Slow Growth and Thermodynamic Integration

one may imagine taking X intervals so small that AE on any given interval is arbitrarily close to zero. In that case, we may represent the exponential as a truncated power series, deriving 

This expression may be further simplified by noting that ln(l + x) is well approximated by X for sufficiently small values of x, so that we may write 

The removal of the ensemble average over the A, ensemble in the final line on the r.h.s. reflects the protocol of this technique, the so-called slow-growth method. It is assumed that if the Hamiltonian is infinitesimally perturbed at every step in the simulation, then the system will constantly be at equilibrium (following some initial period of equilibration), so separate ensemble averages need not be acquired. 

In practice, then, the slow-growth technique is rather different from FEP when it comes to evaluating AE. Since each change in A is also a step in the simulation, all of the intrasolvent energy terms change in addition to the solvent-solute interaction terms. With respect to the latter terms, however, the evaluation is similar to FEP in that chimeric molecules are involved. 

A third simulation protocol for determining Helmholtz free-energy differences can be illustrated from further manipulation of Eq. (12.19). Thus we may write 

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