Dynamical Shell Model

An atom or ion is polarisable if it develops a dipole moment when placed in an electric field. It is commonly expressed by the equation

$$\mbox{$\underline{\mu}$}=\alpha \mbox{$\underline{E}$},$$ (2.170)

where $\mbox{$\underline{\mu}$}$ is the induced dipole and $\mbox{$\underline{E}$}$ is the electric field. The constant $\alpha$ is the polarisability.

The dynamical shell model is a method of incorporating polarisability into a molecular dynamics simulation. The method used in DL_POLY_2 is that devised by Fincham et al [34] and is known as the adiabatic shell model. An alternative model is presented in [35].

In the static shell model a polarisable atom is represented by a massive core and massless shell, connected by a harmonic spring, hereafter called the core-shell unit. The core and shell carry different electric charges, the sum of which equals the charge on the original atom. There is no 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:

$$\alpha=(2q_{s}^{2}-q_{c}^{2})/k$$ (2.171)

where $q_{s}$ and $q_{c}$ are the shell and core charges and $k$ is the force constant of the harmonic spring.

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 $\nu$ of the harmonic spring (i.e.

$$\nu=\frac{1}{2\pi} \left [ \frac{k}{x(1-x)m} \right ]^{1/2},$$ (2.172)

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.)

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 (the kinetic energy of these is regarded as zero).

In DL_POLY_2 the shell forces are handled by the routine SHLFRC. The kinetic energy is calculated by CORSHL and the routine SHQNCH performs the temperature scaling. The dynamical shell model is used in conjunction with the methods for long range forces described above.