Umbrella sampling 3432.0 2.5 3.0 3.5
0.0
10.0
20.0
30.0
ξ (Α)∆
A(kcal/mol)Butadiene
2,F-ButadieneFig. 8.6Free energy profiles for the addition of 1,3-butadiene and 2-F-1,3-butadiene to a
Si(100)-2×1 surface.
8-10 kcal/mol (Minary and Tuckerman, 2004; Iftimieet al., 2005). These calculations
were performed using a molecular dynamics protocol in which forcesare obtained “on
the fly” from an electronic structure calculation (this is calledab initiomolecular
dynamics (Car and Parrinello, 1985; Tuckerman, 2002; Marx and Hutter, 2009)). The
calculations are performed at 300 K using 13 different values forξseparated by a
distance of 0.15 ̊A.
8.8 Umbrella sampling and weighted histogram methods
In this section, we will discuss another free energy technique known asumbrella sam-
pling(Torrie and Valleau, 1974; Torrie and Valleau, 1977). This method bears some
similarity to the blue moon ensemble approach; however, rather than constraining the
reaction coordinate, the latter isrestrainedwith a biasing potential. The bias is usually
taken to be a harmonic potential of the form
W(f 1 (r 1 ,...,rN),s) =1
2
κ(f 1 (r 1 ,...,rN)−s)^2 , (8.8.1)which is also known as anumbrella potential. Eqn. (8.8.1) is added toU(r) so that
the total potential isU(r) +W(f 1 (r),s). This potential is then used in a molecular
dynamics or Monte Carlo calculation. As is done in the blue moon ensemble approach,