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]

See also

• Rotational-vibrational spectroscopy

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; 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.