  A thesis providing a pedagogical introduction to the problem of achieving
self-consistency in density functional theory. Contained is an introduction to
the framework of Kohn-Sham density functional theory, leading then to the
considerations required to solve the equations of Kohn-Sham density functional
theory. Specifically, a focus is placed on where current self-consistent field
methodology is inefficient and/or fails to converge to a solution. As such,
this review spans sub-disciplines such as numerical analysis of linear and
non-linear systems, linear response theory, and general electronic structure
theory. Toward the end of the thesis, certain contemporary methods for
achieving self-consistency from literature are outlined, and a novel,
computationally efficient preconditioning strategy is proposed. This work is
implemented in the CASTEP software.
