Essentials of Computational Chemistry

440 12 EXPLICIT MODELS FOR CONDENSED PHASES
might want this least populated bin to contain at least 100 points, so we would require a
sample of some 1 000 000 snapshots. An ensemble of this size is accessible with current
computational technology, but represents a reasonably significant investment of resources.
Now, consider if the highest energy point on the curve were to be 6 kcal mol−1 above the
lowest at 298 K. Because the probability involves the exponential of the energy difference,
doubling the difference squares the sampling ratio (i.e., the highest energy region is now
sampled 10 000 times less frequently than the lowest energy region). Obtaining a statistically
meaningful sample of low probability regions now becomes a significantly more difficult
prospect, and statistically reliable PMFs cannot be obtained in this fashion.
The problem of low probability regions is even more severe when it comes to chemical
reaction coordinates, where free energies of activation for chemically viable processes may
range well above 20 kcal mol−1 . The probability of obtaining a snapshot in the region of a
transition state structure having so high an energy (assuming for the moment that we have
some Hamiltonian capable of describing bond-making/bond-breaking) is so remote that no
brute force simulation can legitimately expect to capture even one relevant point, much less
a statistically meaningful sample. This is the problem of sampling ‘rare events’.
One approach to overcoming this problem is to apply a so-called ‘umbrella potential’ or
biasing potential. This potential, a function of the coordinate of interest q, is added to the
force-field energy with the aim of forcing q to be sampled heavily within a certain range
of values that would not otherwise be statistically accessible. An ideal umbrella potential is
one that is the exact negative of the PMF, since then the probability of sampling any value
of q should be uniform. However, one rarely knows the PMF ahead of time (otherwise why
would one be trying to calculate it?), so instead one typically applies rather simple biasing
potentials (e.g., a quadratic potential) to force q to be sampled over some interval including
a particular value q0.
Consider, for instance, the SN2 reactionofBr−with CH3Br in aqueous solution, which
has an activation free energy on the order of 20 kcal mol−1 . If we define our reaction
coordinate as
q = rC–BrA − rC–BrB
(12.24)
where A and B are the incoming and outgoing bromide ions, respectively, we see that the
reactants correspond to large positive values of q, products to large negative values of q,
and from our knowledge of bimolecular nucleophilic substitution reactions, we know that
the transition state region will have values of q very near zero. Let us assume that we
have a force field that provides an accurate potential energy curve in the gas phase for this
SN2 process – in spite of this, in a normal MC or MD simulation in a box of water we
would be very unlikely to sample in regions anywhere near the TS because of the very
low probabilities associated with such high-energy structures. However, if we apply biasing
potentials of the form
U(q) = 1
2
k(q − q0) (12.25)
2
where q0 is the particular value near which we want to sample, and we select the force
constant k to be suitably large, we can ensure that the simulation will sample heavily within