Numerical integration techniques are used to investigate the quantum dynamical behavior of the Morse, Fues, and truncated squarehyphen;well oscillators. These procedures, unlike the methods involving expansion in terms of a finite basis set of functions, readily permit the study of dissociating systems. Stability problems arise when the timehyphen;dependent Schrouml;dinger equation is solved by difference equation methods; an extension of a difference equation scheme reported by Mazur and Rubin is found to yield a conditionally stable system. A method based on the approximation of the time evolution operator by a finite symmetric matrix is found to yield a simple, rapid, and quite accurate algorithm for integrating the timehyphen;dependent Schrouml;dinger equation; this method is also applicable to other partial differential equations of physical interest. The agreement between the results of numerical integration methods and those of expansion in a basis set of functions is found to be excellent for systems in which bound states only are of importance. Numerical integration results for dissociating systems are presented and discussed. Application to the vibrations of linear harmonic triatomic molecules is discussed.
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