vibronic coupling in dimer – a convenient approximation revisited

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Chemical Physics 335 (2007) 155–163

Vibronic coupling in dimer – A convenient approximation revisited

Marcin Andrzejak, Piotr Petelenz *

K. Guminski Department of Theoretical Chemistry, Jagiellonian University, Ingardena 3, 30-060 Cracow, Poland

Received 7 February 2007; accepted 12 April 2007Available online 21 April 2007

Abstract

The problem of linear vibronic coupling in model dimer is revisited with the objective of testing a version of the strong vibronic-cou-pling approximation that is recently used in crystal-optics applications. It is shown that in the limit of strong intermolecular resonanceinteraction (intense electronic transitions) this approximation fails in a spectacular way, predicting a set of artefact absorption bands.The consequences of this finding for the interpretation of the absorption spectra of oligothiophene crystals are discussed.� 2007 Elsevier B.V. All rights reserved.

Keywords: Vibronic coupling; Dimer; Strong-coupling approximation

1. Introduction

There is abundant literature on vibronic coupling inmolecular aggregates. Comprehensive studies of this sub-ject started in 1960s. The limiting cases of strong and weakvibronic coupling were tentatively defined [1] and investi-gated for the simplest case of a molecular dimer [2]. Subse-quently, a numerical scheme to solve the dimer vibronicequation was proposed and implemented [3,4]. The prob-lem was later reformulated by Siebrand and collaborators[5–7].

The trimer [8] and tetramer [9] cases were also studiedand the results were used to interpret the absorption spec-tra of some molecular crystals [9,10]. Independently, theo-retical description of vibronic coupling in molecularcrystals was gradually developed [11–14], but in this casegeneral solutions were never obtained because of the com-plexity of the problem.

The recent numerical studies [15,16] offer a new perspec-tive of vibronic coupling. They explicitly deal with molecu-lar aggregates of a size that was intractable in the past;some of the conclusions can be naturally extrapolated tothe limit of infinite crystal. The approach is very realistic

0301-0104/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2007.04.007

* Corresponding author.E-mail address: [email protected] (P. Petelenz).

in reproducing the physics of the system in hand. However,the size of the system, as well as the amount of system-spe-cific detail involved make it difficult to extract the salientgeneral features of the underlying physical mechanisms.

The need of revisiting vibronic coupling problems ismotivated by new experiments that are interpreted in termsof a simplified approach [17–20]. The model applied thereis based on the assumption that the vibrational excitationalways strictly accompanies the electronic excitation (limitof strong vibronic coupling). In effect, the couplingbetween the exciton residing in one unit cell and the intra-molecular vibrations located in another unit cell is dis-carded from the outset, and the Hamiltonian can bereadily diagonalized by Fourier transformation applied tothe composite quasiparticle (neutral polaron) consistingof the Frenkel exciton and the intramolecular phonons(s)of the electronically excited molecule. For the oligothioph-ene crystals containing four molecules in the unit cell thisansatz allows one to reduce the vibronic problem to theinteraction between two optically active Davydov compo-nents, which is formally equivalent to vibronic couplingin a dimer. However, even for a dimer the curtailed basisset of the strong vibronic coupling limit is a serious approx-imation. As the specific version of this approximation(hereafter referred to as modified strong-couplingapproach, MSC) that was recently used [17–20] has never

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156 M. Andrzejak, P. Petelenz / Chemical Physics 335 (2007) 155–163

(to our knowledge) been tested for a dimer, this is theobjective of the present paper. MSC will be shown to haveconsiderable advantages over the standard version of thestrong vibronic-coupling approximation that was com-monly used in the dimer context, but to be inapplicablein the case of oligothiophenes.

2. The dimer model

Consider the classic model [2,3] of a dimer consisting oftwo identical molecules (A and B); their relative positionsand orientation are fixed. In order to facilitate the analysisof the results let us assume that there is a symmetry oper-ation that physically interchanges the dimer moieties. The

H ¼Ee þ

P 2A

2l þlx2

2Q2

A þ LQA þP 2

B

2l þlx2

2Q2

B v

v Ee þP 2

A

2l þlx2

2Q2

A þP 2

B

2l þlx2

2Q2

B þ LQB

24 35; ð4Þ

coupling between the molecules is weak compared to intra-molecular interactions. Each of the two molecules isassumed to have a single non-degenerate, electric-dipole-allowed excited state (in addition to the ground electronicstate). The wave functions for both states are assumed tobe Born–Oppenheimer-separable and the potential energyfor the nuclear motion is harmonic for both the groundand excited state of the molecule. We take into accountone totally-symmetric normal mode per molecule andassume that the only change introduced by the electronicexcitation to the shape of the potential is linear in thenuclear coordinate. The derived formulas may be readilygeneralised for any number of molecular vibrations [21].

According to the standard procedure [2,22], the elec-tronic wave function for the ground state of the dimercan be represented as a product of the monomer electronicground state wave functions:

W0ðq; Q0Þ ¼ wA0 ðqA; QA

0 ÞwB0 ðqB; QB

0 Þ: ð1Þ

The lowest electronic excited state is doubly degenerate,with either of the two molecules excited and the other in the

h ¼C þ 1

2ðp2þ þ q2

þÞ þ bqþ þ 12ðp2� þ q2

�Þ bq�bq� �C þ 1

2ðp2þ þ q2

þÞ þ bqþ þ 12ðp2� þ q2

�Þ

" #; ð6Þ

ground state, which violates the applicability conditions ofthe Born–Oppenheimer approximation. The electronicwave functions for the two excited states of the dimer:

WAe ðq; Q0Þ ¼ wA

e ðqA; QA0 Þw

B0 ðqB; QB

0 Þ;

WBe ðq; Q0Þ ¼ wA

0 ðqA; QA0 Þw

Be ðqB; QB

0 Þ;ð2Þ

form a basis for the two-dimensional space of vibronicstates:

Wðq;QÞ ¼ aðQÞWAe ðq; Q0Þ þ bðQÞWB

e ðq; Q0Þ; ð3Þwhere the coefficients a and b depend on nuclear coordi-nates and are represented as products of vibrational wavefunctions for the totally symmetric normal modes of thetwo monomers.

It should be emphasized for future reference that theabove ansatz for the wavefunction is general in the sensethat it encompasses any distribution of vibrational excita-tion between the two moieties; the phonons are allowed toreside not only on the electronically excited molecule butalso on its neighbour which is in the electronic ground state.

The unknown functions a, b are determined from theeigenvalue problem with the 2 · 2 vibronic Hamiltonian[22]

where Ee is the vertical excitation energy of the moleculeand L is proportional to the shift of the equilibrium posi-tion in the molecular normal mode upon electronic excita-tion. v ¼ hWA

e ðq; Q0ÞjV jWBe ðq; Q0Þi is the excitation-

resonance integral.

2.1. Exact numerical solution (ENS)

Further treatment is facilitated by introducing the sym-metry-adapted electronic basis functions and normalcoordinates

W�e ¼1ffiffiffi2p ðWA

e �WBe Þ;

q� ¼1ffiffiffi2p ðQA � QBÞ:

ð5Þ

Then, according to the standard derivation [22], theHamiltonian matrix can be conveniently written in dimen-sionless units

where all the energies are expressed in terms of the vibra-tional quantum �hx and all the displacements in units of(�h/2lx)1/2, x being the frequency of the molecular vibra-tion. The intramolecular potential energy shift L is re-placed by the dimer parameter b defined as (2�hlx3)�1/2L,and C = v/�hx. From now on the zero of energy is takenat the vertical excitation energy of the monomer.

In the vibronic basis described above the Hamiltonianmatrix elements are given by the following formulas:

M. Andrzejak, P. Petelenz / Chemical Physics 335 (2007) 155–163 157

hWþe vlðqþÞvvðq�ÞjhjWþe vl0 ðqþÞvv0 ðq�Þi

¼ dv;v0

�dl;l0 ðC þ lþ vþ 1Þ

þ bffiffiffi2p ðdl;l0þ1

ffiffiffiffiffiffiffiffiffiffiffiffil0 þ 1

pþ dl;l0�1

ffiffiffiffil0

pÞ�;

hW�e vlðqþÞvvðq�ÞjhjW�e vl0 ðqþÞvv0 ðq�Þi

¼ dv;v0

�dl;l0 ð�C þ lþ vþ 1Þ

þ bffiffiffi2p ðdl;l0þ1

ffiffiffiffiffiffiffiffiffiffiffiffil0 þ 1

pþ dl;l0�1

ffiffiffiffil0

pÞ�;

hWþe vlðqþÞvvðq�ÞjhjW�e vl0 ðqþÞvv0 ðq�Þi

¼ dl;l0bffiffiffi2p ðdv;v0þ1

ffiffiffiffiffiffiffiffiffiffiffiffiv0 þ 1p

þ dv;v0�1

ffiffiffiffiv0pÞ:

ð7Þ

In the limit of infinite vibrational basis numerical diag-onalization of the Hamiltonian matrix yields the exact eige-nenergies and eigenvectors of the vibronic Schrodingerequation. For practical applications the basis set has tobe limited to a certain maximum number of excited vibra-tional quanta. This presents no difficulties as long as onlyone mode per monomer is taken into account: when thebasis set is gradually increased, in most cases the approxi-mate spectra rather quickly converge to the exact solution.

2.2. Approximate solutions

The vibronic problem is determined by the interplaybetween the parameters b, C and the vibrational quantum(=1 in the dimensionless units). In special instances whereone of the parameters is small, perturbational approxima-tions may be used instead of the exact solution, which savesnumerical effort and greatly simplifies interpretation.

2.2.1. Weak vibronic coupling

In the limit C� b2, C� 1 the Born–Oppenheimerapproach is approximately valid for the dimer as a whole.The wavefunction is then factorized into the electronicpart, which is one of the symmetry-adapted componentsdefined by Eq. (5), and the vibrational part. Some correc-tions to this picture are due to the fact that b is not strictlyzero and in consequence the perturbation linear in the q�symmetry-adapted mode [the off-diagonal term in Eq. (6)]couples the two electronic states of the dimer, which resultsin intensity borrowing by one state from the other. Theeffect closely resembles the Herzberg–Teller coupling in amolecule, with the only difference that the coupled statesare dimer electronic states, analogues of the Davydov com-ponents. A few years ago this approach was successfullyapplied to interpret the low-energy part of the absorptionspectrum of the sexithiophene crystal; the details of thetreatment may be found in Ref. [23].

2.2.2. Strong vibronic coupling – conventional approach

(CSC)

The opposite limit is b2� C. In that case the local vib-ronic interaction in a moiety is the dominant term and it is

preferable to use the local basis of Eq. (2) instead of thesymmetry-adapted one. It is also expedient to shift the ori-gin of the vibrational coordinate in the excited moiety tothe minimum of the excited state potential curve, whichis done by introducing eQ ¼ Qþ L=lx2. Then the zeroth-order wavefunctions assume the form

WA�Bl ðqA; qB; eQA;QBÞ¼ WA

e ðqA; eQA0 ÞWB

0 ðqB; QB0 Þvlð eQAÞv0ðQBÞ;

WAB�l0 ðqA; qB;QA; eQBÞ¼ WA

e ðqA; QA0 ÞWB

0 ðqB; eQB0 Þv0ðQAÞvl0 ð eQBÞ:

ð8Þ

These states are coupled by the intermolecular interac-tion C which is now modulated by the vibrational overlapintegrals. It is assumed that the vibrational excitationalways accompanies the electronic excitation; all otherstates are eliminated from the basis set. As in this limit Cis by assumption small, the coupling between the stateswith a different number of phonons is neglected. In effect,the only non-zero elements of the intermolecular interac-tion matrix couple (pair-wise) those of the above basisfunctions where l = l 0, so that the Hamiltonian matrix isa set of 2 · 2 blocks arranged along the diagonal.

Accordingly, the first-order vibronic wavefunctions arereadily obtained by diagonalising these blocks, which yields

W�l ¼1ffiffiffi2p ðWA�B

l �WAB�l Þ: ð9Þ

In this limit, each vibrational state individually splitsinto two Davydov-like components, corresponding to thedifferent irreducible representations of the dimer symmetrygroup. The approximate eigenvalues are

h�l;l ¼ l� C � ðSl0Þ

2; ð10Þ

where as previously, the zero of energy is set at the energyof the molecular electronic excitation, C denotes the inter-molecular resonance integral and Sl

0 ¼ ðv0ðQÞjvlð eQÞÞ is theoverlap integral of the harmonic oscillator functions for theelectronic ground state and excited state normal modes at0th and lth vibrational level, respectively. All the energiesare consistently dimensionless and expressed in units of �hx.Based on the closure property for vibrational wavefunc-tions, it can be easily shown that in this approximationthe resulting splittings of all vibronic states add up to twicethe purely electronic overlap integral C.

2.2.3. Strong vibronic coupling – modified approach (MSC)

The approach [24] that has been recently used in themolecular crystal context [17–20] is a modified version ofthat presented above. To our knowledge, it has never beenapplied for a dimer. The difference with respect to the stan-dard approach consists in the fact that the interactionbetween the states with different number of phonons isnot neglected. The symmetry-adapted states of Eq. (9) thatin CSC were the approximate eigenstates of the system, inMSC are still coupled by the matrix elements

h�l;l0 ¼ �C � Sl0Sl0

0 : ð11Þ


The Hamiltonian matrix splits into two blocks corre-sponding to different irreducible representations of thedimer symmetry group, and the eigenstates have to besought by numerical diagonalization of these blocks.Clearly, the number of interactions thus accounted for isby far larger than in the standard scheme, which seems tosuggest that this approximation is more realistic, althoughthe size and nature of the vibrational basis remainunchanged. However, in some respects the approach isnot entirely consistent: by assuming that the vibronic exci-tation always accompanies the electronic excitation, somevectors belonging to the degenerate vibronic subspacespanned by the states characterized by a given value ofthe vibrational quantum number are arbitrarily discarded.From the vibronic manifold with v vibrational quanta,included are only those vectors where all the quanta arelocated at the electronically excited molecule, while the‘‘two-particle’’ states [16,33,34] where at least one quantumresides at the electronically unexcited moiety do not appearat all. In this way, some of the interactions operative withineach degenerate subspace are neglected, while their inclu-sion should have priority over the couplings that operatebeyond the degenerate subspace according to Eq. (11). Itis our present aim to test to what extent this inconsistencyimpairs the physical validity of the results.

3. Parametrization

The vibronic spectra of the dimer, simulated using theexact numerical solution (ENS), the conventional strong-coupling scheme (CSC) and the modified approach out-lined in the preceding chapter (MSC), are reported below.The results will eventually be referred to the vibronic spec-trum of the a-sexithiophene (T6) crystal, where the simpledimer model provides a reasonable first approximation forstudying the absorption spectra [23]. In that context, themodel parameters will have to be regarded as effective val-ues characterising the crystal rather than an isolated pair ofmolecules (just as it is done in the recent papers on crystal-optics [20]). Accordingly, in the present paper the inputparameters are kept close to the values characterising thesexithiophene crystal; we deviate from this strategy in thecases where a relevant salient feature of the solutions wasdetected in a different parameter range. The treatmentwould be identical for quaterthiophene [20] or other oli-gothiophenes, with almost the same numerical values ofthe input parameters.

In the sexithiophene molecule, the S1 S0 excitationenergy is about 21,500 cm�1. In our figures the zero ofenergy is set at the molecular vertical electronic transition.For the vibrational frequency we assume �x ¼ 1500 cm�1 tomimic the normal mode that dominates the absorptionspectra of oligothiophenes; its dimensionless molecularFranck-Condon parameter l (defined as 21/2b) is about 1.2.

In view of the model nature of this paper, the resonanceintegral C governing inter-moiety excitation-transfer isconsidered a free parameter, and the results will be exam-

ined for its several values. As stated above, the focus ofour treatment will be on the sexithiophene case. The Dav-ydov splitting of the lowest excited molecular singlet in thiscrystal was evaluated by Ewald summation [25,26] in sub-molecule approximation [27,28]. The sum is only condi-tionally convergent and depends on the direction of thek-vector. For the most common experimental geometrywhere the probing light propagates in the direction perpen-dicular to the crystal bc plane, the calculated Davydovsplitting is about 6000–8000 cm�1, which gives C ffi 3000–4000 cm�1 [29].

For clarity, in our figures the total intensity is assumedthe same for both (x and y) polarizations, which corre-sponds to the perpendicular relative orientation of the moi-ety transition moments and disagrees with the actualsituation in the a-sexithiophene crystal, where the intensi-ties of the two Davydov components are drastically differ-ent. Should need arise, the spectra may be scaled asappropriate for any given geometry of the dimer.

In our plots, each individual transition is represented bya Lorenz function with the position and intensity deter-mined by the respective energy and oscillator strengthobtained from the calculations. For all the transitions thewidth (at half maximum) is set at 30 cm�1.

4. The model spectra

Fig. 1 shows the spectra calculated for one vibronicallyactive mode with l = 1.2 and a set of different values of theelectronic excitation-transfer integral. The limit of strongvibronic coupling is represented by C = 0.2; C = 0.5approaches the intermediate-coupling regime, fully devel-oped at C = 1.0, while C = 2.0 corresponds to the limitof weak vibronic (strong intermolecular) coupling. In eachcase the first panel compares the absorption spectraobtained within the standard strong vibronic-couplingapproximation (CSC) with the exact numerical solution(ENS), the second panel presents an analogous comparisonfor the modified strong-coupling approximation. For thesake of clarity the y-polarized transitions (built on thelower-energy electronic state of the dimer) are displayedin the same plot with reversed intensity scale.

As should have been expected, when the electronic cou-pling is weak enough (C = 0.2), both versions of the strongvibronic-coupling approximation plausibly reproduce theexact results. Obviously the splittings of the third line ofthe exact spectrum (x polarization) cannot be achieved ineither of the approximate approaches, because of the lim-ited basis set. Apart from that, the peak positions andintensities for both polarizations do not deviate signifi-cantly from the exact solution. The x-polarized intensitiesare better reproduced by the standard approximation(CSC), whereas in the y polarization the modified treat-ment (MSC) seems to be slightly more successful.

For C = 0.5 the harbingers of intermediate-couplingbehaviour are becoming apparent. In the exact spectrumthe familiar Franck-Condon progression begins to change

Fig. 1. Vibronic spectra of the dimer calculated exactly (solid line) and in the approximate way (broken line) with two versions of the strong-couplingapproximation (MSC and CSC), for a series of increasing values of the resonance integral C. The x-polarized and y-polarized transitions are displayedabove and below the horizontal energy axis, respectively, so that in the latter case the intensity scale is reversed.


into a more complicated pattern. The performance of CSCdrastically deteriorates. The discrepancies in peak positionsare already discernible and the intensity distribution is sub-stantially affected. The modified approximation (MSC) isdoing much better, with the intensities reasonably wellreproduced. While the breakdown of CSC is no surprisesince the criterion for the strong-coupling limit b2

C � 1 isno longer satisfied, it is worth noting that the modifiedapproach (MSC) still provides a reasonable picture of thevibronic spectrum. Of course, the consequences of insuffi-cient basis set (vide supra) are evident in the inherent inabil-ity of both approximate approaches to reproduce the high-

energy multiplets. This flaw, though, as well as the gener-ally larger errors in the positions of high-energy bands,might well escape attention when compared with an exper-imental spectrum, where the resolution is typically poorerin that region.

The full-fledged erratic intensity pattern of the interme-diate-coupling regime develops at C = 1. Both approxi-mate approaches produce considerable errors. Especiallyconspicuous is the intensity of the first band in x polariza-tion, largely overestimated in CSC. The advantage of themodified approach, yielding reasonable intensities of thedominant peaks in both polarizations, is quite remarkable.


Yet, the errors in peak energies are large and there is ageneral tendency to underestimate the spacing betweenthe peaks in the x-polarized spectrum, whereas in they-polarization the spacing is significantly overestimated.In phenomenological fits of experimental spectra these dis-crepancies might pass unnoticed if the resolution is suffi-ciently poor and the multiplets are not resolved.

For C = 2 the regular progressions in the exact spec-trum, built upon the two Davydov-like electronic compo-nents, reconstitute the familiar Born–Oppenheimer resultfor the dimer as a whole. The small peaks at negative ener-gies in x polarization are due to intensity borrowing by the

Fig. 2. Vibronic spectra of the dimer calculated exactly (solid line) and in theapproximation (MSC and CSC), for a series different values of the moleculartransitions are displayed above and below the horizontal energy axis, respecti

lower component from the upper component, induced byone quantum of the out-of-phase mode q�. The exact the-ory predicts no other x-polarized bands below the upperDavydov component at 2500 cm�1 (27,800 cm�1 [30]). Incontrast, both approximate treatments (rooted in the limitof strong vibronic coupling) predict in this region a set ofspurious vibronic bands, while failing to account for thelow-energy vibronically induced features. The modifiedapproach (MSC) does still seem to work somewhat better.At least the artefact peaks in the low-energy region of thex-polarized spectrum are less prominent, so that the globalsplitting between the two polarizations is closer to the exact

approximate way (broken line) with two versions of the strong-couplingFranck-Condon parameter. As in Fig. 1, the x-polarized and y-polarized

vely.


value than that produced by the standard approximation.However, the peak positions and intensities largely deviatefrom the exact values, so that essentially both approxima-tions fail in describing the actual vibronic states. Evidently,the MSC basis, appropriate for strong vibronic coupling, isbound to be ineffective in the opposite limit of weak vib-ronic coupling where the Born–Oppenheimer approxima-tion is valid with only marginal corrections. This casecorresponds to the situation in oligothiophenes.

It should be noted in passing that exact numerical calcu-lations may be alternatively performed in a different(Lang–Firsov [31,32]) basis set, where the vibrational wave-functions on the excited moiety are expressed in terms ofthe coordinate shifted to the excited state potential mini-mum. This feature is analogous to the trick used in Eq.(8); the difference is that the single term of Eq. (8) is supple-mented by a set of basis functions with one, two, three etc.vibrational quanta residing on the other, electronically

unexcited moiety. The MSC wavefunctions represent thusan incomplete form of the Lang–Firsov basis simplifiedby elimination of the vibrational excitations on the othermolecule. It is reassuring to note that when the MSC basisis extended by including the two-particle contributions[16,33,34], the eigenstates very rapidly converge to theexact result upon increase of the total number of quantaon both molecules. In other words, the Lang–Firsov basisis very effective in approximating the actual eigenstates ofthe dimer even in the case of strong intermolecular interac-tion, in contrast to the MSC basis, which in these circum-stances is incapable of achieving the correct result at all.This shows explicitly that it is indeed the neglect of vibra-tional excitations on the electronically unexcited moleculethat is responsible for the failure of the MSC approxima-tion for large resonance interaction.

Based on a more detailed scan of the parameter space,for l = 1.2 the value of C ffi 0.4 turns out to be the limitof practical applicability for the conventional strong-cou-pling approach (CSC); the modified approach (MSC) isfaring reasonably well for C ffi 0.5, and even for C ffi 1.0some of its qualitative conclusions remain valid.

Fig. 2 shows a test (for the critical value of C ffi 0.5) ofthe sensitivity of these tendencies to a minor change ofthe displacement parameter l, which is slightly reduced orincreased with respect to the base value (l = 0.9, 1.2 and1.5). As expected, the exact results indicate that the largerthe shift of the normal mode equilibrium position l, themore developed the vibrational structure in the spectra ofboth polarizations. Not only do the progressions becomelonger, but also the satellite bands grow to become compa-rable with the dominating peaks; the overall patterns aregetting increasingly irregular. Owing to the limitationsimposed on the basis set, the approximate schemes cannotreproduce the multiplet structure. From this point of view,the larger the displacement parameter l, the worse is thematch between the approximate and exact spectra. Theresult is seemingly paradoxical: when l is large, the criterionfor strong vibronic coupling (b2

C � 1) is strictly satisfied and

the corresponding limiting approximation should workbetter. The apparent inconsistency results from the factthat the vibrational quantum (=1 in dimensionless units)sets an independent scale for parameter values. It is a tacitpremise of the strong vibronic coupling approach that b2

must not be too large with respect to unity. The directcause of the problem is readily explained by inspectingthe matrix elements of the Hamiltonian: when l is large,the basic assumption that the vibrational excitation accom-panies the electronic excitation is not valid, since the cou-pling between the basis functions that are taken intoaccount and those that are discarded is no longer negligi-ble. In effect, (especially for the 0–0 line) the best energiesobtainable with the ansatz of Eqs. (10), (11) are indeed clo-ser to the exact ones, but for the vibrational replicas thisvery ansatz becomes insufficient, as the vibrational excita-tion may spread onto the electronically unexcited moiety.If the bands are so broad that the multiplets are notresolved, the discrepancies are likely to be ironed out andthe vibrational structure predicted by the approximatetreatments may turn out to be in misleading agreementwith experiment.

5. Discussion

Strictly speaking, there is presently no straightforwardway to test the quality of the MSC approximation for acrystal, since complete solution of the vibronic problemfor a three-dimensional lattice with a few molecules inthe unit cell is a formidable task and has not yet beenachieved. However, the representation used in the recentpractical implementation [20] formally reduces the crystal(after the Fourier transformation) to an effective dimer,and for this reason the results obtained in the present paperare expected to be relevant in that case. Also, one tends toexpect on intuitive grounds that if the approximation givesrise to artefacts for a mere dimer, its shortcomings should a

fortiori show up for a crystal.Altogether, the modified approach (MSC) applied for a

dimer is evidently numerically superior to the standardstrong vibronic coupling approach (CSC); it consistentlyyields a significantly better match with the exact resultand its range of validity is wider. However, its constrainedbasis set is not flexible enough to handle the limit of weakvibronic coupling where the Born–Oppenheimer approxi-mation is valid with only marginal corrections. This wasto be expected.

On the other hand, in spite of its being manifestly appli-cable for very intense electronic transitions (where C islarge), the Born–Oppenheimer approximation is reluc-tantly used for molecular crystals whenever the polaritoneffects are to be included, because the adiabatic picture isincompatible with the commonly used formalism wherethe excitons are represented as a set of classical oscillatingdipoles which interact with each other. Instead, MSC ispreferred, despite its questionable validity. This is the casefor oligothiophenes [20].


In the simulated sexithiophene spectrum the breakdownof the MSC approximation is manifested by a series ofartefact absorption bands that provide a low-energy onsetof the upper Davydov component. They have the polariza-tion of this Davydov component, and have no counterpartsin the exact solution. Based on the observed Davydov split-ting, the lowest exciton transition of the T6 crystal seems aprime example where the adiabatic description should cer-tainly be valid [35]. According to our exact results, the ac-polarized spectrum should then begin with a weak systemof bands built upon the 0–1 line in the q� mode of severalprogression-forming intramolecular modes (of which onewas explicitly considered in our present calculations), fol-lowed by an essentially blank interval. Then, at a muchhigher energy, the upper Davydov component shouldappear.

Yet, the experimental spectrum is drastically different.While the low-energy false origins appear as expected(and their detailed interpretation was already given[23,16]),instead of the blank interval (possibly with some back-ground absorption) they are followed by two evident peaksat about 21,000 cm�1 and 22,250 cm�1, well below theupper Davydov component at 29,000 cm�1[35]. Seemingly,these peaks correspond to the bands predicted by the MSCapproach. However, as demonstrated above, these verybands have emerged in the simulated spectrum only as anartefact: for a dimer the comparison with exact solutionsreveals that this prediction is blatantly incorrect. There isno possibility of the approximate treatment being ‘‘morevalid’’ than the exact result; hence, the agreement of theMSC result with this particular feature of the experimentalspectrum of the T6 crystal has to be viewed as purely acci-dental. The same would apply to other oligothiophenes.

This opens the question regarding the true origin of thesexithiophene ac-polarized bands, observed experimentallyat 21,000 cm�1 and 22,250 cm�1. Some time ago [36] theywere tentatively attributed to charge-transfer (CT) states;this interpretation was recently upheld for quaterthiophene[37]. However, detailed calculations on CT states [38] sug-gest the ac-polarized absorption intensity of CT prove-nance to be much smaller in the relevant energy range.At the same time, advanced calculations of vibronic cou-pling in finite aggregates [15,34] predict substantial Frenkelstate contribution due to vibronic intensity borrowing bythe replicas of the lower Davydov component, althoughit seems to be located at somewhat lower energy. Alto-gether, the available evidence is suggestive rather than con-clusive, since the ultimate interpretation will have to takeinto account the energy contributions due to macroscopicpolarization, non-analytic at the centre of the Brillouinzone, and for this reason would have to be based on amodel valid for a single crystal rather than for a finiteaggregate. Moreover, the model would also have to consis-tently include the interaction between the Frenkel and CTexcitons. Nevertheless, in view of the results available todate it seems likely that the bands under consideration con-tain some Frenkel contribution, and (as follows from our

present calculations) this contribution is not interpretablein terms of the dimer model. Accordingly, the peaks mustbe due to some physical mechanism which is not operativein a dimer.

As long as the vibrational excitation is assumed tostrictly accompany the electronic excitation, the vibronicproblem for the crystal under consideration is reduced tothe dimer case where the two electronic states involvedare those located at the different sites of the unit cell, or(in a different representation) those that correspond to thedifferent Davydov components. Evidently, the effect whichis not included in the dimer model is the vibronic couplingengaging the molecules outside the unit cell where the elec-tronic excitation is located, e.g. in the neighbouring unitcells. The pertinent states are not included in the dimermodel, because the dimer represents only one unit cell. Evenin the approaches where the whole crystal is explicitly takeninto account, these states are not included in the basis setcorresponding to the limit of strong vibronic couplingbecause of the constraint that the electronic and vibrationalexcitation reside at the same molecule. It is our conjecturethat the peaks observed in the sexithiophene spectrum at21,000 cm�1 and 22,250 cm�1 should be attributed to thevibronic states where the electronic and the vibrational exci-tation are located in different (probably adjacent) unit cells.In the near future, this conjecture will be tested on a largeraggregate model (M. Andrzejak, in preparation).

Referring to the MSC approach, our present resultshighlight the risk of indiscriminate use of ad hoc

approaches where a successful phenomenological fit ofthe experimental spectrum is viewed as the only and ulti-mate test of the physical validity of a model which is notsafely based on a systematic set of approximations.Approximations are inherent to any interpretation, butmay produce artefacts (such as those shown in this paper)when used beyond their range of validity. Considerable dis-crepancies between the simulated and experimental spec-trum may be masked by low resolution or by theinherent spectral width of the observed transitions, leadingto deceptive impression of nearly perfect agreement. It issometimes worthwhile to stretch a model beyond its strictrange of applicability, but in that case it is necessary toexercise utmost caution in interpreting the results thatmay be misleading.

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vibronic coupling in dimer – a convenient approximation revisited

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