336 Free energy calculations
constantKw(Trout and Parrinello, 1998). A clearer picture of this reaction waspro-
vided by Chandler and coworkers (2001) using the transition path sampling (Dellago
et al., 1998; Bolhuiset al., 2002; Dellagoet al., 2002) discussed in Section 7.7. From
their calculations, these authors posited that that the dissociation reaction is complete
only when the H 3 O+and OH−ions separate in such a way that no contiguous path
of hydrogen bonds exists between them that would allow them to recombine through
a series of proton transfer steps. In order to describe such a process correctly, a very
different type of reaction coordinate that involves many solvent degrees of freedom
would be needed. Later, in Section 8.12, we will describe a technique for judging the
quality of a reaction coordinate.
Keeping in mind our caveats about the use of reaction coordinates,we now describe
a number of popular methods designed to enhance sampling along preselected reaction
coordinates. All of these methods are designed to generate, either directly or indirectly,
the probability distribution function of a subset ofnreaction coordinates of interest
in a system. If these reaction coordinates are obtained from a transformation of the
Cartesian coordinates to generalized coordinatesqα =fα(r 1 ,...,rN),α = 1,...,n,
then the probability density that thesencoordinates will have valuesqα=sαin the
canonical ensemble is
P(s 1 ,...,sn) =CN
Q(N,V,T)
∫
dNpdNre−βH(r,p)∏nα=1δ(fα(r 1 ,...,rN)−sα), (8.6.4)where theδ-functions are introduced to fix the reaction coordinatesq 1 ,...,qnats 1 ,...,sn.
OnceP(s 1 ,...,sn) is known, the free energy hypersurface in these coordinates is given
by
A(s 1 ,...,sn) =−kTlnP(s 1 ,...,sn). (8.6.5)
8.7 The blue moon ensemble approach
The term “blue moon,” as the name implies, colloquially describes a rare event.^4 The
blue moon ensemble approach was introduced by Carteret al.(1989) and Sprik and
Ciccotti (1998) for computing the free energy profile of a reaction coordinate when
one or more high barriers along this coordinate direction lead to a rare-event problem
in an ordinary thermostatted molecular dynamics calculation.
Suppose a process of interest can be monitored by a single reactioncoordinate
q 1 =f 1 (r 1 ,...,rN). Then according to eqns. (8.6.4) and (8.6.5), the probability that
f 1 (r 1 ,...,rN) has the valuesand the associated free energy profile, are given by
P(s) =CN
Q(N,V,T)
∫
dNpdNre−βH(r,p)δ(f 1 (r 1 ,...,rN)−s)=
1
N!λ^3 NQ(N,V,T)∫
dNre−βU(r)δ(f 1 (r 1 ,...,rN)−s)(^4) A “blue moon” usually refers to the occurrence of a second full moon in a calendar month. The
extra full moon, occurs roughly every 2.7 years and is, therefore, a rare event.