# Calculate deprotonation energy or Gibbs free energy of acids

I'm delving into the world of superacids, acids with an acid strength greater than that of sulfuric acid at 100%. I've recently encountered some computational papers which suggest that the deprotonation energy of an acid is a useful measure of its strength.

Assuming the following equilibrium

$$\ce{HA -> H+ + A-}$$

one can calculate the deprotonation energy (DPE) by $$\ce{DPE = E_{H+} + E_{A-} - E_{HA}}$$. It is my understanding that the energy of $$\ce{H+}$$ derived from DFT should be zero because of the absence of electrons, and that the positive and negative charges of the unit cell should introduce some errors.

While some paper refer to DPE, others mention deprotonation Gibb's free energy as measure of acid strength. There seems to be some correlation between the two but I can not identify the difference.

Despite the fact I'm not sure what your question is, I'll try to answer it :-)

the energy of H+ derived from DFT should be zero

The idea is that a pure $$\ce{H+}$$ ion exists in a gas phase only, whereas in a liquid state it's bound by a base, which is usually a solvent. For water solutions usually $$\ce{H3O+}$$, $$\ce{H5O2+}$$, and $$\ce{H3O+(H2O)3}$$ ions are considered. This approach is called the "explicit solvent approach". In addition, you should add an implicit solvent model to your computations, e.g. PCM, CPCM, etc. Check this paper  as an example.

negative charges of the unit cell should introduce some errors

Usually, such systems are computed without boundary conditions using non-periodic codes like Orca, Gaussian, GAMESS, etc. Therefore you can compute isolated ions without technical or theoretical problems.

While some paper refer to DPE other mention deprotonation Gibbs free energy as measure of acid strength. There seems to be some correlation between the two but I can not identify the difference.

Deprotonation energy is computed using electronic energies, and deprotonation Gibbs free energy is computed via Gibbs energies. To obtain Gibbs energies and other thermodynamic parameters from DFT calculations, one needs to run frequency computation for the optimized structure.

The final note: in some approaches, DFT is used to compute $$E(\ce{AH})$$ and $$E(\ce{A-})$$ only, whereas for $$E(\ce{H+})$$ (which actually is a proton solvation energy) the experimental value is used. Check this paper  for more details.

#### References

1. Steenken, S.; Reynisson, J. DFT Calculations on the Deprotonation Site of the One-Electron Oxidised Guanine–Cytosine Base Pair. Physical Chemistry Chemical Physics, 2010, 12, 9088. https://doi.org/10.1039/c002528c.
2. Dutra, F. R.; Silva, C. de S.; Custodio, R. On the Accuracy of the Direct Method to Calculate pKa from Electronic Structure Calculations. The Journal of Physical Chemistry A, 2020, 125, 65–73. https://doi.org/10.1021/acs.jpca.0c08283.
• +1 you're on a roll! Mar 16 at 15:06
• @NikeDattani why thank you! though judging by the profile it's you who are in a roll ;-) Mar 16 at 15:15
• Hi @IvanChernyshov thanks for the answer. So I guess most of the energy contribution for gas-phase acidity which seems to be approximated to Gibbs energy, most of the energy contribution are coming from the DPE and then there should be some sort of correction obtained from the vibrational analysis. I'm modelling periodic system so I need to handle that error also Mar 16 at 15:41
• Something I would like to add is that I'm assuming that my material has not solvent, so the proton species is the isolated H+ not the solvated one. So I guess is more similar to gas-phase acidity Mar 16 at 15:43
• @manuelpb I can think of two approaches to this: 1) cut part of your slab or crystal and compute it in a gas phase (cluster approach, example); 2) add a weak base (e.g. CO) to the acidic group and analyze change in X-H bond frequency, example Mar 17 at 6:45