Analytic second derivatives in restricted Hartree–Fock theory. A method for high-spin open-shell molecular wave functions

Theoretical methods have been developed for the analytic determination of second energy derivatives (i.e., force constants) from restricted Hartree–Fock wave functions for certain types of open-shell systems. Specifically treated are systems for which all electrons outside closed shells have their spins parallel. Although the open-shell formalism is somewhat complicated, its application once implemented, is not greatly more arduous than the closed-shell theory presented in 1979 by Pople and co-workers. Like previous procedures for the evaluation of the second derivatives of electrons repulsion integrals, the present method exploits the Rys polynomial concept. Beyond this general framework, however, significant departures appear, and these differences are described. Preliminary application of the new method has been to the two lowest triplet states of the formaldehyde molecule, for which both equilibrium geometry and harmonic vibrational frequencies have been evaluated.