1. science
2. chemistry
3. organic chemistry

Journal of Molecular Liquids, 32 (1986) 173-181
Elsevier Science Publishers B.V., Amsterdam
- Printed
MOLECULAR DYNAMICAL SIMULATION
173
in The Netherlands
OF NEW AUTO AND CROSS CORRELATIONS
IN LIQUID
WATER
M. W. EVANS
Department
Swansea
of Physics,
University
College
of Swansea,
Singleton
Park,
SA2 8PP
(Received
21 January
1986)
ABSTRACT
This Letter
centripetal,
liquid water,
functions
reports
together with
involving
velocity.
the first simulation
and non-uniform
auto-correlation
of the classical
functions
strong single molecule
its centre of mass position,
These results were obtained
based on the ST2 of Stillinger
Coriolis,
of the Hz0 molecule
in
cross correlation
linear velocity
with a new empirical,
and angular
pair potential
111
and Rahman.
INTRODUCTION
This Letter
reports
types of dynamical
translation
obtained
the first detailed
numerical
auto and cross correlations
and position
in liquid water.
and a brief account
atom type potential
between
Several
is given of each.
investigation
molecular
original
of new
rotation,
results have been
Using a new empirical
based on the ST2 of Stillinger
and Rahman
Cl1
atom
it is
found that:
a)
autocorrelation
Coriolis,
centripetal,
laboratory
molecular
b)
functions
and non-uniform
moments
various
motion
accelerations
(x,y,z) frame and <n the moving
types of single molecule
in water
frame,
classical
exist.both
in the
(1,2,3) of the principal
of inertia;
showing
exist in both frames,
diffusion
(a.c.f. 's) of the water molecule's
as being
cross correlation
that it is impossible
independent
functions
to consider
of the centre of mass
(c.c.f.'s)
rotational
translational
of the same molecule.
The vast aqueous
diffusional
dynamics
0167s7322/86/$03.50
literature
C21 leaves room for detailed
of the Hz0 molecule,
0
1986 Elsevier Science
but still within
Publishers
B.V.
studies
the severe
of the
174
limitations
gauged.by
of classical
choosing
dynamics.
and the other by Clementi
to be that the evolution
ab initio methods
(in frame
Clementi
of computer
obtained
functions
(p.d.f. 's) ofclementi
(classical)
which
simulations
is entirely
in agreement
reference
(1,2,3) by
pair distribution
Monte Carlo
disappearing
Empiricism
therefore
from
but the recent paper by Morse and
potentials,
none of
satisfactory.
equations
written
been obtained
in rotating
analytically
and moving
- the former for the centre of mass molecular
latter for the same molecule's
need for friction
seems
3 and 4 body
with experiment.
that there are many empirical
Some new insight has recently
use of Langevin
difference
and in frame
The atom-atom
of liquid water,
shows clearly
in 1974
et al. to introduce
and to introduce
and Rahman
seem to be gradually
111
for the centre of mass velocity
et al., from a separate
131, seems to be broadly
and two body constraints
Clementi
the pair potential
in the two papers
(x,y,z) by Stillinger
in this field may be
and Rahman
The essential
power allows
et al.) do not differ markedly.
simulation
Rice [4]
one by Stillinger
et al. 13; in 1985.
of computing
The results
terms.
a.c.f.
A decade of progress
two key papers:
rotational
cross terms, C7-91
the three dimensional
diffusion
movement.
translation
This method
and provides
of an asymmetric
[5,61
by the
frames of
a natural
top molecule
and the
removes
description
the
of
in the liquid
state.
Computer
Simulation
Methods
With the restricted
computer
the well known ST2 potential
the oxygen and hydrogen
described
power available
by including
atoms.
The empirical
terms centred both on
pair potential
is then
as follows:
E/k(O-0)
= 58.4 K
;
0(0-o) = 2.8 2
;
E/k(H-H)
= 21.1 K
;
o(H-H) = 2.25 2
;
c/k(O-H)
= (d (O-O) ; (H-H)));
where 5 and o are the usual atom-atom
k
the tetrahedral arrangement of charges
paper,
to me, I chose to modify
atom-atom
Cl1 together with the original
of using separate
hydrogen
and oxygen
;
a(O-H) = ;(a(O-0) + o(H-H))
Lennard-Jones
parameters.
In addition,
in ST2 was used as in the original
ST2 geometry
atom-atom
(fig. (I)).
One advantage
terms is that the artificial
175
swithing
atom-atom
function
of Stillinger
parameters
estimates.
were based
and Rahman
closely
Cl1
is no longer required.
on independent,
The
experimental,
Cl01
9
Q
L
Fig.
Illustration
1
of the ST2 geometry,
With a time step of 5.0 x 10
no Ewald corrections,
configurational
obtained
the mean pressure
energy
by Stillinger
corrections.
an experimental
These results
equations
can be compared
and Rahman
of motion
on each molecule
this internal
estimate
boundary
conditions,
C31
the
of about
with a value
The
of -34.3 kJ/mole
Cl], also in the absence
energy
1000 time steps,
bar.
at 314K
of Ewald
the use of Ewald sums by Clementi
to -38.6 kJ/mole,
which
compares
with
of -41.0 kJ/nole.
were obtained
numerically
segment
- 137 ? (1.15 x 106)~
For two body interactions
et al. 131 reduced
(1,2,3).
energy at 300 K; p = 0.997 gm/cm, molar
was, for a typical
= 18.08 cm/mole;
- 35.5 kJ/mole;
of frame
sec., a sample of only 108 molecules,
and simple cubic periodic
total mean configurational
volume
-16
and definition
by integrating
in two stages;
was first computed
the classical
C11,121
from the atom-atom
rotational
the total torque
and charge-charge
forces,
176
and the angular momentum
interpolation
translational
numerically
respect
equation
of motion
with the Verlet
corrections
to be constant
were applied
for Lennard
NUMERICAL
RESULTS
Atom-Atom
P.D.F.'s
values
- Newton's
equation
- was integrated
applied
with
The total energy
separation.
a few parts
to the virial
cubic
The
and the cut off criterion
centre of mass
within
in time sequence.
in a thousand.
Long range
sum and to the total configurational
Jones terms only.
Atom atom p.d.f. 's were averaged
(4000 time steps).
The oxygen
Table 1 with equivalent
Clementi
by a numerical
(1,2,3) then
algorithm,
to the intermolecular
was observed
energy
in frame
over the four previous
et al. 131.
over about four consecutive
to oxygen
results
(0-0)p.d.f.
from Stillinger
(The notation
is compared
and Rahman
111
segments
in
and from
is that of the former paper.)
TABLE 1
Oxygen
T/K
-
oxygen
p.d.f.'s
Potential
RI(~)
Rz(%
R3(%
2.63
3.21
4.31
r(Ml)/a
2.86
ML
3.02
r(M2)/8
4.74
M2
1.08
314
STZ1
300
Clementi
et al.
300
This work
2.72
3.70
5.25
3.10
2.18
5.66
1.11
323
Exptl.ly3
2.64
3.31
4.21
2.85
2.28
4.75
1.11
As usual
%2.8
141, the empirical
and weaknesses.
O-O p.d.f.
maximum
potential
In table 1, for example,
(at 323 K) is reproduced
intensity
2.45
s4.0
'L1.1
for this letter has its strengths
the first peak of the experimental
much better
in intensity
than ST2;
of the second peak is also close to the experimental
the
value,
177
but the position
of the second peak is too far out.
In this respect
et al. 131 seems to be too far in, by about the same amount,
Clementi
their two body potential
this work
are broadly
and Monte Carlo simulation.
similar
in detail
with
The H-H p.d.f.'s
of
III to STL, and will be illustrated
in full elsewhere.
Time Correlation
Functions
The more original
dynamical
1)
properties
Coriolis,
molecule
exist
19
for the Hz0 Molecule
results
of this Letter
are given
in this section
on
I - a.c.f. 's and c.c.f.' s of the Hz0 molecule.
centripetal
and non-uniform
in liquid water
accelerations
1131
of the Hz0
at 300 K in both frames of reference,
V
0.5
Fig. 2
a)
The a.c.f.
b)
A.c.f.
of the centripetal
acceleration,
frame
(x,y,z).
of the non-uniform
acceleration,
frame
(x,y,z).
of the H20 Coriolis
c)
A.c.f.
d)
As for a), frame
(1,2,3).
e)
As for b), frame
(1,2,3).
f)
As for c), frame
(1,2,3).
acceleration,
frame
(x,y,z).
and can
178
be observed
speaking,
through
because
their respective
frame
respect
to frame
fig. 2
is the first illustration
molecule's
a.c.f.'s.
(x,y,z) and vice-versa.
Coriolis
acceleration,
the linear
results
2)
Ryckaert
<t(t)xT(o)>
et al. 1141
in frame
off-diagonal
difference
the basic
or translational
elements
between
is a measure
molecular
inapplicability
for liquid water.
two different
segments,
"noise".
Here v is
?i
of the molecular
velocity.
of theories
of purely
the simple
the existence
The hatched
each of about
The intensity
Non-vanishing
(i,j) : <u.(t)v
1
j
elements
(o)> /(<w.~>'
1
of <e(t)xT(o)>
<v.2>1).
J
c.c.f.
of two of its
area denotes
the
1000 time steps,
and
of these elements
PS
Fig. 3
These
in liquid water.
(1,2,3), and fig. 3 confirms
of my computer
vector
angular
were the first to observe
111,121
acceleration,
k(t) x c(t).
;F.is the position
diffusion
with
- that of the Hz0
centripetal
acceleration
and t is the resultant
show clearly
frame of reference
As far as I am aware,
of these a.c.f.'s
centre of mass velocity,
rotational
15,61
2&(t) x x(t);
t(t) x (t(t) x ,r(t)); and non-uniform
centre of mass,
This is true, essentially
(1,2,3) is a non-inertial
in frame
(1,2,3).
shows
179
the strength
of cross-correlation
in this dynamical
3)
context.
The development
recently
15,61
frame c.c.f.'s,
of rotating
to reyeal
typified
confirming,
and the existence
the dominant
frame Langevin
equations
the existence
of numerous
by the diagonal
elements
where 4 : x,g x 6 or 6.
water
and indicates
role of H-bonding
121
I have checked
new types of moving
of <k(t) x t(t) kT(o)>
that these all exist for liquid
the validity
inter alia,
of strong dynamical
has been used
of the new Langevin
cross-correlation.
equations
The c.c.f.
C61
for
A-V
is shown in Fig. 4.
%
?r
0.04 m
O.l2Q
0.04
ps_
_
JJ
!
2,2
1
-
Fig. 4.
The non-vanishing
in frame
(1,2,3);
4)
C.c.f.'s
frame
(also C
is confirmed
2v
of <x(t) x t(t)xT(o)>
C61
asymmetric
top CHzC12.
for the Hz0 molecule
1)
in the frame
of the type <g(t) x x(t)tT(o)>
in this Letter
) in Fig. 5.
/<(v2>02>
(2) and (3) above all vanish
elements
(x,y,z) for the C2v symmetry
result
elements
(i,i) = <(x(t) x t(t));vi(o)>
of types
but of off-diagonal
(diagonal)
(x,y,z)
exist 1151
in
This simulation
in liquid water
180
Fig. 5.
The non-vanishing
in frame
(x,y,z);
(i,i) =
'(I
(off-diagonal)
elements
of <2(t) x
$t)tT(o)>
x ~(t))iwi(o)'
I
<$>5 <J>
CONCLUSIONS
The existence
accelerations
of the classical
of the Hz0 molecule
for the first time, together
Coriolis,
centripetal,
in liquid water
with strong dynamical
and non-uniform
is demonstrated
explicitly
cross-correlation
on the
181
single molecule
potential,
These results
level.
based on the well-known
were obtained
with a new empirical
ST2 of Stillinger
and Rahman.
pair
Cl1
ACKNOWLEDGEMENTS
The University
of Wales
is thanked
for the award of the Pilcher
Senior
Fellowship.
REFERENCE
1
s
F.H. Stillinger
(see also
and A. Rahman,
J. Chem. Phys.,
: ibid., 55 (1971) 3336;
61 (1974) 1545 ;
57 (1972) 1281; 61 (1974) 4973
;
68 (1978) 666).
2
e.g. D. Bertolini,
C. Salvetti,
in "Advances
P. Grigolini,
S.A. Rice,
3
in Chemical
(Wiley/Interscience,
G. Corongiu,
L. Domingo,
A. Laaksonen
Dynamics",
10
and J.S. Dahler,
N.L. Allinger,
Forces
11
M. Ferrario
12
M.W. Evans
13
M.R. Spiegel,
14
J.P. Ryckaert,
15
M.W. Evans,
S. Chin,
74 (1985).
55 (1985) 818.
55 (1985) 1551
W.T. Coffey
44 (1966) 3988
23 (1982) 296
and P. Grigolini,
New York,
and Scattering
S.J. Angyal
(Wiley, New York,
131B&C,
76 (1982) 650.
J. Chem. Phys.,
(Wiley/Interscience,
"Intermolecular
M.W. Evans,
Physica,
and
5.
H. Kahnmohammadbaigi,
and R.F. Fox, J. Math. Phys.,
G.J. Evans,
Vol. 62, ed. M.W. Evans,
1985), Chapter
Phys. Rev. Letters,
Phys. Rev. Letters,
D.W. Condiff
M.W. Evans,
New York,
J.H. Detrich,
and
ser. ed., I. Prigogine
and S.A. Rice, 3. Chem. Phys.,
M.W. Evans,
P. Grigolini
Physics",
and N.L. Nguyen,
M.W. Evans and C.J. Evans,
U. Steiger
M. Ferrario,
and G. Pastori-Parravicini;
E. Clementi,
M.D. Morse
M. Cassettari,
"Molecular
1982), Chapter
Phenomena",
and G.A. Morrison,
5.
in E. Eliel,
"Conformational
Analysis",
1965).
and M.W. Evans,
Adv. Mol. Rel. Int. Proc.,
Phys. Rev. Letters,
and G.J. Evans,
"Theory
22 (1982) 245;
50 (1983) 371
J. Chem. Sot., Faraday
and Problems
Trans II, 79 (1983) 767
of Vector Analysis",
(Schaum, New York,
1959), p. 53.
Trans.
A. Bellemans
and C. Ciccotti,
Phys. Rev. Letters,
II, in press.
in press;
Mol. Phys.,
44 (1981) 979
J. Chem. Sot., Faraday