We present a diatomic potential which closely resembles the standard Morse function but incorporates additional flexibility for fitting experimental vibrational energy-gap data. This flexibility is accommodated by introducing a continuously variable radially dependent change in the exponent of the Morse function, which in practice is adequately realized via a relatively small number of constant parameters. As an illustration, the method is applied to calculate the quantum vibrational levels of the X1Sigma+g ground electronic state of the N2 molecule. Copyright 1998 Academic Press.