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.,

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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)

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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

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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