IV) Materials calculations with dynamical mean field theory (DMFT)
IV) Materials calculations with dynamical mean field theory (DMFT) IV) Materials calculations with dynamical mean field theory (DMFT)
Ab-initio electronic Hamiltonian(non-relativistic/Born-Oppenheimer approximation)kinetic energy lattice potential Coulomb interactionH = X i"− 2 ∆ i2m e+ X l−e 24πɛ 0Z l|r i − R l |#+ 1 2Xi≠je 24πɛ 01|r i − r j |LDA bandstructure corresponds toH LDA = X i"− 2 ∆ i2m e+ X l−e 24πɛ 01|r i − R l | + Zd 3 r#e2 1LDAρ(r) + Vxc (ρ(r i ))4πɛ 0 |r i − r|LDA + local Coulomb interactionĤ = X ɛ LDAklm ĉ†klσĉkmσ+ 1 XU σσ′lm2 ˆn ilσ ˆn imσ ′ − ∆ɛ X ˆn imσU’ U’−Jklmσi lσmσ| {z }′ imσm=1H LDA U• non-local Coulomb interaction → Hartree term (to leading order in 1/Z; contained in LDA)• constrained LDA ⇒ U, J, V =U−2J, ∆ɛ(McMahan et al.’88, Gunnarsson et al.’89)m=2
Constrained LDA1. Calculate LDA energies E(n d ) for constrained problem, i.e., without hopping of d (orf) electrons, for differnet numbers n d of d (or f) electrons.2. Adjust U (J) and ∆ɛ inĤ that these LDA energies are obtained.E000 111000 111000 111000 111n −1 n n +1dd
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- Page 3 and 4: Ab-initio electronic Hamiltonian(no
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- Page 10 and 11: LDA+UAnisimov et al.’91LDA+U: Sol
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- Page 23 and 24: LDA+DMFT — overviewMetzner, Vollh
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- Page 37 and 38: DMFT self-consistency cycleG(ω) =
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- Page 41 and 42: Experimental motivation — (Cr x V
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- Page 45 and 46: One-band Hubbard modelĤ = − t∗
- Page 47 and 48: Numerical results - spectrumSpectru
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Constrained LDA1. Calculate LDA energies E(n d ) for constrained problem, i.e., <strong>with</strong>out hopping of d (orf) electrons, for differnet numbers n d of d (or f) electrons.2. Adjust U (J) and ∆ɛ inĤ that these LDA energies are obtained.E000 111000 111000 111000 111n −1 n n +1dd
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