The PDT and Thermodynamic Integration for Exact Quantum Free Energy Changes

10 The PDT and Thermodynamic Integration for Exact Quantum Free Energy Changes 

Say you have performed a classical calculation to determine the excess chemical potential from the first two terms on the right side of (11.22) followed by another classical calculation to obtain an estimate of the quantum correction from the expression (11.29), and the estimated correction is large. This suggests that a full quantum treatment is necessary. In this section, we derive the appropriate formulas for changes in the excess chemical potential due to mutating masses. If the original mass is very large, which corresponds to the classical limit, the derived expressions yield the quantum correction. 

Consider a problem, in which we are interested in the mass-dependent partitioning of a solute between an ideal gas phase and a condensed phase. The ratio of the densities in the two phases or the partition coefficient for species a is then 

Let us first examine changes in the quantum correction of (11.22) due to a change in mass. For that change we obtain 

We will use (11.5) yet again, but we should be careful to note that the normalization integrals for the numerator and denominator are slightly different. Taking care of 

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