Complex explicitly correlated one-center all-particle Gaussian functions (CECGs) are tested as basis functions for molecular non-Born-Oppenheimer (non-BO) calculations. The tests concern the complete spectrum of the 23 bound pure vibrational states of the HD+ ion. The tests show that CECGs are very effective in representing both the nuclear-nuclear correlation and the vibrational oscillation of the wave functions in terms of the internuclear distance. This finding is important because it paves the way for high-accuracy non-BO calculations of bound vibrational states of molecules with more than two nuclei that have never been performed before. (C) 2016 Elsevier B.V. All rights reserved.