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Strictly speaking, in numerical simulations the Hamiltonian should be calculated from the quantum wave‐function equation, Hψ = Eψ, where ψ and E are the wave‐function and energy of the system, respectively. As expounded in Chapter 2.9, such calculations require so much computing work that they are currently restricted to smaller systems typically made up of a few tens of atoms. In the present chapter, simulations made within a classical framework will thus be considered instead. They rely on the fact that the differences between vibrational and electronic energies and frequencies are so large that atomic vibrations may be considered to take place within a fixed electronic configuration. This is the celebrated Born–Oppenheimer approximation whereby the Hamiltonian of a system is expressed as the sum of kinetic and potential energies, which are functions of the selected set of coordinates q(i) and momenta p(i):

(3)

In Eq. (3) the kinetic energy, K(p), is simply expressed as a function of the mass and of the velocity calculated for each atom. The potential energy, U_p(q), is much less readily determined because it strongly depends on the specific interactions between the various kinds of atoms present in the system. Quite generally, however, interaction potentials are markedly asymmetrical in terms of interatomic distances because they rise extremely steeply when atoms get mutually very close whereas they reach a constant value – the dissociation energy – when atoms become so distant that they can be considered as no longer bonded. As introduced in 1929 to represent isolated diatomic molecules, the Morse potential accounts well for this asymmetry:

(4)

where E₀, k, and r₀ are the dissociation energy, a measure of the bond strength, and the equilibrium interatomic distance of the molecule, respectively, three parameters that are determined from vibrational spectroscopy data.

In a condensed phase, potentials are much more complicated since a given atom interacts with a great many others over distances that can be large. To keep the number of parameters as small as possible in the expression of potential energies, one thus groups into the same term all interactions between given pairs of like or unlike atoms regardless of their mutual distances. Although the Morse potential remains a good starting point for systems where bonding is covalent, other kinds of analytical expressions are generally used for potential energies in the MC and MD simulations dealt with in this chapter. As borne out by the variations with composition of macroscopic properties, atomic interactions have the simplifying feature that they are primarily pairwise in oxide or salt systems. This feature is embodied in the most popular potentials used for these systems, namely, the Buckingham,

(5)

and the Born–Mayer–Huggins potentials,

(6)

where r_ij is the distance between two atoms, and A, B, r₀, C, D are parameters inherent to each interaction (Figure 2).

In both potentials, the first, second, and third terms represent repulsive interaction, dipole–dipole dispersion, and dipole–quadrupole dispersion, respectively. In more precise formulations, ternary and higher‐order effects must be accounted for so that the potential energy is made up of terms depending on the coordinates of individual atoms, pairs, triplets, etc.

Figure 2 Examples of potential energy models: Morse and Buckingham potentials used in B₂O₃ simulations for B─O and for O─O and B─B, respectively [6].

(7)

The first term u₁ is discarded in standard simulations because it accounts for external fields (i.e. wall, electrical field, gravity, magnetic field, and centrifugal force). The second term u₂ is the most important since it represents the relevant pair potentials. When determined empirically, it actually includes three‐body and many‐body effects, which is why models relying on simple pair potential model reproduce reasonably well liquid or glass structures, and why it is better in this case to denote it by the term “effective” pair potential.

As illustrated by Eq. (7), empirical potentials have a great flexibility since specific terms may be added if needed. When electrostatic interactions are important, a Coulomb charge–charge interaction may, for instance, be included in the form:

(8)

where z_i, z_j, and ε₀ are the charge on atom i and j and the permittivity of free space. Because the Coulombic series converges very slowly, the Ewald, particle‐mesh, or multi‐pole techniques are used in periodic systems. Likewise, more complex models implement three‐ and four‐body terms to reproduce bond‐bending and torsional forces, respectively. Alternatively, polarizable or shell models are employed to consider ions as nonrigid entities and, thus, to account for the effects of the deformation of electron clouds with suitable additional parameters.

From a practical standpoint, the parameters of equations such as (4–7) can be estimated in two different ways depending on the nature of the data to which they are fitted [4]. In the most rigorous way, one relies on energy profiles determined in first‐principles calculations or quantum‐mechanical simulations of appropriate reference systems (cf. Chapter 2.7). Alternatively, potential energy parameters are fitted through MD or lattice dynamics calculations to some selected physical properties. Structural and elastic data are generally chosen because they are most directly related to interatomic potentials. Thermal properties may also be used, but they are sensitive to second‐order effects such as anharmonicity and are in turn generally predicted less accurately.