IV. Calculations of X-ray Spectra in Real-space and Real-time. J. J. Rehr

Transcription

1 TIMES Lecture Series SLAC-Stanford U March 2, 2017 IV. Calculations of X-ray Spectra in Real-space and Real-time J. J. Rehr

2 Calculations of X-ray Spectra in Real-space and Real-time Goal: Real-space, real time Theory of XAS Talk: Mostly real-time I. RT-TDDFT and XAS II. Many-body effects S 2 0 and satellites III. Vibrational & non-equilibrium properties 4

3 The real-time formalism is a gift of god W. Kohn

4 Motivationn Why real-time? XFEL pulsed x-ray sources (FLASH, LCLS) Pump-probe experiments Interest in time-dependent response Non-equilibrium systems

5 Theoretical challenge: many length & time-scales ν BSE True Molecular Crystals Crystals IR V/UV X-Rays RSGF Molecules Clusters Liquids k r DFT/MD s ns ps fs as RT-TDDFT t

6 A. Real-time approach for XAS I. Real-time approach for XAS * Time-correlation function formalism ¹(!) = 1 Z 1 ¼ Re dt e i!t G c (t)hã(t)jã(0)iµ(! + ² c E F ): (1) 0 Equivalent to single-particle Fermi golden rule

7 Real-time engine: RT-TDDFT *K. Yabana and G. F. Bertsch, Phys. Rev. B 54,

8 RT-TDDFT Formalism Yabana and Bertsch Phys. Rev. B 54, 4484 (1996) 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: Can be more powerful and more EFFICIENT than Frequency space

9 Numerical Real-time Evolution c(t) Ground state density ρ 0, overlap matrix S, and H(t) at each time-step evaluated with SIESTA Coefficients of Orbitals Crank-Nicholson time-evolution: unitary, time-reversible Stable for long time-steps! _, t = t + Δ t/2 Adiabatic GGA exchange-correlation (PBE) functional

10 Example: CO molecule Linear Response Delta Function (Unit Impulse at t=0) Step Function (Turn-off Constant E at t=0) Time (fs) E(t) Dipole p z (t) (a.u.) Im α(ω) 0 Ground state without field E(t) Re α(ω) 0 Ground state with constant field Time (fs) Evolution for t>0 Evolution for t>0 Energy (ev) 12

11 Shaped pulses and non-linear response Nonlinear expansion in field E(t) including time lag in response Time (fs)? How to invert the equation for nonlinear response? 15

12 Shaped pulses and non-linear polarizability Define E j (t) = F(t)E j and response p i (E) where p (1) represents linear response, p (2) non-linear quadratic response,. Quadratic response function χ (2) 17

13 Shaped pulses and non-linear response F δ (t) Sine wave enveloped by another sine wave or Gaussian F(t) p (1) ij p (2) ijk Time (fs) Time Time (fs) (fs) Time (fs) Re F(ω) Frequency (ev) Im F(ω) Frequency (ev) SHG OR Linear and Nonlinear response of CO 18

14 Example: non-linear SHG in pna Comparison with other methods Real-time results β k (-2ω,ω,ω) (au) Expt Energy (ev) 25

15 Application to XAS: RTXS Equations XAS Absorption Fermi golden rule, ΔSCF, FSR) FT Core Hole Green s Function Autocorrelation Function Crank-Nicolson RT-TDDFT

16 Interpretation of correlation function pdos = FT of for seed state ½ Ã (!) = 1 ¼jÃj 2 j Im R 1 0 dt e i!t hã(t)jã(0)i

17 Example C K-edge of CO

18 Example: C K-edge XAS of Diamond cluster C 47 H 60 clusters Expt: Fister et al., Phys. Rev. B 75, (2007)

19 B. Real-time approach for many-body effects 1. Intrinsic losses - due to sudden core-hole Formalism: Cumulant expansion for core-hole Green s function G + c (t) = e i² ct e C(t) µ(t) Many body effects implicit in cumulant C(t) *D. Langreth Phys Rev B 1, 471 (1970)

20 Reviews and references

21 Cumulant expansion properties Landau formula for C(t) Excitation spectra GW Σ Spectral Function *For diagrammatic expansion of higher order terms, see e.g. O. Gunnarsson et al., Phys. Rev. B 50, (1994)

22 Intrinsic losses: CT excitations RT-TDDFT cumulant Intrinsic losses: real-time TDDFT cumulant satellites Langreth cumulant in time-domain* TiO 2 *D. C. Langreth, Phys. Rev. B 1, 471 (1970)

23 Real-space interpretation RT-TDDFT cumulant RT TDDFT Cumulant Theory vs XPS Charge transfer fluctuations ω ct Interpretation: satellites arise from oscillatory charge density fluctuations between ligand and metal at frequency ~ ω CT due to turned-on core-hole

24 C. Real-time approach for vibrations and non-equilibrium systems 1. XAS Debye-Waller factors 2. DFT/MD approach for nanocatalysts

25 1. Real-time EXAFS Debye-Waller factors Displacement-displacement Autocorrelation function

26 2. Non-equilibrium systems A theoretical horror story Starring Fernando Vila & Anatoly Frenkel with J. Kas, S. Bare & S. Kelly Directed by J. J. Rehr a Sequel a DOE CSGB Production

27 *J.H. Kang, L. D. Menard, R. G. Nuzzo, and A. I. Frenkel. JACS 2006, 128, Theoretical Challenge: Anomalous properties of Pt 10 /γal 2 O 3 Pt-Pt nn Negative Thermal Expansion & Bond expansion in H 2 Anomalous Pt-Pt disorder NOT bulk-like!

28 *J.H. Kang, L. D. Menard, R. G. Nuzzo, and A. I. Frenkel. JACS 2006, 128, More Anomalous properties* Pt 10 /γal 2 O 3 Increased edge intensity Redshift of XANES with increasing T (charge effects) Standard theory fails!

29 What s going on? Blob 573 K Breakthrough: DFT- MD The BLOB Cluster center of mass

30 Fuzzy structure DFT/MD nn distance

31 Dynamic Disorder: Anomalous Behavior Decomposition into Vibrational and Disorder components Vibrational Normal behavior (THz fs periods) Dynamic disorder Large, chaotic, sub THz

32 CONCLUSIONS Real-time, real-space formalism - powerful alternative to frequency and k-space methods Linear and non-linear optical and x-ray response to monochromatic and pulsed sources Many-body effects: multi-electron excitations, plasmon and charge-transfer satellites Vibrational and non-equilibrium effects

33 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