RKR potentials

Tellinghuisen J A 1974 A fast quadrature method for computing diatomic RKR potential energy curves Comput. Phys. Commun. 6 221-8... 

An extensive compilation of RKR potential curves for NO+ has recently been reported by Albritton, et. al.87 Their studies include all of the experimentally known electronic states of NO+ below 24 eV. Potential energy curves and spectroscopic constants are tabulated and estimates of accuracy in the location of the excited states are reported. 

The Ukt are isotopically invariant parameters. Our discussion of the RKR potential in section 6.13.3 showed that the potential Vn(R) can be determined from a knowledge of Gv and Bv. Since the parameters Uu with l > 2 can be calculated from Vn(R), it follows that they are exactly determined by the values of Uu with t = 0 and 1. The simplest case of Uq2 is a familiar one. The relationship in question,... 

For the purpose of outlining the results which we have obtained In studying supersonic sodium expansions, the data presented in Figure 2 will be useful. Here we consider calculated RKR potential curves for the X Zg" , A Zy" ", and 8 11 states of Na2 and a rough schematic of two cuts through the Naa potential surface based, in part, on the recent quantum chemical calculations of Martin and Davidson (40). For Na2, the location of the... 

The recently available spectroscopic data and the RKR potentials of the alkali hydrides allow us to determine the "experimental" values of the parameters relevant to the transition probability of the charge transfer processes. In the Landau-Zener model these parameters are the energy gap between the and X S adiabatic potentials at the avoided crossing distance and the coupling matrix elements. In this paper the coupling matrix elements are evaluated in a two-state ionic-covalent interaction model. The systema-tic trends found in the alkali hydride series for their X e potentials are presented. This leads to a simple model for the ionic potentials. 

In this study we first examine the systematics that exist in the RKR potentials of the state to establish that the potential curves are strongly ionic for R R. (R. is the distance of the pseudocrossing point). Next we evaluate an essentially experimental value of the parameters relevant to the cross section for the charge transfer process in the Landau-Zener approximation, We also construct a model ionic potential which can be used to describe the charge transfer process in the ionic region. 

Our intention is to use as much of the experimental RKR potentials as possible to evaluate the relevant quantities Tic Hii(Rj-). 

New spectroscopic measurements of the alkali hydrides have provided information related to the dynamical charge transfer process for these systems. We have examined the RKR potentials derived from these spectra. Several striking regularities for the X l potentials are presented along with an interpretation based on a simple model of ionic potentials for internuclear distances shorter than the crossing distance R. . 

In the avoided crossing region the experimental RKR potential of the state and the "essentially experimental" potential of the state are used to determine the crossing distance R(- and the coupling matrix element T -. in the two state approximation. These quantities are relevant to the evaluation of the total charge transfer cross section at high energy (e.g. in the Landau-Zener model). 

In this study we used the scaled theoretical potential curves of Olson and Liu (21), Stevens, Karo and Hiskes ( ), and Laskowski and Stall cop (2T) to make a short extrapolation of the X E" RKR potentials into the avoided crossing region. Experimental measurements to obtain strictly experimental RKR potentials for this region are in progress. New optical measurements on the dipole moments (, ), the transition moments (38> nd radiative lifetimes (, ) of the alkali hydrides are also becoming available. This type of information will soon provide additional details about the ionic-covalent interactions in these molecules. 

At wavelengths longer than 100 nm, near the LIF transmission limit, CO appears to be completely stable with respect to dissociation in a primary step. The reactivity of optically excited CO in this region is thus due to the reactions of its excited states. Tilford and Simmons (24) have presented a set of RKR potential energy curves for CO calculated by R.H. Howard and this is given in Fig. 5 for the excited states. 

The RKR potential may be tested against the input G(v) and B(v) values by exact solution of the nuclear Schrodinger equation [see Wicke and Harris, 1976, review and compare various procedures, e.g., Numerov-Cooley numerical integration (Cooley, 1961), finite difference boundary value matrix diagonaliza-tion (Shore, 1973), and the discrete variable representation (DVR) (Harris, et al., 1965)]. G(v) + y00 typically deviates from EVjj=o by < 1 cm-1 except near dissociation. Bv may be computed from Xv,J=o(R) by... 

Figures (a) Morse potential-energy functions for ICl (6) RKR potential-energy curves for BrF. oken lines are schematic, not calculated [(a) After ref. 70 and b) after ref. 103]...

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. 

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. 

De = 4.48 eV ( 36000 cm ) was estimated [14] by approximating the RKR potential curve for NF(X 2 ) calculated from the spectroscopic constants [3] by a five-parameter Hulburt-Hirschfelder potential function [15]. 

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. 

Fig. 1. RKR potential energy curves for the lowest lying valence and ion-pair...

Once RKR potential energy curves have been generated for the lower and... 

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. 

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