Running molecular dynamics with constraints included

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Running molecular dynamics with constraints
included
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Restraint or Constraint Dynamics?
Constraint Must be satisfied (eg fixed bond
length) Restraint penalties are imposed for a
derivation from the reference value (eg
increased force constant) (Chapter 9.2)
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What is Constraints Dynamics?
One of more internal coordinate is kept fixed
during the computation.
Why Constraints Dynamics?
The time step is dictated by the highest
frequency. Therefore, using constraint can
reduce the time step and make the computation
faster. It reduces the numbers of degrees of
freedom to 3N - k .
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Homolytic Constraints
(q1,q2,q3,,t)0 eg. r2 - a2 0
Non-Homolytic Constraints
(q1,q2,q3,,t)gt0 eg. r2 - a2 gt 0
The SHAKE Algorithm uses Homolytic Constraints.
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dij
1
2
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The Verlet Algorithm gives (from last week)
eqn 7.8
Applying the expressions for the forces in the
Verlet Algorithm...
Combining both...
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For the case with two atoms
What else do we know?
This is a problem with 3 equations and 3
variables ( r1(tdt), r2(tdt), ?12 ) !!!
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Let
After painful manipulation of variables...
This can be solved algebraically!
However, this is not always the case...
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For the case with 3 atoms, we get the following
equations
This will become more complex very rapidly!
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One way of solving this system of equation is to
drop the ?2 terms, to form linear equations with
respect to ?. We then get a k x k matrix, were k
is the number of constraints.
The SHAKE algorithm doesnt do that... It
satisfies one constraint at one time and
reiterate around the constraints until a
convergence is obtained. The tolerance needs to
be tight enough that the fluctuations in this
algorithm dont affect the remainder of the
simulation.
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So, Whats the catch?
You need to make sure that the constraints dont
prevent a full exploration of all the
possibilities (eg by preventing a rotation or a
torsion). In other words, the constrained
degrees of freedom must be weakly coupled to the
remaining degrees of freedom.
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But I dont want to use the Verlet Algorithm...
This method has also been applied to the other
algorithms, ie Velocity Verlet (renamed to
RATTLE) leap-frog predictor-corrector
What about angles and torsions?
Angles can be constrained with an additional
distance constraint (eg SPC water).
Theres also a method by Tobias and Brooks that
enable constraints to be applied to arbitrary
internal coordinates.