DERIVATION OF POTENTIAL PARAMETERS

The availability and accuracy of suitable interatomic potentials is crucial to the sim ulation of atomistic properties of materials. There are num erous approaches to the derivation of such param eters which shall be outlined here. Derivation of m olecular mechanics param eters is usually achieved by empirical fitting to the properties of a set of representative molecules and functional groups, or by

fitting to an ab-initio energy hypersurface. Only potentials pertaining to Born m odel simulations were derived for use in the w ork presented in this thesis and the following discussion on param eter derivation will be restricted to

procedures related to such models. However, the m ethods for deriving

param eters for molecular mechanics forcefields are very similar - the reader is referred elsewhere for details, for example Burkett and Allinger^ and Maple et al}^

3.4.1 Empirical Fitting

The potential param eters are fitted to known structural properties of systems. Other physical properties such as lattice energy, dielectric constants and elastic constants can also be used. The param eters are adjusted in an iterative

procedure to achieve the best fit to the experimental properties. The resulting potentials often model very accurately the systems to which they were fitted (naturally) and also point defects in this lattice and closely related systems. However, care m ust be taken w hen transferring potentials to other systems, particularly if the equilibrium interionic distances differ significantly betw een the system in which the potential is fitted and that in which it is being used. This is a consequence of the fitting procedure only sam pling a small portion of the interaction energy surface. Simultaneous fitting of potentials to a range of systems is often used to reduce this problem and to obtain self consistent param eters.^

3.4.2 Ab-initio Energy Surface Fitting

Energy hypersurfaces generated from ab-initio calculations are increasingly being used to generate potential param eters for sim ulation t e c h n i q u e s . F o r solids, these calculations can be based on infinite solids (utilising periodic ab- initio codes such as CRYSTAL^^) or on finite clusters.^^ These potentials have the advantage that not only are they derived from first principles b u t can also investigate non-equilibrium geometries to improve the perform ance of the potentials w hen investigating systems in which there is significant deviation from perfect lattice equilibrium spacings as in e.g. defects.

3.4.3 Electron Gas Interactions

This m ethod was used extensively in this thesis and will be discussed in more detail. As in the above method, the interactions of two atomic species are calculated at a range of interionic distances, to which a potential is fitted. The m ethodology used is based on the work of Wedepohl^^ which was developed by G ordon and Kim^® and for which an useful implem entation is available from H arding and Harker.^^ The electron density is treated as a degenerate fermi-gas and the electron density is assum ed to be the sum of the individual spherical atomic densities. It is also considered to change slowly w ith distance and can therefore be treated as a homogeneous electron gas. A lthough this is clearly not valid for regions close to the nuclei of the ions, these regions contribute little to the interaction energy. However, this approxim ation does hold for the valence electrons which are the major contributors to the interaction energy. In

application to solids the electron densities of the isolated species are calculated in the presence of a suitable M adelung well to simulate the lattice. W hilst this has little effect on the electron density of small cations, it is im portant w hen

considering highly polarisable anions. For example, the ion is unstable as a free ion; the second electron is unbound w ith respect to the O ion. How ever, this ion is stabilised in the presence of a suitable crystal field.

The energy of two interacting ions can be expressed as

E,

w here the first term is the energy of the interacting ions and the second and third terms the energy of the isolated ions; p is the electron density. The energy of the interacting ions can be expanded as;

^[p,- (r) + p y (r)] = Eelec + Ekin + Eexch + Ecorr + Edisp (3.18)

Eelec is the electrostatic interaction. Ekin is the kinetic energy and is proportional to ^ p where p is the electron density. The exchange energy Eexch is

proportional to ^ p . The correlation energy (Ecorr) and the dispersive terms

( E d i s p ) are similarly approxim ated (see H arding and Harker^^ for further details).

Although the method is relatively crude and tends to overestimate lattice param eters and interatomic distances, several studies using this m ethod have obtained good results w hen applied to both perfect and defective s y s t e m s . I t has the advantage that potentials for impurities and lattice ions w ith different oxidation states can be calculated in the host lattice M adelung field. Thus, the param eters are all self-consistent, in contrast to empirically derived potentials w hich may be unreliable w hen transferred to other systems.

3.5. Simulation Techniques based on the Born