XSPECTRA A tool for X-ray absorption spectra (XAS) calculations G. - PowerPoint PPT Presentation

XSPECTRA A tool for X-ray absorption spectra (XAS) calculations G. Radtke * & N. Mas *, � * Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie, Université Pierre et Marie Curie, Sorbonne Universités � Synchrotron SOLEIL Slides courtesy of O. Bun ă u, M. Calandra and D. Cabaret EWinS 2016 - EUSpec Winter School on Core level spectroscopies Ajdov š č ina, Slovenia, February 5th 2016

Outline 1. 1. Introduction on XAS Introduction on XAS 2. 2. The PAW theory The PAW theory 3. 3. Hands on (I) : Si-K edge in � -quartz SiO 2 Hands on (I) : Si-K edge in � -quartz SiO 2 -GIPAW pseudopotential generation for Si and Si1s * -GIPAW pseudopotential generation for Si and Si1s * -SCF calculation for � -quartz SiO 2 unit cell without/with core-hole -SCF calculation for � -quartz SiO 2 unit cell without/with core-hole -core wave-function extraction -core wave-function extraction 4. 4. The Lanczos algorithm The Lanczos algorithm 5. Hands on (II) : Si-K edge in � -quartz SiO 2 5. Hands on (II) : Si-K edge in � -quartz SiO 2 - XSpectra calculations for � -quartz SiO 2 unit cell without/with core-hole - XSpectra calculations for � -quartz SiO 2 unit cell without/with core-hole - Supercell calculation : 2x2x2 cell for � -quartz SiO 2 - Supercell calculation : 2x2x2 cell for � -quartz SiO 2 6. 6. Hands on (III) : Dipolar and Quadrupolar contributions at the Ni-K edge in NiO Hands on (III) : Dipolar and Quadrupolar contributions at the Ni-K edge in NiO 7. 7. Summary Summary

A probe of the projected density of unoccupied electronic states The absorption cross-section reads : conduction X σ ( ω ) = 4 π 2 α 0 ~ ω | h ψ f |O| ψ i i | 2 δ ( E f � E i � ~ ω ) i,f Prefactor Final one-electron state (atomic state, molecular orbital, Bloch state, …) Transition operator (electric dipole, ...) valence Initial one-electron (core-state) Energy conservation of the system (specimen + photon) Assuming a transition operator of the form (electric dipole approximation) : O ≈ ˆ ✏ · r 2p The angular part of the matrix element leads to : 2s ħω ` f = ` i ± 1 In a monoelectronic picture (weak e - - e - interaction) : 1s K (or L 1 ) edges probe p -LDOS, L 23 edges probe s + d -LDOS, M 45 edges probe p + f -LDOS, ....

XSpectra What is it ? XSpectra calculates X-ray absorption dipolar and quadrupolar cross sections in the pre- edge to near-edge region within the single particle approximation (i.e. mostly K or L 1 edges but also certain L 23 edges) Where can I find XSpectra ? XSpectra is distributed in the QUANTUM-ESPRESSO package (http://www.quantum-espresso.org/) Can I use it ? XSpectra is distributed under the GNU licence, so you can use it for free. Please remember to cite the following papers to acknowledge people building the software: P. Giannozzi et al. , J. Phys. Condens. Matter 21 , 395502 (2009). C. Gougoussis, M. Calandra, A. P. Seitsonen and F. Mauri, Phys. Rev. B 80 , 075102 (2009) M. Taillefumier, D. Cabaret, A. M. Flank and F. Mauri, Phys. Rev. B 66 , 195107 (2002)

Outline 1. 1. Introduction on XAS Introduction on XAS 2. 2. The PAW theory The PAW theory 3. 3. Hands on (I) : Si-K edge in � -quartz SiO 2 Hands on (I) : Si-K edge in � -quartz SiO 2 -GIPAW pseudopotential generation for Si and Si1s * -GIPAW pseudopotential generation for Si and Si1s * -SCF calculation for � -quartz SiO 2 unit cell without/with core-hole -SCF calculation for � -quartz SiO 2 unit cell without/with core-hole -core wave-function extraction -core wave-function extraction 4. 4. The Lanczos algorithm The Lanczos algorithm 5. 5. Hands on (II) : Si-K edge in � -quartz SiO 2 Hands on (II) : Si-K edge in � -quartz SiO 2 - XSpectra calculations for � -quartz SiO 2 unit cell without/with core-hole - XSpectra calculations for � -quartz SiO 2 unit cell without/with core-hole - Supercell calculation : 2x2x2 cell for � -quartz SiO 2 - Supercell calculation : 2x2x2 cell for � -quartz SiO 2 6. 6. Hands on (III) : Dipolar and Quadrupolar contributions at the Ni-K edge in NiO Hands on (III) : Dipolar and Quadrupolar contributions at the Ni-K edge in NiO 7. 7. Summary Summary

The PAW (Projected Augmented Wave) method We need to reconstruct the all electron (AE) states from pseudo (PS) states (Note : AE states do not mean many-electron states !!!) The problem arises for in the absorption cross-section, as we only | ψ f i get from the PP-PW calculation. is an atomic core-state. | ˜ | ψ i i ψ f i We admit that a linear transformation exists, such that : T mapping | ψ i = T | ˜ | ˜ ψ i ψ i P. E. Blöchl, Phys. Rev. B 50 , 17953 (1994)

The PAW (Projected Augmented Wave) method T is linear differs from identity in the core (augmentation) regions only : T Ω R ⇣ ⌘ X X | φ R ,n i � | ˜ T = 1 + T R = 1 + φ R ,n i h ˜ p R ,n | R R ,n sum over atomic sites, ( ` , m ) and eventually, a channel index j all electron partial waves A B pseudo partial waves Ω R B Projection functions (one per channel) ( ` , m ) Ω R (i) are 0 outside | ˜ p R ,n i R B C p R ,n | ˜ Ω R h ˜ φ R 0 ,n 0 i = δ RR 0 δ nn 0 (ii) Inside , D X p R ,n ih ˜ or | ˜ φ R ,n | = 1 n Notes : (i) and coincide outside | φ R ,n i | ˜ Ω R φ R ,n i (ii) A natural choice for are the solution of the radial Schrödinger equation for the | φ R ,n i isolated atom

Cross-section in the PAW formalism | ψ f i = T | ˜ Using , we get : ψ f i h ψ f |O| ψ i i = h ˜ X h ˜ X h ˜ p R ,n ih ˜ ψ f |O| ψ i i + ψ f | ˜ p R ,n ih φ R ,n |O| ψ i i � ψ f | ˜ φ R ,n |O| ψ i i R ,n R ,n Here, is localized on the site of the absorbing atom so that all matrix elements will be | ψ i i R 0 negligible except for R = R 0 h ψ f |O| ψ i i ⇡ h ˜ X h ˜ X h ˜ p R 0 ,n ih ˜ ψ f |O| ψ i i + ψ f | ˜ p R 0 ,n ih φ R 0 ,n |O| ψ i i � ψ f | ˜ φ R 0 ,n |O| ψ i i n n X p R ,n ih ˜ Using , the first and last terms cancel, so that : | ˜ φ R ,n | = 1 n X h ˜ h ψ f |O| ψ i i ⇡ ψ f | ˜ p R 0 ,n ih φ R 0 ,n |O| ψ i i n X Defining , the cross-section finally reads : | ˜ ϕ R 0 i = | ˜ p R 0 ,n ih φ R 0 ,n |O| ψ i i n | h ˜ X σ ( ω ) = 4 π 2 α 0 ~ ω ϕ R 0 i | 2 δ ( E f � E i � ~ ω ) ψ f | ˜ f M. Taillefumier et al. , Phys. Rev. B 66 , 195107 (2002)

Practical PAW for XAS In the expression : X | ˜ ϕ R 0 i = | ˜ p R 0 ,n ih φ R 0 ,n |O| ψ i i n The sum runs over a complete set, i.e. an infinite number of projectors !! In practice a finite number of projectors is enough : 1 projector/channel ( ) • ` generally yields wrong intensities wrong dipole/quadrupole ratio 2 projectors/channel ( ) • ` correct intensities in the near edge region (~ 50 eV above the edge, in most of the cases) need to be linearly independent (i.e. , span a 2 x 2 subspace) Finally, we need the core WF without the core-hole and the AE partial-waves (from PP generation) to fully determine | ˜ ϕ R 0 i Let’s go for an example : the Si and the Si1s* ultrasoft PPs

Outline 1. 1. Introduction on XAS Introduction on XAS 2. 2. The PAW theory The PAW theory 3. 3. Hands on (I) : Si-K edge in � -quartz SiO 2 Hands on (I) : Si-K edge in � -quartz SiO 2 -GIPAW pseudopotential generation for Si and Si1s * -GIPAW pseudopotential generation for Si and Si1s * -SCF calculation for � -quartz SiO 2 unit cell without/with core-hole -SCF calculation for � -quartz SiO 2 unit cell without/with core-hole -core wave-function extraction -core wave-function extraction 4. 4. The Lanczos algorithm The Lanczos algorithm 5. 5. Hands on (II) : Si-K edge in � -quartz SiO 2 Hands on (II) : Si-K edge in � -quartz SiO 2 - XSpectra calculations for � -quartz SiO 2 unit cell without/with core-hole - XSpectra calculations for � -quartz SiO 2 unit cell without/with core-hole - Supercell calculation : 2x2x2 cell for � -quartz SiO 2 - Supercell calculation : 2x2x2 cell for � -quartz SiO 2 6. 6. Hands on (III) : Dipolar and Quadrupolar contributions at the Ni-K edge in NiO Hands on (III) : Dipolar and Quadrupolar contributions at the Ni-K edge in NiO 7. 7. Summary Summary

Practical details cp -r ~/TutorialXSpectra /scratch/ cd /scratch/TutorialXSpectra/ Directory structure: ./pseudo / pseudopotentials for this tutorial input files necessary to generate a GIPAW pseudopotential for Si ./PPgeneration/ input files to be modified for the examples ./SiO2 ./SiO2h ./SiOsupercell ./NiO ./*/outdir/ tmp output reference inputs and outputs ./solutions/ relevant papers, the .pdf of these lecture and manual INPUT_XSPECTRA ./references/

Calculation flow for XAS with XSpectra Prepare the GIPAW pseudopotentials Extract the core wavefunction Prepare input file and run SCF calculation Prepare input file and run XSpectra

GIPAW (Gauge independent PAW pseudopotentials) The GIPAW pseudopotential includes all the reconstruction information needed to run XSPECTRA needed for the absorbing atom only (non-absorbing atoms accept any kind of pseudopotential) contains the following information on the absorbing atom: the core wavefunction without hole the all electron partial waves atomic states the Blöchl projectors can be obtained with the atomic code ld1.x