A Method for Obtaining the Vibrational Wave Functions Appropriate to the True Potential Energy Curves of Diatomic Molecules

A method has been suggested for obtaining the vibrational wave functions appropriate to the true U ( r ) curves of diatomic molecules by the application of the Wentzel-Kramers-Brillouin (WKB) method. It consists in dividing the true U ( r ) curve into a number of segments and determining a quadratic expression in r , representing the U ( r ) curve, for each segment. These quadratic expressions are then used for evaluating the phase integrals. For checking the accuracy of the wave functions thus obtained, the Morse wave functions of a number of vibrational levels of various molecular electronic states, computed by the WKB method, have been compared with their exact Morse wave functions and the agreement has been found to be quite good. The proposed method has been applied to obtain the true wave functions of the fourth and sixth vibrational levels of the A 3 II g state of C 2 and of the zeroth, first and tenth vibrational levels of the A 1 Σ+ state of RbH. The general shape and anharmonicity of these wave functions are observed to be as expected from the nature of their true U ( r ) curves. Further, the deviation of the true wave functions of the fourth and sixth vibrational levels of the A 3 II g state of C 2 from the exact Morse wave functions has been found to be large in comparison with the error in the WKB wave functions.