# Solving the Redundant Coordinate Problem

At any given geometric configuration of an *N*-atom molecule there are 3*N* – 6 linearly independent local coordinates. To describe the molecule at *any** *point in configuration space, however, we must use a redundant coordinate description, describing the molecule in terms of some set of primitive redundant coordinates. Such descriptions are used in the optimisation routines available in electronic structure codes and in descriptions of global molecular potential energy surfaces. For *normal** *molecules, for example methane, CH_{4}, the ground vibrational state *can** *be described using 3*N* – 6 local coordinates from this set– for example, the CH_{4} normal coordinates. This choice, however, is particularly problematic for the protonated water dimer, H_{5}O_{2}^{+}: even large sets of primitive redundant coordinates suffer from linear. Here we introduce a general solution to the redundant coordinate problem and demonstrate that it can be used to develop accurate interpolated molecular potential energy surfaces. We assess our potential energy surface using quantum diffusion Monte Carlo (QDMC) simulations of the H_{5}O_{2}^{+} ground state, determining its quality of in terms of (i) a test set of configurations (ii) the number of QDMC walkers “killed” as a result of artefacts in the potential, (iii) the H_{5}O_{2}^{+} zero-point energy and (iv) approximate H_{5}O_{2}^{+} anharmonic vibrational frequencies. In particular, our solution to the redundant coordinate problem enables us to retain inverse atom-atom distances as primitive redundant coordinates, with a number of inherent advantages. Moreover, our strategy is generally applicable to *any** *case where redundant coordinates are used to describe a system. Time permitting, application to hydrogen absorption materials will also be discussed.