Experimental molecular hyperpolarizabilities from vibrational spectra in systems with large electron-phonon coupling

SYlITHETIIC lilnl|TALS ELSEVIER

Synthetic Metals 74 (1995) 171-177

Experimental molecular hyperpolarizabilities from vibrational spectra in systems with large electron-phonon coupling C. Castiglioni, M. Del Zoppo, P. Zuliani, G. Zerbi Dipartimentodi ChimicaIndustriale e Ingegneria Chimica,Politecnicodi Milano,piazzaLeonardoda Vinci32, 20133 Milan, Italy Received 15 May 1995; accepted 6 June 1995

Abstract Experimental data are presented and discussed on the relaxation contribution (/3 ~and ,),r) tO first- and second-order molecular hyperpolarizabilities of classes of polyconjugated organic materials. These data are obtained from the measure of classical vibrational observables, i.e., vibrational frequencies, infrared intensities and Raman cross sections. It turns out that the values of/3 r and ~'~are very similar (both in values and trends) to the electronic molecular hyperpolarizabilities/3 and y obtained from traditional direct experiments. It is concluded that for the molecular systems considered,/3~ and y~ may be a good estimate of static, non-resonant/3 and % The justification for such conclusions is that in polyconjugated systems very large electron-phonon coupling takes place such that vibrational displacements are strictly related to and cannot be separated from electronic excitations.

Keywords:Electron-phonon coupling; Molecular hyperpolarizabilities;Spectra

1. I n t r o d u c t i o n The recent interest in new organic materials for photonics aims at finding criteria for the preparation of the best materials with large non-linear optical ( N L O ) response. From the microscopic viewpoint the intrinsic properties which have to be optimized and which are essential in the comparison of different compounds are the molecular hyperpolarizabilities /3 and 3'. In this work we collect several results which we have obtained on the determination (both by experiments and by 'ab initio' calculations) of the relaxation contribution to molecular hyperpolarizabilities. The discussion of these results shows the relevant role of vibrational spectroscopy in the study of N L O responses of organic molecules. Expressions for/3 ~ and 3'r have been derived in a previous paper [ 1 ] where a 'semiclassical' model which describes the nuclear relaxation process is illustrated. According to this model, the relaxation contribution to the molecular electric dipole induced by an external static electric field is described as a charge rearrangement within the molecule which follows the nuclear displacements under the action of the applied field. The model allows us to obtain analytic expressions for the relaxation contribution to/3 and 3' (hereafter referred to as/3* and 3"*). Notice that/3~ and 3'* are not purely nuclear contributions, since electronic charge fluxes, which may be 0379-6779/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved

SSDI0379-6779(95)O3364-P

very relevant in conjugated ~electrons [ 2], take place during the distortions of the nuclear geometry. For this reason fl~ and 3'r are very large in those systems where the electronphonon coupling is strong. Expressions for a r,/3 r and yr are 1 1 O/x~ a/g,, ot,~= 4,n.2c2 ~(7~k2)[(~-~--£)(~--~'k)]

(1)

1 _ [ 1 ~[[O~][O__~e_] f l ~ ' = 47flc2 ~[-~k)t[-~QJ[ OQk)

t Jt oed*t dt-rdTJ ]

)

1 _ [ 1 ~[-[Otz.~Cafl_.~ ~ 3":~s= 4 ~ c z ~[-~JLt-~Q3t aQk )

~OQk]\ OQk] \aQdt aQk ] \OQd\ OQ~ ]

/aa.,,,a/aa3 (aa,,,,~faa~ (aa,~faa_V!

taQ,)t0e) t0eJt0C t0Ct0o )J

where the following parameters appear: (i) Vibrational frequencies ~'k, i.e., energies of the vibrational transitions. (ii) Terms like O#~/aQk, i.e., the nth Cartesian component of the derivative of the molecular dipole with respect to the

C. Castiglioniet al. / Synthetic Metals 74 (1995) 171-177

172

CH~,,.N,. O4 ~

vibrational coordinate Q,. O/z,/OQk can be easily obtained from infrared absorption intensities, according to

A m = N~'/3c2~_, I otx,/~a, 12

(a)

(4)

n

I

where N is the Avogadro number, and c is the light velocity. It has been shown with full analytical quantum treatment [ 3 ] (and also from empirical findings [4] ) that it is impossible to account for infrared intensities without considering the electronic charge rearrangement during nuclear vibrational displacements. (iii) Terms like 0a,,,/OQ,, i.e., the derivative of the molecular linear polarizability with respect to Q,. This quantity is directly related to the Raman cross section of the kth normal mode. It is well known that Raman cross sections are closely related to the virtual electronic transitions involved in the process [3]. (iv) Terms like O/3,,,p/~Qk, i.e., the derivative of the first hyperpolarizability with respect to Q,, which is related to hyper-Raman intensities. Eq. (2) shows that/3 ~depends on cross terms from infrared and Raman intensities. This implies that in order to have nonvanishing/3 ~it is necessary that some normal modes Qk have non-vanishing intensity both in the infrared and in the Raman spectra. According to the mutual exclusion principle between infrared and Raman activity/3 r = 0 for all molecules with an inversion centre, as expected by symmetry considerations applied directly on the/3 tensor. Moreover, according to Eq. (2), large /3r are attained with vibrational normal modes strong both in the infrared and in the Raman. This requirement is not fulfilled in most of the cases studied by vibrational spectroscopy. It is indeed commonly known that some complementarity between infrared and Raman intensities exists also in the case of molecules with very low symmetry. When a normal mode shows a very large activity in the Raman it is generally weak in the infrared and vice versa. We were surprised to find that this empirical rule is not followed in the case of organic molecules which show large /3 values. A very nice example of this unusual behaviour is illustrated in Fig. 1 where the infrared and Raman spectra of Crystal Violet are reported. The two spectra practically show the same intensity pattern; from Eq. (2) it follows that the occurrence of simultaneous large ~tz,/~Qk and ~a,,,/~Qk yields a very large/3r value. As far as 31is concerned, we can predict from Eq. (3) that molecules with large Raman intensities will have also large ~/r since the last three terms in Eq. (3) depend only on Raman cross sections. For this reason, in the case of polyconjugated compounds such as polyenes, polythiophenes, polypyrroles, etc., we expect large 3,~. As shown in detail in Section 2, this turns out to be true. In the case of molecules with large Raman cross sections (as in all the cases mentioned above) we can neglect the infrared and hyper-Raman cross terms which appear in Eq. (3). With this approximation a very simple expression for the mean value of 7 ~can be written:

I

zo

¢.,~.

____...~ i" 1

so

~Sso l~iso l~so l~so l~,so 1lso NRVENUPtBER

tOso

~so

~so

N

(b) oo

--m u m

E~ J ~9

1~9 NRVENUPIBER

Fig. 1. Comparisonbetweeninfrared (a) and Raman (b) spectraof Crystal Violet.

~f = 115(1147r2c2) ~, IR~m='/V~

(5)

i

where ~ = 1/5Ei,~T~i~. In Eq. (5) the polarization of the scattered light (li) is parallel to that of the exciting laser. Eq. (5) shows clearly that ~r can be obtained in a very simple way. To have a quick estimate of ~/r all one has to do is to measure the area of a few Raman bands (normalized to a suitable absolute standard) and their peak frequencies. These operations can be routinely performed with any Raman spectrometer. The method becomes only slightly more complicated for the measure of/3r: in this case measurements of both infrared and Raman absolute intensities are required. Eqs. (2) and (3) also provide the way to obtain theoretical predictions for/3 r and 7 r. When spectroscopic experiments are not available it is possible to use 'ab initio' computed vibrational spectra which may be routinely obtained with quantum mechanical calculations. It is important to stress that Eqs. (2) and (3) are valid when Raman cross sections are measured in the absence of resonances. An experiment must then be made with a suitable choice of the exciting wavelength. For the systems considered in this work the Raman experiments were made with excita-

C. Castiglioni et al. /Synthetic Metals 74 (1995) 171-177

tion at A = 1.064/xm which is safely away from resonances, thus allowing comparison with the NLO coefficients theoretically computed in the limit of static fields.

173

the case of the third-orderNLO response (7), the compounds studied are polyenes and alkylthiophenes. 2.1. 3'

2. Results The relevance of the relaxation contributions in determining the NLO response was studied by doing experiments and calculations for different classes of organic molecules that are known to have large NLO responses. In Table 1 the compounds with large second-order NLO response (/3) considered in this work are indicated with suitable labelling. In

The first class of prototypical compounds is the series of oligoene molecules. ,yr values were evaluated on a completely theoretical basis from 'ab initio' 6-31 G computed Raman cross sections for oligoenes with increasing chain length [5a] ; moreover experimental absolute Raman cross sections [5b,c] were measured for polyene systems (diphenylpolyenes, fl-carotene, polyenovanillins, etc.) and experimental ,~r were evaluated according to Eq. (5). In Fig. 2 both theoretical and experimental ,~r are reported. In the same figure

Table I Comparison between/3 and/3 r values (units of 10 -30 e.s.u.) for some selected organic molecules obtained both theoretically (with ab initio 3-21 G basis set) and experimentally Molecule

I [I

III

/3~,( 3-21 G)

H\ ~ /O H / N ' J ~ N\O I'~N~'~N/O H/ / ~ /k---(k \o H / O",, H--N N-- O O\ _ ~ /H o/N N,,H H--N\

/ H 0

/3~,( 3-21 G)

10.67

9.55

28.24

30.96

8.27

11.08

/3~,(exp.)

/3, (exp.)

12.6 b

10"

24 a

N--O

CH3.~N/ CH3

IV CH3.~N ~ C ~ I ~ ] . , , . N I

46 c

10.2 34.6 50.9

3.8 a 39.8 a 42.1 d

jC H3 I

CH3

Cle

/

V

50

CH3

H

MeS " " ' ~ ~ " O n= 1 ,~,~ ~ " ~n " n= 3 MeO n =4

11.6 34.7

a From electric field-induced second-harmonic generation (EFISH) experiments [ 13]. b /3yyy.

c From hyper-Rayleigh scattering (HLS) experiments [ 14]. a From EFISH experiments [ 15].

10.2 32.0

174

C. Castiglioni et al. / Synthetic Metals 74 (1995) 171-177

O IOO(

W

~t 100

-

[]

W

| ~, Ioo

[] ©

t

I

I

t

t

I

2

4

6

8

10

12

N

Fig. 2. Comparison among ~ values obtained from different methods for polyene systems of increasing chain length: (©) ~/r from calculated ('ab initio' 6-31G) Raman intensities [5a]; (&) ~ from 'ab initio' calculations (6-31G) [6] ; (O) ~r from experimental Raman cross sections [5b] ; ('k) from THG measurements [7]; ([]) ~ from experimental Raman cross sections of polyenovanillins [ 5 c ].

•

[]

•

[]

[]

| 10-

A

I

I

I

q

!

i

2

4

6

8

10

12

view of the fact that 'ab initio' 3' from standard derivative methods have a purely electronic origin since nuclear relaxations are not explicitly taken into account [ 8 ]. Also, in the case of data from third-harmonic generation experiments (THG), since experiments are made with an incident laser line well above all vibrational transitions (A= 1.9 /xm), relaxation contributions are not allowed, at least in principle. Similar plots have been obtained in the case of experimental ,~r from measured Raman intensities for a series of oligoalkylthiophenes with increasing chain length [9]. These data are compared (Fig. 3) with the theoretical estimates obtained with different methods, namely 'ab initio' ~r from computed Raman spectra (3-21 G basis set) [ 10] and theoretical ~ obtained with a different theoretical approach based on a sum-over-states (SOS) method with INDO/MRD CI semi-empirical calculations [ 11 ]. The results presented in Fig. 3 indicate that also in the case of oligothiophenes the values of ~r are similar, if not equal (within the experimental uncertainties) to those from other independent determinations of ~.

2.2./3 Next, we consider first-order hyperpolarizabilities. A test of the model is obtained when values of/3r are calculated from 'ab initio' theoretical spectra [ 12] or measured from experimental vibrational spectra. In Table 1 we report/3r data for different molecules well known for their high first-order hyperpolarizability. The compounds studied belong to three main groups: (i) classical push-pull systems (I, II), whose prototype is p-nitroaniline (I); (ii) high symmetry apolar molecules (III, IV), recently presented by Zyss [ 16], whose /3 tensor has octupolar symmetry; (iii) push-pull polyenes (V) with different chain length. The relevant fact which stems again from these data is that /3r and/3 obtained with the derivative method are extremely similar. Moreover, the experimentally determined/3r are in good agreement with independent and direct measurements of/3 and with the theoretical predictions.

3. D i s c u s s i o n : the r e l e v a n t r e l a x a t i o n m e c h a n i s m

N

Fig. 3. Comparison between ~' values obtained from experimental ([]) [9] and computed (A) [ 10] Raman cross sections, and ~ values ( 0 ) calculated with SOS methods [ 11 ] for oligothiophenes with increasing chain length.

we compare ~r with the electronic ~/determined either theoretically with standard derivative methods from quantum mechanical 'ab initio' calculations (6-31 G basis set [6] ), or with third-harmonic generation experiments [7]. In Fig. 2 all values of ~/r and -7/show an identical superlinear increase with increasing chain length. Moreover, we notice that "7 values obtained with our model are very similar, if not equal (within the experimental uncertainties) with the values of electronic ~. This observation is very surprising just in

The values calculated and/or measured for /3' and 3,r clearly indicate that the relaxation mechanism plays a relevant role in determining the NLO response of organic conjugated molecules. The similarity between relaxation hyperpolarizabilities and other completely independent determinations of /3 and 3Isuggests that for polyconjugated materials the use of the vibrational spectra may lead to the same physical observables which are directly measured with more traditional standard techniques. Indeed this similarity, observed in so many cases, cannot be taken as casual and indicates the existence of a particular physical situation which needs further theoretical studies.

175

C. Castiglioni et al. / Synthetic Metals 74 (1995) 171-177

A closer examination of the relaxation processes in polyconjugated materials suggests a possible interpretation of the phenomenon. Raman spectra give useful information on these processes. Raman intensities of polyene systems have been thoroughly discussed [ 17] in the light of the theory of the effective conjugation coordinate (ECC) [18,19]. The Raman spectra of polyenes (and also of most polyconjugated organic systems) show a characteristic pattern with very few, but very strong bands. This pattern can be accounted for with a simple two-state model in which the electronic ground state and only one strongly dipole allowed excited electronic state are assumed to exist. With this hypothesis the Raman activity of vibrational normal modes follows a simple rule: Raman active are only those normal modes Qk which have a nonvanishing projection along the direction of the path of geometry variation which nuclei follow to reach their equilibrium position in the allowed excited electronic state. Moreover, the Raman activity of Qk depends on the value of the displacement of the equilibrium position of the normal coordinate Qg in going from the ground to the excited state. If a vibrational coordinate (R) is chosen in the direction of this relevant geometry change it is possible to write an expression for the Raman tensor where the dependence from ~ appears explicitly: oa/aQi o~oalalr~,

(6)

where LRi is the value of the eigenvector of the ith normal mode projected along the R direction. IfEq. (6) holds, Eq. (3) can be further simplified:

,~r= 1 /15(aax~/o~) 2F~-'

(7)

where FR is the diagonal force constant associated with the collective coordinate ~. According to Eq. (7), from the knowledge of R, we also know the direction along which the relevant relaxation process takes place. In the case of polyenes it is known that R is exactly the dimerization amplitude [ 18 ] ; in other words ~t represents the oscillations of the value of the bond alternation parameter. As a consequence, in the case of polyenes, relaxation under the action of an external field induces a change in the degree of bond alternation, which in turn is responsible for large .yr. It has been shown elsewhere [20] that in the case of polythiophene oligomers or polymers ~ can be defined as an oscillation between a more aromatic (structure A) and a more quinoid structure (B):

(A)

(B)

In this case the relevant relaxation process leads to a more quinoid-like structure. At least in the cases where a two-state model holds, the ~ coordinate can be defined in terms of the difference of the equilibrium structures between the ground and the only one relevant excited electronic state. It follows that during the relaxation process induced by an external field,

nuclei probe the excited state, w e then expect that during this process the polarization of the ~- electrons reflects the charge distribution in the relevant excited state. Even in the case of push-pull molecules, such asp-nitroaniline, only a few normal modes are relevant in determining /3~ [ 12]. Examination of the eigenvectors relative to these relevant normal modes shows that they contain a large contribution from the vibrational coordinate that describes the oscillation between a more aromatic and a more quinoid structure. Notice that the electronic state which is found to be the only one relevant in an SOS approach [21] is the charge transfer state to which the quinoid structure contributes in a significant way. In the case of molecules with octupolar/3 tensor the situation is slightly more complex. However, it has been shown [ 12] that also for these molecules it is possible to identify more than one relevant vibrational mode which can be made to correspond to relevant electronic excited states. Finally, in the case of push-pull polyenes such as polyenovanillins (V), it is apparent from Fig. 4 that Raman spectra are very similar to those of polyenes without polar end groups. We can conclude, as in the case of oligoenes, that the equilibrium structure of the relevant excited state differs from that of the ground state in the degree of the bond alternation of the polyenic chain. It must be pointed out that a charge transfer state would stabilize a structure in which bond alternation is reversed with respect to the (polyene-like) structure of the ground state. A striking support of the role played by the ~ coordinate in determining the NLO properties of polyconjugated molecules has been recently found in the study of the solvatochromism and of the solvent effect on/3 of push-pull polyenes with suitably chosen donor (D) and acceptor (A) groups (e.g. A = C(CN)2, D =N(CH3)2, as in VI):

Me\ ~

~

Me

/CN CN

VI A sizeable modulation of/3 by the solvent has been experimentally observed [22] (by electric field second-harmonic generation experiments (EFISH)) and theoretically predicted [23]. In the same works the modulation of NLO response has been ascribed to large changes in the degree of bond alternation of the polyenic chain due to the effect of the solvent [22,23]. With our approach this bond alternation modulation can be described as a displacement of the equilibrium structure along the effective conjugation coordinate R. Structural changes along • induce sizeable variations of the vibrational (infrared and Raman) band intensities. These variations have been rationalized and discussed [24]. The obvious consequence is that solvents can modulate not only the electronic fl but also/3 ~. It follows that solvent effects can be correctly studied by the only measure of vibrational intensities. The results obtained are fully parallel in values

176

C. Castiglioni et al. / Synthetic Metals 74 (1995) 171-177 H

(a)

"B b

c

8a

d

n

DO

~90

I~eo

,~7o

1~6o l~so

i~o

153o

62o

~1o

o!2

o!,

o!8

d8

~

~

NAVENUMBER

Fig. 5./3 r dependence from solvent polarity of molecule VI obtained from experimental infrared and Raman intensities [24]. The scale of the x-axis corresponds to the empirical solvent polarity parameter E~ [25].

(b)

including also methods based on purely electronic excitations (such as third-harmonic generation (THG)).

n=3

800

I~90

i~0o

t~7o

1§so

t~5o

t[~o

163o

~zo

@:o

t l ~ o tSso

§zo

~io

WRVENUMBER

(c)

~oo

igso 1~eo

I~0

l~so

1~so

NRVENUMBER

Fig. 4, FT-Raman spectra ( A ~ = 1,064/Lm) o f polyenovanillins of increasing chain length.

and trends with the results on /3 of Refs. [22] and [23] (Fig. 5). As a conclusion we wish to stress the physical meaning of the R coordinate, namely, it describes the direction along which electron-phonon coupling is most effective. This is the reason why relaxation along ~ induces large electron polarization; it is impossible to move nuclei in the R direction without the excitation of ~r electrons. If this principle is true the opposite must also be true, namely, the relaxation process cannot be separated from the electronic excitation. This accounts for the close resemblance, shown above, between /3~ and 3"~ and /3 and 3" obtained with different methods,

4. Conclusions In this paper we have presented several results on experimentally determined and theoretically calculated relaxation contributions/3 r and 7 r to the first- and second-order molecular hyperpolarizabilities of polyconjugated organic molecules. The relevant conclusions are the following: (i) In the many cases studied/3 r and 3'r are very similar to /3 and 7 values obtained from experiments which exploit electronic excitations (THG. EFISH .... ). This coincidence strongly suggests that there may exist some sort of redundancy between/3r and/3, as well as between 3'r and 7. The origin of this redundancy can be found in the existence of a large electron-phonon coupling which prevents a complete separation of nuclear relaxation and electronic excitation. A complete theoretical proof of the above statement must be provided. (ii) The measurement of/3r and 3'r is experimentally very simple since it requires only the knowledge of classical vibrational observables, i.e., Raman and infrared intensities and vibrational frequencies. These measurements can be made routinely on large classes of compounds. (iii) Due to the close resemblance between/3~ and 3'~ and molecular hyperpolarizabilities evaluated in a direct way with standard methods, we propose the spectroscopic evaluation of/3r and 7' as an independent measure of the NLO responses in the case of systems with large electron-phonon coupling. This method directly yields molecular hyperpolarizabilities. On the contrary, most of the methods usually employed measure bulk susceptibilities. This offers a criterion of choice in selecting among very many molecules the most suitable ones for science and technology.

C. Castiglioni et aL / Synthetic Metals 74 (1995) 171-177

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[13] L.T. Chang, W. Tam, S.R. Marder, A,E. Stiegman, G. Rikken and C.W. Spangler, J. Chem. Phys., 15 ( 1991 ) 10 603. [ 14] J. Zyss, T. Chau Van, C, Dhenaut and I. Ledoux, J. Chem. Phys., 177 (1993) 281. [ 15 ] C. Andraud and A. Collet, personal communication. [16] J. Zyss, J. Chem. Phys., 98 (1993) 6583. [ 17] C. Castiglioni, M. Del Zoppo and G. Zerbi, J. Raman Spectrosc., 24 (1993) 485. [18] C. Castiglioni, M. Gussoni, J.T. Lopez Navarrete and G. Zerbi, Solid State Commun., 65 (1988) 625. [19] M. Gussoni, C. Castiglioni and G. Zerbi, in R.J.H. Clark and R.E. Hester (eds.), Vibrational Spectroscopy of New Materials, Wiley, New York, 1991, p. 251. [20] G. Zerbi, M. Gussoni and C. Castiglioni, in J.L. Br&las and R. Silbey (eds.), Conjugated Polymers, Kluwer, Dordrecht, 1991, p. 435. [21 ] M. Joffre, D. Yaron, R. Silbey and J. Zyss, J. Chem. Phys., 97 (1992) 5607. [22] (a) S.R. Marder, J.W. Perry, G. Bourhill, C.B. Gorman, B.G. Tiemann and K. Mansour, Science, 261 (1993) 186; (b) S.R. Marder, J.W. Perry, B.G. Tiemann, C.B. Gorman, S. Gilmour, S.L. Biddle and G. Bourhill, J. Am. Chem. Soc., 115 (1993) 2524. [23] (a) F. Meyers, S.R. Marder, J.W. Perry, G. Bourhill, S. Gilmour, L.T. Cheng, B.M. Pierce and J.L. Br6das, Nonlinear Opt., in press; (b) F. Meyers, S.R. Marder, B.M. Pierce and J.L. Br6das, Phys. Rev. Lett., in press. [24] P. Zuliani, M. Del Zoppo, C. Castiglioni, G. Zerbi, S.M. Marder and J.W. Perry, J. Chem. Phys., submitted for publication. [25] Y. Marcus, J. Solution Chem., 20 (1991) 929.