Nature of the optical transition in polymethine dyes and J-aggregates

Nature of the optical transition in polymethine dyes and J-aggregatesVladimir V. Egorov Citation: J. Chem. Phys. 116, 3090 (2002); doi: 10.1063/1.1436076 View online: http://dx.doi.org/10.1063/1.1436076 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v116/i7 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 7 15 FEBRUARY 2002

Nature of the optical transition in polymethine dyes and J-aggregatesVladimir V. Egorova)

Photochemistry Center, Russian Academy of Sciences, 7A Novatorov Street, Moscow 117421, Russia

~Received 20 April 2001; accepted 26 November 2001!

Using a new theory of photoinduced electron transfer beyond the Landau–Zenner ideology or, moreprecisely, electrodynamics of extended multiphonon transitions@V. V. Egorov, Chem. Phys.269, 251~2001!#, we give an explanation for the well-known experimental data@L. G. S. Brookeret al., J.Am. Chem. Soc.62, 1116~1940!# for the absorption line shape in the vinylogous series of an idealpolymethine dye represented by a thiacarbocyanine. Then, using this explanation together with ourearlier explanation@V. V. Egorov, Chem. Phys. Lett.336, 284 ~2001!# for the nature of thewell-known intense narrowJ-band due to an aggregation of polymethine dyes, we predict veryintense narrow absorption lines for short optical transitions. Interpretation of these results is givenon a basis of the Heisenberg uncertainty relation. A process of creation of the pure~quantum-mechanical! electron-transfer state is considered for the two complementary cases: theelectron-transfer state is determined by interaction of the electron with its environment throughspontaneous pumping of this state by an ordered or disordered environmental motion. The latter casecorresponds to the Landau–Zenner-type picture of adiabatic and nonadiabatic electron transfers.The former case is used to account for the nature of the intense narrow bands. ©2002 AmericanInstitute of Physics.@DOI: 10.1063/1.1436076#

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I. INTRODUCTION

Polymethine~cyanine! dyes and theirJ-aggregates havewidespread application as converters of light energy,these dyes have been theoretically investigated througthe past~see, e.g., Refs. 1–4, and references therein!. Nev-ertheless, up to now, the nature of the optical transitionpolymethine dyes remains to be discovered. This conclusfollows from the fact that the nature of a resonantlike behior of the well-known absorption line shape in the vinylgous series of an ideal polymethine dye represented bthiacarbocyanine5 ~see also Ref. 2! is still not understood.The same is also true for the nature of the optical transithat results in the well-known intense narrowJ-band6–9 dueto an aggregation of polymethine dyes.10,11

Recently, the current picture of elementary electrtransfer in condensed matter, derived from the LandaZenner-type theories, was revised to incorporate, amother things, an unusual resonance between the electronenvironmental nuclear reorganization motion.10,11 In Ref. 11the author constructed a new theory of photoinduced electransfer~beyond the Landau–Zenner ideology! or, more pre-cisely, a semiclassical electrodynamics of extended mtiphonon transitions in light of which the nature of thJ-band is due to just this resonance, not to averagingenvironmental statistical disorder by the quickly moviFrenkel exciton10,11 as is generally believed. Also presenthere is a theoretical explanation for the familiar experimendata of Herz12–14~see also Ref. 2! on the transmutation of theabsorption line shape for a benzimidazolocarbocyanineJ-aggregation.

In this paper, the new theoretical approach10,11 is dem-

a!Electronic mail: [email protected], fax:117-095-936-1255.

3090021-9606/2002/116(7)/3090/14/$19.00

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onstrated to find an application in treating experimental don absorption line shapes for polymethine dye monomertheir own right, namely, in treating the mentioned expemental data on the absorption line shape in the vinylogseries of a thiacarbocyanine that obtained by Brookeret al.5

back in 1940~see Fig. 1!. This is the first goal of this paperWhen our numerical results of theoretical explanation forexperimental data of Brookeret al. are compared with thoseof theoretical explanation for the experimental dataHerz12–14~see Fig. 2!, it is apparent that very intense narroabsorption lines can be predicted. In other words, the theleaves room for vastly more strong effects as comparedthe J-band effect. The demonstration of this fact is the sond goal of this paper.

In relation to the strongJ-band effect and the predictioof the stronger optical effects, the question again arises awhether there is the physical interpretation in its simplform of the dynamics of elementary electron transfer in codensed matter, which is calculated in Ref. 11. Presentatiosuch an interpretation of the dynamic organization of phoinduced electron transfer is the third goal of this paper.

II. IDEAL POLYMETHINE STATE

It is common knowledge that the heart of the polymthine dye chromophore is a strongly extendedp-electrondensity alternating along the polymethine chain~see Fig. 3!and is redistributed alternatively on light excitation.4 As earlyas 1978, Daehne introduced the concept of an ideal polythine state~IPS!3 ~see also Ref. 4!.

For IPS, thep-electron density distribution along thpolymethine chain is of maximum alternation and tp-bond orders level off, in the ground state as well as inexcited state. On light excitation of IPS, the change in

0 © 2002 American Institute of Physics

license or copyright; see http://jcp.aip.org/about/rights_and_permissions

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3091J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Optical transitions in polymethine dyes and J-aggregates

p-electron density andp-bond orders is at its maximum anat its minimum, respectively.4 In line with these facts, wedevise a model for the electronic structure of IPS wheredistribution of the atomic charge in absolute value is unifoalong the polymethine chain in the ground state as well aan excited state with an equal linear charge density to giveenvironmental nuclear reorganization of equal energy in bstates.10,11 Then thep-electron transition in IPS can be described by our electrodynamics of extended multiphontransitions11 which in its present form has been constructunder the assumption that reorganization energies for thetial and final states were equal in magnitude.

FIG. 1. Experimental data of Brookeret al. ~Ref. 5! ~see also Ref. 2! on theabsorption dependence in the vinylogous series of a thiacarbocyaninmethanol at room temperature (@«#5M 21 cm21).

FIG. 2. Experimental data of Herz~Refs. 12–14! ~see also Ref. 2! on theabsorption dependence of a benzimidazolocarbocyanine at 25 °C onconcentrations (@«#5M 21 cm21). In part A, these are in micromoles/liteof aqueous 0.001M NaOH: 0.5~1!, 1.0 ~2!, 5.0 ~3!, 10 ~4!, 100~5!, and 400~6!. In part B, molar concentrations of monomeric (CM) and aggregated dye(CP), which were derived fromA, are plotted to obtain, from mass actioconsiderations, the association number,n, of theJ-aggregate.


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For IPS, in the neighborhood of the Fermi level thp-electron energy levels are evenly distributed and symmric about it~see Fig. 4!. Quantum-chemical calculations~seeRef. 4 and references therein! show that on light excitation ofIPS from the ground state to the first excited~singlet! statethe contribution of the transition from the highest occupimolecular orbital and with the moment directed along tpolytmethine chain is the greatest~. 90%!. This allows theuse of a single-electron approximation to the electronic wfunction and hence our electron-transfer result for the optline shape@see below, Sec. V, Eqs.~6!–~8!# in devising thetheoretical model of light excitation of thep-electron systemat hand.

FIG. 3. Ideal polymethine state~IPS!. Charges on atoms of the polymethinchain in the ground state. Charge:~1! is positive,~2! is negative. The dia-gram is borrowed from Ref. 4.

FIG. 4. Electronic levels, single-excited configurationsx i ~a!, and electronicstates and transitions~b! in IPS. The diagram is borrowed from Ref. 4Quantum-chemical calculations~see Ref. 4 and references therein! showthat in symmetrical polymethine dyes with equalized bonds, electronicels in the vicinity of the Fermi level are arranged uniformly, and the fiexcited state is mainly described by thex1 configuration. The correspondingtransition moment is directed along the polymethine chain.

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3092 J. Chem. Phys., Vol. 116, No. 7, 15 February 2002 Vladimir V. Egorov

Experimentally, polymethine dyes with an odnumbered methine chain symmetrically terminated by nitgen atoms are particularly close to IPS.3 As will readily beobserved, a thiacarbocyanine is among these dyes~see thelabel in Fig. 1!.

In summary thep-electron charge transfer along thpolytmethine chain determines the optical transition in Iand hence in the vinylogous series of a thiacarbocyanBecause thep-electron density strongly alternates along tpolymethine chain~see Fig. 3! and redistributes alternativelon light excitation, there is a strong overlap of the grouand excited state wave functions. Therefore, the tunnelfects are small for excitation of the extendedp-electron sys-tem in polymethine dyes. Notice that this conclusion follofrom the single fact that their chromophoric properties cancrudely estimated from Kuhn’s free-electron model~see Ref.1 and references therein!. Thus, to describe photoexcitatioof the extendedp-electron system, in a theory of photoinduced electron transfer the tunnel effects must be neglecAs this takes place, we are bound to ascribe a new physmeaning to the extent of the electron transfer, namely, aneglect of the tunnel effects it will be considered as thetent of thep-electron system.10,11 This is how matters standin the arguments in favor of the fact that the nature ofoptical transition in polymethine dyes ultimately reducesthe nature of the electron transfer.

III. STANDARD THEORY OF THE ELECTRONTRANSFER AND FAILURE OF ITS APPLICATION TOTHE CHARGE TRANSFER IN THE DYECHROMOPHORE

In a simplified form the up-to-date picture of elementaelectron transfer in condensed matter~see, e.g., Refs. 15–18!can be represented as follows.

Formation of a transition state in an electron-transferaction involves changes in the stochastic environment~outer-sphere reorganization!. In thermal reactions it is fluctuationin the environment that bring about occasional equalizaof the energies of the reactant and product states. The rtion coordinate corresponds to a displacement of the enviment, such as might be represented in the simplest casemeasure of the environmental polarization fluctuation. Tfree energy increases as the environment deviates frommost probable configuration. The well-known two displacintersecting, curves~diabatic terms! represent the free energassociated with each of the reference placements of thetron, on either the donor or acceptor. At the instant in tiwhen the environmental fluctuation brings the two electrostates into energy coincidence, the electron can transfer wout energetic penalty. This coupling of the motion of telectron to the environmental nuclear degrees of freedomin accord with the usual Born–Oppenheimer approximatiIf the coupling is weak~the nonadiabatic limit!, the probabil-ity for electron transfer is low. Here the electronic couplinrather than environmental dynamics, will control the reactrate. In the limit of strong coupling~the adiabatic limit!, thereaction can be viewed as proceeding on a single electrsurface. Here the electron perfectly follows the nuclemovements of the environment.


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The applicability of these general theoretical concehave been validated by computer simulations of the relevfree-energy surfaces. In particular, for the prototypical ferrferrous electron-transfer reaction in water, the calculatedtivation energy, 20 kcal mol21, was in good agreement withthe estimated experimental value of 15–20 kcal mol21. Moreimportantly, the data were quantitatively consistent with tMarcus picture19 of parabolic free-energy surfaces foelectron-transfer processes.~See Ref. 17 and referencetherein.!

Both limits, nonadiabatic and adiabatic, give the saexponent in rate constant and vary in preexponential fathat for the former limit is far less than for the latter one15

For the case of reactions intermediate between nonadiaband adiabatic, expressions for preexponential factor19 of theLandau–Zenner20–23type are available, but this case, thusfined, constitutes mainly a refinement only infrequently us

The up-to-date theories are applicable to both putaand photoinduced electron-transfer reactions by proper dnition of the initial and final electronic states and the mecnistic scheme. For photoinduced transfer the initial statesimply an excited state for the reacting molecules andfinal state could be either of another electronic excited sor a ground state. By putative transfer is meant that the inelectronic states are the lowest energy states possible foreacting molecules. In either case the electron transfertake place by thermal activation~Arrhenius type! or by tun-neling mechanisms. In both cases the transfer is driventhe free-energy difference of the final and initial states; thisto be contrasted with the situation for intervalence trans‘‘reactions’’ which require the presence of an external eletromagnetic field to drive the transfer and so fall underaegis of the theory of electronic spectroscopy.~See Ref. 24.!

In the case of photoinduced electron transfer, the Martype equation for absorbencyK is as follows:11

K5\3

2mJ1expS 2

2R

a D 1

A4pl rkBTexpF2

~D2l r !2

4l rkBT G ,~1!

wherem is the effective mass of the electron,J1 is the bind-ing energy of the electron in the initial state~J1.0 by defi-nition!, andR is the width of the electronic potential barrieseparating the initial and final states in space or, in otwords, R is the extent of the electron transfer;a[\/A2mJ1; l r is the Marcus’s reorganization energy of thenvironmental nuclear motion, associated with the electtransfer;T is the absolute temperature~kBT.\v/2, v is thephonon frequency!;

D5\V1J22J1 , ~1a!

whereD is the enthalpy pertaining to the heat take~or re-lease! in electron transfers,V is the photon frequency, andJ2

is the binding energy of the electron in the final state~J1

.J2.0 by definition!. As discussed above~see Sec. II!, anelectron-transfer result is adapted to tackle the challengelight excitation of the extendedp-electron chromophores oa polymethine dye by neglecting the tunnel effects, i.e.,making in a corresponding result@here in Eq. ~1!# theGamow exponential exp(22R/a) equal to unity, and subse


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quently ascribing the sense of the extent of thep-electronsystem to the extent of electron transferR. @Recall that thep-electron extent is the length along which most of tp-electron charge of a polymethine dye chromophore trafers on light excitation of it~see Sec. II!.# Evidently Eq.~1!is the Gaussian distribution versus photon frequencyV withhalfwidth

w1/252A2 ln 2A2l rkBT. ~1b!

Using the experimental data of Brookeret al. for n53~see Fig. 1 and the discussion below!, let us elucidate thenumerical value of reorganization energyl r that results fromEq. ~1b!. From Fig. 1 it follows that halfwidth

w1/252p\c

nrefrS 1

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1

λ2D'0.09 eV

~λ15730 nm, λ25786 nm,nrefr51.33!,

wherel1,2 are the corresponding wavelengths, andnrefr is therefractive index of the light. Then, substitutingw1/2

50.09 eV and T5300 K into Eq. ~1b!, we obtain l r

'0.03 eV.On the other hand, as we have mentioned in Sec. I

Refs. 10 and 11 we show that the nature of theJ-band is dueto an unusual resonance between the electron and envmental nuclear reorganization motion. This very unusresonance occurs for a weak dissipation when a charactefrequency of the extended electron motion becomes apprmately equal to a characteristic frequency of the environmtal nuclear reorganization motion:

~2te!21't21, ~2!

wherete5R(2J1 /m)21/2 and t5\/E, where in its turnE[l r /2 is the reorganization energy of the environmennuclear motion, associated with the presence of the elecin the initial or final state. The nature of this electron-nucleresonance is associated with the fact that in one-half revtion the periodiclike extended electronic motion under thconditions enters into a state resonant with the vibrationlike motion of the ordered environmental nuclear reorganition that is just produced by the change in this electromotion during the elementary electron transfer. To maklong story short the electron-nuclear coupling may enspontaneous pumping of the electron-transfer state by andered environmental motion. One can concede that the natuof the abovementioned resonantlike behavior of the abstion line shape in the vinylogous series of a thiacarbocyanwith a resonant state forn53 ~see Fig. 1! is also due to thisresonance.11 Thus, let us elucidate now the numerical valof reorganization energyl r that results from Eq.~2!. Substi-tuting the representative parameters,R(n53)[R* >1.4 Å310514 Å, J1>5 eV ~see Ref. 11!, andm5me , into Eq.~2! gives: l r'0.63 eV. Thus, as will readily be observethis numerical value of reorganization energyl r is manytimes that resulted from Eq.~1b!: 0.63 eV/0.03 eV521(!).

The reason of such huge discrepancy between the ll r and the formerl r , which is in line with popular belief~see, e.g., Ref. 25!, is that Eq.~1! represents a particularesult from the standard theory of multiphonon transitio


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where the electron transitions are considered as instaneous. In other words, the application of Eq.~1! to the ex-tended system, such as the system ‘‘polymethine1environment,’’ offers a local instead of the total reorganzation energy. In summary the Marcus type equation is aunsuitable for treating thep-electron charge transfer alonthe dye chromophore and hence should be changed forproper one.

IV. DEFINITION OF THE PROBLEM ON THE NATUREOF ELEMENTARY ELECTRON TRANSFER INCONDENSED MATTER

There is reason to think that the starting point for tderivation of a microscopic theory revealing the natureelementary electron transfer in condensed matter is the sdard theory of multiphonon transitions~TMT! ~see Ref. 26!which has been developed back in the 1950–60s to accfor optical and thermal electron transitions in impurity ceters of dielectrics or semiconductors. In this theory the eltron transition attended with an environmental nuclear reganization motion is considered as instantaneous. Evidethis idealization is in conflict with a gross inertia of the evironmental motion and so dictates a limited characterconsideration of a dissipation in the system ‘‘electr1environment’’: in TMT the dissipation is considered in aimplicit form only, namely, by way of the enthalpy that enters into the equation of the conservation of energy forelectron transition. This in turn suggests that the instanneous electron transition can not be considered as irrevible. Nevertheless, these contradictions in terms are no baan explanation for the electron transitions in the abovemtioned impurity centers. What this means is TMT would costitute a certain limiting case of the microscopic theory of telectron transfer, which falls in the region at the top of tclear dissipation providing the irreversibility of the electrotransitions where their rates depend only slightly on the dsipation. Similar conclusions hold for my former theoryelectron/proton transfer27,28 ~see also Ref. 29 and referenctherein! which, like TMT, includes implicitly a reasonablystrong dissipation, however, unlike TMT, it takes into acount the extended character of the transitions~see details inRef. 11!. Interestingly in the framework of this simplifiedtheory alone it has been possible to explain30–32for the well-known Bronsted relationships33 as well as the temperaturedependent electron transfer34 in Langmuir–Blodgett films.

Thus, the problem on the inclusion of the dissipationthe theory in an explicit form is of paramount importance fan understanding of the real electron~multiphonon! transi-tions that are extended with space and time. This very dipation is conditioned for the most part by the environmendynamic viscosity and governs the relaxation betweenelectron and nuclear reorganization motion during thetended transitions.

It should be noted that TMT has its good point: TMprovides a description for the environmental nuclear reornization so that it is self-consistent with the electron densdistribution. Therefore, the microscopic theory of the eletron transfer must provide the self-consistent description


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the system ‘‘electron1environment’’ at steady state~in theinitial and final states! as well as during the electron transfe

In summary the theory of photoinduced electron transbeyond the Landau–Zenner ideology must be constructegeneralization of TMT to take into account the extendcharacter of a multiphonon transition and hence the actiodissipative properties of the environment on the dynamican elementary electron transfer and the self-consistent rtion of the electron on the environmental~virtual11! nuclearreorganization motion during the transfer, or more simpthe fact that the electron motion goes hand in hand withenvironmental motion and vice versa during an elemenelectron transfer. Attacking the problem on the electrtransfer by this strategy leads us to the discovery10,11 of thevery unusual resonance between the electron and ordenvironmental nuclear reorganization motion or, in a woelectron-nuclear resonance~see above, Sec. III! which, as weshall see later, is responsible for the nature of a resonanbehavior of the well-known absorption line shape in thenylogous series of a thiacarbocyanine5 ~see also Ref. 2 andFig. 1! and the nature10,11 of the well-known intense narrowJ-band6–9 for a benzimidazolocarbocyanine12–14 ~see alsoRef. 2 and Fig. 2!.

V. BROAD OUTLINES OF THE MICROSCOPICTHEORY OF PHOTOINDUCED ELECTRON TRANSFER

The microscopic theory of photoinduced electron trafer, or, more precisely, electrodynamics of extended mtiphonon transitions,11 is governed by the following parameters:te is the characteristic time of an extended electromotion @see above, Sec. III, Eq.~2!#, t is the characteristictime of a virtual11 nuclear reorganization motion@see above,Sec. III, Eq.~2!#, t85\/D is the characteristic time pertaining to the heat take~or release! in electron transfers, whereDis the corresponding enthalpy@see above, Sec. III, Eq.~1a!#;t05\/g is the characteristic time pertaining to dissipatienergyg5g i1gv'gv , whereg i is the intrinsic dissipationenergy associated with an intrinsic-electron-nuclerelaxation dynamics andgv is the dissipation energy assocated with an environmental-relaxation dynamics that arifrom the environmental dynamic viscosity or any othdamping mechanism; andj5(12E/J1)1/2 is the quantitycharacterizing the relative height of the electronic potenbarrier ~J1.E by definition!.

It is well known that the following methods have a widdistribution in the standard theory of multiphonon transitio~see Ref. 26!: the method of the generating polynomia~generating functions! of Krivoglaz and Pekar,35,36 theFeynman–Lax method of operational calculus,37,38 themethod of the density matrix of Kubo and Toyozawa,39,40

and the quantum field techniques~see Ref. 26!. The methodof the generating polynomials is the simplest one and allofor the main effect of the interaction of the optical electrwith a vibrational motion of nuclei of environment, namethe shifts of the normal phonon coordinates due to an etron transition. The changes in the phonon frequenciesthe other effects of higher orders can be taken into consiation by the rest of the mentioned techniques. Because


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case of extended multiphonon transitions~electron transfer!as a more general case of electron-phonon motion is far mcomplex when compared to the case of the usual~con-strained! multiphonon transitions, it is clear that a generazation of the theory of multiphonon transitions to the caseelementary electron transfer must be initiated by evolvthe simplest technique of Krivoglaz and Pekar.

Compared to the standard Hamiltonian~see Ref. 26!, inour theory the Hamiltonian is only complicated by an adtional electronic potential wellV2(r2R) separated from theinitial well V1(r ) by distanceR[uRu:

H52\2

2m∆r1V1~r !1V2~r2R!1(

kVk~r !qk

11

2 (k

\vkS qk22

]2

]qk2D , ~3!

wherer is the radius vector of the electron,qk are the realnormal phonon coordinates,vk are the eigenfrequencies othe normal vibrations, andk is the phonon index; termSkVk(r )qk is due to the electron-phonon coupling.

For calculating the initial and final states of the syste‘‘electron1environment,’’ we follow the Pekar adiabatitheory41,42 that uses an amplitude of nuclear vibrationsenvironment as a small parameter and restrict our consiation, like Pekar, to the first order of this perturbatiotheory. We refer to this approximation as adiabatic appromation. To take into account the strong virtual11 electron-nuclear~-phonon! coupling that results in narrowing of thoptical absorption line, we must grasp the fact that the inistate stands out, against the final state, as being unique inthe resonance between the electron and ordered environtal nuclear reorganization motion can be produced onlythe initial change in the extended electron motion duringelementary process of the extended multiphonon transitSimply stated, only before absorbing the photon, the elecmotion may be tuned to resonance with the ordered nucreorganization motion.11 Because of this, in the equation

A125^C2~r2R,q!uVuC1~r ,q!& ~4!

for the amplitude of an extended multiphonon transitionA12,the wave function of the system for the electron localizedthe final stateC2(r2R,q) obeys the Schro¨dinger equationand so is taken in the Pekar approximation, in a literal senand the extended character of the multiphonon transitiototally enclosed in the wave function of the system for telectron localized in the initial stateC1(r ,q).43 This wavefunction obeys the integral Lippmann–Schwinger equatio

C15GVC1 , ~5!

where G is the Green function of the system ‘‘electro1environment,’’ constructed on the basis of the wave funtions of HamiltonianH2V; operatorV is given by equationV5SkVk(r )(qk2qk) ~qk are the normal phonon coordnates corresponding to equilibrium positions of the nucwhen the electron is localized in the initial or final state!.Equation~5! is solved by applying adiabatic approximatioto its integral side in perfect analogy to the Pekar appro


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mation. Notice that although adiabatic approximation is sused for solving the problem, the result for amplitudeA12

thus obtained is beyond the Born–OppenheimerFranck–Condon approximations.

Using the methods of the semiclassical radiation the~see Ref. 26!, as applied to the case of extended multiphontransitions,27,28 we obtain the following expression for thlight extinction:

«54p2e2NAV

3\cnrefrK, ~6!

wheree is the amount of electronic charge transferred inextended multiphonon transition,NA is the Avogadro num-ber; in Ref. 11 we show that absorbencyK in the frameworkof the approximation of zero radius44 for the electronic po-tential wellsV1(r ) andV2(r2R) @see Eq.~3!# and the Ein-stein model of nuclear vibrationsvk5const[v is repre-sented as follows:

K5K0 exp~W!, W51

2lnFvt sinh~b!

4p cosh~ t ! G22

vt Fcoth~b!

2cosh~ t !

sinh~b!G1~b2t !1

vtQ2

sinh~b!

4vtQ2 cosh~ t !,

1!1

vtQ<

2 cosh~ t !

vt sinh~b!, ~7!

whereb[\v/2kBT,

t5vte

u FAC1BD

A21B2 12Q~Q21!

~Q21!21~Q/u0!2 1u0

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E

g,

and whereE[\vSkqk2 is the abovementioned reorganiz

tion energy of nuclear vibrations,g.0 is the standard constant originating in the energy denominator of the Grefunction G and which can be treated as a dissipation ene~see above and also below, Secs. IX and X!. ExpressionsA5A(u,Q,u0 ,j), B5B(u,Q,u0 ,j), C5C(u,Q,u0 ,j), andD5D(u,Q,u0 ,j) are formulated from elementary functions:

A5cosS u

u0D1L1S 1

u0D 2

N, B5sinS u

u0D1

1

u0M,

C5uFcosS u

u0D2

12j2

2u0sinS u

u0D G1M,

D5uFsinS u

u0D1

12j2

2u0cosS u

u0D G2

2

u0N;

where functions L5L(u,Q,j), M5M(u,Q,j), and N5N(u,Q,j) are as follows:


ll

d

yn

e

ny

L52~Q21!2E1F ~Q21!u

r1Q~Q22!GE~12r!/~12j!,

M52Q~Q21!E2F ~2Q21!u

r12Q~Q21!GE~12r!/~12j!,

N5QFQE2S u

r1Q DE~12r!/~12j!G ;

E[expS 2u

11j D , r[Aj2112j2

Q.

In Eq. ~7! factor K0 takes the form:

K052t3J1

m

~A21B2!r3Q4j

u2F ~Q21!21S Q

u0D 2G2F11S 1

u0D 2G

3H 1

vt F11sinh~b22t !

sinh~b! G2

1cosh~b22t !

sinh~b! J h, ~8!

where the electronic tunnel effects have been neglectedmaking the Gamow exponential exp(22R/a) equal to unityand so ascribing the sense of the extent of thep-electrontransfer along the dye chromophore to the extent of electransferR ~see above, Secs. II and III!. Notice that this viewleaves room for a weak tunnel effect which is herein takinto account phenomenologically by factorh<1 @see Eq.~8!#.

From the absorption line shape given by Eqs.~6!–~8! itfollows that the unusual very resonance occurs for a wdissipation~g,E, see details in Secs. IX and X! when acharacteristic frequency of the extended electron motioncomes approximately equal to a characteristic frequencythe ordered environmental nuclear reorganization mot@see Eq.~2!#. The nature of this electron-nuclear resonancediscussed above~see Sec. III!. For a strong dissipation~g.E, see details in Secs. IX and X!, from Eqs.~6!–~8! fol-lows the standard Landau–Zenner20–23 type picture of adia-batic and nonadiabatic electron transfers with nonadiabone answering a somewhat stronger dissipation.11

VI. NATURE OF THE OPTICAL LINE SHAPE IN THEVINYLOGOUS SERIES OF A THIACARBOCYANINE

Our theoretical absorption line shapes45 fitted to the ex-perimental data5 ~see also Ref. 2! of Brookeret al. ~see Fig.1! are shown in Fig. 5.

Going from the polymethine dye chromophore ofn50to it of n51 and so on up ton55 ~see Fig. 1! is associatedwith a lengthening of the extendedp-electron system and, aa consequence, with an increase in the environmental rganization energy E. Thus, product RE and actionER/AJ1/2m, related to this product, governs a transmutatof the optical absorption band in the vinylogous series ofideal polymethine dye represented here by a thiacarbonine ~see Fig. 5! and so the extendedp-electron state ischanged from a nonresonant state (n50) to the resonantstate~(2te)

215t21 or ER/AJ1/2m5\ or n53!. In the lat-ter state the environmental dynamic viscosity has a minim~see the series of dissipation energyg in the caption of Fig.


ro

on

ere

thi

thethR

ta

l,

th-

n

nd

is-

ntal

ol

Eq

,

nine

a-

vens-

:


5! which is caused by a synchronism11 in the environmentalnuclear motion during the charge transfer along the chmophore.

The wing of the resonant band (n53) is determined bythe whole reorganization energy~some 2E! in an elementaryprocess of the optical absorption whereas the peak isdetermined by a part of it~some E! corresponding to anuclear reorganization in an intermediate state of the elemtary process. That is why the wing is blue and the peak iswith respect to one another~see Fig. 5!. The resonant band(n53) has been shifted to the red range with respect tononresonant band (n50) because of the standard decreasethe energy level gap that is associated with a rise inextent of the system: the dynamic effects are found insence to be canceled out. A similar situation related toJ-aggregate and monomer bands is discussed in detail in11.

Our theory is in a good agreement with the experimendata at hand~cf. Figs. 1 and 5!. We notice that for statesn54 andn55 factorh is less than unity~see Fig. 5! and thusfor the long transitions weak tunnel effects become valid~seeabove, Sec. V!.

VII. NATURE OF THE J-BAND

As to the intense narrow J-band6–9 for abenzimidazolocarbocyanine12–14~see also Ref. 2 and Fig. 2!,in Refs. 10 and 11 we assume precisely the same modedescribed above~see Sec. II!, which does nothing but takeinto account a lengthening of the dye chromophore up toresonant state by ap-p electron interaction of the heterocyclic rings in theJ-aggregate~see Refs. 14, 46, 47 and Fig. 6!.This results in an increase in the amount of charge tra

FIG. 5. Theoretical absorption dependence (l52pc/Vnrefr) of anideal polymethine dye represented by a thiacarbocyanine in methan25 °C on the length of the polymethine chain 2(n12)d, where d is aunitary bond length in the chain. The line shapes are calculated from~6!–~8! by fitting them to the experimental data of Brookeret al. ~see Fig. 1!in wavelengthlmax, extinction«max and halfwidthw1/2 with a high degreeof accuracy. The parameters of the system ‘‘dye1environment’’ in useare as follows:e5e where e is the electronic charge,m5mc , v5531013 s21, d51.4 Å, nrefr51.33; for n50, 1, 2, 3, 4, 5 we have,respectively,J15~5.63, 5.4, 4.25, 3.9, 3.735, 3.4! eV, J12J25~1.705,1.31, 1.111,0.9, 0.735, 0.4! eV, E5~0.245, 0.248, 0.256, 0.275, 0.2970.496! eV, andg5~0.402, 0.205, 0.139, 0.120, 0.129, 0.131! eV; for n50, 1,2, 3 factorh51 and forn54, 5 factorh50.55, 0.1 respectively.


-

ly

n-d

ene

s-eef.

l

as

e

s-

ferred along the chromophore on light excitation of it ahence in the environmental nuclear reorganization energy~inthe electron-nuclear coupling! as well as in a substantial~seebelow, Sec. VIII! decrease in the environmental dynamic vcosity during the charge transfer~cf. eJ , EJ , gJ and eM ,EM , gM , respectively, in the caption of Fig. 7!. Our theoryis in a good agreement with the representative experime

at

s.

FIG. 6. A cyanine dye molecule and its brickstoneJ-aggregate~Ref. 1!.Lengthening of the dye chromophore on theJ-aggregation by ap-p electroninteraction of the heterocyclic rings, whered is a unitary bond length ineither chromophore~Refs. 10, 11!.

FIG. 7. Theoretical absorption dependence of a benzimidazolocarbocyaat 25 °C on dye concentrations. In part A, the dye monomer~M! andJ-aggregate~J! line shapes are calculated from Eqs.~6!–~8! by fitting themto the experimental data of Herz~see Fig. 2! in wavelengthlmax, extinction«max and halfwidthw1/2 with a high degree of accuracy, when 1%~1! and99% ~6! of dye are converted toJ-state respectively. The intermediate reltive dye concentrations are as follows: 9%~2!, 53% ~3!, 66% ~4! and 82%~5!. In part B, molar concentrations of monomeric (CM) and aggregated dye(CJ[CP), which were derived from the absolute dye concentrations giby Herz ~see the caption of Fig. 2! and the relative dye concentrationobtained from our fitting~see A!, are plotted to obtain the association number, n, of the J-aggregate. The parameters of the systems ‘‘J-aggregate1environment’’ and ‘‘dye monomer1environment’’ in use are as followsmJ50.86me and mM50.97me , v5531013 s21, d51.4 Å, nrefr51.33;J1J5J1M55 eV, J1J2J2J51.114 eV and J1M2J2M51.371 eV, EJ

50.420 eV andEM50.315 eV, gJ50.0674 eV andgM50.2309 eV; eJ

.A2«DRJEJ'1.28e and eM>A2«DRMEM'0.96e where dielectric con-stant«D52.5 ~s-electron portion, solvent! ~Ref. 1!; h51.


o

aar

na

ou

taroon

nfo

hegld

er

a

rasi

d as

icalarycancia-the

.

el-on-ing

d or-lec-ron

nor

ket.

re-

n

nA,h

s


data on the transmutation of the absorption line shapeJ-aggregation~cf. Figs. 2 and 7!.10,11 We notice@Fig. 7~b!#that the experimental data at high concentrations@Fig. 2~a!~5, 6!# have poor precision which is explained byJ-aggregate lability towards the occurrence of colloidal pticles.

VIII. PREDICTION OF VERY INTENSE NARROW LINESFOR SHORT OPTICAL TRANSITIONS

As will readily be observed, for the resonantJ-state of abenzimidazolocarbocyanine as compared to the resostate in the vinylogous series of a thiacarbocyanine (n53)~cf. the captions of Figs. 7 and 5!, the dissipation energy isless approximately by half:gJ /g50.0674 eV/0.120 eV'0.56. This fact is due to the superior electron-nuclear cpling in the J-state (EJ /E50.420 eV/0.275 eV'1.53),which provides a superior synchronism in the environmennuclear motion during the charge transfer along the chmophore. Thus, as illustrated in Fig. 8, the strong electrnuclear coupling in the resonant state@(2te)

215t21 orER/AJ1/2m5\, see Eq.~2!# entails a waste of dissipatio@Fig. 8~a!# and so gives rise to very intense narrow bandsthe short optical transitions@Fig. 8~b!# ~n51, 0!#.45

It might be well to point out that, generally speaking, tintensity and the width of the absorption band are strondependent on a specific type of the relationship betweensipation energyg and electron-nuclear coupling~reorganiza-tion energy! E. However, in the energy range betweenE50.275 eV andEJ50.420 eV this dependence is rathweak. This is evident from the fact that any suitablea priorifunction g5g(E) can be approximated in this range bylinear one. Here, we assume thatg5g(E) is a hyperbolicfunction @Fig. 8~a!#. The true functiong5g(E) can be es-tablished only in the framework of a future more genetheory that involves the electromagnetic field in the ba

FIG. 8. Strong electron-nuclear coupling in electron-nuclear resonaAJ1/2m/R5E/\ @see Eq.~2!# gives rise to intense narrow bands. In partdissipation energyg5g(E) is approximated by a hyperbolic function witthe use of our theoretical treatment@Fig. 5 ~E50.275 eV,g50.120 eV! andFig. 7 ~EJ50.420 eV, gJ50.0674 eV!# of the experimental data for theresonant states of Brookeret al. ~Fig. 1,n53! and Herz~Fig. 2,J-band!. Inpart B, the electron-nuclear resonance for the data of Brookeret al., n53, isenhanced by strengthening the electron-nuclear coupling and this followthe same fashion that the nonresonant states,n52, 1, 0 ~Fig. 5!, are con-verted one by one to the resonant states, n52, 1, 0, as the coupling isstrengthened.


n

-

nt

-

l--

r

yis-

lc

Hamiltonian of the system. Such a theory can be classifiequantum electrodynamics of extended transitions.

IX. DYNAMIC ORGANIZATION OF ELEMENTARYELECTRON TRANSFER IN CONDENSED MATTER

A start has been made in Refs. 10 and 11 on the physinterpretation of the dynamic organization of an elementelectron transfer in condensed matter. This interpretationnow be extended to include, among other things, the assotion between our unusual electron-nuclear resonance andHeisenberg uncertainty relation

DpDx5\

2. ~9!

It is easily seen that Eq.~2! can be written as Eq.~9! whenwe place

Dp5E

A2J1 /m, ~10!

where reorganization energyE[\vSkqk2 ~see above, Sec

V!, and

Dx5R. ~11!

It is common knowledge that equality~9! relates to the wavefunction packet of the Gaussian form of widthDx, describ-ing a free-moving quantum particle. Therefore, for theementary electron transfer, this quantum particle can be csidered as a free electron-phononic quasi-particle havmomentum

p0'Dp5E

A2J1 /m~12!

@see Eq.~10!# and wave number

k05p0

\'

v(kqk2

A2J1 /m. ~13!

This quasi-particle will be termed thetransferonfor short.Thus, in the case of resonance between the electron andered environmental nuclear reorganization motion, the etron transfer can be considered as a motion of the transfewhich is created by the electron detachment from the doand annihilated by its attachment to the acceptor.

Let us estimate massmtr and lifetimet tr of the transf-eron, and the widening of the corresponding wave pacExpression of lifetimet tr follows immediately from the defi-nition of the transferon when we rewrite the uncertaintylation given by Eqs.~9!–~11! in the form of an uncertaintyrelation for the total energy of the system ‘‘electro1environment’’Etotal, namely, in the form

dEtotaldt5\

2, ~14!

where

dEtotal5E ~15!

and

ce

in


veth

ti

nis

e

sts

ted

on

te

e-

avth

da

ery

me

ones

t

adsk

viorentor-

ays--

ape

en

sfree


dt5R

A2J1 /m[te5t tr . ~16!

It is common knowledge that the maximum of the wapacket moves with the group velocity that is equal tovelocity of quantum particlen0 ~see, e.g., Ref. 48!, i.e.,

n05p0 /mtr'E

mtrA2J1 /m~17!

for the transferon. Substitutingn0 andt tr from Eqs.~17! and~16!, respectively, into equation

R5n0t tr ~18!

gives:

mtr'E

2J1m. ~19!

The value of reorganization energyE, as a rule, is no morethan 1 eV, and for polymethine dyes, binding energyJ1

>5 eV ~see Ref. 11 and references therein!. Therefore,E/2J1!1 and

mtr!m, ~20!

i.e., the mass of the transferon is far less than is the effecmass of the electron.

It is common knowledge that the wave packet widewhen moving. The width of the wave packet shows a rfrom the value ofb at the instant of time t50 to the value of

b85bF11S \t

mpb2D 2G1/2

'\

mpbt ~21!

at instantt5t ~see, e.g., Ref. 48!, wheremp is the mass ofthe quantum particle. By applying this equation to the casthe transferon motion, we obtain

R85RF 11S \R

A2J1 /m

E

2J1mR2

D 2G 1/2

5A5R'2R. ~22!

Thus, we can see that the wave packet of widthR describingthe free-moving electron-phononic quasi-particle, the traneron, widens in its life time by a factor of 2 or thereabouno matter what the parameters of the system.

In our theory~see above, Sec. V, and also Ref. 11! thequantum state of the transferon as well as any other stathe elementary electron transfer in condensed matter isscribed by the wave function of the system ‘‘electr1environment’’C15C1(r ,q;R,E,g). As indicated earlier,the transferon state is defined by two parameters~exclusiveof massm and binding energyJ1!: by extentR[uRu of elec-tron transfer and reorganization energyE. Let us find outnow what is the physical sense of the third parameter ening into wave functionC1(r ,q;R,E,g), namely, of dissipa-tion energyg. Before proceeding to that, it should be rcorded again that the transferon life timet tr discussed aboveis related to the displacement of the electron-phononic wpacket from the donor to the acceptor, i.e., it is related to


e

ve

se

of

f-,

ofe-

r-

ee

free-moving transferon. Besides this free lifetimet tr , there isa pumping life timet tr

p of the transferon what is associatewith the dynamics of its creation and life on the donor inbound state~before its transition to the free state! and whatdetermines the dynamic stability of the transferon. This vtime is just the one defined by dissipation energyg as fol-lows:

t trp5

\

2g[

t0

2, ~23!

and may be considerably more time than the free lifetit tr5R/A2J1 /m.

The function of the pumping lifetimet trp can be illus-

trated by dependence of the optical absorption line shapethis time. In Fig. 9 it is intimated how the line shape variwith parameteru0[E/g5Et0 /\52Et tr

p/\ which is in di-rect proportion to timet tr

p . From this figure we notice thathe optical absorption band having the pronounced peak~so-calledL-peak11!, associated with the transferon state, spreout with decreasing timet tr

p through the take-up of the peaby its own blue wing~so-calledD-band11! and ultimatelyforms the wide band of near-Gaussian shape. Such behais determined by decay of the transferon into its constituelectron and phonons, or, in other words, by dynamic disganization of the elementary electron transfer.~It is well tobear in mind that an alternative dynamic organization mappear here, see below.! Thus, as a rough guide the tranferon state has meaning whenL-peak stands out on the absorption band, i.e., forg,E ~see Fig. 9!. For g.E, evi-dently the pumping lifetimet tr

p5\/2g of the transferon is

FIG. 9. Transferon and dissipon: the variation of the absorption line sh

F5F(V) @F[K31013/(2t3J1 /m), see Eqs.~7! and ~8!# with decreasingdissipation energyg ~with increase in parameteru0[E/g!. Here, we for-

mally put J22J150 for simplicity; then Q215\V/E[V. The twicepumping life time 2t tr

p5\/g is equal to or greater than the free life timt tr5R/A2J1 /m5\/2E. The parameters of the system ‘‘electro1environment’’ in use are as follows:J155 eV, E50.4 eV, m5me , v5531013 s21 andT5300 K. The line shapes foru054, 3, 2 correspond toa free motion of the transferon. The line shape foru050.5 corresponds to afinite motion of the dissipon. Foru051, the quasi-particle motion could aappropriately be considered as the free transferon motion as well as adissipon motion.


tan

e

uco

r-dnsthent

il

snsth

en

in

esi-

bytionentac-om

i-forthe

ass

thethe

e.

me

one ofrghetyty

ionree

the

forof-


less than its free lifetimet tr5te5t/25\/2E, and we havethe wide band of near-Gaussian shape. In other words,transferon in this situation is so much unstable that decbefore it takes to reach the acceptor. Conversely, wheg!E, the dynamic stability of the transferon following fromlarge timet tr

p@t tr is so much high that it gives rise to thvery intense narrow lines~see Fig. 10 and also Fig. 8!. It isinteresting to note in this connection that the dynamic strture of the transferon is determined by the large valuereorganization energyE and thus is associated with an odered constituent of the condensed matter motion inducethe electron motion, whereas the detachment of the traeron from the donor to move as free quasi-particle towardacceptor is determined by the small value of dissipationergy g and so is associated with a disordered constituenthis very motion of condensed matter.@The dynamics of theattachment of the transferon to the acceptor and its annihtion on it can be neglected~see Discussion in Ref. 11!.#

The question now arises as to whether there is a pobility to interpret the dynamics of elementary electron trafer and the corresponding optical absorption line shape incase ofg.E as easily as in the case ofg,E. For an answerto be given, we call attention to the fact that from the depdence of the wave functionC15C1(r ,q;R,E,g) on param-etersR,E,g hand in hand with resonance (2te)

215t21 @seeEq. ~2!# pertaining to the transferon state is possiblea priorithe resonance as follows:

~2te!215t0

21. ~24!

This equation can be written as the Heisenberg uncertarelation, much as Eq.~2! can be done~see above!. To deter-mine if this is truly valid, let us write Eq.~24! in the form ofEq. ~9!, where now

Dp5Dpg[g

A2J1 /m~25!

FIG. 10. Transferon: the variation of the absorption line shapeF5F(V)@F[K31013/(2t3J1 /m)# with decreasing dissipation energyg ~with in-crease in parameteru0[E/g!. The pumping life timet tr

p5\/2g is far inexcess of the free life timet tr5R/A2J1 /m5\/2E. The parameters in useare the same as in Fig. 9.


heys

-f

byf-e-

of

a-

si--e

-

ty

@i.e., in the former expression forDp ~see Eq.~10!! quantityg is substituted for quantityE#, and evidentlyDx5R willcontinue in use@see Eq.~11!#. Thus, we have now the wavpacket describing the motion of a free quantum quaparticle having momentum

p0g'Dpg5g

A2J1 /m~26!

and wave number

k0g5p0g

\'

g

\A2J1 /m, ~27!

respectively. This quasi-particle will be termed thedissiponfor short. In summary, in the case of the resonance givenEq. ~24!, the electron transfer can be considered as a moof the dissipon which is created by the electron detachmfrom the donor and annihilated by its attachment to theceptor. Because the dissipon can be formally obtained frthe transferon through the substitution of parameterg forparameterE in the Heisenberg uncertainty relation, it is evdent that the overall picture of motion of the wave packetthe dissipon will be the same as for the transferon andexpression for the dissipon massmdiss takes the form:

mdiss'g

2J1m ~28!

@cf. this equation and Eq.~19!#. Our theory11 assumesg/2J1!1 @see inequalityu0@E/2J1 after Eq. ~7!#. There-fore, the dissipon mass is much less than the effective mof the electron:

mdiss!m, ~29!

as in the case of the transferon@see Eq.~20!#.In the case of the dissipon as well as in the case of

transferon considered above, the dynamic stability ofquasi-particle state is characterized by a pumping lifetimThis time,t diss

p , is now defined as

t dissp 5

\

2E[

t

2~30!

and may be considerably more time than the free life titdiss5t tr5R/A2J1 /m @see below, Eq.~33!#.

It should be pointed out that a family of the absorpticurves what presented in Figs. 9 and 10 falls into the casE5const andg5var. As this takes place, the Heisenbeuncertainty relation for the transferon is in the form of texact equalityDpDx5\/2, and the Heisenberg uncertainrelation for the dissipon is in the form of the inequaliDpgDx>\/2, where the exact equality is the case withg5E. Thus, one can say that the family of the absorptcurves shown in Figs. 10 and 9 falls into the cases of a fmotion of the transferon (u056,5,4,3,2) and a finite motionof the dissipon (u050.5). Forg5E ~or u051!, the quasi-particle motion could as appropriately be considered asfree transferon motion as well as a free dissipon motion.

On the other hand, a family of the absorption curvesg5const andE5var in Figs. 11 and 12 falls into the casesa free motion of the dissipon~u050.3,0.4,0.5,0.6; the dissi


s

diothrtr

-

yhe

en

ipo-

a

ent

on

in it

ns-

sf-s-rgythe

ion,r toined

ofy-and.

ednce

nyweely

ape

-

ssianfer:freeline

f

-

-orrm


pon massmdiss is taken to be equal to the transferon mamtr50.04me adopted in Figs. 9 and 10! and a finite motionof the transferon (u051.6, 2). For E5g ~or u051!, thequasi-particle motion could as appropriately be considerethe free dissipon motion as well as a free transferon motAs can be readily appreciated, this dissiponic picture ofelementary electron transfer can be considered as a cepicture which is complementary to the transferonic pictuconsidered above and illustrated by Figs. 10 and 9.

At this point, it is worth noting that the uncertainty relation for the dissipon@see Eqs.~9!, ~11!, and ~25!# can berepresented as an uncertainty relation for the total energthe systemEtotal, just as was done earlier in the case of ttransferon@see Eqs.~14!–~16!#, namely, as follows:

d8Etotaldt5\

2, ~31!

where

d8Etotal5g ~32!

and

dt5R

A2J1 /m[te5tdiss. ~33!

This uncertainty relation can be considered as a certain rtion which is complementary to the uncertainty relation cosidered above for the transferon: in the case of the dissthe width of the energy levelEtotal is determined by a dissipation process in the environment@see Eq.~32!#, whereas inthe case of the transferon this width is determined by

FIG. 11. Dissipon: the variation of the absorption line shapeF5F(Vg)@F[K31014/(2t3J1 /m), see Eqs.~7! and~8!# with decreasing reorganiza

tion energyE ~with reduction in parameteru0[E/g!; Vg[\V/g ~here, asin Figs. 9 and 10, we formally putJ22J150 for simplicity!. The pumpinglife time tdiss

p 5\/2E is greater than the free life timetdiss5R/A2J1 /m5\/2g. The parameters of the system ‘‘electron1environment’’ in use areas follows: J155 eV, g50.4 eV, m5me , v5531013 s21 and T5300 K. The near-Gaussian line shape foru050.3 corresponds to the standard nonadiabatic electron transfer. The near-Gaussian line shapes fu0

50.4, 0.5, 0.6 correspond to the case of electron-transfer reactions intediate between nonadiabatic and adiabatic~u051, see Fig. 12!.


s

asn.eaine

of

la--n

n

ordered nuclear reorganization process in the environm@see Eq.~15!#. To put it differently, the pure~quantum-mechanical! electron-transfer state~e.g., the dissipon or thetransferon! may be determined by interaction of the electrwith its environment throughspontaneous pumpingof thisstate by adisordered environmental motion~dissipon! aswell as by anordered environmental motion~transferon!. Putvery simply, the environment in its simplest form49 may bebarrier to the electron transfer as well as can be a help~cf. Figs. 11 and 10!.

In summary the picture of the elementary electron trafer in the language of the dissipon~see just below! iscomplementary to the picture in the language of the traneron ~see above!. Namely, the dynamic structure of the disipon is determined by the large value of dissipation eneg and so is associated with a disordered constituent ofcondensed matter motion induced by the electron motwhereas the detachment of the dissipon from the donomove as free quasi-particle toward the acceptor is determby the small value ofV reorganization energyE and so isassociated with an ordered constituent of this very motioncondensed matter.@As in the case of the transferon, the dnamics of the attachment of the dissipon to the acceptorits annihilation on it can be neglected~see Discussion in Ref11!.#

It is significant that the electron-transfer results obtaintheoretically by the author in Ref. 11 in the case of resona(2te)

215t21 @see Eq.~2!# andg,E are absolutely novel.Therefore, there is no way of comparing them with aformer theoretical electron-transfer results. In this respectcan state with assurance that the new quasi-particle, nam

FIG. 12. Dissipon and transferon: the variation of the absorption line sh

F5F(Vg) @F[K31013/(2t3J1 /m)# with decreasing reorganization energyE ~with reduction in parameteru0[E/g!. The twice pumping life time2tdiss

p 5\/E is equal to or greater than the free life timetdiss5R/A2J1 /m5\/2g. The parameters in use are the same as in Fig. 11. The near-Gauline shape foru051 corresponds to the standard adiabatic electron transthe quasi-particle motion could as appropriately be considered as thedissipon motion as well as a free transferon motion. The near-Gaussianshape foru050.6 ~a free motion of the dissipon! corresponds to the case oelectron-transfer reactions intermediate between nonadiabatic~u050.3, seeFig. 11! and adiabatic. The line shapes foru051.6, 2 correspond to a finitemotion of the transferon.

e-


o

ua.e

es

er

-n

u–sth

ss

icens

Ithgianrieoo

pa

e-no

roon

trat

an

u

th

pathia

tee

odarth

ated

y by

-

thearyipa-m-mu-nly

thealnlyxed

gyental

f the

an-nceasandhef-

o-is

ionions-he

n-issf-

the

onayure

e

ess, our


the transferon, is discovered in theory. As to the caseresonance (2te)

215t021 @see Eq.~24!#50 andg.E, we have

arrived here at the electron-transfer picture which is simlated by the standard Landau–Zenner picture of the adiab(g'E) and nonadiabatic (g@E) electron-transfer reactionsIn this respect the dissipon was already known in the formtheoretical considerations~see, e.g., Ref. 51 and referenctherein! as a certain particle, or ‘‘coordinate’’~e.g., the sol-vent polarization!, moving on a potential energy surfac~adiabatic reactions! in the region of an avoided crossing ohopping from one surface to another~nonadiabatic reactions!. However, it should be pointed out that there is a fudamental distinction of the dissipon from the LandaZenner particle. That lies in the fact that the dissipon iquantum particle, not a classical one, as believed inLandau–Zenner picture. This essential circumstance hatimately associated with him a distinction in physical senof dissipation ~or friction! in these two pictures. In theLandau–Zenner type theories the moving classical partexperiences friction on the source side of an environmwhereas in our theory friction or dissipation representproperty of the internal state of the quantum dissipon.summary the Landau–Zenner type theories do no moresimulate the respective proper result and this does notan adequate physical picture of the elementary electron trfer. What this means is the Landau–Zenner type theoexploit in essence the fact that in the region at the topdissipation the electron-transfer rates depend only slightlythe specific transition dynamics and depend for the moston the properties of the initial and final states.

X. DISCUSSION AND CONCLUSIONS

In this paper we study the optical transition in polymthine dyes andJ-aggregates as an extended electron-photransition along the polymethine chain orJ-aggregate chro-mophore. In other nomenclature, the extended electphonon transition is the elementary electron transfer in cdensed matter~see details of this definition below!. Theextended electron-phonon state corresponding to the electransfer is described by a wave function of the initial stC15C1(r ,q;R,E,g) ~see Sec. V and also Ref. 11! whichdepends on the electron-phonon coupling through reorgzation energyE and dissipation energyg. Introduction of thedissipation states here is to some extent similar to introdtion of quasi-steady states in quantum mechanics~see, e.g.,Ref. 52!. Namely, a quasi-steady state has its origin insolution of the time-dependent Schro¨dinger equation inwhich the total energy is supposed to have an imaginaryin addition to the real one, whereas in our problem onelementary electron transfer a dissipation state is assocwith the standard imaginary additionig (g.0) entering intothe energy denominator of the Green function of the sys‘‘electron1environment’’ and has its origin in changing thinfinitesimally small quantityg.0 by any one of positivenumbers. Such procedure suggests the validity of a mgeneral dynamic equation as compared to the stanLippmann–Schwinger or Schro¨dinger equations and so oudescription of the elementary electron transfer is beyond


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scope of standard quantum mechanics. The postulLippmann–Schwinger equation with 0,g,` suggests thatthe pure quantum-mechanical state is determined not onlthe electron coordinatesr , the phonon coordinatesq, theelectron-transfer distanceR, and the electron-phonon coupling through reorganization energyE, but by the electron-phonon coupling through dissipation energyg as well. To putit differently, one can say that a correct consideration ofelectron-phonon coupling in the problem on the elementelectron transfer is possible only in the event that a disstion ~relaxation! process is introduced into the pure quantumechanical state. As can be readily appreciated, such forlation of the problem is beyond the scope of the commoaccepted picture of the elementary electron transfer~see, e.g.,Ref. 53! where relaxation processes are introduced intoequation for the density matrix in the form of an additionterm associated with relaxation, i.e., they are introduced oat the stage of the phenomenom description by the mi~quantum-statistical! state.

The quantity\/2g[t trp can be treated as a pumpin

lifetime of the quasi-particle ‘‘transferon’’ what is created ba resonance between the electron and ordered environmnuclear reorganization motion~see Sec. III and also Ref. 11!and whereby progress was made towards the treatment oelementary electron transfer in condensed matter; timet tr

p isthe characteristics of a stability of the transferon as a qutum particle. In the case of this electron-nuclear resona@see Sec. III, Eq.~2!#, the electron transfer can be treateda spontaneous pumping of the transferon on the donorthen its free motion from the donor to the acceptor in tevent that timet tr

p is above the free lifetime of the transeron,t tr , which is the time the transferon~or the electron!takes to propagate the donor-acceptor distanceR: t tr5te

5R/A2J1 /m. For the case of equality 2te5\/g @see Sec.IX, Eq. ~24!#, the electron transfer can be treated as the mtion of a certain other quasi-particle, the dissipon, whatcreated by a ‘‘resonance’’~broadresonance! between theelectron motion and the ‘‘environmental nuclear dissipatmotion’’ ~disordered environmental nuclear reorganizatmotion! that arises from the environmental dynamic viscoity or any other damping mechanism. The free lifetime of tdissipon is, by definition,tdiss5t tr5te5R/A2J1 /m. Thepumping lifetime of the dissipon is defined ast diss

p 5\/2E,with t diss

p .tdiss. Thus, the physical picture of the elemetary electron transfer in the language of the dissiponcomplementary to the picture in the language of the traneron. Such complementary description is evident fromform of the Heisenberg uncertainty relation,DpDx>\/2, asapplied to the dynamics of elementary electron transfer~seedetails in Sec. IX!.

In summary from our theory of the elementary electrtransfer it follows that the electron-phonon coupling mentail the process of spontaneous pumping of the pelectron-transfer state~by an ordered~transferon! or disor-dered~dissipon! environmental motion!. This physical pic-ture differs fundamentally from the picture given by thquasi-steady-state quantum mechanics~see, e.g., Ref. 52!mentioned above, where, for electron transitions, the procof spontaneous decay alone may be tolerated. Besides


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picture of narrowing the optical absorption line differs fudamentally from the well-known Kubo’s picture54 of the mo-tional narrowing55–57 of it. Indeed, in our picture based othe microscopic theory the narrowing of the absorption l@see Sec. VIII, Fig. 8~b!# is due to the effect of spontaneoupumping of the pure quantum-mechanical state~of the pureelectron-transfer state by an ordered environmental motsee above and details in Sec. IX!, whereas in the Kubo’sphenomenological picture the narrowing is due to a cermixed quantum-statistical effect.55–57

It should be noted that the transferon is a new quaparticle which has been discovered in our theory and hasany analog in the former theories of the elementary electransfer. To put it in reverse, the dissipon is a certain anaof the classical particle, or ‘‘coordinate’’~e.g., the solventpolarization!, appearing in the Landau–Zenner type theorof the electron transfer. The fundamental distinction ofdissipon from the Landau–Zenner quasi-particle is ccerned with its quantum nature~see details in Sec. IX!. Thus,our picture bearing the transferon-dissipon duality ofelectron transfer is far beyond the Landau–Zenner pictTherefore, our dualistic theory of the electron transfer difffundamentally from the monistic polaron theory~see, e.g.,Ref. 58! that, like TMT ~see Sec. IV!, is confined in essencto the Landau–Zenner picture.

Closing the discussion on the dynamic organizationelementary electron transfer in condensed matter, one ational comment is necessary. Namely, the considerationthe elementary electron transfer as a dualistic transfedissipon motion goes far beyond the discussed here casresonance (2te)

215t21 and resonance (2te)215t0

21 thatcorrespond to the Heisenberg uncertainty relation in the foof the exact equalityDpDx5\/2. For example, in Ref. 11the transferonic peak~designated formerly asL-peak11! isdemonstrated to survive for the nonresonant casete

215t21

that corresponds to the Heisenberg uncertainty relation inform of inequalityDpDx.\/2 ~namely,DpDx5\!. Closerexamination of this matter will be made elsewhere.

Let us now discuss and conclude on the optical transias such in polymethine dyes andJ-aggregates. This opticatransition is the extended electron-phonon transition althe polymethine chain orJ-aggregate chromophore; or, moprecisely, it is the electron-phonon transition in the extendelectronic potential of the dye chromophore. Evidently tconstruction of a theory of quantum transitions of this sfor any potential surfaces of the general shape is a challeing task. Therefore, it would appear reasonable that onthe first modes of attacking the problem is that of simplifyithe form of the potential surface. As can be readily appreated, one of the most simple potentials having a physmeaning is the potential in the form of the sum of two shorange potential wells that is considered in the paper at hanamely,V1(r )1V2(r2R), whereR[uRu is the distance between the wells~see Sec. V!. The irreversible electronphonon transition in this potential is referred by us toelementary electron transfer in condensed matter. As careadily appreciated, there is a price to be paid for changfrom consideration of the optical transition in the dye chmophore having a real potential surface to consideration


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the elementary electron transfer. This price must be paidloss of the tunnel effects in the final result, i.e., in making tGamow exponential exp(22R/a) equal to unity~or nearly so!and subsequently ascribing the sense of the extent ofp-electron system to the extent of electron transferR ~seeSecs. II, III, and V!. As things now stand, the greater partthe warrant of this procedure is in the good agreement oftheory with the familiar experimental data of Brooket al.2,5 and Herz2,12–14~see Secs. VI and VII!.

The short-range potential wells,V1(r ) and V2(r2R),require further simplification before the desired result for trate constant~here, the absorbency! can be obtained in areasonably simple analytical form. This is accomplishedthe use of the Fermi’s zero-radius potentials.11 Theoretically,as has already been intimated in Ref. 11, our electrtransfer result for the optical absorption line shape@see Sec.V, Eqs. ~6!–~8!# as applied to polymethine dyes anJ-aggregates can be warranted only in the framework omore general theory that involves a far more detailas compared to the Fermi’s zero-radius-potenapproximation,44 description for the extended electron statesay, the Hueckel’s molecular orbital approximation.

Closing this section and the overall paper, I will enlaron the subject of our prediction~see Sec. VIII!. When theresult of our theoretical treatment of the experimental dataBrooker et al.2,5 is compared with that of Herz,2,12–14 it isapparent that, in the event of the transferon motion, disstion energyg will fall with a rise in reorganization energyE.This offers scope for predictions of the very intense narrabsorption lines. To put it differently, our theory leaves roofor vastly more strong effects as compared to theJ-bandeffect. At the moment it is not clear how these strong effemight be arrived at a particular matter. Thus, we restrictdiscussion to general observations. First and foremostwell to bear in mind that these effects may be discoveexperimentally in both natural and manmade substanAny artificial organization of the hard transferon state, or,other words, of the strong effect of spontaneous pumpingthe electron-transfer state by an ordered environmentaltion assumes the prior comprehensive quantum-chemanalysis of the material for purposes to estimate the pareters entering into our dynamic theory.

It may be said that the effect of spontaneous pumpingthe electron-transfer state in hand provides an exampledynamic self-organization of the electron transitions. Ocan concede that self-organizations of this sort are atheart of the nature of biological functions. Therefore, amonatural substances, it seems likely that the biological systmay exhibit this very effect of spontaneous pumping.

1H. Kuhn and C. Kuhn, inJ-Aggregates, edited by T. Kobayashi~WorldScientific, Singapore, 1996!, p. 1.

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approximation: V5p, wherep is the momentum operator of the electrolocalized in the final state.


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1971!.49With no intermediate bridge states.50Characteristic timeste5R(2J1 /m)21/2, t5\/E, and t05\/g, along

with the other characteristic parameters of the system, appear naturathe original theory~Ref. 11! during the progress of the formal deduction othe electron-transfer result for the optical absorption line shape@see Eqs.~6!–~8!#. For the physical interpretations in the present paper, we thusentitled to treat the equations for the characteristic times as thoughfollow from general considerations.

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Nature of the optical transition in polymethine dyes and J-aggregates

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