lecture march25

7/24/2019 Lecture March25

1/14

Module-3

Ab Initio MolecularDynamics

March 25


2/14

Liouvilles Theorem

x

pxdXt

dXt+

Phase space volume occupied by a collection of members ofthe ensemble will be preserved during the time evolution


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Time Reversibility

x

px +t

t

Hamiltons equation ofmotion is invariant undertime reversal :

t t

PI= MIRI

dt MI

RI

d(t) =PI

dRI

d(t)

= H

PI

, d(PI)

d(t)

= H

RI

original

Hamiltonseqns. of motion

RI =

H

PI

PI = H

RI

External forces makesystem out of equilibrium,

then the resultingdynamics is irreversible


4/14

Position Verlet Algorithm Velocity Verlet Algorithm

Leap-frog algorithm

Reversible multiple time step algorithms

time reversible

numerical stability (Energy cons.) symplectic: phase space volumeconservation

Properties of integrators:

Homework:Read about the first three integrators (above) from the bookUnderstanding Molecular Simulation, Frenkel & Smith


5/14

Derivation of velocity Verlet

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

2t

2RI(t)

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

2t

2RI(t+t)

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

2MI[FI(t) + FI(t+t)]

Let us now make a reverse step from t+t t

Substituting the first eqn. in the second, we get,


6/14

History

Alder & Wainwright(JCP, 1957)

five-term virial expression.

Some further investigation for both 32 molecule and

larger systems will be made on the present calculators,

but a satisfactory determination of the detailed behavior

in the apparent transition region will require higher

speed equipment. The possibility

that

a similar phe-

nomenon for hard spheres in two dimensions may have

been missed in the original Monte Carlo calculations

l

will also be investigated.

Work performed under the auspices of the U.

S.

Atomic

Energy Commission.

1

M. N. Rosenbluth and A.

W.

Rosenbluth,

J.

Chern. Phys. 22,

881

1954).

2

Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller,

J.

Chern. Phys. 21, 1087 1953).

3 B. J. Alder and T. Wainwright, J. Chern. Phys. 27,1208 1957).

4 W. W.

Wood and F. R. Parker,

J.

Chern. Phys. 27,

720

1957).

This paper discusses the Monte Carlo method in some detail, as

well as giving computational results for Lennard-Jones molecules.

6

Kirkwood, Maun, and Alder, J. Chern. Phys. 18, 1040 1950).

Phase Transition for a Hard

Sphere System

B.

J. ALDER AND

T.

E. WAINWRIGHT

University of California Radiation LalJoratory

Livermore

California

Received August 12, 1957

A

CALCULATION of molecular dynamic motion

has been designed principally to study the re-

laxations accompanying various nonequilibrium phe-

nomena. The method consists of solving exactly to the

number of significant figures carried) the simultaneous

classical equations of motion of several hundred par-

ticles

by

means of fast electronic computors. Some of the

details as they relate to hard spheres and to particles

having square well potentials of attraction have been

described.

l

2

The method has been used also to calculate

equilibrium properties, particularly the equation of

state of hard spheres where differences with previous

Monte Carl0

3

results appeared.

detected.

t

became apparent

that

some long runs were

necessary at intermediate densities, accordingly the

IBM-704 was utilized where, for 32 particles, an hour is

required for 7000 collisions. Larger systems

of

108, 256,

and 500 particles can also conveniently be handled; in

an

hour 2000, 1000, and 500 collisions, respectively, can

be calculated. The results for 256 and 500 particles are

not now presented due to inadequate statistics.

The equation of state shown in Fig. 1 of the ac-

companying paper6 for

32

and

108

particles

is

for the

intermediate region

of

density, where disagreement was

found with the previous Monte Carlo results. The

volume, v

is

given relative to the volume of close

packing, Vo Plotted also are the more extended Monte

Carlo results; the agreement between these three sys-

tems is within the present accuracy of the pressure

determination. This agreement provides an interesting

confirmation

of

the postulates of statistical mechanics

for this system.

Figure 1 of the accompanying paper shows two

separate and overlapping branches.

In

the overlapping

region the system can,

at

a given density, exist in two

states with considerably different pressures. As the

calculation proceeds the pressure is seen to jump sud-

denly from one level to the other. A study of the posi-

tions of the particles reveals that as long as the system

stays on the lower branch

of

the curve the particles are

all confined to the narrow region in space determined

by

their neighbors, while on the upper branch of the curve

the particles have acquired enough freedom to exchange

with the surrounding particles. Since the spheres are

originally in ordered positions, the system starts out on

the lower branch; the first jump to the upper branch can

require very many collisions. The trend, as expected,

is

that at

higher densities more collisions are necessary for

the first transition, however, there are large deviations.

At vlvo 1.60, 5000 collisions were required;

at

1.55,

25000; while at 1.54 only 400; at 1.535, 7000;

at

1.53,

75 000; and at 1.525,

95

000. Runs in excess of 200 000

collisions at vivo of 1.55 and 1.53 have not shown any

P

HVSI CA I

REVIE '

VOLUME

f36,

NLJM8RR

2A l9

OCTOB

ER 1964

Correlations in

the

Motion

of

Atoms

in

Liquid

Argon*

A.

RAHMAN

Argonee

National Laboratory, Argo@me,

I/Blois

(Received 6

May

1964)

A

system

of

864

particles

interacting with a

I.ennard-Jones

potential

and

obeying

classical

equations

of

motion

has been studied on

a

digital

computer (CDC 3600)

to simulate molecular

dynamics

in

l i qu id a rgon

at 94 .

4'K

and a

density

of

1.374

g

cm

'.

The

pair-correlation f un ct io n a n d

t he c on st an t o f

self-di6usion are

found to

agree

well with

experiment;

the latter is

15

lower than

the

experimental

value. Th e

spectrum

of

the

velocity

au tocor rela tion func t ion shows a broad

maximum

in

the

frequency

region &o

=

0.25

(h&T/h).

The

shape

of

the

Van Hove

function

G,

(r,

t)

attains a

maximum

departure

from a

Gaussian

at about

s=3

0.

X1 0

sec

and

becomes a Gaussian

again

at about 10

sec.

The

Van

Hove function

Gg(r,

t)

h as b ee n

com-

pared

with

the

convolution

approximation

of

Vineyard,

showing

that

this

approximation

gives

a too

rapid

decay

of

Gd,

(r,

t)

with time. A

delayed-convolution

approximation

has

been

suggested

which

gives

a

better 6t

with

Gz(r,

I);

this delayed

convolution

makes

Gz(r, t}

decay

as

t4

at

short

times

and

as

I

at

long

times.

I.

INTRODUCTION

'N

recent

years

considerable use has been made

of

~ ~

large digi tal computers to

study

various

aspects

of

molecular dynamics

in solids, liquids,

and

gases.

The

following

is

a

description

of a

compu te r e xper iment

on

liquid

argon

(using

the CDC

3600)

to

study

the

space

and

time

dependence of

two-body cor re la ti on s

which

system,

namely,

94 .

4'K

and

1.374

g

cm

'.

A less

exhaustive

study,

at

130'IZ

and

1.

16

g

cm

',

is

men-

tioned

briefly

at the end.

II.

METHOD

OF COMPUTATION

The

calculations

reported

here

were

based on

the

following

ingredients.

A. Rahman (Phys. Rev. 1964)

first MD first realistic MD


7/14

e

o

g

e

n

s

e

v

a

.

1

0

I

0

5

IO

A

I

I

FIG.

Theoretical

(solid line)

, and experim

ental(dotte

d

line),x-ray

scatteringi

ntensitiesfo

rliquidwa

ter.Thelatte

rare

takenfrom

Narten,Ref

.29,andref

erto25C.

gooo)

,

onc2

,

nn2

o

n

s

.

2

g


8/14


9/14

MD Algorithm

a) Initialize coordinates and velocities

b) Velocity Verlet-1: R(t) !R(t+"t)

c) Compute force (acceleration) for R(t+"t)

d) Velocity Verlet-2: V(t)!V(t+"t)

e) Go to step (b) or stop

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

2

hRI(t) + RI(t+ t)

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

1

2RI(t)t

2

& compute force


10/14

Time Step = 1

=c

= 4000 cm1

= 8

.33 10

15s 10 fs

t /10 = 1 f s

0 10 20 30 40 50Time (fs)

0.5

0.6

0.7

0.8

0.9

1

r[H-H](Ang.)

t

fastest motion

determines thetime step

vibrational freq. of H2:

period of oscillation of H2:

Time step should at least (equally) divide a periodfor better capture of the features of dynamics


11/14

Initial State

Initial coordinates:X-ray structure,chemical intuition, arbitrary/random

structures

Initial velocities: from Boltzmanndistribution of velocities for a given

initial temperature T0=> this can be

generated from Box-Muller sampling(see next page)


12/14

uniform

rand. number

between 0 & 1

Box-Muller Sampling

f(v) =

M

2kBT

1/2

exp

Mv2

2kBT

X=p2 ln 0

2cos(21)

Y =p2 ln 0

2sin(21)

=pkBT/M

uniform rand.

numbers

between 0 & 1

This is a gaussian with

standard deviation

Maxwell distribution of velocities:

velocity

f(

v)

0X Y

X and Y are two random velocities obtained bypicking randomly from the distribution f(v)

all quantities areknown for a givensystem; M is themass of an atom


13/14

If you have two atoms, we require 6 values

of velocities (each for a degree of freedom) For this, we generate 6 uniform random

numbers from computers:

Substituting these random numbers in thetwo Box-Muller equations above [a pair ata time], we obtain, 6 values for velocities

(with a probability according to Boltzman

distribution):

Assign these values of X and Y to eachdegree of freedom (in any order that you

like)

1, , 6

X1, Y1, X2, Y2,X3, Y3


14/14

Temperature

Equipartition principle

Instantaneous

temperature T(t) = 1

3NkB

N

XI

MIR2

I

limt

1

t

tX

NX

I=1

1

2MI

R2I()

T= limt

1

t

tX

T()Macroscopic

temperature

1

3N*

N

XI=1

1

2

MIR2

I+ = 12

kBT

1

3N

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