324 Free energy calculations
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f (l)(a) (b)Fig. 8.2(a) Free energy profiles from eqn. (8.3.7). The solid line indicates switches
f(λ) = (λ^2 −1)^2 andg(λ) = ((λ−1)^2 −1)^2 , and the dashed line indicatesf(λ) = (λ^2 −1)^4
andg(λ) = ((λ−1)^2 −1)^4. (b) Corresponding switchf(λ).
λ-path for the problem of crossing a high barrier. That is, we have created what is
commonly referred to as arare-eventproblem (see also Section 8.5).
In order to overcome the rare-event problem, we introduce an approach in this
section known asadiabatic free energy dynamics(AFED) (Rossoet al., 2002; Rosso
et al., 2005; Abramset al., 2006). A related method known as canonical adiabatic free
energy samples was proposed by VandeVondele and R ̈othlisberger(2002). LetU‡be
the value of the potential energy at the top of the barrier. In thecanonical ensemble, the
probability that the system will visit a configuration whose potentialenergy isU‡at
temperatureTis proportional to exp[−U‡/kT], which is exceedingly small whenkT≪
U‡. The exponential form of the probability suggests that we could promote barrier
crossing by simply raising the temperature. If we do this na ̈ıvely, however, we broaden
the ensemble distribution and change the thermodynamics. On the other hand, suppose
we raise the “temperature” of just theλdegree of freedom. We can achieve this by
couplingλto a thermostat designed to keep the average〈p^2 λ/ 2 mλ〉=kTλ, where
Tλ > T. In general, the thermodynamics would still be affected. However,under
certain conditions, we can still recover the correct free energy.In particular, we must
also increase the “mass” parametermλto a value high enough thatλisadiabatically
decoupled from all other degrees of freedom. When this is done, it can be shown that
the adiabatically decoupled dynamics generates the correct free energy profile even
though the phase space distribution is not the true canonical one.Here, we will give
a heuristic argument showing how the modified ensemble distribution and free energy
can be predicted and corrected. (Later, in Section 8.10, we will analyze the adiabatic
dynamics more thoroughly and derive the phase space distribution rigorously.)
Under the assumption of adiabatic decoupling betweenλand the physical degrees
of freedom,λevolves very slowly, thereby allowing the physical degrees of freedom to
sample large portions of their available phase space whileλsamples only a very local-
ized region of its part of phase space. Since the physical degrees of freedom are coupled
to a thermostat at the physical temperatureT, we expect the adiabatic dynamics to