In this paper we present new formulas related to the parametrization of molecular orbitals by exponential transformations. We show that it is not necessary to use series expansions of the exponentials and that the first derivative of the energy with respect to independent parameters can be calculated at arbitrary positions in an efficient manner. These formulas can be implemented such that computational costs are of order N(2)M and storage requirements of order NM, where N is the number of occupied orbitals and M the size of the basis set. We discuss possible applications to Hartree-Fock calculations and ab initio molecular dynamics.