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c CCLRC
Section 2.5
electrostatic interaction (i.e. self interaction) between the core and shell of the same atom. Non-
coulombic interactions arise from the shell alone. The effect of an electric field is to separate the
core and shell, giving rise to a polarisation dipole. The condition of static equilibrium gives the
polarisability as:
= (2q
2
s
- q
2
c
)/k ,
(2.159)
where q
s
and q
c
are the shell and core charges and k is the force constant of the harmonic spring.
The calculation of the virial and stress tensor in this model is based on that for a diatomic molecule
with charged atoms. The electrostatic and short ranged forces are calculated as described above.
The forces of the harmonic springs are calculated as described for intramolecular harmonic bonds.
The relationship between the kinetic energy and the temperature is different however, as the core-
shell unit is permitted only three translational degrees of freedom, and the degrees of freedom
corresponding to rotation and vibration of the unit are discounted as if the kinetic energy of these
is regarded as zero (
3.11
).
2.5.1
Dynamical (Adiabatic Shells)
The dynamical shell model is a method of incorporating polarisability into a molecular dynamics
simulation. The method used in DL POLY 3 is that devised by Fincham et al [
35
] and is known
as the adiabatic shell model.
In the adiabatic method, a fraction of the atomic mass is assigned to the shell to permit a dynamical
description. The fraction of mass is chosen to ensure that the natural frequency of vibration of
the harmonic spring (i.e.
=
1
2
k
x(1 - x)m
1/2
,
(2.160)
with m the atomic mass,) is well above the frequency of vibration of the whole atom in the bulk
system. Dynamically the core-shell unit resembles a diatomic molecule with a harmonic bond,
however, the high vibrational frequency of the bond prevents effective exchange of kinetic energy
between the core-shell unit and the remaining system. Therefore, from an initial condition in which
the core-shell units have negligible internal vibrational energy, the units will remain close to this
condition throughout the simulation. This is essential if the core-shell unit is to maintain a net
polarisation. (In practice there is a slow leakage of kinetic energy into the core-shell units, but this
should should not amount to more than a few percent of the total kinetic energy.)
2.5.2
Relaxed (Massless Shells)
The relaxed shell model is presented in [
36
], where shells have no mass and as such their motion
is not governed by the usual Newtonian equation, whereas their cores' motion is. Because of that,
shells respond instantaneously to the motion of the cores: for any set of core positions, the positions
of the shells are such that the force on every shell is zero. The energy is thus a minimum with
respect to the shell positions. This represents the physical fact that the system is always in the
ground state with respect to the electronic degrees of freedom.
Relaxation of the shells is caried out at each time step and involves a search in the multidimensional
space of shell configurations. The search in DL POLY 3 is based on the powerful conjugate-
gradients technique [
37
] in an adaptation as shown in [
36
]. Each time step a few iterations (5÷15)
are needed to achieve convergence to zero net force.
In DL POLY 3 the shell forces are handled by the routine core shell forces. In case of the
adiabatic shell model the kinetic energy is calculated by core shell kinetic and temperature
40