where ∇ is the kinetic energy operator, V ion is the ionic potential, V H is the Hartree potential, Σ is the nonlocal, energy-dependent self energy within the GW approximation, and E QP nk and ψ QP nk are the quasiparticle energies and wavefunctions, respectively. Equation 9 is in Hartree atomic units. In the GW formalism, G refers to the one-body Green's function and W the dynamically screened Coulomb interaction. We use the single-particle wavefunctions from DFT as our quasiparticle wavefunctions (the diagonal approximation) and can iterate to self-consistency in the quasiparticle energies that enter into the Green's function G in the GW approximation if desired.