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DFT Calculation of Electronic Structure: an Introduction Application to K-Edge

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DFT Calculation of Electronic Structure: an Introduction Application to K-Edge DFT calculation of electronic structure: an introduction Application to K-edge XAS Amélie Juhin Sorbonne Université-CNRS (Paris) Slide courtesy : Delphine Cabaret Quanty School 2019 1 About XANES spectroscopy at the K-edge probe of structural and electronic properties of materials local probe: up to 5-10 Å around the absorbing element isotropic spectra isotropic spectra polarized spectra 9 Al K edge Al K edge Al K edge [6] beryl Al 6 6 [6] Al 6 corundum corundum α-Al O 2 3 α-Al2O3 ε //c 3 3 beryl 3 berlinite [4] (BeO)3(Al2O3)(SiO2)6 Normalized absorption (arbit. units) Normalized absorption (arbit. units) Normalized absorption (arbit. units) c Al AlPO4 ε ⊥ 0 0 0 1560 1565 1570 1575 1580 1585 1590 1560 1565 1570 1575 1580 1585 1590 1560 1565 1570 1575 1580 1585 1590 incident X-ray energy (eV) incident X-ray energy (eV) incident X-ray energy (eV) 1 eV shift tremendous amount of information coordination number fingerprint needs for calculations Quanty School 2019 2 About XANES spectroscopy at the K-edge probe of structural and electronic properties of materials selection rules selective probe of empty states localized on the absorber K edge: 1s p transitions (mainly) isotropic spectra isotropic spectra polarized spectra 8 3 Al K edge Cr K edge ruby ε // c Ca K edge corundum 3+ 4 CaC α-Al2O3:Cr 6 α-Al2O3 6 ε ⊥ c 2 exp. 4 exp. with Ca 1s hole exp. 2 no Ca 1s hole with Al 1s hole no Al 1s hole 1 2 calc. Normalized absorption (arbit. units) Normalized absorption (arbit. units) Normalized absorption (arbit. units) nowith Cr Cr 1s 1s hole hole Normalized absorption (arbit. units) no Cr 1s hole calc. calc. 0 0 0 1560 1565 1570 1575 1580 1585 1590 5995 6000 6005 6010 6015 6020 6025 6030 4030 4040 4050 4060 4070 4080 incident X-ray energy (eV) incident X-ray energy (eV) incident X-ray energy (eV) tremendous amount of information with variable core-hole effects Quanty School 2019 3 Modeling of XANES spectroscopy Basic issue: calculation of the absorption cross section for a material, i.e., a system of N electrons + Nat nuclei 2 1 2 σ(~!)=4⇡ ↵ ~! f i δ(Ef Ei ~!) di |h |O| i| − − Xi,f incident x-ray energy operator of the interaction between x-rays and the system initial state: final state: ground state of the system excited state of the system with energy , Ei degenerescence di of energy Ef. weak e- - e- interaction - - strong e - e interaction monoelectronic approach multielectronic approach (LFM) (Density Functional Theory) ex : L2,3 edges of 3d elements ex : K edges Quanty School 2019 4 Part 1 Introduction to electronic structure calculation using DFT Quanty School 2019 5 Basic issue System composed of Nat atoms: N electrons, mass m, charge q < 0, position r (i = 1, …, N) + i Nat nuclei, mass MI, charge qI , position RI (I = 1, …, Nat) Hsyst. = Te + Tnucl + Vnucl nucl + Ve nucl + Ve e − − − N ~2 T = − 2 e 2m ri i=1 X 1 1 q2 Nat 2 Ve e = ~ 2 − 2 4⇡"0 ri rj Tnucl = I i j=i | − | −2MI r X X6 I=1 X 1 qqI Ve nucl = − 4⇡"0 ri RI 1 1 qI qJ i I | − | Vnucl nucl = X X − 2 4⇡"0 RI RJ I J=I X X6 | − | Quanty School 2019 6 Born-Oppenheimer approximation m < MI (m / mproton ≈ 1836) Order of magnitude of the velocity ratio 1 Electron : mv2 = 1Ry ≈ 10 eV 2 e ve MI 10eV vnoy ⇡ m 10meV 1 2 r Nucleus : MI v ~!vib ≈ 10 meV 2 I ⇡ ≈ 103 The nuclear motion is much slower than the electronic motion è separation The nuclei are considered at fixed positions in the electronic motion study Positions RI è parameter Quanty School 2019 7 Born-Oppenheimer approximation Electronic Hamiltonian parameter RI He{ } = Te + Ve e + Ve nucl + Vnucl nucl − − − constant R R R H{ I } { I } = E ( R ) { I } e | n i n { I } | n i Quanty School 2019 8 Multielectronic Schrödinger equation q H n = En n e = | | p4⇡"0 2 2 2 2 ~ ZI e 1 e H = − ri + − + 2m ri RI 2 ri rj i I i j=i ⇣ | − | ⌘ 6 | − | X (i) X X X T (i) V Ve e | {z } nucl − (r , r ,...,|r ; s{z, s ,...,} s ) | {z } n ⌘ n 1 2 N 1 2 N antisymmetric (electrons are fermions) Quanty School 2019 9 Multielectronic Schrödinger equation H n = En n We can solve it for a system of non-interacting electrons ! N (i) (i) (i) (i) H = Hmono with Hmono = T + Vnucl i=1 solution X Hmono φ↵ = ✏↵ φ↵ ↵ =1,..., 1 spinorbital n : Slater determinant (SD) - 1929 build from N spinorbital functions φ↵ Quanty School 2019 10 Slater Determinant Ground state (1) (1) (1) φ1 φ2 ... φN (2) (2) (2) φ1 φ2 ... φN N 1 gs = .... Egs = ✏i pN! .... i=1 (N) (N) (N) X φ φ ... φ 1 2 N Properties : - antisymmetric - Pauli exclusion principle satisfied Quanty School 2019 11 Multielectronic Schrödinger equation H n = En n 2 2 2 2 ~ ZI e 1 e H = − ri + − + 2m ri RI 2 ri rj i I i j=i ⇣ | − | ⌘ 6 | − | X (i) X X X T (i) V Ve e solution | {z } nucl − | {z } | {z } (r , r ,...,r ; s , s ,...,s ) n ⌘ n 1 2 N 1 2 N The Slater determinants form a basis in the N-electrons Hilbert space n: Linear combination of an infinite number of Slater determinants configuration interaction (“brute force”) ↵ ! αmax= 10 è 252 SD CN = max ↵max N = 5 α = 20 è 15, 504 SD N = 20 αmax= 30 è 3, 004, 015 SD N!(↵max N)! max − αmax= 50 è 2, 118, 760 SD Quanty School 2019 12 Mean-field approximation Non-interacting electrons Mean-field approximation Interacting electrons Exact solution : SD Approximative solutions Exact solution : ∞ LC of SD Exact multilelectronic Hamiltonian 2 (i) (i) 1 e H = T + Vnucl + 2 ri rj i i j=i X h i X X6 | − | Ve e − | {z } Effective multielectronic Hamiltonian : mean-field Hamiltonian N e↵ e↵,(i) H = Hmono Solutions : i=1 X Slater ~2 2 determinants with He↵ = − r + V (r)+V e↵ (r) mono 2m nucl Quanty School 2019 13 Mean-field approximation Monoelectronic Schrödinger solved self-consistently (SCF) ~2 2 − r + V (r)+V e↵ (r) φe↵ (r)=✏e↵ φe↵ (r) 2m nucl [⇢] i i i e↵ e↵ with φ φ = δ h i | j i ij N electron density ⇢(r)= φe↵ (r) 2 | i | i=1 X Quanty School 2019 14 Mean-field methods: examples Hartree (H) Hartree-Fock (HF) Density Functional Theory (DFT) Quanty School 2019 15 Hartree (H) - 1927 ~2 2 − r + V (r)+V e↵ (r) φe↵ (r)=✏e↵ φe↵ (r) 2m nucl [⇢] i i i Coulomb interaction 2 ⇢(r0) VH(r)=e dr0 between one electron and r r the electron density Z | − 0| 2 2 N e↵ 2 ~ e↵ e↵ 2 φj (r0) e↵ e↵ e↵ − r φ (r)+V (r)φ (r)+e | | dr0φ (r)=✏ φ (r) 2m i nucl i r r i i i j=1 0 X Z | − | includes a self-interaction term (if i = j) Hartree product Pauli exclusion principle multielectronic = of monoelectronic exchange not taken into account wave function wave functions Quanty School 2019 16 Hartree-Fock (HF) - 1935 exchange included: the multielectronic wave function as a (unique) SD additional term that depends on the spin state and the antisymmetric character of the SD : N e↵ e↵ φj (r0)⇤φi (r0) e↵ δ dr0 φ (r) − sisj r r j j=1 0 X Z | − | strictly compensates the self-interaction term when electrons i and j have the same spin state BUT no correlation between electrons with opposite spins Correlation energy: difference between the exact total energy of N-electrons system and the total energy obtained using HF Quanty School 2019 17 Density functional theory (DFT): definition exact theory for systems of N interacting electrons within an external potential 2 2 2 ~ i 1 e H = − r + Vnucl(ri) + 2m 2 ri rj i i i j=i X X X X6 | − | ext T V (r) Ve e − | {z } | {z } | {z } H n = En n ⇢(r) (r , r ,...,r ; s , s ,...,s ) n ⌘ n 1 2 N 1 2 N Quanty School 2019 18 DFT: Hohenberg and Kohn theorems (1964) The (non-degenerate) ground state electron density uniquely determines (to a constant) ext the external potential ⇢gs(r) V (r) (and thus all the properties of the system). ⇢(r) basic variable There is a universal density functional F [ ⇢ ], not depending explicitly on V ext ( r ) , defined as : F [⇢]=T [⇢]+Ve e[⇢] with ⇢(r)dr = N − Z Quanty School 2019 19 DFT: Hohenberg and Kohn (1964) Total energy E E[⇢]=F [⇢]+ V ext(r)⇢(r)dr ⌘ Z E[ρ] Variational principle Egs ρ ρgs Exact theory but F [⇢]=T [⇢]+Ve e[⇢]=??? question 1 − Quanty School 2019 20 DFT: Kohn and Sham (1965) Hohenberg - Kohn theorems valid for any N-electrons system also valid for N non-interacting electrons (Ve-e= 0) Kohn – Sham ansatz We can find an external potential VKS(r) for a fictitious system of non-interacting electrons, giving the same ground state electron density as the real system Exact theory BUT question 2 VKS(r)=??? Quanty School 2019 21 DFT: Kohn and Sham (1965) Kohn-Sham system of N non-interacting electrons, fictitious in an external potential VKS(r), system: which gives the same gs electron density as the real system Ground state : SD build from N orbitals (monoelectronic) Kohn-Sham orbitals ~2 2 − r + V KS(r) φKS(r) = ✏KS φKS(r) (1) solutions of: 2m ↵ ↵ ↵ ✓ ◆ Kinetic energy of the fictitious system electron density 2 N N ~ KS 2 KS T [⇢]=− φ (r)⇤ φ (r)dr ⇢(r)= φKS(r) 2 s 2m i r i | i | i=1 Z i=1 X X (indice s : « single particle ») F [⇢]=T [⇢]+Ve e[⇢]=Ts[⇢]+Ve e[⇢]+(T [⇢] Ts[⇢]) − − − Quanty School 2019 22 DFT: Kohn and Sham (1965) What we can write as a functional of the electron density in Ve-e : 2 e ⇢(r)⇢(r0) Hartree energy: E [⇢]= dr dr0 H 2 r r Z | − 0| F [⇢]=Ts[⇢]+Ve e[⇢]+(T [⇢] Ts[⇢]) − − = Ts[⇢]+EH[⇢]+(T [⇢] Ts[⇢]+Ve e[⇢] EH[⇢]) − − − Exc[⇢] exchange and correlation energy we partially answered question 1 ! Quanty School 2019 23 DFT: Kohn and Sham (1965) Total energy ext E[⇢]=Ts[⇢]+VH[⇢]+Vxc[⇢]+ V (r)⇢(r)dr Z minimization Ground state total energy variational principle applied to E[ρ] with respect to KS orbitals with the constraint: KS KS φi (r)⇤ φj (r)dr = δij Z ~2 2 − r + V (r)+V (r)+V ext(r) φKS(r) = ✏KS φKS(r) (2) 2m H xc ↵ ↵ ↵ ✓ ◆ exchange and correlation potential δE V = xc xc ⇢(r) Quanty School
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