Force in SCF theories. Second derivative of potential energy

The analytic second derivative of a potential energy hypersurface is studied with the use of the Hellmann–Feynman theorem at the stage of the first derivative. As shown previously, the use of the {r, r′} basis in the SCF calculation guarantees that the Hellmann–Feynman theorem is essentially satisfied. (r′ is a derivative of the AO r = χr, r′ = ∂χr/∂xr.) In comparison with the straightforward second derivative of the SCF energy, the number of terms and the labors in calculations are much reduced in the present method. Furthermore, the added {r′} basis improves the quality of the calculated force constant, and an intuitive physical picture is associated with the calculated second derivative, as the intuitive Hellmann–Feynman force picture is associated with the first derivative. The applications are given for the force constants of the molecules N2, CO, and H2O. Some interesting features of the electron density reorganizations during vibrations are reported.