Real-space multiple-scattering theory of EXAFS and XANES
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1 Nordita School on Photon-Matter Interaction Stockholm, Sweden Oct 3-7, 2016 Real-space multiple-scattering theory of EXAFS and XANES J. J. Rehr, J. J. Kas and F. D. Vila
2 Outline Goals: -Real-space multiple-scattering (RSMS) Theory aka Real-space Green s function (RSGF) theory -Implementation of RSMS in FEFF Key approximations and limitations Effects of structure and disorder A few Advanced methods
3 Full spectrum XAS: Expt. Vs Theory fcc Al UV x-ray
4 The devil is in the details: edges, fine-structure fcc Al UV x-ray
5 Historical interpretation of EXAFS* *Stern Sayers Lytle, UW 1971 Short range order theory EXAFS EXAFS Fourier Transform Cu Shifted Radial Distribution shift X-ray Microscope! R nn BUT need to calibrate experiment with Standard
6 EXAFS Theory Quantitative theory of EXAFS: Theory behind FEFF6 & general summary J. J. Rehr & R.C. Albers Rev. Mod. Phys. 72, 621 (2000)
7 Advances in Theory FEFF9 Update of Rehr & Albers: Advanced techniques and ab initio treatment of many-body effects
8 RSGF in the hierarchy of spectroscopy methods sophistication Atomic models: e.g. de Groot. Atomic cross-sections, multiplet s theory with fitted parameters, crystal field model Hamiltonians DFT (Density Functional Theory): WIEN2k, ABINIT, VASP, CASTEP, StoBe, Orca, Accurate for ground-state properties, not reliable for excited states, Delta-SCF Final State Rule with core-hole Quasi-particle Green s Function Theory: FEFF9 Appropriate for excited states, NOT full potential BSE (Bethe-Salpeter Equation): Exc!ting, OCEAN, AI2NBSE. Accurate but demanding. Less user friendly. Misses excitations & satellites QC methods: MRCI, MRCC, CASPT2, QMC, etc, highly accurate but completely intractable
9 FEFF development philosophy Pretty good spectra Advantages: Real-space Fully relativistic, all-electron Semi-automated, user-friendly, easy to use Built for EXAFS and related x-ray spectroscopies Applicable to materials throughout the periodic table Disadvantages: Not always the best tool: Spherical potentials can lose accuracy near edges Quasi-particle theory only ignores multiplets, satellites
10 FEFF quantitative XANES theory in one diagram Inelastic Losses Self-energy Σ Real-Space Green s Function x-ray + Screened core-hole
11 Can you write an equation for the theory? P.A.M. Dirac
12 Answer: Exact EXAFS Equation* Effective Scattering Amplitude f eff S 0 2 Many body amplitude factor EXAFS measures local structure & disorder Mean free path λ k σ 2 Mean square vib amplitude Distance R Coordination N Disorder σ 2 *JJR, RC Albers, CR Natoli, EA Stern, Phys Rev B34, 4350 (1986)
13 BUT: need many parameters! Question: Can the EXAFS parameters k f eff Φ k σ 2 λ k S 0 2 be calculated theoretically?
14 FEFF: Many-body effective single particle Many-body Fermi s Golden Rule XAS absorption coefficient Effective Single particle Fermi s Golden Rule
15 FEFF: From sum-over-states to Green s function Effective Single particle Fermi s Golden Rule
16 FEFF: From sum-over-states to Green s function Effective Single particle Fermi s Golden Rule Density Matrix
17 FEFF: From sum-over-states to Green s function Effective Single particle Fermi s Golden Rule Density matrix from Green s function
18 FEFF: From sum-over-states to Green s function Effective Single particle Fermi s Golden Rule Substitute sum over final states with Green s function
19 FEFF: Local basis and matrix elements
20 What s a Green s function? Wave function in QM H Ψ = E Ψ Ψ(r) = Amplitude to find particle at r Green s function (H E) G = - δ(r-r ) G(r,r,E) = aka Propagator = Amplitude to go from r to r
21 FEFF: Local basis and matrix elements Insert complete set of states
22 FEFF: Local basis and matrix elements Insert complete set of states
23 FEFF: Local basis and matrix elements Insert complete set of states Matrix elements Green s Function matrix
24 Getting G: Multiple Scattering Theory Dyson s equation: Iterating:
25 Getting G: Multiple Scattering Dyson s equation: Iterating:
26 Getting G: Multiple Scattering Dyson s equation: Iterating: Atomic pot. partition
27 Getting G: Multiple Scattering Dyson s equation: Iterating: Atomic pot. partition Site scatt. matrix
28 Getting G: Multiple Scattering Dyson s equation: Iterating: Atomic pot. partition Site scatt. matrix
29 Getting G: Multiple Scattering
30 Getting G: Multiple Scattering Central atom contrib. EXAFS
31 Getting G: Multiple Scattering Central atom contrib. EXAFS Graphically: Path expansion
32 Getting G: Full Multiple Scattering
33 Getting G: Full Multiple Scattering Total scatt. matrix
34 Getting G: Full Multiple Scattering Total scatt. matrix Sum and invert XANES
35 Implementation: FEFF Code 89 atom cluster Core-hole, SCF potentials Essential! BN
36 Example: Pt EXAFS path expansion Phase Corrected EXAFS Fourier Transform * Rnn= fcc Pt χ(r) Path Expansion 15 paths No peak shift! R (Å) *Theoretical phases accurate distances to < 0.01 Å
37 Example: Pt XANES full multiple-scattering Pt L 3 -edge 1.4 Pt L 2 -edge (S. Bare, UOP) Normalized Absorption FEFF calculation Experiment' Normalized Absorption FEFF calculation Experiment PtL3_xmu '98feb002_xmu' PtL2_xmu '98feb004_xm 0.0 PtL3edge Photon Energy, ev PtL2edge Photon Energy, ev Good agreement: Relativistic FEFF8 code reproduces all spectral features, including absence of white line at L 2 -edge. Self-consistency essential: position of Fermi level strongly affects white line intensity
38 Green s Functions and Parallel Computation Energy E is just a parameter! 1/N CPU Natural parallelization Each CPU does one energy
39 Self-consistent Densities and Potentials Spectrum: Golden Rule
40 Key approximations in FEFF Dirac-Fock relativistic atomic states ; semi-relativistic scattering states Spherical overlapped muffin-tin potentials: Huge simplification of the problem Quasi-particle approximation: Electron propagates in lossy medium Approximate self-energy Core-hole treatment: RPA or DFT-Screened core-hole
41 The muffin-tin potential Scattering potential partition
42 The overlapped muffin-tin potential Better density: Resembles bonding Charge redistribution Overlap Region Nuclei Muffin-tin potential
43 FEFF Density of States
44 FEFF electron density
45 Disorder and Debye-Waller factors in XANES DW factors: Crucial for EXAFS Very little effect in XANES region Can be included anyway in single-scattering approx. Both ab initio and model forms Disorder: Can be crucial in XANES Need external input for FEFF simulations MD trajectories MC sampling
46 Quick intro to Ab Initio DW factors Multiple Scattering Path XAFS DW Factor Average commonly expressed in terms of the cumulant expansion FT of Ge EXAFS χ(k) Leading cumulants χ(r) 2 σ (1) R + σ R (Å) J. Kas et al. (2007)
47 EXAFS near-neighbor DW Factor of Cu CD (Correlated Debye): Standard FEFF LDA, hgga: Ab initio DW Isotropic bonding: Good CD results Expt: Fornasini et al. (2004)
48 EXAFS near-neighbor DW Factor of Ge CD (Correlated Debye): Default in FEFF LDA, hgga: Ab initio DW Directional bonding: Needs AIDW Expt: Dalba et al. (1999)
49 Ab Initio DW factors: Lanczos algorithm XAFS DW Factor for path R: VDOS expressed as imaginary part of the phonon propagator Seed state: Displacement along path Dynamical Matrix: Calculated using ab initio methods (abinit, Gaussian, VASP, etc)
50 Ab Initio DW Factors in Metal-Ligand Complexes Ru(bpy) 2 (AP)(H2O) ++ N(AP) Good agreement for tight ligands (bpy) Useful agreement for weak ligands (AP and H2O) Still within error margin N(bpy) Ru O R M-L (in Å) σ 2 (in 10-3 Å 2 ) Path Theory Exp Theory Exp Ru-N(bpy) ± ± Ru-N(AP) ± ±3 Ru-O ± ±7 Expt: Salassa et al., J. of Physics: Conference Series 190, (2009)
51 Beyond DFT: Quasi-particle Self-Energy Effects Quasi-particle (QP) effects: BN 89 atom cluster Ground state potential: Usually insufficient Need QP effects and SCF potentials
52 Improvements to the theory: key many body effects Treatment of the core hole: Screening DFT or RPA Chemical shifts Self-energy approximations: Need more than single-pole self-energy Many-body effects: Charge transfer excitations: Transition metal oxides, cuprates,
53 Core-hole issues: RPA Screened Core-hole Linear response: Comparison of the core-hole in H 2 O
54 Core-hole issues: Chemical shifts N in NH4 + and NO3 - : Extreme case of chemical shift (4.3 ev)
55 Self-energy : HL plasmon-pole model Based* on GW approx.: W W: Screened Coulomb interaction G Dielectric function: 1 st approximation: single plasmon pole *B.I. Lundqvist Phys. Kondens Materie, 6, pp. 206, (1967)
56 Self-energy model: many-pole model Based* on GW approx.: W W: Screened Coulomb interaction G Dielectric function: Key ingredient Usually external source FEFF OPCONS: Semi-quantitative approximation J. Kas et al., PRB 76, (2007)
57 Self-energy issues: Many-pole model Loss Function Cu Many-pole (full) better than Hedin-Lundqvist (dashed) vs even better theory (dot-dashed) Kas et al., PRB 76, (2007)
58 Other FEFF capabilities Quick overview of other FEFF capabilities: XES RIXS Compton Profiles Reciprocal space: impurity GF model Hubbard U method
59 Non-resonant X-ray Emission (XES) RDX (High explosive) Vila et al., J. Phys. Chem. A 2011, 115, 3243
60 Compton Profiles
61 Resonant Inelastic RIXS X-ray and Scattering COMPTON (RIXS) TiO 2 (Ti Kα) FEFF Expt.
62 Reciprocal space capability Faster/better for some crystals No supercell needed for core hole Use cif file for input: ** feff.inp CIF w-gan.cif KMESH 1000 CONTROL GaN: N K edge EELS, 001 orientation Still experimental Not fully tested K. Jorissen et al. Phys. Rev. B 81, (2010)
63 Strongly correlated systems: Hubbard GW+U U calculated using constrained RPA within RSMS Not yet fully tested (Nearly) parameter free MnO Phys. Rev. B (2012)
64 Other references:
65 Further information The FEFF Project website: URL: feffproject.org The FEFF Users Guide: URL: feffproject.org/feffproject-feff-documentation.html Developers contact: URL: feffproject.org/feffproject-contact.html
66 Summary Take away messages Know the basics of RSMS/RSGF theory Understand the key approximations in FEFF Know some of FEFF s advanced capabilities
67 The FEFF group: Seattle and beyond From left to right: Ken Nagle Yoshi Takimoto Kevin Jorissen Towfiq Ahmed Hadley Lawler Aleksi Soininen Fernando Vila Adam Sorini Alex Ankudinov Micah Prange John Vinson (Shauna Story) John Rehr Josh Kas (Egor Clevac)
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