Module-3

Ab Initio Molecular Dynamics

March 23

Velocity Verlet Algorithm

RI(t+�t) = RI(t) + RI(t)�t+1

2RI(t)�t2 +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))

derivative

derivative

deri

vativ

e

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

t�t

t�t

RI RI

velocity Verletn

RI(0), RI(0)o

n

RI(�t), RI(�t)o

n

RI(2�t), RI(2�t)o

Molecular Dynamics (MD)

trajectory

Trajectory & Phase Spaceis trajectory

x

x

phase space

x

x

E =1

2mx(t)2 +

1

2mx(t)2

>>

constant energy curvesshowing traj. in phase space

Home Work• Highlight the accessible phase space regions

of a one dimensional particles moving under the potential as shown in the figure for

(a) if the particle has not sufficient energy to pass the barrier at X = 0

(b) if the particle has sufficient energy to pass the barrier at X = 0

Ensemble

• N number of system with same coordinates, but different momenta => follows different trajectory

• A large number of them may have the same macroscopic property

• A collection of systems with same macroscopic property, but different microscopic property is called an ensemble

...

x

x

Ensemble Average & Time Average

...

ensemble average

a1 a2 a3

hai = 1

N

NX

i

ai

...

at at+Δt at+2Δt

Ergodic Hypothesis

provided, that all the important microscopic states are visited during the dynamics!

a = limt!1

1

t

tX

⌧

a⌧

Lagrangian Formulation of Classical Mechanics

Euler-Lagrange equation:

L(RN , RN ) = K(RN )� U(RN )

d

dt

✓@L@RI

◆� @L

@RI= 0

Verification: @L@RI

=@

@RI

1

2

NX

I

MIR2I

!

=MIRI

d

dt

✓@L@RI

◆=MIRI

@L@RI

=� @U

@RI

=FI

d

dt

✓@L@RI

◆� @L

@RI=MIRI � FI

=0

Home Work

• Write Lagrangian for a 1-D simple harmonic oscillator

• Derive EOM

Hamiltonian FormulationH(RN ,PN ) = K(PN ) + U(RN )

RI =@H@PI

PI =� @H@RI

• Write Hamiltonian for a 1-D simple harmonic oscillator and derive EOM

• Hamilton’s equations of motion:

Home work

Hamiltonian vs Lagrangian equation of motion

•

H ⌘ H(RN ,PN )

L ⌘ L(RN , RN )

Useful if momentum representation is req. (for e.g. QM)

Useful if a coordinate system has a conjugate momentum representation that is complicated

(Cartesian=> linear mom.; angular => ang. mom. etc. )

L , H (Legendre transform)

Hamiltonian Conservation

dHdt

= 0

dHdt

=3NX

I=1

@H

@RI· RI +

@H

@PIPI

�

=3NX

I=1

@H

@RI· @H

@PI� @H

@PI· @H

@RI

�

=0

Total energy is also conserved

0 20000 40000 60000 80000 1e+05

Time Step

-200

0

200

400

En

erg

y (r

ed

. u

nit)

K.E.P.E.T.E.

Total Energy

E =NX

I=1

P2I

2MI+ U(RN )