Adiabatic vs non-adiabatic potential surfaces

We begin by postulating the existence of two harmonic potential energy surfaces. The potential energy function for surface 1 is U₁ = (k/2)q². Surface one is centered at the origin. The potential energy for surface 2 is U₂ = (k/2)(q-q₀)². Potential surface 2 is centered at q₀. If these two potential energy surfaces are coupled by J then the eigenvalues for the adiabatic surfaces are given by the solution to

The determinant can be expanded

so that

and

or

To illustrate the effect of increasing the coupling J on the shape of the potential energy surfaces we plot two surfaces below for various values of J.

An expansion of the curve crossing region is shown below.

The initial and final states represented here are the reactant and product in a self-exchange reaction. For example, this might electron transfer between two iron atoms in solution. In such a case the coupling is almost surely weak J << 0.01 and the curves are nearly identical to the diabatic representation. Thus, in such a case we speak of non-adiabatic electron transfer. The symmetrical potential energy surfaces shown can also represent the self-exchange process in a mixed-valence complex. In such a case the coupling can be strong (e.g. Cu dimer, class III mixed valent complex), intermediate (Ru-4,4'-bipyridine, class II mixed valent) or weak. The difference between the various classes can be expressed in terms of the relative magnitude of the splitting between the potential energy surfaces, 2J and the transition energy from one surface to another, E. If J > E/4, then we are approximately in the strong coupling of class III limit. This is shown by the blue curve in the figures above.