Biomedical Engineering Reference
In-Depth Information
degrees of freedom) rather than explicitly treating the three nuclei separately (nine degrees
of freedom). The classical energy for a collection of
N
rigid rotor molecules consists of the
kinetic energy for translation and rotation, together with the intermolecular potential. Each
water molecule is described by six coordinates: three specify the centre of mass and three
angles that fix the spatial orientation about the centre of mass. In this section, I will denote
these coordinates by the six-dimensional vector
X
. In terms of the linear velocities
v
i
, the
angular velocity vectors
ω
i
, the moments of inertia
I
i
and the coordinates
X
i
, the energy
turns out to be
N
mv
i
ω
i
I
i
ω
i
+
1
2
ε
=
+
U
(
X
1
,
X
2
, ...
X
N
)
i
=
1
The question then is the extent to which the intermolecular potential is pair wise additive;
such functions may always be resolved into contributions from pairs, triples and higher
contributions:
N
N
U
(2)
X
i
,
X
j
+
U
(3)
X
i
,
X
j
,
X
k
+···+
U
(
X
1
,
X
2
, ...
X
N
)
=
U
(
N
)
(
X
1
,
X
2
, ...
X
N
)
i
<
j
i
<
j
<
k
(10.14)
In the case of simple fluids such as liquid argon it is widely believed that
U
is practically pair
wise additive. In other words, the contributions
U
(
p
)
for
p
> 2 are negligible. However, the
local structure in liquid water is thought to depend on just these higher order contributions
and so it is unrealistic in principle to terminate the expansion with the pair contributions.
It is legitimate to write the potential as a sum of pair potentials, provided one understands
that they are effective pair potentials that somehow take higher terms into account.
The classic molecular dynamics study of liquid water is that of Rahman and Stillinger
(1971). They wrote the effective pair potential as a sum of two contributions: a L-J 12-6
potential
U
L-J
for the two oxygen atoms
4ε
σ
R
6
12
σ
R
U
LJ
(
R
)
=
−
and a function
U
el
, modulated by a function
S
(
R
ij
) that depends sensitively on the molecular
orientations about the oxygen atoms
U
(2)
X
i
,
X
j
=
U
LJ
R
ij
+
S
R
ij
U
el
X
i
,
X
j
(10.15)
The L-J parameters were taken to be those appropriate to neon, σ
=
282 pm and
ε
10
−
22
J, on the grounds that neon is isoelectronic with water.
Four point charges
Q
each of magnitude 0.19
e
and each 100 pm from the oxygen nucleus
are embedded in the water molecule in order to give
U
el
. Two charges are positive, to
simulate the hydrogen atoms, whilst the other two are negative and simulate the lone pairs.
The four charges are arranged tetrahedrally about the oxygen.
The set of 16 Coulomb interactions between two water molecules gives
U
el
. Figure 10.7
shows the minimum energy conformation, with the two 'molecules' 76 pm apart (i.e. an
oxygen-oxygen distance of 276 pm).
=
5.01
×
