Second-order Rydberg-Klein-Dunham curves

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 one-dimensional 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 Rydberg-Klein 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 one-dimensional 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 Rydberg-Klein and the Dunham potentials were aligned through second-order 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 Rydberg-Klein equation was transformed from action to energy. The simple second-order Rydberg-Klein-Dunham (RKD) equations were evaluated for the ground state of hydrogen using Jacobi-Gauss 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_rdr. Second-order 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 Davies-Vanderslice corrections open the curve up. Finally, the value of Y₀₀ is stable for the RKD corrections but varied up to 1.2 cm ^-1 for the Davies-Vanderslice corrections as the number of parameters increased from 6 to 9 in an intermediate least-squares fit of the derivative of the effective potential energy to a polynomial in powers of its square root.