1 The potential energy surface

All computational modelling of materials rests on a common foundation: for any arrangement of atoms, there is an associated energy. This mapping — configuration to energy — defines the potential energy surface (PES).

1.1 Configuration and energy

Consider a crystal with two types of atoms in a square unit cell. Call this structure \(S_1\), with atomic positions \(\mathbf{r}_1\) and energy \(E_1\).

Swap some atoms: sites that held A atoms now hold B atoms, and vice versa. The atoms sit at different positions, \(\mathbf{r}_2\), and this structure \(S_2\) has a different energy \(E_2\). We can consider different changes — shearing the unit cell into a parallelogram, for example, gives structure \(S_3\), with another different set of positions \(\mathbf{r}_3\) and another different energy \(E_3\). This behaviour generalises: a structure is defined by a set of atomic positions, and each set of positions maps to a corresponding energy.

The shear transformation reveals something more. The transition from \(S_1\) to \(S_3\) is continuous: gradually shearing the cell, every point along the way has positions slightly different from the last, with a correspondingly different energy. The mapping from positions to energy is not just defined at discrete structures — it is a continuous surface.

This is the potential energy surface: \(E(\mathbf{r})\), where \(\mathbf{r}\) represents the positions of all atoms. For \(N\) atoms in three dimensions, \(\mathbf{r}\) has \(3N\) coordinates. Any point in this \(3N\)-dimensional space is a configuration — a complete specification of all atomic positions. And every configuration has an energy \(E(\mathbf{r})\).

The potential energy surface is a mathematical object. A direct physical pathway between two points is not required for both to exist on the surface. The transition from \(S_1\) to \(S_2\) has no direct pathway — exchanging atom types would require atoms to pass through each other. Yet both are well-defined points with well-defined energies. In real materials, such changes happen by indirect routes: vacancy diffusion, interstitial migration, or other mechanisms that avoid atomic overlap. These indirect routes are themselves paths on the surface.

1.2 Features of the PES

The features of the potential energy surface have direct physical significance.

Minima are stable structures. A minimum is a configuration where small displacements in any direction increase the energy — forces on all atoms are zero, and the structure is locally stable. These are the crystal structures determined from diffraction: the system sits at a minimum (or near one, at finite temperature).

Relative energies of minima determine thermodynamics. If \(S_1\) has lower energy than \(S_2\), then \(S_1\) is thermodynamically preferred — at least at zero temperature. At finite temperature, entropy also matters, and the relevant quantity becomes the free energy rather than the energy.

Barriers control kinetics. Along the pathway from \(S_1\) to \(S_2\), there is typically a maximum — an energy barrier. The height of this barrier determines how fast the transformation happens. A high barrier means the process is slow; a low barrier means it is fast. This applies to phase transitions, ionic diffusion, and any process where the system moves from one configuration to another. Diffraction reveals which structure is present; the barrier height explains why a system may be trapped in a metastable minimum.

The shape of the PES determines what is physically possible, not just what is energetically preferred. Consider a direct atom swap — two atoms exchanging positions. A continuous pathway for this exists on the PES, but it requires atoms to pass through each other, where the energy diverges. The barrier is infinite; the pathway is kinetically forbidden. In real materials, atoms change places by other mechanisms — via vacancies, interstitials, or correlated motion — pathways through configuration space that avoid catastrophic overlap.

Curvature at minima determines vibrational properties. The energy change for small displacements around a minimum depends on the shape of the potential well. A steep, narrow well produces strong restoring forces and high vibrational frequencies. A shallow, broad well produces softer vibrations at lower frequencies. This curvature is what phonon calculations determine — and what inelastic neutron scattering measures. The shape of the PES around a minimum appears directly in INS spectra.

1.3 Two questions

How is \(E(r)\) calculated? Given an arrangement of atoms, how is its energy determined? Different methods — classical potentials, density functional theory, machine-learned potentials — give different answers with different tradeoffs of accuracy and computational cost.

What is done with \(E(r)\)? Given the ability to calculate energies, what questions can be answered? Finding the lowest-energy structure is one application. Characterising vibrations around that structure is another. Simulating atomic motion at finite temperature, or sampling over different configurations — these require different approaches.

These are independent choices. Molecular dynamics can use a classical potential, DFT, or a machine-learned potential — the choice of how to calculate \(E(r)\) is independent of the choice to do MD. A common misconception is that “molecular dynamics” implies classical potentials, or that “DFT” implies static calculations. Neither is true. Any method for calculating \(E(r)\) can be combined with any method for using it.

The remainder of Part I covers three approaches to calculating E(r): classical potentials, density functional theory, and machine-learned interatomic potentials. Parts II and III turn to what is done with that ability — geometry optimisation, phonon calculations, molecular dynamics, and Monte Carlo sampling.