R

e’βV (r) dr,

AP = ’kB T ln QP QP = c (7.53)

P

where the integrations are now carried out over the parts of con¬guration

space de¬ned as the R and P regions, respectively. We may assume that

these regions encompass all local minima and that the integration over space

outside the R and P regions does not contribute signi¬cantly to the overall

Q.

We immediately see that, although Q = QR + QP , A = AR + AP . Instead,

de¬ning the relative probabilities to be in the R and P state, respectively,

230 Free energy, entropy and potential of mean force

as wR and wP :

QR QP

R Q

w= and w = , (7.54)

Q Q

it is straightforward to show that

A = wR AR + wP AP + kB T (wR ln wR + wP ln wP ). (7.55)

The latter term is due to the mixing entropy resulting from the distribution

of the system over two states. Note that the zero point for the energy must

be the same for both R and P.

Now de¬ne a reaction coordinate ξ(r) as a function of particle coordinates,

chosen in such a way that it connects the R and P regions of con¬guration

space. There are many choices, and in general ξ may be a complicated

nonlinear function of coordinates. For example, a reaction coordinate that

will describe the transfer of a proton over a hydrogen bond X-H· · ·Y may

be de¬ned as ξ = rXH /rXY ; ξ will encompass the R state around a value

of 0.3 and the P state around 0.7. One may also choose several reaction

coordinates that make up a reduced con¬guration space; thus ξ becomes

a multidimensional vector. Only in rare cases can we de¬ne the relevant

degrees of freedom as a subset of cartesian particle coordinates.

We ¬rst separate integration over the reaction coordinate from the integral

in Q:

dre’βV (r) δ(ξ(r) ’ ξ).

Q=c dξ (7.56)

Here ξ(r) is a function of r de¬ning the reaction coordinate, while ξ is a

value of the reaction coordinate (here the integration variable).10 In the case

of multidimensional reaction coordinate spaces, the delta-function should be

replaced by a product of delta-functions for each of the reaction coordinates.

Now de¬ne the potential of mean force V mf (ξ) as

dre’βV (r) δ(ξ(r) ’ ξ) ,

def

V mf (ξ) = ’kB T ln c (7.57)

so that

e’βV

mf (ξ)

Q= dξ, (7.58)

and

e’βV

mf (ξ)

A = ’kB T ln dξ . (7.59)

10 Use of the same notation ξ for both the function and the variable gives no confusion as long

as we write the function explicitly with its argument.

7.6 Reconstruction of free energy from PMF 231

Note that the potential of mean force is an integral over multidimensional

hyperspace. Note also that the integral in (7.57) is not dimensionless and

therefore the PMF depends on the choice of the unit of length. After integra-

tion, as in (7.58), this dependency vanishes again. Such inconsistencies can

be avoided by scaling both components with respect to a standard multidi-

mensional volume, but we rather omit such complicating factors and always

keep in mind that the absolute value of PMFs have no meaning without

specifying the underlying metric.

It is generally not possible to evaluate such integrals from simulations.

The only tractable cases are homogeneous distributions (ideal gases) and

distribution functions that can be approximated by (multivariate) Gaussian

distributions (harmonic potentials). As we shall see, however, it will be

possible to evaluate derivatives of V mf from ensemble averages. Therefore,

we shall be able to compute V mf by integration over multiple simulation

results, up to an unknown additive constant.

7.6 Reconstruction of free energy from PMF

Once the PMF is known, the Helmholtz free energy of a thermodynamic

state can be computed from (7.59) by integration over the relevant part

of the reaction coordinate. Thus the PMF is a free energy for the system

excluding the reaction coordinates as degrees of freedom. In the following we

consider a few practical examples: the harmonic case, both one- and multi-

dimensional and including quantum e¬ects; reconstruction from observed

probability densities with dihedral angle distributions as example; the PMF

between two particles in a liquid and its relation to the pair distribution

function; the relation between the partition coe¬cient of a solute in two

immiscible liquids to the PMF.

7.6.1 Harmonic wells

Consider the simple example of a PMF that is quadratic in the (single)

reaction coordinate in the region of interest, e.g in the reactant region R (as

sketched in Fig. 7.3):

1

V mf ≈ V0mf + k R ξ 2 . (7.60)

2

Then the Helmholtz free energy of the reactant state is given by integration

kR

1

’βV mf (ξ)

A ≈ ’kB T ln

R

V0mf

e dξ = + kB T ln . (7.61)

2 2πkB T

’∞+∞

232 Free energy, entropy and potential of mean force

Beware that the term under the logarithm is not dimensionless, but that the

metric dependence is compensated in V0mf . We see that A becomes lower

when the force constant decreases; the potential well then is broader and

the entropy increases.

In the multidimensional harmonic case the PMF is given by a quadratic

term involving a symmetric matrix KR of force constants, which is equal to

the Hessian of the potential well, i.e., the matrix of second derivatives:

V mf ≈ V0mf + 1 ξ T KR ξ. (7.62)

2

Integration according to (7.59) now involves ¬rst an orthogonal transforma-

tion to diagonalize the matrix, which yields a product of one-dimensional

integrals; carrying out the integrations yields a product of eigenvalues of the

matrix, which equals the determinant of the diagonalized matrix. But the

determinant of a matrix does not change under orthogonal transformations

and we obtain

det KR

1

A ≈ V0 + kB T ln

R mf

. (7.63)

2 2πkB T

Thus far we have considered the system to behave classically. However, we

know that particles in harmonic wells (especially protons!), as they occur

in molecular systems at ordinary temperature, are not at all close to the

classical limit and often even reside in the quantum ground state. The

classical expressions for the free energy are very wrong in such cases. The

PMF well itself is generally determined from simulations or computations

with constrained reaction coordinates in which the quantum character of the

motion in the reaction coordinate does not appear. It is therefore relevant

to ask what quantum e¬ects can be expected in the reconstruction of free

energies from harmonic PMF wells.

Quantum corrections to harmonic oscillator free energies can be easily

made, if the frequencies of the normal modes are known (see Chapter 3,