but not the other. Feb 12 at 18:24
• Whenever dealing with a imaginary frequency there are two different approaches: i) if small and it's not the wanted; ii) "discarded" if the structure represent a TS for instance. As far I know this is the way to treat the imaginary frequency. Unfortunately I don't know if it's the correct way to treat them so I didn't answer the question Feb 13 at 8:43
• In molecular calculations we just discard them. However I'm not sure if this is also the generally accepted practice in periodic calculations. Feb 13 at 10:15

The two most common approaches for dealing with imaginary frequencies in free energy computations are:

1. Compute the free energy using only the modes with real valued frequencies
2. Reoptimize the structure to remove any imaginary frequencies

Neither of these are ideal: 1 is not necessarily a reasonable approximation and 2 will likely distort the geometry relative to the configuration that you aim to study.

The Grimme group recently developed the Single-Point (or Biased) Hessian approach for computing free energies of non-equilibrium structures [1]. This approach reoptimizes the geometry under a constraining potential, aiming to remove imaginary frequencies while retaining the initial structure as much as possible. From the paper, this can lead to more accurate calculations of free energy and is a more rigorous way of handling non-equilibrium structures.

Off-hand, I only know of this method being implemented in xTB, but the paper gives a description of a general implementation.

#### References

1. Sebastian Spicher and Stefan Grimme Journal of Chemical Theory and Computation 2021 17 (3), 1701-1714 DOI: 10.1021/acs.jctc.0c01306
• Thank you! I was more curious about the justifications for ignoring the imaginary modes since I am likely to go that direction. But if it is common practice then maybe it's sufficient for me. Feb 14 at 1:55
• Just one comment: when the Gibbs free energy of a transition state is sought for, and there is only one imaginary frequency, then approach 1 is completely reasonable, and is in fact exact under the RRHO framework. It is only a not necessarily accurate approximation if you are interested in an energy minimum, got a (possibly high-order) saddle point, believe that the imaginary frequencies are spurious, and cannot remove them by e.g. tightening numerical thresholds. Also see mattermodeling.stackexchange.com/questions/3942/… Feb 14 at 13:09
• @HexagonReversal That's a very good point, if you are confident you are actually at an n-th order saddle point, its reasonable to remove the n imaginary frequencies from your free energy calculation. But if you are just considering, say, an arbitrary structure along the minimum energy path for a reaction, removing the imaginary frequency may not work as well. Definitely check out the Q/A that wzchem5 linked.
– Tyberius
Feb 14 at 18:07
• Thanks for the linking your informative answer @wzkchem5. I am interested to read more about your statement "Approach 1 is completely reasonable, and is in fact exact under the RRHO framework". Would you mind to explain a bit more on this and link any literature for further study? Feb 15 at 14:17
• @HexagonReversal See the derivation of Eq, (10) in doi: 10.1063/1.1749604. From there you can see that the omission of the imaginary mode from the free energy is due to a fortunate cancellation of the contributions of the imaginary mode. However, the derivation uses the RRHO approximation and assumes negligible tunneling. For anharmonic potential energy surfaces and/or reactions with non-negligible tunneling, the imaginary mode typically has a non-zero contribution to the Gibbs free energy of the transition state, although the contribution is often small. Feb 15 at 15:21