Potential energy surfaces

In the statistical mechanics chapter we saw that the potential energy function determines the equilibrium properties of a system via the Boltzmann distribution . In this chapter, we take a closer look at the potential energy surface (PES) itself — the landscape that sampling algorithms like nested sampling must navigate.

Understanding the topology of the PES — its minima, barriers, and symmetries — is essential for designing efficient sampling strategies and interpreting simulation results.

What is a potential energy surface?

A potential energy surface maps every configuration of a system to a potential energy value . For a system with degrees of freedom, the PES is a scalar function on an -dimensional configuration space:

Key features of a PES include:

• Minima — stable configurations where the system tends to reside at low temperature
• Saddle points — transition states connecting different minima
• Barriers — energy differences that separate minima, determining transition rates
• Basins of attraction — regions of configuration space that drain toward a particular minimum

In practice, the PES can have an astronomically large number of minima and saddle points, especially for many-particle systems. This is what makes sampling so challenging.

A gallery of 2D energy landscapes

To build intuition, let's explore a collection of 2D energy functions. Each surface has a different character — from simple unimodal to highly multimodal — and poses different challenges for sampling algorithms.

Some observations:

• Hosaki has a local and a global minimum — a common situation where sampling can get trapped in the wrong basin
• Himmelblau has four equivalent minima, testing the algorithm's ability to explore symmetric landscapes
• Rastrigin and Ackley are highly multimodal with many local minima — a worst-case scenario for local search methods
• Rosenbrock has a narrow curved valley — easy to find the minimum, hard to sample the distribution
• Double well has two equivalent minima separated by a barrier — a minimal model for phase transitions

Non-periodic systems: molecules

For an isolated molecule with atoms, the configuration is specified by Cartesian coordinates (or internal degrees of freedom after removing translations and rotations). The PES describes how the total potential energy — arising from bond stretching, angle bending, torsional rotation, and non-bonded interactions — depends on the atomic arrangement.

For a diatomic molecule, the PES reduces to a simple 1D curve where is the interatomic distance. For a triatomic molecule, we already have a 3D surface (three internal coordinates). The dimensionality grows rapidly with system size, making visualization impossible for realistic molecules — but the concepts of minima, barriers, and basins remain.

The Ramachandran plot

A classic example from structural biology is the Ramachandran plot. A protein backbone is a repeating chain of N–C–C units. The conformation of each residue is determined by two dihedral (torsional) angles: (rotation around the N–C bond) and (rotation around the C–C bond). Not all combinations are physically accessible — steric clashes between backbone atoms and side chains create forbidden regions, leaving only a few allowed basins corresponding to known secondary structures.

The energy surface is computed from steric repulsion between non-bonded atoms in the peptide fragment: when atoms overlap (high terms), the energy rises sharply. The blue regions — where backbone atoms point away from each other — correspond to sterically allowed conformations like the -helix () and extended -sheet regions. The red/yellow regions are sterically forbidden: atoms are forced too close together. Drag the sliders to rotate around the N–C () and C–C () bonds and watch the 3D structure change.

Periodic systems: crystals and the toy model

In condensed matter, systems are typically described using periodic boundary conditions (PBC). Instead of tracking isolated molecules, we define a repeating unit cell and tile space periodically. The configuration is then specified by:

1. The cell parameters — shape and size of the unit cell (related to volume)
2. The atomic positions within the cell — usually given as fractional coordinates

This introduces a new degree of freedom compared to isolated systems: the volume (or more generally, the cell shape). The PES now depends on both atomic positions and cell parameters, and the volume becomes a thermodynamic variable that can fluctuate in the NPT ensemble.

A toy model for periodic systems

To illustrate these ideas concretely, we use a toy model: two particles in a one-dimensional periodic box, interacting via a pair potential with a repulsive core and an attractive well. The system has just two degrees of freedom:

• — the interparticle distance
• — the lattice parameter (box length), playing the role of volume

Due to periodic boundary conditions, particles that move beyond the box are wrapped back. This means that configurations with are unphysical — they would correspond to a distance larger than the box itself. The physical region is therefore the triangle .

The interactive visualization above shows the PES alongside a schematic of the simulation box. The red dot on the PES marks the current configuration. Try varying:

• Lattice parameter — changes the box size (volume). At small , the particles are forced close together, hitting the repulsive wall.
• Distance — moves the particles within the box. The attractive well creates a preferred separation.
• Pressure — adds a term, tilting the surface to penalize large volumes. This turns the energy surface into an enthalpy surface.

The effect of pressure

At zero pressure, the PES has an extended minimum valley at the preferred interparticle distance, and the system is free to expand to any box size. Applying pressure adds a linear term to the energy, creating a well-defined enthalpy minimum at a specific volume. The competition between the attractive well (favoring a preferred distance) and the pressure term (favoring small volumes) determines the equilibrium structure.

This is exactly the physics that nested sampling explores in real materials: by systematically lowering the energy threshold, NS maps out how the system transitions between expanded and compact phases as a function of pressure and temperature — as we saw in the statistical mechanics chapter.

The challenge of high-dimensional PES

The 2D examples above are useful for building intuition, but real systems have to degrees of freedom. In high dimensions, the PES has qualitatively different properties:

These challenges motivate the development of advanced sampling methods. Nested sampling addresses them by systematically compressing the accessible volume, avoiding the need to cross barriers directly. The MCMC methods used within NS — Metropolis-Hastings, Hamiltonian Monte Carlo, and Galilean Monte Carlo — each have different strengths for navigating complex landscapes.