The methodology for a vibrational analysis within the nuclear-electronic orbital (NEO) framework is presented. In the NEO approach, specified nuclei are treated quantum mechanically on the same level as the electrons, and mixed nuclear-electronic wave functions are calculated variationally with molecular orbital methods. Both electronic and nuclear molecular orbitals are expressed as linear combinations of Gaussian basis functions. The NEO potential energy surface depends on only the classical nuclei, and each point on this surface is optimized variationally with respect to all molecular orbitals as well as the centers of the nuclear basis functions. The NEO vibrational analysis involves the calculation, projection, and diagonalization of a numerical Hessian to obtain the harmonic vibrational frequencies corresponding to the classical nuclei. This analysis allows the characterization of stationary points on the NEO potential energy surface. It also enables the calculation of zero point energy corrections and thermodynamic properties such as enthalpy, entropy, and free energy for chemical reactions on the NEO potential energy surface. Illustrative applications of this vibrational analysis to a series of molecules and to a nucleophilic substitution reaction are presented. (C) 2003 American Institute of Physics.