Rotational RKR methods

Fig.2 Test of the rotational RKR method. Solid and open dots refer to inner and outer turning points for J=0-30 in steps of 5. Residuals are riven for (a) Lennard-Jones (b) Morse (c) no iterative refinement. (Taken irom ref.5 with permission).

A test of this method for 1-D radial potentials is shown in Fig. 2. Here we have taken an assumed intermolecular potential (shown in solid line) for a weakly bound diatomic molecule (which for realism had well depths and radial minima similar to a psuedo diatomic treatment of Ar-HF), and solved exactly for the J = 0, 5, 10, 15, 20, 25, and 30 eigenvalues for v=0 by exact close coupling quantum calculations. These eigenvalues are then taken as input into the rotational RKR method, which attempts to regenerate the initial potential. 

To capture the flavor of the method, consider the limit of a M-HX complex in which the intermolecular potential is mostly isotropic, but with sufficient anisotropy so the nearly free internal HX rotor motion is oriented in the body fixed frame. The ground state of such a complex would look like a j=0 HX rotor (i.e. an "s" orbital) bound to M, whereas the three lowest excited HX bending states would approximate the three j l rotor wave functions (i.e. three "p" orbitals) oriented with respect to the end-over-end plane of rotation of the M-HX centers of mass, one in a S and two in a II configuration. The j=0 HX rotor state probes predominantly the isotropic part of the intermolecular radial potential, whereas j=l HX rotor states (the S and either one of the II configurations) begin to sample in addition the lowest order anisotropic parts of the potential. The radial dependence of the intermolecular potential for each of these three states can be determined from rotational RKR method. In principle, these curves contain sufficient information to determine the three lowest... 

For a diatomic molecule the Rydberg-Klein-Rees (RKR) method may be used to determine the potential energy curve V r) from the experimental vibrational/rotational energy levels (Hirst, 1985). This method is based on the Einstein-Brillouin-Keller... 

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,... 

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. 

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. 

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