D E R I V A T I O N OF T H E S I N K M O D E L V. C/(PEK
Institute of Physics, Charles University, Ke Karlovu 5, 121 16 Praha 2, Czechoslovakia R. PELESKA
Institute of Physics, Czechosl. Acad. Sci., Na Slovance 2, 180 40 Praha 8, Czechoslovakia Received 10 December 1991 Standard two-site Generalized (to nonperiodic systems and finite temperature) Stochastic Liouville Equation model is rigorously reduced, using a partitioning projector, to a single-site model with a sink. Under some additional restrictive assumptions, the usual single-site sink model is reproduced. Repeating the reasoning under the same assumptions for a more-site model, the resulting sink model is identified with that in the version by C~pek and SzScs. It is argued that the original sink model by Kenkre cannot be derived in this or any other rigorous way. 1. I n t r o d u c t i o n In m a n y situations, inclusion of finite life-time effects in investigation of dynamics of extended systems is a m a t t e r of necessity. As an example, one can mention dynamics of excitons in molecular crystals with traps or a n t e n n a systems with reaction centres. The most direct way to do t h a t is to include the s t a t e without~ e.g., the exciton in the complete set (basis) of states used to describe the kinetic problem in question. On the other hand, this way often implies technical complications which are not in principle necessary when the exciton-less state is (for a given problem) of no importance. Another way to handle the problem is to introduce a so called sink t e r m into kinetic equations used which means in particular to resort to the sink model known in several modifications, On the level of the time-convolution Generalized Master Equations ( G M E ) theory, Kenkre [1-3] introduced the finite excitation life-time effects adding the sink t e r m (the first t e r m on the right hand side of (1.1)) to a specific t y p e of G M E for probabilities Pm (t) of finding the excitation at site (molecule) m. Thus, one gets
_a Pro(t) ~t
+ E ~( :r m )
= -
$2
E P (t) Tr
[Wm,~(t--r) P , ~ ( r ) - - W ~ , ~ ( t - - T ) P m ( r ) ] d r + I m ( t ) .
(1.1)
One must say, however, t h a t the sink p a r a m e t e r s ~_-1 in (1.1) were originally assumed to have no influence on b o t h the m e m o r y functions w m , ( t ) and the initial condition t e r m I,~(t), i.e., wm~(t) as well as Ira(t) were originally assumed to be the same as those in the case of the infinite life-time (~-~ --~ +co). This assumption Czechoslovak Journal of Physics, Vol. 42 (1992), No. 7
65,~
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R. Pelegka
was an inherent feature of the original sink model by Kenkre [1-3] which we now call the restricted sink model [4]. (Notice that none of the values of indices m, n in (1.1) correspond to the state without exciton.) Cgpek and Szhcs [5] were the first who turned attention to the fact that in the case of the restricted sink model, the basic requirement for probabilities
P (t) > o
(1.2)
might well become disturbed by the solution to (1.1). Simultaneously, (at least on the level of the Stochastic or Generalized Stochastic Liouville Equation model [6, 7] or in general time-convolution GME for the whole single-exciton density matrix) another form of the sink model (full sink model [4])was suggested [57]. It consists in adding, to the right hand side of the just mentioned kinetic equations for the exciton density matrix p,~n in the local basis, the sink term --0.5(1/rm-b 1/'gn) pmn(t). Thus, in particular the Generalized Stochastic Liouville Equation (GSLE) model with this sink term and in the Haken-Strobl-Reineker parametrization then consists in equations [6, 7]
O--t prnn(t) = -0.5
~mm+ ~nn
Pmn(t) -- -~ ([Hexc, p(t)])mn
+6~n ~--~[27mppvv(t)-
27v.~p~m(t)l
p -(1 2r
p.,,,(t) - i,,,,,, p,,,,.(t)], . =
+ >d
= 2r.
.
(1.3)
7"
(One should notice that, in contradistinction to the original Stochastic Liouville Equation model in the Haken-Strobl-Reineker parametrization [8], the reservoirassisted transfer rates 27ran , m # n might differ from 2%,~ as predicted by the detailed balance condition for systems of nonequivalent sites and finite temperature, and ~'mn = 5*m might become complex.) Nt~merica! studies then indicated that from (1.3), one always obtains
pmm(t)>__O,
t > 0
(1.4)
provided that this condition was satisfied for all m at t -= 0 [5-7]. From (1.3), one can extract a closed time-convolution equation for diagonal elements p,~m(t) = P~(t) only, using the standard projection methods [9, 10]. This reads as (1.1) with, however, one additional feature (as Compared with original Kenkre's GME sink model): The memory functions (and also the initial condition term) become strongljr influenced by the sink parameters 7-~1 [5, 4, 11]. This influence is nontrivial, in general nonlocal [11] and is in principle able to yield, in tl~e limit of one r r ] --+ +co, even the dynamical split-off effect of the sink (the fear-of-death effect [4]). As numerical studies based on the latter equations or, 651~
Czech. J. Phys. 42 (1992)
Derivation of the sink model equivalently, on" (1.3) have shown, the full sink model is capable of describing well the finite life-time effects in various situations [6, 7, 4, 12-15]. Its derivation is, on the other hand, still lacking. For a simple model of two (or, in Section 3, N + 1) sites (one of them being identified with the sink), this is the aim of the present contribution. 2. Sink m o d e l f r o m t h e G S L E for a two-site m o d e l Let us start from the Generalized Stochastic Liouville Equation (GSLE) model for sites 0 and 1 in form
[.oo {,oo)
60 / flO1/ = //901 ~plO ] - i L ~Pl~ \pll / \pll -2i71o -J/h J/h 2i710
L=
-J/h - A w - 2iF 2i~10 J/h
J/h 2i%1 Aw - 2iF -J/h
'
2i7o1 ) J/h -J/h -2i701
"
(2.2)
This is (with F = F10 = F01 and Aw = (~1 - eo)/h) just the set of GSLE model equations (1.3) without any sink (T, ~ +oo); Hexc has been chosen as H~xc----s0a0ta0 + ~1a~al -{- J(atoal -{-a~ao) .
(2.3)
Site 0 can be identified with, e-g-, a deep trap for the case of a carrier or exciton or excitationless state (sink in our and standard terminology). Thus, 2701 and 2710 have the meaning of the reservoir- (e.g. phonon-) assisted transfer rates to and from the sink. Site 1 then corresponds to our system in a narrower sense (i.e. without the sink). Let us choose our projector
D=
(i oo 0 0 0 0 0 0
(2.4)
Multiplying (2.i) by D and 1 - D, determining
Poo (1 - D )
Czech. J. Phys. 42 (1992)
pol I =- (1 - D ) p
Plo ) P11/
(2.5) 657
V. Cdpek, R. Pelegka
from the latter equation (upon expressing Lp as LDp + L(1 - D) p) and putting the result into the former equation yields the Nakajima & Zwanzig [9, 10] equation 0 Dp(t) = - i D L D p ( t ) - f0 t DL exp(-i(1 - D) L(t - T)) (1 -- D) LDp(T) dT O--t
-iDi
exp(-i(1 - D) Lt) (1 - D) p(0).
(2.6)
Because of the form of (2.4), matrix equation (2.6) has only one (11-) element nonzero which reads [16] 0 oq--~ P n (t) = - 2 7 o l P n (t) +
~n0rOl-x
~1
-
2i ~
"rl0(U 2
j2
j
)
-2i ~ ~0~(~ ~) -.~(J)'~=l(J) - ~ (4 ~) - 4 j) ) (~J)- ~J)) (2.7) Here, for simplicity, we assume initial condition
fl11(0)-- 1,
(2.8)
i.e., (1 - D)p(0) = 0 and the last (initial condition) term on the right hand side of (2.6) disappears. Further, u~.0 (or-(i), vj ) are the j - t h component (j = 1, 2, 3) of the i-th left (or of the i-th right) eigenvector of the 3 x 3 matrix (entering -i(1 - D) L(1 - D)) -2710 (i/h)J -(i/h)J
(i/h)J i A w - 2F 2~10
-(i/h)J ) 2%1 - i A w - 2F
(2.9)
while )~j are the corresponding (complex) eigenvalues. For the eigenvectors, the normalization (and orthogonality) condition
3
uj(i)vj(k) = ~,~
(2.10)
j=l has been used. One should now compare (2.7) with (1.3) for the special case where m, n (in (1.3)) take just one value (the system is composed of only site 1). We then see that all the right hs side of (2.7) has the meaning of a generalized sink term. In order to get a better insight into the structure of (2.10), let us first realize that according to (1.3), 2F -- 2Fol = 700 + 3'1o + "~01 + 711 9 (2.11) 658
Czech. 3. Phys. 42 (1992)
Derivation o f the sink model
From the microscopic theory [6, 7], it follows, however, that the diagonal 7-parameters (i.e. 700 and 711) disappear (for A w r 0) in the lowest order in J. (Compare also the second alternative treated in Ref. [7].) Thus, let us assume that 700 ~ 0 ,
711 ~ 0 ,
701 '~ '~10 ~ 7 ,
~10 :
(2.12)
~/01 -----0 9
Then (2.7) can be made more explicit since A1 = __27 , )t2, 3 = _ _ 2 7 . 4 -
~r
r
i x/(hAw)2 + 2 j e
2 --~ 2 3 2
1 = 2 v / ( h / ' ~ ) ~ + 2J~
~_ ( ~ ( 1 ) ) T , .
"
/
haw :t: ~/(hAw) 2 + 2J ~ = (u'(2)'(3))T. h / , ~ ~: ~/(h/,~)~ + 2J~ /
(2.13)
Thus, (2.7) yields
O--t pll(t) = -27p11(t) + -2J 2 [1-
4~/2 (hAw)2 + 25 ~
(h~)~4"/2+ 2z~] cos[(t- ~)v/(~)2 + 2j2]
fL.
27 sin[(t - T)v/(hAw) 2 + 2J2]} e -~(~-~) pll(T) dT. +4j2 v/(hAw)2 + 2 J 2 (2.14) It is not difficult to see that (2.14) yields lim P l l r 0 as it should be (owing to the t---*-boo
assumed inequality 710 ) 0) but in contradiction with the restricted sink model as well as its full (consistent) form mentioned above. (In the last two cases, we always get t-~q-oo lim Pll = 0.) Moreover, in (2.14), we always obtain a long-range oscillatory memory term which is not encountered in the sink model. So, we see that in order to recover the sink model, one must suppress all mechanisms responsible for return of the excitation back from the sink (site 0) to the system (site 1). Thus, instead of (2.12), we must take J --~ 710 ~---0 . (2.15) Then (2.7) yields 0 0--t P l l (t) -~- --2701 P l l (t)
(2.16)
which is the restricted (original Kenkre [1-3]) as well as full (as suggested by 0s and Szbcs [5-7]) sink model for our situation (compare with (1.3) for m , n being Czech. J. Phys. 42 (1992)
659
V. (?,~pek, R. Pelegka
limited to just one value, i.e. 1 only). In our oversimplified situation with only one site (molecule) in the system, we unfortunately cannot distinguish between the above mentioned modifications of'the sink model. Thus, in the next section, we resort to a more complicated system containing, in general, N sites. 3. F r o m ( N + l ) - s i t e G S L E t o N - s i t e G S L E m o d e l w i t h t h e s i n k t e r m In agreement with the above reasoning, let us assume t h a t site 0 (zero) corresponds to the sink while sites 1 to N form the system. Let us assume t h a t in (1.3) (for m, n = 0, 1,... N), there are no finite life-time effects (T71 --* 0). Let us as usual assume the exciton (excitation) Hamiltonian in form N
N
Hexc = E
r
+
rn=O
E Jmnatman" rnrfin;m,n..=O
(3.1)
Finally, corresponding with the above discussion, let us suppress all the channels leading to return of the excitation back from the sink to the system, i.e. J,~o ( = Jo,~ ) = Tmo = O ,
m = l, 2, . . . N .
(3.2)
Then one can proceed as in (2.6) making use of the projector D . . . = E Dmnpqlm) (Pl...Iq)(nl,Dm,~pq =hmvhpq(1--hmO) (1--5,~0). mnpq
Instead of this, one can specify set (1.3) (with " r r 1 ~ 1, 2 , . . . N. Both these ways lead to the set 0 i N O--t titan(t) = ---~ [(gm -- gn) W E [ J m v r-~l: N
+emo
(3.3)
0) to jUSt indices m, n =
prn(t) -- flmr(t)Jrn]
-
prom(t)]
p=l
~r -(1 -
+
-(~'o~ +'yo,)o,~,~(t),
-
. re, n =
25'm~
1,2,...N
(3.4)
provided that initially, the excitation was fully in the system (P00(0) = 0). This is, however, exactly the full sink model as suggested by Cs and Szbcs [5-7]. Excluding the off-diagonal matrix elements pm~(t), m ~ n as mentioned above, one obtains GME for the site-diagonM elements prom(t) = Pro(t) (i.e. probabilities) (1.1) but with inclusion of the influence of the sink parameters 70m on the memory 660
Czech. J. Phys. 42 (1992)
Derivation o f the sink model
functions [5] (and, similarly, the initial condition t e r m ) . So, we c a n n o t derive the original (i.e. restricted in our terminology) K e n k r e sink m o d e l in this way. As the l a t t e r version of the sink m o d e l does not preserve (1.2), we assert t h a t it is impossible to derive it in a n y rigorous way in general.
Re[eren ces
[1] Kenkre V: M., Wong Y. M.: Phys. Rev. B 23 (1981) 3748. [2] Kenkre V. M., Parris P. E.: Phys. R.ev. B 27 (1983) 3221. [3] Kenkre V. M.: in Exciton Dynamics in Molecular Crystals and Aggregates. Springer tracts in modern physics, Vol. 94. Springer-Verlag, Berlin-Heidelberg-New York, 1982, p. 1. [4] Chvosta P., Barvfk I.: Z. Physik B 85 (1991) 227. [5] (~s V., Sz6cs V.: Phys. Status Solidi B 125 (1984) K 137. [6] (~s V.: Z. Physik B 60 (1985) 101. [7] (~.pek V., Sz6cs V.: Phys. Status Solidi B 131 (1985) 667. [8] R.eineker P.: in Exciton Dynamics in Molecular Crystals and Aggregates. Springer tracts in modern physics, Vol. 94. Springer-Verlag, Berlin-Heidelberg-New York, 1982, p. 111. [9] Nakajima S.: Progr. Theor. Phys. 20 (1958)948. [10] Zwanzig R.: Physica 30 (1964) 109. [11] Barvfk I., He~man P.: Phys. Rev. B 4 5 (1992) 2772. [12] Nedbal L., SzScs V.: J. Theor. Biol. 120 (1986) 411. [13] Sz6cs V., Barvfk I.: J. Theor. Biol. 122 (1986) 179. [14] Barvfk I.: J. Theor. Biol. - in press. [15] Barvfk I., Nedbat L.: J. Theor. Biol. 154 (1992) 303. [16] Pele~ka 1~.: Diploma work. Faculty of Mathematics and Physics, Charles University, Prague, 1991.
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