Unformatted text preview: i2 H 0 = ∑ −
+ v(ri ) + vH (ri ) + v XC (ri ) 2m
i =1 KohnSham Hamiltonian for each particle
N N = ∑ h(ri ), where hφi = ε iφi .
i =1 Now, ∆V=UHVhxc is the full, bare, longrange Coulomb repulsion between
pairs, minus a onebody potential vH+vXC.
∆V does not change the charge density from H0!
However, it gives a big change in the energy: ψ 0 H 0 ψ 0 =
while ε=<ΨHΨ> is much different.
21 N ∑ε
i =1 i , DFT: Progression/Generalization
Nonrelativistic, non magnetic: ε[n], n(r)
Nonrelativistic, magnetic: ε[nαβ], t
sr
1
n(r )1 + m • σ
2 Relativistic, magnetic: ε[jμ], 4current jμσ(r)
Nonrelativistic, B field: ε[jμ], charge density, current
density
tdependent vext(r,t): ε[n(t′), t′<t]
Superconducting states: ε[n,Δ], charge density, pair density
22 DFT for excited states (orthogonal to ground state)
DFT for other properties: momentum distribution function,
Compton profile
DFT for ferroelectrics (densitypolarization functional
theory)
Temperaturedependent (finite T) DFT
DFT for the Hubbard model or practically any other model,
exists. 23 Proper ties Calculated Using DFT
ε({Ri}), {Ri}= atom positions in solid (or molecule); “relaxation of
structure”
• ground state structure: bcc, fcc, hcp, Pnma, etc.
(but with many independent coordinates there are many
local minima)
• pressure: P=∂ε/∂V (V=volume)
• pressuredriven structure transformations
• phonons: ∂2ε/∂Ri∂Rj 24 • elastic constants
• defect energies
Slope:
P=∂ε/∂V
Slope gives
pressure at which
transformation
occurs 25 ε ({R }) : ground state density
0
i • elastic field gradients
• ionization potential/work function
• Fermi surfaces: usually very good! (although not exact)
Magnetic properties from Spin DFT
• magnetic order of ground states: magnetic impurities
• hyperfine fields at nuclei
...
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 Spring '12
 WarrenE.Pickett
 Physics, DFT, density functional theory, ground state density

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