Dunham expansion molecules

Quite often, rotational-vibrational spectra of molecules are analyzed by means of empirical formulas. A convenient formula for diatomic molecules is the Dunham expansion (Dunham, 1932 Ogilvie and Tipping, 1983)... 

Calculations of vibrational spectra of bent triatomic molecules with second order Hamiltonians produce results with accuracies of the order of 1-5 cm-1. An example is shown in Table 4.9. These results should again be compared with those of a Dunham expansion with cubic terms [Eq. (0.1)]. An example of such an expansion for the bent S02 molecule is given in Table 0.1. Note that because the Hamiltonian (4.96) has fewer parameters, it establishes definite numerical relations between the many Dunham coefficients similar to the so-called x — K relations (Mills and Robiette, 1985). For example, to the lowest order in l/N one has for the symmetric XY2 case the energies E(vu v2, V3) given by... 

Even if one restricts one s attention to vibrations and rotations of molecules, there are a variety of Lie algebras one can use. In some applications, the algebras associated with the harmonic oscillator are used. We mention these briefly in Chapter 1. We prefer, however, even in zeroth order to use algebras associated with anharmonic oscillators. Since an understanding of the algebraic methods requires a comparison with more traditional methods, we present in several parts of the book a direct comparison with both the Dunham expansion and the solution of the Schrodinger equation. 

The analysis of spectroscopic data for bound states of diatomic molecules gives accurate potential curves if one follows the semi-classical Rydberg-Klein-Rees method. For a review of this see Ref. 126). It is sufficient to note that this gives the two values of r as a function of potential energy by considering the dependence of the total spectroscopic energy on the vibrational and rotational quantum numbers n and J. A somewhat simpler procedure, and the only one plicable to polyatomic molecule, is to use the Dunham expansion of the potential 127). 

In the energy range 0-16,000 cm-1, the vibrational Hamiltonian of this molecule can be modeled by a Dunham expansion without anharmonic resonances of the classical form [112]... 

Watson s treatment applies to vibration-rotation effects in diatomic molecules in singlet electronic states it has not yet been extended to include spin-dependent phenomena. It is based on the Dunham expansion [23] which we met earlier in the previous chapter,... 

At higher levels of excitation anharmonicity has to be included to obtain accurate energy levels. Perturbation theory has been used to derive the following expression, often called a Dunham expansion (Hirst, 1985), for polyatomic anharmonic vibrational energy levels, which is similar to the Morse energy level expression Eq. (2.59), for a diatomic molecule ... 

In order to understand these features, we will analyze the Arnold web of acetylene using the effective Hamiltonian that is constructed by the Dunham expansion [22]. In our study, our main focus is to obtain a universal understanding for the mechanism of IVR based on the general properties of dynamical systems, rather than to study detailed aspects of specific molecules. 

As the molecule is vibrationally excited, couplings among these harmonic modes become important. These couplings are taken into account by the Dunham expansion. The Dunham expansion for the vibration-rotation energy of a linear polyatomic molecule above the zero-point level is given by... 

The calculated harmonic frequencies (cy ) for CH, HF, N2, CO, F2, 2, and CIF are shown in Table 18 as a function of the correlation consistent basis sets (cc-pVnZ and aug-cc-pVnZ). The corresponding basis set errors (cc-pV Z sets) are shown in Figure 10 for CH, HF, N2, and CO. These molecules are representative of a broad range of bond types in strongly bound molecules. In each case, spectroscopic constants were obtained from polynomial fits to 7-9 total energies using the familiar Dunham 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... 

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) ... 

Users should note that higher order terms in the above energy expressions are required for very precise calculations constants for many of these terms can be found in the references. Also, if the ground electronic state is not Z, additional terms are needed to account for the interaction between electronic and pure rotational angular momentum. For some molecules in the table the data have been analyzed in terms of the Dunham series expansion ... 

The calculated and experimental equilibrium distance, dissociation energy, and harmonic frequency for the ground state of the O2 molecule are listed in Table 5. The calculations predict an R which is too long by 0.03 A, which as we shall see below carries directly over to the HO2 molecule, a which is too low by 7 kcal/mol, and a harmonic frequency which is too small by 230 cm (if cubic and quartic terms are included by a Dunham analysis in the energy expansion this error is reduced to 50 cm"" ). 

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