The role of solitons in charge and energy transfer in 1D molecular chains

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&IVWld

Operator Ajt(A,) group. hi(h,)

describes

the presence

creates (annihilates)

(absence)

of the excitation

with the energy A on the nth molecular

wy. while FL, denotes exciton-phonon

phonon quanta with frequency

coupling

parameter. It is given by Fq = 2ix(h/2Mw,,) ‘/’ sin qRo in the case of coupling with acoustic phonons with frequency wy = 08 sin I4R0/21. wg = ~(K/M) ‘/’ denotes maximal phonon frequency. K is the spring constant. M denotes the mass of the molecular group, Ro denotes lattice constant and finally J and x represent intersite dipoledipole transfer integral and coupling strength. respectively. In the case of coupling with dispersionless optical modes (Holstein’s molecular crystal model), we have wq = wo = const and F = ~(i7/2Mwo)‘~‘. DST had caused a lot of controversy concerning both the theoretical and experimental foundations of the idea [3,6-l I] and in spite of all the efforts, the problem of the DS existence in realistic conditions still remains wide open [ 31. This especially concerns its applicability at physiologically relevant (T - 300 K) temperatures for which different theoretical

approaches

results in different.

some authors find that thermal fluctuations

sometimes

destabilize

mutually

soliton resulting

quite opposite. predictions.

Thuc while

in its finite lifetime being too short to he

relevant in realistic conditions [7.8], others find that thermal effects even support soliton creation and its stability [5,9-l I]. The origin of these discrepancies lies in the unjustified application of different approximative approaches without precise understanding

of their domain of applicability.

Namely. as known from the general theory of ST

phenomena [ 12-l 51, in dependence of the values of the three basic parameters determining the energy spectrum of the system: 2J -the width of the exciton band, 11w~ the maximal phonon energy and small polaron binding energy ,EB = ( I /N) CL, (1Fc/j’/hw,). only two cases should be considered as well understood on the basis of the existing theories. parameter), lattice distortion with large inertia Thus in the adiabatic limit B > I (B - 2J/?jwe_adiabaticity fails to follow the exciton motion and it forms an essentially static potential well where that particle may be trapped. Depending on the value of coupling constant (S - Ee/hw~), the spatial extent of an entity created in this way the polaron - may vary from being concentrated around one site only (S >> I ) - ctrlicrbtrric mall p~tltrtuttt (ASP) _ to be spread over the large number of lattice sites (S < I) - lrrrgr M/~&L&C polarott (LAP) - or soliton in the one-dimensional

media. Under these conditions

phonon subsystem

behaves as a classical one and the theoretical

description is possible on the basis of the time-dependent version of Pekar’s semiclassical variational method: Davydov’s D2 ansatz. System properties and the character of polaron states in the nonadiabatic limit are dominated by the quantum nature of phonons. In this case “dressing” effect occurs and exciton is surrounded

by the cloud of virtual phonons engaged

[ 12-171. This causes in the creation of short ranged lattice distortion which follows its motion instantaneously the decrease of the effective transfer matrix element (J). Such excitations are usually called rzorlrrdiahcrric, .sr~rrl/ pd~trotts

(NSP) [ l2- 17). In many particle systems “dressing”

causes the effective attractive interaction

between

the different particles which may lead to the creation of their bound state - exciton drop [ 181. These solutions may attain soliton form depending

on the values of the system parameters

of ST-states in this limit is possible on the basis of the Lang-Firsov

(S and B). Satisfactory unitary transformation

theoretical description (LFUT) method 1IS] or

by its variational extensions [ 16.171. In the present paper we shall examine the possibility of the soliton formation within the framework of model ( I ) and both single and multiexciton systems will be considered. In the single-particle systems term exciton has quite general a meaning and may denote electron. hole, exciton vibron. etc., while in the multiquanta systems it will he related to vibrational

quanta (vibrons) only.

2. Model Hamiltonian

of the system

For that purpose we shall apply mean-field widely used in the small polaron theories, theories of exciton transport. etc. [ 12-l S.I9,20]. The primary task is to find the so-called effective or model Hamiltonian of the system which then

Z. IviL:/Physica

220

D 113 (1998) 21X-227

serves as a basis for further analysis. In practical calculations by the following

we shall use modified or incomplete

LFUT specified

unitary operator: +RoAn+A,z(b_q

Here 0 < S < 1 is the variational

- bq’)

parameter measuring

the relative extent of the induced lattice distortion.

The first step is to rewrite

(1) in terms of new polaron operators, i.e. B, = U’A,U = A,@,, (8,, = - uTq )I), describing the “dressed” particle - polaron, consisting of the original one surrounded by the phonon cloud and uq = U+b,U representing new phonons in the chain with shifted equilibrium positions of the molecular groups. exp( (6/a)

‘& F~ednRo(uy

The second step consists transfer

integral

in the substitution

of the so-obtained

Hamiltonian

by the effective one in which the

(J) is replaced

by its mean field value: Jeff = J (@,~8,*1jPt, = JePGzSCT), where S(T) = equilibrium phonon distribution) denotes temperaturesin2(yRo/2)(W~ + 1) (Vq -

(VW ~,~IFy12/(~w,)‘1 dependent coupling constant

introduced

in [16]. The above choice of the effective Hamiltonian

is not arbitrary

but follows straightforwardly by virtue of Bogolyubov’s theorem. For details, e.g. see [ 191. After adding and subtracting the term H’ = J C,, (Bz B ,,+I (@$@,+I) + h.c) to the transformed Hamiltonian, it can be represented as a sum of two terms: H = Heff + Hint where

f&f = [A

-w - WBI C B,+B,,- JeffC il

+ (1 -6) a

+

~F;,eiY”RoB~B,,(~y

plaJquy+u4- a(2 -

B,~(B,+,

@EB

+ufq)

c

B,t’B,2 + ; EWdBlt,, &+I&,

+

B,+B,+_, B,,-I&,)

,I

n

q

+ B,_~)

I,

9

(3)

1

and Hint =

J xiBzBn+~ [(S,fO,+,) - O,f8,+,] + h.c.).

(4)

n In the spirit of the mean-field the smallness

method we shall base our further analysis upon the effective Hamiltonian

of the interaction

term. Its influence

is usually accounted

perturbatively.

(3) assuming

For the present purpose

however, it could be neglected. The main advantage of the Hamiltonian (3) with respect to the original one is that it includes the effective dynamical inter-exciton interaction. Now it describes the system of the “dressed” excitons (exciton polarons) interacting mutually

with each other and with lattice vibrations.

In such a way it could be the basis for the examination

of the

whole scale of the many-particle effects in multiexciton system such as the creation of two, three or multiexciton bound states. In the present context, however, we are only examining the possibility of the soliton creation. In the single-exciton systems, it could arise on account of ST mechanism only, while in the multiexciton ones two competing mechanisms give rise to soliton formation: exciton-phonon and exciton-exciton interaction. If the first mechanism prevails, soliton formation is analogous to the one in the single-particle systems so that such soliton, if any arise, comes on account of the ST of JV excitons in the common potential well formed by the static lattice deformation. Such solitons were examined by Lomdahl and Ker [4] who, in their numerical simulations, found that the increasing of the number of quanta up to N = 6 leads to the enhanced stability of solitons even at physiological temperatures. If the second mechanism prevails, soliton represents the bound state of N “dressed” excitons. The character of the eigenstates of system, both in the single and multiparticle systems, depends on the value of S. Its choice is not arbitrary, and for the given set of system parameters, there should exist an optimal value of 6

corresponding

to the minimum

energy state. We shall find this optimized

value by minimizing

system in the static (V = 0) limit. Therefore moving (u # 0) and nonstationary represent the evolution near these minimal energy states. In the analysis of the single-exciton

solitons standard &-trial

should be modified and therefore the excitonic

the GS-energy

(time-dependent)

state will be used. For the multiexciton

part of the total wave function

of the

soliton solutions solitons it

will be defined, in accordance

with

[ 25-29 1. as a product of the coherent states of operators B,, To see how the value of S is related to the nature of the phonons involved in the creation of the lattice distortion. it is necessary

to find expectation

phonon operators,

value of the lattice displacement

in the corresponding

trial state (&

operator (li,,). expressed

for the single-exciton

in terms of the new

case). It turns out that it consists of

two competetive contributions: the first one measured by the value of coherent phonon amlitudes defines slow or classical part of lattice distortion, while the remaining one, proportional to the value of S, comes on account of the quantum corrections. The explicit expression for the classical phonon amplitudes may be found following the standard procedure (see e.g. [ lb,17 1) and it is proportional to ( I - 6) F,*/fzwy .It follows that 6 is limited to the range: 0 < 6 < I. Thus when S vanishes. phonon subsystem

behaves as a classical one and soliton solution. if any exist, arises on account of ST mechanism exclusively. On the other hand if S + I, classical part of lattice distortion vanishes and the character of the system eigenstates is determined by the quantum nature of phonons. In the intermediate region both the effects

contribute

competing

each other. It depends on the value of 6 which will be determined

in Sections 3 and 3.

3. Single-particle solitons In order to find criteria for the existence find 6 by minimizing &;s(S)

the GS-energy

of the solitons arising on account of the single exciton ST we have to

of the system:

= d - 6(2 - S)EB - 2JeP

6’S(T) --?--J

’

which was obtained after the self-consistent elimination of the polaron wave function and phonon amplitudes from the expectation value of the Hamiltonian (3) in the D2 trial state. In deriving this expression we have used the well-known result: 9(.x)

= JE1/2sech(p/Ro)x

(6)

for the soliton solution. Here I* = (EB/J)( 1 - 6) 2e S’S(T) is the so-called soliton parameter related to its width as 1s - R()/p. Applicability of the continuum approximation demands p < I. The stable eigenstates of the system correspond to a EGs/tlG = 0 and a’EGs/aS2 > 0. As it was shown in [ 161 the essential information concerning the optimization of dressing parameter may be obtained considering the linear part of the EGS (i.e. neglecting the last term in expression (5)). Equation LEGS/&? = 0 cannot be solved explicitly for 6 = 6(.S(T). B(T)). However we can solve it to find S(T) = S(6. B(T)) with B(T) = ZJS(T)/EB being the parameter (7) This expression can be simply inverted to find desired dependence B(T) from the stability condition we obtain stability line S(T) = [2&i

- 6)]_‘.

of dressing parameter on S and B. Eliminating

(8)

Z. h.iC/Ph?:c.icu D 113 (199X) 21X-227

222

30

/

20-

\

I ‘/ \ \ \

lo-

\

\ \\

\

:(

Fig.

I. Coupling

following

constant (S) vs. dressing fraction 6 for a single exciton self-trapping.

values of B:

IO.5,

Bc = e”/‘/Z.

Btj = e/2.0.5

From left to right these curves correspond to the

and 0. I respectively. The dotted line represents so-called confinuum houndaq

while the dashed one corresponds to the s~uhiliq line.

separating continuum S(T)

stable solutions, boundary = [2S( 1 -

and continuum

lying below it, from the unstable

ones. In the same manner

we find the so-called

S)f .

approximation

Our results are visualized

(9) is valid for the points of (S(T), Q-plane

lying below it.

in Fig. 1 where we, together with the stability

line and continuum

boundary,

have

plotted the set of adiabatic curves (S(T) = S(6, B(T))) for a few chosen values of B(T) spanning the whole range of adiabaticity. Each of these “adiabats” represents the set of points in (S(T), 6)-plane corresponding to the extrema of the GS energy. Physically meaningful region corresponds to all the points lying below the stability line, while the soliton ones correspond to the part of (S(T), b)-plane lying below the continuum boundary. Looking at the adiabatic curves in Fig. 1 as a functional dependence of 6 on the coupling constant with B being the parameter, we may discuss the character of the ST states. In the nonadiabatic region, i.e. as far as B(T) < B,( B, = (1/2)e’/‘), these curves include only minima while 6 shows continuous growth with the increase of the coupling strength (S(T)). Furthermore, for each adiabaticity parameter less than Bo = e/2 all these curves lie completely below the continuum boundary. Thus our method predicts the existence of the solitons even in the highly nonadiabatic regime (B(T) < I), where other approaches predict the occurrence of the NSP-band states which, due to the reduction of the effective tunneling integral, continuously turns into the localized ones, as the coupling constant increases. Therefore, in order to determine to which extent the present variational method is applicable in the nonadiabatic limit

Z. IviC/Phyica we

D 113 (199X) 218-227

have to compare GS energy of the system (Eq. (6)) substituting

6 -

223

I - ems(r), with that of NSP obtained by

means of the purely linear analysis. Due to the variational character of the present method, soliton picture survives as long as its energy is lower than that of NSP. Following [ 171 we found, after some calculations, that the GS-energy of the linear (NSP-band) states is lower with respect to the soliton ones as long as B(T) < 1. This means that in this limit continuum approximation is not valid and the values of S as determined by Eq. (7) characterize the partially dressed excitations: NSP-band states. Increase of the coupling strength causes the continuous increase of the small polaron effective mass which tends to infinity when 6 approaches unity. The divergence of polaron mass is the consequence of the vanishing of the effective hopping integral (Jew = J ( 1- S)/2B( T)6) in the nonadiabatic strong coupling limit where S 2 I - eCscT) + I. Consequently in this limit NSP-band breaks down and localization transition occurs. When B(T) exceeds critical adiabaticity, and between the two critical values of coupling by the crossing points of a particular adiabatic curve and stability line, 6 becomes multivalued

constant determined function of coupling

constant since for each value of S(T) there appear two values of 6 corresponding to the EGS minima. Therefore, there occur two kinds of (meta-) stable eigenstates of the system with an abrupt transition between them. The first of these values of 6 falling onto small 6 region (i.e. S * l/(B(T) + I)) characterizes the soliton, if continuum approximation

holds or (ASP), in the opposite case. The second one, having the value close to unity, corresponds

to the NSP. Comparing the GS energy of the system corresponding to these values of dressing parameter we find that the soliton or ASP are always energetically more favourable than NSP. For S(T) < $1 (S,t the lower critical coupling

constant)

only soliton solution,

which continuously

constant increases, exists. If S > $2, only nonadiabatic

collapses

into adiabatic

small polaron

if coupling

small polaron exists.

Concerning the thermal effects we note that, within this mean-field method, they could be manifested only through the temperature-dependence of coupling constant and adiabaticity which both increase with the growth of the temperature. Thus its increasing drives the system towards the adiabatic and strong coupling limit supporting the applicability of the semiclassical approximation and favours DS formation. Note that the present method, i.e. the combination of the Davydov D2 ansatz mean-field method, is fully equivalent to the so-called Davydov’s DI -ansatz. so it explains the conclusions about the enhanced stability of DS due to the thermal effects obtained previously [5.9,1 I]. Our predictions, obtained within purely static analysis. should be taken with certain reserves. Namely. as shown recently, taking into account of the dynamical effects [ 191 results in the finite lifetime of the soliton: t1/2 = 9.2(TS)-’ lop8 s. Thus the temperature plays the twofold role and soliton existence and stability demands the delicate balance between these two mutually

4. Multiquanta

quite opposite tendencies.

solitons

In order to find multiquanta

soliton solution we shall apply slightly modified standard Davydov’s Dz ansatz which

is based upon the following trial state of the system: ]9 (t)) = n, lpn(t)) 63 nq lay(t)), where I/3,,(t )) and ]aq (t )) are coherent states of polaron (B,) and phonon operators (Q,), respectively. If the number of “dressed” vibrons in the system is N, the following the GS-energy

normalization

condition

holds: N = (@Ix,,

E$ B,, IS) = c,,

i#L1’. Minimizing

of the system Ez&3es’sm

EGS = N[A which is determined

- 6(2 - S)EB - 2Je-S2S(T’] in the same way as expression

2B(T)e-SZs(7Y s=

I+ [

l+K

-

12J ’ (6), we obtain self-consistent

1’ h/W.

( IO)

equation for the dressing parameter

(11)

-’

It is written in the form which we found the most convenient

for the numerical

procedure.

Z. hYf/Physica

224

As well as in the single-exciton

D 113 (1998) 218-227

case we found stability line and continuum

boundary:

(12) (13) To find 6 for the particular values of S(T) and B(T) we have to solve Eq. (12). It could be done only numerically and our results are visualized in Figs. 2 and 3 where the set of adiabatic curves each corresponding was plotted for a two-fixed exciton populations the nonadiabatic

(B(T)

<

(ti

1) to the adiabatic (B(T)

= 5 and 10, respectively). >

1) limit. According

to particular B(T)

The B(T) is chosen to vary from

to (12) and (I 3) physically

meaningful

region of (S(T), 6)plane corresponds to those points lying below stability lines - solid lines in Figs. 2 and 3; while the soliton solutions correspond to those points lying below continuum boundary -the long-dashed curves in $%) exp( fi/Zm, Figs. 2 and 3. It follows that, as long as B(T) is less than some critical value - Bc = (m/2 all these “adiabats” lie below stability line and therefore include only minima of GS energy. When B(T) exceeds this critical value, each adiabatic curve intersects stability line at one point. Consequently

each adiabatic curve for

each S(T) which is higher than the critical one - SC = $%/2fl, has two solutions for 6 corresponding to GS extrema. Clearly, due to the stability condition (I 2) only the lower one defines stable eigenstate of the system. Critical adiabaticity parameter corresponds to that adiabatic curve which has a single common point with the stability line. Critical coupling constant corresponds to the crossing point of the critical “adiabat” and line 6 = 1. Concerning the validity of soliton picture we see from Figs. 2 and 3 that, since continuum boundary lies always below stability line, condition for the applicability of continuum approximation is also the condition for soliton existence. To Iind the range of the coupling constant in which soliton may exist for particular adiabaticity, one must find crossing points of each adiabatic curve with continuum boundary. So we find SO x l/( I + 2 B(T)) which separates those values of S corresponding (ISM). Inserting 60 into the continuum

to the soliton solutions from those which define second type of solution

boundary we find simple condition

B(T) N

S(T) < -

Thus soliton may exist for each adiabaticity if 6 < 60 and if (14) holds. When S exceeds this value, soliton collapses into extremely narrow exciton complex corresponding to the so-called intrinsic self-localized modes (ISM) recently examined by Takeno and co-workers [22-241. As one can see from Figs. 2 and 3 in the extreme adiabatic limit 6 x [l/( I + B(T)] -+ 0, semiclassical

nature

of phonon prevails and soliton arises as a result of ST of N excitons in common potential well. In the intermediate range of adiabaticity, but still higher than Bc, both mechanisms give rise to soliton formation but still ST is more dominant. Lowering of the adiabaticity disturbs this balance in the benefit of exciton-exciton interaction. In the nonadiabatic limit soliton can exist in the weak coupling limit, but ST mechanism could play a certain role in soliton formation in the extremely weak case only if B is not too small. Increasing very small amount rapidly turns S to unity, and soliton formation is practically exciton-exciton interaction.

of the coupling constant, even for exclusively influenced by effective

The variational character of our partial dressing method imposes certain restrictions on the domain of applicability of our results. Thus in order to determine in which degree our results are reliable for the understanding of the soliton properties in realistic biological context, we must compare them with those obtained on the ground of some other methods. As to our best knowledge, multiexciton soliton states, arising on account of the exciton-phonon interaction, were examined in a few papers only [25-301. So Wiedlich and Heudorfer [25], Kislukha [28] and finally Lomdahl and Ker [4] in the context of DT investigated such solitons using semiclassical analysis. which corresponds to our

225

0.00

0.75

0.50

0.25

1.00 6

Fig. 2. Adiabatic

curves - S =

correspond to the following

S(S) (N

= 5) for a few chosen values of adiabaticity

values of B(T):

10. 5. Bc z

1.014. 0.5 and 0.

while long-dashed one corresponds to the cor~irzu~rn boundary.

Short-dashed

I

parameter

respectively.

(B). These curves. from left to right.

Solid line represents so-called stuhili/yv line

part of each adiabatic curve for B(T)

I > B(. corresponds

to the unstable region.

choice 6 = 0, while Nakamura

[27] and Skrinjar with coworkers 1301 utilized UT techniques

- full dressing approximation. Superiority of the present approach corresponds where it predicts lower estimates for the GS energy of the system. Substituting

with the choice 6 = I

to that part of the parameter space 6 = 0 in (10) and S = I/( 1 + B).

which is good approximation for S in the range where soliton existence condition (14) is satisfied, we estimate which ensures better applicability of partial dressing method with respect to &is(S) - EGS(6 = 0) x -(~S/B)EB the strict semiclassical analysis practically in the whole parameter space of system. Note that in the limit of infinite adiabaticity (B + co) difference between these two methods disappears, the result which was to be expected. :> BO (Bo = Repeating this procedure for 8 = 1 and 6 z l/(1 + B) we found that &s(B) - EGS( 1) -C 0 if B(T) 0.65).

This means that when adiabaticity

lies below this value (i.e. B(T)

< 0.65)

one must take 6 = 1 while for

the rest our results are unaffected. Let us now estimate how the variations of the exciton number and the temperature could modify above conclusions. As follows from Figs. 2 and 3 dependence of 6 on adiabaticity and coupling constant is qualitatively the same for both values of N; however, its increasing significantly reduces the allowed range of the coupling constant in which (for given B(T)) soliton may exist. This could be seen especially from the above-mentioned condition for soliton existence (S(T) < B(T)/n/). As for the influence of temperature it follows that, due to the increase of adiabaticity temperature, it favours ST mechanism of the soliton formation.

parameter with the rise in

Z. IviC/Physica D 113 (1998) 218-227

226

Fig. 3. Adiabatic

curves for

N = IO and same set of adiabatic parameters

(Bc 2 1.294).

5. Conclusions On the basis of previous

analysis

it follows that by calculating

S for the particular

values of S(T) and B(T)

and locating it in the particular part of the S(T)&plane, one may determine the character of the eigen-states of system, and to examine possibility for their existence. Using the often cited values of the parameters for the a-helix and acetanilide

(ACN) [2-l 1,16,17], in the context of intramolecular

B z 0.1 - 0.16 and S % 0.01 - 0.1. On the basis of these estimates Bloch band states which rapidly turn into localized

vibrational

energy transfer, we found that

it seems that we deal with small poluron

ones if temperature increases. Therefore, we conclude that the solitonic mechanism cannot be relevant for the vibrational energy transfer in a-helix and ACN. On the contrary, it seems that a-helix supports the existence of DS consisting of an elecrtron and lattice distortion since the model independent estimates give J = 1 eV [31] which implies B x IO2 while S falls onto intermediate coupling region which enables one to apply DA. Concerning the multiexciton systems we find that, even for the above-quoted values of system parameters, one can expect soliton existence but on account of quite different mechanism than the one proposed by Lomdahl and Ker [4] in their generalizations of DST. Thus for these values of adiabaticity and coupling constant quantum nature of the phonon subsystem dominates system properties and soliton can be formed due to the effective exciton-exciton interaction only. At 300 K, kB T/AwB - 1.95 - 2.27 resulting in B - 0.28 - 0.32 still belonging to nonadiabatic limit which does not change the character of soliton.

2. h~ic’/Phyicu

D 113 (1998)

218-227

227

Acknowledgements This work was supported by the Serbian Ministry of Science and Technology.

Grant No 0 1El 5

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