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

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?