The Projector Augmented Wave method

Electron density (again)n(⃗r) = ∑ an a c (|⃗r − ⃗R a |) + ∑ nf n |ψ n (⃗r)| 2= ∑ an a c + ∑ nf n∣ ∣∣∣∣˜ψn + ∑ ai〈˜p a i | ˜ψ n 〉(φ a i − ˜φ a i )∣2= ∑ n a c + ∑ f n | ˜ψ n | 2a n+ ∑ ∑f n 〈˜p a i | ˜ψ n 〉(φ a i − ˜φ a i )〈 ˜ψ n |˜p a j 〉(φ a j − ˜φ a j )n aij⎧⎫⎪⎨ ∑ ∑ ∑+2Re f n 〈˜p a i |⎪⎩˜ψ n 〉 ˜φ∑⎪⎬ai 〈 ˜ψ n |˜p a j 〉(φ a j − ˜φ a j )n a ij⎪⎭} {{ }˜ψ n10