C &t)

JournolofMolecularLiquids, 31 (1986)231-249
Elsevier Science Publishers B.V., Amsterdam -Printed
231
in The Netherlands
CORRELATION OF INTERMOLECULARCORIOLIS
THE STATISTICAL
AND CENTRIFUGAL FORCES
M. W. EVANS
Department
of Physics,
University
College
of North Wales,
Bangor,Gwynedd
LL572UW
(Received 29 July 1985)
ABSTRACT
Correlation
molecular
static
functions
coriolis
are
and centrifugal
in the laboratory.
this laboratory
motion,
laboratory
Coriolis
and correlate
or translational
the enantiomers
the
forces with respect
The molecular
They are not usually
rotational
in
frame result from a combination
and translational
motion.
computed
the
to a frame of reference
forces in
of the molecule's
rotational
these two types of
in the theory of decoupled
diffusion.
and racemic mixture
for dichloromethane
for
and centrifugal
statistically
considered
frame
Correlation
functions
are computed
of liquid bromochlorofluoromethane,
liquid subjected
to a strong, uniaxial,
force in the z axis of the laboratory
electric
for
and
field of
frame.
INTRODUCTION
This paper describes
inter-molecular
The Coriolis
reference.
v,(t)
Coriolis
is the molecular
C
&t)
symmetries.
in
and translation,
molecular
x 8
the laboratory
frame,
0167-7322/86/$03.50
of the
and translation.
force is:
x 8 (0) >
a natural
it correlates
of
frame of
the angular velocity
rotation
frame of reference
the statistical
because
Coriolis
. x(o)
functions
to x(t) x e(t), where
and g
of molecular
of the Coriolis
(t)
C Car is therefore
the laboratory
of correlation
ir. the laboratory
it is a natural measure
nature
function
= < y, (t)
and also exists
forces
centre of mass velocity
interdependent
The time-correlation
simulation
force Cl,21 is proportional
Therefore
of the same molecule.
statistically
a computer
and centrifugal
method
for all molecular
of expressing,
correlation
directly
between molecular
in
rotation
in this frame a real force, the
force at t = 0 and a time t later.
0 1986 Elsevier SciencePub1ishersB.V.
Coriolis
forces have a
232
real effect
on measurable
spectra,
theory of vibration-rotation
quantised
vibrational
quantised
rotational
also classical
resultant
intermolecular
its own rotation,
as much detail
translation
therefore
Section
is analysed
the treatment
the computer
Section
subjected
mechanics,
rotation
is extended
of reference.
131
forces
with respect
The statistical
and the Onsager
reciprocal
that the theory of 'rotational'
rotates
for the (R) enantiomer
inversion
considered
diffusion
and
of
between
frame
molecular
in the theory of linear
It is pointed
statistically
Coriolis
the isotropy
in the laboratory
does not describe
forces correlated
frame of reference
moments
and translates
in terms of the time-correlation
Force,
out, de facto,
the molecular
in this paper.
2m&(t)
$(Q
mu(t) x (y(t) x x(t)).
an inertial
molecular
under
and for liquid dichloromethane
relations.
The Origin of the Molecular
Consider
principal
time-correlation
used and the conditions
cross-correlation
response
Force
functions.
frame of reference.
field, which removes
to parity
is not usually
1:
The correlation
to the statistical
are provided
and translation
Section
and centrifugal
correlation
in the laboratory
of some results,
Results
rotation
and Centrifugal
of the Coriolis
of the algorithms
z-axis electric
and centrifugal
to solve
of 108 molecules.
and translation.
of bromochlorofluoromethane
ensemble
simulation
were pursued.
3 is a discussion
the molecular
like
co treat
the centre of mass
using computer
for an assembly
on
in this paper is
in terms of its component
simulations
to an intense
Coriolis
to the
superimposed
It is difficult
they involve
of the nature
centrifugal
and their spectra.
racemic mixture
molecule
diffusion.
Their computation
of molecular
? is a discussion
Section
on the
there are
as follows.
1 is a discussion
of molecular
functions
of motion
is arranged
In this context
which
because
to classical
equations
CCor(t)
However,
forces of the type $ x t, due
of molecular
mechanically
in the context
function
Coriolis
superimposed
molecule.
and these do not seem to have been treated in anything
restricted
The paper
function
of atoms in a molecule
of the complete
of the whole molecule.
the classical
in the quantum
In this case they arise from the
traLlslatiou of the complete
in the context
these forces quantum
forces
movement
movement
centre of mass
and this is well known
coupling.
of inertia.
with respect
defined by the frame of the three
A laboratory
frame of reference
to this molecule-fixed
frame, and is
then
233
therefore
a non-inertial
fixed frame.
frame of reference,
It follows
in this non-inertial
that Newton's
frame,
with respect
equation
the laboratory
to the molecule-
of motion
for the molecule
is: CSI
frame,
(2)
where J$ is the translational
other;
acceleration
the energy U is defined
2
Lo=;""
expressed
mass, x
0
(3)
is the molecular
t is the angular velocity
position
vector
the laboratory
defined
in the laboratory
of the one frame with respect
adapted
in the laboratory
to the original
Note that theories
0
inertial
frame as
is
frame; and 5 is the
to the origin of
linear velocity
frame of reference,
is the Lagrangian
of decoupled
eqn (2) in the laboratory
fixed frame,
centre of mass with respect
In eqn. 3 x is the centre of mass
fixed frame, and L
frame,
to the other, and
above for the molecule
angular velocity
frame.
fixed frame; m is the molecular
centre of mass velocity
of the molecular
with respect
molecule
inertial molecule
to the definition
also the molecular
to the
-u
in the original
according
of the one frame with respect
by the Lagrangian:
i.e. the
in this same frame.
translational
the equivalent
diffusion
simply write
:
of
m dx
=E
x
with no account
(4)
taken of the terms involving
in the laboratory
The molecular
example
eqn (2).
vector
Coriolis
force in the laboratory
of a real force neglected
of decoupled
molecular
translational
in theories
diffusion
151.
It
x, this theory neglects
is also absent
centrifugal
in theories
frame of reference
both of decoupled
is the term 29
In so far as the theory of rotational
is the molecular
angular velocity,
&
frame.
diffusion
the real force defined
force
in the laboratory
of translational
In this paper, we use the technique
diffusion
of "molecular
to construct laboratory frame time-correlation
is an
rotation
x q
and
in
does not involve
by rnt x (6 x @,
the
which
This
frame of reference.
which do not involve e.
dynamics"
functions
computer
simulation
of each force.
It
234
follows
that these c.f.' s measure
molecular
rotation
laboratory
spectra,
frame of reference.
and molecular
on far infrared
dependent
and dielectric
rotation
rotational
(or translational)
Component
1)
between
in the
of these c.f.'s are
forces therefore
have an effect
171 and indeed on all the spectra
Conversely,
statistical
molecular
description
spectra
correlation
and position
transforms
and centrifugal
diffusion.
on the natural
statistical
translation
The Fourier
Coriolis
on molecular
information
the natural
and centre of mass
these spectra
correlation
and centre of mass translation.
diffusion
all contain
in the lab frame, between
is therefore
The theory of decoupled
an incomplete
of these spectra.
Correlation
The vector
Functions
of the Coriolis
and Centrifugal
Forces
identify
(5)
immediately
provides
<(x(t)
x t(t))
the result:
. (x(o)
= <(~(t).~(o))(pJ(t)
The Coriolis
therefore
x $0))’
. $L$o))> - <(x(t)
force Zm(g(t) x t(t))exists
. *(o))($$(t)
* X(O))’
in the laboratory
# <(~(t).~(o))(~(t).~(o))>
(7)
and all three types of natural mixed rotation/translation
exist in the laboratory
this frame.
<(VW
xx
+
vYwY
translation
coupling
+
laboratory
frame is:
in eqn (7) becomes
Y
in the laboratory
(8)
Y
natural measures
centrifugal
in
i.e.:
of rotation-
frame of reference.
of the molecular
functions
time dependence
clearer,
vz”z)2’ # <(v2
+ v2+“i) (UZ+ o2+ cl?,>
x
in eqn (6) are, therefore
the time-correlation
correlation
frame and have a different
At t = 0, the inequality
The time correlations
2)
frame, and
by eqn (6)
<(~(t).~(o))(~(t).~(o))>
therefore
(6)
force in the
235
C
cent.
=
y(t)
x
This is the natural
frame molecular
(z(t)
K(t))
x
way of expressing
angular velocity,
the same molecule
. t(o)
x ($fgo) x E(O))>
the correlation
t, and the position
in the laboratory
frame, expressed
this point in the 3-D space of the laboratory
frame.
In order to analyse
two vector
identities
x
(8
x R,
between
by the vector &, linking
function
(a), the following
are useful:
- A3B3Cl
+ A3BlC3),i
+ (A3B2C3 - A3B3C2
- A1BlC2
+ A1B2Cl)i
+
-
+ A2B3C2)k
(A1B3C1
. g x
the laboratory
of the centre of mass of
frame to the origin of this
the time-correlation
4 x (8 x $, = (A2BlC2 - A2B2Cl
4
(9)
(2
-
A1B1C3
A2B2C3
(10)
x @
@
(11)
9
Using eqn
C
(11) gives
=
cent
i!~(t).~(o))(~(t).r(t))(~(o).~(o))>
- <(~(t).~(o))(~(t).~(t))(~(o).~(o))'
(17-j
- <(~(t).~(o))(~(t).~(t))(~(o).~(o)j>
+ <(~(t).~(o))($t).$(t))(~(o).~(o)'
which
therefore
exist in the laboratory
It is interesting
frame.
to point out that the force proportional
where t. is the time derivative
Coriolis -and a centrifugal
kinematic relation:
of the molecular
force.
This follows
dipole moment
immediately
to # x 1,
[71, is both a
from the
(13)
236
In this context
laboratory
thevectork,
time-correlation
frame is the Fouriertransform
coefficient,
can always be written
= i,(t)
k(t)
whose
function
in the
of the far infra-red
power absorption
as:
- &W
(14)
where r and r
are the position vectors in the laboratory frame.
The
""x
QY
rotational velocity described by the molecular vector t(t) is therefore the
difference
follows
of two translational
that the Coriolis
velocities
at the same instant
force y(t) x G(t) exists
and that is also a centrifugal
force, as mentioned
t.
It
in the laboratory
already,
frame,
because,
evidently:
J(J(t) = &Cd
- cy(t)
(15)
The equality:
provides
the following
involving
considered
<(k(t)
the vectors
relations
$,k and k
in the literature
x
k(t))
. (y(o)
between
which,
laboratory
frame correlation
functions
again, do not seem to have been
on the theory of molecular
diffusion:
x ,Q(o))
= <(t(t)
x
(t(t)
= <(Q(t)
. k(o))(k(t)
. &CO))’
- <(g(t)
. Q(o))Q(t)
. &CO))’
= c+(t)
.pJ(o))($+J(t)
. ,l$(t))(t(o)
- k(O))>
- <(e(t)
.t(o))(g(t)
. g(t))(gb)
. l$(O))’
- c($!j(t)
f t(o))
+ <Q(t)
. k(o))@(t)
x
e(t)))
(x(t)
. (&CO) x (y(o) x e(o)))>
. e(t))(&(O)
. &CO))>
. $f$t))(t(o)
* k(O))’
(17)
237
The existence
of all these correlation
of the fact that a molecule
If this rotational
are present,
motion
time correlation
is non-uniform,
This means
functions
. i(o)>
- <z(t)
. qco, k(t)
. z(o)>
It becomes
molecule,
the term
that the following
relation
between
in the
gases,
gas where
These correlations
the vectors
Furthermore,
x and LI, -in
correlations
case of the infinitely
torque is infinitely
present
of
dilute
small.
in all spectra
as a natural
the natural
existence
in the laboratory
frame of the
functions:
<x(t) x e(t)
. t(o)
x &CO)’
= <t(t)
. g(o) $ (t)
- <e(t)
. k(o) k(t)
Furthermore,
and solids.
are therefore
and
non-uniformly
and translation.
(15) implies
correlation
for a translating
This leads us to the conclusion
only in the abstract
the intermolecular
of rotation
frame.
of this type, involving
as well as liquids
the type (18) disappear
(6) to (17) are valid
and eqn (18) for a translating.and
in the laboratory
that there are correlations
. @Co)>
(19)
. ~kJ)>
we have the kinematic
relation
(by differentiation
of eqn 13).
(20)
$xg=;-Kxk
Eqn
torques
(18)
clear now that eqns
rotating
rotating molecule
Eqn
frame.
. g(o) x cJ(o,>
. z(o) k(t)
outcome
i.e. if intermolecular
of this force exists naturally
= <x(t)
dilute
consequence
in the laboratory
frame:
c:(t) x k(t)
uniformly
is a natural
and translate
then we must also take into account
in eqn (2).
mt(t) x k(t)
laboratory
functions
must rotate
(20) leads directly
to the following
result,
important
from an
238
experimental
<$t)
point of view:
. j_lJo,> = <k(t)
x
IA:(t) . ($0,
+ <t(t)
x
Q(t)
. t(o)
+ <k(t)
x
t(t)
. @CO)x Q(O)’
+ y(t)
x
k(t)
. &Co) x g(o)>
x t(o)>
x k(o)>
(21)
A we--known
theorem on spectral moments
ii;(t)
GE-
. i(o)>
[71 provides
w2a(o)
0:
where P stands for Fourier
(22)
transform
and n(w) is the far infrared
absorption
coefficient.
The higher
observable
in favourable
cases by careful
and eqn 21 therefore
of correlations
between
laboratory
frame
of the molecule-fixed
(or vice-versa)
have seen, are both Coriolis
Therefore,
w%(w)
rotation
for uniform
($J= $
rotation
is the time correlation
Note that this result
accelerations
x
k(t)
transform
is a combination
limit of the 'Debye plateau',
to the
the Fourier
function
because
x Q(o)>
transform
of the second moment
of the Coriolis
diffusion,
force t(t) x k(t).
181 'inertial
where angular
in this case w's(w)
the area beneath
is infinite.
(The meaning
(and indeed
a(w) is the so-called
of k(t) in the
is obscure).
Eqn 23 would be valid for an ensemble
as we
to:
is also true in the so-called
transform
on the non-
in nature.
. t(o)
are ignored, but paradoxically
'inertial approximation'
is directly
spectroscopy,
frame with respect
eqn 21 reduces
in the theory of molecular
a(w)) has no Fourier
power
and terms such as k(t) x k(t), which,
and centrifugal
= ygt)
. $o)>,,
approximation'
moment w2z(w)
interferometric
forces such as k(t) x t(t), dependent
of the relation
$t)
spectral
shows that its time Fourier
uniformity
For uniform
the result:
of 'free rotors'.
239
The Spectral
Observations
of Direct Coriolis
By 'direct' Coriolis
on the centre of mass
frame.
linear velocity
In this respect,
dipole moment
Coriolis
vector
forces, we mean
Forces
terms of the type x x 8 dependent
x of a molecule
in the laboratory
terms such as !J x 8, etc, involving
for example)
fixed in the molecule
a vector
(the
are indirect
forces).
It is impossible
a diffusing
molecule
and translational
without
a theory for direct
taking into account
diffusion
[9-191.
in a purely
such as k x k
treatment
to construct
However,
'rotational'
exists at present
Coriolis
explicitly
both rotational
it is possible
context,
although
apart from the subject
forces in
to treat forces
no satisfactory
of this paper,
computer
simulation.
It is possible
to observe
direct Coriolis
forces of the type y, x & in
at least two ways.
1)
In Anisotropic
Materials
The application
molecular
of an electric
liquid imparts
molecule
.
CZOI
If E
in the laboratory
field
is uniform
frame, and its refractive
frame) differs
well-known
C211.
Kerr effect
field
(the Faraday
Note in this context
whole
range of frequencies
It therefore
absorption
presence
follows
coefficient
effect)
index in the axis of g (eg, the z
birefringence
This is the
can be induced by a
and an electromagnetic
that birefringence
from static
field.
of this type occurs across
to the visible
from the Kramers-Kronig
to each
anisotropic
relations
o(w) must also be different
through
the
the infrared.
that the power
in the z and x axes in the
of ;E.
It has been shown recently
of the matrix
<x(t)eT(o)>
liquid subjected
[31 (by computer
i.e.
simulation)
exist in the laboratory
to a uniform
electric
There are the (x,y) and (y,x) elements,
image,
the liquid becomes
from that in the x axis.
Similarly,
dipolar
frame torque -8 x g
and uniaxial
axis of the laboratory
magnetic
(,$) to a non-polarisable
the extra laboratory
that two elements
frame in a birefringent
field g in the laboratory
one being
the other's
frame z axis.
exact
mirror
240
~“,WyJ)
It follows
= -
<vy(t)wx(o)>
(24)
from eqn 24 that the 'time-offset'
does not average
to zero in the laboratory
direct
Coriolis
force x(t) x z(o)
frame in the presence
of E
z'
In other words:
<x(t)
x t(o)>
#
(Ez # o)
2
= 2k
< vx(t)wy(o)>
- <“y(t)wx(o)~l)
p(vx(t)wy(o)>
( =
In eqn. 25 k is a unit vector
denote
the usual ensemble
(equivalent
is always,
in the z axis; of the laboratory
averaging
in a statistically
The presence
existence
(25)
used to construct
stationary
frame and <>
a correlation
sample to running-time
of the vector k on the r.h.s. of eqn 25 implies
of the correlation
dependent
function
on anisotropy.
<x(t) x k(o)>
function
averaging
that the
in the laboratory
A diffusional
C5 I).
frame
theory for
can be developed from the generalised Langevin equation governing
x(t)
and is described elsewhere 1211. This theory
t(t)
shows that the 'time-offset' Coriolis force correlation <v,(t)wy(o)>
is
<vx(t)wy(o)>
the complete
column vector
observable
approximately
absorption
coefficient
from the correlation
IX(W) perpendicular
Note that <x(t) x t(t)
= Q for all t and &
seems to be the only means at present
natural
direct Coriolis
of the far infra-red
and parallel
so that computer
of calculating
force proportional
to
power
to EZ.
correlations
,v(t) x t(t).
simulation
between
Results
the
of this
type are given for the first time in this paper.
2)
Far Infra-red
Collision
Collision
induced absorption
and intramolecular
dependent
Induced Absorption
separation.
on centrifugal
centre of mass distance.
1231
is dependent
on the molecular
The induced dipole moment
terms such as 8 x (& x ,rij), where &ij
If the relative
velocity
palarisability
is therefore
directly
is the inter-
of two molecules
in the
frame is v.. , then the Coriolis forces v.. x g also play a part
$lJ
%=J
The translation part of the far
in the dynamics of the induction process.
laboratory
infra-red
identified
spectrum
1231
as
of simple
linear molecules
the low frequency
such as N2 and CO2 has been
component
of a broad
infra-red
band.
241
Section
2:
Computer
Simulation
Method
The Chiral Bromochlorofluoromethanes
Standard
methods
C67
constant
volume molecular
were employed
and for the racemic mixture
for 108 molecules
parameters
a(Br-Br)=3.9
computer
of the R enantiomer
of CHBrClF
The input temperature
26 *3
296 K and 1.2 x 10
. Atom-atom
were used as follows:
8; o(Cl-Cl)
= 3.6 8; o(F-F) = 2.7
o(C-C) = 3.4 #; t/k(Br-Br)
= 218.OK;
e/k@+)
= 10.0 K; e/k(C-C)
= 54.9K;
The electrostatic
simulation
of 54 R and S molecules.
and molar volume were, respectively,
Lennard-Jones
dynamics
e/k(H-H)
8;
o(H-H) = 2.8 8;
e/k (Cl-Cl) = 158.0 K;
= 35.8 K,
part of the force field was simulated
with partial
charges
on each atom as follows:
qBr =
qF
- 0.16
= - 0.22
The complete
le/
le/
;
qc1
;
[el
qH = 0.225 lel
pair-potential
25 (5x5) site-site
= - 0.18
between
;
;
two molecules
terms with the above parameters.
the same for R-R and R-S interactions
qc = 0.335
because
le/.
therefore
consisted
This was assumed
of the mirror-image
of
to be
symmetry
of the R and S enantiomers.
The system was equilibrated
and time correlation
(500 records,
in building
each separated
Field
over about 2000 time-steps
then computed
by 0.01 ps).
up the correlation
Liquid Dichloromethane
Electric
functions
functions
in the presence
Running
time averages were used
in the laboratory
of a Strong,
frame of reference.
z-Axis External
1201
This is the same algorithm
the functions
of 0.005 ps each
with about 1000 time steps
<vx(t)wy(o)>
as used in the first description
and <vy(t)wx(o)>
external
torque -gxE
is applied
as described
external
field used in this algorithm
referred
13,221 of
to already.
in the literature
can be strong enough
The
C20al.
almost
The
to saturate
242
the Langevin
function,
equilibrium.
(However,
the sample remaining
At present,
experimentally
measure
<vx(t)wy(o)>
remains
to be done).
charges
0
or
(of mass
potential
only with computer
Kerr effect apparatus
using far infra-red
representing
-CH2 group
this is possibly
available
The intermolecular
as a liquid and in anisotropic
the electrostatic
to
data and this is an experiment
is a 3 x 3 Lennard-Jones
14) is represented
simulation.
can be adopted
part of the complete
atom-atom
that
with point
potential.
The
by
(CH2 - CH2)/k = 70.5K;
a(CH2 - CH2) = 3.96 A,
qCH2
=
0.302(e/
The Cl group is represented
partial-charge
by a mass of 35.5 and Lennard-Jones
and
parameters:
c(C1 - Cl) / k = 173.5K;
o(C1 - Cl)
qc1 =
=
;
3.35 1
- 0.151
lel
The input temperature
Autocorrelation
m3/mole.
was 296K and the input molar volume
functions
were evaluated
steps) of 0.015 ps each after field-on
strength
equivalent
pronounced
RESULTS
An electric
on the direct
Coriolis
(2700 time
field
the effect of
forces.
AND DISCUSSION
(R) - Bromochlorofluoro
Figure
in Eqn
with 900 records
equilibration.
to about 14.0 kT was used to investigate
liquid anisotropy
-5
8.0 x 10
(6);
Methane
(1) illustrates
and the Racemic
the (normalised)
a) for the R enantiomer,
three correlation
functions
similar but not identical
of Eqn
Mixture
correlation
functions
defined
and b) for the racemic mixture.
(6) exist in the laboratory
time dependencies.
All
frame with
243
(a)
Auto-correlation
functions
<x(t) x y(t)
<(x(t)
. &CO))
.........
<(x(t)
.
(b)
force in (R)-CHBrClF.
. J( (0) x y(o)> / <(x (0) . y(o))
---------
x
of the Coriolis
e(t)
(
(0))
(4 (t)
Co))>
”
2r
.
/ <(x
Q(O))’
/
(0)
.
. (x (0) . y(o))>
!_!$o)P>
<v2(0) cI12
Co)>
As for (a). racemic mixture.
The three correlation
mixture,
is a useful
lead
and vice-versa.
elements
seem to be different
into the nature of the statistical
This has been pointed
of the matrix
Some Fourier
the R enantiomer
a.c.f.
functions
so that comparison of spectra for an enantiomer
<$t)wT(o)>
transforms
in enantiomers
of these correlation
and angular
velocity
be seen from fig. 2(a) that the frequency
function
is similar
to those of the
This fits in with the general
rotational
a similar
dependence
out in the literature
functions
and translational
time scale.
motion
(However,
a.c.f.
<@t).?(o)
/<w2>.
of the Coriolis
simulation
molecular
1211).
It can
correlation
functions.
that
ensembles
this is not the case in anisotropic
(see fig 5(b)) or in liquid crystals
for
of the velocity
two auto correlation
from computer
in isotropic
for
are compared
transforms
dependence
above
finding
of x upon t
C20bl
and racemic mixtures.
in fig (Z), together with Fourier
<$t).x(o)>/<v2>
for the racemic
and racemic mixture
evolve on
liquids
244
10
Freq.
Figure
(a)
5
units
Freq. umts
‘0
(2)
Fourier
transform
of the a.c.f. of
x(t) x v(t)
for (R) - CHBrClF.
+
l
(b)
Fourier
transform
of
< e(t)
. y(o)> / <u2>.
Fourier
transform
of
< x(t)
. x(o)> / <v2>.
Fourier
transforms
Therefore,
experimentally
of the curves
to a first approximation,
to being roughly
in turn is approximately
The rotational
velocity
of the far infra-red
the same as the rotational
a.c.f.
<$,(t).@o)>
approximately
a.c.f. may be estimated
velocity
is related
coefficient.
a.c.f., which
a.c.f.
(fig. (3)).
to the Fourier
Therefore
from the far infra-red
function
of the centrifugal
in fig (4) for the R enantiomer
clear that this correlation
is again different
This property
cA
the Coriolis
to the angular velocity
transform
the Coriolis
power absorption
liquids.
The correlation
illustrated
similar
power absorption
a.c.f. can be estimated
of isotropic
in Fig l(a).
function
for enantiomer
is echoed
= <(i*(t) x t(t))
reaches
(Ccent of Eqn (9)) is
a constant
It is
level as t -t m, and
and racemic mixture.
in fig (3),
. ($(o)
force
and racemic mixture.
for the correlation
function:
x ~(O))><($*(O) x k(O) . gA(0) 2-ct(o))>
245
is a unit vector in the A axis of the molecular principal
$
As discussed already this is both a centrifugal
of inertia frame.
where
correlation
amenable
function
to calculation
satisfactory
Figure
and a Coriolis
function;
with a theory of 'purely rotational'
theoretical
description
exists,
however,
force
and is
diffusion.
No
at present.
3
(R)
CHBrClF,
the a.c.f.:
< ,y* (t) x
gj(t)
“&
the same c.f. computed
accuracy
y(o)P>
in the principal
The dipole moment
computational
. iA (0) x y(o)>
(0) x
where tA is a unit vector
molecule.
-+y
force correlation
moment
vector,
moment
of inertia frame of the
n, is a combination
of eA, tB and EC.
with 4 x r;A for iA as a check of the
and overall
self-consistency
in the correlation
function.
_------function
........
function.
‘iAW
.
iA
(o)> / <ez>, the rotational
velocity
autocorrelation
of iA.
< Q(t)
. q(o)>
/
<u2>,the
angular
velocity
autocorrelation
246
Figure
(4)
The a.c.f. of the molecular
(1)
___
(R) CHBrClF;
(2)
------
the racemic mixture.
Dichloromethane
in the Presence
Fig (5a) illustrates
simulated
field.
function
(eqn (6) are likewise
Fig 5(b) illustrates
velocity
a.c.f.
fig 5(a).
A complete
Model Theory
but clearly different
(R.M.T.) of Grigolini
laboratory
frame of reference
of inter-molecular
of the molecular
Finally,
of molecular
rotation
1241.
time derivative
function
diffusion
torques,
of the vector & x @
and racemic mixture
function
but clearly
in both enantiomer
exists
in the
and racemic mixture
due to
non-uniformity
frame.
the a.c.f. of the Coriolis
for the anisotropic
of
is not usually
and the subsequent
in the laboratory
fig (7) illustrates
vector t(t) x ,$A(t)
of motion.
seems to be the Reduced
This autocorrelation
in theories
as those of
should be able to describe
from the basic equations
the correlation
considered
a.c.f.'s
. g(o)> and linear
<t(t)
diffusion
at present
(1.h.s. of eqn (18)) for the R enantiomer
the presence
a.c.f.
z axis electric
in time dependence.
under the same conditions
line of approach
bromochlorofluoromethane.
of eqn (6) computer
and the component
theory of molecular
Fig (6) illustrates
Field
of an intense
figs 5(a) and 5(b) self-consistently
The most promising
functions
x &), for:
in the presence
the angular velocity
. x(o)>
rnk x (K
is oscillatory,
oscillatory,
<x(t)
force:
of a z Axis Electric
the correlation
for dichloromethane
The Coriolis
centrifugal
CH2C12
liquid.
of a unit vector ,$A, in the principal
cum centrifugal
HerGA(t)
moment
is the
of inertia
247
0.5
0
-0.5
0
Figure
(a)
1.2
DS
(5)
As for Fig.
electric
(b)
0.6
(l), liquid CH2C12
subjected
to an intense uniaxial
field of force in the lab frame z axis.
(1) <x (t)
F
(2) ‘8 (t)
. x (0)) / <“;>
. & (0)) / 0.c >
Note that the time-dependence
The time dependence
of these two functions
of the Coriolis
functions
is very different.
of Fig. 5(a) is intermediate.
I
_.
I
I
1.0 ps
Figure
-
(6)
CR) ___
normalised
angular motion
respect
CHBrClF and -------,
the racemic mixture:
a.c.f. of the vector & x i, due to the non-uniformity
of one frame of references
to the other
(the laboratory
(the molecule
frame).
the
of the
fixed frame) with
248
1.0
0.5
--I;;:._
OO
0.6
1.2
PS
Figure
(7)
CH2C12 + field, the auto-correlation
&
(t)
function
of the force vector
(t)
x i*
frame of the CH2C12 molecule.
is oscillatory
level at t+
computed
in this paper will show the Grigolini
fall-transient
m.
This function
constant
The Coriolis
acceleration
and centrifugal
1261
effect
decoupling
with varying
and reaches
a
force a.c.f.'s
effect
1251, and
field-strength
E
z'
CONCLUSIONS
Autocorrelation
frame of reference
molecular
Coriolis
5 x &
functions
which are not usually
diffusion.
generated
these vectors
centrifugal
in the laboratory
as natural measures
of mass translation
It follows
;
by the non-uniformity
torques
or position
and translates
of molecular
due to inter-
of Wales
of the inter-relation
of a molecule
to consider
between
the centre
and its own rotational
broadening
to the laboratory
is thanked
motion
It is appropriate
frame.
for a Fellowship.
motion.
of spectra must take these
the molecular
ACKNOWLEDGEMENTS
The University
of
of the
and the vector
forces into account whenever
with respect
in theories
functions
force rnt x (E x z)
frame.
that a theory of collisional
basic non-inertial
fully considered
These include autocorrelation
force, 2mx x t
molecular
have been shown to exist in the laboratory
frame rotates
249
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