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23 proper ties calculated using dft ri ri atom

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Unformatted text preview: i2 H 0 = ∑ − + v(ri ) + vH (ri ) + v XC (ri ) 2m i =1 Kohn-Sham Hamiltonian for each particle N N = ∑ h(ri ), where hφi = ε iφi . i =1 Now, ∆V=UH-Vhxc is the full, bare, long-range Coulomb repulsion between pairs, minus a one-body 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=<Ψ|H|Ψ> is much different. 21 N ∑ε i =1 i , DFT: Progression/Generalization Non-relativistic, non magnetic: ε[n], n(r) Non-relativistic, magnetic: ε[nαβ], t sr 1 n(r )1 + m • σ 2 Relativistic, magnetic: ε[jμ], 4-current jμσ(r) Non-relativistic, B field: ε[jμ], charge density, current density t-dependent 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 (density-polarization functional theory) Temperature-dependent (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) • pressure-driven 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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