# Quantum calculation of free energy change in neat conditions

I am slightly baffled by a seemingly simple situation: I want to calculate the free energy of association between a molecule $$\ce{A}$$ and the solvent $$\ce{B}$$ using quantum methods (e.g. DFT).

From what I've understood, the basic way to do this in most situation is to optimize the geometry of each compound, then perform frequency calculations to obtain free energy corrections to the SCF. Let the SCF plus the corrections for specie $$\ce{X}$$ be $$\ce{G_X}$$. This yields the free energy of each compound at 1 atm, which can be converted into the free energy at 1 M via a factor of 1.89 kcal/mol. Thus, for the complexation $$\ce{A + B \rightleftharpoons AB}$$, the free energy of complexation would be

\begin{align} \Delta G_{comp}^0 &= (G_{AB}^0 + \pu{1.89 kcal/mol}) - (G_{A}^0 + \pu{1.89 kcal/mol}) - (G_{B}^0 + \pu{1.89 kcal/mol}) \\ &= G_{AB}^0 - G_{A}^0 - G_{B}^0 - \pu{1.89 kcal/mol} \end{align}

However, in the case of association between molecule $$\ce{A}$$ with concentration $$\ce{C_A}$$ and solvent $$\ce{B}$$ with concentration $$\ce{C_B}$$, how could one obtain the free energy of complexation?

There is the equation $$\Delta G = \Delta G^0 - RT \ln\left(\frac{C_{AB}}{C_A*C_B}\right)$$, but the "initial" concentration of the adduct is null. There seems to be a $$RT \ln(C_B)$$ term floating around in the literature, but I'm not too sure where it's coming from and how to handle this situation rigorously.

One needs to distinguish between the standard free energy (of reaction) for a process, $$\Delta _r G^\circ$$, and the free energy in general, $$\Delta _r G$$. You show how to use quantum methods to compute $$\Delta _r G^\circ$$. This tells you how much the free energy changes if you take 1 mole of each reagent at 1 M concentration and generate one mole of complex at 1 M concentration. At other concentrations one uses
$$\Delta_r G = \Delta_r G^0 + RT \ln\left(\frac{C_{AB}/C^\circ}{(C_A/C^\circ )(C_B/C^\circ)}\right) \tag{Eq. 1}$$
for the reaction $$\ce{A +B<=>AB}$$ (assuming all components behave ideally; $$\pu{C^\circ= 1 M}$$ is the standard concentration). In fact, you use an analogous expression to compute the free energy difference between 1 atm (at $$\pu{0^\circ C}$$) and 1 M concentrations for each reagent or product:
\begin{align}\Delta_r G(\pu{1 atm}\rightarrow\pu{1 M}) &= RT \ln\left(\frac{C_X(\pu{1 M})}{C_X(\pu{1 atm})}\right)\\ &= RT \ln\left(\frac{\pu{1 M}}{(\pu{1e-3 \pu{L/m^3}})\cdot(1/(RT/\pu{1 atm})}\right)\end{align}
If you are interested in computing the equilibrium concentration of complex given specified concentrations of A and B (assumed constant) then you solve for $$C_{AB}$$ in Eq. (1) after plugging in values for $$C_A$$ and $$C_B$$, remembering again that you are assuming your solution behaves ideally.