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In [[quantum chemistry]], the '''Dunham expansion''' is an expression for the rotational-vibrational [[energy level]]s of a [[diatomic molecule]]:
<ref name="dunham1932">
{{cite journal| last=Dunham|first=J. L.|title=The Energy Levels of a Rotating Vibrator|journal=Phys.Rev.|year=1932|volume=41|pages=721–731|doi=10.1103/PhysRev.41.721}}</ref>
:<math>
E(v,J) = \sum_{k,l} Y_{k,l} (v+1/2)^k [J(J+1)]^l,
</math>
where ''v'' and ''J'' are the [[Rigid_rotor#Quantum_mechanical_linear_rigid_rotor|rotational]] and [[Molecular vibration#Quantum mechanics|vibrational quantum numbers]].
The constant coefficients <math>Y_{k,l}</math> are called '''Dunham parameters''' with <math>Y_{0,0}</math> representing the electronic energy. The expression derives from a [[semiclassical physics|semiclassical]] treatment of a [[perturbation theory|perturbational approach]] to deriving the energy levels.<ref name=Inostroza2010>{{cite journal|last=Inostroza|first=N.|coauthors=J.R. Letelier and 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}}</ref> The Dunham parameters are typically calculated by a least-squares fitting procedure of energy levels with the quantum numbers.

==Relation to conventional band spectrum constants==

{| class="wikitable"
|
| <math>Y_{1,0} = \omega_e</math>
| <math>Y_{2,0} = \omega_ex_e</math>
| <math>Y_{3,0} = \omega_ey_e</math>
| <math>Y_{3,0} = \omega_ez_e</math>
|-
| <math>Y_{0,1} = B_e</math>
| <math>Y_{1,1} = -\alpha_e</math>
| <math>Y_{2,1} = \gamma_e</math>
|-
| <math>Y_{0,2} = D_e</math>
| <math>Y_{1,2} = \beta_e</math>
|-
| <math>Y_{0,3} = F_e</math>
|-
| <math>Y_{0,4} = H_e</math>
|}

==See also==
* [[Rotational-vibrational spectroscopy]]

==References==

{{Reflist}}

[[Category:Spectroscopy]]
{{applied-math-stub}}