where τ is the collection of Morse variables, c i are linear coefficients, p i are PIPs, and n p is the total number of polynomials (and linear coefficients c i ) for a given maximum polynomial order. The Morse variables are transformed internuclear distances r ij between atoms i and j, i.e., y ij = exp(-r ij /a). The range (hyper)parameter a is typically between 2 and 3 bohr; however, it can be optimized for special applications. A more revealing expression for a PIPs potential for a for a tetra-atomic molecule is given by