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Dissipation of vibronic energy in a dimer
JOURNALOF
Journal of Luminescence 53(1992) 195—197 North-FloHand
Dissipation of vibronic energy in a dimer Michael Schreiber
a
Jörg Sabczynski
a
and Volkhard May
h
Institut fr~rPhysikalische Chemie, Johannes-Gutenberg-Unit’ersitat, Jakob-Welder- Weg 11, W-6500 Mainz, Germany “Institute of Molecular Biology, Robert-Roessle-Str. 10, 0-1115 Berlin, Germany “
The density matrix theory is used for the study of the dissipative quantum dynamics of electron transfer in a dimer. The vibrational modes of the dimer are divided into a single interaction coordinate coupling to the transfered electron and the remaining modes which form a dissipative environment. To correlate the dissipative dynamics with the exact eigenlevels computed for the model system without dissipative environment we analyse the time dependence of the expectation value of the number of vibrational quanta. We analyse the renormalisation of the eigenvalues due to the damping and the relaxation of an excitation into these states.
1. Introduction The investigation of electronic properties of dimers as the simplest examples of molecular aggregates received much attention in recent years. On the one hand the coupled electron— vibration spectrum is of interest. It has been calculated by various authors [1—6].On the other hand the dynamic properties have been investigated concentrating on different aspects like the calculation of rate constants for the charge transfer [7]. the search for chaotic transfer scenarios [8—10],and the inclusion of a dissipative environment [11,12]. In this paper we are concerned with the latter. The treatment of the problem in [11] contains an approximative consideration of the electron— vibration coupling using the classical description of the vibrational degrees of freedom. This implies the incorporation of low-frequency modes which is in contradiction to the assumption that the vibrational modes follow the electron motion instantaneously (the so-called self-trapping approximation). In order to avoid this inconsistency we developed an appropriate density matrix the-
Correspondence to: Dr. M. Schreiber, institut fur Physikalische Chemie, Johannes-Gutenberg-Universitat, JakobWelder-Weg 11, W-6500 Mainz, Germany.
ory [12,13] which allows an exact treatment of the strong coupling between the electron and the vibrational modes of the molecular constituents and, in addition, the microscopic desCription of the interaction with the dissipative environment.
2. The model In the following, we investigate the transfer dynamics of an excess electron in a dimer model with a single vibrational mode characterized by the interaction coordinate Q. For the actual calculations we employ the representation of the Mth vibronic excitation of the electron on the tnth isolated monomer: m, M) p.). Then the dimer Hamiltonian reads H0
=
~
{[e,~ —
+VF
FC
.
hw,~g,?~ + hw,,(M +
(p. ~)ljp.)KiH
~-)]
~,,
(1)
‘ .
with the electronic energies e
,
the vibrational
frequency w~,the dimensionless coupling constants g,,, of the linear interaction between the electron density and the interaction coordinate and the electronic inter-centre transfer renormalized by the Franck—Condon factor FFC.
0022-2313/92/105.00 © 1992 — Elsevier Science Publishers B.V. All rights reserved
/tI. Si/ircihcr Ct (1/.
IOU
J)o opatunI of ,bro,iu c,iergv in a don,,r
Here we restrict our study to the case of a symmetric dimer by taking e1 e~and g, g-, g. The environment will he considered as an ensemble of harmonic oscillators with coordinates q, in thermal equilibrium. Their coupling to the dimer is taken in the linear form
Energy
—
—
H,
—
~KqQ.
2
0
(2) -2
The corresponding equations of motion for the densit~matrix elements can he derived in applying the non-equilibrium Green’s fLinction technique (for details see ref. [131). The dissipative terms in the equations of motion are governed by the damping rates of the vibronic levels
~
y(w)
I)otentaI surfaces and
~.
=
2 ~a( w ) [I
+ n(
~
)
( ~)
J
determined by the density of states a( w) of the environment and their respective thermal occupa— tion u( w). y( w~)and y( w,.) reflect the emission and the absorption of one environmental quantum, respectively,
_______
_____________
-8
-6
-4 2 0 2 4 6 8 Interaction Coordinate Q I ,\diahaiie (solid curse) :utd diahatie (hmkcn curse)
respective vihronic lesels with (F liw thin solid lines) and without (F’ I). thin broken lines) inter centre coupling for the dimer I lamiltonian (I). flie energ\ is measured in uniis of li,,
—
quanta. This quantity can he expressed in terms of the density matrix elements in the following form [13]
3. Results and discussion A pronounced dynamic behaviour can he cxpected tor strong electron—vibration coupling and a moderate inter—centre transfer interaction [14]. Therefore, v~econcentrate our analysis on g 2 and V hwy. As a basis for the further discussion we present in fig. I the spectrum of the dimer obtained from a diagonalization of H,, using 50 oscillatory states, in comparison with the trivial V I) case, i.e.,2g,,,harmonic displaced .~Q and ~ oscillators ~ Due to the by renormal— iLation of the inter-centre coupling the lowest states shift slightly and split. hut the splitting cannot he resolved in fig. 1 for the low-lying states. The spectrum of the higher vibrationally excited states in the energy range of the harrier of the adiabatic potential in fig. 1 shows no more resemblance to the monomer spectrum. It is just in this energy region where we prepare the initial state 1, 3), the time-evolution of which we intend to pursue. To correlate the respective dissipative dynamics with the computed eigenlevels it is most ap=
=
=
propriate to consider the time—dependence of the expectation value 14’ of the number of vibrational
—
14’
g
+
( Mp~,
~
mM —2g(
—
1 )
+
I Re p 01,,+
(4)
i:i,iti)•
It should be noted that 14’ has been defined for an undisplaced harmonic oscillator. Since we are using the representation of displaced oscillators the energy shift .X E is reflected in the first term of eq. (4). As mentioned, in fig. 2 we have used state I,on3) the so that I 0) one + 3 the 7. initial Accordingly. first W( instance would expect an asymptotic value of W 4 to he reached for all values of the damping rate. eq. (3). However, this holds only for sufficiently small intercentre coupling V (see ref. [13]). In our case of V ttw, shown in fig. 2 a contrasting behavior can he observed. For this relatively large value of V the considerable changes in the spectrum as shown in fig. 1 influence the relaxation. In particular, the lowest eigenstate reached by the relaxation process in the here considered low temperature case is characterized by W(i 3~).This value is asymptotically reached for small =
—
—
=
—*
—
M. Schreiher et al.
/ Dissipation of tibronic energy in
w 8
=
~
~ 4
\ ...‘~—=
3 0
10
20
30
40
50 w~t
Fig. 2. Time-dependence of the number of vibrational quanta for different damping rates y, determined by a/hw~= l0~ (solid line), 10—2 (long dashes), 10_i (short dashes), and I (dash-dotted line) and k~T= 0.lhu,,. The initially prepared state is II, 3). The values of W corresponding to the exact eigenstates in fig. 1 are denoted by arrows.
a dimer
197
On the other hand, the time-dependence of the electron density on the different centres (see refs. [12,131) is governed by the Frank—Condon renormalized inter-centre coupling V yielding slower oscillations. This coherent motion and its alteration to incoherent (hopping-like) motion is discussed elsewhere [13]. Here we have been concerned with the dissipation of the vibronic energy of the dimer. Besides the discussed change of the asymptotically reached states an important result of our analysis is the non-exponential relaxation in the overdamped case.
References values of the damping constant, but this behavior can only be conceived on much larger time-scales than presented in fig. 2. For larger damping the expectation value W(t —s x) is renormalized reaching values significantly even below g2 4. This is caused by the shift as well as the broadening of the energy levels of the dimer spectrum. It is not clear at the moment, why for the second largest damping rate the relaxation ends at W 4.6, corresponding to the excited level in fig. 1 with W= 4.75 without renormalization. It is obvious from fig. 2 that the asymptotic values are quickly reached for larger damping. In the case of the lowest damping value used this behaviour is masked by oscillations with frequency which are due to the fact that the initially prepared state is no eigenstate. It is interesting to note that in the case of the largest =
=
~,
considered damping the relaxation takes comparatively long, here we are in the regime of overdamping.
[1] R. Friesner and K. Silbey, J. Chem. Phys. 74 (1981) 1166; 75 (1981) 3925; 5630. 121 H. Köppel, W. Domcke and L.S. Cederhaum, Adv. Chem. Phys. 57 (1984) 59. 131 S. Weber and H. Biittner, Solid State Commun. 56(1985) 395. [4] S.F. Fischer and N. Heider, in: Tunneling, eds. J. Jortner and B. Pullmann (Reidel, Dordrecht, 1986) p. 315. [5] C. Durst, E. Sigmund, P. Reineker and A. Scheuing. J. Phys. C 19 (1986) 2701. 161 W. Schmidt and M. Schreiber, J. Chem. Phys. 86 (1987) 953. 171 J.N. Onuchic, D.N. Beratan and J.J. Hopfield, J. Phys. Chem. 90 (1986) 3707 and references therein. 181 D. Feinberg and J. Ranninger, Physica D 14 (1984) 29. [9] P X. Tran and T. Tamura, Phys. Rev B 37 (1988) 2270. 1101 L. MOller, J. Stoize, H. Leschke and P. Nagel, Phys. Rev. A 44 (1991) 1022. [11] G.P. Tsironis, V.M. Kenkre and D. Finley, Phys. Rev. A 37 (1988) 4474. 112] V. May and M. Schreiber, Chem. Phys. Lett. 181 (1991)
267. 1131 V. May and M. Schreiber, Phys. Rev. A, to be published. 1141 V. May, Chem. Phys. 143 (1990) 437.
Journal of Luminescence 53(1992) 195—197 North-FloHand
Dissipation of vibronic energy in a dimer Michael Schreiber
a
Jörg Sabczynski
a
and Volkhard May
h
Institut fr~rPhysikalische Chemie, Johannes-Gutenberg-Unit’ersitat, Jakob-Welder- Weg 11, W-6500 Mainz, Germany “Institute of Molecular Biology, Robert-Roessle-Str. 10, 0-1115 Berlin, Germany “
The density matrix theory is used for the study of the dissipative quantum dynamics of electron transfer in a dimer. The vibrational modes of the dimer are divided into a single interaction coordinate coupling to the transfered electron and the remaining modes which form a dissipative environment. To correlate the dissipative dynamics with the exact eigenlevels computed for the model system without dissipative environment we analyse the time dependence of the expectation value of the number of vibrational quanta. We analyse the renormalisation of the eigenvalues due to the damping and the relaxation of an excitation into these states.
1. Introduction The investigation of electronic properties of dimers as the simplest examples of molecular aggregates received much attention in recent years. On the one hand the coupled electron— vibration spectrum is of interest. It has been calculated by various authors [1—6].On the other hand the dynamic properties have been investigated concentrating on different aspects like the calculation of rate constants for the charge transfer [7]. the search for chaotic transfer scenarios [8—10],and the inclusion of a dissipative environment [11,12]. In this paper we are concerned with the latter. The treatment of the problem in [11] contains an approximative consideration of the electron— vibration coupling using the classical description of the vibrational degrees of freedom. This implies the incorporation of low-frequency modes which is in contradiction to the assumption that the vibrational modes follow the electron motion instantaneously (the so-called self-trapping approximation). In order to avoid this inconsistency we developed an appropriate density matrix the-
Correspondence to: Dr. M. Schreiber, institut fur Physikalische Chemie, Johannes-Gutenberg-Universitat, JakobWelder-Weg 11, W-6500 Mainz, Germany.
ory [12,13] which allows an exact treatment of the strong coupling between the electron and the vibrational modes of the molecular constituents and, in addition, the microscopic desCription of the interaction with the dissipative environment.
2. The model In the following, we investigate the transfer dynamics of an excess electron in a dimer model with a single vibrational mode characterized by the interaction coordinate Q. For the actual calculations we employ the representation of the Mth vibronic excitation of the electron on the tnth isolated monomer: m, M) p.). Then the dimer Hamiltonian reads H0
=
~
{[e,~ —
+VF
FC
.
hw,~g,?~ + hw,,(M +
(p. ~)ljp.)KiH
~-)]
~,,
(1)
‘ .
with the electronic energies e
,
the vibrational
frequency w~,the dimensionless coupling constants g,,, of the linear interaction between the electron density and the interaction coordinate and the electronic inter-centre transfer renormalized by the Franck—Condon factor FFC.
0022-2313/92/105.00 © 1992 — Elsevier Science Publishers B.V. All rights reserved
/tI. Si/ircihcr Ct (1/.
IOU
J)o opatunI of ,bro,iu c,iergv in a don,,r
Here we restrict our study to the case of a symmetric dimer by taking e1 e~and g, g-, g. The environment will he considered as an ensemble of harmonic oscillators with coordinates q, in thermal equilibrium. Their coupling to the dimer is taken in the linear form
Energy
—
—
H,
—
~KqQ.
2
0
(2) -2
The corresponding equations of motion for the densit~matrix elements can he derived in applying the non-equilibrium Green’s fLinction technique (for details see ref. [131). The dissipative terms in the equations of motion are governed by the damping rates of the vibronic levels
~
y(w)
I)otentaI surfaces and
~.
=
2 ~a( w ) [I
+ n(
~
)
( ~)
J
determined by the density of states a( w) of the environment and their respective thermal occupa— tion u( w). y( w~)and y( w,.) reflect the emission and the absorption of one environmental quantum, respectively,
_______
_____________
-8
-6
-4 2 0 2 4 6 8 Interaction Coordinate Q I ,\diahaiie (solid curse) :utd diahatie (hmkcn curse)
respective vihronic lesels with (F liw thin solid lines) and without (F’ I). thin broken lines) inter centre coupling for the dimer I lamiltonian (I). flie energ\ is measured in uniis of li,,
—
quanta. This quantity can he expressed in terms of the density matrix elements in the following form [13]
3. Results and discussion A pronounced dynamic behaviour can he cxpected tor strong electron—vibration coupling and a moderate inter—centre transfer interaction [14]. Therefore, v~econcentrate our analysis on g 2 and V hwy. As a basis for the further discussion we present in fig. I the spectrum of the dimer obtained from a diagonalization of H,, using 50 oscillatory states, in comparison with the trivial V I) case, i.e.,2g,,,harmonic displaced .~Q and ~ oscillators ~ Due to the by renormal— iLation of the inter-centre coupling the lowest states shift slightly and split. hut the splitting cannot he resolved in fig. 1 for the low-lying states. The spectrum of the higher vibrationally excited states in the energy range of the harrier of the adiabatic potential in fig. 1 shows no more resemblance to the monomer spectrum. It is just in this energy region where we prepare the initial state 1, 3), the time-evolution of which we intend to pursue. To correlate the respective dissipative dynamics with the computed eigenlevels it is most ap=
=
=
propriate to consider the time—dependence of the expectation value 14’ of the number of vibrational
—
14’
g
+
( Mp~,
~
mM —2g(
—
1 )
+
I Re p 01,,+
(4)
i:i,iti)•
It should be noted that 14’ has been defined for an undisplaced harmonic oscillator. Since we are using the representation of displaced oscillators the energy shift .X E is reflected in the first term of eq. (4). As mentioned, in fig. 2 we have used state I,on3) the so that I 0) one + 3 the 7. initial Accordingly. first W( instance would expect an asymptotic value of W 4 to he reached for all values of the damping rate. eq. (3). However, this holds only for sufficiently small intercentre coupling V (see ref. [13]). In our case of V ttw, shown in fig. 2 a contrasting behavior can he observed. For this relatively large value of V the considerable changes in the spectrum as shown in fig. 1 influence the relaxation. In particular, the lowest eigenstate reached by the relaxation process in the here considered low temperature case is characterized by W(i 3~).This value is asymptotically reached for small =
—
—
=
—*
—
M. Schreiher et al.
/ Dissipation of tibronic energy in
w 8
=
~
~ 4
\ ...‘~—=
3 0
10
20
30
40
50 w~t
Fig. 2. Time-dependence of the number of vibrational quanta for different damping rates y, determined by a/hw~= l0~ (solid line), 10—2 (long dashes), 10_i (short dashes), and I (dash-dotted line) and k~T= 0.lhu,,. The initially prepared state is II, 3). The values of W corresponding to the exact eigenstates in fig. 1 are denoted by arrows.
a dimer
197
On the other hand, the time-dependence of the electron density on the different centres (see refs. [12,131) is governed by the Frank—Condon renormalized inter-centre coupling V yielding slower oscillations. This coherent motion and its alteration to incoherent (hopping-like) motion is discussed elsewhere [13]. Here we have been concerned with the dissipation of the vibronic energy of the dimer. Besides the discussed change of the asymptotically reached states an important result of our analysis is the non-exponential relaxation in the overdamped case.
References values of the damping constant, but this behavior can only be conceived on much larger time-scales than presented in fig. 2. For larger damping the expectation value W(t —s x) is renormalized reaching values significantly even below g2 4. This is caused by the shift as well as the broadening of the energy levels of the dimer spectrum. It is not clear at the moment, why for the second largest damping rate the relaxation ends at W 4.6, corresponding to the excited level in fig. 1 with W= 4.75 without renormalization. It is obvious from fig. 2 that the asymptotic values are quickly reached for larger damping. In the case of the lowest damping value used this behaviour is masked by oscillations with frequency which are due to the fact that the initially prepared state is no eigenstate. It is interesting to note that in the case of the largest =
=
~,
considered damping the relaxation takes comparatively long, here we are in the regime of overdamping.
[1] R. Friesner and K. Silbey, J. Chem. Phys. 74 (1981) 1166; 75 (1981) 3925; 5630. 121 H. Köppel, W. Domcke and L.S. Cederhaum, Adv. Chem. Phys. 57 (1984) 59. 131 S. Weber and H. Biittner, Solid State Commun. 56(1985) 395. [4] S.F. Fischer and N. Heider, in: Tunneling, eds. J. Jortner and B. Pullmann (Reidel, Dordrecht, 1986) p. 315. [5] C. Durst, E. Sigmund, P. Reineker and A. Scheuing. J. Phys. C 19 (1986) 2701. 161 W. Schmidt and M. Schreiber, J. Chem. Phys. 86 (1987) 953. 171 J.N. Onuchic, D.N. Beratan and J.J. Hopfield, J. Phys. Chem. 90 (1986) 3707 and references therein. 181 D. Feinberg and J. Ranninger, Physica D 14 (1984) 29. [9] P X. Tran and T. Tamura, Phys. Rev B 37 (1988) 2270. 1101 L. MOller, J. Stoize, H. Leschke and P. Nagel, Phys. Rev. A 44 (1991) 1022. [11] G.P. Tsironis, V.M. Kenkre and D. Finley, Phys. Rev. A 37 (1988) 4474. 112] V. May and M. Schreiber, Chem. Phys. Lett. 181 (1991)
267. 1131 V. May and M. Schreiber, Phys. Rev. A, to be published. 1141 V. May, Chem. Phys. 143 (1990) 437.