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In general, constrained molecular dynamics generates biased
It can be shown that the correct average for a quantity
can be obtained using the formula:
stands for the statistical average of the quantity
enclosed in angular parentheses computed for a constrained ensemble and
is a mass metric tensor defined as:
It can be shown [66,68,67,70] that the free energy gradient
can be computed using the equation:
is the Lagrange multiplier associated with the parameter
used in the SHAKE algorithm .
The free-energy difference between states (1) and (2) can be computed
by integrating the free-energy gradients over a connecting path:
Note that as free-energy is a state quantity, the choice of path connecting (1)
with (2) is irrelevant.
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