A trust-region augmented Hessian implementation for restricted and unrestricted Hartree-Fock and Kohn-Sham methods

We present a new implementation of a trust-region augmented Hessian approach (TRAH-SCF) for restricted and unrestricted Hartree-Fock and Kohn-Sham methods. With TRAH-SCF, convergence can always be achieved even with tight convergence thresholds, which requires just a modest number of iterations. Our convergence benchmark study and our illustrative applications focus on open-shell molecules, also antiferromagnetically coupled systems, for which it is notoriously complicated to converge the Roothaan-Hall self-consistent field (SCF) equations. We compare the number of TRAH iterations to reach convergence with those of Pulay's original and Kollmar's (K) variants of the direct inversion of the iterative subspace (DIIS) method and also analyze the obtained SCF solutions. Often, TRAH-SCF finds a symmetry-broken solution with a lower energy than DIIS and KDIIS. For unrestricted calculations, this is accompanied by a larger spin contamination, i.e., larger deviation from the desired spin-restricted ⟨S²⟩ expectation value. However, there are also rare cases in which DIIS finds a solution with a lower energy than KDIIS and TRAH. In rare cases, both TRAH-SCF and KDIIS may also converge to a non-aufbau solution. For those calculations, standard DIIS always diverges. For cases that converge smoothly with either method, TRAH usually needs more iterations to converge than DIIS and KDIIS because for every new set of orbitals, the level-shifted Newton-Raphson equations are solved approximately and iteratively. In such cases, the total runtime of TRAH-SCF is still competitive with the DIIS-based approaches even if extended basis sets are employed, which is illustrated for a large hemocyanin model complex.