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Calculations of X-ray Spectra in Real-space and Real-time
J. J. Rehr, F. Vila, Y. Takimoto
Department of PhysicsUniversity of Washington
Seattle, WA USA
Time (s)
X-Ray Science in the 21st Century
KITP, UCSB Aug 2-6, 2010
Goals: Real-space & Real-time response beyond linear response & harmonic approx
Talk: Two approaches:
• I. Linear & Non-linear Response RT-TDDFT
• II. Real space & time XAS of non-equlibrium systemFinite Temperature DFT/MD + Real-Space Green’s Function XAS
4
Calculations of X-ray Spectra in Real-space and Real-time
``If I can’t calculate it,
I don’t understand it.”
R.P. Feynman
I. Real-Space & Real-Time Linear and Non-linear Response
• Difficulty: frequency-space is computationallydemanding - too-many excited states
• Strategy: extend RT-TDDFT/ SIESTA approach**Sanchez-Portal, Tsolakidis, and Martin, Phys. Rev. B66, 235416 (2002)
5
Approach I: RT-TDDFT
6
J. Chem. Phys. 127, 154114 (2007)
RT-TDDFT Formalism
• Yabana and Bertsch Phys. Rev. B54, 4484 (1996)
• Direct numerical integration of TD Kohn-Sham equations
• The response to external field is determined by applying atime-dependent electric field ΔH(t) = −E(t)·x.
• Optical properties determined from total dipole moment:
8MORE EFFICIENT THAN FREQUENCY –SPACE METHODS !
Numerical Real-time Evolution
• Ground state density ρ0, overlap matrix S, and H(t) at each time-step evaluated with SIESTA
• Crank-Nicholson time-evolution: unitary, time-reversibleStable for long time-steps !
• Adiabatic GGA exchange-correlation (PBE) functional
Coefficients of Orbitals
10
__
, t = t + Δ t/2 _
Real time Linear Response
Induced Dipole Moment
Linear Response Function
Optical Absorption
Linear Dielectric Function
11
Time (fs)
Dip
ole
p z(t)
(a.u
.)
• Delta Function (Unit Impulse at t=0)
• Step Function (Turn-off Constant E at t=0)
Example: CO Linear Response pz(t) response due to applied Ez(t)
Time (fs) Energy (eV)
Im α
(ω)
Re α
(ω)
12
E(t)
E(t)
0
0Ground state without field
Ground state with constant field
Evolution for t>0
Evolution for t>0
Example: Small molecule p-Nitroaniline (pNA)
• Linear absorption
• Sum rule
Energy (eV)
Energy (eV)
fsum
Ele
ctro
n C
ount
s Abs
orpt
ion
(au)
(in chloroform)
Total 52 valence electrons
13
Nonlinear Polarizabilities
• Second order nonlinearities
Second Harmonic Generation (SHG)
Optical Rectification (OR)
Electro-Optic effect (Pockel’s effect) 7
Extraction of Static Nonlinear Polarizabilities
• Standard technique: static nonlinearity
Finite-difference or polynomial fitting pi(E) e.g.,
16
Example: Static CHCl3 Hyperpolarizability*
*J. Chem Phys 133, 034111 (2010)
3 methods
Difficult case: β very small!
All agree with large
diffusebasis sets!
Local Response Densities
Non
-line
ar R
espo
nse
Hyp
erpo
lariz
abili
ty
GTO RS
Note: Contributions from Cl and HC are ofopposite sign – Explains smallness of β
Real time Dynamic Nonlinear Response
• The nonlinear expansion in field strength
• Accounting for time lag in system response
How can we invert the equation to get nonlinear response function?15
?
Dynamic Nonlinear Polarizabilities
• Set Ej(t) = F(t)Ej and define expansion pi(t)
where p(1) yields linear response, p(2) first non-linear (quadratic) response, ….
• Quadratic response χ(2)
17
Time (fs) Time (fs)
p(1) ij
p(2) ijk
Time (fs)
Frequency (eV) Frequency (eV)
Re
F(ω
)
Im F
(ω)
Time (fs)
Dynamic Nonlinear Response with Quasi-monochromatic Field Fδ(t)
• Sine wave enveloped by another sine wave or Gaussian
SHG
OR
F(t)
Linear andNonlinearresponseof CO
18
Real time vs Frequency spaceNonlinear Response
• Operation cost– Sternheimer equation (frequency space)
– Real time
• Memory cost– Sternheimer equation (frequency space)
– Real time
19
Example pNA: Nonlinear SHG
• Comparison with other methods
Energy (eV)
β k(-
2ω,ω
,ω) (
au) Expt.
25
PBE
Extension to high fields: High Harmonic Generation in Ar
Dip
ole
Res
pons
e
Pulse Shape
RT-TDDFT High Harmonic Generation in ArOdd Harmonic Magnitude
II. Real-space & Real-time calculations of X–ray Response*
*Phys Rev B, Rapid Commun. 78, 121404(R), (2008)
Real-space Green’sFunction theory
XAS, XES, IXS, XMCD…
FEFF9
JJR et al., Comptes RendusPhysique 10, 548 (2009)
in Theoretical SpectroscopyL. Reining (Ed) (2009)
Paradigm shift:
Use Green’s functions not wave functions!
Efficient!
Ψ
FAST! Parallel Computation FEFFMPI
MPI: “Natural parallelization”
Each CPU does few energies
Lanczos: Iterative matrix inverse
1/NCPU
Experiment vs Theory: Full spectrumX-ray Absorption Spectra (XAS)
Photon energy (eV)
fcc Al
UV X-ray
arXiv:cond-mat/0601242
http://leonardo.phys.washington.edu/feff/opcons
theory vs expt
Pt10 Cluster on [110] γ-Al2O3
Example: Finite T Nano-scale Pt Clusters
metallic Pt oxidized PtAl O
Alternative to conventional paradigm!
MYSTERY: Unusual properties of
Pt10 /γ-Al2O3
Negative thermal expansion,
large disorder, …
Goals: Understand structure
Explain all properties
Method: Real-time DFT/MD
Energy (eV)11550 11560 11570 11580 11590
Nor
mal
ized
Abs
orpt
ion
0.0
0.2
0.4
0.6
0.8
1.0
1.2
165 K200 K293 K423 K573 K
4 Red shift
3 Enhanced σ2
1 H bond expansion 2 NTE
4 Anomalies
*Kang, Menard, Frenkel, Nuzzo., JACS Commun. 128, 12068 (2006)
Experimental Observations (X-ray Absorption Expt)*
XAS
3.0 4.0 5.0 6.0 7.0 8.0Time (ps)
2.56
2.58
2.60
2.62
2.64
R(P
t-P
t) (
Å)
165 K573 K
Calculation – Finite-T DFT/MD10 atom Pt/ γ-Al2O3
Mean nn distance RPt-Pt
2500 3 fs steps
~ 104 cpu-hrs (VASP)
NTE
time-elapsed rendering
Non-equlilibrium Finite temperature
Computational Details
DFT/MDVASPPBE Functional396 eV Cutoff3 fs Step3 ps Equilibration5 ps Runs (3)165 K & 573 K
XASFEFF8Full Multiple Scattering32 Configurations from MD7 Å Clusters (~150 atoms)
Prototypical Pt10cluster
on [110] surface of γ-Al2O3
Bond expansion in H2atmosphere
2.534
2.563
2.658
2.529
2.589
2.559
Adding H increases bond lengthBond expansion in H2atmosphere
Adding H increases bond lengths
Negative Thermal Expansion
2.585 Å
2.596 Å- 0.011 Å
(- 0.027 Å expt)
R
2.5 3.0 3.5 4.0 4.5
Pt-Pt Distance (Å)
0
5
10
15
Pair
Dis
trib
utio
n Fu
nctio
n
165 K573 K
2.4 2.6 2.8 3.0 [ ]2)( 1)( 0 −=Φ −− rreDr αβ
)()( rAerg Φ−=
Morse-potential Fits to PDFs
PtPtNote: increased low r width at HTimplies PDF is non-vibrational.
Φ effective pairpotential
g(r)
High Pt-Pt Disorder
10×10-3 Ų (10×10-3 Ų)
5×10-3 Ų (8×10-3 Ų)σ
Center of Mass MotionPhysical Interpretation
Librational motionof center of mass
Period ~ 2 psAmplitude ~ 1 Ǻ
Hindered Brownian motion
Librational motion: long time-scale fluctuations of the center of mass
Fluxional behavior in tetrahedral clusters with carbonyl ligands
Y Roberts, BFG Johnson, RE Benfield, Inorg. Chim. Acta 1995
Co4(CO12)
Librational motion
Cluster footprint @ 573 K
Increased intensity and redshift at high T
32 configuration average over last 5.5 ps
Pt L3 XANES
3.0 4.0 5.0 6.0 7.0 8.0
Time (ps)
-5.7
-5.6
-5.5
-5.4
EF (
eV)
165 K574 K
XANES
Interpretation of red shift:Charge fluctuations due to transient bonding
Fermi energy vs time
11550 11560 11570 11580 11590
Energy (eV)
0.0
0.5
1.0
1.5
Abs
oprt
ion
(au)
Expt. (165 K)Expt. (573 K)Theor. (165 K)Theor. (573 K)
Surface Pt-O bonds & charge fluctuate!
Conclusions
1. RT-TDDFT explains linear and non-linear response & high harmonic generation
Challenge: extension to core-XAS
(e.g. time-correlation function methods - in progress … )
2. RT-DFT/MD + RSGF XAS explains dynamic structure & experimental XAS of Pt nanoclusters
Novel nano-scale behavior: Brownian-like motion
Challenge: Extension to Faster, Hotter, Denser …
Acknowledgments
• J. Kas (UW)• F. Vila (UW)• Y. Takimoto (ISSP,UW)• J. Vinson (UW)
Collaborators
Supported by DOE-BES and NSF
A.L. Ankudinov (APD)A. Frenkel (Yeshiva)R. Nuzzo (UI)R. Albers (LANL)
Rehr Group
That’s all folks