20/01/2010 Quantum Master Equations for the Electron Transfer Problem Seminarvortrag Dekohaerenz und Dissipation in Quantensystemen Antonio A. Gentile
The general transport problem in micro/mesoscopic physics
The Electron Transfer problem Working definition: spontaneous charge redistribution between well-defined reactant and product state (bound states) Strong: adiabatic case DONOR state V coupling between ACCEPTOR state Single-charge transition Charge redistribution Weak: nonadiabatic case
The Electron Transfer problem: classification Concerning the involved units : Concerning the excess electron mechanism: Intra-molecular Inter-molecular Injection (lead, e-beam, ) Excitation (scattering, exciton transfer, optical absorption, ) Concerning the bridge assistance DA complex 2 state model Through-space transfer Through-bond transfer Superexchange (off-resonant) Sequential (resonant)
Adiabatic VS Diabatic representation Suppose we know the molecular Donor-Acceptor part of the Hamiltonian, and its eigenstates: they define the adiabatic electron-vibrational states ψ Through adiabatic states ραβ (t ) Adiabatic states can be thought as referring to electronic states in fixed-nuclei configurations, these states experience possible transitions due to the nonadiabatic coupling because of the motion of nuclei. Is there an alternative? Thinking about how the Born-Oppenheimer approx. is carried on: ϕ a (r ; R ) ϕ a (r ; R0 ) H el ( R) = H 0 ( R0 ) + V ( R, R0 ) Vab ( R, R0 ) = ϕ a V ( R, R0 ) ϕb
Adiabatic VS Diabatic representation So that now the coupling is included in the V interaction, while all of the nonadiabaticity elements vanish identically (diabatic states χ ϕ are not R dependent!) Then, due to completeness assumption, a basis change is always possible: So that for the density matrix we have the diabatic representation: That can be used to straightforwardly compute the diabatic electronic state populations:
The Electron Transfer problem Theoretical modeling: splitting up electronic & nuclear contribution We introduce an effective potential experienced by the excess injected electron: V (r ) = Vm (r ) m Where the states can be considered analogue of those diabatic states that we are going to introduce later. Through such an expansion: Approx.: we neglect overlap & 3-centers integrals
The Electron Transfer problem If we now express the off-diagonal part as: We have just rigorously defined the transfer integral or inter-state coupling and the full electronic hamiltonian is: Similarly we can adopt such a solution for photoinduced ET, simply replacing the 1e states with the many-electron wavefunction Φ m ( r, σ ) (that describes spatial charge localization and includes Coulomb-interaction) Also, vibrational degrees of freedom are straightforward to include: Where Tnuc is the kinetic energy of vibrations, Um includes terms due to motion and intercoupling of the nuclei, and we have exploited dependencies on the vibrational coordinate R
The Electron Transfer problem Regimes of electron transfer Let s introduce some typical times for electronic quantum motion and vibrational motion (in absence of strong damping): We have two limiting cases: tel << tvib Adiabatic case: before any changes in the nuclear configuration occurs, the charges are able to move many times Delocalization: gain of the corresponding energy Rearrangement of the vibrational degree of freedom configuration: (Arrhenius ET rate)
tel >> tvib Nonadiabatic case: electronic wave function has not enough time to move completely from the donor to the acceptor Localization energy Eloc gained by the presence of the excess electron Since VDA is small, perturbational treatment is possible: in lowest order (Golden Rule ET rate)
Photoinduced ultrafast Electron Transfer Def: the ET process is so fast that no complete vibrational relaxation is possible Optical pulses required in the same region: ultrafast laser technology required for experiments No longer of non-adiabatic type! τ vib (10 12 s ) > tel (very high value for VDA)
Photoinduced ultrafast Electron Transfer Beyond the limit of nonadiabatic ET Improve lowest order perturbational theory Non-perturbative description of the ET Path-integral approach Density matrix time evolution approach Small number of active vibrational coordinates All remaining are assumed to form a heat-bath
Photoinduced ultrafast Electron Transfer System Reservoir separation H= Molecular + ext field Vibrational ground state + ET term + H S (t ) = H mol + H F (t ) = H g ϕ g ϕ g + H DA H S R + HR = K m ( s )Φ m ( Z ) m Multiple factorized form In the bi-linear form of the coupling potential obtained neglecting transfer integrals dependencies from (s,z), we have: Reservoir part System part Note the dependence on the equilibrium position!
QME for PU Electron Transfer reactions Markov approximation supposed to be valid Can be read as assuming a coarse grain time step: information on the reduced density matrix not required/obtainable for t < τ mem τ mem But we need an external field! Whose pulse duration can be evaluated in relationship with τ mem τ pulse : τ impulse V τ mem τ long? V
QME for PU Electron Transfer reactions Choosing the basis: diabatic Adiab: Diab: Easy to compute (starting with the density matrix) the diabatic electronic state populations These last ones are directly related to the ET rate But at first sight not appropriate if the inter-state coupling VDA is strong (adiabatic state representation permits non-perturbative description) (directly from von Neumann eq., note the appearance of interstate couplings)
QME for PU Electron Transfer reactions Problem: the diabatic representation is no energy representation! Hint: think about which kind of states defines it No possibility to make use of the Redfield tensor: to express non-perturbatively the real part (dissipative processes) of the RDM time-evolution:
Expressed in terms of the damping matrix related to irreversible redistributions in the RDM populations amplitude; in Markov approx. it reads: (u ) M ab,cd (t ) = Cuv (t ) K ab K cd( v ) u,v Where the memory matrices Mab,cd determine the time-span for correlations and depend on both: (system-reservoir coupling) Fluctuations with respect to average (thermal equilibrium) Correlation functions Cuv (t ) (remember the fluctuation dissipation theorem) for the reservoir
QME for PU Electron Transfer reactions Let s move one step back: operator dynamics i ρˆ (t ) = H Seff ρˆ (t ) ρˆ (t ) H Seff + + K u ρˆ (t )Λ(u+) + Λu ρˆ (t ) K u t ρ u ( ) ( ) * dissipative contributions acting only on the left or on the right side can be comprised in a non-hermitian hamiltonian where the others acting on both left and right size compensate this change in the vector state norm
The derivation of * was made based on the following steps/assumptions: Introduction of projector operator Perturbation treatment of the RDO of 2nd order in HS-R Usage of the multiple factorized form for S-R coupling and of mean field results for the expectation values of the reservoir part
Going back to Schroedinger representation: And finally applying the Markov approximation to the dissipative part after have switched again to the interaction representation
QME for PU Electron Transfer reactions For the system contribution to the interaction hamiltonian we assume a single harmonic reaction coordinate: So that for the respective diabatic matrix elements Where we have made use of the operator-based definition for the states derived from the Born-Oppenheimer wavefunctions: χ an = D + ( g a ( s )) N ShiftedHamiltonian eigenstate Displacement intensity in dimensionless coordinates g Non-shifted eigenstate of oscillators Hamiltonian
The bi-linear coupling is responsible of the form of the matrix elements: ( s s ( m ) ) 2ms Ω s ( Ds + Ds+ ) So that we are left with (avoiding the numerical factors) an overlap integral of the form: χ k, K χ l, L 1 = K D( sk ) D + ( sl ) L 1 And then we stop to the zeroth order (the only one non-identically 0, since we g ab )=the 0 expansion of the exponential displacement are dealing with operators on the oscillator states (similar derivation can be found e.g. for the Franck-Condon factor).
This means, intuitively, that the system contribution to the interaction HS-R couples only firstneighbors shifted-oscillators in the system coordinate
QME for PU Electron Transfer reactions Now we want to find an expression for the operator: Starting with the adiabatic states we obtain: And we remember we can always give this result in terms of the diabatic basis: As well as for the Λ operator (Cmn(ω ) are the half sided Fourier transforms of the correlation functions seen before):
QME for PU Electron Transfer reactions Now we have an expression also for the dissipative term contained in the diabatic basis representation of the QME. Some remarks: We have by-passed the problems concerning a big VDA: the result has to be considered exact for all kinds of the transfer integral; So far, we should only concern about the 2nd-order perturbational treatment subtended: high values of the coupling strength to the reservoir should be considered with care.
QME for PU Electron Transfer reactions What have we actually done? schroedinger Redfield tensor calculation adiabatic QME for RDO QME for RDM Final result! ρµν (t ) = ρµν (t ) + ρµν (t ) t t non diss t diss diabatic
Rate expressions Intent: from the solution of the general density matrix equation to a transfer rate approach Ansatz: from Pauli form we expect exponential decay of excited populations in the donor state PD(t) Leads to (at least) two possible definitions: asintotic like lifetime like
Rate expressions : QME 1 : QME 2 : Golden Rule (reference) Single relevant coordinate: E = 100 /cm Notice: high VDA adiabatic case expected progressive independence from the transfer integral
Bibliography: Charge and energy transfer dynamics in molecular systems, V. May-O. Kuehn, Wiley (2004) Quantum Transport: Atom to Transistor, S. Datta, Cambridge Univ. Press (2005) A special thank to dr. Lars Kecke for useful discussions and suggestions
Mache es einfach wie moeglich, aber nicht einfacher. (A. Einstein)