group (changing benzene into toluene), the carbon“particle distance will

change from 0.110 to 0.152 nm. In a simulation with bond constraints,

the constraint length is modi¬ed as a function of the coupling parameter.

Each length modi¬cation in the presence of a constraint force involves a

change in free energy, as work is done against (or with) the constraint

force. So the work done by the constraint force must be monitored. The

constraint force Fc follows from the procedure used to reset the constraints

(see Section 15.8 on page 417); if the constraint distance rc is changed by

a small increment ”rc = (drc /d») ”», the energy increases with Fc ”rc .

Thus there is a contribution from every bond length constraint to the

ensemble average of ‚V /‚»:

‚V drc

= Fc . (7.45)

‚» d»

constr

In addition, there may be a contribution from the Jacobian of the trans-

formation from cartesian to generalized coordinates, or “ equivalently “

from the mass-metric tensor (see Section 17.9.3 and speci¬cally (17.199)

on page 501). The extra weight factor |Z|’1/2 in the constrained ensemble

8 See van Gunsteren et al. (1993), pp 335“40.

7.5 Free energy and potentials of mean force 227

may well be a function of » and contribute a term in dA/d»:

dA 1 ‚|Z|

|Z|1/2

= kB T . (7.46)

d» 2 ‚»

metric

The same arguments that are given in Section 17.9.3 to show that the

metric e¬ects of constraints are often negligible (see page 502) are also

valid for its »-dependence. Even more so: in closed thermodynamic cycles

the e¬ect may cancel.

• A large improvement of the e¬ciency to compute free energy changes for

many di¬erent end states (e.g., ¬nding the binding constants to a pro-

tein for many compounds) can be obtained by using a soft intermediate

(Liu et al., 1996; Oostenbrink and van Gunsteren, 2003). Such an inter-

mediate compound does not have to be physically realistic, but should

be constructed such that it covers a broad part of con¬gurational space

and allows overlap with the many real compounds one is interested in. If

well chosen, the change from this intermediate to the real compound may

consist of a single perturbation step only.

7.5 Free energy and potentials of mean force

In this section the potential of mean force (PMF) will be de¬ned and a

few remarks will be made on the relations between PMF, free energy, and

chemical potential. The potential of mean force is a free energy with respect

to certain de¬ned variables, which are functions of the particle coordinates,

and which are in general indicated as reaction coordinates because they are

often applied to describe reactions or transitions between di¬erent potential

wells. What does that exactly mean, and what is the di¬erence between a

free energy and a potential of mean force? What is the relation of both to

the chemical potential?

The potential energy as a function of all coordinates, often referred to as

the energy landscape, has one global minimum, but can have a very com-

plex structure with multiple local minima, separated by barriers of various

heights. If the system has ergodic behavior, it visits in an equilibrium state

at temperature T all regions of con¬guration space that have an energy

within a range of the order of kB T with respect to the global minimum. It

is generally assumed that realistic systems with non-idealized potentials in

principle have ergodic behavior,9 but whether all relevant regions of con¬g-

9 Idealized systems may well be non-ergodic, e.g., an isolated system of coupled harmonic oscilla-

tors will remain forever in the combination of eigenstates that make up its initial con¬guration

and velocities; it will never undergo any transitions to originally unoccupied eigenstates unless

there are external disturbances or non-harmonic terms in the interaction function.

228 Free energy, entropy and potential of mean force

V mf

P

R

ck T

B

T

reaction coordinate ξ

Figure 7.3 Potential of mean force in one “reaction coordinate” ξ. There are two

regions of con¬gurational space (R and P) that can be designated as con¬ning a

thermodynamic state.

uration space will indeed be accessed in the course of the observation time

is another matter. If the barriers between di¬erent local minima are large,

and the observation time is limited, the system may easily remain trapped in

certain regions of con¬guration space. Notable examples are metastable sys-

tems (as a mixture of hydrogen and oxygen), or polymers below the melting

temperature or glasses below the glass transition temperature. Thus ergod-

icity becomes an academic problem, and the thermodynamic state of the

system is de¬ned by the region of con¬gurational space actually visited in

the observation time considered.

Consider a system that can undergo a slow reversible reaction, and in the

observation time is either in the reactant state R or in the product state P. In

the complete multidimensional energy landscape there are two local minima,

one for the R and one for the P state, separated by a barrier large enough

to observe each state as metastable equilibrium. Let the local minima be

given by the potentials V0R and V0P . Then the Helmholtz free energy A (see

Chapter 17) is given “ for classical systems in cartesian coordinates “ by

A = ’kB T ln Q, (7.47)

7.5 Free energy and potentials of mean force 229

with

e’βV (r) dr,

Q=c (7.48)

where the integration is carried out for all particle coordinates over all space.

The constant c derives from integration over momenta, which for N particles

consisting of species s of Ns indistinguishable particles (with mass ms ) equals

3N /2

3N/2

ms s

2πkB T

c= Πs . (7.49)

h2 Ns !

Expressed in de Broglie wavelengths Λs for species s:

h

Λs = √ , (7.50)

2πms kB T

and using Stirling™s approximation for Ns !, the constant c becomes

Ns

e

c = Πs . (7.51)

Ns Λ3

s

Note that c has the dimension of an inverse volume in 3N -dimensional space

V ’N , and the integral in (7.48) has the dimension of a volume to the power

N . Thus, taking logarithms, we cannot split Q in c and an integral without

loosing the metric independence of the parts. It is irrelevant what zero point

is chosen to express the potential; addition of an arbitrary value V0 to the

potential will result in multiplying Q with a factor exp(’βV0 ), and adding

V0 to A.

When each of the states R and P have long life times, and have local

ergodic behavior, they can be considered as separate thermodynamic states,

with Helmholtz free energies

e’βV (r) dr