Essentials of Computational Chemistry

436 12 EXPLICIT MODELS FOR CONDENSED PHASES
The removal of the ensemble average over the λ 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 E. Since each change in λ 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
〈A〉B −〈A〉A = lim
1
〈(Eλ+dλ − Eλ)〉 λ
dλ→0
λ=0
1
(Eλ+λ − Eλ)
= lim
λ→0
λ=0
1
∂E
= dλ
0 ∂λ λ
λ
λ
λ
1
∂E
≈
∂λ
λ (12.20)
λ=0
where we first recognize the calculus relationship between the sum appearing on the r.h.s. in
the second line and the definite integral in the third line (and simultaneously the definition
of the partial derivative), and we then approximate the definite integral as a sum over small
intervals. While the transformation from line 3 to line 4 may appear to simply reverse the
transformation from line 2 to line 3, this is not the case, because the partial derivative
remains in its analytic form; this is possible because most simulations evaluate the energy
using E(λ) functions that are trivially differentiated. Moreover, λ in the final line is no
longer infinitesimally small, i.e., this is a standard estimation of an integral by division of the
integration range into discrete intervals with the function approximated over each interval
by a single value, in this case the value at the start of the interval. This process defines the
thermodynamic integration (TI) method. [TI can be derived in a much more rigorous and
general way, and indeed, FEP may be regarded as a special case of TI; interested readers
are referred to the bibliography at the end of the chapter.]
It is evident that TI and FEP are similar in that they involve multiple simulations over
different windows λ, with accuracy expected to increase when more and smaller windows
are employed. However, there are key differences as well. In TI, the ensemble average for
one value of λ is not used to evaluate any energies involving a different value of λ; only
the ensemble average of the energy derivative is accumulated. Moreover, different forms
of E(λ) may be conveniently evaluated, corresponding to different mutation paths from A
λ