Molecular symmetry. II. Gradient of electronic energy with respect to nuclear coordinates

Symmetry methods employed in the HONDO ab initio polyatomic SCF program are extended to the analytic computation of energy gradients. Validity of the Hellmann–Feynman theorem is not assumed, i.e., all two-electron contributions to the gradient are included explicitly. The method is geared to the efficient computation of entire blocks of two-electron integrals. Just one of a set of symmetrically related blocks must be computed. The gradient contribution from each unique block is multiplied by q4, the number of equivalent blocks, and added into a ’’skeleton gradient vector,’’ all other blocks are simply omitted. After processing molecular integrals, the true gradient vector is generated by projecting the symmetric component out of the skeleton vector. The analysis is based on Eqs. (26) and (33) which are valid for many variational wavefunctions including restricted closed shell and unrestricted open shell self-consistent field functions. We also extend the use of translational symmetry proposed previously by Morokuma et al. To illustrate the method, the gradient of the restricted SCF energy is computed for eclipsed ethane using a Pople-type 63lG** basis and D3h symmetry. The same calculation is repeated using various subgroups of D3h. Computation times for SCF and for the gradient are each roughly inversely proportional to the order of the group, and for a given symmetry, the gradient computation takes about two and a half times as long as SCF.