Fundamentals of TDDFT for nonlinear phenomena of light

OUTLINE 

Introduction: TDDFT some Open questions 

Monitor excitations; photoemission; Correlations ( H 2 ; H-chain ) 

Four linked concepts: 

* Molecular dissociation: bumps in Vxc 

* Electron many-body tunneling 

* Rabi Oscillations population analysis 

* Photochemistry... 

Application: Organic photovoltaics: triads “carotene-porphyrine-C 60 ” 

Publications: 

Quantum coherence controls the charge separation in a prototypical artificial light harvesting system, C. A. Rozzi, S. M. Falke, N. 

Spallanzani, AR, E. Molinari, D. Brida, M. Maiuri, G. Cerullo, H. Schramm, J. Christoffers, C. Lienau Nature Communications (2013) 

Simulating pump-probe photo-electron and absorption spectroscopy on the attosecond time-scale with time-dependent density-functional 

theory, U. De Giovannini, G. Brunetto, A. Castro, J. Walkenhorst, A.R, Chemphyschem (2013) 

Universal Dynamical Steps in the Exact Time-Dependent Exchange-Correlation Potential 

Peter Elliott, Johanna I. Fuks, Angel Rubio, Neepa T. Maitra, Physical Review Letters 109, 266404 (2012) 

Fundamentals of Time-Resolved Charge-Transfer in Time-Dependent Density Functional Theory, 

Johanna I. Fuks, Peter Elliott, AR, Neepa T. Maitra Journal Of Physical Chemistry Letters (2013) 

Fundamentals of TDDFT for nonlinear phenomena of light-matter interactions 

Workshop: Calculation of Optical Properties of Nanostructures from First Principles (20 th Feb.2013)

E.g. 

Beyond linear response: Real-time dynamics 

High-harmonic generation 

Multiple-ionization 

Photo-dissociation 

Photo-isomerization 

Enhanced ionization 

Electronic quantum control 

Optimizing a single 

harmonic (in Ar) 

Light-driven chiral molecular 

motors, Yamaki, Nakayama, Hoki, 

Kono, Fujimura, Phys. Chem. Chem. 

Phys. 11, 1649 (2009) Dye-Sensitized Solar Cell 

But how well do the TDDFT approximations work ? 

…importance of exactly-solvable models… 

Murnane & 

Kapteyn et al, 

Nature 2000 



Photovoltaics 

Absorption of ligh 

(matching solar spectra) 

“Level alignment” 

Exciton dynamics and 

splitting 

Transport at the interface 

Extracting charges 

Efficiency 



Theoretical Framework(s) 

DFT & TDDFT 

MBPT 

Octopus Code 



TDDFT formulation for BO-MD dynamics: Ehrenfest 

dynamics (non adiabatic) 

The equation of motion without external field for simplicity 

i ∂ 

∂t i=[ −ℏ2 

2m ∇ 2 V ionV HV xc] i 

J.L. Alonso, X. Andrade, P. Echenique, F. Falceto, D. Prada, AR PRL (2008), JCTC (2009), NJP (2010) 

Non-adiabatic couplings: 

M I ¨R I =−∇ I E[ ,R] 

See for example review by N.L Doltsinis and D. Marx, 

J.Theo.Comp.Chem 1, 319 (2002) and the books on 

TDDFT Springer Lecture Notes in Physics Vol. 706 

(2006) and Vol. 837 (2012) 



TDDFT: All observables are functionals of n(r,t) (Runge&Gross 1984) 

Absorption spectra: from the power spectrum of the TD dipole moment 

d(t) = < r n(r,t) > ------> d(w) = FT[d(t)] 

........... and current/magnetic and spin responses 

Ionisation yields: 

Linear and non linear phenomena accessible 

Octopus Code http://www.tddft.org 

Time-Dependent Density Functional Theory, Lecture Notes in Physics, Springer Vols. 837, 706 (2012, 2006) 

PES ( p)=∑ i ∣Ψ i( p , t →∞)∣ 2 

Excited state densities and transition probabilities 

....................... and many more observables!!!!!! 



Photoelectron spectroscopy: spatial and energy resolved 

U. De Giovannini, D. Varsano, H. Appel, M. A. L. Marques,, E.K.U. Gross and and A. Rubio (2012) 







Transient Photoelectron Spectroscopy – 

time resolved pump-probe spectroscopy 



TRPES for C 2 H 4 

A pump 

A p probe 



TRPES for C 2 H 4 

Simulating pump-probe photo-electron and absorption spectroscopy on the attosecond time-scale with time-dependent density-functional theory, U. D 

Giovannini, G. Brunetto, A. Castro, J. Walkenhorst, A.R, Chemphyschem (2013) 



Charge Transfer 

excitations? 

Photovoltaic Hybrids: Grätzell cells 

Road to Improving Efciency: Controlling position of the frontier orbitals 



Photovoltaics 

Absorption of ligh 

(matching solar spectra) 

“Level alignment” 

Exciton dynamics and 

splitting 

Transport at the interface 

Extracting charges 

Efficiency 



Can we modify/optimise 

and control the spectra of 

a molecular system 

(”dye”) at will? 

Non-equilibrium and non- 

linear response functions 



Transient Absorption Spectroscopy 

“Unperturbed” Unperturbed” Time-dependent Hamiltonian 

: static Hamiltonian 

: “pump” laser pulse couple 

: initially the system is at an equilibrium 

state

Response 

Measured by the variation of the 

expectation value of some observable Â. 

Apply a weak probe pulse: 

The non-equilibrium response function is: 

The response will be a functional of both pump and probe 

pulses

Transient Absorption Spectroscopy: a test case 



Helium 



Ethylene molecule C 2 H 4 

Transient Absorption (TAS) 

The molecule is excited by a 45 cycle sin 2 envelope laser pulse polarized along 

the x- axis with carrier w = 0.297 a.u. of intensity I=1.38 10 11 W/cm 2 . 



Transient Absorption C2H2 

Laser Pump 

15 cycles with a Sine 

square envelope function 

Frequency: 0.264 Ha 

Intensity: 5x10 13 W/cm 2 

? 

To be complemented by an optimal control loop to 

match better a given spectral range (“solar”) 



Let's increase complexity ??!!!! 

H + 

H2 

2 

and Dimers 

Illustrate the problem of SIC and beyond from a different perspective 



The step in KS potential: open shell fragments 

●Step aligns the atomic HOMOs 

●Kohn-Sham system builds a ”wall“ to mimick the repulsion due to 

interaction and prevent tunneling 

Prevents dissociation to unphysical fractional charges 

●Step-height = difference between highest eigenvalues of the two wells 

Step in exact KS potential that 

exactly realigns the two atomic 

HOMOs: Step Δ = |ID – IA| 

Early work of Perdew, Almbladh& von Barth, Gritsenko & Baerends 

Recent work of Tempel, Martinez, Maitra, JCTC (2009) and N. Helbig, I. Tokatly, A. Rubio, JCP (2009) 



H 2 

Diference between the KS-efective potential 

and the external (ionic) one 



V KS- V ionic 

H-chain 

Mott insulator : in DFT !!! 

Clearly all local functionals and most orbital dependent 

functionals do not capture the step in the potential 

The KS systems is metallic : fxc responsible for the gap 



Finding the exact time-dependent xc potential 

for a given known density-evolution 

• In general, not easy. 

• But for 2 electrons, starting in a doubly-occupied orbital, it’s easy 

Must have 

Insert φ(x,t) into TDKS equation, and solve for : 

Further, we have 

so we can extract 

(↑↓ -↓↑)/√2 



We found a wide range of different types of two-electron dynamics 

display non-adiabatic and spatially non-local step-like features. 

What´s new? 

● our steps typically appear: no need for fractional charges, nor 

even applied fields (see shortly), and 

● unlike previous, ours cannot be captured by adiabatic approx 

Outline 

Dynamical steps and peaks 

Time-Resolved Charge Transfer 

Another kind of non-local, non-adiabatic step also generically 

develops in time… 



Example 1: Field-free evolution of a non-stationary state He-atom 

exact vc 

vc adia-ex 

∂u/dt 



Example 1: Field-free evolution of a non-stationary state He-atom 

exact vc 

vc adia-ex 

∂u/dt 



Example 2: Weak on-resonant driving of a “He atom” 

Zooming in to an optical 

period around TR/4 

exact vc 

“exact-adiab.” 

density 

A = 0.00667 au 

w = 0.533 au (1st excn.) 

Snapshots over a 

Rabi period 



Example 2: Weak on-resonant driving of a “He atom” 

Zooming in to an optical 

period around TR/4 

exact vc 

“exact-adiab.” 

density 

A = 0.00667 au 

w = 0.533 au (1st excn.) 

Snapshots over a 

Rabi period 



Example 3: Strong off-resonant driven He atom 

exact vc 

exact-adiab 

density 



Example 4: Time resolved Charge transfer 

CT-resonant excitation of two closed shell fragments 

(2pi/T R ) = 0.00136 Ha 



Example 4: Time resolved Charge transfer 

CT-resonant excitation of two closed shell fragments 

(2pi/T R ) = 0.00136 Ha 



We have derived a new (memory) 

functional for 2e that reproduces 

all this phenomena 

Work in progress results soon! 



Charge Transfer 

excitations? 

Photovoltaic Hybrids: Grätzell cells 

Road to Improving Efciency: Controlling position of the frontier orbitals 



Photo-excitation of the light-harvesting 

carotenoid-porphyrin-C 60 

C.A. Rozzi et al, Nat. Comm. (2013) 



Photo-excitation of the light-harvesting 

carotenoid-porphyrin-C60 



Carotenoid-porphyrin-C 60 photodynamics 

Charge on the C60 

Dynamical ions 

Clamped ions 

Clamped ions 





(Some) Open questions 

* Tunneling and/or CT processes at interfaces ??? transport 

Open questions 

* Related concepts: 

* Molecular dissociation: 

* Electron tunneling 

* Rabi Oscillations (light induced e-h spliting) 

* Dissipation; lifetimes; de-coherence, Non adiabatic couplings 

* Quantum control and open quantum systems 

Work to be done: “new spatial and w-dependent functionals” 

. 



Acknowledgements 

N. Helbig, J. Fuks, A. Crawford, L. Stella, I. Tokatly 

ETSF , Dpto Material Physics, Centro Mixto CSIC-UPV, Donostia, Spain 

A. Castro BIFI, Zaragoza H. Appel FHI-Berlin 

E.K.U. Gross ; P. Elliot 

MPI-Halle Modena 

C.A. Rozzi 

N. Maitra Hunter College NY E. Molinari 



For more details see: 

http://nano-bio.ehu.es 

http://etsf.eu 

Thank you!!!!

Fundamentals of TDDFT for nonlinear phenomena of light

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