Secondorder RydbergKleinDunham curves
Abstract
The exact analytical formulas for radial action of the Kratzer and Davidson rotating vibrators and the WKB approximation for the radial action of Dunham's onedimensional rotating vibrator have indicated that the form of the action is I _{r} = h[v+ {1}/{2}+∊{J(J+1)}] The correctness of this form was verified through use in the RydbergKlein method which was analytically applied to energy eigenvalue formulas of both the Kratzer and Davidson rotating vibrators. Exact Kratzer and Davidson potentials were extracted. Dunham's treatment of the onedimensional oscillator has been shown to produce the same phase integrals through terms of order ( {h̵}/{(2m) ^{{1}/{2}}}) ^{3} when applied to the radial part of the Schroedinger wave equation. Dunham's analysis is therefore applicable to diatomic molecules through terms of order {h̵}/{(2m) ^{{1}/{2}}}) ^{3}. Furthermore, comparison of Dunham's WKB solution of the Kratzer and Davidson potentials with their quantum mechanically calculated energy eigenvalue formulas indicated a worst case energy difference of about 0.003 cm ^{1} for hydrogen. Thus, Dunham's WKB method is highly accurate. The RydbergKlein and the Dunham potentials were aligned through secondorder terms by using Dunham's method to approximate the classical action. The result was a simple correction to the Klein g function, defined in the text, which came about when the independent variable of the RydbergKlein equation was transformed from action to energy. The simple secondorder RydbergKleinDunham (RKD) equations were evaluated for the ground state of hydrogen using JacobiGauss quadrature. Results are compared with those of Davies and Vanderslice who use the semiclassical radial action, I _{r} = h(v + {1}/{2}) , instead of the classical radial action, I_{r} = ∮ p_{r}dr. Secondorder RKD corrections to the turning points are a factor of 2 to 4 smaller than those of Davies and Vanderslice and move the potential curve to the left whereas the DaviesVanderslice corrections open the curve up. Finally, the value of Y_{00} is stable for the RKD corrections but varied up to 1.2 cm ^{1} for the DaviesVanderslice corrections as the number of parameters increased from 6 to 9 in an intermediate leastsquares fit of the derivative of the effective potential energy to a polynomial in powers of its square root.
 Publication:

Journal of Molecular Spectroscopy
 Pub Date:
 May 1974
 DOI:
 10.1016/00222852(74)900587
 Bibcode:
 1974JMoSp..51..306M