Orbital optimization in electronic wave functions; equations for quadratic and cubic convergence of general multiconfiguration wave functions

We derive variational equations for optimization of the orbitals of arbitrary multiconfiguration wave functions. Expressing the transformation matrix connecting the set of orthonormal trial vectors and the set of final optimal orbitals as an exponential matrix of independent rotation angles allows a simple derivation of the coupled variational equations to arbitrary order. We include the explicit results through third order which are sufficient for cubic convergence in the iterative solution of the optimum orbitals. The equations were programmed (including all terms), and applications to Hartree-Fock and generalized valence bond wave functions of the carbon atom, ${\mathrm{FeO}}_2$, and ${\mathrm{NiCH}}_2$ are reported.