The anharmonic vibrational spectrum of UF6 is computed in full dimensionality directly from ab initio data, i.e., bypassing the construction of a potential energy surface (PES). The vibrational Schrodinger equation is solved by fitting parameters of an adaptable basis using a modified version of the rectangular collocation algorithm of Manzhos and Carrington (J. Chem. Phys. 2013, 139, 051101). The basis functions are products of parametrized Hermite polynomials that impose approximate nodal structure. The Schrodinger equation is solved in normal coordinates. The results show that anharmonicity and coupling do noticeably affect the vibrational transitions, shifting them by several cm(-1). Although UF6 has 15 coordinates, we compute hundreds of levels with fewer than 1000 basis functions and about 50 000 ab initio points. It is the efficiency of the basis that makes it possible to forego a PES.