<![CDATA[This study explored the energy states of a nonrelativistic system by applying the Schrödinger equation to a particular potential. Specifically, energy levels were examined using a modified Pöschl–Teller potential. This study analyzed the impact of varying quantum numbers on energy levels and investigated the vibrational mean energy and specific heat capacity of diatomic molecules. The focus was on nitric oxide, nickel monocarbide, and scandium nitride molecules to assess their behavior at different temperatures. When the screening parameter is present, the potential becomes weaker and less confining, leading to a lower energy eigenvalue compared to when there is no screening and the potential is deeper or more confining. A screened potential results in a particle having a reduced kinetic energy because it is less confined and less tightly bound to the well. An increase in quantum number leads to a decrease in the energy level of the system. The vibrational mean energy increased at high temperatures and then steadily decreased. The vibrational specific heat capacity decreased to a negative value as the temperature decreased, indicating that the system converted kinetic energy into potential energy within the bonds. These findings offer insights into molecular stability and thermodynamic behavior, which could be beneficial for predicting and managing molecules in high-temperature environments.]]>
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