Physics:Dunham expansion

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

$E(v,J,\mathrm{\Omega })=\sum _{k,l}{Y}_{k,l}(v+1/2{)}^k[J(J+1)-{\mathrm{\Omega }}^2{]}^l,$

where $v$ and $J$ are the vibrational and rotational quantum numbers, and $\mathrm{\Omega }$ 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. 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

	$Y_{0,1}=B_e$	$Y_{0,2}=-D_e$	$Y_{0,3}=H_e$	$Y_{0,4}=L_e$
$Y_{1,0}={\omega }_e$	$Y_{1,1}=-{\alpha }_e$	$Y_{1,2}=-{\beta }_e$
$Y_{2,0}=-{\omega }_e{x}_e$	$Y_{2,1}={\gamma }_e$
$Y_{3,0}={\omega }_e{y}_e$
$Y_{4,0}={\omega }_e{z}_e$

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

See also

• Rotational-vibrational spectroscopy