Essentials of Computational Chemistry

12.2 COMPUTING FREE-ENERGY DIFFERENCES 441
some distance of q0, since structures having values of q significantly different from q0 will
be heavily penalized by addition of U.
When this procedure is followed, a different probability function π ∗ (q) will be obtained
over the sampled region. The correct PMF (i.e., for the unbiased potential) is related to the
new probability function according to
W(q) =−kBT ln π ∗ (q) − U(q)− kBT ln
−U(q)/kBT
e ∗
(12.26)
where the ensemble average is accumulated with the biasing function added to the system
Hamiltonian. This function is quite simple to evaluate for a typical selection of U. However,
it is often the case for an unknown PMF that a single choice of functional form for U
will not lead to a statistically useful sample over the entire range of interest for q. Instead,
one carries out several simulations, with different choices for U (for instance, by varying
choice of q0 in Eq. (12.25)), and then patches together the relevant regions of the PMF to
generate a single curve. This process is illustrated in Figure 12.5. Obtaining a good overlap
of the individual pieces can be difficult in some instances, which contributes to error in the
method when overlap is required. Indeed, Figure 12.5 is somewhat misleading, since each
individual PMF fragment actually rises to infinitely positive free energy values at either end
(that is, the probability of finding the system far to the right or left becomes so small that
the corresponding free energy is very large). As these PMF walls have no physical meaning,
but are artifacts of the umbrella function, they have been left out of Figure 12.5 for clarity,
but in practice they can add to the difficulty associated with reliably overlapping different
segments of the full reaction coordinate. The weighted histogram analysis method (WHAM;
∆G
R q P
Figure 12.5 A reaction coordinate q constructed piecewise from reactants R to products P as a
series of PMFs determined using different umbrella functions. The individual PMFs determined using
Eq. (12.26), shown below the dashed line and taking each left endpoint as the relative zero, are held
within their respective regions of the reaction coordinate by the umbrella function. Their overlap on a
common energy scale generates the complete PMF shown above the dashed line