Journal of Molecular Structure 993 (2011) 26–37

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Journal of Molecular Structure

journal homepage: www.elsevier .com/locate /molstruc

Modulation of the electronic structure of polyconjugated organic moleculesby geometry relaxation: A discussion based on local Raman parameters

Chiara Castiglioni a,b,⇑, Alberto Milani a, Daniele Fazzi b, Fabrizia Negri c

a Politecnico di Milano, Dipartimento di Chimica, Materiali e Ing. Chimica ‘‘G. Natta’’, p.zza Leonardo da Vinci 32, I-20133 Milano, Italyb Center for Nano Science and Technology CNST, IIT@Polimi, via Pascoli 70/3, I-20133 Milano, Italyc Università degli Studi di Bologna, Dipartimento di Chimica ‘‘G. Ciamician’’, Via F. Selmi, 2 and INSTM, UdR Bologna, I, 40126 Bologna, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Available online 17 December 2010

Keywords:Raman activityElectron–phonon couplingDFT calculationsVibrational dynamics

0022-2860/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.molstruc.2010.12.021

⇑ Corresponding author at: Politecnico di MilanoMateriali e Ing. Chimica ‘‘G. Natta’’, p.zza LeonardoItaly.

E-mail address: chiara.castiglioni@polimi.it (C. Cas

The Raman response of some polyconjugated materials is analyzed on the basis of local Raman parame-ters h@a/@RCCi determined by Density Functional Theory calculations and discussed in the frame of theEffective Conjugation Coordinate Theory. This approach allows to explain Raman spectra even when adeviation from the common behaviour is observed, thus providing a key for the interpretation of the mainspectroscopic features in terms of structural parameters. The examples reported demonstrate that anychange of the hBLAi parameter, describing the average degree of CC bond alternation along a conjugatedsequence, is reflected by a remarkable modulation of the Raman spectrum, which indeed shows the sig-nature of the molecular and electronic structure resulting from different chemical substitutions, confor-mation changes, chain length and environment effects.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

A wide variety of polyconjugated organic materials and amongthem the so called conducting polymers and oligomers, share pe-culiar features in the Raman spectrum, where few strong linesare observed also in those cases where the vibrational degrees offreedom are several and symmetry selection rules are notrestrictive.

This behaviour cannot be simply justified as a direct conse-quence of resonance or pre-resonance effects (often invoked to ex-plain the occurrence of a strong, very selective Raman spectrum)since it lasts in FT-Raman spectra recorded at relatively low excita-tion energy (1064 nm). Moreover, the main features are correctlymodelled by ab initio or Density Functional Theory calculations ofthe Raman activities in the limit of a static exciting field (i.e. com-pletely out of resonance).

The origin of the strong Raman activity of polyconjugated mate-rials has been ascribed since long time to the remarkable electron–phonon coupling [1–3]. This phenomenon has been studied as afunction of chain length [2–14], and its modulation has been corre-lated to chemical substitutions [14] and to environmental effects,such as for instance changes of the solvent polarity [15], inter-molecular interaction in crystalline phases [16,17] and even chem-

ll rights reserved.

, Dipartimento di Chimica,da Vinci 32, I-20133 Milano,

tiglioni).

ical doping or charge injection [3,18]. A relevant fact from the pointof view of the structural characterization of conjugated moleculesis the noticeable sensitivity of their spectroscopic response (and inparticular of the Raman spectrum) to specific structural changes,especially those involving changes in the degree of CC bond lengthalternation (BLA) along a well defined conjugation path[1,3,10,12,15,19–21].

Moreover, the vibrational normal modes showing relevant Ra-man activities always have a remarkable projection along a pecu-liar collective vibrational coordinate, describing the oscillation ofthe BLA parameter. Once a sequence of conjugated CC bonds isidentified in the molecule, this vibrational coordinate is describedas simultaneous stretching and shrinking of adjacent CC bonds Ri,according to the definition of a totally symmetric Z coordinate[3,22]:

z ¼ ð1=N1=2ÞX

i¼2j�1

Ri �Xi¼2k

Ri

" #ð1Þ

where j (k) = 1, 2, . . . , N/2 in the case of a sequence of an even num-ber N of CC bonds; j = 1, 2, . . . , (N + 1)/2 and k = 1, 2, . . . , (N � 1)/2for chains containing an odd number N of CC bonds.

Such a definition of the vibrational coordinate Z is at the basis ofthe Effective Conjugation Coordinate Theory (ECCT), developedabout 20 years ago and firstly exploited for the interpretation ofthe Raman spectrum of polyacetylene in its pristine and doped(or photoexcited) states [3,22].

C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37 27

In the early applications of ECCT, the focus was mostly on thepeculiar frequency dispersion shown by the strongest Ramanmodes (Z modes), which in some cases cover a frequency rangeof about 200 cm�1 while varying the number of CC bonds in theconjugated backbone. This is observed for instance in the case ofall trans-polyenes and polyacetylene [3,4] and the case of linearcarbon chains (see Fig. 1) [11,12].

It was shown [3,22] that a strong modulation of the generalizeddiagonal force constant (FZ) associated to the Z coordinate, due tothe increasing (negative) contribution of long range interactionsbetween CC bonds along the chain, is the main responsible of sucha marked frequency shifts.

While the first works dealing with ECCT were based on empir-ical force fields, the results were later supported by several differ-ent theoretical approaches such as: (i) semi-empirical calculationsbased on the Hückel theory followed by Kakitani [23] and appliedby Piseri et al. [2] to polyacetylene; (ii) the QCFFPI approach, firstlyproposed and exploited for the determination of force fields ofshort polyenes [24] and then extended to polyheteroaromatic sys-tems as oligothiophenes [25]; and (iii) ab initio and DFT calcula-tions applied to a large number of polyconjugated systems[13,26–28].

The molecular approach, based on force constants calculations,provides a simple description of the frequency dispersion of the Zmodes in terms of stretching/stretching interactions; as an alterna-tive, the noticeable frequency modulation of the strongly Ramanactive Z modes can be justified based on a solid state physics ap-proach. On this ground, the frequency softening of Z modes ob-served for chains of increasing length is ascribed to areminiscence of the Kohn anomaly which is expected to occur inthe infinite limit of a metallic chain, with perfectly equalized CCbonds. This observation is particularly important since on its basisthe presence of a strong electron–phonon interaction along the Zmode can be stated [12,29–31].

On the other hand, a direct relationship between solid state andmolecular approaches can be immediately found considering that,in the framework of the Kakitani valence force field [23], it is pos-sible to explicitly introduce an electron–phonon interactionparameters (@b/@R) which indeed describes the modulation ofthe electronic hopping integral (b) between adjacent carbon sites,with the CC stretching coordinate.

Fig. 1. Left panel: Experimental Raman frequency of the strongest Raman line (Z mode(red, full circles) [5,16], oligofuranes (blue, stars) [8], oligopyrroles (cyano, open squares) [oligo-phenylene-vinylenes (black, triangles) [9], with increasing number of chemical unRaman frequency of the strongest Raman line (Z mode) for polyynes: adamantyl-cappeinterpretation of the references to colour in this figure legend, the reader is referred to

Because of the difficulties related to the modelling and to obtainreliable predictions of the Raman activity, a discussion of the originof the strong Raman intensity of Z modes is less frequently re-ported in the literature.

In the frame of the ECC theory this issue has been addressedaccording to a perturbative approach [3,32], following theAlbrecht’s theory [33] and models derived from it [34,35]. Fromthe analysis of the expression of the Raman polarizability tensorit results that the leading term involves the first strongly dipoleallowed excited state (with large character of HOMO–LUMO one-electron transition), showing an equilibrium molecular geometrydisplaced along the Z coordinate. This displacement correspondsto a change from the dimerized structure of the electronic groundstate to a more equalized one in the excited state. This observationallowed to nicely explain the origin of the large Raman activity ofthose normal modes showing large Z content [3,10], but also pro-vided an alternative definition of the Z coordinate (the effectiveconjugation coordinate) as the vibrational coordinate describingthe trajectory along which nuclei must relax when the moleculeis kept in this relevant excited state [3,10,13,32].

There is a second way to analyze the intensity of the stronglyRaman active Z modes, namely considering Raman parameters ob-tained by DFT Raman activities calculated in the limit of a staticexciting field, which are indeed based on the description of theground state wave-function.

Following this approach, we present in this paper a systematicstudy of Raman intensities of the Z modes for several representa-tive conjugated species, sketched in Fig. 2.

Most of the molecules considered follow the behaviour de-scribed above, showing strong Raman active Z modes; moreover,few cases which deviate from these general finding will be pre-sented and rationalized under a unified perspective.

Raman intensities of trans and cis polyenes (hereafter referredas P16T and P16C respectively), linear carbon chains showing bothpolyynic (Py16) and cumulenic (Cu16) structure, aromatic andquinoid oligothiophenes [T4(ar) and T4(qu)] and a cyanine willbe discussed through the analysis of local Raman parameters,namely on the derivatives of the molecular polarizability with re-spect to CC stretching coordinates. It will be shown that theseparameters are strongly correlated to the molecular structurethrough BLA. Moreover, general trends will be found and the

) for: tertbutyl-end-capped polyenes [3,4,13] (green, open circles), oligothiophenes7], protected (tertbutoxycarbonyl capped) oligopyrroles (light blue, full squares) [7],its (N). See also reference [6] and papers cited therein. Right panel: Experimentald polyynes (red circles) and hydrogen-capped polyynes (black squares) [11]. (For

the web version of this article.)

T4 (ar) C16S4H10

T4 (qu) C16S4O2H8

P16T C16H18

P16C C16H18

Py16 C16H2

Cu16 C16

Cyanine C15N2H19

Fig. 2. Sketches of the molecular structures of the eight model systems consideredin this work.

Fig. 4. Bond lengths differences between aromatic T4 and its quinoid counterpart,according to optimized DFT geometries of T4(ar) and T4(qu) models.

28 C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37

existence of a strong electron–phonon interaction driven by BLAmodulation will be demonstrated.

2. Methods

Optimized structures and vibrational spectra of the moleculesdescribed in Fig. 2, and for the two different diphenyldithiophenesdiscussed in Section 3.3 have been obtained based on DensityFunctional Theory (DFT) calculations. The hybrid Becke’s three-parameters exchange correlation functional B3LYP with a split-va-lence Pople basis set (6-31G��) was used [36]. All the calculationswere carried out by means of the Gaussian 03 code [37].

3. Results and discussion

3.1. Molecular structures

The molecules reported in Fig. 2 have been selected according tothe following criteria:

Fig. 3. Optimized CC bond lengths of the model molecules illustrated in Fig. 2.

1. The length of the sequence of CC conjugated bonds is the same:it corresponds to 16 carbon atoms, i.e. to 15 conjugated CCbonds, except for the case of the (positively charged) cyanine,where the number of CC bonds is necessarily even (N = 14).

2. Effects induced by the CC chain configuration (compare P16T toP16C and T4) are modelled.

3. Modulation of the BLA pattern is induced by suitable choice ofthe end groups. For instance, thanks to the presence of a car-bonyl unit directly linked to the peripheral thiophene rings,we expect to found a sign inversion in the BLA parameter fromT4(ar) to T4(qu); moreover we expect a marked decrease of BLAwhile passing from the hydrogen capped polyyne Py16 to thecorresponding Cn chain (Cu16), which is indeed characterizedby an equalized cumulenic structure [38–40].

Our study aims at giving an interpretation to the changes of theRaman response and in particular to correlate trends shown by Ra-man intensities to well defined changes of the molecular structure,determined by equilibrium geometry optimization. Equilibrium CCbond distances for the molecules reported in Fig. 2 are shown inFig. 3.

Looking to Fig. 3 (see also the numerical data reported in theSupplementary data, Table SD1) bond length values can be

CC bonds are numbered along the molecular backbone, from left to right.

Table 1Average bond length alternation hBLAi parameter, Z mode frequency and Raman activity (IZ), derivative of the molecular polarizability tensors with respect to the Z vibrationalcoordinate (average polarizability derivative hoa/oZi and oaxx/oZ component, see text), obtained from DFT calculations on the model molecules sketched in Fig. 2. In the lastcolumn: Raman activities (Icalc

z ) of Z modes as estimated on the basis of oaxx/oZ parameter according to the procedure illustrated in Section 3.2.

hBLAi (Å) tZ (cm�1) IDFTz (Å4/amu) hoa/oZi (bohr2) oaxx/oZ (bohr2) Icalc

z (from oaxx/oZ) (Å4/amu)

T4 (ar) 0.0539 1506 52559 157.36 466.62 33427T4 (qu) �0.0743 1491 109088 �271.17 �801.02 79586P16T 0.0793 1608 294820 443.96 1330.48 254247P16C 0.0832 1622 130270 330.28 986.29 123003Py16 0.1135 2087 447904 403.73 1210.86 381610Cu16 0.0063 1846 3184 �5.04 �4.34 52Cyanine 0.0182 1642 17698 114.43 334.52 15916

Fig. 5. Experimental Raman spectrum of T4 (solid state) and its theoreticalsimulation (B3LYP/6-31G�� calculation, scale factor 0.9614). In the inset: sketch ofatoms displacements as described by the vibrational eigenvector of the strongestRaman mode (Z mode).

1 For the case of cyanine, the correct definition of average BLA parameter is given inSection 3.2.

C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37 29

grouped into two families, characterized by the different hybrid-ization state of carbon atoms, with the shorter bond distancesbelonging to molecules with sp carbon atoms (Py16 and Cu16).Py16 shows a clear bond length alternation (BLA � 0.1 Å), follow-ing the conventional structure formula which describes the molec-ular structure as an alternated sequence of quasi triple and singleCC bonds. Notice however that the bond length of these quasi sin-gle bonds is RCC � 1.35 A, i.e. their length are very close to that ofthe standard CC double bond. As expected [38,39], the uncappedCn chain (Cu 16) shows a very equalized sequence of CC bonds,where the largest difference of bond length (of only 0.02 Å) in-volves bonds 2 and 3 (and bonds 13 and 14 for symmetry reasons).

sp2 carbons chains are grouped in the upper part of the plot ofFig. 3. Three characteristic trends can be found:

– T4(ar), P16T and P16C show a similar BLA pattern which can bedescribed as an alternated sequence of quasi double and quasisingle CC bonds, starting from a quasi double bond in positioni = 1.

– The BLA pattern is completely inverted for the case of T4(qu),showing a quasi single CC bond at position i = 1, i.e. the positionimmediately adjacent to the carbonyl (double) bond.

– Cyanine presents a remarkable bond length equalization espe-cially in the inner region.

Bond length differences between the aromatic T4(ar) and itsquinoid counterpart T4(qu) are reported in Fig. 4, showing thatthe C@O substitution induces a quinoidal structure on the twoperipheral thiophene rings which extends also to the two centralrings. This behaviour, however, cannot be extrapolated to longerthiophene chains, where a competition between a quinoidal struc-ture induced by end groups and an aromatic structure occurs,affecting the inner units.

This effect has been demonstrated by Ponce Ortiz et al. [41],considering a series of oligothiophenes with dicyanomethyleneend groups and it has been confirmed more recently also in thecase of thiophene based hetero-phenoquinones [42]. Thanks to thiseffect, ground state structures show some biradicaloid character,which becomes more and more relevant as far as the number ofrings in the chain increases [41,42]. As a consequence the equilib-rium geometry is affected, resulting in a more equalized sequenceof CC bonds; in the case of relatively long thiophene chains (e.g. sixthiophene rings), also a sign inversion of the average BLA parame-ter is found for the inner thiophene units thus indicating a transi-tion toward an aromatic structure [42].

Another interesting feature is found in the case of T4, where thearomatic character of the thiophene moiety determines a sequenceof three bonds (those belonging to a given ring, e.g. 5-6-7), showinga smaller degree of bond alternation with respect to the subse-quent triplet of bonds involving the inter-ring CC bond (e.g. 7-8-9). In other words, the quasi single bond of the ring (e.g. bond 6)is always shorter than the quasi single inter-ring bond (bond 8).

This effect can be ascribed to a stabilizing effect of resonance en-ergy in the five members ring.

A useful parameter which summarizes the relevant behaviourscommented above is the average BLA (hBLAi), reported in Table 1and obtained as the difference of the average bond length of CCbonds in even position along the chain (R2, R4, . . . , R14) and theaverage bond length of CC bond in odd position (R1, R3, . . . , R15).1

3.2. Raman intensities of Z modes

Our computational results can be safely used for the discussionof the correlation between different molecular equilibrium struc-tures and Raman spectra, provided that sufficiently accurate Ra-man activities are predicted.

As an example, in Fig. 5 we compare the experimental FT-Ra-man spectrum (kexc = 1064 nm) of T4 with its prediction by DFTcalculation (B3LYP/6-31G⁄⁄): the main spectral features are nicelyreproduced both in terms of vibrational frequencies and intensityratios. Eigenvectors analysis (see the sketch in the inset of Fig. 5)allows to assign the strongest Raman transition to a normal modeshowing a relevant contribution from the Z vibrational coordinate.

In order to make the following analysis and comparisons more

30 C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37

effective, we have selected model molecules (Fig. 2) showing thesame number of conjugated CC bonds and characterized by simpleand small end groups. For this reason, it is impossible to providedirect comparison between calculations and experimental datafor most cases, since experimental Raman spectra are not available.On the other hand, for several chemical species closely related tothe molecules here considered, Raman experiments and theoreticalpredictions have been carefully analyzed and compared, showingthat DFT modelling allows to capture the essential features of theexperimental Raman spectra (see for instance Refs. [11,13,27] forthe cases of polyenes and polyynes).

While a comprehensive description of the predicted vibrationalspectra of the molecules sketched in Fig. 2 is given in the Supple-mentary data, Tables SD8-SD14, in Table 1 we report the calculatedRaman activity of the Z mode, which in all cases (except for cumu-lene (Cu) [38]), corresponds to the strongest Raman line. The vibra-tional assignment can be obtained looking to the CC stretchingcontributions in the vibrational eigenvectors (Fig. 6). It should benoticed that, with the exception of the trivial cases of polyynesand cumulenes, these normal modes show a remarkable couplingwith other vibrational displacements such as CH waggingoscillations.

To better emphasize the key role of the Z coordinate in deter-mining the (often huge) Raman intensity of the Z modes, we havecalculated the Raman activity (see Table 1) as due to the only con-tribution of Z oscillation to the polarizability change, according tothe following equations:

@axx=@Q i ¼ @axx=@zLzi ð2Þ

IRamani ¼ 45�aþ 7c2 ¼ 12ð@axx=@Q iÞ2 ð3Þ

Fig. 6. Local Raman parameters and Z mode description as obtained from DFT calculatiooRCCi = 1/3 Trace (oa/oRCC); individual CC stretching content of the Z mode eigenvector;bonds are numbered following the conjugation path, from left to right.

The parameters needed in Eqs. (2) and (3) have been obtainedby suitable transformation of the Raman tensors and vibrationaleigenvectors referred to the cartesian displacement coordinates(which are indeed provided as standard output of DFT calcula-tions). The new set of parameters, referred to internal coordinatesdisplacements, are indeed derived according to the followingrelationships:

@a@R¼ @a@x� A ð4Þ

R ¼ B � Lx ð5Þ

where the B matrix describes the relationship between cartesianand internal coordinates chosen according to Wilson’s definitions[43], namely

R ¼ B � X ð6Þ

and the A matrix is the generalized inverse of B [44].In turns, the parameters @axx/@Z and LZi are immediately ob-

tained as scalar product of the vector @axx/@R (and LRi) with a U vec-tor which provides the definition of the Z coordinate on the basis ofthe valence (internal) ones, namely Z = U0 R. U is obtained accord-ing to Eq. (1).

Notice that Eq. (3) makes use of the xx component of the @a/@Ztensor only: this choice is related to the peculiar characteristic ofthis tensor, showing (for all the molecules considered in this work)negligible values of the components different from the diagonalone referred to the long molecular axis (x). This observation isbased on the data reported in Tables SD2–SD7 of the Supplemen-tary data, showing that CC stretching Raman parameters, namely

ns: for each molecule the three panel represent (from the top to the bottom): hoa/contribution to hoa/oQi (Q = Z mode) by each CC stretching coordinate (see text). CC

Fig. 6 (continued)

C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37 31

the tensors @a/@RCC, are always characterized by a marked anisot-ropy, with a predominant element @axx/@RCC (Tables SD2–SD7). On

the other hand, looking to the @a/@Qi tensors directly taken fromthe output of DFT calculations, it is possible to verify that the xx

Fig. 6 (continued)

3 Indeed, in order to obtain a totally symmetric coordinate, the two adjacent central

32 C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37

component largely exceeds the other ones when Qi corresponds tothe Z mode.

The fact that the @a/@Z tensor can be nicely approximated by its@axx/@Z component (i.e. putting to 0 all the other terms) also ex-plains why the value of its invariant h@a/@Zi = 1/3 Trace (@a/@Z)(see Table 1) is practically one third of the (@axx/@Z) value. The pe-culiar behaviour discussed above allows to analyze local CCstretching polarizability tensors (Section 3.3) simply consideringtheir scalar invariants h@a/@RCCi.

Comparing Raman intensities obtained according to Eq. (3) tothe DFT computed intensities of the Z modes (Table 1), we can ver-ify that in most cases the polarizability change associated to the Zdisplacement contributes to more than 80% of the total Ramanintensity.

It is worth noticing that the use of Eq. (2) does not provide aprediction of the whole spectrum, giving rise to large errors inthe intensity estimate of the minor transitions, that indeed cannotbe described as Z modes. On the other hand, the fact that theseweak bands show negligible Z content is confirmed by the findingthat their Raman activity, when predicted according to Eq. (2), re-mains still negligible in comparison to the (very strong) Z modes.

Among the model molecules here considered, two cases deservea deeper discussion: namely Cu16 and cyanine.

As already discussed in a previous paper [38], cumulenes showa Raman response which strongly deviates from what is observedfor polyynes. Based on theoretical predictions,2 we can see thatthe Raman spectra of cumulenes are not selective, but they are char-acterized by several Raman bands of comparable intensity (see

2 Unfortunately no experimental Raman data are available for cumulenes, whichare hard to measure due to their very high reactivity; moreover long cumulenicchains are shown to be unstable versus cyclization [40].

Table SD13). Indeed, the Z mode of Cu16 (see Table 1) does not cor-responds in this case to the strongest Raman line (found at1967 cm�1 with a Raman intensity of 3309 Å4/amu); in additionother four Raman lines of similar intensity are predicted. On theother hand, the intensity of the Z mode of cumulene is very weak,especially if compared to the intensity data of the other moleculeslisted in Table 1 (for instance the intensity of the Z mode of Py16is two order of magnitude larger). Moreover, in the case of Cu16,the Raman intensity of the Z mode is dramatically under-estimatedif calculated according to Eq. (3), still suggesting that we are in pres-ence of some peculiar effect, related to the high degree of bondequalization in the molecule.

In spite of its equalized structure, the case of cyanine is quitedifferent from cumulene.

Its Z mode intensity is low (compare with P16T showing anintensity 10 times larger), but in this case we obtain a very goodintensity prediction according to Eq. (3).

On the other hand, it should be stressed that cyanine differsfrom the other molecules here considered under several perspec-tives: the CC sequence in cyanine is indeed formed by an odd num-ber of carbon atoms (15 atoms) reflecting a C2v point symmety. Inthis case a relatively strong Raman active Z0 coordinate can be de-fined (belonging to the A1 symmetry species), which however doesnot obeys to the general form of Eq. (1)3: this Z0 mode can be con-sidered as a true Z mode with a node on the central C atom of themolecule. Notice that, for the same symmetry reasons which deter-mines the definition of Z0, also the hBLAi parameter reported in Ta-ble 1 has been calculated considering only one half of themolecule (otherwise it would be equal to zero, by symmetry).

The data reported in Table 1 deserve another important obser-vation. Normal modes listed as Z modes have been selectedaccording to their content of Z coordinate as shown by eigenvec-tors and on the basis of their Raman activity. As already mentionedin the introduction, sometimes more than one normal mode fulfilsthe definition of Z mode. For instance this occurs in the case ofpolyenes, where two strong Raman features are found, both relatedto the Z oscillation coupled to CH wagging vibrations in a differentphase relation [3,10,13]. Moreover, while in short and mediumlength polyenes the higher frequency feature shows the strongestRaman activity (and the largest Z content), the lower frequencycomponent become more and more relevant and acquires largerand larger Z content as far as the chain length increases. Indeed,in the case of polyacetylene the intensity ratio between thesetwo strong Raman lines is reversed. For the sake of simplicity, inthis work we restrict ourselves to the discussion of the higher fre-quency Z mode of P16T (and P16C): it is however important tostress that similar observations and a parallel discussion apply alsoto the lower frequency Z mode.

3.3. Local Raman parameters: comparison and discussion

In this Section we would like to discuss in detail the origin ofthe Raman intensity of Z modes considering the individual internalparameters determining its value.

We have already stated that, with the only exception of Cu16,the Raman activity of the Z modes is determined by polarizationfluctuation associated to the Z coordinate, namely @axx/@Z. In turns@axx/@Z is obtained (Eqs. (2) and (4)) by the contributions of the CCstretching parameters @axx/@RCC, which are responsible of its

onds of the chain must stretch in phase, thus reversing the phase of the alternatedC stretching/CC shrinking in the two molecular moieties; in other words Z0mode ofe cyanine can be seen as the out of phase combination of two ‘‘true’’ Z coordinateshrinking and stretching of adjacent CC bonds) defined on the two halves of theolecule.

bCth(sm

C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37 33

(often) huge value. The trends illustrated in Fig. 6 explain why thelocal CC stretching parameters are so effective in determining theRaman activity of the Z modes. For each molecule three panelsare reported where the values of the individual parameters (rela-tive to each CC stretching coordinate along the molecular chain)are compared. For each CC bond the scalar parameter h@a/@RCCi =1/3 Trace (@a/@RCC), the individual CC stretching content in the Zmode eigenvector and the contribution to h@a/@Qi (Q = Z mode)by each CC stretching coordinate are reported.

First of all, looking to eigenvector components, we can see thatall the selected Z modes show the alternate stretching/shrinkingcharacter of adjacent CC bonds: the patterns describing the contentof each CC stretching coordinate are quite similar for carbon chains(P16T, P16C, Py16, Cu16) while non-negligible difference is shownby the two T4 models, especially if one considers the weight of CCstretching belonging to the peripheral tiophene units. The case ofcyanine is markedly different, and reflects the definition of Z0 coor-dinate as illustrated in detail in Section 3.2.

Let us now consider the intensity parameters h@a/@RCCi: T4(ar),P16T, P16C and Py16 show a very similar trend characterized bypositive h@a/@RCCi values at the odd sites (corresponding to quasidouble or quasi triple CC bonds) and negative values at even sites

Fig. 7. Local Raman parameters and Z mode description as obtained from DFT calculatio(from the top to the bottom): hoa/oRCCi = 1/3 Trace (oa/oRCC); individual CC stretching costretching coordinate (see text). CC bonds are numbered following the conjugation path

(corresponding to quasi single CC bonds). It is indeed this ‘‘signrule’’ that determines a cooperative effect of these parameterswhen adjacent CC bonds stretch out of phase during the Z oscilla-tion. This is illustrated by the plot reporting contributions of eachbond to h@a/@Qi, namely h@a=@Rk

CCiLkQ . For the four molecules men-tioned above, all these contributions have the same sign so thatthey add up to give a large total h@a/@Qi.

A similar result is found in the case of T4(qu): in this case allsigns of the individual stretching parameters are reversed, so theyare still alternated; thanks to this sign change the relationship pre-viously found between the bond length and h@a/@RCCi still holds(remember that the bond length pattern is reversed passing froman aromatic to a more quinoidal structure). The final result is, alsoin the case of T4(qu), a large intensity of the Z mode, since contri-butions from each bond add up, all with the same negative sign.

Carbonyl substitution on the end thiophene rings is the simplestway to obtain a quinoid model which can be immediately com-pared to T4 aromatic case. However, to our knowledge, this mole-cule has not been so far synthesized, making impossible anyvalidation of our theoretical results by experimental findings.

To demonstrate the reliability of our approach, we report inFig. 7 DFT computed parameters of an aromatic biphenylthiophene

ns for the two molecules sketched on the top of the Figure. The three panel reportntent of the Z mode eigenvector; contribution to hoa/oQi (Q = Z mode) by each CC, from left to right.

34 C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37

and of its quinoid counterpart, widely studied in a recent work [42]where theoretically predicted and experimental Raman spectra areanalyzed and compared in details.

The plots reported in Fig. 7 show the existence of a Z modemainly localized on the thiophene unit, coupled with CC stretch-ings of bonds belonging to the end phenyl rings.

Looking to the h@a/@RCCi parameters belonging to the thiophenering (bond 4–10) we can find the same pattern already found for T4models, showing the characteristic change of sign moving fromshorter to (adjacent) longer bond. As expected, the sign rule isopposite while passing from the aromatic to the quinoid case, inagreement with the ‘‘displacement’’ of the CC double bondspattern.

According to the above discussion, we can conclude that thesign rule shown by h@a/@RCCi, strictly related to the nature(short/long) of the CC bond considered, is the origin of the hugeh@a/@Zi values, characteristic of a chain with non-negligible BLAvalues (dimerized chain).

Let us now consider the two cases (Cu16 and cyanine) charac-terized by a vanishing hBLAi: the sign inversion of h@a/@RCCiparameters relative to adjacent sites is lost in these cases. In bothmolecules h@a/@RCCi are always positive; moreover they show rel-atively low values with respect to the other (dimerized) molecules.If these positive contributions are weighted by the eigenvectorcomponents of the Z mode, they give rise to contributions withopposite sign which determine a partial cancellation when theyare added up to obtain the total @a/@Q. The small Raman intensityof the Z mode of Cu16 is then directly related to CC bondsequalization.

The situation is much more complicated in the case of cyaninebecause of the peculiar definition of Z0 mode. Notice however thatalso in this case contributions with opposite sign are included inthe sum which gives the total value of h@a/@Qi; on the other hand,the intensity of the Z mode is still quite large because of the verydifferent values of the positive h@a/@RCCi parameters at adjacentsites, which prevents an effective cancellation of their contributionto h@a/@Qi.

3.4. Modulation of Raman parameters by geometry relaxation

According to the data commented in Section 3.3 we can con-clude that there is a tight correlation between the equilibrium valueof the BLA parameter of a conjugated molecule and the degree ofelectrons polarization following the excitation of Z mode. This factis a direct consequence of the existence of a relationship betweenequilibrium CC bond length and the associated local Raman param-eters through a ‘‘sign rule’’. The sign rule and the absolute value ofthe h@a/@RCCi parameters are determined by the molecular struc-ture which in turns is strongly sensitive to chain length [13,38],conformation [45], end groups and molecular symmetry [14].

Accordingly, we would like to verify whether it is possible tomodulate the Raman response by a simple relaxation of the molec-ular geometry. To this aim we decided to carry out DFT calculationsof the Raman spectrum of a polyconjugated molecule (we will con-sider here only the case of P16T) while modifying its equilibriumgeometry by nuclei displacements along the Z coordinate. Thisprocedure corresponds to a change of the hBLAi parameter of themolecule by setting the molecular geometry in an out-of-equilib-rium configuration. This ‘‘frozen’’ out-of-equilibrium geometry isadopted for subsequent simulation of the Raman spectrum.

Cartesian nuclei displacements are obtained according to thefollowing relationship:

Dx ¼ A � DR

DR ¼ U0Dz

where the values of DZ have been chosen in the range from 0 (opti-mized structure) to about �0.54 Å with steps of about 0.04 Å. Thesedisplacements determine a decrease of the hBLAi toward an equal-ized structure and an inversion of the alternation pattern of CCbonds characteristic of a polyene.

The results are reported in Fig. 8: a remarkable frequency soft-ening of the Z mode frequency (m1) is observed in the range 1610–1500 cm�1, till a turning point and a subsequent frequency upshift. This trend is accompanied by a drastic lowering of the Ramanactivity.

As already mentioned, in the case of polyenes there is a secondZ mode showing a large Raman activity: it indeed appears in thepredicted spectra reported in Fig. 8, at lower frequency (m2) withrespect to m1. While increasing |DZ!, this second line becomesstronger and stronger at expenses of the higher frequency line,showing the same trend observed for experimental Raman spectraof polyenes of increasing length. However, after the turning pointof m1, also m2 starts to loose intensity.

While the predicted frequency softening reveals the well recog-nized relationship occurring between the effective force constantFZ and the molecular structure [3], the intensity behaviour is re-lated to the evolution of the local Raman parameters.

In Fig. 8b the values of h@a/@Zi while changing hBLAi are re-ported. After a first, modest increase, we find a monotonic decreas-ing trend: the minimum value of h@a/@Zi is reached at the largestvalue of the displacement. The two panels reported in Fig. 8.b showthe local stretching Raman parameters in the two extreme situa-tions: at the equilibrium structure the well known sign rule forh@a/@RCCi occurs, while the end of the relaxation path the situationis quite similar to the case of Cu16 (see Section 3.3) showing posi-tive values for all h@a/@RCCi. This trend suggests that, at larger dis-placements, a different pattern could be reached, showingalternate signs and characterized by a systematic sign inversionwith respect to the values obtained at the equilibrium geometry.

Moreover, the artificial way adopted to modulate the electronicstructure through nuclei displacements would require relativelylarge DhBLAi in order to observe such a dramatic change of theh@a/@RCCi pattern. Indeed, as shown by Fig. 8b, we find the behav-iour typical of an equalized structure only when an inversion of thesequence of single and double bonds along the polyene chain is ob-tained. This can be justified considering that we are perturbing theelectronic cloud simply through a displacement of the nuclei, whilechanges occurring in a real molecule (as those induced by modify-ing end groups or by polarization effects due to inter-molecularinteractions) directly affects electrons (i.e. the electronic wave-function), which in turns determine the suitable equilibriumconfiguration.

This mismatch between the electronic wave-function (and theelectronic parameters, such as h@a/@RCCi) and the artificially in-duced (out-of-equilibrium) nuclear relaxation has been already no-ticed in [46], where the modulation of infrared and Raman spectraof push–pull polyenes in solvents of different polarity was studied.Also in this case the correct frequency and intensity trends werepredicted by calculations at out-of-equilibrium geometries, butthey were obtained at unrealistic, strongly modified BLA values.

We can conclude that in conjugated materials electrons are sostrongly coupled to the molecular geometry that changes of nucleiconfigurations along the Z trajectory can induce a huge perturba-tion of the electronic cloud, which in turns shows a clear signaturein the Raman spectrum.

Interestingly, the trends found with an artificial modulation ofthe BLA parameter show the same qualitative effects (in terms ofRaman parameters) which can be observed for molecular speciescharacterized by different equilibrium BLA patterns: this demon-strates that the relevant Raman features are mainly determinedby the degree of bond length alternation shown by a given

Fig. 8. (a) Simulated Raman spectra of P16T at different out-of-equilibrium geometries, obtained by displacing nuclei along the Z trajectory (see text). Values of thedisplacement corresponding to each spectrum are indicated. (b) Evolution of the parameter hoa/oZi of P16T according to different hBLAi values adopted.

C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37 35

molecule, independently on the fact that it is the result of a specificchemical substitution or it is introduced by specific interactions.

In this concern, it is worth mentioning the paper by Cuff andKertesz [47], which provides a nice evidence of the fact that the Ra-man spectrum knows intra-molecular and inter-molecular interac-tions (included charge injection), through the resulting relaxationof the equilibrium BLA parameter. The authors [47] were indeedable to predict (by DFT simulations) the Raman response of dopedpolythiophenes, considering simple neutral model molecules char-acterized by a quinoidal ground state structure, induced by prop-erly chosen end groups.

4. Conclusions

On the basis of local Raman parameters h@a/@RCCi computed byDFT calculations we proposed a rationalization of the Raman fea-tures of some molecules, characterized by a linear sequence of15 conjugated CC bonds. In particular, we explained why polycon-jugated systems showing a dimerized structure (i.e. hBLAi– 0)

present strong and selective non-resonant Raman spectra: the signinversion of h@a/@RCCi when moving from a given CC site to theadjacent one, determines indeed huge values of the polarizationfluctuation h@a/@Zi with BLA oscillation (i.e. during Z modes) andhence large Raman intensities.

This behaviour is quite general and takes place whenever pat-terns characterized by alternate longer and shorter CC bonds occur.For instance, molecules made by condensed aromatic units,namely polycyclic aromatic hydrocarbons, behave in a similarway [48]. In this case the occurrence of a dimerized structure isdetermined by the confinement of p electrons due to the finite sizeof molecules and it is responsible for very strong Raman transitionsin the range 1300–1200 cm�1. The above observation allowed toextend a ‘‘molecular approach’’ to the study of carbon materialsand contributed to the assignment of the strong Raman band(the so called D line), which indeed shows up in nano-structuredgraphites and in disordered carbon materials [48].

It is pleasant to notice that, based on a different theoretical ap-proach exploiting an Hubbard Hamiltonian, namely the Essential

36 C. Castiglioni et al. / Journal of Molecular Structure 993 (2011) 26–37

State Model [49], Painelli et al. [50] recently predicted the rise ofanomalously high non-resonant Raman activity in correspondenceof the transition from the equalized phase to the charge transferdimerized phase of a p conjugated systems made by stacks of alter-nated donor and acceptor molecules.

The analysis of the local Raman parameters h@a/@RCCi allowsalso to rationalize the Raman response of some polyconjugatedmaterials which seems to deviate from the most common behav-iour, since they show a Raman spectrum scarcely selective, withmoderate or weak band intensities. This is for instance the caseof linear chains characterized by a cumulenic structure and of cya-nines: their Raman features are representative of a vanishing BLAand, accordingly, the characteristic sign rule which commonlyholds for h@a/@RCCi is lost. This is indeed the reason why these mol-ecules don’t exhibit a peculiar coordinate (i.e. the Z coordinate)able to lead a huge polarization fluctuation.

The analysis reported in this paper is necessarily restricted tofew molecular species; we are however confident that the exam-ples here presented, along with the large amount of experimentaldata and simulations reported in the literature, demonstrate thatany modulation of the hBLAi parameter (induced by chemical sub-stitutions, conformation changes, peculiar end groups, chain lengthand environmental effects) is ‘‘amplified’’ by a remarkable modula-tion of the Raman spectrum, which indeed shows clear signaturesof the specific molecular and electronic structure.

Our recent study on thiophene based hetero-phenoquinones[42] represents a nice demonstration of this conclusion. Since long-time the Raman spectrum of a quinoid molecule, namely dithio-phenediphenoquinone, was a puzzle from the computationalpoint of view: in particular DFT calculations dramatically failedin the frequency calculation of the strongest Raman transition as-signed to the Z mode of the dithiophene moiety, predicted indeedabout 100 cm�1 above the experimental one. The reason of thiswrong prediction was the lack of accuracy in the description ofthe molecular structure obtained at the B3LYP closed shell level.Indeed, it has been recently demonstrated [42] that the true equi-librium ground state structure of this molecule is the result of acompromise between a quinoidal and an aromatic structure, whichin turns derive from a non-negligible contribution from a biradic-aloid electronic structure, not explicitly taken into account by theclosed shell wave-function. When the suitable geometry is chosen,showing a non-negligible displacement of hBLAi of the bitiopheneunit, as predicted by a UB3LYP broken symmetry approach, thesubsequent calculation of the Raman spectrum gives a very gooddescription of the major experimental features (in particular theZ mode observed at 1304 cm�1 is predicted at 1300 cm�1). This re-sult proves that the Raman spectrum can provide unique indica-tion relative to the molecular structure.

In this framework, we have shown in this paper that it is possi-ble to follows the effect of hBLAi modulation on the Raman re-sponse by adopting out-of-equilibrium geometries. This approachshed light on some peculiar spectroscopic features, such as theturning points in the frequency evolution of the Z modes, redistri-bution of Raman intensity among bands, increase (or decrease) ofthe Raman activity.

Some of these behaviours were indeed experimentally ob-served: for instance, the Raman spectrum of push–pull polyenesshows a marked evolution when molecules are dissolved in sol-vents with increasing acidity [15]. On the other hand, it is wellknown that the increase of the chain length often gives rise to aremarkable modulation of the Raman spectrum of conjugated olig-omers and polymers [3,10,13].

As a final remark we can mention an example were the exis-tence of a turning point in the frequency evolution of the Z modewith chain length has been experimentally observed: this happensin the case of oligothiophenes with dicyanomethylene end groups,

characterized by a mainly quinoid structure and can be correlatedto the increasingly important contribution of a biradicaloid struc-ture in the description of the ground state wave-function [41,42].

Acknowledgment

This work was supported by PRIN Project 2008 JKBBK4 ‘‘Track-ing ultrafast photoinduced intra- and inter-molecular processes innatural and artificial photosensors’’.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.molstruc.2010.12.021.

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