Module-3

Ab Initio Molecular Dynamics

March 18

Molecular Dynamics (MD) Simulation

Ab initio MD

PLAN

R. Car and M. Parrinello Phys. Rev. Lett. 55, 2471, 1985

Molecular Dynamics

We like to move....atoms!How can I do it?

The Story of Newtons Apple...Equations of motion:

Solving (integrating) D.E.

F = md2x

dt2

We may use numerical or analytical integration

trusting truly on the

classical equations of

motion

F=ma

Mangalyaan Mission

also used to create your favorite characters!

gradient

how to get forces?

Fi = @U(x1, x2, , xn)@xi

Born-Oppenheimer Molecular Dynamics

U(RN ) = minD |Hel|

E+ ENN

{i}

Ab Initio Molecular Dynamics

Demonstration: 1D-Harmonic Oscillator

U(x) =1

2kx2 ) md

2x

dt2= kx

F = dUdx

= kx Analytical solution obtainableReq. two initial conditions:

x(0) x(0)&

Analytical integration of differential equation:

! =

rk

m

x(t) = A cos(!t)

v(t) = Aw sin(!t)

x(0) = A v(0) = 0For the initial conditions:

time(t)

x(t)

; v(t

)

2D-Harmonic OscillatorU(x, y) =

1

2kx2 +

1

2ky2

U(x,y)

x y

U(x,y)

xy

forces direct the system to the minimum PE

Fx = @U@x

= kx Fy = @U@y

= ky

F = Fxi+ Fyj

PE

KE

How about atoms? Classical mechanics? or quantum mechanics? It is a good assumption to treat the atomic

motions classically

atom 2

atom 3

atom 1v1

v2

v3

each degree of freedom of an atom will be having unique

positions, velocities, and

forces

F = md2x

dt2i ii i=1,3N

N=number of particles

Each degree of freedom has an equation of motion (classical) as you have seen for a 2D-harmonic oscillator (before)

For e.g. consider two atoms: 1 2

x1

y1

z1

x2

y2

z2

Fx1 =M1d2x1dt2

Fy1 =M1d2y1dt2

Fz1 =M1d2z1dt2

For atom 1 For atom 2

Fx2 =M2d2x2dt2

Fy2 =M2d2y2dt2

Fz2 =M2d2z2dt2

6 Eq. of motion to

solve independently

The force acting on each degree of freedom of every atom has the information about the inter-atomic interactions

Interatomic interactions are governed by the potential energy surface (as negative of the gradient of the potential energy is the force)

Lennard Jones Potential

ULJ(R1, ,RN ) =XI

XJ>I

4

"

RIJ

12

RIJ

6#

FI = @ULJ(R1, ,RN )@RII = 1, , N

RIJ = |RI RJ |

LJ potential is an example of interatomic interaction

Note: it is a 3D-vector(x,y,z components)

Numerical Integration

VI(t) = VI(0) +

Z t0dFI()

MI

RI(t) = RI(0) +

Z t0

dVI() atom 2 atom 3

atom 1

many body interactions within the

force: numerical integration is required

R

tt

Velocity Verlet Algorithm

RI(t +t) = RI(t) + RI(t)t +1

2RI(t)t

2 +O(t3)

ULJ (R1(t), ,RN (t))

RI(t +t) = RI(t) +t

2

hRI(t) + RI(t +t)

i+O(t2)

ULJ (R1(t+t), ,RN (t+t))

derivat

ive

derivative

deri

vativ

e

Originally by Carl Stoermer (in 1907, particles in electric field)

tt

tt

RI RI

velocity VerletnRI(0), RI(0)

o nRI(t), RI(t)

o nRI(2t), RI(2t)

o

Molecular Dynamics (MD)

trajectory