This thesis is a study of electronic excitations at metal surfaces, as described within the context of density functional theory (DFT). Before presenting any physics, the document develops an adaptive spline collocation method that is the workhorse for most of the numerical computations presented afterwards. The background and mathematical foundations of DFT are briefly explored. From there, two different techniques for computing ground state densities---orbital-free density functional theory and Kohn-Sham density functional theory---are fully realized. Development of algorithms for numerical computation is the primary component of the presentation, but both techniques are also given rigorous theoretical treatment in full proofs of asymptotic results. What follows next is a rigorous derivation of linear response theory from first principles. Great effort is then spent to develop a scheme for numerical computation of excited responses via linear response theory. Finally, the thesis concludes by demonstrating the application of the techniques developed in the previous chapters to a nonlinear optical phenomenon called second harmonic generation.
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