Chemistry Labs

Theoretical and computational chemistry

Potential-energy surfaces and free-energy calculations

Connect molecular geometry to potential-energy landscapes, then use statistical mechanics and enhanced sampling to estimate free-energy differences and barriers.

1 Lower secondary2 Upper secondary3 Undergraduate4 Advanced5 Research

IntuitionLandscapes and accessible states

A potential-energy surface assigns an energy to each molecular geometry. Its slopes indicate forces and its valleys suggest stable structures, but temperature changes which regions are statistically populated: free energy, not potential energy alone, governs equilibrium probabilities.

Compare a molecular energy landscape with a projected free-energy profile. A low-energy path can still be kinetically inaccessible, and a free-energy profile depends on the chosen collective variables and adequate sampling.

SchoolReading an energy diagram

Definition: Potential-energy surface

For fixed electronic state and nuclear coordinates R, V(R) is the potential energy. A stationary point has zero gradient; a local minimum has positive curvature in all unconstrained directions, while a first-order saddle has one downhill direction.

F(R)=−∇RV(R)F(R) = −∇_R V(R)

Example: A barrier on a reaction coordinate

Along a simplified one-dimensional coordinate, a reactant minimum is 12 kJ mol⁻¹ below a transition-state maximum. What is the potential-energy barrier from reactant to transition state?

Solution

The barrier is V‡ − Vreactant = 12 kJ mol⁻¹. It is a potential-energy difference, not automatically the activation free energy at finite temperature.

UndergraduateFrom microscopic states to free energy

Definition: Potential of mean force

For a collective variable ξ, the free-energy profile W(ξ) is defined by the equilibrium probability density p(ξ): W(ξ)=−kBT ln p(ξ)+constant. It integrates out all coordinates not explicitly retained in ξ.

ΔG(A→B)=−kBTln(PB/PA)ΔG(A→B) = −k_B T ln(P_B/P_A)

The equilibrium populations of two basins determine their free-energy difference, not just the energy at each minimum. Entropic contributions reflect how much configuration space each basin occupies.

MethodStrategyCaution
Umbrella samplingBias overlapping windowsOverlap and reweighting matter
MetadynamicsAccumulate bias along collective variablesPoor variables hide slow modes
Free-energy perturbationReweight samples between statesNeeds phase-space overlap

AdvancedBarriers, coordinates, and uncertainty

A reaction coordinate is a reduced description; a poor choice can hide orthogonal slow rearrangements and produce hysteresis. Free-energy methods must demonstrate overlap, reversibility or convergence across windows, and uncertainty estimates; a visually smooth profile alone is insufficient evidence.

ResearchFrontier: rare events and reliable free energies

References