A density matrix approach to the convergence of the self-consistent field iteration
For researchers in computational quantum chemistry and materials science, this work provides a deeper theoretical understanding of SCF convergence, though it is an incremental improvement over existing analyses.
The paper provides a local convergence analysis of the self-consistent field (SCF) iteration using density matrices, deriving sufficient and almost necessary conditions in terms of the spectral radius of the Jacobian. It introduces upper bounds expressed via higher eigenvalue gaps, offering more insight into convergence behavior than standard results.
In this paper, we present a local convergence analysis of the self-consistent field (SCF) iteration using the density matrix as the state of a fixed-point iteration. Sufficient and almost necessary conditions for local convergence are formulated in terms of the spectral radius of the Jacobian of a fixed-point map. The relationship between convergence and certain properties of the problem is explored by deriving upper bounds expressed in terms of higher gaps. This gives more information regarding how the gaps between eigenvalues of the problem affect the convergence, and hence these bounds are more insightful on the convergence behaviour than standard convergence results. We also provide a detailed analysis to describe the difference between the bounds and the exact convergence factor for an illustrative example. Finally we present numerical examples and compare the exact value of the convergence factor with the observed behaviour of SCF, along with our new bounds and the characterization using the higher gaps. We provide heuristic convergence factor estimates in situations where the bounds fail to well capture the convergence.