Potential Energy Surfaces

A potential-energy surface (PES) is the mathematical object that connects a structural model to energies, forces, stationary points, vibrations, reaction paths, and classical trajectories. It is a function supplied by a particular calculator under a particular electronic, boundary, and model state, not a property of a molecular formula alone.

For $N$ atoms with Cartesian coordinates collected in $\mathbf R$, and a periodic cell $\mathbf h$ when present, write

$$E = E(\mathbf R,\mathbf h;\mathcal M),$$

where $\mathcal M$ includes the calculator family, parameter or weight set, charge and spin state where the model recognizes them, dispersion and solvation choices, and numerical settings that affect the evaluated energy. Changing $\mathcal M$ changes the surface. A minimum or barrier does not exist independently of that choice.

This chapter establishes the shared language used by the method chapters. It does not replace their algorithmic details.

Coordinates and dimensionality

A nonperiodic structure of $N$ unconstrained atoms has $3N$ Cartesian coordinates. Overall translation and rotation do not change the internal energy of an isolated nonlinear molecule, leaving $3N-6$ internal vibrational degrees of freedom; a linear molecule has $3N-5$. Periodic calculations may also treat the cell vectors as variables. Constraints remove or couple degrees of freedom and thereby define a reduced search space.

A PES is often illustrated in one or two dimensions, but an actual molecular or crystal surface is high-dimensional. A plotted reaction coordinate is a projection through that space. Two structures with the same plotted coordinate can differ in every coordinate that the plot hides. This is why a smooth one-dimensional energy curve cannot by itself prove that a path is continuous or chemically meaningful.

Forces are negative Cartesian derivatives:

$$\mathbf F_i=-\frac{\partial E}{\partial \mathbf R_i}.$$

The gradient $\mathbf g=\nabla_{\mathbf R}E$ and force contain the same local slope with opposite sign. A single-point calculation that returns forces therefore describes both the energy at one geometry and the local direction of steepest decrease in Cartesian space. It does not move the atoms.

For periodic systems, the derivative with respect to strain gives a stress convention. Schematically,

$$\sigma_{\alpha\beta}=\frac{1}{V} \frac{\partial E}{\partial\varepsilon_{\alpha\beta}},$$

although sign and stress/virial conventions must be checked against the calculator contract. Atomic forces can relax positions; stress information is required when an operation changes the cell.

Stationary points

A stationary point satisfies

$$\nabla E(\mathbf R^\ast)=\mathbf 0$$

in the coordinates that are free to vary. This definition is local. It does not say whether the point is a minimum, a transition state, or a higher-order saddle.

Local curvature is described by the Hessian,

$$H_{i\alpha,j\beta}= \frac{\partial^2 E} {\partial R_{i\alpha}\partial R_{j\beta}}.$$

After the appropriate mass weighting and removal of external motions, the Hessian eigenvalues distinguish types of stationary point:

• a local minimum has no physically meaningful negative-curvature mode;
• a first-order saddle has one negative-curvature direction;
• a higher-order saddle has more than one.

In vibrational language, a negative Hessian eigenvalue is reported through an imaginary frequency convention. Numerical noise, incomplete optimization, constraints, finite-difference error, periodic acoustic modes, and an unsuitable calculator can all complicate this classification. The mode displacement must be inspected, not merely counted.

Constrained stationary points

With constraints, the optimizer seeks stationarity only in the allowed subspace. If $\mathbf P$ projects onto free coordinates, the relevant condition is

$$\mathbf P\nabla E(\mathbf R^\ast)=\mathbf 0.$$

The full gradient may remain nonzero. A frozen substrate, fixed bond, or selective-dynamics mask can therefore yield a valid constrained minimum without being an unconstrained minimum of the same PES.

Constraints define the modelled experiment. They are appropriate when representing a deliberate approximation—such as immobilized deep slab layers—but can hide an instability when they remove the direction in which the structure would otherwise relax. Report the constraint policy with the result and interpret convergence in the reduced coordinate space.

Local models and finite steps

Knowing the force at one point does not reveal the entire surface. Optimization algorithms construct a local approximation and take finite steps. Near $\mathbf R_k$, a quadratic model is

$$E(\mathbf R_k+\Delta\mathbf R)\approx E_k+\mathbf g_k^\mathsf T\Delta\mathbf R+ \frac12\Delta\mathbf R^\mathsf T\mathbf H_k\Delta\mathbf R.$$

Newton-like methods use or approximate curvature; first-order methods use gradient history and step control. Far from the local region where the model is useful, a nominal downhill step can overshoot, encounter a different basin, or expose an unphysical geometry to the calculator. Trust radii, line searches, maximum displacements, and optimizer-specific damping control this numerical risk.

A converged optimizer establishes a threshold condition on the chosen surface and coordinate space. It does not establish that the located basin is the intended conformer, crystal phase, adsorption pose, protonation state, or chemical product.

Paths between basins

Suppose two local minima $A$ and $B$ are known. A continuous path $\mathbf R(s)$, $0\le s\le1$, connects them, but infinitely many such paths exist. A minimum-energy path is characterized locally by vanishing physical force perpendicular to its tangent:

$$[\nabla E(\mathbf R(s))]_\perp=\mathbf 0.$$

NEB approximates this object with a discrete band and projected forces. The highest-energy band image is a saddle candidate, not a verified transition state. A dimer or other saddle refinement follows local negative curvature, while a subsequent Hessian or vibrational analysis tests the order and physical character of the stationary point.

The path coordinate used in a plot may be normalized image index, cumulative Cartesian displacement, a mass-weighted distance, or a chemically chosen coordinate. These are different quantities. An energy maximum at the same horizontal position on two differently parameterized plots does not imply that the atomic paths are identical.

Dynamics is not an optimization path

Classical molecular dynamics solves an initial-value problem:

$$m_i\frac{d^2\mathbf R_i}{dt^2}=\mathbf F_i(\mathbf R),$$

modified as required by thermostat, barostat, or constraint algorithms. The trajectory has momenta, a timestep, and an ensemble protocol. It can cross, revisit, or remain trapped within regions of the PES according to the available energy and coupling model.

An optimization trajectory is an algorithmic history toward stationarity; it has no physical time interpretation. An NEB band is a discretized path; it is not a dynamical trajectory or an intrinsic reaction coordinate. A molecular-dynamics trajectory samples a distribution only after equilibration and with sufficient duration. These three sequences may all appear as frames in a viewer, but their scientific meanings are not interchangeable.

Electronic energy, effective energy, and free energy

The scalar returned by a calculator must be named according to its method. A semi-empirical self-consistent calculation may include an electronic free-energy contribution associated with finite electronic temperature or occupation smearing. An MLIP predicts the energy convention represented by its training labels. A dispersion or implicit-solvent term changes the effective surface. None of these values equals a thermodynamic Gibbs free energy for the molecular or condensed-phase process of interest.

Thermal and entropic corrections require additional statistical-mechanical assumptions. A potential of mean force is obtained after averaging over degrees of freedom under a sampling protocol; it is not the same object as the bare potential energy along one optimized coordinate. When an application reports a reaction, binding, adsorption, or voltage result, it must state which energy level enters the cycle and which terms are omitted.

Derived observables

Not every reported property is a derivative of the same scalar PES in the same sense.

The shared runtime can store all these outputs, but shared storage does not make them equivalent observables.

Calculator choice changes the landscape

GFN2-type methods solve an approximate electronic problem and respond to supported charge, spin, dispersion, and solvation settings. Machine-learned potentials evaluate a learned mapping from atomic environments to energy and derivatives. Their surface reflects the reference data and training domain rather than an explicit electronic state available to the user.

A model can fail noisily through nonconvergence or fail silently by returning smooth values outside its domain. Smooth forces, a completed optimization, or an attractive-looking energy curve are numerical observations that do not on their own establish chemical validity. Validation should probe representative structures, forces, relative energies, and, where important, curvature or stress against an appropriate higher-level or experimental reference.

One surface, several questions

Each operation asks for different derivatives, algorithms, and acceptance evidence. The Calculation Foundations and Execution Model explains how those requests are normalized, executed, and stored. The method chapters explain how the corresponding numerical questions are answered.

References

1. D. J. Wales, Energy Landscapes: Applications to Clusters, Biomolecules and Glasses, Cambridge University Press (2003).
2. F. Jensen, Introduction to Computational Chemistry, 3rd ed., Wiley (2017), chapters on potential-energy surfaces, geometry optimization, and vibrational analysis.
3. H. B. Schlegel, “Geometry optimization,” WIREs Computational Molecular Science 1, 790–809 (2011).
4. A. Hjorth Larsen et al., “The atomic simulation environment—a Python library for working with atoms,” Journal of Physics: Condensed Matter 29, 273002 (2017).