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The probability density for a geometric parameter
of the system driven by a Hamiltonian:
 |
(6.10) |
with
, and
being kinetic, and potential energies, respectively,
can be written as:
 |
(6.11) |
The term
stands for a thermal average of quantity
evaluated
for the system driven by the Hamiltonian
.
If the system is modified by adding
a bias potential
acting only on a selected internal parameter
of the system
, the Hamiltonian takes a form:
 |
(6.12) |
and the probability density of
in the biased ensemble is:
 |
(6.13) |
It can be shown that the biased and unbiased averages are related via a simple formula:
 |
(6.14) |
More generally, an observable
:
 |
(6.15) |
can be expressed in terms of thermal averages within the biased ensemble:
 |
(6.16) |
Simulation methods such as umbrella sampling [82] use a bias potential to enhance
sampling of
in regions with low
such as transition regions of chemical
reactions. The correct distributions are recovered afterwards using the equation
for
above.
A more detailed description of the method can be found in Ref. [71]. Biased
molecular dynamics can be used also to introduce soft geometric constraints in which the controlled
geometric parameter is not strictly constant, instead it oscillates in a narrow interval
of values.
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