Molecular symmetry. III. Second derivatives of electronic energy with respect to nuclear coordinates

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Abstract
Symmetry methods employed in the ab initio polyatomic program HONDO are extended to the analytic computation of the energy Hessian matrix. A ’’skeleton’’ Hessian matrix is calculated from the unique blocks of electron repulsion integrals. The true Hessian matrix is generated by projecting the symmetric component out of the skeleton Hessian. The analysis is valid for many wave functions, including closed- or open-shell restricted and unrestricted Hartree–Fock wave functions, multiconfiguration Hartree–Fock wave functions, and configuration interaction wave functions. We also extend the use of translational invariance previously used for energy gradient calculations. To illustrate the method, we compare the computer time required for the two-electron contribution to the Hessian matrix of eclipsed ethane, using Pople’s 6-31G basis set and D3h symmetry and various subgroups of D3h. Computational times are roughly inversely proportional to the order of the point group.