# How to treat pressure in computational chemistry?

I have a reaction where a gas at high pressure and a solution with all kinds of species is involved. How do I take pressure correctly into account to get reasonable values for $$G$$?

Do I calculate thermochemical corrections at the given temperature and pressure (frequency calculations) and then add additional corrections for concentration ($$\Delta G = RT \ln(p)$$, where $$p$$ is (partial) pressure or mole fraction)? Or should I calculate the thermochemical corrections at the given temperature, but standard pressure? For the additional term, should I use the overall pressure of the gas, and the mole fraction of the species in solution, or should I check for the solubility of the gas at given temperature and pressure in the solvent?

Usually, I would approach it this way:

Geometry optimization at lower level of theory, then computation of $$G$$ as the sum of:

1. single point energy at higher level
2. $$G$$ thermodynamic corrections (at $$\pu{298.15 K}$$ and $$\pu{1 bar}$$)
3. $$G_\mathrm{solv}$$ corrections from COSMOtherm (at infinite dilution)
4. $$\Delta G_p = RT \ln(p)$$, where $$p$$ is either the molar fraction of the compound in solution or the partial pressure in gas phase

This works quite well, however, I'm not sure how to adjust the different terms when dealing with non-standard conditions. I think that as only dissolved gas participates in the reaction, I should be interested in this as a reference point for $$\Delta G_\mathrm{r}$$, etc., but with what temperature/pressure do I then calculate 2? And do I take the pressure of gas above the solution in 4 or the molar fraction of gas in solution?

• I am not sure to understand in detail your question, can you expand a little more?. Normally standard thermodynamic functions in gas phase is what we trivially calculate, but if there is a strong interaction between molecules (different from chemical reaction) it can be not useful. Apr 6 '16 at 14:56
• There are two complexities to this. 1. Determining the free energy of a pure component at a specified temperature and a high pressure beyond the ideal gas region and 2. Starting with the free energy of the pure components at the same total temperature and pressure and determining the free energy of the mixture. Are you familiar with the concept of fugacity? Do you know how to do the first step using dG=VdP at constant T. Apr 6 '16 at 19:45
• I tried to clarify the problem - does that help? It's true that strong interactions between the molecules have an impact, especially at higher pressures/concentrations, but I think this can be neglected in my case. Apr 7 '16 at 6:57
• That's a very interesting question. What about using a dynamical approach? Jan 30 '17 at 15:45

I recommend calculating the thermal correction for $$G$$ at the temperature of interest and the default pressure ($$e.g.$$ 1 atm for Gaussian). You can then add the correction for the isothermal expansion/compression to the actual concentration $$c_f$$ / partial pressure $$p_f$$: $$\int_{p_i}^{p_f}\bigg(\frac{\partial G}{\partial p}\bigg)_T dp = RT\ln\bigg(\frac{p_f}{p_i}\bigg) = RT\ln\bigg(\frac{c_f RT}{p_i}\bigg)$$ where $$p_i$$ is the pressure for the thermal correction ($$e.g.$$ 1 atm). Note that these formulas are based on the ideal gas approximation, but that's what software like Gaussian does under the hood anyway. Hence, you could in principle directly set the correct pressure and should get the same result, but I find it more convenient to have the same route section for all calculations.