First and second energy derivative analyses for open-shell self-consistent field wavefunctions

A study of first and second derivatives of the orbital, electronic, nuclear and total energies for the self-consistent field (SCF) wavefunction has been applied to general open-shell SCF systems. The diagonal elements of the Lagrangian matrix for the general open-shell SCF wavefunction are adapted as the ‘orbital’ energies. The first and second derivatives of the orbital energies in terms of the normal coordinates are determined via the finite difference method, while those of the electronic, nuclear and total energies are obtained by analytical techniques. Using three low lying states of the CH₂ and H₂CO molecules as examples, it is demonstrated that the derivatives of the SCF energetic quantities with respect to the normal coordinates provide useful chemical information concerning the respective molecular structures and reactivities. The conventional concept of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) has been extended to the molecular vibrational motion, and the terminology of vibrationally active MOs (va-MOs), va-FIOMO and va- LUMO has been introduced for each normal coordinate. The energy derivative analysis method may be used as a powerful semi-quantitative model in understanding and interpreting various chemical phenomena.