Vibronic quantum dynamics of exciton relaxation/trapping in molecular aggregates

Symposium, Bordeaux Vibronic quantum dynamics of exciton relaxation/trapping in molecular aggregates Alexander Schubert Institute of Physical and Theoretical Chemistry, University of Würzburg November 7, 2012 Alexander Schubert Exciton trapping in aggregates 1 / 17

Perylene Bisimide (PBI) 3,4,9,10-perylene tetracarboxylic bisimide acid PBI 1,2 : n-type semi-conductor intense photoluminescence robust & air-stable functionalizing of up to 12 positions self-aggregating 1 F. Würthner: Chem. Commun., 11 1564, (2004). 2 C. Li & H. Wonneberger: Adv. Mat., 24(5), (2012). Alexander Schubert Exciton trapping in aggregates 2 / 17

Perylene Bisimide (PBI) 3,4,9,10-perylene tetracarboxylic bisimide acid PBI 1,2 : n-type semi-conductor intense photoluminescence robust & air-stable functionalizing of up to 12 positions self-aggregating Self-assembled π π-stack (H-aggregate) 1 F. Würthner: Chem. Commun., 11 1564, (2004). 2 C. Li & H. Wonneberger: Adv. Mat., 24(5), (2012). Alexander Schubert Exciton trapping in aggregates 2 / 17

Perylene Bisimide (PBI) 3,4,9,10-perylene tetracarboxylic bisimide acid PBI 1,2 : n-type semi-conductor intense photoluminescence robust & air-stable functionalizing of up to 12 positions self-aggregating Self-assembled π π-stack (H-aggregate) Drawback: disappointing aggregat properties 1 F. Würthner: Chem. Commun., 11 1564, (2004). 2 C. Li & H. Wonneberger: Adv. Mat., 24(5), (2012). Alexander Schubert Exciton trapping in aggregates 2 / 17

Photoinduced Processes 1 Photoabsorption delocalized state 1 2 Localization dimer state 2 3 competing processes Exciton Energy Transfer Emission Self-Trapping 1 B. A. West et al.: J. Phys. Chem. B, 115, (2010). 2 I. A. Howard et al.: J. Phys. Chem. C, 113, (2009). Alexander Schubert Exciton trapping in aggregates 3 / 17

Photoinduced Processes 1 Photoabsorption delocalized state 1 2 Localization dimer state 2 3 competing processes Exciton Energy Transfer Emission Self-Trapping 1 B. A. West et al.: J. Phys. Chem. B, 115, (2010). 2 I. A. Howard et al.: J. Phys. Chem. C, 113, (2009). Alexander Schubert Exciton trapping in aggregates 3 / 17

Photoinduced Processes 1 Photoabsorption delocalized state 1 2 Localization dimer state 2 3 competing processes Exciton Energy Transfer Emission Self-Trapping 1 B. A. West et al.: J. Phys. Chem. B, 115, (2010). 2 I. A. Howard et al.: J. Phys. Chem. C, 113, (2009). Alexander Schubert Exciton trapping in aggregates 3 / 17

Absorption & emission spectra Monomer model 1 single vibrational mode (harmonic) ω, q eq., E electronic state transition? 1 J. Seibt et al.: Chem. Phys., 328, (2006). Alexander Schubert Exciton trapping in aggregates 4 / 17

Absorption & emission spectra Monomer model 1 single vibrational mode (harmonic) ω, q eq., E Dimer model Absorption Emission dipole-dipole coupling (ĤM M H ex = Ĵ Ĵ Ĥ M M geometry dependence (static) ) red-shifted, broad emission band Quantum chemical (TD-HF) calculation 2 : torsional mode φ adiabatic, ab-initio PECs red-shift long radiative lifetime 3 electronic state transition? 1 J. Seibt et al.: Chem. Phys., 328, (2006). Alexander Schubert Exciton trapping in aggregates 4 / 17

Absorption & emission spectra Monomer model 1 single vibrational mode (harmonic) ω, q eq., E Dimer model Absorption Emission dipole-dipole coupling (ĤM M H ex = Ĵ Ĵ Ĥ M M geometry dependence (static) ) red-shifted, broad emission band Quantum chemical (TD-HF) calculation 2 : torsional mode φ adiabatic, ab-initio PECs red-shift long radiative lifetime 3 electronic state transition? 1 J. Seibt et al.: Chem. Phys., 328, (2006). 2 R. F. Fink et al.: J. Am. Chem. Soc., 130, (2008). 3 I. A. Howard et al.: J. Phys. Chem. C, 113, (2009). Alexander Schubert Exciton trapping in aggregates 4 / 17

Absorption & emission spectra Monomer model 1 single vibrational mode (harmonic) ω, q eq., E Dimer model Absorption Emission dipole-dipole coupling (ĤM M H ex = Ĵ Ĵ Ĥ M M geometry dependence (static) ) red-shifted, broad emission band Quantum chemical (TD-HF) calculation 2 : torsional mode φ adiabatic, ab-initio PECs red-shift long radiative lifetime 3 electronic state transition? 1 J. Seibt et al.: Chem. Phys., 328, (2006). 2 R. F. Fink et al.: J. Am. Chem. Soc., 130, (2008). 3 I. A. Howard et al.: J. Phys. Chem. C, 113, (2009). Alexander Schubert Exciton trapping in aggregates 4 / 17

Transient absorption spectroscopy aggregate signal OD [%] 0,0-0,5 0,0 Delay time [ps] 0,5 PBI Fit 1,0 Solvent monomer signal Fit components: oscillation τ=215 fs step function 0,0 0,5 Delay time [ps] 1,0 S. Lochbrunner, Uni Rostock E. Riedle, I. Pugliesi, LMU Munich exponential decay of stimulated emission signal (τ = 215 fs) = ultrafast de-population of the excited state Alexander Schubert Exciton trapping in aggregates 5 / 17

The dimer model The role of charge transfer states quantum chemical description 1 HOMO-LUMO transitions 4 electronic states 2 find induced nuclear motion 3 high-level ab initio computations 4 (mixed) adiabatic PECs and electronic properties 4 diabatization 4 coupled diabatic states 5 set up exact kinetic energy operator 6 include dissipation effects Excitation energy φ{1,...,4} g Orbital energy M M* ζ1 M* M ζ2 M + M ζ3 adiabatic energy levels Frenkel states Charge transfer states LUMO HOMO M M + ζ4 Alexander Schubert Exciton trapping in aggregates 6 / 17

The dimer model The role of charge transfer states quantum chemical description 1 HOMO-LUMO transitions 4 electronic states 2 find induced nuclear motion 3 high-level ab initio computations 4 (mixed) adiabatic PECs and electronic properties 4 diabatization 4 coupled diabatic states 5 set up exact kinetic energy operator 6 include dissipation effects Excitation energy φ{1,...,4} g Orbital energy M M* ζ1 M* M ζ2 M + M ζ3 adiabatic energy levels Frenkel states Charge transfer states LUMO HOMO M M + ζ4 Alexander Schubert Exciton trapping in aggregates 6 / 17

The dimer model The role of charge transfer states quantum chemical description 1 HOMO-LUMO transitions 4 electronic states 2 find induced nuclear motion 3 high-level ab initio computations 4 (mixed) adiabatic PECs and electronic properties excited state dynamics 4 diabatization 4 coupled diabatic states 5 set up exact kinetic energy operator 6 include dissipation effects Excitation energy φ{1,...,4} g Orbital energy M M* ζ1 M* M ζ2 M + M ζ3 adiabatic energy levels Frenkel states Charge transfer states LUMO HOMO M M + ζ4 Alexander Schubert Exciton trapping in aggregates 6 / 17

The effective mode groundstate structure q = 0 deformational mode q q = 0: neutral (ground state) geometry monomer distance: 3.3Å, torsional angle: 30 q = 1: combined dimer: anionic and cationic structure else: linear inter- and extrapolation Alexander Schubert Exciton trapping in aggregates 7 / 17

The effective mode groundstate structure relaxed anion structure ( 20) relaxed cation structure ( 20) q = 0 q = 1 q = 1 deformational mode q q = 0: neutral (ground state) geometry monomer distance: 3.3Å, torsional angle: 30 q = 1: combined dimer: anionic and cationic structure stabilization of charge transfer states? else: linear inter- and extrapolation Alexander Schubert Exciton trapping in aggregates 7 / 17

Potential Energy Curves 3,5 V4 ad 1,0 A ad 4 potential energy V ad [ev] na 3,0 V2 ad V1 ad V3 ad CT character A ad na 0,8 0,6 0,4 0,2 A ad 2 A ad 1 A ad 3 2,5-0,5 0,0 0,5 1,0 1,5 2,0 coordinate q ab initio computations adiabatic Potential Energy Curves groundstate (SCS-MP2) 4 excited states (SCS-CC2) avoided crossings 0,0-0,5 0,0 0,5 1,0 1,5 2,0 coordinate q Alexander Schubert Exciton trapping in aggregates 8 / 17

Potential Energy Curves 3,5 V4 ad 1,0 A ad 4 potential energy V ad [ev] na 3,0 V2 ad V1 ad V3 ad CT character A ad na 0,8 0,6 0,4 0,2 A ad 2 A ad 1 A ad 3 2,5-0,5 0,0 0,5 1,0 1,5 2,0 coordinate q ab initio computations adiabatic Potential Energy Curves groundstate (SCS-MP2) 4 excited states (SCS-CC2) avoided crossings 0,0-0,5 0,0 0,5 1,0 1,5 2,0 coordinate q Character analysis via projection in monomer MO basis (Löwdin orthogonalization) electronical structure Alexander Schubert Exciton trapping in aggregates 8 / 17

Diabatization Hamilton-Operator Ĥ = ˆV (R) + ˆT R in electronic bases: {φ n(r)}: adiabatic basis (ab initio basis) H ad nm H ad = n,m φ {}}{ n φ n ˆV + ˆT φ m r φ m {ξ n f(r)}: diabatic basis T d diagonal Unitary transformation UT ad U 1 = T d UV ad U 1 = V d non-diagonal coordinate coupling Alexander Schubert Exciton trapping in aggregates 9 / 17

Diabatization Hamilton-Operator Ĥ = ˆV (R) + ˆT R in electronic bases: {φ n(r)}: adiabatic basis (ab initio basis) H ad nm H ad = n,m φ {}}{ n φ n ˆV + ˆT φ m r φ m {ξ n f(r)}: diabatic basis T d diagonal Unitary transformation UT ad U 1 = T d UV ad U 1 = V d non-diagonal coordinate coupling Alexander Schubert Exciton trapping in aggregates 9 / 17

Diabatization Hamilton-Operator Ĥ = ˆV (R) + ˆT R in electronic bases: {φ n(r)}: adiabatic basis (ab initio basis) H ad nm H ad = n,m φ {}}{ n φ n ˆV + ˆT φ m r φ m V ad nn = φ n Ĥel φ n r (R) diagonal T ad nm = δ nm ˆTR + 2 φ n R φ m r (R) R + φ n 2 R φm r (R) non-adiabatic, derivative (kinetic) coupling {ξ n f(r)}: diabatic basis T d diagonal Unitary transformation UT ad U 1 = T d UV ad U 1 = V d non-diagonal coordinate coupling non-diagonal Alexander Schubert Exciton trapping in aggregates 9 / 17

Diabatization Hamilton-Operator Ĥ = ˆV (R) + ˆT R in electronic bases: {φ n(r)}: adiabatic basis (ab initio basis) H ad nm H ad = n,m φ {}}{ n φ n ˆV + ˆT φ m r φ m V ad nn = φ n Ĥel φ n r (R) diagonal T ad nm = δ nm ˆTR + 2 φ n R φ m r (R) R + φ n 2 R φm r (R) non-adiabatic, derivative (kinetic) coupling {ξ n f(r)}: diabatic basis T d diagonal Unitary transformation UT ad U 1 = T d UV ad U 1 = V d non-diagonal coordinate coupling non-diagonal Alexander Schubert Exciton trapping in aggregates 9 / 17

Diabatization Hamilton-Operator Ĥ = ˆV (R) + ˆT R in electronic bases: {φ n(r)}: adiabatic basis (ab initio basis) H ad nm H ad = n,m φ {}}{ n φ n ˆV + ˆT φ m r φ m V ad nn = φ n Ĥel φ n r (R) diagonal T ad nm = δ nm ˆTR + 2 φ n R φ m r (R) R + φ n 2 R φm r (R) non-adiabatic, derivative (kinetic) coupling {ξ n f(r)}: diabatic basis T d diagonal Unitary transformation UT ad U 1 = T d UV ad U 1 = V d non-diagonal coordinate coupling Task: eigenvalues of V d resemble adiabatic PECs Problem: not unambiguous non-diagonal Alexander Schubert Exciton trapping in aggregates 9 / 17

Diabatization {ζ n}: diabatic basis of localized MO orbitals to define CT-character operator  = ζ CT 1 ζ CT 1 + ζ CT 2 ζ CT 2 A ad n (R) = ζ 3 φ n r (R) 2 + ζ 4 φ n r (R) 2 A d n (R 0) = ζ 3 ξ n r 2 + ζ 4 ξ n r 2 constant known Alexander Schubert Exciton trapping in aggregates 10 / 17

Diabatization {ζ n}: diabatic basis of localized MO orbitals to define CT-character operator conditions for V d : Â = ζ CT 1 ζ CT 1 + ζ CT 2 ζ CT 2 A ad n (R) = ζ 3 φ n r (R) 2 + ζ 4 φ n r (R) 2 A d n (R 0) = ζ 3 ξ n r 2 + ζ 4 ξ n r 2 1 U 1 V d U = V ad 2 U 1 A d }{{} const. U = A ad constant known Alexander Schubert Exciton trapping in aggregates 10 / 17

Diabatization conditions for V d : U 1 V d U = V ad U 1 A d U = A ad 3,5 100 1,0 A ad 4 potential energy [ev] 3,0 V4 d V3 d V2 d V1 d V4 ad V3 ad V2 ad V1 ad coupling elements V d nd 4 [mev] 80 60 40 20 V d 14 V d 34 na CT character A ad 0,8 0,6 0,4 0,2 A ad 2 A ad 1 2,5-0,5 0,0 0,5 1,0 1,5 2,0 coordinate q dashed line: constructed diabatic curves (V d n ) solid lines: eigenvalues of V d (V ad n ) dots: ab initio computations V24 d 0-0,5 0,0 0,5 1,0 1,5 2,0 coordinate q 0,0 A ad 3-0,5 0,0 0,5 1,0 1,5 2,0 coordinate q solid lines: unitary transformation of constant CT elements dots: ab initio CT characters Alexander Schubert Exciton trapping in aggregates 11 / 17

kinetic energy operator + dissipation kinetic energy exact treatment of ˆT R B. Podolsky, Phys. Rev., 32(5), (1928). linear, one-dimensional motion effective mass approach ˆT q = 2 2 j,k g 1 4 [ g 1 2 g jk ] g 1 4 q j q k Alexander Schubert Exciton trapping in aggregates 12 / 17

kinetic energy operator + dissipation kinetic energy exact treatment of ˆT R B. Podolsky, Phys. Rev., 32(5), (1928). linear, one-dimensional motion effective mass approach ˆT q = 2 2 j,k = 1 2 m eff [ g jk q j q k ( ) 2, where m eff = ( m α q α ] x (q=1) α ) x (q=0) 2 α Alexander Schubert Exciton trapping in aggregates 12 / 17

kinetic energy operator + dissipation kinetic energy exact treatment of ˆT R B. Podolsky, Phys. Rev., 32(5), (1928). linear, one-dimensional motion effective mass approach Dissipation ˆT q = 2 2 j,k = 1 2 m eff [ g jk q j q k ( ) 2, where m eff = ( m α q α simplified stochastic Hamiltonian: energy-dependent damping term Ĥn diss d = i λ Ĥd n d and renormalization Ψ d n d (q, t + t) = Ψ d n d (q,t+ t) 4n=1 Ψ d n (q,t+ t) 2 dq ] x (q=1) α ) x (q=0) 2 α Alexander Schubert Exciton trapping in aggregates 12 / 17

kinetic energy operator + dissipation kinetic energy exact treatment of ˆT R B. Podolsky, Phys. Rev., 32(5), (1928). linear, one-dimensional motion effective mass approach Dissipation ˆT q = 2 2 j,k = 1 2 m eff [ g jk q j q k ( ) 2, where m eff = ( m α q α simplified stochastic Hamiltonian: energy-dependent damping term Ĥn diss d = i λ Ĥd n d and renormalization Ψ d n d (q, t + t) = Ψ d n d (q,t+ t) 4n=1 Ψ d n (q,t+ t) 2 dq ] x (q=1) α ) x (q=0) 2 α Alexander Schubert Exciton trapping in aggregates 12 / 17

resulting dynamics population 1,0 0,5 0,0 1,0 nd = 1 nd = 2 nd = 4 Ψ d nd Ψd nd q (t) Ψ d nd (q, t) 2 coordinate q 0,0-1,0 1,0 0,0-1,0 nd = 2 nd = 1 0,0 0,2 0,4 0,6 0,8 1,0 Time t [ps] Alexander Schubert Exciton trapping in aggregates 13 / 17

summarized picture Alexander Schubert Exciton trapping in aggregates 14 / 17

summarized picture OD [%] 0,0-0,5 0,0 Delay time [ps] 0,5 PBI Fit 1,0 Solvent Fit components: oscillation τ=215 fs step function 0,0 0,5 Delay time [ps] 1,0 intensity [arb. units.] Ex = 0, 11 ev Ex = 0, 01 ev 0,0 0,5 1,0 time t [ps] Alexander Schubert Exciton trapping in aggregates 14 / 17

quantum beating simultaneous excitation of two modes E q, E x Ψ, t = c 00 Ψ 00 e i E 00t intensity [arb. units.] E x = 0, 11 ev E x = 0, 01 ev 0,0 0,5 1,0 time t [ps] beating signal +c 10 Ψ 10 e i E 10t +c 01 Ψ 01 e i E 01t +c 11 Ψ 11 e i E 11t Ψ, t  Ψ, t 2 cos [ Ex t/ ] +2 cos [ E q t/ ] + cos [ ( E x + E q) t/ ] + cos [ E x E q t/ ] Alexander Schubert Exciton trapping in aggregates 15 / 17

first insights in localization processes Mr1 M r2 M r1 Mr2 energy ω r2 W (t) r1 M-M excited Dimer time-dependent perturbation Localization J. Wehner et al.: Chem. Phys. Lett., 541, (2012). Alexander Schubert Exciton trapping in aggregates 16 / 17

first insights in localization processes Mr1 M r2 M r1 Mr2 energy resonant r2 W (t) r1 M-M excited Dimer time-dependent perturbation Localization J. Wehner et al.: Chem. Phys. Lett., 541, (2012). Alexander Schubert Exciton trapping in aggregates 16 / 17

first insights in localization processes M-M excited Dimer M-M-M excited Trimer M-M-M time-dependent perturbation Localization J. Wehner et al.: Chem. Phys. Lett., 541, (2012). Alexander Schubert Exciton trapping in aggregates 16 / 17

PBI Spectroscopy Quantum Chemistry Diabatization Trapping dynamics Localization Acknowledgement Acknowledgement Workgroup Engel (W urzburg) Prof. Dr. Volker Engel Johannes Wehner Workgroup Engels (W urzburg) Prof. Dr. Bernd Engels Prof. Dr. Reinhold F. Fink Volker Settels Dr. Wenlan Liu Prof. Dr. Stefan Lochbrunner (Rostock) Prof. Dr. Frank W urthner (W urzburg) Alexander Schubert GRK 1221 FOR1809 Exciton trapping in aggregates 17 / 17