The Projector Augmented Wave method

The PAW transformationψ n (⃗r) = ˆτ ˜ψ n (⃗r)ˆτ = 1 + ∑ a∑(|φ a i 〉 − | ˜φ a i 〉)〈˜p a i |,iwhere ˜p a i (⃗r) = 0 and ˜φ a i (⃗r) = φa i (⃗r) for r > ra c , and 〈˜p a i | ˜φ a j 〉 = δ ij.|˜p a i 〉 = ˜p a nlm(⃗r − ⃗R a ) = ˜p a nl(r)Y lm ( ⃗r − ̂ ⃗R a )ˆτ ˜φ a i = φ a i8