The Projector Augmented Wave method

Compensation charges (continued)Using ˜ρ = ñ + ∑ a ˜Z a , ˜ρ a = ñ a + ˜Z a and ρ a = n a + Z a , we get:E C = (ñ + ∑ a˜Z a + ∑ a[n a − ñ a + Z a − ˜Z a ]) 2= (˜ρ + ∑ a[ρ a − ˜ρ a ]) 2= ˜ρ 2 + 2˜ρ ∑ a(ρ a − ˜ρ a ) + ∑ ab(ρ a − ˜ρ a )(ρ b − ˜ρ b )Since ρ a − ˜ρ a has no multipole moments, we get:E C = ˜ρ 2 + 2 ∑ a˜ρ a (ρ a − ˜ρ a ) + ∑ a(ρ a − ˜ρ a ) 2= ˜ρ 2 + ∑ a(ρ a ) 2 − ∑ a(˜ρ a ) 2 (1)15