Studies of molecular dynamics and molecular spectroscopy frequently require some form of analytical potential energy surface (total potential energy expressed as function of the configuration of nuclei), and sometimes a dipole moment surface as well. In work in the Bowman group at Emory University we have used computational invariant theory and the MAGMA computer algebra system as an aid to develop representations for the potential energy and dipole moment surfaces that are fully invariant under permutations of like nuclei. We express the potential energy in terms of internuclear distances using basis functions that are manifestly invariant, with coefficients fitted to the results of ab initio calculations. The resulting full-dimensional surface is then used for quasiclassical trajectory calculations, for diffusion Monte Carlo or path integral calculations, or for quantum mechanical calculations of a rovibrational spectrum. A dipole moment is represented with use of effective charges at positions of the nuclei, and they must transform as a covariant, rather than as an invariant, under permutations of like nuclei. The talk will describe the mathematical procedure and some recent applications: potential energy surfaces for acetaldehyde (CH₃CHO, with Benjamin Shepler), malonaldehyde (CHOHCHCHO, with Yimin Wang), and the water trimer.

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