The Projector Augmented Wave method

More documents

Recommendations

Info

Compensation charges (continued)<strong>The</strong> compensation charges are constructed like this:˜Z a (⃗r) = ∑ lmQ a lm˜g a lm(⃗r),where ˜g a lm (⃗r) = 0 for r > ra c :˜g a lm(⃗r) = C l r l exp(−α a r 2 )Y lm (ˆr),<strong>The</strong> Q a lm ’s are chosen such that na − ñ a + Z a − ˜Z a has no multipolemoments: ∫d⃗rr l Y lm (ˆr)(n a − ñ a + Z a − ˜Z a ) = 014
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
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 and 12: Electron density (continued)n = ∑
Page 13: Compensation chargesLet Z a (⃗r)
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

Create successful ePaper yourself

Delete template?

Save as template?