Dunham expansion

 ( Spectroscopy ) 

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

$$

E(v,J,\Omega) = \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, and $\backslash Omega$ 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.

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Relation to conventional band spectrum constants

	$Y\_\{0,1\}\; =\; B\_e$	$Y\_\{0,2\}\; =\; -D\_e$	$Y\_\{0,3\}\; =\; H\_e$	$Y\_\{0,4\}\; =\; L\_e$
$Y\_\{1,0\}\; =\; \backslash omega\_e$	$Y\_\{1,1\}\; =\; -\backslash alpha\_e$	$Y\_\{1,2\}\; =\; -\backslash beta\_e$
$Y\_\{2,0\}\; =\; -\backslash omega\_ex\_e$	$Y\_\{2,1\}\; =\; \backslash gamma\_e$
$Y\_\{3,0\}\; =\; \backslash omega\_ey\_e$
$Y\_\{4,0\}\; =\; \backslash omega\_ez\_e$

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

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See also

• Rotational-vibrational spectroscopy

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