:Dunham expansion

In quantum chemistry, the Dunham expansion is an expression for the rotational-vibrational energy levels of a diatomic molecule:

{{cite journal| last=Dunham|first=J. L.|title=The Energy Levels of a Rotating Vibrator|journal=Phys. Rev. |year=1932|volume=41|issue=6|pages=721–731|doi=10.1103/PhysRev.41.721|bibcode=1932PhRv...41..721D}}

:$$

E(v,J,\Omega) = \sum_{k,l} Y_{k,l} (v+1/2)^k [J(J+1) - \Omega^2]^l,

where $v$ and $J$ are the vibrational and rotational quantum numbers, and $\backslash Omega$ is the projection of $J$ along the internuclear axis in the body-fixed frame.

The constant coefficients $Y\_\{k,l\}$ are called Dunham parameters with $Y\_\{0,0\}$ representing the electronic energy. The expression derives from a semiclassical treatment of a perturbational approach to deriving the energy levels.{{cite journal|last=Inostroza|first=N. |author2=J.R. Letelier |author3=M.L. Senent|title=On the numerical determination of Dunham's coefficients: An application to X1 R + HCl isotopomers|journal=Journal of Molecular Structure: THEOCHEM|year=2010|volume=947|pages=40–44|doi=10.1016/j.theochem.2010.01.037}} The Dunham parameters are typically calculated by a least-squares fitting procedure of energy levels with the quantum numbers.