Non-inertial accelerations in the theory of molecular dynamics

Journal of Molecular Liquids, 33 (1987) 119-126

119

ElsevierSciencePublishersB.V.,Amsterdam-Printed

NON-INERTIAL

ACCELERATIONS

inTheNetherlands

IN THE THEORY OF MOLECULAR

DYNAMICS

M.W. EVANS

Dept. of Physics, Swansea

University

SA2 SPP, WAles,

(Received

College

of Swansea,

Singleton

Park,

(UK).

14 July 1986)

ABSTRACT

By repeated reference

non-inertial

of computer

acceleration

simulation

of some of these new statistical the conventional

methods

in the rotating

ensemble.

can be used to illustrate which

correlations,

built

frame of

of ways exist of correlating

terms in a molecular

using diffusion

fields can be conveniently revealing

operators

it is shown that a large number

fundamental technique

use of differential

equations.

in to the rotating

The

the time dependence

cannot be described

with

The effect of external frame analysis,

many new correlations.

INTRODUCTION

The new rotating to the molecular

frame theory of molecular

dynamics

transformation possible realised.

Cl1

Using

the result

No spectral method computer

the simplest,

the influence fields

correlations

(c.c.f. 's) is increased

0167-7322/87/$03.50

here to include new

of

has not been explicitly on any of these

Standard C2,31

simplifying

diffusion

cannot be used to assumption.

and electromagnetic

When

force

the extra torque due to the the number of possible

considerably.

between molecules

frame method,

equations

magnetic,

of liquid crystals,

of cross-correlations rotating

electric,

applied

of frame

information

on a very few.

[41, or alternatively

potentials

direct

and Kramers

recently

of a very large number

whose existence

can provide

even with considerable

of external

is included

director

is the generation

simulation

theory, based on Fokker-Planck describe

the simple mathematics

auto and cross-correlations

and contemporary

top, is extended

of an asymmetric

types of cross-correlation.

diffusion,

results

of new possibilities

Similarly,

crossconsideration

in the emergence,

from the

when dealing with the collective

0 1987 Elsevier Science PublishersB.V.

120 correlations

of the molecular

liquid state.

all emerge from a very simple mathematical number of possible mathematically, exists.

auto and cross-correlation

the majority

Therefore,

infinite,

because

the minority

physical

implications

describe

molecular

vanishes

generated

by syrmnetry for all t;

the total number

of possible

diffusion

Neither

nor the hydrodynamical

things stand at present.

approach

theories

which may be used to approximate functions

simulation

The general

of external

force fields

Cll,lZl

.

function

and its quantum mechanical [ll], then the existence

experimental

under favourable

equivalent,

conditions,

a broad band,

quantum

theoretically,

features,

DEVELOPMENT

Consider

This defines higher

wave functions

Let gfs to

the differential

derivatives

m = (1,2,3)',

Let c be the position

in each frame.

This

of a very large number

frame f = (x,y,z) at the angular

centre of mass relative

types of

produces, work of

is associated,

are mutually

influential

than meets

the eye.

FRAME THEORY

a frame of reference,

has the same origin.

which

type of lab. frame auto and cross-correlation

and translational

OF THE ROTATING

to the laboratory

molecule

is composed

Clll.

There is much more to a broad band, therefore,

113,141.

and c.c.f's

features,

normally

infra-red

sets of lines, each of which

with a particular

in which rotational

value of a wave

new a.c.f.'s

spectral

fields

correlation

could be linked to specific

such as that in the far

shows that the broad band absorption discrete

an expectation

of discrete

to

of molecular

the classical

The liquid state at room temperature

cross-correlation. however,

C41, and also of internal

of the many different

here would mean the emergence

method

with the specific

and the interaction

If we make the analogy between

function described

fields

of this as

of the cross - and

effect on ensemble

potential

or

laws that may be expected

approach may be those concerned

dynamics

capable

as a numerical

emerge from this fundmental

such as those caused by the director

(a.c.f.'s)

[2,31 based on Kramers

the time dependence

CS-101.

functions

141 seems remotely

This leaves computer

The

to

the time dependence

and auto-correlation

the diffusion

is

in number.

is intended

must be capable of describing

self-consistently.

but a minority

correlations

infinity

of this are that any theory which

cross-correlation

auto-correlation

Out of the infinite

functions

that exists also approaches

of all the existing

Langevin

These new cross-correlations treatment.

vector

which rotates with respect frequency,

be the linear velocity

of & in both frames.

Rf and $.

Each frame

of the molecular

(x,y,z) and Qrn$ that relative operators

w.

of the centre of mass of a

to (1,2,3)'.

Let us now examine

the

121 VELOCITIES

The velocities

Em6 = Qf’

where

-

in the fixed and rotating

x

l$

(2)

5

it is understood

respect

to frame

that all the terms on the r.h.s. are defined with

(1,2,3)',

The angular

(x,y,s).

frames are linked through:

and that on the 1.h.s. with respect

frequency

to frame

w is the same in both frames because

the rate at which one frame rotates with respect

it is

to the other.

ACCELERATIONS

The frame relations gf and $

providing

the forward

The eqn (4) introduces centripetal

the Newton

molecular

dynamics

of existence

conditions

C13,141,

other spectral

DERIVATIVES

equations

are written

they are important diffusion

counterparts, result

of the operators

transformations.

acceleration,

acceleration.

in a non-inertial

C6-91 theory

,

the

These appear context

Lll

.

For

and cannot be approximated C2,31

.

Naturally,

which may, under "particle

in discrete,

observable

out

they have

in the box"

infra red absorptions

and

features.

OF ACCELERATION

In this case the operators transforms,

and backward

and the non-uniform

as in standard

quantum mechanical

two applications

into the lab frame the Coriolis

acceleration

whenever

now involve

are applied

three times in the forward

and back

producing:

D3 = 2;~ + &( 28 x Frn@ + ~m!,(c&&) x f) + Qrn(!+x (& x z)) %fr +8x

I&+

* x (2% x grn:, + * x ((&&)

x f) + t x (g x (b x z))

(5)

122 D3 = & 2rmr +Ex Eqn

x 6) + gf(&

- ;f(2& x 2fZ) - 2f((Qf#) (2KX?f9

+tx

(6) shows the existence

((Rf&) x5)

of many higher

in the laboratory

in frame

(x,y,z) and some of the c.c.f. 's between Furthermore,

(x,y,z).

& (‘3)

for each

The a.c.f. 's of all of these exist the terms are allowed by

making use of the results:

<&W

- p&3(0)> #

0;

t#o

‘g,(t)

. l&O,’

0;

t

it becomes

- * x

(w x (&XX))

@X

order accelerations

molecule

symmetry.

frame

-

x (8 x z))

+

#o

clear that many new cross-correlations

are generated

in frame

(1,2,3)' and (x,y,s).

EXTERNAL

TORQUES

If the molecular acceleration intrinsic

ELECTRIC

term gft

acceleration

ensemble

is treated with an external

in frame

(x,y,z) must be rewritten

and a term due to the effect

field C2,4,121

the

as the sum of the

of the field.

FIELD

The extra angular

acceleration

where x is the molecular in the denominator

dipole,

DfK -

ti

and E the external

is the equivalent

The total angular

molecule.

due to the electric

inertia

acceleration

field is:

field strength.

Here gE

due to the effect of g on the

is therefore:

“B

(9)

Bg Substituting

this in eqns

(2), (4) and (6) immediately

lab. frame cross-correlation Taking

into account

molecule

functions

polarisability

involving

effects

C2,31

produces

a range of new

the field acceleration. the effective

field on each

is:

g = E, + E .p . E + EEE where 9 is the molecular

i y- +............

polarisability,

1 the hyper-polarisability,

(10) and so on.

123 Similar

considerations

external

magnetic

inevitable

of Lorentz

E = ,E, exp(i(wt 2.em

correlations

field vector

for example,

generated

go nearly


generate effect,

new cross

i.e. a torque and

field applied

Some diagonal

However

0.50 elements

x ($L$t) x F(t)).gl(O)

3. (3,3) element.

diffusion,

to produce

the relatively

to a molecular

hydrodynamical

these new correlation

primitive

methods

t(ps) of the cross-correlation

(1,2,3)

of

can be used to illustrate

I x V(O)>/[<,2>Z] %

frame of reference

2. (2,2) element.

of molecular

we now have available

0.25

in the moving

theories

into enough detail

analytically. simulation

cl Fig. 1.

automatically

the Buckingham

Cl21.

otherwise, functions

(11)

by an electromagnetic

None of the available

computer

is, as usual,

- k . ,R))

(2), (4), and (6) would through,

birefringence ensemble

forces.

FIELD

The electromagnetic

and from eqns

x H generated by a static % in this case the situation is affected by the

field, although

presence

ELECTROMAGNETIC

hold for the torque -t

function

their

or

124 existence

numerically,

correlation increase made

and also to pinpoint

of an external

in speed and power of computers

C15,16,171,

selenide,

illustrates

the results

in fig.

to see how the results

scheme.

In the meantime,

of a simulation

(I) the existence

to the sample.

. qgo)

can be

of this letter

for gaseous

with only 108 molecules

of a simple cross-correlation

one out of many which has not seen the light of day before.
With the

the link up with hydrodynamics

and it would be possible

fit in to the overall hydrodynamic hydrogen

[41 on cross-

the exact effects

field of force applied

function,

This is:

x X(O)’ (12)

4<,2,,,2,~<,2, This cross-correlation, frame defined vanishes

by symmetry

components

in frame

exist however

I

0.25

0 Fig. 2.

Autocorrelation

CROSS-CORRELATIONS

(x,y,z) for all t.

in the moving

of inertia,

The a.c.f.'s

but

of the

(x,y,z) and are illustrated

functions

of the force 2#m&(t) x (t(t) x ,$t),

frame and (2) the moving

DIFFERENT

leading to eqns

intermolecular

e(t) = cl(t) we obtain varieties vectors

moments

in frame

BETWEEN

If the analysis time-dependent

(5) and (3), exists

molecular

in fig. (2).

Ups)

in (1) the laboratory

position

implied by eqns

by the three principal

frame of reference.

MOLECULES

(2) to (6) is repeated

separation

for the

of two molecules: (13)

:2(t)

of cross-correlation

$1 and 52

contain

which depend on e and

of the two molecules

in question.

A

the

125 straightforward correlation

piece of algebra

functions

<4(k(t) x

leads to the implied existence

of cross-

such as

~2(Q)

.

(~(0)

x

$0)

x

$b))>

and

(14) <4(&(t) x x1(t))

which

correlate

. (t(o) x t(o) x x2(0))>

and define

molecule

(I) depend

explored

in any detail,

construction

expanding

theoretical

the way that the dynamics

although

the number

existence

use of supercomputers could be explored hydrodynamic

to extend

of a.c.f.'s

play a role in the

diffusion.

Having

got as far as thus

and c.c.f's which have a

condensed

states of matter.

the nature of these cross-correlation

in considerable

and molecular

of

None of these seems to have been

the logic to three and more particles,

in the molecular Cl51

(2).

they implicitly

of the theory of N particle

eqn (14) it is natural greatly

statistically

on those of molecule

detail, with a view to unifying

theory of the condensed

With the functions the

phases.

ACKNOWLEDGEMENTS

The University

of Wales

is thanked

and the staff of the University thanked

of Dublin

for the award of a Visiting

for a Pilcher

Senior Fellowship

for many discussions.

Professorship

IBM is

at Kingston.

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1

M.R. Spiegel,

2

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in Condensed

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Phys. Rev

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