Model-Kernel Corrected DIIS: Cheap Response Information for the Roothaan Step

Nearly all Kohn–Sham density functional theory (DFT) and Hartree–Fock calculations solve the self-consistent field (SCF) equations with the direct inversion in the iterative subspace (DIIS). Their cost is set by the number of Fock builds that DIIS needs. Each DIIS cycle diagonalizes an extrapolated Fock matrix as if this matrix stayed fixed while the electrons move. The charge that the step moves, however, changes the Coulomb potential, and the next cycle has to undo part of the step. This charge sloshing makes DIIS slower as molecules grow. We correct every step with a model kernel, an inexpensive approximation of this Coulomb response that is built from a few Gaussian functions on each atom and contains no fitted parameters. The resulting method, MK-DIIS, needs no extra Fock build, converges to the same solution as DIIS and adds about 1% to the run time. On 170 transition-metal complexes and organic molecules it needs about one third fewer Fock builds than DIIS, fewer than any SCF algorithm of a production code that we compared with. On molecules with up to 300 atoms it needs about half as many, and on linear acenes its number of Fock builds does not grow with length. The saving is smaller with more exact exchange and in a continuum solvent, both of which screen the Coulomb response. For open-shell molecules, a delayed start keeps the reliability of DIIS. Because only the diagonalization step changes, the correction can be added to existing SCF codes and combined with other convergence aids.