Running molecular dynamics with constraints included

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)

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 .

What is Constraints Dynamics?

One of more internal coordinate is kept fixed during the computation.

Homolytic Constraints

ƒ(q1,q2,q3,…,t)=0

eg. r2 - a2 = 0

Non-Homolytic Constraints

ƒ(q1,q2,q3,…,t)>0

eg. r2 - a2 > 0

The SHAKE Algorithm uses Homolytic Constraints.

1 2dij

σ ij i j ijr r d= − − =( )2 2 0 Fxx k=λ

ƒσƒ

F r rk i k i j, ( )= −λ F r rk j k i j, ( )=− −λ

Applying the expressions for the forces in the Verlet Algorithm...

r t t r t r t tt

mF t

t

mr ti i i

ii

k

ikij( ) ( ) ( ) ( ) ( )+ = − − + + ƒ ƒ

ƒ λ ƒ2

2 2

The Verlet Algorithm gives (from last week):

r t t r t r t tt

mF ti i i

ii' ( ) ( ) ( ) ( )+ = − − +ƒ ƒ

ƒ2

2

Combining both...

r t t r t tt

mr ti i

k

iij( ) ' ( ) ( )+ = + + ƒ ƒ

λ ƒ 2

eqn 7.8

For the case with two atoms:

r t t r t tt

mr t r t1 1 12

2

11 2( ) ' ( ) ( )( ( ) ( ))+ = + + −ƒ ƒ λ

ƒ

r t t r t tt

mr t r t2 2 12

2

21 2( ) ' ( ) ( )( ( ) ( ))+ = + + −ƒ ƒ λ

ƒ

What else do we know?

r t t r t t d r t r t1 2

2

122

1 2

2( ) ( ) ( ) ( )+ − + = = −ƒ ƒ

This is a problem with 3 equations and 3 variables ( r1(t+δt), r2(t+δt), λ12 ) !!!

r t t tm m

r t tm m

r t d122

122

1 212 12

2 4

1 2

212

212

221 1 1 1

' ( ) ( ) ( ) ( ) ( )+ + + + + =ƒ λ ƒ λ ƒ

After painful manipulation of variables...

r t r t r t12 1 2( ) ( ) ( )= −Let

This can be solved algebraically!

However, this is not always the case...

For the case with 3 atoms, we get the following equations:

r t t r t t tm m

r tt

mr t12 12

2

1 212 12

2

223 23

1 1( ) ' ( ) ( ) ( ) ( )+ = + + + −ƒ ƒ ƒ λ

ƒλ

r t t r t t tm m

r tt

mr t23 23

2

2 323 23

2

212 12

1 1( ) ' ( ) ( ) ( ) ( )+ = + + + −ƒ ƒ ƒ λ

ƒλ

This will become more complex very rapidly!

One way of solving this system of equation is to drop the λ2 terms, to form linear equations with respect to λ.

We then get a k x k matrix, were k is the number of constraints.

The SHAKE algorithm doesn’t 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 don’t affect the remainder of the simulation.

So, What’s the catch?

You need to make sure that the constraints don’t 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.

But I don’t want to use the Verlet Algorithm...

This method has also been applied to the other algorithms, ie:

Velocity Verlet (renamed to RATTLE)leap-frogpredictor-corrector

What about angles and torsions?

Angles can be constrained with an additional distance constraint (eg SPC water).

There’s also a method by Tobias and Brooks that enable constraints to be applied to arbitrary internal coordinates.