Dunham expansion

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

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

where ${\displaystyle v}$ and ${\displaystyle J}$ are the vibrational and rotational quantum numbers, and ${\displaystyle \Omega }$ is the projection of ${\displaystyle J}$ along the internuclear axis in the body-fixed frame. The constant coefficients ${\displaystyle Y_{k,l}}$ are called Dunham parameters with ${\displaystyle Y_{0,0}}$ 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

	${\displaystyle Y_{0,1}=B_{e}}$	${\displaystyle Y_{0,2}=-D_{e}}$	${\displaystyle Y_{0,3}=H_{e}}$	${\displaystyle Y_{0,4}=L_{e}}$
${\displaystyle Y_{1,0}=\omega _{e}}$	${\displaystyle Y_{1,1}=-\alpha _{e}}$	${\displaystyle Y_{1,2}=-\beta _{e}}$
${\displaystyle Y_{2,0}=-\omega _{e}x_{e}}$	${\displaystyle Y_{2,1}=\gamma _{e}}$
${\displaystyle Y_{3,0}=\omega _{e}y_{e}}$
${\displaystyle Y_{4,0}=\omega _{e}z_{e}}$

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

See also

• Rotational-vibrational spectroscopy