Advanced · 25 min

A potential energy surface: minima, saddle, reaction path

Explore a computed-style surface with two wells joined by a saddle point, follow the minimum-energy path and connect it to the activation energy of transition-state theory.

Goal

Identify reactant, product and transition-state regions on a 2D potential energy surface and trace the minimum-energy path between them.

Apparatus and reagents

A model two-coordinate potential energy surface (isomerisation A → B) rendered as a rotatable 3D landscape.

Procedure

Rotate the surface until you can see both wells (species A and B) and the ridge between them.
Locate the saddle point at the centre: a maximum along the reaction path but a minimum across it.
Follow the dashed path from the A well over the saddle to the B well and estimate the barrier height relative to each well.
Tilt the view sideways to turn the landscape into the familiar 1D reaction profile: A – barrier – B.

What to observe

The two wells sit at the same depth: the reaction is thermoneutral, but the path still must climb the saddle.
The saddle is flat along the transverse direction and steep along the path — the signature of a first-order saddle (one imaginary frequency).
Deviating from the dashed path costs energy on both sides — the minimum-energy path is the route a slow reaction actually prefers.

Explanation

A potential energy surface gives the electronic energy as a function of the nuclear coordinates (Born–Oppenheimer). Minima are stable molecules; first-order saddle points are transition states. Transition-state theory converts the barrier height into a rate constant, $k = \frac{k_B T}{h}\,e^{-\Delta G^{\ddagger}/RT}$ , so each extra kJ·mol⁻¹ of barrier slows the reaction exponentially. Free-energy methods recompute this surface including entropy at finite temperature.

Chemists behind it

John Pople