NLO response of derivatives of benzene,stilbene and diphenylacetylene:MP2 and DFT calculations

A.M.Elhorri*,M.Zouaoui-Rabah

Laboratoire de Microscopie,Microanalyse et Spectroscopie Moléculaire,Faculty of Sciences,Djillali Liabes University of Sidi-Bel-Abbes,22000 Sidi-Bel-Abbes,Algeria

1.Introduction

Because of their numerous applications in the areas of photonics and optoelectronic[1,2],various types of active materials in NLO have been developed over the recent years.These materials are used in technical sophistication of lasers[3].Among these materials,organic and organometallic materials[4]are acquiring more and more interest because of to their relatively easy syntheses and their interestingly cost.

In general,Organic materials are constituted of a π-conjugated system end capped with electron donor and electron acceptor groups to facilitate the intra-molecular charge transfer(ICT)which is directly related to the magnitude of the NLO response[5-7].This later,is highly impacted by parameters such as the energetic gap(ΔE= εLUMO-εHOMO),the occupation quotients π*/π,the distance of the central bond in the chromophore,the bond length alternation(BLA),the stabilization energyE2,the electronic transition parameters,the first hyperpolarizabilty and the induced dipole moment[8-11].

So far,many papers in this field were devoted to the comparison between experimental and calculated parameters.For example,perfect agreements;giving regression coefficients of about0.99 between experimental and calculated parameters cited here above have been highlighted by Kleinpeter and Frank[12].These authors have reported for a series of disubstituted alkynes the evolution of the theoretical occupation π*/π,the central C═C bond length and the13C chemical shiftvs.the experimental ones using as theoretical method the B3LYP/6-311G(d)level of theory.A further study[13]with the same objective and the same methodology was carried out on the azine chromophore.Linearity between experimental and theoretical has been demonstrated for occupations quotients,central distances,13C chemical shifts and also for hyperpolarizabilities.Moreover,the push-pull character of 71 azo dyes compounds was examined in order to determine their potential for NLO applications[14].Regression coefficients of about 0.8927 between experimental and calculated values were found for13C chemical shifts of the N═N bond,while between occupation π*/π and central bond length and hyper polariz abilities,these coefficients were of 0.627 and 0.756 respectively.

On the other hand,the parameters related to the excited states provide information on the behavior of the molecule in solution,when the solvent effect is considered,as well as on the nature of the intramolecular charge transfer.In this context,comparisons between calculated and experimental maximum absorptions λmaxof oxazolones derivatives have been reported;large deviations of 10 to 66 nm have been found out,this is probably due to the fact that calculations have been performed in the gas phase using B3LYP/6-311++g(2d,2p)level of theory[15].However,in another work,on quinoline chalcone derivatives,using the PBE0 functional combined with the 6-31g(d)basis set in a PCM medium[16],regression coefficients about 0.9322 et 0.9727(after the rejection of some points)have been determined.

The great challenge for quantum mechanical methods remains the estimation of the first hyperpolarizability coefficient β.Since two decades,a number of papers focused their objectives on finding out methods which could reproduce molecular hyperpolarizability with high accuracy.Many of these papers based their work on comparison between calculated and experimental hyperpolarizabilites.K.Y.Suponitskyet al.have addressed a thorough analysis,based on the results of experimental measurements,in the predicting capability of hybrid DFT methods of hyperplarizabilities ofp-NA derivatives[17].More recently,M.Medved and D.Jacquemin[18]obtained a deviation between calculated and experimental hyperpolarizabilities of 43×10-30esu for the azo-benzene chromophore.Another investigation[19],reported a deviation of80×10-30esu between the calculated and experimental values of the β coefficient for the stilbene molecule substituted by a nitro group and a halogen at each end.

The second-order M?ller-Plesset correlation energy correction,truncated at second order MP2 method[20],which has been already considered among the most efficient method in calculating hyperpolarizabilities[21],has been used with the Def2-TZVPP basis set and has yielded very close first hyperpolarizabilities to the experimental ones for benzene derivatives in solvent[22].However,for derivatives of hymecyanine,deviations of 4×10-30to 24×10-30esu have been obtained when MP2/6-31g(d,p)has been employed[23],while with MP2/6-311G(d)on arylvinyldiazine derivatives,deviations from experimental data of 12×10-48to 300×10-48esu have been obtained[24].

Thereby,in this paperthe two main objectives were; firstto establish from a range of basis sets,combined with MP2 method,which one performs better in calculating the closest static hyperpolarizabilities β(0)to experiments with a reasonable computationaltime.The second objective was still to test the reliability,relatively to experiments,ofa variety of DFT methods incorporating different XC percentages,such as B3LYP,CAMB3LYP,PBE0,M06-HF[25]and M06-2X[26],in producing the β(0)parameters.In parallel,energy gaps(ΔE),NBO(Natural Bond Orbitals)parameters and electronic transitions are also examined.For this aim,a series of eight push-pull molecules(Fig.1)for which experimental data are available in the literature are considered.

1.1.Computational details

All calculations were carried out using the Gaussian 09 suite of programs[27].Ground-state structures optimizations in the gas phase as well as static firststatic hyperpolarizabilities have been first computed using M?ller-Plesset methods second order MP2 using 6-31g(d),6-31g(d,p),6-31+g(d,p),6-31++g(d,p),cc-pvdz,Aug.-cc-pvdz and Cc-pvtz basis sets.

Calculations of structures and energies were also carried out using the cc-pvdz basis set with hybrid meta DFT methods based upon the Khon-Sham et Hohenberg formalism[28]and incorporating a given percentage of exchange(XC,%).These functional are namely;the M06L[29],the most popular functional B3LYP[30],its corresponding long-range corrected using Coulomb Attenuation Method the CAMB3LYP[31],the Perdew-Burke-Ernzerhof PBE0(pbe1pbe)[32],the Minnesota functional with a full HF exchange M06-HF[24]and the M06-2X[25].Their XC percentages are respectively:0%,20%,65%,25%,100%,54%[33,34].

Under a strong external electric field,the induced dipole moment is described by the following equation

Eiis the electric field,is the permanent dipole moments,αijis the linear polarizability, βijkand ?ijklare the first and second order hyperpolarizabilities.

The dipole moment[35]and the first hyperpolarizability[36,37]are calculated as follows:

In order to determine the structure which matches best with experiment,we also performed NBO calculations as well as BLA,E2,dc═c,ΔEπ*/π as defined here above by the chosen functional which matches best with experiment.

In the last part of this paper,ground state structures were optimized in different solvents[38]using the M06-2X functional combined with the aug-cc-pvdz basis set which has been recommended as the most suitable basis set for describing excitation energy transfer[39].The IEFPCM[40,41]model was used to simulate the implicit solvent represented by its dielectric constant ε.Preliminary calculations of Solvation Gibbs free energies ΔGsolvhave been performed for three molecules in order to determine their specific solvent.In the end,the maximum absorption wavelengths λmax,the corresponding oscillator strengths and the associated HOMO LUMO transitions have been calculated using TD-M062X.

2.Results and Discussio n

Fig.1.Considered molecules in this work.

Table 1First hyperpolarizabilities(×10-30 esu):calculated using different basis set with MP2 method compared with the experimental ones

On Table 1 are reported static first hyperpolarizabilities for the eight considered molecules presented on Fig.1 using the M?ller-Plesset methods MP2 combined with different basis sets.The corresponding structures were first optimized in the gas phase.Experimental hyperpolarizabilities[38,42]are also listed on Table 1.The calculated β(0)using the cc-pvdz basis set fit the best the experimental values for the four molecules 1a,2a,3a and 1b while for the remaining molecules,the 6-31g(d,p)basis set gives the best results,nonetheless those obtained with cc-pvdz are still very close to the experimental ones.On the other side,on the addition of diffuse functions enhance sharply the magnitudes of hyperpolarizabilities and therefore leads to very erroneous results.It is then notappropriate to add diffuse functions when carrying this kind of calculations on this kind of molecules.Overall,it is preferable to use the cc-pvdz basis set.

On Table 2 are reported calculated static first hyperpolarizabilities for the same eight molecules using M06L,B3LYP,CAM-B3LYP,PBE0,M06HF and M062X functional all combined with the cc-pvdz basis set.Here also,all structures have been first optimized.As quoted above,different exchange-correlation XC amounts are incorporated in these functionals,for instance we expect to observe their effects on hyperpolarizability values(Table 2)and be able to determine among the six functionals which reproduces the most closely the experimental values.According to Table 2,the eight molecules can be classified into two groups;a group constituted by the molecules 1a,2a,3a and 1b and a group by the molecules 1c,2c,3c and 4c.The molecules of the first group are characterized by their low hyperpolarizabilities,probably due to a less important charge transfer,in comparison to those of the second group and the functionals M0-2X and CAM-B3LYP give,for this group,similar results to those achieved by MP2 and very close to those recorded experimentally.It is noted besides that B3LYP and PBE0 yield also acceptable results in this case.Therefore we suggest that increasing the XC amount contributes in an efficient way to improve the calculated values of hyperpolarizabilities when these later are relatively low.While for the second group,the three first functionals;M06-L,B3LYP and PBE0 overestimate excessively the hyperpolarizabilities and although M06-2X and CAM-B3LYPgive results in a good agreement with those obtained from MP2,they remain far from the experimental values.On the other hand,M06-HF method,sometimes overestimating and sometimes underestimating the polarizabilities,gives a good average error relatively to experiment when considering both groups.However,in the further part of this paper,we will use the M06-2X functional rather than the M06-HF because of the chaotic results of this later and the closer ones to MP2 of the former.

On Table 3 are presented the calculated with M06-2X/cc-pvdz different components of hyperpolarizabilities and those of dipole moments as well.It is clearly observed from these results for the eightmoleculesthat the π-electrons delocalization is one dimensional along theXaxis and this is con firmed from the μxand the βxcomponents values which are significant compared to the other components of dipole moments and hyperpolarizabilities respectively.

Table 3Total static dipole moment(μ)in Debye,mean first-order hyperpolarizabilities(β)in esu(×10-30)for all molecules with M062X/cc-pvdz

3.NBO Analysis

NBO analysis has been revealed to be very effective and useful for understanding intra-molecular π-electron delocalization in push-pull molecules.This,with considering donor(i)-attractor(j)interactions,the related stabilization energyE(2)which is evaluated as the secondorder perturbation,it is estimated[43-45]as:

Whereqiis the orbital occupancy,εietεjare the diagonal elements(orbital energies)andF(i,j)are the off-diagonal elements of the NBO Fock matrix[46,47].

NBO results are reported within Table 4 among other parameters such as the central bond lengths(C═C or C≡C)or central bond lengths occupation π*/π quotient or summation over the benzene bond occupations when there is no central bond.Attention is drawn on thefact that in Table 4 results are presented in an increasing order of studied molecules hyperpolarizabilities.

Table 2Obtained deviations between calculated and experimental first hyperpolarizabilities(×10-30 esu)using different functional and MP2 methods

Table 4R c═c or R c≡c(nm),BLA(nm),μtot(Debye),Gap ΔE(eV),π */π quotients,E(2)(kJ·mol-1),βexp(10-30 esu),calculated hyperpolarizability βtheo with M062X/cc-pvdz.(10-30 esu)

We note,for molecules 1c,2c,3c and 4c,a lengthening of the central bonds due to the substitution of molecules by different groups.The more bulky groups,the longerRc═cdistance and the larger β value.

On the other hand,although it has been reported in the literature[48-50]that the occupation π*/π quotients increase together with the bond lengths,this fact is not noticeable in our results and lead us to suggest that the central bond length is not a good indicator for the intra-molecular charge transfer(ICT).Nevertheless,we note a shortening of the central bond in molecule 1b(for which a low β value is noticed)since in this case it is a triple bond while in all other cases the central bond is a double bond.

The most representative parameter of the lntramolecular Charge Transfer(ICT)is by far the HOMO-LUMO energy gaps ΔE[51-54]which in turn is closely related to the first hyperpolarizabilities β.To confirm this later fact,we plotted the experimental values of the eight moleculesvs.the corresponding calculatedΔEand we found outafaithfulcon firmation since the plotted graph(Fig.2)with a regression coefficient of0.9015.

This properly suggests that weaker energy gaps give rise to better ICT.

The BLA is the calculated average between adjacent bond lengths differences in the chromophore.So far,the relationship between this entity and the first static hyperpolarizability has been considered to predict accurately the structure/property relationship,although,N.A.Muruganet al.[55]have reported recently that this averment is limited to the no- field,vacuumcase.From Table 4,we observe for molecules 1b,1c,2c,3c and 4c a decrease in BLA values when those of first hyperpolarizabilities increase.This means that a strong electron delocalization generates a weak BLA value[11].This is correctly reproduced in Fig.3,for which a regression coefficient of 0.9826 is obtained when BLA is plottedvs.ΔE.

On the other hand,concerning the(π*/π)occupation quotients shown on Table 4,they yielded many useful information.Particularly for molecules 1a,2a and 3a for which the occupation quotients were calculated as the average of the benzene double bonds occupations,a reasonable consistency between(π*/π)values and the electron donating/acceptor groups power was noted;the more powerful groups,the larger(π*/π)values.Analog behaviors are observed for molecules 1b,1c,2c,3c and 4c.Moreover,Fig.4 shows that(π*/π)values progress in the same way as the hyperpolarizabilities with respect to the energy gaps.The regression coefficient for the graph(π*/π)vs.ΔEis of 0.9967.

Fig.2.Relationship between βexp and calculated with…energy gaps corresponding to the eight considered molecules with M062X/cc-pvdz.

Fig.3.Relationship between the calculated with….BLA and the energy gaps(ΔE)for molecules 1b,1c,2c,3c and 4c with M062X/cc-pvdz.

The last parameter to study is the delocalization energyE(2).This parameter helped us to identify precisely the delocalization mechanism in the considered molecules.We noted for all molecules that mainly four types of delocalizations occur;the first one when the donor group donates electrons to the chromophore(LP(donor)π*(chromophore)).The second one,occurs within the chromophore in the spontaneous direction(from the donor side to the acceptor one)and is identified as π(chromophore)π*(chromophore).While the third delocalization consists in the charge transfer from the chromophore towards the acceptor group:π*(chromophore)π*(attractor).Up to here,our reasoning is consistent with that of F?rster[56,57]proving that when the donor group is excited,the resulting excited charge moves towards the attractor group making the donor group positive and the attractor group negative according to the following scheme:

Fig.4.Relationship between π*/π quotients and the calculated with… energy gaps(ΔE)for molecules 1b,1c,2c,3c and 4c with M062X/cc-pvdz.

NBO calculations(Table 2)show,for molecules 1a,2a and 3a,more importantE(2)energies from D A than those corresponding to the reverse direction(A D).In fact,for molecule 1a,2a and 3a the delocalization energies are of 172.8,174.9 and 178.2 kJ·mol-1in the forward direction respectively while values of 91.21,88.3 and 86.6 kJ·mol-1are registered for the opposite direction.This in turn leads us to assume that the donor is intrinsically stronger than the attractor in the case of this chromophore.In the same way,for these three molecules,more importantE(2)values are observed for LP(donor)π*(chromophore)compared to those observed for π (chromophore)π*(attractor).Values of250.6,143.93 and 256.5 kJ·mol-1are calculated for the former and of 90.37,118.4 and 136.4 kJ·mol-1for the later.Regarding,the five other molecules;theE(2)in one direction or in the other are slightly different.

Finally,it is noteworthy that an agreement is recorded between βexpand βtheo(Fig.5),the regression coefficient for the graph βexpvs.βtheois of0.8434 suggesting that some further parameters have to be improved,either in the method itself or in surrounding middle of the considered molecules.

Fig.5.Relationship between βexp and βtheo corresponding to the eight considered molecules with M062X/cc-pvdz.

4.Molecular Electrostatic Potential Surface Analysis(MEPS)

The Molecular Electrostatic Potential Surfaces of the eight molecules have been calculated for the ground and the excited states.The results are shown in Fig.6.The method is helpful when determining the polarity of a given molecule[58]and it also provides information about the reactivity taking into account the electrophile and nucleophile sites[59].The MEPS have been calculated using a value of0.02 u.a for the isodensity.On the figure,the red color is representative for a negative charge,while blue refers to a positive charge and green to neutral charge[60].According to Fig.6,the positive charge is concentrated on the donor group and the negative charge on the acceptor group,the chromophore and the hydrogen atoms are neutral.This means that the electron delocalization causes a depletion of the donor group and an enrichment of the acceptor group in negative charge.Nevertheless,for molecules 1b and 1c we have seen an apparent negative charge on the donor group,this can be explained by the fact that only one anti-bonding doublet belonging to the oxygen atom is involved in the delocalization process.This is consistent with the aforementioned F?rster's affirmation.

5.Electronic Transitions and Solvent Effect

Fig.6.MEPS of the eight molecules with M062X/cc-pvdz.

Table 5Maximum absorption wavelengths;experimental λmax exp.(nm)and calculated using TD-M062X/aug-cc-pvdz with IEF-PCM λmax(nm),the corresponding oscillator strengths ? and excitations contributions(%)

All results concerning vertical transitions such as their maximal wavelength λmax,oscillator strength as well as their excitation percentage in different solvents for seven molecules are gathered on Table 5.Calculations corresponding to this section are carried out using TD-DFT method[61,62]combining the M06-2X functional with the aug-cc-pvdz basis set with convergence criteria equal to 6.Structures have been all first optimized in different solvents;those used in experiments[38],using the IEF-PCM model.From Table 5,deviations bet weenλmax(exp)andλmax(theo)from4 to 54 nm are observed.In the same way,the plotofthe experimentalλmax(exp)vs.those calculatedλmax(theo)(Fig.7)and gives a regression coefficientof0.82.Here also,a low failure of calculating methods is raised suggesting that additional parameters should be taken into account.

Fig.7.Relationship between experimental and calculated λmax with TD-M062X/aug-ccpvdz(IEFPCM).

Nevertheless,the role of the solvent is often very important in determining some physical properties of molecules and when experimental data are lacking,it is useful to have available a way which solvent is appropriate for a molecule.Therefore,the second part of this section is devoted to find an explanation and confirm the reason that led the authors of ref.[38],to choose specifically a given solvent for a given molecule.For this purpose,we computed the ΔGsolv(solvation Gibbs Energy),the λmaxand the oscillator strength in different solvents using the SMD solvation model[63],in the cases of molecules 2a,2c and 4c.The corresponding results are reported in Table 6.Actually,results demonstrate a good consistency between the choice of specific solvents in ref.[46]and the calculated values of ΔGsolv.As a matter of fact,ΔGsolvhas been calculated for each molecule in different solvents and the lowest values have been found to correspond to the solvent considered in experiment.For example,in the case of molecule 2a,acetone yielded a ΔGsolv=-54.40 kJ·mol-1,which the lowest among those calculated for all other solvents.In experiment acetone has been chosen as a specific solvent for this molecule.Similarly,the lowestΔGsolvvalues compared to other solvents have been registered in acetone for molecules 2c and 4c;-74.68 and-72.59 kJ·mol-1respectively.We conclude that the thermodynamic parameter ΔGsolvgives a quite accurate picture on the choice of the specific solvent over the frequently used settings which involve solvent dielectric constants[64],solvent polarity[65,66],bathochromic shifts[67]or oscillator strengths[68].The ΔGsolvvalue is evidently directly related to the solubility of a given molecule in a given solvent.

6.Conclusions

This paper deals with computing the static first hyperpolarizability β(0)for eight push-pull molecules.The first results are related to the most suitable basis set with MP2 method in determining the magnitude of this parameter.Considering the computing time and the reliability in producing the closer values to experiments,the cc-pvdz basis set has been definitely selected as the most appropriate.Then,the goal was to constrict the number of reliable functionals yielding the most accurate results in this field.For that purpose,M06-L,B3LYP,PBE0,M06-2X,CAM-B3LYP and M06-HF have been used.Owing to the different XC amounts these functional incorporate,significant deviations from one method to the other were obtained.It was demonstrated that as long as the value of the hyperpolarizability is low,CAM-B3LYP,M06-2X and even B3LYP give satisfactory results compared to experiments.However,for larger NLO responses,the M06-HF performs better but the magnitudes of the deviations remain unpredictable.While M06-2X and CAM-B3LYP give acceptable deviations consistent with those obtained with MP2 and closer to experiments.The M06-2X was used for further purposes in the paper.

On the other hand,calculations of BLA as well as of NBO parameters such as π*/π quotients andE(2)revealed that these parameters are closely related and evolve exponentially over the HOMO-LUMO energy gaps.In addition,a very useful information con firming F?ster's affirmation could be obtained fromE(2)values;it comes down to the fact that the conjugation moves from the donor group towards the attractor then in the reverse direction.This fact was confirmed by MEPS calculations.

Incidentally,TD-DFT calculations were performed in order to calculate the λmaxfor the height molecules in a concern to compare them with the experimental ones.When the M062X combined with aug-cc-pvdz basis set is used,relatively appreciable results are achieved.

Table 6ΔG solv(kJ·mol-1),λmax(nm)and the corresponding oscillator strength ? calculated withTD-M062X/aug-cc-pvdz with IEF-PCM in different solvents for molecules 2a,2c and 4c

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