INTERATOMIC POTENTIAL MODEL

M any studies have been reported concerning interatom ic potentials for oxide m aterials (see Catlow and M ackrodt eds. 1982; Catlow et al 1988). We now discuss the functional form s and the m ethods u sed to derive appropriate param eters.

5.3.1 POTENTIAL FUNCTION

Oxides have been m ostly described by the ionic m odel w ith form al or partial ionic charges assigned to point entities w hich also interact w ith short-range term s. The interactions betw een p o in t charges come from long-range electrostatic Coulombic forces betw een ions, w hile the short- range interactions come from the overlap of the electron charge clouds of the interacting ions.

The sim plest and m ost w idely used short-range form is the central- force pair-potential:

V (ri,r2,...,rn) = S Vij ( I ri-rj I ) (5.1)

The total p o ten tial energy V is sum m ed over all the p a ir interaction term s, each of w hich is dependent only on the distance betw een the ions.

The m ost w idely used function form of this pair-potential for ionic solids is the Born-M ayer or Buckingham potential:

V(rij) = A exp(-rij/pij) - Crij-6 (5.2)

The second term is often ad d ed to express d ispersion an d attractive term s.

The o th er fu n ctio n al form , w hich is said to be su itab le for m odelling the effect of the covalent bonding is the Morse potential:

V(rij) = Dij {exp[-2Pij(rij-ro)] - 2exp[-Pij(rij-ro)]} (5.3) A lth o u g h these p air-p o ten tials have reasonably rep ro d u c ed n o t only structures b u t also properties of oxide m aterials, m ore sophisticated

m odels are used to include polarization or covalent bonding effect m ore precisely.

Polarizability is described effectively using the shell m odel (Dick an d O verhauser 1958). An ion is described as a massless shell of charge Y, a core in w hich m ass is concentrated and a harm onic sp rin g w hich connect the shell w ith the core. This m odel has im proved the calculation of dielectric, lattice-dynamical and defect properties of ionic solids.

On the other hand, in order to express covalent effects, a three-body term is added. The m ost com m on form is the sim ple-harm onic, bond- b en d in g form;

V(0) = l/2 K B(0-eo)2 (5.4)

w here Kg is the bond-bending force constant and 0 0 is the equilibrium b ond angle.

For crystalline silicates, pair-potential m odels (Tsuneyuki et al 1988; Beest et al 1990), and shell m odels w ith three-body potential (Sanders et al 1984; Price et Parker 1984) have successfully been em ployed to m odel structures and properties.

5.3.2 DERIVATION OF INTERATOMIC POTENTIAL

Interatom ic potentials have been derived by tw o m ain procedures. The first is the so-called em pirical m ethod. The p aram eters in the p otential m odel are fitted so that they can reproduce the experim ental structures a n d /o r properties (e.g. elastic constants, dielectric constants or vibrational properties) as well as possible. This m ethod m ay be applied even w hen the only data available are the crystallographic param eters. But care m u st be taken w h en such potentials are ap p lied to related structures w hich are different from the fitted structure. A nd in general it

is crucial to use as m any data as possible for fitting and testing potential m odels.

The o th er ap p ro ach is to use non-em pirical or sem i-em pirical m eth o d s, em p lo y in g q u an tum -m echanically calculated d a ta for the p otential energy surface. In the electron gas m ethod (G ordon and Kim 1972), electron densities are calculated for the isolated interacting atom s, an d th en the C oulom b interactions, the kinetic energy, exchange and correlation contributions to the interacting energy are calculated. O n the other hand, ab-initio m ethods m ay be em ployed on clusters or periodic array s of atom s. For exam ple, using ab-initio, periodic H artree-Fock tech n iq u es. Gale et al (1992) rep ro d u c ed the stru c tu re a n d elastic constants of a -A l2 0 3 from using the calculated potential energy surface obtained using the CRYSTAL code. In the case of cluster calculations, the im p o rtan ce of crystal field effects m u st be stressed . For exam ple, M ack ro d t an d S tew art (1979) in tro d u c e d the M ad elu n g p o te n tia l a p p ro p ria te to the crystal w hen the w avefunctions w ere solved, and T suneyuki et al (1988) an d Beest et al (1990) w ere obliged to use experim ental d ata on elastic constants to determ ine the p artial charge valu es.

One of the m ost im portant aspects concerning a potential m odel is its transferability. Some potentials (for exam ple. Price and Parker 1984; Tsuneyuki and al 1988) successfully reproduced the structures of several polym orphs using the sam e potentials. H ow ever, several potentials are not transferable betw een polym orphs. H ow ever, potentials fitted to the crystal structure and properties of Si02 w ere applied to vitreous states and they reproduced successfully the experim ental RDFs (Vessal et al 1989). H ow ever, it is still not clear to w h at extent such potentials can

rep ro d u ce hig h ly d istorted structures u sing the po ten tials w hich are fitted to a particular crystal structure and its properties.

5.4 APPLICATION OF PREVIOUSLY REPORTED POTENTIALS