pKa values
**********

COSMO-RS pKa

In the literature one can find   several strategies to calculate pK_a
values.   Some of these strategies involve the inclusion of explicit
solvent molecules, in combination with a continuum model, like
COSMO. The strategies described in the next examples do not take
into account explicit solvent molecules, only the continuum model   is
included.


Empirical pKa calculation method
================================

In Refs. [511,512] one can find   several strategies to calculate pK_a
values.   The strategy described in this example does not take into
account   explicit solvent molecules, only the continuum model is
included.   Like in Ref. [512] an empirical fit is   used to correlate
calculated values with experimental   pK_a  values, to account for
systematic errors.   The largest errors are probably present in the
calculation of the   solvation energies of the charged species. In
these cases one   probably should include explicit solvent molecules,
however, that   may introduce other sources of errors, and this will
not be   investigated here. The calculation of pK_a    values of acids
(HA) and bases (B) is based on the following   reaction model:

      (acids) HA (aq, 1M) + H2O (l, 1M) → H3O^+ (aq, 1M) + A^- (aq, 1M)        (5.1.1)

      (bases) HB^+ (aq, 1M) + H2O (l, 1M) → H3O^+ (aq, 1M) + B (aq, 1M)        (5.1.2)

The free energy of dissociation for acids and bases is calculated as

      (acids) \DeltaG^*_diss = G(A^-) - G(HA) + G(H3O^+) - G(H2O)        (5.1.3)

      (bases) \DeltaG^*_diss = G(B) - G(HB^+) + G(H3O^+) - G(H2O)        (5.1.4)

The ^*  denotes a standard state of 1M (1 mol/L). The   pK_a  can then
be calculated as (see for   example Refs. [511,512])

      pK_a = \DeltaG^*_diss/(RT ln(10)) - 1.74        (5.1.5)

At T = 298.15, 1/(RT ln(10)) = 0.733 mol/kcal. The term -1.74 is   to
correct for the standard state of liquid water, which is 55   mol/L.

**Emperical fit**

Like in Ref. [512] instead of this   equation (5.1.5), a linear fit
has been made by correlating the   calculated \DeltaG^* _diss  values
with experimental   pK_a  values, to account for systematic errors
that are present in this method. For acids and bases a different
empirical adjusted equation will be used, optimized for the ADF
COSMO-RS implementation:

      (acids) pK_a = 0.62 \DeltaG^*_diss/(RT ln(10)) + 2.10        (5.1.6)

      (bases) pK_a = 0.67 \DeltaG^*_diss/(RT ln(10)) - 2.00        (5.1.7)

These fitted parameters are not so far from the fitted parameters in
Ref. [512].   Zero-point vibrational energies have not been taken into
account   in the calculation of the free energy of dissociation.   H_3
O^+  (Hydronium ion) is the conjugate acid   of Water. Molecules can
have two or more equivalent sites for   protonation or deprotonation
are also not taken into account,   which can have an effect on the
pK_a  value.   However, such effects are not taken into account here.
Like for   neutral compounds, one should optimize the anions and
cations in   the gas phase, and use this geometry also in the COSMO
calculation. It is important to choose the lowest energy   conformer.
In the example below the molecules have a single   relevant
conformation for the protonated and deprotonated form.

**Acids**

Add the compounds in "$AMSHOME/examples/COSMO-
RS/pKa/tutorial5.1_acid.compoundlist" using the **Add compounds**
function in the GUI (**Compounds → List of Added Compounds**).   The
tutorial5.1_acid.compoundlist is a file with a list of acids   and
their conjugate bases that is limited to the compounds needed   in
this example. In these .coskf files already the correct number   of
ring atoms is included.

   Open a new COSMO-RS GUI window
   Add the compounds listed in tutorial5.1_acid.compoundlist
   Select **Properties → Activity coefficients**
   Select 'Water' for the first component in Solvent
   Enter '298.15' for 'Temperature Kelvin'
   Check the '+' button to add the first 12 compounds from the list
   Press 'Run'
 [image]

Using equation 5.1.6 (pK_a  =   0.62*0.733* \DeltaG_diss +2.10), with
\DeltaG_diss  =   G(conjugate_base_acid) - G(acid) +
G(conjugate_acid_Water) -   G(Water) in kcal/mol, results can be put
in a table.

+---------------------------+---------------------------+---------------------------+---------------------------+
|                           | Acid                      | experimental pK _a [512]  | calculated pK _a          |
|===========================|===========================|===========================|===========================|
| 1                         | Methanol                  | 15.5                      | 16.27                     |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 2                         | Ethanol                   | 15.9                      | 15.97                     |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 3                         | Phenol                    | 9.82                      | 10.32                     |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 4                         | Acetic acid               | 4.75                      | 3.64                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 5                         | Benzoic acid              | 4.27                      | 4.28                      |
+---------------------------+---------------------------+---------------------------+---------------------------+

**Bases**

Copy the .coskf files which are listed in
$AMSHOME/examples/crs/Tutorial5/tutorial5.1_base.compoundlist and
the file tutorial5.1_base.compoundlist the directory Tutorial.   The
tutorial5.1_base.compoundlist is a file with a list of bases   and
their conjugate acids that is limited to the compounds needed   in
this example. In these .coskf files already the correct number   of
ring atoms is included.

   Open a new COSMO-RS GUI window
   Add the compounds listed in tutorial5.1_base.compoundlist
   Select **Properties → Activity coefficients**
   Select 'Water' for the first component in Solvent
   Enter '298.15' for 'Temperature Kelvin'
   Check the '+' button to add the first 16 compounds from the list
   Press 'Run'
 [image]

Using equation 5.1.7 ((pK_a  =   0.67*0.733* \DeltaG_diss -2.00), with
\DeltaG_diss  =   G(base) - G(conjugate_acid_base) +
G(conjugate_acid_Water) -   G(Water) in kcal/mol, results can be put
in a table.

+---------------------------+---------------------------+---------------------------+---------------------------+
|                           | Base                      | experimental pK _a [512]  | calculated pK _a          |
|===========================|===========================|===========================|===========================|
| 1                         | Aniline                   | 4.6                       | 5.75                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 2                         | 1H-Imidazole              | 7                         | 5.65                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 3                         | Pyrazine                  | 0.7                       | 0.85                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 4                         | Pyrazole                  | 2.5                       | 2.55                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 5                         | Pyridine                  | 5.14                      | 4.62                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 6                         | Quinoline                 | 4.80                      | 4.43                      |
+---------------------------+---------------------------+---------------------------+---------------------------+
| 7                         | Guanidine                 | 13.8                      | 14.09                     |
+---------------------------+---------------------------+---------------------------+---------------------------+

511. J. Ho and M.L. Coote, A   universal approach for continuum
     solvent pK_a    calculations: are we there yet?,  Theoretical
     Chemistry Accounts 125, 3 (2010)

512. F. Eckert, M. Diedenhofen, and   A. Klamt, Towards a first
     principles prediction of   pK_a : COSMO-RS and the cluster-
     continuum approach,    Molecular   Physics 108, 229 (2010)


Relative pKa calculation method
===============================

The method described in this example is based on one of the
strategies in Ref. [521]. This method uses   the experimental pK_a
value of a reference   compound, experimental gas phase deprotonation
energies, and   COSMO-RS solvation free energies. A suitable reference
compound   (HRef) should be chosen, which is similar to the actual
compound   (HA) one is interested in. For example, the deprotonation
reaction could be similar in both compounds. The calculation of   pK_a
values is based on the following reaction   model:

[image]

The ^*  denotes a standard state of 1M (1 mol/L). The   free energy of
solvation is then calculated as

      \DeltaG^* _soln  = \DeltaG^* _gas  +   \DeltaG^* _solv (HRef) +   \DeltaG^* _solv (A^- ) -   \DeltaG^* _solv (HA) -   \DeltaG^* _solv (Ref^- )

The pK_a  can then be calculated as (see for   example Ref. [521])

      pK_a  = \DeltaG^* _soln /(RT   ln(10)) + pK_a (HRef)

The success of this method relies on the availability of a   suitable
reference compound with an accurately known experimental   pK_a
value. Instead of experimental gas phase   deprotonation energies one
might calculate the gas phase reaction   free energy \DeltaG^* _gas ,
using DFT or some high   level ab initio method.

**Example Ethanol**

In this example the pK_a  value of Ethanol will   be calculated. As
reference compound Methanol is chosen, which   has an experimental
pK_a  value of 15.5.   Experimental deprotonation energies are taken
from Ref. [521, supporting   information], for Methanol this is
\Delta_r G^0  =   1569 kJ/mol, and for Ethanol \Delta_r G^0  = 1555
kJ/mol, see also, for example, Ref. [522]. The COSMO-RS   solvation
free energies,  \Delta\text{G(gas->solute)}, of Methanol, Methoxide
(the conjugate   base of Methanol), Ethanol, and Ethoxide (the
conjugate base of   Ethanol) have already been calculated in the
previous tutorial   5.1. At T = 298.15, 1/(RT ln(10)) = 0.733
mol/kcal. The free   energy of solvation (kcal/mol) and the pK_a  of
Ethanol are then calculated as

      \DeltaG^* _soln  = (1555-1569)/4.184 - 4.97 - 86.56   + 4.79 + 90.14 = 0.05 kcal/mol

      pK_a  (Ethanol) = 0.733*0.05 + 15.5 \approx 15.5

The calculated pK_a  of Ethanol of 15.5 is   close to the experimental
value of 15.9.

521. J. Ho and M.L. Coote, A   universal approach for continuum
     solvent pK_a    calculations: are we there yet?,  Theoretical
     Chemistry Accounts 125, 3 (2010)

522. NIST Chemistry WebBook
