|pi⟩ ϕi v 1 eff ϕj -φi ṽ1 eff φj ⟨pj| (24) where ṽeff is the potential energy defined outside the augmentation regions and which act directly on the plane wave expansion, whereas v 1 eff and ṽ1 eff are the atom-centered potential operators defined within the augmentation regions and acting directly on the all-electron and pseudo partial wave functions, respectively. We add an external potential ∆Vi where i = (α, l, m l ) is defined by the atomic site α, azimuthal quantum number l, and magnetic quantum number m l to the all-electron atom-centered potential v 1 eff such that