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Is the coupled coherent and incoherent exciton motion characterized by its mean square displacement?
Volume 69A, number 2
PHYSICS LETTERS
27 November 1978
IS THE COUPLED COHERENT AND INCOHERENT EXCITON MOTION CHARACTERIZED BY ITS MEAN SQUARE DISPLACEMENT? P. REINEKER and R. KUHNE Abteilung ft~rTheoretische Physik I, Universitat Ulm, 7900 VIm, Germany Received 13 September 1978
The relevance of the mean square displacement for the comparison between different microscopic models is discussed starting from the Nakajima—Zwanzig generalized master equation. The time dependences of its kernel and of the mean square displacement are explicitly given for the Haken—Strobl model of exciton motion.
In recent years several methods for describing the coupled coherent and incoherent motion of excitons have been proposed [1—3]. For the comparison of the various models the mean square displacement, i.e. the second moment of the motion, has been calculated in each of the models [2—5]. In this paper the question is discussed, how far the coupled coherent and incoherent exciton motion is characterized by this quantity. To that end we start from the equation of motion for the density operators p of excitons L 1~ p “ ‘ where L is the Liouvile operator of the system. Using the Nakajima—Zwanzig projection formalism [6,7], we arrive at —
—
d ~9p(t)
=
t
exp[(i— ~)Lt’] (1
+
~L exp[(1
—
—
~)L}~p(t
~)Lt] (1
—
~Pn(t)
J’ dt’ E ~
=
0
(t’)p~,(t — t’)
÷1.
(4)
‘~
The inhomogeneity I results from the last term in eq. (2). In the following discussion we shall assume that the kernel Gnn(t’) is invariant under translation and inversion: G =G G =G (5) nfl
fl—n”
m
—m’
and that the density operator p at the initial time is diagonal (O~= (6) ann’
/
~~n’ /
With this initial condition the inhomogeneous term disappears and from translational invariance we arrive at
dt’ ~PL{2~(t’)
+
d
—
t’)
~)~(O).
~p~(t)
(2)
In order to obtain an equation of motion for the probability p,~of finding the exciton at site n, we use as projection operator ~ the following expression
~..EIn>
(3)
=
f dt’ ~
G~ ~~(t’)p~(t
—
t’).
(7)
The Fourier transform of p~(r)and G~_~(t) are given by .
p~(t)=
Z~e11~~(t), Gm(t)
=
N~
E ehlcm Gk(t), (8)
where N is the number of molecules in a linear chain and periodic boundary conditions are assumed. Eq. (7) may then be written as
133
Volume 69A, number 2
~~t)
=
f dt’
PHYSICS LETTERS
and together with eq. (12)we have from eq. (15) G~(t)(t
r’).
-
(9)
0
After summation over n in eq. (7) and using the normalization
E p,~
1,
=
i.e.
N75~= 1,
(10)
from eqs. (7) and (9) we arrive at t
G
,(t’)
—
=
0,
f’ dt’
~
0(t’) = 0. (11) 0 Expressions conclude (11) are valid for all t, and therefore we 0
‘~
~ G,(r)
=
0,
(t)
=
(d/dt) (n(t)) = 0,
Z~np~(t)
(n(t)) =
=
c=
E np~(O).
(18)
If the initial distribution is symmetric with respect to the origin eq. (18) results in (n(t)) = 0.
t
f dt’ ~
27 November 1978
0.
(12)
(19)
The second moment of the density operator, i.e. the mean square displacement is 2defined by (n2(t)) = n2p~(t)= —N d (20) n dK 1K0 and its equation of motion is given by
E
—~
,
t
In the sum of eq. (7) the term with n’ n may be considered separately, and with the help of the first of eqs. (12), which has been obtained by using only the translational invariance of the kernel ~ (eq. (5)) and the normalization (10) of the density operator, we arrive at the following generalized rate equation: d
~p~(t)
~
=
n(n)
f dt’ G~~~(t’)
t
.4(fl2(t))
=
f dt’(E m2Gm(t?))( Ep~(t 0
(13)
(t)IL~=o
np~(t)= iN~-~ dg”
have —
t’))
m
f dt’(E mGm(t’)) (n(t
—
t’))
0
The first moment of the density operator is defined by ~
0
+ 2
0 X (p~,(t— t’) — p~(t— t’)).
(n(t)) =
(n~(t))=
f
dt’ ~ ~ n2G (t’)p~(t— t’). (21) ~ ~‘ Introducing again m = n — n’ as a new variable, we
~
(14)
,
t
+
f dt’ (E G 0
2(t
—
t’)).
(22)
(t’)) (n
Using eqs. (10), (12) and (17), we finally arrive at ~(n2(t))
=
f dt’ 0
Em2Gm(t~) m
and from the equation of motion (7) we have (n(t))
=f dt’ Z~Z~ nG n
0
Introducing m
=
n
n’
—
—
t’).
(15)
~
)
f
E p~,(t’
—
—
fdt’_-~GK(t’)J
is completely determined by ~m m2Gm(t’). In terms of the Fourier transform Gg(t) this means that every microscopic model having for the single value ~c= 0 the same second derivative of ~k(t) with respect to results in the same mean square displacement. As an example we consider the stochastic model for the coupled coherent and incoherent exciton motion [1] The mean square displacement has been calculated in refs. [41and from there we have
(16)
.
134
=
0,
(23)
~,
The assumed inversion symmetry of Gm (t) gives EmGm(t)
.
0 This result shows that the mean square displacement
n’, eq. (15) transforms into
(n(r)) = dt’ mGm(t’) + G(t’)(n(t t’))).
~
=
,(t’)p,(t
(17)
Volume 69A, number 2
PHYSICS LETTERS
27 November
1978
+ 2H~/(F+ ‘~~) (d/dt~n2(t))= [2H~/(r’ ÷‘~~)] exp {—2(r + ~
terms of the last line of eq. (27) and using the follow-
(24) 2’y~,where For nearest-neighbour interaction F = ~‘y0 and + Yo measures the strength of local and the strengths of non-local fluctuations; H 1 is the interaction matrix element between neighbouring molecules and is responsible for the coherent part of the exciton motion [1]. The comparison of eqs. (23) and (24) gives
ing relation for Bessel functions [9] : ~m J~(2H1t’x) = get1, in the case of nearest-neighbour interaction we
—
1)t}.
~
_-~—~
2K
=
=Em2G
(t’)~ -
m
m
a2 a
— —
=
2Gm(t’)
2
~
K (i”)II
=
E m m
8~y 2rt’ 1~t’)+ 4H~e
f
(t’)
~ {_27 2)}]. (28) The integral in eq. (28) may be calculated 1t~l_x easily and
87 16(t’) + 4H~exp {—2(F +
(25)
~1)t’},
demonstrating that the mean square displacement of the above model [1] would be reproduced if the time-dependence of each Gm(t’) were proportional to the right side of eq. (25). The exact evaluation of the kernel of the Nakajima— Zwanzig equation (2) for the HRS-model of the coupled coherent and incoherent exciton motion [1], however, results in the following expression [8]: 4~6m,O 7m)6(t’) + exp {—2Ft’} Gth(t’) = 1 X 4H~ dx [(x2—1) exp {—2’yi t’( 1 —x2 —
f
)}])
-~-(~
0
we again arrive at eq. (25). This result clearly shows that the rather involved time dependence of the kernel Gm (r) according to eq. (26), which determines the evolution in time of the probability of finding an exciton at site n (see eq. (7)), reduces to a 6-function and an exponential term, when the mean square displacement is calculated. Therefore the mean square displacement is not a very sensitive quantity in order to compare different microscopic models. The same reasoning naturally holds when the experiments. theories are applied for the descriptionmay, of diffusion Much more information however, be drawn from experiments whose characteristic time constant is comparable or shorter than the decay time of the phase of the exciton, such as
X [J~ +1 (2H 1 t’x) + J~1(2Ht’x)
—
2.J,~(211i t’x)].
(26) The ~m’~ are Bessel functions of integer order. Using this result G(t’) may be calculated according to eq. (25) 2Gm(t’) aK G(t) JK=O = m m
E
=
—~
E (m2F6m m
27m)6(t’) + exp {—2Fr’} 0
—
m
ESR of triplet excitons at low temperatures or time resolved optical spectroscopy. References [1] H. Haken and G. Strobi, The triplet state, ed. A.B. Zahlan (Cambridge U.P., 1967); Excitons, magnons and phoH. Haken and P. Reineker, nons, ed. A.B. Zahlan (Cambridge U.P., 1968); H. Haken and P. Reineker, Z. Phys. 249 (1972) 253; H. Haken and G. Strobl, Z. Phys. 262 (1973) 135. [2] M. Grover and R. Silbey, i. Chem. Phys. 54 (1971) 4843. [31 V.M. Kenkre and R.S. Knox, Phys. Rev. B9 (1974) 5279.
1
f dx-~-(~[(x2—1)exp
X 4H~
{_27i t’( l_X2)}])
0
P. Reineker, Z. Phys. 261 (1973) 187.
[5] V.M. Kenkre, Phys. Lett. 47A (1974) 119; Phys. Rev. B!!
X ~ m2 [J2÷ 1(2H1t’x) +J~1(2H1t’x) —
[41 P. E. Reineker, SchwarzerPhys. and H. Lett. Haken, 42A Phys. (1973) Lett. 389; 42A (1972) 317;
m 2J~(2H1t’x)].
(1975) 1741;B12 (1975) 2150. (27)
Shifting the summation variable in the first and second
[6] S. Nakajima, Prog. Theor. Phys. 20 (1958) 948. [71 R. Zwanzig, J. Chem. Phys. 33 (1960) 1338. [8] R. Kühne and P. Reineker, to be published. [9] Handbook of mathematical functions, eds. M. Abramowitz and IA. Stegun (Dover Publications, New York, 1970).
135
PHYSICS LETTERS
27 November 1978
IS THE COUPLED COHERENT AND INCOHERENT EXCITON MOTION CHARACTERIZED BY ITS MEAN SQUARE DISPLACEMENT? P. REINEKER and R. KUHNE Abteilung ft~rTheoretische Physik I, Universitat Ulm, 7900 VIm, Germany Received 13 September 1978
The relevance of the mean square displacement for the comparison between different microscopic models is discussed starting from the Nakajima—Zwanzig generalized master equation. The time dependences of its kernel and of the mean square displacement are explicitly given for the Haken—Strobl model of exciton motion.
In recent years several methods for describing the coupled coherent and incoherent motion of excitons have been proposed [1—3]. For the comparison of the various models the mean square displacement, i.e. the second moment of the motion, has been calculated in each of the models [2—5]. In this paper the question is discussed, how far the coupled coherent and incoherent exciton motion is characterized by this quantity. To that end we start from the equation of motion for the density operators p of excitons L 1~ p “ ‘ where L is the Liouvile operator of the system. Using the Nakajima—Zwanzig projection formalism [6,7], we arrive at —
—
d ~9p(t)
=
t
exp[(i— ~)Lt’] (1
+
~L exp[(1
—
—
~)L}~p(t
~)Lt] (1
—
~Pn(t)
J’ dt’ E ~
=
0
(t’)p~,(t — t’)
÷1.
(4)
‘~
The inhomogeneity I results from the last term in eq. (2). In the following discussion we shall assume that the kernel Gnn(t’) is invariant under translation and inversion: G =G G =G (5) nfl
fl—n”
m
—m’
and that the density operator p at the initial time is diagonal (O~= (6) ann’
/
~~n’ /
With this initial condition the inhomogeneous term disappears and from translational invariance we arrive at
dt’ ~PL{2~(t’)
+
d
—
t’)
~)~(O).
~p~(t)
(2)
In order to obtain an equation of motion for the probability p,~of finding the exciton at site n, we use as projection operator ~ the following expression
~..EIn>
(3)
=
f dt’ ~
G~ ~~(t’)p~(t
—
t’).
(7)
The Fourier transform of p~(r)and G~_~(t) are given by .
p~(t)=
Z~e11~~(t), Gm(t)
=
N~
E ehlcm Gk(t), (8)
where N is the number of molecules in a linear chain and periodic boundary conditions are assumed. Eq. (7) may then be written as
133
Volume 69A, number 2
~~t)
=
f dt’
PHYSICS LETTERS
and together with eq. (12)we have from eq. (15) G~(t)(t
r’).
-
(9)
0
After summation over n in eq. (7) and using the normalization
E p,~
1,
=
i.e.
N75~= 1,
(10)
from eqs. (7) and (9) we arrive at t
G
,(t’)
—
=
0,
f’ dt’
~
0(t’) = 0. (11) 0 Expressions conclude (11) are valid for all t, and therefore we 0
‘~
~ G,(r)
=
0,
(t)
=
(d/dt) (n(t)) = 0,
Z~np~(t)
(n(t)) =
=
c=
E np~(O).
(18)
If the initial distribution is symmetric with respect to the origin eq. (18) results in (n(t)) = 0.
t
f dt’ ~
27 November 1978
0.
(12)
(19)
The second moment of the density operator, i.e. the mean square displacement is 2defined by (n2(t)) = n2p~(t)= —N d (20) n dK 1K0 and its equation of motion is given by
E
—~
,
t
In the sum of eq. (7) the term with n’ n may be considered separately, and with the help of the first of eqs. (12), which has been obtained by using only the translational invariance of the kernel ~ (eq. (5)) and the normalization (10) of the density operator, we arrive at the following generalized rate equation: d
~p~(t)
~
=
n(n)
f dt’ G~~~(t’)
t
.4(fl2(t))
=
f dt’(E m2Gm(t?))( Ep~(t 0
(13)
(t)IL~=o
np~(t)= iN~-~ dg”
have —
t’))
m
f dt’(E mGm(t’)) (n(t
—
t’))
0
The first moment of the density operator is defined by ~
0
+ 2
0 X (p~,(t— t’) — p~(t— t’)).
(n(t)) =
(n~(t))=
f
dt’ ~ ~ n2G (t’)p~(t— t’). (21) ~ ~‘ Introducing again m = n — n’ as a new variable, we
~
(14)
,
t
+
f dt’ (E G 0
2(t
—
t’)).
(22)
(t’)) (n
Using eqs. (10), (12) and (17), we finally arrive at ~(n2(t))
=
f dt’ 0
Em2Gm(t~) m
and from the equation of motion (7) we have (n(t))
=f dt’ Z~Z~ nG n
0
Introducing m
=
n
n’
—
—
t’).
(15)
~
)
f
E p~,(t’
—
—
fdt’_-~GK(t’)J
is completely determined by ~m m2Gm(t’). In terms of the Fourier transform Gg(t) this means that every microscopic model having for the single value ~c= 0 the same second derivative of ~k(t) with respect to results in the same mean square displacement. As an example we consider the stochastic model for the coupled coherent and incoherent exciton motion [1] The mean square displacement has been calculated in refs. [41and from there we have
(16)
.
134
=
0,
(23)
~,
The assumed inversion symmetry of Gm (t) gives EmGm(t)
.
0 This result shows that the mean square displacement
n’, eq. (15) transforms into
(n(r)) = dt’ mGm(t’) + G(t’)(n(t t’))).
~
=
,(t’)p,(t
(17)
Volume 69A, number 2
PHYSICS LETTERS
27 November
1978
+ 2H~/(F+ ‘~~) (d/dt~n2(t))= [2H~/(r’ ÷‘~~)] exp {—2(r + ~
terms of the last line of eq. (27) and using the follow-
(24) 2’y~,where For nearest-neighbour interaction F = ~‘y0 and + Yo measures the strength of local and the strengths of non-local fluctuations; H 1 is the interaction matrix element between neighbouring molecules and is responsible for the coherent part of the exciton motion [1]. The comparison of eqs. (23) and (24) gives
ing relation for Bessel functions [9] : ~m J~(2H1t’x) = get1, in the case of nearest-neighbour interaction we
—
1)t}.
~
_-~—~
2K
=
=Em2G
(t’)~ -
m
m
a2 a
— —
=
2Gm(t’)
2
~
K (i”)II
=
E m m
8~y 2rt’ 1~t’)+ 4H~e
f
(t’)
~ {_27 2)}]. (28) The integral in eq. (28) may be calculated 1t~l_x easily and
87 16(t’) + 4H~exp {—2(F +
(25)
~1)t’},
demonstrating that the mean square displacement of the above model [1] would be reproduced if the time-dependence of each Gm(t’) were proportional to the right side of eq. (25). The exact evaluation of the kernel of the Nakajima— Zwanzig equation (2) for the HRS-model of the coupled coherent and incoherent exciton motion [1], however, results in the following expression [8]: 4~6m,O 7m)6(t’) + exp {—2Ft’} Gth(t’) = 1 X 4H~ dx [(x2—1) exp {—2’yi t’( 1 —x2 —
f
)}])
-~-(~
0
we again arrive at eq. (25). This result clearly shows that the rather involved time dependence of the kernel Gm (r) according to eq. (26), which determines the evolution in time of the probability of finding an exciton at site n (see eq. (7)), reduces to a 6-function and an exponential term, when the mean square displacement is calculated. Therefore the mean square displacement is not a very sensitive quantity in order to compare different microscopic models. The same reasoning naturally holds when the experiments. theories are applied for the descriptionmay, of diffusion Much more information however, be drawn from experiments whose characteristic time constant is comparable or shorter than the decay time of the phase of the exciton, such as
X [J~ +1 (2H 1 t’x) + J~1(2Ht’x)
—
2.J,~(211i t’x)].
(26) The ~m’~ are Bessel functions of integer order. Using this result G(t’) may be calculated according to eq. (25) 2Gm(t’) aK G(t) JK=O = m m
E
=
—~
E (m2F6m m
27m)6(t’) + exp {—2Fr’} 0
—
m
ESR of triplet excitons at low temperatures or time resolved optical spectroscopy. References [1] H. Haken and G. Strobi, The triplet state, ed. A.B. Zahlan (Cambridge U.P., 1967); Excitons, magnons and phoH. Haken and P. Reineker, nons, ed. A.B. Zahlan (Cambridge U.P., 1968); H. Haken and P. Reineker, Z. Phys. 249 (1972) 253; H. Haken and G. Strobl, Z. Phys. 262 (1973) 135. [2] M. Grover and R. Silbey, i. Chem. Phys. 54 (1971) 4843. [31 V.M. Kenkre and R.S. Knox, Phys. Rev. B9 (1974) 5279.
1
f dx-~-(~[(x2—1)exp
X 4H~
{_27i t’( l_X2)}])
0
P. Reineker, Z. Phys. 261 (1973) 187.
[5] V.M. Kenkre, Phys. Lett. 47A (1974) 119; Phys. Rev. B!!
X ~ m2 [J2÷ 1(2H1t’x) +J~1(2H1t’x) —
[41 P. E. Reineker, SchwarzerPhys. and H. Lett. Haken, 42A Phys. (1973) Lett. 389; 42A (1972) 317;
m 2J~(2H1t’x)].
(1975) 1741;B12 (1975) 2150. (27)
Shifting the summation variable in the first and second
[6] S. Nakajima, Prog. Theor. Phys. 20 (1958) 948. [71 R. Zwanzig, J. Chem. Phys. 33 (1960) 1338. [8] R. Kühne and P. Reineker, to be published. [9] Handbook of mathematical functions, eds. M. Abramowitz and IA. Stegun (Dover Publications, New York, 1970).
135