Dunham expansion
# Dunham expansion

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

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

where $v$ and $J$ are the vibrational and rotational quantum numbers, and $\Omega$ 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

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

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

