A new modified Morse potential is introduced to describe the vibrational motion of diatomic molecules. The vibrational energy eigenvalues for the ground state of the H2 and N2 molecules are obtained by numerically solving the time-independent Schrödinger equation using a Python program and the Matrix Numerov method for the case of the Morse potential, Hulbert-Hirschfelder potential, and the modified Morse potential functions. The comparison of the vibrational energy eigenvalues and the anharmonicity constant obtained for the modified Morse potential function reveals better agreement with the experimental data than the other potential functions. Also, the proposed modified Morse potential function fits the RKR potential energy curve more closely than the Morse potential function and Hulbert-Hirschfelder potential function, especially in the region where the potential extends to near the dissociation limit.