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Hybrid Models



\(pK_a\) (Born-Haber cycle)

\(pK_a\) is typically calculated using the Born-Haber cycle with the equation:

Where \(\Delta G^{\circ\prime}=\Delta G^{\circ\prime}_{aq}\)

\[ \begin{align} pK_a&=-\log\bigg[\exp{\bigg(\frac{-\Delta G^{\circ\prime}}{RT}\bigg)}\bigg]\\ &=\frac{\Delta G^{\circ\prime}}{2.303RT} \end{align} \]

Using the free energy cycle:



  1. For the anion, I need diffuse functions, a big basis set and a good level of theory
  2. QM packages won’t let you calculate the electronic energy of \(\ce{H+}\), since there’s no electron to calculate
  3. \(\Delta G^\circ_{(s)}=-264.0\:kcal\cdot mol^{-1}\) (experimentally derived)
  4. Standard-state concentration-change free energy must be included
  5. each non-cancelled error of \(1.4\:kcal\cdot mol^{-1}\) in any step will lead to an error of 1 \(pK_a\) unit
    • Errors in ionic solvation free energies can be much larger than that
  6. Function-group systematic errors can be corrected for

The overall equation

We can condense this all into the one line equation:

\[ 2.303RT\:pK_a=\Delta G_g^\circ(AH)-\Delta G^*_{aq}(AH)+\Delta G^*_{aq}(A^-)+\Delta G_{aq}^*(H^+) \]

Using this free energy cycle:


Free energy cycles and ion structures

Here we’re treating the ion as a cluster, in an attempt to try and reduce the amount error on the cluster:

\[ 2.303RT \: pK_a = \Delta G_g^\circ (AH) - \Delta G^*_{aq}(AH) - \Delta G^*_{aq}(\ce{H2O})+\Delta G^*_{aq}(\ce{H2O}\cdot A^-)+\Delta G_{aq}^*(H^+) \]

Utilising this free energy cycle:


Comparison (Experimental Data \(pK_a=15.5\))

Experimental data

Method 1 \(\ce{MeOH/MeO-}\) Method 2 \(\ce{MeOH/H2O.MeO-}\)
\(\Delta G^\circ_g=375.0\:kcal\cdot mol^{-1}\) \(\Delta G^\circ_g=358.0\:kcal\cdot mol^{-1}\)
\(\Delta G_{aq}^*(\ce{H+})=-265.9\:kcal\cdot mol^{-1}\) \(\Delta G_{aq}^*(\ce{H+})=-265.9\:kcal\cdot mol^{-1}\)
\(\Delta G_{aq}^*(\ce{MeOH})=-5.11\:kcal\cdot mol^{-1}\) \(\Delta G_{aq}^*(\ce{MeOH})=-5.11\:kcal\cdot mol^{-1}\)
\(\Delta G_{aq}^*(\ce{H2O})=-6.32\:kcal\cdot mol^{-1}\)

Computed data (SM6)

Method 1 \(\ce{MeOH/MeO-}\) Method 2 \(\ce{MeOH/H2O.MeO-}\)
\(\Delta G^*_{aq}(\ce{MeO-})=-88.3\:kcal\cdot mol^{-1}\) \(\Delta G^*_{aq}(\ce{H2O.MeO-})=-81.8\:kcal\cdot mol^{-1}\)
\(pK_a=20.4\) \(pK_a=16.0\)

Adding more waters

This is the experimental \(pK_a\) data


\[ \ce{H2CO3 <=>[pK_{a_1}=6.4] HCO3- <=>[pK_{a_2}=10.3] CO3^{2-}} \]

Without adding explicit waters of solvation, the results are insanely inaccurate, however when we add water molecules in, the structure becomes much more stabilised and the \(pK_a\) is much more in line with experiment.

No. \(\ce{H2O}\) \(pK_{a_1}\) \(pK_{a_2}\)
0 -0.6 1.6
1 1.3 5.0
2 2.3 7.8
3 4.2 9.0