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Umbrella sampling attempts to overcome the sampling problem by modifying
the potential function so that the unfavorable states are sampled
adequately. The modification of the potential function can
be written as

(24) 
where is a weighting function, which often takes a quadratic
form;

(25) 
For configurations far from the equilibrium state the weighting
function will be large and so a simulation using the method energy
function will be biased along some relevant reaction coordinate
away from the configuration . The resulting distribution will,
of course, be nonBoltzmann. The corresponding Boltzmann averages
can be extracted from the nonBoltzmann distribution by [28]

(26) 
The subscript indicates that the average is based on the probability
, which in turn is determined by the modified
energy function . As shown in Fig. 12, we can use
the distribution function with the weighting function which would
be determined and then corrected to give the true radial distribution
function. The free energy can be calculated as a function of the separation.
It is usual to perform an umbrella sampling calculation in a series
of stages, each of which is characterized by a particular value of
the coordinate and an appropriate value of t. However, if the weighting
function is too large, the denominator in Eq. (26) is dominated
by contributions from only a few configurations with especially large
values of so that the averages may take too long to
converge.
Figure 12:
We can calculate the real distributions by several umbrella sampling
windows.

In our case, we add a weighting function to the distance between two
methane molecules, to simulate and get many piece of PMF curve. Then,
we reduce the weighting function and combine these pieces of curve
to get the PMF.
Next: Constrained Molecular Dynamics Method
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JeLuen Li
20070717