Dunham expression

Thus, we have two expressions for E(v, J), Eqs. (II.7) and (2), as functions of v and J. The first, the Dunham expression, is valid for v near the bottom of the potential well. The second, the Le Roy-Bemstein-Stwalley expression, is applicable in the dissociation limit. Naturally, it would be expected to derive a formula for E(u, J) in the complete space, whose limiting cases coincide with Eqs. (II.7) and (2). Evidently, this formula, if it exists in general, may have a more complex expression. One of the possible ways to construct this formula is based on the application of the Padd approximant technique. This way was exploited by Beckel and co-workers (Hashemi-Attar et al., 1979 Hashemi-Attar and Beckel, 1979), who suggested... 

Extending previous accurate laser studies of the IR spectra of metal hydrides in gas phase, including NaH, CsH, BaH, SrH, RbH, GaH, InH, TIH CdH, ZnH, KH, MgH, CaH and LiH and also of diatomic hydride radicals, for instance SnH, NiH, FeH, CoH and GeH, the infrared spectra of two isotopic forms of silver hydride [ Ag (51.35%) and Ag (48.65%)] in its ground electronic state ( S) have been recorded . Accurately known absorption hnes of nitrogen dioxide sulphur dioxide and formaldehyde served for wavenumber calibration 21 transitions in the bands from v=l -0uptov = 3 2 have been measured for AgH and AgH respectively, their assignments and equation wavenumbers were presented explicitly and fitted to the Dunham expression (equation 38) for the energy levels of diatomic molecules, and the parameters Yij for two isotopic forms of AgH were determined. ... 

The new data permitted one to obtain a new value of the equihbrium bond length of the AgH, which was found to be Vg = 1.617798 (5) A. The expected Ag/ Ag isotope effect in the effective bond length of AgH isotopomers, caused by the anharmonicity of the Ag—H vibrations, is not discussed by Birk and Jones as was done in the case of C—H/C—D bond distances and vibrations. Subsequently, AgD was obtained from silver vapour with deuterium and 33 transitions have been recorded for AgD and AgD. As before, the rovibrational spectra have been fitted to the Dunham expression in equation 38 and the parameters Yij of two isotopic AgD species were determined, when Tio was found to equal 1250.8912(15) for AgD and 1250.676(30) for AgD. The particular values of Y are greater for the hghter isotope species as required by equation 39, in which p and p are reduced masses of the two molecules. The values of Tio in silver deuterides are smaller by a factor of 1.407 than the Tio values for silver hydrides. The mean amphtudes of vibrations of Ag—D and Ag—H stretches and the isotope effects on the size of isotopic Ag H molecules have not been presented in... 

A, accurate line positions are important and were calculated from the Dunham expression (56). Appendix I contains a nummary of the Einstein coefficients and line positions for the Av=l and Av=2 HF transitions originating from levels up to v =5, J =18. 

For a balanced historical record I should add that the late W. E. Blumberg has been cited to state (W. R. Dunham, personal communication) that One does not need the Aasa factor if one does not make the Aasa mistake, by which Bill meant to say that if one simulates powder spectra with proper energy matrix diagonalization (as he apparently did in the late 1960s in the Bell Telephone Laboratories in Murray Hill, New Jersey), instead of with an analytical expression from perturbation theory, then the correction factor does not apply. What this all means I hope to make clear later in the course of this book. 

Dunham [5] derived these expressions Y ((t)e,Be,a0, necessarily manually, through a JBKW procedure, which he claimed to make more general [4] than what had appeared in previous literature. Dunham reported expressions F containing coefficients aj up to a, and Sandeman [19] and Woolley [20] extended manually these results according to a roughly analogous procedure. Kilpatrick [21] applied perturbation theory in successive orders to derive expressions for 1, and Bouanich [22] applied Rayleigh-Ritz perturbation theory for solution of... 

Apart from the distance variable x that Dunham used in his function V(x) for potential energy, other variables are amenable to production of term coefficients in symbolic form as functions of the corresponding coefficients in a power series of exactly the same form as in formula 16. Through any method to derive algebraic expressions for Dunham coefficients l j, the hamiltonian might have x as its distance variable, but after those expressions are produced they are convertible to contain coefficients of other variables possessing more convenient properties. To replace x, two defined variables are y [38],... 

Dunham s consideration of available spectra of HCl in 1930 [1,2] resulted in production of a radial function for electric dipolar moment that we express in a contemporary form, similar to that in formula 56 but in terms of variable x, defined in formula 15, instead of z, defined in formula 21 ... 

Although authors [74,75] have claimed that a generator-coordinate theory yields an expression for spectral terms exactly of Dunham s form, as in formula 8, in which however term coefficients encompass intrinsically elfects of at... 

Basic to much of the work in which calculated points are fitted to a polynomial is the Dunham analysis. Dunham31-32 showed that if a potential energy curve is expressible as a power series,... 

Commonly used polynomials are the cubic56-58 and, most usually, the quartic.29 30-30 59-76 The quartic has the advantage that it gives analytical expressions for the most commonly recorded constants, toe, x0a>c, Be, o. Probably a high proportion of the work quoting a Dunham analysis uses this function too. 

Relationships connecting ae, //., ye with coe, coexe and Be have been given by Pekeris [67] and Dunham [65], In a very detailed study of vibration-rotation interactions, Dunham [65] has shown that the term values for a vibrating rotor should actually be expressed as a double power series, given by... 

The ultimate objective of the analysis is to express the vibrational and rotational dependence of the various molecular parameters as power series in their derivatives with respect to the intemuclear distance. In chapter 6 we discussed various analytic representations of the potential energy in terms of the intemuclear distance. Perhaps the most general representation is that due to Dunham [101] who assumed a power series for the potential of the form... 

Dunham then showed that for this potential the vibration rotation energies could be expressed by... 

The results given in table 10.3 constitute a good data set with which to test theoretical relationships [86, 87] relating to breakdown of the Born-Oppenheimer approximation. As we have described elsewhere, the vibration-rotation term values may be expressed as power series using the Dunham parameters T ... 

The remaining coefficients listed in table 10.5 arise from an analysis originally given by Herman and Asgharian [102] and Watson [103], Dunham showed that T0i may be expressed in terms of the Born-Oppenheimer equilibrium constant Bf° and an additive correction term AY0i,... 

Ju, Z., R.A. Dunham and Z. Liu. Differential gene expression in the brain of channel catfish (Ictalurus punctatus) in response to cold acclimation. Mol. Genet. Genomics 268 87—95, 2002. 

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