Real-time approaches to x-ray and optical spectra

Real-time approaches to x-ray and optical spectra

J. J. Rehr, F. Vila, J. Kas

Department of Physics University of Washington

Seattle, WA USA

Nordita School on Photon-Matter Interaction Stockholm, Sweden Oct 3-7, 2016

Goal:

Talk:

•

•

•

4

Real-time Approaches for Optical and X-ray Spectra

I. Real-Space & Real-Time Linear and Non-linear Optical Response

● Difficulty: frequency-space

is computationally demanding too-many excited states

•

*Sanchez-Portal, Tsolakidis, and Martin, Phys. Rev. B66, 235416 (2002)

5

Real-Time-TDDFT*

6

*K. Yabana and G. F. Bertsch, Phys. Rev. B 54, 4484 1996.

Real time Linear Response

Induced Dipole Moment

Linear Response Function

Optical Absorption

Linear Dielectric Function

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RT-TDDFT Formalism

•

• Direct numerical integration of TD Kohn-Sham equations

• The response to external field is determined by applying a time-dependent electric field ΔH(t) = −E(t)·x.

• Optical properties determined from total dipole moment:

Sometimes more EFFICIENT than Frequency space methods 8

Numerical Real-time Evolution

•

•

•

Coefficients of Orbitals

10 _

_

, t = t + _ Δ t/2

Time (fs) Dipole p z(t) (a.u.)

•

(Unit Impulse at t=0)

•

(Turn-off Constant E at t=0)

Example: CO Linear Response

p_z(t) response due to applied E_z(t)

Time (fs) Energy (eV)

Im α(ω) Re α(ω)

12 E(t)

E(t)

0 0

Ground state without field

Ground state with constant field

Evolution for t>0

Evolution for t>0

Real time Nonlinear Response

•

•

How can we invert the equation to get

nonlinear response function? ₁₅

?

Extraction of Dynamic Nonlinear Polarizabilities

• ^{Set E}

(t) = F(t)E

, and define expansion

where p⁽¹⁾ yields linear response, p⁽²⁾ first non-linear quadratic response, ….

• Quadratic response χ

is then

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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 F

(t)

•

SHG

OR

F(t)

Linear and Nonlinear response of CO

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Example pNA: Nonlinear SHG

•

Energy (eV) β k(-2ω,ω,ω) (au)

Expt.

25

PBE

II. Real-time core-level XAS

jÃ(0)i = djbi

*

¹(!) = 1

¼Re

Z ₁

0

dt e^i!tG_c(t)hÃ(t)jÃ(0)iµ(! + ²^c ¡ E^F): (1)

Time-correlation function approach

XANES with time-dependent DFT

Goal:

Time-dependent x-ray response Include core hole dynamics

Why use a real-time approach?

New experimental pulsed sources (XFEL, LCLS) Pump-probe experiments

Increased interest in time-dependent (TD) response

RTXS: The cartoon view

Atom GS

CH PP Screened CH

SCF Init TD

PAW

RTXS equations

XAS Absorption (FGR, ΔSCF,

FSR)

Core Hole Green’s Function Autocorrelation Function FT

Crank-Nicolson

Physical interpretation

Projected density of states p-DOS

given by autocorrelation function for Seed state of p-symmetry:

½

(!) = ¡

Im R

0

dt e

hÃ(t)jÃ(0)i

Connection with Fermi golden rule agrees in limit t ∞

G(!) = R

0

dt e

U (t; 0)

¹(!) = ¡ _¼¹ Imhcjd^yG(E)djciµ(E ¡ E^F )

G(E) = [E ¡ H + i¡]^¡1

¹(!) = P

k jhcjdjkij²±_¡(! + ²_c ¡ ²^k)µ(E ¡ E^F )

Check:

C K-edge of CO

Example: C K-edge XES of Benzene

Expt: Fister et al., Phys. Rev. B 75, 174106 (2007)

C K-Edge XAS of Diamond (C₄₇H₆₀ cluster)

G

(t) = e

e

µ(t)

Cumulant expansion for core-hole Green’s function*

III. Many-body effects:

Intrinsic losses

*D. Langreth Phys Rev B 1, 471 (1970)

Particle-hole cumulant for XAS*

* cf. L. Campbell, L. Hedin, J. J. Rehr, and W.

Bardyszewski, Phys. Rev. B 65, 064107 (2002)

All losses in particle-hole spectral function A_K

NiO

Intrinsic losses: real-time TDDFT cumulant satellites

Langreth cumulant in time-domain*

TiO₂

*D. C. Langreth, Phys. Rev. B 1, 471 (1970)

Real-space interpretation: RT-TDDFT cumulant explains intrinsic excitations in TiO₂

RT TDDFT Cumulant

Theory vs XPS

Interpretation: satellites arise from oscillatory charge density fluctuations between ligand and metal at frequency ~ ωCT due to turned-on core-hole

Charge transfer fluctuations

ω_ct

XPS

F. Fossard, K. Gilmore, G. Hug, J J. Kas, J J Rehr, E L Shirley and F D Vila

NIST Preprint; submitted to PRB 2016

RT-TDDFT cumulant Particle-hole cumulant

X-ray Edge Singularities

Low energy particle-hole excitations in cumulant explain edge singularities in XPS and XAS of metals

Excitation spectrum

Question: Does the cumulant method

work for correlated systems ?

Hedin’s answer * MAYBE

“Calculation similar to core case … but with more complicated fluctuation potentials …

… not question of principle, but of computational work...”

* L. Hedin, J. Phys.: Condens. Matter 11, R489 (1999)

Vⁿ → -Im ε^-1(ω_n,q_n)

Ce L₃ XAS of CeO₂

Spectral function

Spectral weights

Particle-hole cumulant for CeO

Ce 5s XPS of CeO₂

Real-time approach for x-ray Debye-Waller factors

Conclusions

Efficient RT-TDDFT approach for frequency dependent nonlinear optical response –

Accuracy comparable to frequency-domain methods for small systems; also applicable to large systems

Similar real-time approaches can be applied to dynamic structure, Debye-Waller factors , etc.

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Acknowledgments:

Supported by DOE BSE DE-FG02-97ER45623

Thanks to

J.J. Kas L. Reining G. Bertsch

J. Vinson K. Gilmore L. Campbell T. Fujikawa F. Vila E. Shirley

S. Story S. Biermann M Guzzo M. Verstraete J. Sky Zhou C. Draxl et al.

& especially the ETSF

Conclusions

Particle-hole cumulant theory yields reasonable approximation for inelastic losses in XPS & XAS

All losses (intrinsic, extrinsic and interference) in

spectral function A_K(ω) – can be added ex post facto

Interference terms explain mysteries in amplitudes

and energy dependence: adiabatic- sudden transition Theory also applicable to some d- and f-systems.

Many-body amplitudes S

( ω) in XAS

•

•

to

sudden

|g_q |²= |g_q^ext |² ₊ | g_q^intrin |² _- 2 g_q^ext g_q^intrin

≈ μ

( ω) S

( ω)

Interference reduces loss!

Dynamic core-hole screening algorithm

Also: Grebennikov, Babanov and Sololov, Phys. Stat. Sol. 79, 423 (1977) and Privalov, Gel’mukhanov & Agren: Phys Rev. B 64, 165115 (2001)

alá Nozieres & De Dominicis

Extrinsic losses and Interference

Satellite strengths

XAS of Al

Particle-hole cumulant explains cancellation of extrinsic and intrinsic losses at threshold and

crossover: adiabatic

to sudden approximation

Extension to:

Extrinsic and Intrinsic losses

Question: How to extend theory to real-time approach?

Energy Dependent Spectral Function A(k,ω) Quasi-boson Model

Interference: Quasi-Boson Approach*

Excitations - plasmons, electron-hole pairs ... are bosons

*W. Bardyszewski and L. Hedin, Physica Scripta 32, 439 (1985)

“

Many-body Model: |e- , h , bosons

>

Many-pole Self-energy Algorithm

Plasmon-pole model many-pole model

-Im ε^-1(ω) Many-pole Dielectric Function

~

* J. Kas et al. PRB 76, 195116(2008)

Effective GW++ Green’s Function g

( ω)

Damped qp Green’s function

Extrinsic + Intrinsic

-

Spectral function: A(ω) = -(1/π) Im

g

( ω)

L. Campbell, L. Hedin, J. J. Rehr, and W. Bardyszewski, Phys. Rev. B 65, 064107 (2002)

+ - -

That’s all folks