10.5 Estimation of free energy and chemical potential 317method for doing this is thermodynamic integration, described inSection 7.1. This
method suffers from the fact that many simulations are needed in order to calculate
free energy differences. We shall now describe a few alternative procedures for this
purpose.
Suppose the two systems have interactions described by the potential functions
U 0 andU 1. We show now that the partition function of system 1 can be rewritten as
a mechanical average with the Boltzmann factor of system 0. Writing the respective
partition functions asZ 0 andZ 1 we have
Z 1
Z 0=
∑
Xe
−βU 1
∑
Xe−βU^0=
∑
Xe−βU (^0) e−β(U 1 −U 0 )
∑
Xe−βU^0
=〈e−β(U^1 −U^0 )〉 0 , (10.43)
where〈...〉 0 represents an ensemble average in system 0.
Applications of (10.43) are restricted to systems for which the regions they tend
to occupy in phase space have appreciable overlap. This can be seen by considering
two systems with the same potential function but kept at different temperatures,
U 1 =αU 0 .Ifα<1, system 1 is at a higher temperature than system 0. If we
evaluate the expectation value in(10.43)using MC or MD, the result contains the
contributions toZ 1 arising from the equilibrium states of the same system at lower
temperature – hence the free energy estimate will be drastically wrong as a result of
not taking the overwhelming majority of high energy states contributing toZ 1 into
account. On the other hand, ifα>1, the system will visit configurations with high
energy and rarely assume one of the lower energy configurations which contribute
to the partition function of system 1. Sufficient overlap between the phase space
volume occupied by the two systems is essential. One way to achieve such overlap
is to simulate achainof systems with potentialsUpthat interpolate betweenU 0
andU 1 , such that subsequent configurational potential functionsUp,Up+ 1 have
sufficient overlap to give reliable results. This means that the amount of computer
time needed is comparable to that of thermodynamic integration, which uses a chain
of different thermodynamic parameters.
Torrie and Valleau have refined the method by adding an extra weight function
W(X)to the average in(10.43), which pushes the system to a different region of
phase space such as to reduce the overlap problem. Their method is called ‘umbrella
sampling’ and we refer to their paper for a description of the method and examples.
[ 34 , 35 ].
Bennett[33]writes the free energy difference in another way, by defining the
‘Metropolis function’M(x)asM(x)=min[1, exp(−x)]. Then we have
exp[−β(U 0 −U 1 )]=M[β(U 0 −U 1 )]/M[β(U 1 −U 0 )] (10.44)
from which it follows that
Z 0
∑
XM[β(U^1 −U^0 )]e
−βU 0
Z 0=Z 1
∑
XM[β(U^0 −U^1 )]e
−βU 1
Z 1