The energy obtained for the harmonic oscillator depends on the development ( ecuación 2 ) that is truncated by the second term. Adding the third and fourth derivatives to the expansion gives a better approximation for the energy. The correction term for the vibrational energy equation is: $-h\nu_e x_e(v+\frac{1}{2})^2$, where $\nu_e x_e$ is known as the anharmonicity constant. The corrected vibrational energy is: $ E_{vib}=(v+\frac{1}{2})h\nu_e-h\nu_e x_e(v+\frac{1}{2})^2$
The inclusion of anharmonicity leaves us the internal energy of the molecule as follows:
\begin{equation}\label{13} E_{int}=B_e h J(J+1)+(v+\frac{1}{2})h\nu_e-h\nu_e x_e(v+\frac{1} {2})^2+E_{ele} \end{equation}