1549380323-Statistical Mechanics Theory and Molecular Simulation

Blue moon ensemble 337

A(s) =−kTlnP(s). (8.7.1)

In the second line, the integration over momenta has been performed giving the thermal
prefactor factorλ^3 N. In the blue moon ensemble approach, a holonomic constraint
(see Section 1.9 and Section 3.9)σ(r 1 ,...,rN) =f 1 (r 1 ,...,rN)−sis introduced in a
molecular dynamics calculation as a means of “driving” the reaction coordinate from
an initial values(i)of the parametersto a final values(f)via a set of intermediate
pointss(1),...,s(n)betweens(i)ands(f). As we saw in Section 3.9, the introduction
of a holonomic constraint does not yield the singleδ-function conditionδ(σ(r)) =
δ(f 1 (r)−s), wherer≡r 1 ,...,rN, as required by eqn. (8.7.1), but rather the product
δ(σ(r))δ( ̇σ(r,p)), since both the constraint and its first time derivative are imposed in
a constrained dynamics calculation. We will return to this point shortly. In addition,
the blue moon ensemble approach does not yieldA(s) directly but rather the derivative

dA

ds

=−

kT

P(s)

dP

ds

, (8.7.2)

from which the free energy profileA(q) along the reaction coordinate and the free
energy difference ∆A=A(sf)−A(si) are given by the integrals

A(q) =A(s(i)) +

∫q

s(i)

dA

ds

ds, ∆A=

∫s(f)

s(i)

dA

ds

ds. (8.7.3)

In the free energy profile expressionA(s(i)) is just an additive constant that can
be left off or adjusted so the minimum value of the profile atqmin corresponds to
A(qmin) = 0. In practice, these integrals are evaluated numerically using the integration
pointss(1),...,s(n). These points can be chosen equally-spaced betweens(i)ands(f), so
that the integrals can be evaluated using a standard numerical quadrature, or they can
be chosen according to a more sophisticated quadrature scheme.If the full profileA(q)
is desired, however, the number of quadrature points should be sufficient to capture
the detailed shape of the profile.
We next show how to evaluate the derivative in eqn. (8.7.2). Noting thatP(s) =
〈δ(f 1 (r)−s)〉, the derivative can be written as

1

P(s)

dP

ds

=

CN

Q(N,V,T)

∫

dNpdNre−βH(r,p)∂s∂δ(f 1 (r)−s)

〈δ(f 1 (r)−s)〉

. (8.7.4)

In order to avoid the derivative of theδ-function, an integration by parts is performed.
We first introduce a complete set of 3Ngeneralized coordinatesqα=fα(r 1 ,...,rN)
and their conjugate momentapα. This transformation, being canonical, has a unit
Jacobian so that dNpdNr= d^3 Npd^3 Nq. Denoting the transformed Hamiltonian as
H ̃(q,p), eqn. (8.7.4) becomes

1

P(s)

dP

ds

=

CN

Q(N,V,T)

∫

d^3 Npd^3 Nqe−β

H ̃(q,p)∂

∂sδ(q^1 −s)

〈δ(q 1 −s)〉

. (8.7.5)

Next, we change the derivative in front of theδ-function from∂/∂sto∂/∂q 1 , using
the fact that