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I am an experimental chemist interested in anharmonicity—I am particularly interested in how to calculate anharmonic frequencies. The method I have the most experience with is (generalised) second order vibrational perturbation theory (G)VPT2.

In most programs where VPT2 is implemented (if not all), the full quadratic (harmonic) and cubic, and semi-quartic force fields are determined. This is because, while all quadratic and cubic force constants ($\phi_{ijk}$) are used to calculate the anharmonicity constants $x_{ii}$ and $x_{ij}$, the only quartic force constants which contribute are $\phi_{iiii}$ and $\phi_{iijj}$.$^1$

Practically speaking, my understanding of this approach is that, following harmonic frequency calculations for a molecule (where the hessian/second derivatives has/have been calculated analytically), two further hessians are calculated for each vibrational mode to either side of the equilibrium geometry along the vibration's normal coordinate [i.e., 2(2N – 6) additional hessians are calculated]. From the analytical hessians, the cubic and semi-quartic force fields would be calculated numerically.

The issue with calculating a semi-quartic force field is that, if one wanted to look at the frequencies of different isotopologues of molecules, a second VPT2 calculation would have to be run; my understanding is that the same force fields could be used if a full quartic force field had been determined initially.

So, if I wanted to calculate the full quartic force field, what additional calculations would need to be run? Would it just be a case of calculating more hessians along the normal coordinates of each vibration, or would implementation of analytical third derivatives be required, or something else entirely?

  1. Wilson Jr., E. B.; Decius, J. C.; Cross, P. C., Molecular Vibrations; Dover Publications, Inc.: New York, 1980.
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  • $\begingroup$ I suppose you are speaking about bond and angle anharmonic terms, since dihedrals usually are already treated anharmonically? As far as I understand, at each displaced geometry you evaluate energy plus all the available analytical derivatives up to order $n$ to use them in your objective function for minimisation of the difference between QM and MM data. Finally, you could maybe consider introducing off-diagonal terms in your force-field to introduce couplings between internal coordinates? $\endgroup$
    – Anon
    Commented Oct 29, 2023 at 11:10
  • $\begingroup$ @Anon I’m not quite sure what you are suggesting. VPT2 doesn’t use any MM data; it’s an entirely QM method (when I use it, at least). And the off-diagonal terms are used in VPT2; in fact, the point of GVPT2 is to correct some of the problems that arise when couplings become too strong between different vibrations (VPT2 does not address these inherently, so some terms can ‘blow up’ and give very strange frequencies). $\endgroup$
    – isolated matrix
    Commented Oct 29, 2023 at 16:54
  • $\begingroup$ Then I'm not sure what you mean by quartic force field. From the question it seems that you want to obtain MM parameters to describe anharmonically some terms of the potential that are usually described as harmonic. I clearly misunderstood your question. $\endgroup$
    – Anon
    Commented Oct 29, 2023 at 17:46
  • $\begingroup$ @Anon perhaps I'm misunderstanding... I'm assuming by 'MM' you mean 'molecular mechanics', as in the context of QM:MM methods, like ONIOM calculations? In which case, MM-style force fields (which is what I assume you are referring to) are not involved. The VPT2 calculations I am talking about are entirely ab initio- or DFT-based. $\endgroup$
    – isolated matrix
    Commented Oct 30, 2023 at 5:39
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    – Tyberius ♦
    Commented Nov 24, 2023 at 18:36

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