Use of the molecular orbital Hessian for self-consistent-field (SCF) wavefunctions

Abstract The molecular orbital (MO) Hessian for the self-consistent-field (SCF) wavefunction is defined as the second derivative of the electronic energy of the system with respect to changes in the MO coefficients. The lowest eigenvalue of the MO Hessian is widely used to determine the stability of SCF wavefunctions. The present study extends the use of the MO Hessian in several directions: prediction of low-lying excited states, qualification of basis sets, identification of MOs involved in chemical reactions, warning of a variational collapse, relative stability of wavefunctions, as well as absolute stability of wavefunctions. An SCF instability index is defined as the number of negative eigenvalues of the MO Hessian, since it is possible that an SCF wavefunction has more than one negative eigenvalue of the corresponding MO Hessian. The MO Hessian is analyzed for the ground and excited states of H 2 O, H 2 CO and NO 2 at equilibrium geometries and/or at transition states for dissociation and for conformational change. Examples are given of stationary values of the electronic energy with respect to orbital rotations representing local minima, maxima and points of inflection. The relevance of the eigenvalues of the MO Hessian to the physical and chemical properties of the system under consideration is discussed.