j. j. rehr department of physics, university of washington...
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Real-space Green's function approach for electronic, vibrational and optical properties
J. J. RehrDepartment of Physics,
University of WashingtonSeattle, WA, USA
Supported by DOE BES and NIST
Excitations in Condensed Matter: From Basic Concepts to Real Materials
KITP, UCSB October 5, 2009 - December 18, 2009
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Real-space Green's function approach
• GOAL: ab initio theory of electronic, vibrational, and optical properties
• “Pretty good theory”• Broad spectrum VIS – X-ray• Accuracy ~ experiment
• TALK:I. RSGF approachII. Parameter free theory No adjustable parameters
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”The chance is high that the truth lies in thefashionable direction. But on the off chance that it isn’t, who will find it?”
R. P. Feynman
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Experiment vs Theory: Full spectrumOptical - X-ray Absorption Spectra
Photon energy (eV)
fcc Al
UV X-ray
http://leonardo.phys.washington.edu/feff/opcons
RSGF theory vs
expt
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J. J. Rehr & R.C. AlbersRev. Mod. Phys. 72, 621 (2000)
http://leonardo.phys.washington.edu/feff/
I. RSGF Approach
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Physical Considerations in XAS
X-ray energy ω (eV)
XANES EXAFS
EF
XANES EXAFSKE: low HighScattering: Strong weakMFP: long ShortDamping: weak StrongRange: LRO (k-space) SRO (real-space)
ωp
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Key Many body effects in XAS
● Self-energy Σ(E) complex● Core-hole effects ? Screening ?
● Debye-Waller factors e-2σ2 k2
● Multi-electron excitations satellites
Σquasi-particle & beyond
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Ground-state vs Quasiparticle vs Expt
Quasiparticle Theory Essential
Cu
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Conventional Quasi-particle Theory of XAS
Fermi Golden Rule for XAS μ(ω)
Quasi-particle final states ψf with core hole
Final state hamiltonianV′coul = Vcoul + Vcore−hole
Non-hermitian Self-energy Σ(E) (replaces Vxc)
Inelastic Mean free paths λ= k/|Im Σ(E)| ≈ 5 − 20 Å
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Green’s functions vs Wave-functions
Golden rule via Green’s Functions G = 1/( E – h – Σ )
Golden rule via Wave Functions
Ψ
Paradigm shift:
Efficient! No sums over final states
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Connection with Electronic Structure
• Spectral function ρ(E,r,r’) = - Im < r |G (E) r’ >
• Density matrix ρ(r, r’) = ∫ EF ρ(E,r,r’) dE
• Density ρ(r) = ∫ EF ρ(E,r,r) dE
• lDOS ρR,L(E) = <R,L| ρ(E,r,r) |R,L>
DFT GFT
Density operator ρ (E) = - Im G (E)
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Real-space Green’s Function Formalismfull-multiple scattering vs MS path expansion
“Real-space KKR”
G0 free propagators t-matrix = ei δl sin δl δRR’δll’
Ingredients:
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Scattering state representation of G
RL (r,E)= Rl(r,E) YL (ȓ)Angular momentum-site basis
GLn,L’n’ (E)= (eik Rnn’ /Rnn’ )Σss’ Yls Y l’s’
Matrix elements - separable representation*
*JJR + R. Albers Phys. Rev. B 41, 8139 (1990)
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Relativistic RSGF
2 Steps
1) Production
Dirac-Fock atomic theory |< f | d | I >|
2) Scattering G= G0 + G0 T GNon-relativistic, no spin-flips, …
*JJR + A. Ankudinov PRB 56, R1712 (1997)
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Implementation: FEFF8Real Space Green’s Function code
BNCore-hole, SCF potentials
Essential!
89 atom cluster
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FAST! Parallel Computation FEFFMPI
MPI: “Natural parallelization”
Each CPU does few energies
Lanczos: Iterative matrix inverse
Smooth crossover between
XANES and EXAFS!1/NCPU
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Example 1: XAS of Pt Pt L3-edge Pt L2-edge (S. Bare, UOP)
• Good agreement: Relativistic FEFF8 code reproduces all spectral features, including absence of white line at L2-edge.
• Self-consistency essential: position of Fermi level strongly affects white line intensity.
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No peak shift!
Path Expansion 15 paths
Rnn= 2.769 fcc Pt
*Theoretical phases accurate distances to < 0.01 Å
χ(R)
R (Å)
Example 2: Pt EXAFS
Phase Corrected EXAFS Fourier Transform *
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Example 3: Electron energy Loss spectra (EELS)
fcc Al
Photon energy (eV)UV X-ray
-Imε-
1
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Example 4: X-ray Raman Spectra (NRIXS) FEFFq
J.A.Soininen, A.L. Ankudinov, JJR, Phys. Rev. B 72, 045136 (2005)
Finite mom transfer q
feff(q,k)
~ Im ε-1(q,ω)
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Expt
BSE*
feff(q)
*
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pDOS vs XES and XANES
Example 5: lDOS and X-ray Spectra
Angular momentum Projected DOS
XANES
XESpDOS
Fermi energy EF Final state electron energy E
Cu
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II. Parameter Free RSGF Theory*
A. Ab initio self-energies and mean free path
B. Multi-electron excitations S02
C. Core-hole and local field effects BSE/TDDFT
D. Ab initio Debye Waller factors
E. Full potential corrections
GOAL: eliminate adjustable parameters
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II. Parameter free RSGF theory
JJR et al., Comptes RendusPhysique 10, 548 (2009)
in Theoretical SpectroscopyL. Reining (Ed) (2009)
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A. Self-energy and Inelastic Losses
• GW Approximation (Hedin 67)
• Screened Coulomb Interaction W
? How to appproximate ε-1 (ω) ?
Need: complex, energy dependent Σ(E)
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Dielectric functionEnergy Loss (EELS)Absorption coefficientRefractive indexReflectivityX-ray scattering factors f = f0 +f1 + if2
Hamaker constants ε(iω)
RSGF approach: VIS-UV optical constants
Long wavelength limit q = 0 + extrapolation to finite q
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FEFF8OP*
*WWW: M. Prange http://leonardo.phys.washington.edu/feff/opcons/
-Im
ε-1
Cu
PRB 80, 155110 (2009)
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Optical Constants FEFFOP vs DESY Tableshttp://www.leonardo.washington.edu/feff/opcons
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Theoretical check: ε2 Sumrule
fcc AlZ=13 13
Photon energy (eV)
neff
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Many-pole Self-energy Algorithm*
Plasmon-pole model many-pole model
-Im ε-1(ω) Many-pole Dielectric Function
~ Σ i gi δ(ω - ω i)
Many-pole GW self-energy Σ(E)
* J. Kas et al. PRB 76, 195116(2008)
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Cu
ab initio q=0 loss function Sum of single poleself-energies
http://leonardo.phys.washington/feff/opcons
Example: Many-pole model for Cu*
Plasmon pole model
Many-pole pole model
*J. Kas et al., Phys. Rev. B. 76, 195116 (2007)
-Im ε-1(ω)
Cu
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Many-pole vs Single Plasmon Pole Self-energy Models
●●●●
Σ =Σ(E) indep of r
Σ =Σ(E,k) Full GW (Lanczos)
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ab initio Inelastic Mean Free Paths
Plasmon-pole model many-pole model
Plasmon-pole model has too much loss!Cu
Inel
astic
mea
n fre
e pa
th
J. Kas et al. 76, 195116 (2008)
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Self-energy Effects in Cu K-edge XAS
J. Kas Phys. Rev. B 76, 195116 (2007)
Σ(E) MPSE Self energy
Re Σ(
E)
Plasmon pole
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Aposteriori Self-energy corrections toDFT and GW codes
µ(E
) (ar
bitra
ry u
nits
)
E (eV) PARATEC: D. Cabaret et al. (2008)
MPSE
MgAl2O4 Spinel
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B. Intrinsic losses: Multi-electron Excitations
Multi-electron excitations
→ satellites in A(k,ω)
Energy Dependent Spectral Function A(k,ω)
Beyond quasiparticles!Quasi-boson Model
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Quasi-Boson Theory of Inelastic Loss*
Excitations - plasmons, electron-hole pairs ... are bosons
*W. Bardyszewski and L. Hedin, Physica Scripta 32, 439 (1985)
“GW++” Same ingredients as GW self-energy Vn → -Im ε-1(ωn,qn) fluctuation potentials
Many-body Model: |e- , h , bosons >
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Effective GW++ Green’s Function geff(ω)
Damped qp Green’s function
Extrinsic + Intrinsic - 2 x Interference
geff(ω)=
Spectral function: A(ω) = -(1/π) Im geff(ω)
L. Campbell, L. Hedin, J. J. Rehr, and W. Bardyszewski, Phys. Rev. B 65, 064107 (2002)
+ - -
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Effect on Spectra: Amplitude reduction
• XAS = Convolution with spectral function A(ω, ω’)
• Explains crossover: adiabatic
to sudden approximation
≈ μqp(ω) S02
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Many-body amplitude reduction in EXAFSQuantitative R-space EXAFS
S02~0.9
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C. Core-hole and local field effects BSE/TDDFT
Two step approach to TDDFT/BSE
χ = (1 – K χ0) -1 χ0 K = V + W
1) χ’ = (1 – V χ0) -1 χ0 χ ~ RPA response
2) χ = (1 – W χ’ ) -1 χ’ W ~ RPA core hole
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Step 1: Local Field Effects
• Transition operator not external x-ray field φext
Must include polarization φ = φext + φinduced
= ε-1 φext
ε = 1 – K χ0 Dielectric matrixK = V RPA (or V+ fxc for TDDFT)
~→ Golden rule with screened matrix elements* M
*A. Zangwill + P Soven, Phys Rev A 21, 1561 (1980)
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Example: local field effects in XAS TDDFT
*Data: Z. Levine et al. J. Research. NIST 108, 1 (2003)
TDDFT
FEFF8
Data*No adjustable parameters
W
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Step 2. Screened Core Hole
Corrections to ad hoc core-hole approxfinal state rule, Z+1, half-core hole …
Stott-Zaremba algorithm (= static RPA) W = ε-1 Vch
Approx: Include W in g’ = 1/( E – h’ – Σ )as in quasi-boson model
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Screened core-hole potential W
0 1 2 3R (bohr)
-6
-4
-2
0
Pote
ntia
l (H
artre
e)
RPA – Stott ZarembaFully screened FEFF8
Unscreened
Tungsten metal (Y. Takimoto)
RPA
W
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( ) ωωβωρμ
σ di 2
coth0
22 ∫∞
=
( ) ( ){ }recursion Lanczos step6
22
−=
−= ii QDQ ωδωρ
D. Ab initio XAS Debye Waller Factors e-2σ2 k2
Replaces correlated Debye and Einstein Models !
= VDOS
D = Dynamical matrix
Phys. Rev. B 76, 014301 (2007)
Ψ
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Ab initio DFT Phonon Green’s function
G = (ω2 – D)-1
( ) ⎩⎨⎧
∂∂∂
=ssian03ABINIT/Gau from
Matrix Dynamical1
''
2
21'
'',βα
βαljjljj
ljjl uuE
mmD
NB: Need functional for both bond lengths and phonons:“hGGA” = “half and half PBE”
expt
expt
Strong correlation between lengths a and <ω>
GGALDA
hGGA
oo
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Lanczos Recursion (Many-pole)
Total VDOS of Cu
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XAFS Debye-Waller Factor of Ge
Expt: Dalba et al. (1999)
F. Vila et al.
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Ab Initio Debye-Waller Factors in Rubredoxin
Convert Full System into Smaller Model
Path Theory Exp.Fe-S1 2.9 2.8±0.5
Fe-S2 2.9 2.8±0.5
Fe-S3 3.3 2.8±0.5
Fe-S4 3.5 2.8±0.5
σ2 (in 10-3 Å2)
Fe
S
SS
S
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Interface with ORCA FP-SCF code (J. Kas)
ORCA FEFF-FP
ElectrostaticPotential
Density XANES
F. Full Potential Multiple Scattering* In progress
*A. L. Ankudinov and J. J. Rehr, Physica Scripta, T115, 24 (2005)
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CONCLUSIONS
• Parameter free ab initio RSGF approach forelectronic & vibrational structure &spectra: VIS - X-ray, EELS, NRIXS, …
● Attractive alternative to k-space approach
● Efficient relativistic all electron code FEFF9
Ψ
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Acknowledgments
• J. Kas (UW)• F. Vila (UW)• K. Jorissen (UW)• M. Prange (ORNL,UW)• A. Sorini (SSRL/SLAC,UW)• Y. Takimoto (ISSP,UW)
Collaborators
Supported by DOE BES & NIST
A.L. Ankudinov (APD)R.C. Albers (Los Alamos))Z. Levine (NIST)E. Shirley (NIST)A. Soininen (U. Helsinki) H. Krappe (HZB)H. Rossner (HZB)
Rehr Group
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That’s all folks