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.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+