# Dunham expansion

In quantum chemistry, the Dunham expansion is an expression for the rotational-vibrational energy levels of a diatomic molecule: [1]

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

where v and J are the vibrational and rotational quantum numbers. 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.[2] 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

 $Y_{0,1} = B_e$ $Y_{0,2} = -D_e$ $Y_{0,3} = H_e$ $Y_{0,4} = L_e$ $Y_{1,0} = \omega_e$ $Y_{1,1} = -\alpha_e$ $Y_{1,2} = -\beta_e$ $Y_{2,0} = -\omega_ex_e$ $Y_{2,1} = \gamma_e$ $Y_{3,0} = \omega_ey_e$ $Y_{4,0} = \omega_ez_e$

This table adapts the sign conventions from the book of Huber and Herzberg. [3]

## References

1. ^ Dunham, J. L. (1932). "The Energy Levels of a Rotating Vibrator". Phys.Rev. 41: 721–731. doi:10.1103/PhysRev.41.721.
2. ^ Inostroza, N.; J.R. Letelier and M.L. Senent (2010). "On the numerical determination of Dunham’s coefﬁcients: An application to X1 R + HCl isotopomers". Journal of Molecular Structure: THEOCHEM 947: 40–44. doi:10.1016/j.theochem.2010.01.037.
3. ^ Huber, K.P.; Herzberg, G. (1979). Molecular Spectra and Molecular Structure IV. Constants of diatomic molecules. New York: van Nostrand. ISBN 0-442-23394-9.