342 Free energy calculations
compute the average ofz−^1 /^2 (r)O(r) and normalize by the average ofz−^1 /^2 (r). That
is,
〈O(r)〉conds =〈z−^1 /^2 (r)O(r)〉constrs
〈z−^1 /^2 (r)〉constrs. (8.7.28)
Given the connection between the conditional and constrained averages, eqn. (8.7.13)
for the free energy profile can be written as
A(q) =A(s(i)) +∫qs(i)ds〈
z−^1 /^2 (r)[
∂U ̃
∂q 1 −kT∂
∂q 1 lnJ(q)]〉constr
s
〈z−^1 /^2 (r)〉constrs. (8.7.29)
Having now demonstrated how to compute a conditional average from constrained
dynamics, we quote an important result of Sprik and Ciccotti (1998) who showed that
a transformation to a complete set of generalized coordinates is not required. Rather,
all we need is the form of transformation functionf 1 (r 1 ,...,rN) associated with the
reaction coordinateq 1. Then, when eqns. (8.7.15) are used to constrainf 1 (r 1 ,...,rN),
the free energy profile can be expressed as
A(q) =A(s(i)) +∫qs(i)ds〈
z−^1 /^2 (r) [λ+kTG]〉constr
s
〈z−^1 /^2 (r)〉constrs, (8.7.30)
whereλis the Lagrange multiplier for the constraint and
G=
1
z^2 (r)∑
i,j1
mimj∂f 1
∂ri·
∂^2 f 1
∂ri∂rj·
∂f 1
∂rj. (8.7.31)
In a constrained molecular dynamics calculation, the Lagrange multiplierλis calcu-
lated “on the fly” at every step and can be used, together with thecalculation of
Gfrom eqn. (8.7.31), to construct the average in eqn. (8.7.30) at each value of the
constraint. An interesting twist on the blue moon method was introduced by Darve
and Pohorille (2001, 2007, 2008) who suggested that the free energy derivative could
also be computed using unconstrained dynamics by connecting the former to the in-
stantaneous force acting on the reaction coordinate.
As an illustration of the use of the blue moon ensemble method, we show Helmholtz
free energy profiles for the addition of two different organic molecules, 1,3-butadiene,
and 2-F-1,3-butadiene to a silicon (100)-2×1 reconstructed surface (see Fig. 8.6). The
surface contains rows of silicon dimers with a strong double-bond character that can
form [4+2] Diels–Alder type adducts with these two molecules (see inset to Fig. 8.6).
An important challenge is the design of molecules that can chemisorb to the surface
but can also be selectively removed for surface patterning, thus requiring a relatively
low free energy for the retro-Diels–Alder reaction. By computing the free energy profile
using a reaction coordinateξ= (1/2)|(rSi 1 +rSi 2 )−(rC 1 +rC 4 )|, where Si 1 and Si 2 are
the two silicon atoms in a surface dimer, and C 1 and C 4 are the outer carbons in the
organic molecule, it was possible to show that a simple modification of the molecule
substantially lowers the free energy of the retro–Diels–Alder reaction. Specifically, if
the hydrogen at the 2 position is replaced by a fluorine, the free energy is lowered by