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

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 (~!)=4⇡ ↵ ~! f i (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 ↵ = ✏↵ ↵ ↵ =1,..., 1 spinorbital

n : Slater determinant (SD) - 1929 build from N spinorbital functions ↵

Quanty School 2019 10 Slater Determinant

Ground state

(1) (1) (1) 1 2 ... N (2) (2) (2) 1 2 ... N N 1 gs = .... Egs = ✏i pN! .... i=1 (N) (N) (N) X ... 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) e↵ (r)=✏e↵ e↵ (r) 2m nucl [⇢] i i i 

e↵ e↵ with = h i | j i ij

N electron density ⇢(r)= e↵ (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) e↵ (r)=✏e↵ e↵ (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 j (r0) e↵ e↵ e↵ r (r)+V (r) (r)+e | | dr0 (r)=✏ (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↵ j (r0)⇤i (r0) e↵ dr0 (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) KS(r) = ✏KS KS(r) (1) solutions of: 2m ↵ ↵ ↵ ✓ ◆

Kinetic energy of the fictitious system electron density 2 N N ~ KS 2 KS T [⇢]= (r)⇤ (r)dr ⇢(r)= KS(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 i (r)⇤ j (r)dr = ij Z

~2 2 r + V (r)+V (r)+V ext(r) KS(r) = ✏KS KS(r) (2) 2m H xc ↵ ↵ ↵ ✓ ◆ exchange and correlation potential E V = xc xc ⇢(r)

Quanty School 2019 24 DFT: Kohn and Sham (1965)

Kohn-Sham equations ~2 2 r + V (r)+V (r)+V ext(r) KS(r) = ✏KS KS(r) (2) 2m H xc ↵ ↵ ↵ ✓ ◆

Schrödinger equations of the fictitious system ~2 2 r + V KS(r) KS(r) = ✏KS KS(r) (1) 2m ↵ ↵ ↵ ✓ ◆

Kohn – Sham potential ext VKS(r)=VH(r)+Vxc(r)+V (r)

We partially answered question 2 !

Quanty School 2019 25 DFT: Kohn and Sham (1965)

ground state : self-consistent field (SCF) resolution of the Kohn – Sham equations ~2 2 r + V (r)+V (r)+V ext(r) KS(r) = ✏KS KS(r) 2m H xc ↵ ↵ ↵ ✓ ◆ Exact theory:

A Vxc potential that compensates the approximations introduced by the mean-field approaches necessarily exist !

In practice: different forms of exchange-correlation functionals

DFT approximate theory exact theory widely used useless

Quanty School 2019 26 DFT: exchange and correlation functionals

The Local Density Approximation (LDA) the simplest ELDA ✏ [⇢(r)] ⇢(r)dr xc ⌘ xc Z exchange - correlation energy for a particle in a homogeneous electron gas (with density ρ)

✏xc[⇢]=✏x[⇢]+✏c[⇢]

known analytically calculated (Dirac exchange energy) using Ceperley-Alder Monte-Carlo method, or other parametrization (ex: Hedin-Lundqvist) In the LDA, DFT is still exact for an homogeneous electron gas.

Quanty School 2019 27 DFT: exchange and correlation functionals

Generalized Gardient Approximation (GGA)

EGGA ✏ [⇢(r), ~ ⇢(r)] ⇢(r)dr xc ⌘ xc r Z the density gradient at r is taken into account

Various parameterizations

example: the PBE functional J.P. Perdew, K. Burke & M. Ernzerhof « Generalized Gradient Approximation Made Simple » Phys. Rev. Lett., 77:3865‎ (1996).

Quanty School 2019 28 DFT: exchange and correlation functionals

Hybrid functionals (H-GGA)

combine the exchange correlation obtained by using GGA methods with a given proportion of the exchange described by Hartree-Fock

The most used: B3LYP (Becke - 3 parameters - Lee, Yang, Parr)

EB3LYP = ELDA + a (EHF ELDA)+a (EGGA ELDA)+a (EGGA ELDA) xc xc 0 x x x x x c c c

Quanty School 2019 29 DFT: SCF cycle

1st step

Crystal atomic calculation structure è atomic ρ

Superimposition of atomic ρ

ρ crystal= ρin

Quanty School 2019 30 DFT: SCF cycle

ρin(r)

Potential calculation Poisson equation

mixing

ρ in(r) & ρ out(r) KS orbitals calculation (1 ↵)⇢ + ↵⇢ in out Kohn and Sham equations

Calculation of ρ out(r)

NO YES Convergence? STOP

Quanty School 2019 31 DFT: softwares

Solid state physics and quantum chemistry

Abinit EXCITING ParaGauss ADF Fireball PARATEC AIMPRO FLEUR PARSEC Augmented Spherical Wave GAMESS (UK) PC GAMESS Atomistix Toolkit GAMESS (US) PLATO CADPAC GAUSSIAN Parallel Quantum Solutions CASTEP GPAW PWscf (Quantum Espresso) CP2K JAGUAR Q-Chem CPMD MOLCAS SIESTA CRYSTAL MOLPRO Socorro DACAPO MPQC Spartan DALTON NWChem SPR-KKR deMon2K OpenMX TURBOMOLE DFT++ ORCA VASP DMol3 WIEN2k

non-exhaustive list

Quanty School 2019 32 DFT: sofwares the main differences between them

real space periodic boundary conditions

cluster infinite number of atoms or

1st Brillouin zone no periodicity constraint (amorphous and crystalline materials)

suited to molecules suited to cristals supercell

Quanty School 2019 33 DFT: sofwares the main differences between them

Basis to expand the Kohn-Sham orbitals

KS i i (r)= cn 'n(r) n X Plane waves Atomic orbitals Pseudopotentials Gaussians LCAO

Mixed basis atomic spheres: LC of radial functions × spherical harmonics Muffin-tin orbitals interstitial region: plane waves spherical waves LAPW LMTO,multiple scattering

Quanty School 2019 34 DFT: sofwares the main differences between them

Solid state physics and quantum chemistry

Quanty School 2019 35 Part 2 Introduction to the calculation of X-ray absorption spectra

X-ray Absorption Near-Edge Structure

Near-Edge X-ray Absorption Fine Structure

Quanty School 2019 36 XANES modeling Basic issue: calculation of the absorption cross section for a material, i.e., a system of N electrons + Nat nuclei

2 1 2 (~!)=4⇡ ↵ ~! f i (Ef Ei ~!) di |h |O| i| Xi,f

weak e- - e- interaction - - strong e - e interaction monoelectronic approach multielectronic approach (Density Functional Theory) ex : L2,3 edges of 3d elements ex : K edges Quanty School 2019 37 Interaction operator between x-rays and electrons

ik r incident X-ray beam: electromagnetic wave treated as a plane-wave e · k: wave vector " : ˆ polarization direction

ik r e · =1+ik r + ... ·

i = "ˆ r + "ˆ r k r O · i 2 · i · i i i X X

electric dipole transitions (E1) electric quadrupole transitions (E2) ` = 1 ` = 2 ± observable in the ±K pre-edge majority electronic transitions of 3d transition elements

Quanty School 2019 38 Interaction operator between x-rays and electrons

K or L Electric dipole (E1) transitions : ` = 1 1 ±

Initial State quantum numbers final state Edge symmetry n  photon energy 1 0 K p L 2 0 L1 p 3

2 1 L2,3 s+d € € L2 3 0 M1 p 3 1 M s+d 2,3 photon energy 3 2 M4,5 p+f

M5 delocalised final state (weak electron repulsion) : p M monoelectronic theories (DFT) 4 localized final state (strong electron) : 3d, 4f photon energy multielectronic theories (multiplets)

Quanty School 2019 39 XANES and DFT

For K edges:

2 2 (~!)=4⇡ ↵ ~! f "ˆ r i (Ef Ei ~!) h | · | i i,f X 1s atomic orbital monoelectronic empty state (KS) calculated with a 1s core-hole in the electronic configuration of the absorbing atom

Quanty School 2019 40 XANES and DFT: the calculation codes 2 types

1- Those written to calculate core-level spectra

cluster, real space

at the beginning: multiple scattering (no self-consistency, muffin-tin potential) continuum, feff, icxanes,….

2- The electronic structure codes, in which a post SCF-process code was added to calculate core-level spectra periodic boundary conditions, reciprocal space

Quanty School 2019 41 Note: muffin-tin (MT) potential

ZONE I

ZONE II

ZONE I: atomic spheres spherical symmetry potentials

ZONE II: interstitial region constant potential Full Potential Quanty School 2019 42 XANES and DFT: the calculation codes 2 types

1- Those written to calculate core-level spectra

cluster, real space

at the beginning: multiple scattering (no self-consistency, muffin-tin potential) continuum, feff, icxanes,….

at the end of the 90’s: finite differences method with fdmnes

currently: fdmnes, feff9

2- The electronic structure codes, in which a post SCF-process code was added to calculate core-level spectra periodic boundary conditions, reciprocal space

Wien2k, CASTEP, Quantum-Espresso (XSpectra), …

Quanty School 2019 43 XANES and DFT: periodic boundary conditions and core hole

With codes using periodic boundary conditions (for crystals), we have to avoid spurious interaction of the excited atom with its periodically repeated images.

Large interaction Weak interaction

2x2 supercell

To restore neutrality : atom with a core-hole (i) Negative background charge atom without a core-hole (ii) Excited electron in conduction band Quanty School 2019 44 XANES and DFT: periodic boundary conditions and core hole Si K edge in α-quartz

unit cell 1×1×1 supercell 2×2×2 supercell a = 4.91Å, c = 5.40Å = unit cell with 1 core-hole a = 9.82Å, c = 10.80Å 3 Si 6 O 1 Si* 2 Si 6 O 1 Si* 23 Si 48 O

without core-hole interaction between core-holes convergence

Quanty School 2019 45 45 XANES and DFT: the drawbacks

well identified

• DFT is dedicated to the calculation of ground state properties but used here for the modeling of excited states… possible underestimation of the excitation energies

• Static modeling of the core-hole-electron interaction through the supercell approach or within a cluster of atoms

• Exchange and correlation functional is not energy-dependent • inelastic losses not calculated, the convolution factor is a parameter

Quanty School 2019 46 XANES and beyond DFT

• G W + Bethe-Salpeter equation (BSE)

Ø Shirley and coll.

OCEAN interface

Ø Olovsson, Puschnig, Ambrosch-Draxl, Lakowski : LAPW

• TD-DFT (real-space approaches)

Ø FDMNES

Ø Quantum-Chemistry codes: Orca, ADF, Q-chem, …

Quanty School 2019 47 3d elements in octahedral site

single-particle picture Cabaret et al. PCCP (2010) 0 3d Ti4+ eg TiO2 rutile, anatase t2g, eg empty favorable case t2g

6 2+ 3d eg Fe LS FeS2 pyrite e empty t2g g acceptable case MbCO protein

3 3+ 3d e Cr e empty g ruby, emerald, g acceptable case t2g spinel:Cr, pyrope:Cr t2g, eg empty

2 3d 3+ t2g partially empty eg V grossular:V eg empty critical case t 2g t2g, eg empty

5 3+ e empty 3d e Fe g g t partially empty critical case LS MbCN protein 2g t2g eg empty

Quanty School 2019 48 Ti4+ in octahedral site

single-particle picture TiO2 rutile, Ti site: centrosymmetric 3d0 pre-edge : E2 + non local E1 eg t2g 0 ε // [001] A2 A3

0 peak A1 local E2 transition 1s 3d t2g A1 peak A2 0 local E2 transition 1s 3d eg E1+E2 + non-local E1 transition 1s p hybrid. 3d t2g (neighb.) 0 A2 Absorption (arbit. units) Absorption (arbit. units) E1 A3 A1 peak A3

0 E2 x 5 0 non-local E1 transition 1s p hybrid. 3d eg (neighb.) 0 50 10 15 520 25 301035 Energy (eV) Cabaret et al. Phys. Chem. Chem. Phys. 12:5619 (2010)

Quanty School 2019 10 Ti4+ in octahedral site

single-particle picture TiO2 rutile, Ti site: centrosymmetric 3d0 pre-edge : E2 + non local E1 eg t2g ε // [001] A2 A2A3 A3 A1 calc. at too high energy ! A1 WHY? E1+E2

Absorption (arbit. units) E1

0 0.5 eV E2E1+E2x 5 0 0 5 5 10 Energy (eV) Cabaret et al. Phys. Chem. Chem. Phys. 12:5619 (2010)

Quanty School 2019 11 Ti4+ in octahedral site

single-particle picture TiO2 rutile, Ti site: centrosymmetric 3d0 pre-edge : E2 + non local E1 eg t2g ε // [001] A2 A3

A1 calc. at too high energy ! A1 The 1s core-hole is not attractive enough…

E1+E2 How to improve ? dashed: no core-hole

Absorption (arbit. units) E1

0 E2 x 5 0 5 10 Energy (eV)

Quanty School 2019 12 Ti4+ in octahedral site

TiO2 rutile, Ti site: centrosymmetric pre-edge : E2 + non local E1

ε // [001] Bethe-Salpeter equation A2 A3

E1+E2

dashed: no core-hole

Absorption (arbit. units) E1

0 E2 x 5 Shirley, J. Electr. Spectr. Rel. Phenom. 136:77 (2004) 0 5 10 Energy (eV)

Quanty School 2019 13 Cr3+ in octahedral site

3+ single-particle picture MgAl2O4:Cr , Cr site: centrosymmetric 3d3 pre-edge : E2 eg

t2g " [1[1 1 1 p2] ; k [[1¯ 1¯ p2] k k 0,1 A1 A2

two E2 peaks (at too high energy…) E1+E2 E1 Absorption (arbit. units)

E2 Origin? 0 -3 -2 -1 0 1 2 3 4 5 6 spin up + spinEnergy down (eV) spin up spin down eg A2 0,02 eg occupied eg states eg t2g

A1 t2g Absorption (arbit. units)

t2g 0 -3 -2 -1 0 1 2 3 4 5 6 Cabaret et al. Phys. Chem. Chem. Phys. 12:5619 (2010) Energy (eV)

Quanty School 2019 14 V3+ in octahedral site

single-particle picture Ca Al (SiO ) :V3+, V site: centrosymmetric 3 2 4 3 3d2 e pre-edge : E2 g t2g

"" [1[1 1 1 pp2]2] ; ;kk [[1¯1¯ 1¯1¯ pp2]2] kk kk 0,1 A2

A1 A3 E1 Three E2 peaks (at 2eV too high energy…) E1+E2 Absorption (arbit. units) E2 Origin? 0 -4 -3 -2 -1 0 1 2 3 4 5 spin up + spinEnergy down (eV) spin up t2g spin down A3 t2g eg occupied 0,02 states e t2g eg g

t2g

Absorption (arbit. units) A2 A1 eg 0 -4 -3 -2 -1 0 1 2 3 4 5 Cabaret et al. Phys. Chem. Chem. Phys. 12:5619 (2010) Energy (eV)

Quanty School 2019 15 Fe2+ in octahedral site

MbCO protein, complex Fe site geometry single-particle picture 3d6 e pre-edge : E2 + local E1 LS g

ε ⊥ heme t2g Exp. A Calc. β=14° 2 A1 E1+E2 β

E1 2 peaks essentially E1, but Absorption (arbit. units) E2 0 • at too high energy peak -5 -4 -3 -2 -1 0 1 2 3 4 5 occupied states 0,05 Energy (eV) • A2 too intense Fe 4p ε z 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 peak A1 Fe 3dxz Fe 3d 2 5 yz 1s pz- 3dz mixing Fe 3dz²

0 -5 -4 -3 -2 -1 0 1 2 3 4 5 local E1 DOS 6 Energy (eV) C 2px 4 C 2p y peak A2 2 C 2pz local and partial DOS (states/eV/cell) 0 1s pz weakly mixed with 3d 6-5 -4 -3 -2 -1 0 1 2 3 4 5

Nhis 2pz Energy (eV) 4 hybridized with π* of CO 2

0 -5 -4 -3 -2 -1 0 1 2 3 4 5 off-site E1 Energy (eV) Quanty School 2019 Cabaret et al. PCCP (2010) 55 Angular dependence of core hole screening in LiCoO2

DFT-GGA DFT-GGA +U

Gap = 1.7 eV Gap =2.3eV

Electronic structure poorly Better described using Hubbard described in GGA parameter (U) on Co 3d states

Quanty School 2019 Juhin et al. PRB 81 (2010) 56 LiCoO2 : the Co K pre-edge

Best agreement is obtained when including U and the 1s core hole

Quanty School 2019 57 LiCoO2 : the Co K pre-edge

pre-edge : E2 + nonlocal E1

Peak α : 1sà3d (Co*) Peak β : 1sà4p + O(2p) + Co (3d) n.n. in plane Peak γ : 1sà4p + O(2p) + Li (2p)+Co (3d) n.n. out of plane 58

Summary

Inherent drawbacks of DFT-LDA/GGA for XANES calculation • electronic repulsion modeling • core hole-electron interaction modeling E2 and E1 transitions toward 3d (abs) systematically at too high energy

0 3d (TiO2) ~ 0.5 eV reduced to 0 using BSE of E. Shirley

6 3d LS (MbCO, FeS2) ~ 0.8 eV 3d3 (Cr-spinel, emerald) 1 ~eV 3d5 LS (MbCN) ~ 1.7 eV

3d3 (ruby) and 3d2 (V-grossular) 2 ~eV

GGA+U reduces 2 eV 1 eV for 3d8 (NiO) Gougoussis et al. PRB 79:045118 (2008) 6 2.4 eV 1.7 eV for 3d LS (LiCoO2) Juhin et al. PRB 81:115115 (2010)

TDDFT+hybrid functionals in Fe complexes : DeBeer et al. J.Phys.Chem.A 112:12936 (2008)

Quanty School 2019 59 Summary

However, DFT in LDA/GGA is useful !

• number of pre-edge peaks well reproduced • relative intensities and positions in rather good agreement with experiment • single-particle description of transitions with Ø E1/E2 character Ø degree of local and non-local hybridization Ø spin polarization

• improved with adding U (NiO, LiCoO2, …)

Quanty School 2019 60