The most straight forward approach is to use the relation between the equilibrium constant

$\mathrm{p}{K}_{\mathrm{a}}$ and the difference in free energy (

$\Delta F$):

with

$\beta =1/{k}_{\mathrm{B}}T$.

In order to reproduce experimental

$\mathrm{p}{K}_{\mathrm{a}}$ values one has to assure that the difference in free energy sufficiently describes the two states. This boils down to the nature of the constraint as well as to the range of sampling the corresponding phase space. In any case a key requirement is the convergence of

$\Delta F(\xi )$ to a constant value towards the limits of the sampled range (

${\xi}_{min}$ to

${\xi}_{max}$), i.e.,

$\underset{\xi \to {\xi}_{max}}{lim}\Delta F(\xi )$ = constant. Further, in order to obtain meaningful results from the thermodynamic integration the free energy at the bound state F

$\left({\xi}_{0}\right)$ has to be set to unity. This method has been successfully applied to several systems using either the d(A−H)−d(O−H), the CN(A−H) or CN(A−H)−CN(O−H) (see Equation (

3)) constraints [

2,

3,

39].

Whether the calculated free energy difference

$\Delta F$ can directly be related to the equilibrium constant is disputed in literature [

4,

6,

31]. Authors who disagree with the previously discussed method commonly refer to Chandler’s derivations of the equilibrium constant based on a classical statistical mechanical description. The basic principle described there is the relation of the free energy difference

$\Delta F$ and the radial distribution function (RDF) according to the reversible work theorem [

40]. The RDF itself might be interpreted as the probability to find a proton within a certain radius of the acidic group. The probability distribution is related to the (inverse) acidity constant according to [

31]:

where

${c}_{0}$ is the standard concentration. Note the difference in free energy

$\Delta F\left(r\right)$ is a function of radius of a sphere around the acidic group and not of the constraint

$\xi $. The latter are only equal in case of the distance constraint d(A−H). In principle

$\Delta F\left(r\right)$ has to be known for infinite separation, in practice however only a finite separation

${R}_{max}\le L/2$ (L is the length of the cubic cell simulation) is accessible. Since

$\Delta F\left(r\right)$ asymptotically approaches a constant value, one often defines

${R}_{c}\le {R}_{max}$, where

${R}_{c}$ is the radius which distinguishes A−H from A

^{−} + H

^{+}, i.e., the distance at which the covalent bond is broken [

31]. The limitations with respect to the simulation cell result in an uncertainty in the

$\mathrm{p}{K}_{\mathrm{a}}$ value which Davies et al. quantified as

$\Delta F\left({R}_{max}\right)/\left(2.3{k}_{\mathrm{B}}T\right)$ [

31].