We have developed accurate Gaussian basis functions obtained with the polynomial generator coordinate Hartree-Fock (p-GCHF) method for H, Zn, and Ga-Kr atoms. These basis sets have been applied in the calculation of nonrelativistic energies for neutral atoms, monovalent cations, monovalent anions, ionization potential (IP), and electron affinity (EA), with the objective of proving the quality of the basis set generated by the p-GCHF method. The total energies calculated for neutral atoms and monovalent cations and respective IP were minimally affected by the addition of polarization functions and their precision was comparable to the values reported in the literature. The relative errors were lower than 6.0 \texttimes\ 10$^{-5}$% and 7.0 \texttimes\ 10$^{-5}$% for neutral atoms and monovalent cations, respectively. The IP results were strictly equal to numerical Hartree-Fock (NHF) calculations and comparable to some experimental values. For monovalent anions, the nonrelativistic total energies were better than the Slater-type functions results and the relative errors were lower than 0.05% when compared to NHF. The EA results were the same as those obtained with NHF calculations reported in the literature for heavier elements. For IP and EA, our results followed the same periodic tendency when compared with experimental data.