Having deduced the functional form of the potential from first principles, it remains to choose the fitting functions and fit their parameters to empirical data. Most papers simply state that ‘the potential was parameterized by fitting to….’ The reality is different.

Firstly, one must decide what functions to use for the various terms. Here, one may be guided by the physics (atomic charge density tails in EAM, square root embedding in FS, Friedel oscillations), by the anticipated usage (short-range potentials will speed up MD, and discontinuities in derivatives may cause spurious behavior), or simply by practicality (Can the potential energy be differentiated to give forces?).

Secondly, one must decide what empirical data to fit. Cohesive energies, elastic moduli, equilibrium lattice parameters, and defect energies are common choices. Accurate ab initio calculations can provide further ‘empirical’ data, notably about relative struc­tural stability, but now increasingly about point defect properties. (It is, of course, possible to calcu­late all the fitting data from ab initio means. Potentials fitted in this way are sometimes referred to as ab initio. While this is pedantically true, the implication that these potentials are ‘better’ than those fitted to exper­imental data is irritating.)

Ab initio MD can also give energy and forces for many different configurations at high temperature. Force matching44 to ab initio data is one of best ways to produce huge amounts of fitting data. There have been many attempts to fully automate this process, but to date, none have produced reliably good poten­tials. This is in part because of the fact that although MD only uses forces (differential of the potential), many essential physical features (barrier heights, structural stability, etc.) do depend on energy, which in MD ultimately comes from integrating the forces. Small systematic errors in the forces, which lead to larger errors in the energies, can then cause major errors in MD predictions. Furthermore, if the poten­tial is being used for kinetic Monte Carlo, the forces are irrelevant.

By using least squares fitting, all the data may be incorporated in the fit, or some data may be fitted exactly and others approximately. However, since the main aim of a potential is transferability to different cases, the stability of the fitting process should be checked. The best way to proceed is to divide the empirical data arbitrarily into groups for fitting and control, to fit using only a part of the data, and then to check the model against the control data. This process can be done many times with different divisions offitting and control. Any parameter whose value is highly sensi­tive to this division should be treated with suspicion.

Structural stability is the most difficult thing to check, since one simply has to check as many struc­tures as possible. In addition to testing the ‘usual suspects,’ fcc, bcc, hcp, A15, o-Ti, MD, or lattice

dynamics can help to check for mechanical instability of trial structures.