  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.
