• Molecular dynamics (MD) is a computer simulation

technique where the time evolution of a set of

interacting atoms is followed by integrating their

equations of motion.

• We follow the laws of classical mechanics, and most

notably Newton's law:

Molecular dynamics - Introduction

A brief description of the molecular dynamics

method

Successive configuration of the molecular system can be obtained by

integrating Newton’s laws of motion. Positions and momenta of the particles

of the given molecular system are described by the trajectories obtained by

the successive integration of the Newton’s equations which are mathematical

description of the following natural rules:

1. A body continues to move in a straight line at a constant velocity

unless a force acts upon it;

2. Force equals the rate of change of momentum;

3. To every action there is an equal and opposite reaction;

The trajectories are obtained by solving the differential equations of the

Newton’s second law:

i

xi

m

F

dt

xdi

2

2

Simple models

Hard sphere potential

Square well potential

MOLECULAR DYNAMICS USING

SIMPLE METHODS

The steps involved in the hard-sphere calculation

as follows:

1. Identify the next pair of spheres to collide and calculate

when the collision will occur.

2. Calculate the positions of all the spheres at the collision

time.

3. Determine the new velocities of the two colliding

spheres after the collision.

4. Repeat from 1 until finished.

The new velocities of the colliding spheres are

calculated by applying the principle conservation

of linear momentum.

MOLECULAR DYNAMICS WITH

CONTINUOUS POTENTIALS

First MD with continuous potentials done in 1964 (simulation of argon

by Rahman).

Finite difference method: the integration is broken down into many small stages, each separated in time by a fixed time dt.

.........)()()(

.........)(2

1)(

. . . . . .)(6

1)(

2

1)()()(

. . . . . .)(24

1)(

6

1)(

2

1)()()(

2

32

432

ttctbttb

tctttba(t)δt)a(t

tcttbtttatvttv

tcttbttatttvtrttr

Verlet algorithm

The most widely used method in molecular dynamics programs is the Verlet algorithm. It uses the positions and accelerations at time t, and the positions from the previous step, r(t-δt) to calculate new positions at t+δt, r(t+δt). Relations between positions and velocities at those two moments in time can be written as:

Those two relations can be added to give:

The velocities do not explicitly appear in the Verlet algorithm. They can be calculated in several ways. A very simple approach is to divide the difference in positions at times t+δt and t-δt by 2δt, i.e.

Another approach calculates velocities at the half step :

Practical application of this algorithm is straightforward and memory requirements are modest, only positions at two time steps have to be recorded r(t), r(t-δt), and the acceleration a(t). The only drawback is that the new position r(t + δt) is obtained by adding small term δ2ta(t) to the difference of two much larger terms 2r(t) and r(t-δt), which requires high precision for r in the numerical calculation.

.....)(2

1)()()(

.....)(2

1)()()(

2

2

tatttvtrttr

tatttvtrttr

)()()(2)( 2 tatttrtrttr

tttrttrtv 2/)]()([)(

tt2

1

ttrttrttv /)]()([)2

1(

Verlet algorithm

The leap-frog method is the variation of Verlet algorithm. It uses the following relations:

The name of this method comes from its nature, i.e., velocities make ‘leap-frog’ jumps over the positions to give their values at

)]2

1()

2

1([

2

1)(

)()2

1()

2

1(

)2

1()()(

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ttattvttv

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tt2

1

Verlet algorithm

The velocity Verlet algorithm gives positions, velocities and accelerations at the same time and does not compromise precision:

)]()([2

1)()(

)(2

1)()()( 2

ttatattvttv

tatttvtrttr

Verlet algorithm

Beeman Algorithm

Better velocities, better energy conservation More expensive to calculate

)(6

1)(

6

5)(

3

1)()(

)(6

1)(

3

2)()()( 22

tttattattatvttv

ttattatttvtrttr

General Predictor-Corrector

Algorithms

Predict the position x(t+dt) and velocity v(t+dt)

at the end of the next step.

Evaluate the forces at t+dt using the predicted

position.

Correct the predictions using some combination

of the predicted and previous values of position

and velocity.

Gear’s Predictor-Corrector methods

Predict ac(t+dt) from the Taylor expansion at the starting point

Begin with a simple prediction, as in any of the previous methods

Initially step to r(t+dt), v (t+dt), a(t+dt),b(t+dt) at that point.

The difference between the a(t+dt) and the predicted ac(t+dt):

a(t t) ac(t t) a(t t)

Estimates the error in the initial step, which is used to correct:

rc(t t) r(t t) c0 a(t t)

vc(t t) v(t t) c1 a(t t)

ac(t t) / 2 a(t t)/ 2 c2 a(t t)

bc(t t)/ 6 b(t t) / 6 c3 a(t t)

Predictor-corrector algorithms

1. Predictor: From the positions and their time derivatives up to a certain order q, all known at time t, one ``predicts'' the same quantities at time by means of a Taylor expansion. Among these quantities are, of course, accelerations .

2. Force evaluation: The force is computed taking the gradient of the potential at the predicted positions. The resulting acceleration will be in general different from the ``predicted acceleration''. The difference between the two constitutes an ``error signal''.

3. Corrector: This error signal is used to ``correct'' positions and their derivatives. All the corrections are proportional to the error signal, the coefficient of proportionality being a ``magic number'' determined to maximize the stability of the algorithm.

Evaluate integration methods

Fast, minimal memory, easy to program

Calculation of force is time consuming

Conservation of energy and momentum

Time-reversible

Long time step can be used

Which algorithm is appropriate

Cost effective

Energy conservation

Root-mean-square fluctuation

Total, 0.02 kcal/mol

KE and PE, 5 kcal/mol

Choosing the time step

Too small: covering small conformation space

Too large: instability

Suggested time steps

Translation, 10 fs

Flexible molecules and rigid bonds, 2fs

Flexible molecules and bonds, 1fs

Multiple time step dynamics

Reversible reference system propagation

algorithm (r-RESPA)

Forces within a system classified into a number of

groups according to how rapidly the force changes

Each group has its own time step, while maintaining

accuracy and numerical stability

Molecular dynamics setup

Initial configuration

Initial velocities (Maxwell-Boltzmann)

Force field

Cutoff: doesn’t save time by itself. But can

combine with neighbor list and speed-up the

simulation

Tk

vm

Tk

mvP

B

ixi

B

iix

22/1

2

1exp

2)(

Running molecular dynamics

Equilibration

Special care is needed for inhomogeneous system

Calculating the temperature

Nc is the number of constraints, so 3N – Nc is the total number of degrees of

freedom

Boundary conditions

No boundary

Periodic boundary condition

Non-periodic: reaction zone, harmonic constraint

boundary atoms

N

i

CB

i

i NNTk

m

pH

1

2

322

Constraint dynamics

High frequency modes takes all the computer time

Low frequency modes correspond to conformational

changes

Constraint: system is forced to satisfy certain

conditions

SHAKE: constraint the bond vibration

Molecular Modelling - Andrew R. Leach

(Principles and Applications)

http:/docjax.com/molecular dynamic simulation methods/

http:/google.com/molecular dynamics/ simulation simple methods/

http:/google.com/molecular dynamic simulation with continuous potential/