Physics:Dunham expansion
In quantum chemistry, the Dunham expansion is an expression for the rotational-vibrational energy levels of a diatomic molecule: [1]
- [math]\displaystyle{ E(v,J,\Omega) = \sum_{k,l} Y_{k,l} (v+1/2)^k [J(J+1) - \Omega^2]^l, }[/math]
where [math]\displaystyle{ v }[/math] and [math]\displaystyle{ J }[/math] are the vibrational and rotational quantum numbers, and [math]\displaystyle{ \Omega }[/math] is the projection of [math]\displaystyle{ J }[/math] along the internuclear axis in the body-fixed frame. The constant coefficients [math]\displaystyle{ Y_{k,l} }[/math] are called Dunham parameters with [math]\displaystyle{ Y_{0,0} }[/math] 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
| [math]\displaystyle{ Y_{0,1} = B_e }[/math] | [math]\displaystyle{ Y_{0,2} = -D_e }[/math] | [math]\displaystyle{ Y_{0,3} = H_e }[/math] | [math]\displaystyle{ Y_{0,4} = L_e }[/math] | |
| [math]\displaystyle{ Y_{1,0} = \omega_e }[/math] | [math]\displaystyle{ Y_{1,1} = -\alpha_e }[/math] | [math]\displaystyle{ Y_{1,2} = -\beta_e }[/math] | ||
| [math]\displaystyle{ Y_{2,0} = -\omega_ex_e }[/math] | [math]\displaystyle{ Y_{2,1} = \gamma_e }[/math] | |||
| [math]\displaystyle{ Y_{3,0} = \omega_ey_e }[/math] | ||||
| [math]\displaystyle{ Y_{4,0} = \omega_ez_e }[/math] |
This table adapts the sign conventions from the book of Huber and Herzberg. [3]
See also
- Rotational-vibrational spectroscopy
References
- ↑ Dunham, J. L. (1932). "The Energy Levels of a Rotating Vibrator". Phys. Rev. 41 (6): 721–731. doi:10.1103/PhysRev.41.721. Bibcode: 1932PhRv...41..721D.
- ↑ 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.
- ↑ 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.
 |  |
