A combined first- and second-order optimization method for improving convergence of Hartree-Fock and Kohn-Sham calculations.

In this work, we investigate the optimization of Hartree-Fock (HF) orbitals with our recently proposed combined first- and second-order (SO-SCI) method, which was originally developed for multi-configuration self-consistent field (MCSCF) and complete active space SCF (CASSCF) calculations. In MCSCF/CASSCF, it unites a second-order optimization of the active orbitals with a Fock-based first-order treatment of the remaining closed-virtual orbital rotations. In the case of the single-determinant wavefunctions, the active space is replaced by a preselected "second-order domain," and all rotations involving orbitals in this subspace are treated at second-order. The method has been implemented for spin-restricted and spin-unrestricted Hartree-Fock (RHF, UHF), configuration-averaged Hartree-Fock (CAHF), as well as Kohn-Sham (KS) density functional theory (RKS, UKS). For each of these cases, various choices of the second-order domain have been tested, and appropriate defaults are proposed. The performance of the method is demonstrated for several transition metal complexes. It is shown that the SO-SCI optimization provides faster and more robust convergence than the standard SCF procedure but requires, in many cases, even less computation time. In difficult cases, the SO-SCI method not only speeds up convergence but also avoids convergence to saddle-points. Furthermore, it helps to find spin-symmetry broken solutions in the cases of UHF or UKS. In the case of CAHF, convergence can also be significantly improved as compared to a previous SCF implementation. This is particularly important for multi-center cases with two or more equal heavy atoms. The performance is demonstrated for various two-center complexes with different lanthanide atoms.

The Molpro quantum chemistry package.

Molpro is a general purpose quantum chemistry software package with a long development history. It was originally focused on accurate wavefunction calculations for small molecules but now has many additional distinctive capabilities that include, inter alia, local correlation approximations combined with explicit correlation, highly efficient implementations of single-reference correlation methods, robust and efficient multireference methods for large molecules, projection embedding, and anharmonic vibrational spectra. In addition to conventional input-file specification of calculations, Molpro calculations can now be specified and analyzed via a new graphical user interface and through a Python framework.

MCSCF optimization revisited. II. Combined first- and second-order orbital optimization for large molecules.

A new orbital optimization for the multiconfiguration self-consistent field method is presented. This method combines a second-order (SO) algorithm for the optimization of the active orbitals with the first-order super configuration interaction (SCI) optimization of the remaining closed-virtual rotations and is denoted as the SO-SCI method. The SO-SCI method significantly improves the convergence as compared to the conventional SCI method. In combination with density fitting, the intermediates from the gradient calculation can be reused to evaluate the two-electron integrals required for the active Hessian without introducing a large computational overhead. The orbitals and CI coefficients are optimized alternately, but the CI-orbital coupling is accounted for by the limited memory Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method. This further improves the speed of convergence. The method is applicable to large molecules. The efficiency and robustness of the presented method is demonstrated in benchmark calculations for 21 aromatic molecules as well as for various transition metal complexes with up to 826 electrons and 5154 basis functions.

Second-order MCSCF optimization revisited. I. Improved algorithms for fast and robust second-order CASSCF convergence.

A new improved implementation of the second-order multiconfiguration self-consistent field optimization method of Werner and Knowles [J. Chem. Phys. 82, 5053 (1985)] is presented. It differs from the original method by more stable and efficient algorithms for minimizing the second-order energy approximation in the so-called microiterations. Conventionally, this proceeds by alternating optimizations of the orbitals and configuration (CI) coefficients and is linearly convergent. The most difficult part is the orbital optimization, which requires solving a system of nonlinear equations that are often strongly coupled. We present a much improved algorithm for solving this problem, using an iterative subspace method that includes part of the orbital Hessian explicitly, and discuss different strategies for performing the uncoupled optimization in a most efficient manner. Second, we present a new solver in which the orbital-CI coupling is treated explicitly. This leads to quadratic convergence of the microiterations but requires many additional evaluations of reduced (transition) density matrices. In difficult optimization problems with a strong coupling of the orbitals and CI coefficients, it leads to much improved convergence of both the macroiterations and the microiterations. Third, the orbital-CI coupling is treated approximately using a quasi-Newton approach with Broyden-Fletcher-Goldfarb-Shanno updates of the orbital Hessian. It is demonstrated that this converges almost as well as the explicitly coupled method but avoids the additional effort for computing many transition density matrices. The performance of the three methods is compared for a set of 21 aromatic molecules, an Fe(ii)-porphine transition metal complex, as well as for the [Cu2O2(NH3) 6]2+, FeCl3, Co2(CO)6C2H2, and Al4O2 complexes. In all cases, faster and more stable convergence than with the original implementation is achieved.