Generalizations of Newton–Raphson and multiplicity independent Newton–Raphson approaches in multiconfigurational Hartree–Fock theory

The application of Newton–Raphson (second order) approaches in multiconfigurational Hartree–Fock (MCSCF) can significantly improve convergence over other MCSCF procedures. When the Hessian (second derivative) matrix has small eigenvalues, convergence of second order procedures may be slowed significantly both far from and closer to convergence. We derive techniques related to the multiplicity independent Newton–Raphson approach which are less affected by these convergence problems. We also formulate and derive generalized Newton–Raphson approaches which show quadratic, cubic, quartic, etc. convergence. We prove that certain fixed Hessian-type Newton–Raphson iterations will show quadratic, cubic, quartic, etc. convergence and demonstrate how these approaches may be advantageously used only for a few iterations. The approaches we describe in both the energy and generalized Brillouin’s theorem formulation have about the same complexity in structure and in actual implementation on a computer as the Newton–Raphson approach. Finally, we examine and discuss conditions for an MCSCF state to be a good approximation to the electronic state.