The high precision spectroscopy of light atomic and molecular systems opens up the possibility for a more in-depth study of fundamental physical constants and laws of nature. This is achieved by juxtaposition of the data from modern experiments with accurate theoretical calculations. In this work, we show our results of precise computations of three-particle systems' electronic and nonrelativistic energy spectra. We are particularly interested in the cases of H$_2^+$ and $\bar{p}$He$^+$ which are presented in detail. In our approach, the electronic wavefunctions are computed by two-dimensional finite difference method in prolate spheroidal coordinates. Uniformly and exponentially spaced grids with varying number of nodes were used in the Schr\"{o}dinger equation's discretization. Accurate molecular quadrupole moments for the studied systems are calculated with the obtained wavefunctions.