RKR potentials and vibrational

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). 

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 Dunham coefficients Yy are related to the spectroscopical parameters as follows 7io = cOe to the fundamental vibrational frequency, Y20 = cOeXe to the anharmonicity constant, Y02 = D to the centrifugal distortion constant, Yn = oie to the vibrational-rotational interaction constant, and Ym = / to the rotational constant. These coefficients can be expressed in terms of different derivatives of U R) at the equilibrium point, r=Re. The derivatives can be either calculated analytically or by using numerical differentiation applied to the PEC points. The numerical differentiation of the total energy of the system, Ecasccsd, point by point is the simplest way to obtain the parameters. In our works we have used the standard five-point numerical differentiation formula. In the comparison of the calculated values with the experimental results we utilize the experimental PECs obtained with the Rydberg-Klein-Rees (RKR) approach [58-60] and with the inverted perturbation approach (IPA) [61,62]. The IPA is method originally intended to improve the RKR potentials. 

Fig. 39. Calculated spin-orbit splitting in the state of O2 as a function of the vibrational quantum numbers by employing the matrix element displayed in Fig. 38. Curve A rrfers to the calculation that employs the ab initio potential curve for vibrational averaging B employs the experimental RKR curve for vibrational averaging, and C shows the experimental points from various measurements.

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. 

Semiclassical techinques have an established place in the analaysis of electronic spectra, to the extent that one often speaks of the "exact RKR" potential curves derived from experimental vibrational and rotational term values 2. Similarly, when applicable, the Le Roy Bernstein - scheme for extrapolation to dissociation limits is far superior to the traditional Birge-Sponer method. 

Direct RKR inversion of vibrationally and rotationally resolved spectroscopic data for diatomics is now a fairly routine procedure. In normal RKR applications, however, the spectral data are exploited in a relatively limited fashion. One simply uses B(v) and E(v), the rotational constant and term value dependence on vibrational quantum v, respectively, to infer the inner and outer classical turning points at each v from a semiclassical analysis. In high resolution spectroscopy of van der Waals complexes, however, there is often far more rotational than vibrational data available. Consequently extensive information exists on very high order centrifugal effects on the radial coordinate, sometimes up to, and by virtue of centrifugal barriers, beyond the dissociation limit The hope is that a simple extension of RKR ideas might be able to extract a 1-D potential directly from rotational data alone. 

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. 

Experimentally deteraiined RKR potentials for the Xi( Il3/2) and XiCllm) states of the halogen monoxides FO, CIO, BrO and 10 are compared to available ab initio potentials. The results suggest that fully relativistic ab initio calculations have the capability to reproduce the experimental bond lengths, harmonic vibrational frequencies and fine structure intervals of the XO series with reasonable accuracy. The quest for spectroscopically accurate XO potentials will provide an excellent benchmark for future theoretical methods. 

We have vibrationally averaged the CAS /daug-cc-pVQZ dipole and quadmpole polarizability tensor radial functions (equation (14)) with two different sets of vibrational wavefunctions j(i )). One was obtained by solving the one-dimensional Schrodinger equation for nuclear motion (equation (16)) with the CAS /daug-cc-pVQZ PEC and the other with an experimental RKR curve [70]. Both potentials provide identical vibrational... 

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... 

---