Many important properties of materials can be ascertained through a description of their electronic-structure—a description of the quantum orbitals available to electrons in the system (in analogy with the 1S, 2S, 2P, etc. orbitals in atoms) and transition probabilities among the orbitals in the presence of an external perturbation (e.g., shining light on the material). Density functional theory (DFT) within the Kohn–Sham approach [1] provides an accurate formalism for computing the ground state properties of materials (those properties of a material where the electrons are occupying the lowest energy orbitals) through the solution of the coupled self-consistent Kohn–Sham equations: p 2 2 m + V ( r ) + V H [ ρ ] ( r ) + V x c [ ρ ] ( r ) φ i ( r ) = ε i φ i ( r ) https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315277400/07e88077-33fb-4fc7-b027-65d6c10b854b/content/math22_1.tif"/> 482with ρ ( r ) = ∑ i o c c φ i ( r ) 2 https://s3-euw1-ap-pe-df-pch-content-public-u.s3.eu-west-1.amazonaws.com/9781315277400/07e88077-33fb-4fc7-b027-65d6c10b854b/content/math22_2.tif"/>