Non-Crossing Rule for Energy Curves in Diatomic Molecules

Let and be energy curves for two different electronic states of a diatomic molecule, both computed within the Born–Oppenheimer approximation. If the two states belong to different symmetry species, say and , and , or singlet and triplet, there is no restriction on whether the curves can cross. If, however, the two states have the same symmetry, a non-crossing rule applies. Close approach of the two curves results in mutual repulsion, known as an anticrossing. For near degeneracy of and , a perturbation , representing higher-order contributions in the Born–Oppenheimer approximation, becomes significant, giving mixed states that do not cross.
In this Demonstration, the lower energy state, , is drawn in blue. It is assumed to be a bonding state, with dissociation energy and equilibrium internuclear distance , which can both be varied with sliders. The upper energy state, , drawn in red, is assumed to be a repulsive state. The mixing parameter can also be varied. In certain cases, the upper state can develop a minimum as a result of the interaction. The dashed curves in the graphic pertain when .


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Snapshot 1: no restriction on crossing for states of different symmetry
Snapshot 2: geometry of an anticrossing
Snapshot 3: stronger interaction between states
Reference: L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-Relativistic Theory, 2nd ed., Reading, MA: Addison–Wesley, 1965 p. 279.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.