Diatomic potential series expansions

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 the present section we discuss the power-series expansions of diatomic potentials, and among these we consider in detail the expansions... 

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) ... 

This choice of p provides the leading term in the T-expansion to reproduce both k2 and k exactly. In particular, we may choose the first correction parameter a] to be zero. As a result we have the correct expression for p in a case when the real diatomic potential is just a Lennard-Jones type. Therefore, the T-expansion for that p is essentially an expansion in a series of Lennard-Jones potentials. 

This pattern of transitions in the rovibrational spectrum of the diatomic is roughly similar to the appearance of a rovibronic spectrum. Equations 9.27 and 9.28 need to be adjusted to include the equilibrium term energy for the difference in energy between the potential minima for the initial and final electronic states and to separate the power series expansions in the vibrational and rotational energies. Parameters such as B (o, and depend on the potential energy curve of each individual electronic state. No simple equation relates these potential energy curves for different electronic states, and therefore distinct values for each of these parameters are given to each electronic state ... 

It has been found useful to represent the interaction potential for a dimer of homonuclear diatomic molecules [4,5,46,58] as a spherical harmonic expansion, separating radial and angular dependencies. The radial coefficients include different types of contributions to the interaction potential (electrostatic, dispersion, repulsion due to overlap, induction, spin-spin coupling). For the three dimers of atmospheric relevance, we provided compact expansions, where the angular dependence is represented by spherical harmonics and truncating the series to a small number of physically motivated terms. The number of terms in the series are six for the N2-O2 systems, corresponding to the number of configurations of the dimer (for N2-N2 and O2-O2 this number of terms is reduced to five and four, respectively). 

When one uses a low-order perturbation expansion, resting on a very crude Hq definition, are the lowest valence potential curves equally correct The large series of papers by Fr and coworkeis devoted to diatoms ° ° ° actually seem to support a positive answer. 

An electronic state of a diatomic molecule is characterized by the potential mergy 17 as a function of internuclear distance r. This is approximated by a power series of an adequate expansion parameter which is regularly chosen as ... 

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