RKR potentials and vibrational levels

The vibrational wave functions can be calculated from an RKR representation of the potential well, which is based on the rotational constants and vibrational levels of the molecule concerned. This leaves the problem of determining the signs of the matrix elements. From equation (8.339) we may write... 

The Xi( Il3y2) and X2CTI1/2) potentials of FO were derived from a combined fit to the existing microwave, LMR and high-resolution FTIR spectroscopic data (2). The data included transitions for vibrational levels up to v = 7 and direct measurements of Xii Hsn) - X2( Ilm) fine structure transitions. The RKR potentials and associate vibrational intervals are shown in Figure 1. Figure 2 compares the X 11 RKR potential with available ab initio potential energy points (5-7). 

Most of the previous discussion has focused on programs that fit rotational lines. Often, in the case of homogeneous perturbations, there are so many mutually interacting vibrational levels that a fit to individual rotational levels (rather than to Gv and Bv) could be contemplated only after developing a model which accounts for the nonrotating molecule vibrational structure. In such a model, the concept of potential-energy curves plays a crucial role. These curves may be either Morse (Lefebvre-Brion, 1969) or RKR curves (Stahel, et al., 1983). One assumes a perturbation interaction of the form... 

If V and the form of

repulsive curve Vi can be determined by an RKR-like method that computes individual turning points (Child, 1973, 1974). This method is useful for obtaining an initial approximation for the repulsive potential. However, if only a few experimental r -values are known, it is difficult to identify unambiguously the oscillatory frequency of T versus v. For example, in Fig. 7.20 the number of vibrational levels sampled is insufficient to determine the actual shape of r . 

Fig. 1. Potential energy curves of PH in the ground and lowest excited states, a) RKR function for the lowest vibrational levels in the ground state X [1]. b) Klein-Dunham-Morse functions for X Z and A ITj [4], Morse functions for a A and b Z [5]. The repulsive curve for is drawn quantitatively [4].

Finally, some papers which carry the theory of vibrational averaging to higher levels of approximation should be mentioned. Bartell, in his 1955 paper " on the Morse oscillator probability distribution, considers the effect of an increase of temperature on r. Bartell s theory is extended by Bartell and Kuchitsu and by Kuchitsu these papers show in particular that the effective mean amplitude obtained by refinement on the molecular intensity curve is not quite equal to the harmonic mean amplitude calculated from the harmonic force field. Bonham and co-workers have calculated the effect of temperature on both a Morse oscillator and an oscillator in an RKR potential energy curve. In their final paper an informative series of diagrams shows how the quantum-mechanical average passes into the classical average at high temperature. 

The classical turning points, and r ax. of the vibrational levels have been calculated for the ground state X Z (v = 0 to 6 molecular constants from [6]) assuming a Rydberg-Klein-Rees-Vanderslice potential [7], for the X Z and A Ili states (v and v" = 0 to 8 molecular constants from [8]) assuming Morse and modified Rydberg-Klein-Rees (RKR) potentials [9], and for the b Z and d Z" states (v and v" = 0 to 3 molecular constants from [10]) assuming modified RKR and simplified potentials [11] and [12], respectively. 

EBK) semiclassical quantization condition given by Eq. (2.72). In contrast to the RKR method for diatomics, a direct method has not been developed for determining potential energy surfaces from experimental anharmonic vibrational/rotational energy levels of polyatomic molecules. Methods which have been used are based on an analytic representation of the potential energy surface (Bowman and Gazdy, 1991). At low levels of excitation the surface may be represented as a sum of quadratic, cubic, and quartic normal mode coordinates (or internal coordinate) terms, that is,... 

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