Should we really worry about anharmonicity?

My colleague, Dr. Matas, asked herself a question. For a very shallow minimum on the potential energy surface (PES), is the quantum harmonic oscillator approximation (or rigid-rotor-harmonic-oscillator (RRHO)) still valid? Should we worry about its error, especially when calculating the entropy and free energy, which is extremely sensitive with respect to very small vibrational frequencies? The small vibrational frequencies often occur in “floppy molecules” or molecules with weak interactions, such as hydrogen bond or van der Waals interactions.

In order to account for the anharmonicity, there are two ways as far as I know:

  • The quasi-harmonic approximation from either Cramer and Truhlar, or Grimme. Basically, all frequencies below the cut-off (typical around 100 cm-1) are uniformly shifted up to the cut-off value before entropy calculation in the RRHO approximation. You can use a nice python program here to calculate the corrections.
  • Explicitly calculate the anharmonic frequencies based on performing numerical differentiation. In the program “You-Know-Who” (YKW), this can be easily enabled by using Freq=Anharmonic.

The guinea pig

Let’s start with the most simple system, the “guinea pig” of weak interaction, aka the water dimer. I used the infamous B3LYP-D3 functional, aug-cc-pVTZ basis set, and a very dense DFT grid. The result is what I expect for the binding energy ΔE (with ZPE corrected) of the water dimer: 3.07 kcal/mol. The standard free energy ΔG is -2.58 kcal/mol.

How about applying the quasi-harmonic approximation (Truhlar or Grimme)? Nothing changes since all the frequencies are larger than 100 cm-1.

The best approximation from Freq=Anharmonic is ΔE = 3.37 kcal/mol and ΔG = -1.30 kcal/mol. I see that ΔE changes by only 0.3 kcal/mol and ΔG changes by 1.28 kcal/mol. The changes are small IMHO, so I think the RRHO works well for this system.

Anti-Electrostatic Hydrogen Bonds

The next system is an interesting one, stolen from the work of Frank Weinhold and Roger A. Klein. This is an interesting paper (shown to me by my FABULOUS friend), in which the authors design imaginary systems involving bonding between two anions in gas phase, e.g. F and HO-CO2. Normally we think that because of the electrostatic force between to anions, the two anions shouldn’t bind, but the authors shows that because of the “anti-electrostatic” hydrogen bond between F … H, there is an extremely shallow minimum on the PES. Interesting, I think that, indeed, the minimum exists! The paper also aims at showing that hydrogen bond is not a simple electrostatic attraction, and we should “rest the superficial quasi-classical conceptions of H-bonding”.
Capture
High-quality picture adapted from low quality picture in the paper of Weinhold and Klein.

Despite the fact that the paper presented an usual and interesting bond between two anions, I have to criticize it, as what Gernot Frenking and Giovanni F. Caramori commented.

  • Frenking and Caramori wrote “It takes much chutzpah to use an energy-minimum structure with a well depth of 0.05(!) kcal/mol at B3LYP/aug-cc-pVTZ.” I have to open a dictionary for the word “chutzpah“. Indeed, Weinhold and Klein should be aware that 0.05 kcal/mol(!) at DFT level shouldn’t be taken seriously? Or did they just ignore that? Again, what’s wrong with the peer-review process, on Angewandte?
  • The NBO analysis shouldn’t be used as an quantitative results. It’s just a model. A model!
  • On the other hand, the comments of Frenking and Caramori are not better, I think even worse. They completely ignored the fact that there is a deep minimum (much more than 0.05 kcal/mol) in other similar systems. The paper mainly “bashed” the NBO method and “demonstrated” that energy decomposition analysis (EDA, a technique extensively used by Frenking) is much better and can give more “quantitative” results. Please, EDA is also just a model!

So, let me now reproduce the extremely shallow minimum on the PES by Weinhold and Klein, and let me add something more, e.g. to answer the question: is anharmonicity important for this extreme system?

Again, I used the B3LYP-D3 functional and aug-cc-pVTZ basis set. I also found a shallow minimum and a reaction barrier of 0.12 kcal/mol(!). The free energy is 0.58(!) kcal/mol and around 0.50 kcal/mol using quasi-harmonic approximation by Truhlar or Grimme. With Freq=Anharmonic, I have 0.17 and 0.76 kcal/mol, respectively.

Conclusion

RRHO works in these two extreme cases, I think it is fine to ignore the harmonicity error in most cases.

According to Betteridge’s law of headlines, the answer to Dr. Matas question is NO!

PS. I’m now not afraid of the small relative energy any more. If famous chemists can publish DFT results of 0.05 kcal/mol on Angewandte, I can publish 0.01 kcal/mol on my blog, right?

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