Stable methods for achieving MCSCF convergence

A composite strategy is presented for systematically achieving convergence of complete active space MCSCF calculations. The method efficiently combines first- and second-order steps. Three aspects are discussed. (1) An approximate matrix of second derivatives with respect to orbital rotations is projected onto a Krylov subspace and diagonalized. If there are small or negative eigenvalues, then a crude search is performed along the damped, steepest descent path (the curved direction of water flowing downhill). One or more of these "opening game" moves overcomes problems associated with poor starting orbitals. (2) For the final approach to convergence, a preconditioned conjugate gradient (cg) search is performed in which the approximate second derivative matrix plays the role of the preconditioning transformation. The cg algorithm corrects for the neglect of CI mixing and any other approximations introduced into the second derivative matrix. (3) Localized and associated oscillator orbitals of Foster and Boys provide reasonable core and active space starting orbitals generated from a preliminary closed-shell SCF calculation. Explicit formulas for all derivatives are given. Tables of computational effort for a wide variety of complete active space calculations are presented.