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The idea of using constrained MD to compute the PMF is simple: by
using a Lagrange multiplier to fix the distance between two solutes,
we are able to compute the required constraint force between two solutes.
After a simple integration, modified by the volume entropy term, we
obtain the PMF.
Starting with Lagrangian formulation of classical dynamics, we introduce
a modified Lagrangian
|
(27) |
where is the Lagrange multiplier, and
are the coordinates of the carbon atom in methanes. Let:
|
(28) |
From Lagrange's equation
|
(29) |
we obtain the equation of motion
|
(30) |
The constraint force is given by
|
(31) |
|
(32) |
By its construction, the following condition holds at any :
|
(33) |
Using Eq. (14), Verlet algorithm, we obtain
|
(34) |
|
(35) |
Subtracting the last two equation, let
|
(36) |
where
|
(37) |
Taking square on both sides, we get
|
(38) |
This tells us how to solve from the known
and :
|
(39) |
There are two roots for Eq. (39), and we shall use the smaller
one. The larger root corresponds to the swap of two atoms, not what
we want in a dynamics. Once one solves , the constraint
force on the first atom is given by Eq. (31).
Next: Results and Discussions
Up: Free Energy Calculations
Previous: Umbrella Sampling
Contents
Je-Luen Li
2007-07-17