Dunham power-series expansion

Diatomic molecules provide a simple introduction to the relation between force constants in the potential energy function, and the observed vibration-rotation spectrum. The essential theory was worked out by Dunham20 as long ago as 1932 however, Dunham used a different notation to that presented here, which is chosen to parallel the notation for polyatomic molecules used in later sections. He also developed the theory to a higher order than that presented here. For a diatomic molecule the energy levels are observed empirically to be well represented by a convergent power-series expansion in the vibrational quantum number v and the rotational quantum number J, the term... 

In 1932, Dunham proposed the following power-series expansion to represent a diatomic potential U(R) about its minimum (Dunham, 1932) ... 

Representing a given diatomic potential as a power-series expansion in terms of the D variable, Eq. (1), Dunham justified via the WKBJ approach (see Section IV) the well-known energy expression for diatomic molecules (Dunham, 1932) ... 

The structure of eq. (2.1a) can be traced back to a power-series expansion of the (bound state) potential energy function U of the intemuclear distance r as first introduced by Dunham [32Dunj ... 

The anharmonicities of the potential contribute by the terms involving the constants x, g, y,. .. as well as the energy shifts AEx = 0(h2),. .. and the frequency shifts Aw, = 0(h2),. These anharmonic constants can be calculated by the Van Vleck contact transformations [20] as well as by a semi-classical method based on an h expansion around the equilibrium point [14], which confirms that the Dunham expansion (2.8) is a series in powers of h. Systematic methods have been developed to carry out the Van Vleck contact transformations, as in the algebraic quantization technique by Ezra and Fried [21]. It should be noted that the constants x and g can also be obtained from the classical-mechanical Birkhoff normal forms [22], The energy shifts AEx,... 

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