A linear third-order differential equation for the diatomic molecular potential-energy function, W n ( R), is formally derived through the differentiation of the quantum mechanical virial theorem with respect to the internuclear distance, R. The general solution of the equation is derived and studied in detail for obtaining an adequate potential-energy function. Since the solution includes an unknown quantity, a reasonable assumption concerning such a quantity is made that leads to a useful, approximate potential-energy function. The approximate function thus obtained is applied to the descriptions of molecular properties such as higher-order potential constants. It is also shown that the above function is closely related to the model potential used in the effective nuclear charge model. Lastly, electronic kinetic ( T) and total potential energy ( V) representative second-order differential equations are derived on the basis of the virial theorem and solved. Their general solutions are subject to investigations of the properties of the general solution for the third-order differential equation in question.