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Vibrational fine structure observed in the published He-I UV photoelectron spectra of H[subscript 2] and N[subscript 2] molecules recorded at room temperature and under microwave excitation or at 700 °C have been reexamined and compared with calculations of the Franck-Condon factors (FCFs) using simple harmonic oscillator wave functions and/or those obtained using Morse potentials. In general, FCFs are used to explain vibronic transitions of molecules observed in the UV-vis and photoelectron spectra of diatomic and polyatomic molecules, which can be computed using differing levels of complexity. We advocate constructing the vibrational wave functions numerically, rather than analytically, with the help of a program such as Excel, Origin, or Python. In this way students can also verify the normalization and orthogonality of the wave functions using numerical integration routes provided within these programs. Special attention is drawn to possible deviations of the FCFs of the vibrationally excited molecules from those of the ground states.