An analysis for the DIIS acceleration method used in quantum chemistry calculations

This work features an analysis for the acceleration technique DIIS that is standardly used in most of the important quantum chemistry codes, e.g. in DFT and Hartree-Fock calculations and in the Coupled Cluster method. Taking up results from Harrison (J Comput Chem 25:328, 2003), we show that for the general nonlinear case, DIIS corresponds to a projected quasi-Newton/secant method. For linear systems, we establish connections to the well-known GMRES solver and transfer according (positive as well as negative) convergence results to DIIS. In particular, we discuss the circumstances under which DIIS exhibits superlinear convergence behaviour. For the general nonlinear case, we then use these results to show that a DIIS step can be interpreted as step of a quasi-Newton method in which the Jacobian used in the Newton step is approximated by finite differences and in which the according linear system is solved by a GMRES procedure, and give according convergence estimates.