The Projector Augmented Wave method

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Electron density (continued)With these definitions:n a = ∑ ijD a ijφ a i φ a j + n a c ,ñ a = ∑ ijD a ij ˜φ a i ˜φ a j + ñ a c ,ñ = ∑ nf n | ˜ψ n | 2 + ∑ añ a c ,we get a very simple expression for the all-electron density:n = ñ + ∑ a(n a − ñ a )12
Compensation chargesLet Z a (⃗r) be the nuclear charge for atom a. <strong>The</strong> Coulomb energy is:(∫ n(⃗r) + ∑ ) (E C = d⃗rd⃗r ′ a Za (⃗r − ⃗R a ) n(⃗r ′ ) + ∑ )a Za (⃗r ′ − ⃗R a )|⃗r − ⃗r ′ |= (n + ∑ aZ a ) 2= (ñ + ∑ a[n a − ñ a + Z a ]) 2We add and subtract compensation charges localized inside theaugmentation spheres:E C = (ñ + ∑ a˜Z a + ∑ a[n a − ñ a + Z a − ˜Z a ]) 213
Page 1 and 2: The Projector AugmentedWave method
Page 3 and 4: LiteratureThe PAW method was invent
Page 5 and 6: Platinum atom43211s2s3s4s5s6s2p3p4p
Page 7 and 8: Electron densityn= ñ plane waves/c
Page 9 and 10: Completeness relationsFor |⃗r −
Page 11: Electron density (continued)n = ∑
Page 15 and 16: Compensation charges (continued)Usi
Page 17 and 18: • Frozen core states.Approximatio
Page 19 and 20: Exchange-correlation energyE xc =
Page 21 and 22: OrthogonalityKeep the wave function
Page 23: Convergencespd 11 22 221 222 321 23

The Projector Augmented Wave method

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