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Dunham expansion

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

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

where v {\displaystyle v} and J {\displaystyle J} are the vibrational and rotational quantum numbers, and Ω {\displaystyle \Omega } is the projection of J {\displaystyle J} along the internuclear axis in the body-fixed frame. The constant coefficients Y k , l {\displaystyle Y_{k,l}} are called Dunham parameters with Y 0 , 0 {\displaystyle 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 {\displaystyle Y_{0,1}=B_{e}} Y 0 , 2 = − D e {\displaystyle Y_{0,2}=-D_{e}} Y 0 , 3 = H e {\displaystyle Y_{0,3}=H_{e}} Y 0 , 4 = L e {\displaystyle Y_{0,4}=L_{e}}
Y 1 , 0 = ω e {\displaystyle Y_{1,0}=\omega _{e}} Y 1 , 1 = − α e {\displaystyle Y_{1,1}=-\alpha _{e}} Y 1 , 2 = − β e {\displaystyle Y_{1,2}=-\beta _{e}}
Y 2 , 0 = − ω e x e {\displaystyle Y_{2,0}=-\omega _{e}x_{e}} Y 2 , 1 = γ e {\displaystyle Y_{2,1}=\gamma _{e}}
Y 3 , 0 = ω e y e {\displaystyle Y_{3,0}=\omega _{e}y_{e}}
Y 4 , 0 = ω e z e {\displaystyle Y_{4,0}=\omega _{e}z_{e}}

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

See also

References

  1. Dunham, J. L. (1932). "The Energy Levels of a Rotating Vibrator". Phys. Rev. 41 (6): 721–731. Bibcode:1932PhRv...41..721D. 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.


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