In this work, vibrations of a diatomic molecule are considered within the framework of quantum mechanics as an anharmonic oscillator with small values of the anharmonicity coefficient. Based on well-known models of quantum harmonic and anharmonic oscillators, a new model with a variable value of the molecule force constant was created here. The force constant depends on the vibrational quantum number n and the anharmonicity coefficient . Mathematical analysis of this model confirmed its physical validity. As an example, the vibrational energy of an oxygen molecule was calculated within the framework of the model presented in the work.