The analytical expression for the matrix elements between Morse's wavefunctions is shown and a modified temperature-dependent Morse potential was developed and validated. The developed formulae indicate that anharmonicity is responsible for a non-null displacement with respect to the equilibrium position at 0 K, that we call Zero Point Position. With their advantage of being computationally inexpensive and fast, the present model can be used to provide highly accurate theoretical estimation in the internuclear distance at vibrational ground state as well as their temperature dependence for not only diatomic but also polyatomic molecules. The present theoretical model was implemented to the development of a simple atomic-level model for the estimation of temperature-dependent thermal expansion coefficients of bulk metals, and was proved to be an efficient and rapid way for the evaluation of material mechanic properties. These models are analytical and are successfully tested on a series of metals.