A systematic way to understand the intricacies of quantum mechanical computations done by a software package known as "Gaussian" is undertaken via an undergraduate research project. These computations involve the evaluation of key parameters in a fitting procedure to express a Slater-type orbital (STO) function in terms of the linear combination of Gaussian-type orbital (GTO) functions. A procedure for the optimization process based on the Newton-Raphson method is developed and is applied to STO-2G and STO-3G basis sets. Satisfactory results obtained by this procedure are used to illustrate the importance of ab initio computations for inclusion in the chemistry or physics undergraduate curriculum. Programming languages such as Python and Maple were employed to obtain the results.