Real Time TDDFT

1. Real Time TDDFT Quantum ESPRESSO Workshop June 25-29, 2012 The Pennsylvania State University University Park, PA Curtesy of Davide Ceresoli Istituto di Scienze e Tecnologie Molecolari CNR c/o Dipartimento di Chimica Fisica ed Elettrochimica Università degli studi di Milano davide.ceresoli@istm.cnr.it

2. Optical excitations: real-time TDDFT In principle, TDDFT is an exact reformulation of time-dependent quantum mechanics. However, in practice we have to adopt some approximations: • Adiabatic approximation: (i.e. without memory effect) • LDA / GGA • self-interaction error • wrong asymptotic limit of XC potential • local/semilocal functional

3. Real time TDDFT Real-time propagation scheme of TDDFT is a simple, efficient, and direct approach to obtain optical properties. Calculate dynamic polarizability and polarization under time-dependent external perturbation Flowchart: (a) apply a perturbative electric field along three directions (b) propagate the TDKS wavefunctions (c) calculate time-dependent dipole moment / angular momentum (d) finally obtain the dipole / rotational strength function via Fourier transform

4. Real time TDDFT: flowchart (a) apply a perturbative electric field along three directions (b) propagate the TDKS wavefunctions First-order Crank-Nicholson integration method A x = b : solved by conjugated-gradient squared method (CGS) Δt = 2 attosecond. Integration for 20 fs (→ 0.2 eV resolution)

5. Real time TDDFT: flowchart (c) calculate time-dependent dipole moment / angular momentum (d) obtain the dipole/rotational strength function via Fourier transform Complex rotatory strength function Dynamic polarizability Dipole strength function Rotational strength function Averaged in 3 directions

6. Real time TDDFT with QE 1- Run SCF calculation (SiH4-scf.in) &control prefix = 'sih4' calculation = 'scf', restart_mode = 'from_scratch' pseudo_dir = './pseudo/', outdir = './scratch/' / &system ibrav = 1, celldm(1) = 20.0 nat = 5, ntyp = 2 ecutwfc = 25 nosym = .true. / &electrons diagonalization = 'david', conv_thr = 1.0d-8 / ATOMIC_SPECIES Si 28.08550 Si.pz-vbc.UPF H 1.00000 H.pz-vbc.UPF ATOMIC_POSITIONS angstrom Si 0.000000000 0.000000000 0.000000000 H 0.859674551 0.859674551 0.859674551 H -0.859674551 -0.859674551 0.859674551 H -0.859674551 0.859674551 -0.859674551 H 0.859674551 -0.859674551 -0.859674551 K_POINTS automatic 1 1 1 0 0 0 ./pw.x <SiH4-scf.in >SiH4-scf.out

7. Real time TDDFT with QE 2- Run TDDFT propagation (SiH4-tddft_x.in) &inputtddft job = 'optical' prefix = 'sih4' tmp_dir = './scratch/' dt = 2.0 ! attoseconds (1e-18 s) nstep = 5000 ! 5000-10000 typically e_direction = 1 ! 1=x, 2=y, 3=z e_strength = 0.01 ! E-field strength (1/Angstrom) / ./tddft.x <SiH4-tddft_x.in >SiH4-tddft_x.out 3- Extract dipole dynamics, calculate optical absorption grep ^DIP SiH4-tddft_x.out >dip_x.dat python plot_optical_absorption.py x Check time-step, and energy range in python script!

8. Let's plot the results (SiH4) gnuplot> plot 'dip_x.dat' us 3:5

9. Let's plot the results (SiH4) 9.4 eV 7.7 eV gnuplot> plot 'sih4_x.dat'

10. Let's take another view gnuplot> set log y gnuplot> plot 'sih4_x.dat' us (1239.8419/$1):2

11. Exercise: aromatic vs. conjugated Aromatic Conjugated C6H6 Fulvene Benzene C10H8 Azulene Naphtalene

12. Aromatic vs. conjugated • Pick one molecule • Run SCF • Run TDDFT // x, run TDDFT // y • (optional) Run TDDFT // z • Plot optical absorption, find maximum λmax of absorption Note: 20 Ry cutoff, 4 Å vacuum!!!

13. Results: benzene 7.33 eV 8.35 eV 6.1 eV $ paste benzene_[xyz].dat >benzene_tot.dat gnuplot> plot 'benzene_tot.dat' us 1:(($2+$4+$6)/3)

14. Results: fulvene 5.32 eV 6.91 eV 3.1 eV

15. Results: naphtalene 6.00 eV 7.99 eV 7.12 eV 4.28 eV

16. Results: azulene 5.22 eV 2.44 eV

17. Comparison to experiments Benzene Naphtalene

18. Comparison to experiments: λmax Benzene Fulvene λmax = 254 nm (expt.) λmax = 203 nm (calc.) λmax = 370 nm (expt.) λmax = 399 nm (calc.) Naphtalene Azulene λmax = 312 nm (expt.) λmax = 289 nm (calc.) λmax = 690 nm (expt.) λmax = 508 nm (calc.)

19. What color is azulene? Let's convert wavelength into approximate RGB: $ python wave2RGB.py “TD-ALDA” azulene “Real” azulene Not absorbed Absorbed Not absorbed Absorbed From wikipedia:

20. Essential bibliography • Yabana and Bertsch, Time-dependent local-density approximation in real time,Phys. Rev. B 54, 4484 (1996) • Yabana and Bertsch, Application of the time-dependent local density approximation to optical activity, Phys. Rev. A 60, 1271 (1999) • Yabana, Nakatsukasa, Iwata and Bertsch, Real-time, real-space implementation of the linear response time-dependent density-functional theory,Physica status solidi (b) 243, 1121 (2006) • Castro, Appel, Oliveira, Rozzi, Andrade, Lorenzen, Marques, Gross and Rubio, Octopus: a tool for the application of time-dependent density functional theory,Physica status solidi (b) 243, 2465 (2006) • Marques, Castro, Bertsch and Rubio, Octopus: a first-principles tool for excited electron-ion dynamics, Comput. Phys. Commun. 151 60-78 (2003) • Castro, Marques and Rubio, Propagators for the time-dependent Kohn-Sham equations,J. Chem. Phys 121, 3425 (2004) • Andrade, Botti, Marques and Rubio, Time-dependent density functional theory scheme for efficient calculations of dynamic (hyper)polarizabilities,J. Chem. Phys 126, 184106 (2007) • Xiaofeng Qian, Ju Li, Xi Lin, and Sidney Yip, Time-dependent density functional theory with ultrasoft pseudopotentials: Real-time electron propagation across a molecular junction,Phys. Rev. B 73, 035408 (2006) Codes: http://www.phys.washington.edu/users/bertsch/V2.1.tar