The ``blue moon'' ensemble approach

The term ``blue moon'' in the present context describes
rare events, *i.e.* events that happen once in a
blue moon. The blue moon ensemble approach was introduced
by Ciccotti and coworkers
as a technique for computing the free energy profile along
a reaction coordinate direction characterized by one or more
barriers high enough that they would not likely be crossed
in a normal thermostatted molecular dynamics calculation.

Suppose a process of interest can be monitored by a single
reaction coordinate
so that
eqns. (29) and (30)
reduce to

The ``1'' subscript on the value of is superfluous and will be dropped throughout this discussion. In the second line, the integration over the momenta has been performed giving the thermal prefactor factor . In the blue moon ensemble approach, a holonomic constraint is introduced in a molecular dynamics calculation as a means of ``driving'' the reaction coordinate from an initial value to a final value via a set of intermediate points between and . Unfortunately, the introduction of a holonomic, constraint does not yield the single -function condition , where required by eqn. (31) but rather the product of -functions , since both the constraint and its first time derivative are imposed in a constrained dynamics calculation. We will return to this point a bit later in this section. In addition to this, the blue moon ensemble approach does not yield directly but rather the derivative

from which the free energy profile along the reaction coordinate and the free energy difference are given by the integrals

In the free-energy profile expression is just an additive constant that can be left off. The values at which the reaction coordinate is constrained can be chosen at equally-spaced intervals between and , in which a standard numerical quadrature can be applied for evaluating the integrals in eqn. (33), or they can be chosen according to a more sophisticated quadrature scheme.

We next turn to the evaluation of the derivative
in eqn. (32). Noting that
,
the derivative can be written as

(35) |

Changing the derivative in front of the -function from to , which introduces an overall minus sign, and then integrating by parts yields

The last line defines a new ensemble average, specifically an average subject to the condition (not constraint) that the coordinate have the particular value . This average will be denoted . Thus, the derivative becomes

Substituting eqn. (38) yields a free energy profile of the form

from which can be computed by letting . Given that - is the expression for the average of the generalized force on when , the integral represents the work done

Although eqn. (39) provides a very useful
insight into the underlying statistical mechanical expression for
the free energy, technically, the need for a full canonical
transformaion of both coordinates and momenta is inconvenient since,
from the chain rule

(40) |

Now, we introduce

where, in the last line, the Jacobian has been exponentiated. Changing the derivative to and performing the integration by parts as was done in eqn. (37), we obtain

Therefore, the free energy profile becomes

Again, the derivative of , the transformed potential, can be computed form the untransformed potential via the chain rule

(45) |