A variational framework for spectral approximations of Kohn-Sham Density Functional Theory X. Wang 1 , T. Blesgen 2 , K. Bhattacharya 1 , and M. Ortiz 1 1 Division of Engineering and Applied Sciences, California Institute of Technology, USA 2 Department of Mathematics, University of Applied Sciences, Bingen, Germany NOT FOR DISTRIBUTION Abstract Kohn-Sham density functional theory (K-S DFT) is widely used to study the electronic structure of materials. The central difficulty in K-S DFT involves the solution of a non-linear eigenvalue problem. This non-linear problem is solved numerically by the self-consistent field method, a fixed point iteration approach, which yields linear eigenvalue problems. Typical solution of the linear eigenvalue problem is to diagonalize the matrix of the differential operator, the Hamiltonian of the system. There have been approximate solutions to K-S DFT that exploit spectral theory of self-adjoint operators, known as the density matrix expansion methods. These methods can avoid diagonalization of the Hamiltonian matrix. They are increasingly used to study the linearized problem because of their computational efficiency. Although these approximations have been verified numerically, the relationship between these approximations of the linearized problem and the original non-linear problem remain incompletely understood. Further, these methods assume smoothness that give rise to errors in conductors. In this paper, we reformulate K-S DFT as a nested variational problem that enables density matrix expansions. We introduce a new approximation, called the spectral binning discretization, which does not require smoothness. We show convergence with respect to both spectral binning discretization and with spatial discretization. 1 Introduction The wave formulation of quantum mechanics proposed by Erwin Schr¨ odinger in 1926 can be used in theory to quantitatively study the electronic structure of materials. However, it is limited to only a handful of electrons due to the high dimensionality of the resulting partial-differential equation. An approximate formulation, called the Hartree-Fock (H-F) method, was introduced in 1930 to reduce the dimensionality of the wave formulation. This reduction is achieved by variationally minimizing the energy over the set of Slater–determinant combinations of independent-electron orbitals, resulting in a non-linear eigenvalue problem in three dimensions [22]. The solution of the H-F equations is nevertheless cumbersome. Density functional theory (DFT) developed by Kohn and Hohenberg in 1964 lay the foundation for the majority of approximate quantum mechanical methods used today. The Kohn-Hohenberg theorem provides a one-to-one correspondence between the ground-state electron density and the ground-state energy; thereby proving the existence of an ground-state energy functional that de- pends only the ground-state electron density. However, the exact form of the energy functional 1