Permutationally invariant potential energy surfaces in high dimensionality

We review recent progress in developing potential energy and dipole moment surfaces for polyatomic systems with up to 10 atoms. The emphasis is on global linear least squares fitting of tens of thousands of scattered ab initio energies using a special, compact fitting basis of permutationally invariant polynomials in Morse-type variables of all the internuclear distances. The computational mathematics underlying this approach is reviewed first, followed by a review of the practical approaches used to obtain the data for the fits. A straightforward symmetrization approach is also given, mainly for pedagogical purposes. The methods are illustrated for potential energy surfaces for CH5+, (H2O)(2) and CH3CHO. The relationship of this approach to other approaches is also briefly reviewed.