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. 2019 Dec 27;5(1):518–528. doi: 10.1021/acsomega.9b03063

Linear and Nonlinear Optical Properties of Functional Groups for Conjugated Polymers. Analysis of the Acceptor–Donor Pair Substituents of Benzene: The Case of meta-Nitroaniline

Cesar A Guarin †,*, Javier Alejandro Díaz Ponce ‡,*
PMCID: PMC6964300  PMID: 31956798

Abstract

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The nonlinear optical (NLO) properties of meta-nitroaniline (m-NA) are evaluated via Hückel–Agrawal’s approximation in a solvent environment. In this context, both the 1B and the intramolecular charge transfer (ICT) electronic transitions are considered. The benzene ring currents on the clockwise or counterclockwise direction and the corresponding Brillouin zone from 0 to π are also considered. Besides, the Bloch equations were applied to a single cell n = 1 defined on the benzene ring. We have considered that the light beam was directed along the ring benzene bonds of m-NA; this topological hypothesis changed the crystal structure to a linear chain and the calculated optical properties were found near the experimental ones. In addition, the Fermi’s golden rule was applied to the crystal state and then the calculated refraction index of m-NA had an error of less than 3% of the experimental one. On the other hand, the molar absorptivity ε of m-NA in acetonitrile for the 1B and intramolecular ICT transitions was experimentally determined to be 11 981 and 1192 L mol–1 cm–1, respectively. With this methodology, we found that the change of the charge in the NO2 group has also a strong influence on the linear and NLO properties. In addition, the dipole transition moments, which are originated from the carbon between the carbons joined to NO2 and NH2, are mainly involved in the NLO properties. Thus, the first hyperpolarizability βz was 1.69 × 10–30 esu at λLaser = 1064 nm, 27% of the experimental value. We attribute this difference to the evaluation of the excited dipole moment. If we attribute a separation of charge of 0.1 e in the excited state, the new dipole moment allows for the simulation of the experimental value. Besides, the calculated value of χ(3) for m-NA in a solution of acetonitrile is 2.9 × 10–13 esu at λLaser = 1064 nm, 158% of the experimental value. The discrepancy between these values is attributed to the influence of the electronic correlation effects, that is, because of resonance structures of the aromatic ring and the zwitterionic pair of nitro and aniline. Besides, we have also evaluated the second hyperpolarizability γ, the second-order susceptibility χ(2) of m-NA and their values have similar differences to the experimental values. This type of approach is important because it reduces computing time and gives insight into the molecular causes responsible for linear and NLO properties in this type of functional groups, which can be used as building blocks in more complex polymer systems.

Introduction

The benzyl moiety is a group that can be added to polyenyne or to the polyene chain to improve the nonlinear optical (NLO) properties of these compounds because the electronic charge can be delocalized in the benzene ring1 with an increase of the conjugation. This increase of the conjugation produces a decrease of the band gap and then an enhancement of the values of the linear and NLO properties. In this way, it is convenient to analyze the spectroscopic properties of benzene, particularly the effect of the disubstituents on its optical properties with the methodology of Hückel–Agrawal’s approximation applied to a linear conjugated closed chain.2−6 In principle, a special feature of benzene is the presence of ring currents,7,8 that is, circular currents in the benzene, which induce a shift in the signal of NMR when a magnetic field is applied. Thus, the Brillouin zone extends from 0 to π in the reciprocal space because the electrons make a higher frequency circular movement around the ring. Then, a pseudoband structure is produced with the variation of θ, with the highest occupied molecular orbital (HOMO) as a pseudovalence band v and with the lowest unoccupied molecular orbital (LUMO) as a pseudoconduction band c. In this context, in the notation of the direction of the ring currents, we have used the Canh–Ingold–Prelog system9 to assign the priority to the substituents NO2 and NH2 of the benzene ring. So the carbon joined to the first-priority group, in this case NO2, is taken as a reference, and the second carbon in hierarchy is the carbon bonded to the group NH2, with only a carbon between both carbons. As a natural fact, the circulation of the ring current can occur from the substituent selected of the first hierarchy to the substituent of the second hierarchy in the shortest path and this direction is named right-rotatory. The opposite direction, that is, the longest way to the substituent of the second hierarchy, is named left-rotatory.

In the evaluation of the optical properties of m-NA, it would be expected that the border electronic interactions at θ = π produce a higher electron–electron interaction and thus a decrease of the linear and NLO properties of this compound. Also, the correlation repulsion effects are expected to go up at θ = π. In this way, an evaluation of the linear and NLO properties disregarding the border and correlation effects would overestimate their values. On the contrary, in the case of the substituted aromatics such as nitroanilines (NAs), it is assumed that a higher number of stable resonance structures with charge separation increase the conjugation and, therefore, the optical properties. The larger amount of resonance structures in ortho-nitroaniline (o-NA) and para-nitroaniline (p-NA) than in m-NA9 indicates that their optical properties must be higher than in m-NA. However, the lesser electronic interaction of the resonance structures of m-NA would decrease the Coulombic on-site electron–electron repulsion potential U,10 which counterbalances this decrease of the optical properties.

In relation to the spectroscopic features of benzene, this compound has three band maxima:11−13 (177 nm 7.0 eV, 1B or lE1u transition); 203.5 nm (3.04 eV, 1La or lB1u transition); 260 nm (2.38 eV, 1Lb or lB2u). These transitions have associated a resonance energy β, which is larger at higher energy of the band maxima. This resonance energy β has a crucial role in the evaluation of the linear and NLO properties. It is important to remember that: (1) the electronic transition 1B is symmetry-allowed14 with no resonance and (2) the next lower energy 1La transition is produced by the resonance energy of the benzene ring.3,15,16 The lowest energy 1Lb transition is produced by ring-distorting vibrations of type e2g14 which diminish the band gap. In addition, the effects of these transitions on the optical properties of benzene are given by the proportion of the molar extinction coefficients:14,17,18 ε(1B) = 66 000, 68 000 L mol–1 cm–1, ε(1La) = 6600 (7400), 8800 L mol–1 cm–1, ε(1Lb) 110 (200, 250) L mol–1 cm–1. Thus, the first band with a transition in 1B has the highest oscillator strength and then this band is the most important band for benzene’s optical properties. Besides, it has been found that if the benzene aromatic ring presents a pair of donor–acceptor intramolecular charge transfer (ICT) substituents, the NLO properties increase because of a higher charge delocalization. In the case of NAs, the third-order NLO properties increase by a factor of 1.4 in the case of o-NA, 1.2 in m-NA, and 2.6 in p-NA in relation to nitrobenzene (γ = 5.7 × 10–30 esu).19 Meanwhile, the dipole moment in these compounds is increased by a factor of 1.03, 1.2, and 1.5, respectively, with respect to nitrobenzene (μ = 4.0 D in dioxane). Thus, the charge redistribution, which produces the variation of the dipole moment, changes less than the NLO properties in o-NA and p-NA. The higher increase of the NLO properties indicates that the polarization is rather local.19 In this fashion, the NLO properties in these compounds are more sensitive to slight band structure dispersions.20 On the other hand, the same proportionality factor (1.2) in the augmentation of the dipole moment and the NLO properties in m-NA predicts that the correlation effects are not as important in this compound.

In this work, we have analyzed the conjugation present in m-NA and the importance of its spectroscopic bands on the linear and NLO properties in the Brillouin zone. The scope of this work is to analyze the optical properties of benzene and its substituent in meta by Hückel–Agrawal’s approximation, and then, simulate the polyenine and polyene polymers with a benzene ring with a substituent in ortho, meta, and para positions.

Hückel–Agrawal’s approximation allowed us to obtain the energies, resonance energies β, and wavefunction coefficients.4,5,21 The aim was to gain a physicochemical insight of the background of the NLO properties of m-NA, which is a question not answered directly, for instance, by ab initio methods.20 On the other hand, NAs have resonance zwitterionic structures22 with charge separation and, hence, an exciton arises and also correlation effects must be considered.6 In principle, it is known that the ICT effect is not present in m-NA via resonance as in o-NA and p-NA19 because the zwitterionic structures of the amine and nitro functionalities are not correlated in m-NA as seen in Schemes S-1 and S-2 of the Supporting Information. Thus, the resonance of the NH2 group to the aromatic ring does not include a resonance from the nitrogen of the group NO2 as seen in Scheme S-1. Indeed, the resonance of the aromatic ring to the NO2 group is not correlated because it causes aromatic instability as seen in Scheme S-2. Then, the ICT from the NH2 to the NO2 group is done via a dispersion of the charge, not by a type of resonance. Bearing this in mind, we have found that the second harmonic generation (SHG) NLO properties are underestimated by this Hückel Agrawal’s method and that the χ(3) NLO properties are higher than the experimental ones as explained below.

Results and Discussion

Energies Calculated for m-NA

As shown in Figure 1, all the energies are constant throughout the Brillouin zone. The actual separation depends on the value of β2 because the normalized energy is ζ = E/β2, where β2 = 5.603 eV for the 1B transition and 3.470 eV for the ICT transition.

Figure 1.

Figure 1

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Normalized energies ζi of m-NA.

Wave-Function Coefficients and Transition Moments

As observed in Figure 2 for m-NA, all the real wavefunction coefficients for the LUMO are different in the Brillouin zone from 0 to π. In this case, 0 indicates a very high displacement because of the high-frequency movement around the benzene ring induced by the ring currents (right-rotatory or left-rotatory).

Figure 2.

Figure 2

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Variation of the real wavefunction coefficients for the LUMO in the m-NA in the Brillouin zone.

As observed in Figure 3 for m-NA, the high dispersion of the π-charge in the HOMO (energy 5) is changed to high charge concentration in the LUMO (energy 1); this produces a decrement of the linear and NLO properties, as seen below. This is explained at length in section S-2.3 of the Supporting Information. Another feature is that the nonzero coefficients of the LUMO (energy 1) are the coefficients c4, c6, and c11. This implies that, for instance, the transition moment Ω51 takes as reference the carbon C5 in the right-rotatory R1C5 current direction (see eq 12 in the Computational Methodology section)

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Figure 3.

Figure 3

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Charge distribution normalized by ci2 for the (a) HOMO (energy 5) and (b) LUMO (energy 1) for m-NA.

Then, the change of the charge distribution in the group NO2 (c4), the carbon C6 (c6), and the group NH2 (c11) in an electronic transition from the HOMO to the LUMO is crucial for determining the linear and NLO properties. In particular, the change of the charge distribution in NO2, i.e., the change of the value of the coefficient c4 from the HOMO to the LUMO is the highest. Then, the change of electronic distribution of the group NO2 represents the most important contribution to the linear and NLO properties.

As seen in Figure 4, the real and imaginary parts of the interband transition moments from the carbon 5 of m-NA between the HOMO (energy 5) and LUMO (energy 1) are higher in the left-rotatory current L1C5 than for the right-rotatory current R1C5. On the other hand, the corresponding difference of the imaginary parts of the intraband transition moments of these currents are similar as seen in Figure S-2 in the Supporting Information. As seen below, the higher shift of the interband transition moment in the left-rotatory current produces a stronger increment of the intraband χ(3) values as observed from eqs 19 and 20 (found in the Computational Methodology section).

Figure 4.

Figure 4

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Real and imaginary parts of the interband transition moments for m-NA in the Brillouin zone for the (a) right-rotatory and (b) left-rotatory currents (see text) disregarding the energy degeneracy.

Linear Optical Properties

We obtained the linear susceptibility χ(1), the molecular polarizability αP, and the dielectric constant for the solution of m-NA with acetonitrile at Wp = 0.0648, which are plotted, respectively, in Figures S-3, S-5, and S-6 of the Supporting Information. We also determined experimentally the molar absorptivity ε of m-NA in acetonitrile for the 1B (11 981 L mol–1 cm–1) and ICT transitions (1192 L mol–1 cm–1). We have simulated these values as seen in Figure S-4. It was necessary to iterate the value of Γ (damping of the optical transition) in order to obtain the correct values of ε. Then, it was possible to determine the value of the relaxation time τR (computed from the phononic factor, η) of these transitions, normalizing with the experimental UV spectra as in ref (6). Thus, the values of η were 0.011 (231 nm) and 0.0105 (373 nm). The relaxation times of m-NA, o-NA, and p-NA are reported in the Supporting Information.

We also simulated the dielectric constant of crystalline m-NA, which is given in Figure S-9. It was necessary to apply the degeneracy factors obtained by the Fermi’s golden rule, as explained above. We have also simulated the refraction index for five experimental data as seen in Table S-1 and Figure S-10 and the error was less than 3% and we consider that this result supports the hypothesis of weighing by the normalized oscillator strengths of the 1B and ICT electronic transitions for χ(1), βz, and χ(3). On the other hand, we analyzed the shift of the derivative of the linear susceptibility χ(1) in the Brillouin zone as was done in ref (6). We found that both the real and imaginary parts of χ(1) for the 1B transition change in a constant way throughout the Brillouin zone as seen in Figure 5 for the left-rotatory current L1C5. This behavior is observed although the real and imaginary parts of the transition moment Ω15 were not constant in the Brillouin zone as observed in Figure 4b, because |Ω15| was constant. Then, the evaluation of χ(1) through eq 13, at λLaser = 231 nm (where the 1B transition happens), with both |Ω15| and band gap constant (see Figure 1), implies that χ(1) changes in a constant way throughout the Brillouin zone for the 1B electronic transition. However, the linear susceptibility changes with the wavelength as seen in Figure S-3 of the Supporting Information.

Figure 5.

Figure 5

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Variation of the derivative of χ(1) in the Brillouin zone for the ring current L1C5 at λLaser = 231 nm (1B transition).

NLO Properties

The overall NLO properties of the m-NA are reported in Table 1 for the transparent region and for λLaser = 1064 nm. The values of χ(3) for the transitions from the six carbons of benzene where the unit cell begins are tabulated for the right-rotatory and left-rotatory directions.

Table 1. NLO Values for m-NAa.

  carbon taken as reference for the beginning of the unitary cell
right-rotatory 5 6 7 8 9 10 average % exp
λ = 1064 nm Re[χ(3)] 10–13 esu –0.00 –11.71 –6.23 –0.11 –0.07 –0.01 –3.02  
  |χ(3)| 10–13 esu 0.00 13.30 7.07 0.12 0.08 0.01 3.43 190
  %|χ(3)|j 0 65 34 1 0 0    
ω = 0 s–1 |χ(3)| 10–13 esu 0.00 1.62 0.86 0.02 0.01 0.00   23
  %|χ(3)|j 10–13 esu 0 65 34 1 0 0    
  %|χ(3)| (λ/ω = 0) 822 822 822 822 822 822 822  
interband |χ(3)| interband/|χ(3)| total 1591 1.69 1.62 0.18 0.64 10.86 268  
left-rotatory 5 10 9 8 7 6    
λ = 1064 nm  Re[χ(3)] 10–13 esu –7.83 –0.56 –0.48 –0.12 –0.00 3.17 –2.03  
  |χ(3)| 10–13 esu 8.90 0.64 0.54 0.13 0.00 3.60 2.30 127
  %|χ(3)|j 64 5 4 1 0 26    
ω = 0 s–1 |χ(3)| 10–13 esu 1.08 0.08 0.01 0.02 0.00 0.44 0.28 15
  %|χ(3)|j 10–13 esu 64 5 4 1 0 26    
  %|χ(3)| (λ/ω = 0) 822 822 822 822 822 822 822  
interband |χ(3)| interband/|χ(3)| total 6.1 138.50 0.31 4.57 84.05 3.24 39  
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a

The experimental value of m-NA at 1064 nm is 1.81 × 10–13 esu.24

In Table 1, it is found that the generation of the NLO properties is more distributed in the reference carbons C6 (65%) and C7 (34%) in the right-rotatory current and in the carbons C5 (64%) and C6 (26%) in the left-rotatory current. The change of importance of the carbon C7 in the right-rotatory current to C5 in the left-rotatory current implies that the direction of the current modifies the importance of the reference carbon in the generation of the χ(3) NLO properties. Indeed, as it was predicted by the behavior of the transition moments from the carbon C5 above, χ(3) is higher in the left-rotatory current. In the same context, from the different constants in the transition moment for the left-rotatory current in eq 1, constant 7 for the linear multiplication of the wavefunction coefficient c4 (i.e., c45*c41) was higher. Then, the influence of the transition moment from the NO2 entity is higher in the overall transition moment and consequently in the NLO properties. This was also predicted above by the change of charge distribution in Figure 3.

It is also seen from Table 1 that the real parts of χ(3) for almost all the reference carbons in m-NA have negative signs, indicating a defocusing behavior.23 We believe that this is due to the higher effect of the band structure through the lattice vibrations. The experimental value of χ(3) for m-NA is 1.81 × 10–13 esu in a solution at Wp = 0.05 of acetone by the method of EFISH (Electric-Field-Induced Second Harmonic Generation) with a laser of λLaser = 1064 nm.24 The calculated value in this work, taking into account the average of the right-rotatory and left-rotatory currents, is 2.87 × 10–13 esu (158% of the experimental value) in a solution at Wp = 0.0648 in acetonitrile. As supplementary data, the values for the corresponding molecular second hyperpolarizability γ are 1.75 × 10–40 esu at ωLaser = 0 s–1, and 1.89 × 10–35 esu at λLaser = 1064 nm. These values were obtained by considering a constant dielectric constant of 35.96 for the wavelengths near ωLaser ≈ 0 s–1 and 1.81 for the wavelengths near λLaser = 1064 nm, respectively,25−27 and γAcetonitrile = 0.09 × 10–35 esu28 (see Supporting Information section S-6 for the considerations to evaluate γ). The experimental value of the second hyperpolarizability γ for the same conditions is 1.2 × 10–35 esu in dc-SHG at λLaser = 1064 nm, with a 50–75% uncertainty.24 In this way, the deviation from the theoretical value (58%) is not so high.

We have also evaluated the shift of χ(3) in the Brillouin zone and found that its real part also remains constant as seen in Figure 6 for the current L1C5 (the imaginary part is negligible as seen in Table 1). Indeed, the terms s1 and s2 of the interband contribution, and s3 and s4 of the intraband contribution (see eqs 19 through 20 in the Computational Methodology section) remain constant. As observed in Figure 6, the absolute values of the intraband terms s3 and s4 are greater than the absolute values of the interband terms s1 and s2. The absolute value of the term s2 has a similar order of magnitude as s3 and s4 because of the higher values of both Ω11 and Ω55 as seen in Figure S-11 of the Supporting Information and eq 19. A larger difference of the intraband moments Ω11 and Ω55 produces a higher increment of χ(3) in the Brillouin zone. On the contrary, polyenes and polyenynes have equal intraband transitions moments6 and then the terms s2 (interband) and s4 (intraband) do not contribute to the generation of the NLO properties in those compounds.

Figure 6.

Figure 6

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Variation of the interband terms (s1, s2) and intraband terms (s3, s4) of χ(3) for m-NA (see eqs 19 and 20 in the Computational Methodology section) in the Brillouin zone for an incident laser ωLaser ≈ 0 s–1.

We have also found that in the evaluation of χ(3), the interband and intraband contributions can be additive or have an opposite sign. As observed in Table 1, the interband contribution has an opposite sign with respect to the intraband contribution to χ(3) in the reference carbon C5 in the right-rotatory direction and for this reason the interband value of |χ(3)| is 1591 times higher than the total value of χ(3). As seen in Table 1, the average interband contribution of χ(3) is higher than the total value of χ(3) by a factor of 268 in the right-rotatory direction and 39 in the left-rotatory direction. Then, the overall effect of the intraband contribution is to decrease the NLO properties in m-NA as also found by Dadsetani and Omidi.20 We consider that the annihilations are due to the interaction 1–3 of the carbons joined to the NO2 and NH2 groups. Then, the main annihilation is from carbon C5 to carbon C7 (interband factor 1591 to the total) in the right rotatory current and from carbon C7 to carbon C5 in the left rotatory current (interband factor 84 to the total), see Table 1. Thus, a higher annihilation is produced when the transition moment takes as reference carbon C5. On the other hand, for the eight permutations of the laser wavelength in the third harmonic generation (THG) experiment at 1064 nm, the contribution in the NLO properties is:29 (1)+ω, +ω, +ω (0%); (2) +ω, +ω, −ω (1%); (3) +ω, −ω, +ω (1%); (4) (SH) +ω, −ω, −ω (8%); (5) −ω, +ω, +ω (0%); (6) (FH) −ω, +ω, −ω (3%); (7) −ω, −ω, +ω (1%); (8) (THG) −ω, −ω, −ω (85%) for the right-rotatory ring currents. It is observed in this list that the THG is the most important. As expected, the SHG and the first harmonic (FH) follow in importance.

In addition, we have simulated the χ(3) NLO properties of o-NA and p-NA by this methodology, and the values are 11 and 34% of the experimental ones: 7.45 × 10–14 esu for o-NA and 2.91 × 10–13 esu for p-NA. The simulated value of m-NA, 2.87 × 10–13 esu, is similar to the value of p-NA and almost four times higher than that of o-NA. We consider that both the resonance structures in the aromatic compound and the correlated electronic resonance of NH2 and NO2 to produce a zwitterionic compound induce an increment of the electronic correlations through the exchange integral K. The resulting effect increases the NLO properties of o-NA and p-NA.

In addition, we have evaluated the SHG coefficient βz using this methodology through eq 13 with the value of μee = 5.6 D obtained by this work. βz is 0.97 × 10–30 esu at ωLaser = 0 s–1 (1.69 × 10–30 esu at λLaser = 1064 nm), 15% (27%) of the experimental value,24 considering the B convention.30 Theoretical calculations using Gaussian 98 with a 3-21 G (p,d) basis set based on the finite field approach give a value of 0.67 × 10–30 esu (11% of the experimental one changing the value to the B convention)30−33 at ωLaser = 0 s–1. We consider that the lower value is due to the gas phase evaluation in the Gaussian methodology instead of an evaluation in the condensed phase,30 which is done in this work. In this way, our methodology gives a better approximation than that of Willetts et al.30 Indeed, other experimental evaluations of βz for m-NA in solution are missing from the literature, as it happens in crystalline state evaluation of χz(2), in order to have more certainty in the evaluation of βz. Thus, the theoretical value in this work for the corresponding χz(2) (in correspondence to βz) in crystalline state (see Supporting Information section S-5) is 4.14 × 10–8 esu at λLaser = 1064 nm. If we consider the effective experimental value of χz(2)34 or the experimental Lorentz approximation of βz,24 the last value is either 84 or 10%, respectively. Besides, the values of Carenco et al. of χz(2)35 normalized by Huang et al.34 give a higher value (153%). Thus, the macroscopic value of χz(2) in the crystalline state found in this work is more accurate than the parameter βz. On the other hand, if we consider the experimental value μee36 of 11.6 D, the value of βz is 5.26 × 10–30 at ωLaser = 0 s–1 (9.14 × 10–30 at λLaser = 1064 nm), 84% (145%) of the experimental one. This corroborates that the experimental value of μee is more exact. Thus, there is a higher charge separation in the functional groups NH2 and NO2 in the LUMO than that predicted by Hückel–Agrawal’s approximation. This higher charge separation in NH2 and NO2 produces an increment of the excited dipole moment μee and, consequently, in the SHG coefficient βz(−2ω;ω,ω). On the other hand, in the evaluation of χ(3), the charge separation in the molecular groups NH2 and NO2 is not important and then the values obtained by Hückel–Agrawal’s approximation are near the experimental ones, as seen above.

Conclusions

Through the analysis of Hückel–Agrawal’s model, we can evaluate the NLO properties and understand the causes behind the generation of the NLO properties through left- and right-rotatory currents circulating in the bonds of the benzene ring. In this way, the transitions from the carbon between the carbons bonded to NO2 and NH2 are more important for the generation of the NLO properties. In addition, the steadiness of the optical properties in m-NA with no resonances in the Brillouin zone does not allow a strong increment of the optical properties by changing the frequency of the laser beam. On the other hand, the border effects in m-NA have almost no importance in the evaluation of the optical properties of this compound. In this way, the macroscopic and molecular linear and NLO properties can be predicted with reasonable accuracy by this methodology.

Experimental Section

Materials and Steady-State Measurements

All compounds were purchased from Sigma-Aldrich. 4-Nitroaniline (para-nitroaniline, p-NA, 99%), 3-nitroaniline (meta-nitroaniline, m-NA, 98%), and 2-nitroaniline (ortho-nitroaniline, o-NA, 98%) were used as received. Acetonitrile (HPLC-plus, >99.9%) was used as the solvent for preparing the stock solution (8.6 × 10–4 M). Electronic absorption spectra for NAs were recorded in a UV 1800 (Shimadzu) spectrophotometer, in a 10 mm path length quartz cell. The aliquots (5 μL) were taken from the stock solution for several solutions with concentrations ranging from 2.0 × 10–5 to 1.9 × 10–6 M. Experimental values of the molar absorptivity ε were determined from these dilute solutions.

Computational Methodology

The values of the linear and NLO properties were obtained by solving the secular equation of Hückel–Agrawal’s method (see the Supporting Information) and then using their corresponding equations by the software Mathematica for the evaluation of the linear and nonlinear properties. The calculations for m-NA were done on a personal computer and the calculations for o-NA and p-NA were done on the supercomputer cluster Yoltla. In the following, we show the concepts and calculations made to determine the optical properties of the NAs.

Resonance Energy β

In Hückel–Agrawal’s approximation, the following parameters are important for the evaluation of the energies and wavefunction coefficients: (1) the resonance energy of the different bonds βj and (2) the degree of conjugation in relation to the double bond vj = βj/βdouble bond = βj/β2. In a previous paper, the values of β2 are higher than the values reported by ab initio calculations.2,37,38 We consider that β2, which is named β2C, includes correlation energies (U + V). In this way, for polyenes without considering electronic correlation effects4,5

graphic file with name ao9b03063_m002.jpg 2a

where v = β1/β2

For polyenes with electronic correlations

graphic file with name ao9b03063_m003.jpg 2b

In the evaluation of the absorption maxima of m-NA, we have analyzed the effect of the three bands of the aromatic ring. In principle, we have considered that the energy ratio of the maxima for the three bands with respect to 1B band indicates the degree of conjugation of the benzene aromatic ring for the three transitions. Thus far, we have assumed that the degree of conjugation for aromatic benzene in relation to the fixed double bond of benzene for the 1B band is vpp = βII/β2 = 177/203.5 = 0.8698, for the 1La band vpp = βII/βII = 203.5/203.5 = 1.0, and for the 1Lb is vpp = βII/β = 260/203.5 = 1.2776. On the other hand, we have hypothesized that the ICT band in m-NA has an influence on the NLO properties given by the proportion of its oscillator strength fi in relation to that of the 1B band. The influence of the ICT transition is due to the resonance structures associated to the charge transfer from the donor NH2 to the acceptor NO2. Besides, for simplicity in the evaluation, we have taken the groups NO2 and NH2 as molecular entities to avoid the complex zwitterionic structures as seen in Schemes S-1 and S-2 of the Supporting Information, which can produce a complex analysis of the optical properties. Indeed, the bond length of the atom N in the group NO2 with the adjacent carbon is higher than a single bond in NAs (common single bond length C–N = 1.395 Å, common double bond length in C–N = 1.355 Å, bond length of C–NO2 in m-NA = 1.4666 Å).39,40 Thus, the ICT effect does not change the characteristic of the single bond in C–NO2 to a double bond to form 4 π-pair bonds and then we can consider that the HOMO is the upper π-pair orbital of 3 π-pair orbitals of occupied electrons.

Evaluation of the Resonance Energies β

We have evaluated the resonance energies β for m-NA with a procedure similar to that featured in ref (6). In this case, the values of the conjugation degrees for NAs were obtained by their linear interpolation in relation to the single (s) and double (d) bond lengths (li) for β-carotene, (ls = 1.4496 Å with v = 0.7188; ld = 1.3502, vd = 1.0000, respectively).6 The data of the bond lengths for NO2 and NH2 joined to benzene were obtained from ref (40). Thus, the values of the ratios of resonance energy of the bond in m-NA are given by v4 = 0.6710, vpp = 0.8698, vr = 0.8844, related to the bonds to NO2, carbons in the aromatic ring, and NH2, respectively. Besides, we have experimentally evaluated the band maxima for the 1B transition and for the ICT transition in m-NA in acetonitrile in the same concentration as in acetone, in which the NLO properties were evaluated. The band maxima and the resonance energies β2C were 1B (λ = 231 nm, β2C = 5.603 eV), ICT (λ = 373 nm, β2C = 3.470 eV). We normalized the oscillator strengths fi reported by Bendazolli et al.41 for 1B and La and their values were 0.60 and 0.40, respectively. These normalized oscillator strengths fi were used as factors for calculating the contribution of the 1B and La bands to the linear and NLO properties, except for the evaluation of the molar absorptivity ε, where the experimental ratio of absorbance was used.

Secular Equations and Wavefunction Coefficients of NAs

The secular equations and the equations of the wavefunction coefficients for m-NA are given in the Supporting Information section S-2, bearing in mind the notation given in Scheme 1, where the ring current in the right-rotatory direction from carbon 5 is labeled R1C5.

Scheme 1. Numbering of the Atoms of m-NA in Which NO2 and NH2 Are Taken as Molecular Entities.

Scheme 1

Open in a new tab

Levine has explained that as the Hückel approximation does not consider the electronic repulsion, the calculated energies are not the experimental energies,3 but rather they are ideal energies. Nevertheless, we have related the experimental absorption band maxima to the evaluation of the resonance energies β as in ref (6) and then the values of β implied that the gap between energies β ζΗΟΜΟ and β ζLUMO in Figure 1 matches the experimental value. It is important to clarify that this benzene Hückel–Agrawal’s approximation does not consider (1) the electronic repulsion between electrons in the on-site bond, (2) the electronic correlations because of the interaction of the electrons on the near bond, and (3) the border effects in the Brillouin zone because of the finiteness of the benzene ring.

Evaluation of the Coulombic Electronic Repulsion through the Wavefunction Coefficients

In addition, we have used the semiempirical molecular orbital theory of Pople and Beveridge42 for the evaluation of the electronic repulsion, which is given by the factors which multiply the wavefunction coefficients

graphic file with name ao9b03063_m004.jpg 3

where

graphic file with name ao9b03063_m005.jpg 4
graphic file with name ao9b03063_m006.jpg 5

where Pμν(43) is the mobile bond order which defines the importance of the electronic repulsion in the bond and is defined by

graphic file with name ao9b03063_m007.jpg 6

cμi and cνi are the wavefunction coefficients, where μ,ν are the indices of the atomic orbitals and i is the index of the molecular orbitals42 of the NAs. The values of Pμμ represent the charge density on atom μ. VAB is the interaction potential of atom B on the electrons of atom A.

γAB is originated from refs (42) and (44)

graphic file with name ao9b03063_m008.jpg 7

J is the electronic coulombic integral and K is the electronic exchange integral, and μ, ν also index the atomic orbitals. We obtain γAB with the approximation of μ = ν, that is, taking the same orbital in atoms A and B.

graphic file with name ao9b03063_m009.jpg 8

We have found that for Hückel–Agrawal’s approximation for m-NA

graphic file with name ao9b03063_m010.jpg 9

The same result was found in o-NA. For p-NA, the only nonzero values were P67,Re = P76,Re = P910,Re = P109,Re = 0.5 because the nonzero wavefunction coefficients for the HOMO orbital are c6, c7, c9, and c10. We have found that, in both o-NA and p-NA the exchange integral K was important, which enhances the electronic correlation effect. For this reason, the linear optical properties for these compounds were simulated, but not the NLO properties, as seen below.

For neutral atoms, PAA = PBB = 1 and with the assumption of VAB = ZA γAB,45 then eqs 4 and 5 for m-NA result in

graphic file with name ao9b03063_m011.jpg 10
graphic file with name ao9b03063_m012.jpg 11

Thus, the electronic repulsion terms are only present in the on-site energies and the repulsion terms are not important in the bond order of m-NA. Besides, we have found that Pμμ has constant values in the range [0.78, 1.27] throughout the Brillouin zone (see section S-3 of the Supporting Information). In this way, the effect of coulombic electronic repulsion J on the bond order Fμνm-NA is negligible in the Pople approximation, and then the resonance energy βμν only influences the mobile bond order, with a negligible interaction of the electrons in m-NA.

Transition Moment and Dipole Moment

In the evaluation of the transition moments by Hückel–Agrawal’s approximation, we have assumed that the transition moments derived from the Bloch equations for n = N unit cells can be applied to the unit cell of the benzene molecule with n = 1, being 3 for one circulation of current in the benzene ring. In Scheme 1, the carbons are numbered from NO2 (coefficient c4) and the right-rotatory moment transitions begin from carbon C5 to the following carbons until carbon C10 in a clockwise (right) direction. The counterclockwise direction in Scheme 1 in the numbering indicates the transition moment in the left direction. Besides, we have used the methodology applied in ref (6). With this background, the transition moment in a right-current which begins from carbon C5 (Scheme 1) of m-NA is from energy 5 (HOMO) to energy 1 (LUMO). In this case, we have considered that from this carbon C5 begins the unit cell

graphic file with name ao9b03063_m013.jpg 12

In addition, the other five carbons in the benzene ring can also be referenced for the right-rotatory and left-rotatory transition moments. For this reason, the transition must be averaged for the six atoms of benzene. However, it was found that the absorption spectra of the 1B transition generated from the six atoms (see Scheme 1) were similar and then the resonance energy of the double bond β2C was almost the same.

In addition, we have estimated the dipole moment of the ground-state HOMO μg through the quantum vector method. This method evaluates the dipole moments of the σ-bonds and the π-bonds to calculate the total dipole moment.3,46−50 We have hypothesized that the π-charge lies in the oxygens, and not in the nitrogen as observed in p-NA in the HOMO frontier orbital of Kaczmarek-Kedziera’s work.51 The configuration of m-NA was the experimental crystalline configuration.40 Thus, the calculated dipole moment for this compound was 4.2 D with no charge separation (Q = 0). We have adjusted the charge in nitrogens and oxygens in the σ-dipole moment and found that a charge separation of 0.01 produces the required experimental value of 4.6 as it is expected for a zwitterionic compound.52 However, the excited dipole moment of 5.60 D differed from the experimental one of 11.6 D reported by electrochromism.36 With a charge separation of 0.11, this experimental value of the dipole was reproduced. Then, it is expected that the NLO properties given by the SHG coefficient βzzz be lower than the experimental one as seen in eq 13,53 with B convention.30 βz = βzzz because the third-order tensor becomes a zero-rank tensor (a complex number) where the direction of the dipole moment is considered the main axis. The notation would be β instead of βz, in accordance with the polarizability α and the second hyperpolarizability γ for having a one-dimensional chain; but in the literature, βz is used more often. In the case of the non-centrosymmetric m-NA, we consider the absolute values of the dipole moments. The approximation of m-NA as a linear chain, with the ring current going along the bonds of the aromatic ring, implies that the approximation of absolute values in the transition and dipole moments be more valid, see below

graphic file with name ao9b03063_m014.jpg 13

where fe is the normalized oscillator strength of the absorption band for the excited state, μgg and μee are the ground and excited moments, ω is the frequency of the laser beam, ωeg and μeg are the excitation energy and the transition moment of the 1B and ICT electronic transitions.

We emphasize that eq 13 results from considering a two-level model, where the 1B electronic and ICT transitions take part in a proportion given by the normalization of the oscillator strengths, see above (the equations for the evaluation of βz for the Hückel–Agrawal approximation are given in the Supporting Information section S-5).

Linear Optical Properties

The evaluation of the linear optical properties is described elsewhere.4,5 Thus, in section S-4 of the Supporting Information, an analysis of these properties is given. We used the following equation of the linear susceptibility χ(1) (esu) in the description of the linear optical properties of m-NA where the degeneracy factors gc and fvc are consistent with the Fermi’s golden rule (FGR),54 see below. More importantly, the FGR was used when the material was in crystalline state and not applied when m-NA was in solution in the evaluation of the transition moments, linear and NLO properties, because the band structure increases the dipole moments.

graphic file with name ao9b03063_m015.jpg 14
graphic file with name ao9b03063_m016.jpg 15

where ωcv is the frequency of the band gap and is given by

graphic file with name ao9b03063_m017.jpg 16
graphic file with name ao9b03063_m018.jpg 17
graphic file with name ao9b03063_m019.jpg 18

σ is the perpendicular surface density of a unit cell (molecule/cm2). V is the volume of the compound. N is the number of unit cells, which include the aromatic ring and the substituents and then N = 4.5. The value of σ depends on the physical state (crystalline, liquid, or solution) and this also influences the value of χ0(1) (or χ(2) and χ(3)). We must clarify that the value of σ also depends on the linear or NLO properties. Then, there is a linear optical perpendicular surface density σχ(1) and a NLO perpendicular surface density σχ(3) as it was tabulated in ref (6). In this way, the linear surface density σχ(1) of a solution of m-NA in acetonitrile at Wp = 0.0648 is 2.8 × 1013 molecules/cm2.

For its part, Γ is the damping in the optical transition55 with Γ = 1/τR (s–1), τR is the relaxation time of an excited state. With these values we can evaluate the phononic factor η = ℏ/β2/τRa. We have assigned the factor 1/2 in eq 14 because of the resonance phenomena as Ward, for example, has assigned56 because of a Lorentzian behavior. In this work, in order to calculate the values of χ(1) and χ(3) of the single aromatic compounds, the limits of integration of the Brillouin zone were cut down from (−π/a, +π/a) to (0, +π/a), that is, the lower limit begins in the unit cell of benzene constituted by a hypothetic single and double bond and continues to infinity because of the ring current, as commented above. With this assumption, we have not considered the border interactions at the end of the unit cell, which diminish the NLO properties because of correlation effects (increment of the electronic interactions). Besides, ζ is a dimensionless variable which needs to be iterated to cover the UV wavelength range.

Because we have considered that m-NA is a linear chain, the second-rank tensor χxx,interband(1) becomes a zero-rank tensor χxx,interband = χ(1) which is a complex number. The same applies for the molecular polarizability α, which is evaluated as a function of wavelength in the Supporting Information section S-4.2 considering the electronic transition 1B and the ICT electronic transition.

Remarkably, the influence of the dielectric constant of the solvent in the solution over χ(1), χ(2), and χ(3) of m-NA is determined by the normalized oscillator strength fi and wavelength of the absorption peaks for the 1B and ICT electronic transitions. The same applied to the polarizability α, first hyperpolarizability β, and second hyperpolarizability γ.

χ(3) NLO Properties

We have also evaluated the NLO properties of m-NA with eqs 16–23 with the values of β2C obtained by fitting the experimental absorption maxima.

For the evaluation of χ(3) (esu) at ωLaser = 0 s–1 (transparent region) the following expressions were used4

graphic file with name ao9b03063_m020.jpg 19
graphic file with name ao9b03063_m021.jpg 20
graphic file with name ao9b03063_m022.jpg 21

where

graphic file with name ao9b03063_m023.jpg 22
graphic file with name ao9b03063_m024.jpg 23

The number of unit cells N is in this case three for the benzene ring in eq 18, because the NLO properties depend mainly on the currents circulating around the aromatic ring. a is the length of the unit cell (cm) associated to the static single and double bonds of benzene. On the other hand, the prefactor χ0(3) changes to χ0 = 8e4σa3/π/β23 when eq 16 and the expression Ωijea are used. As explained above, the FGR is applied to the crystalline state because of the band electronic structure, which increases the dipole moments, then gc and gv can differ from 1 in eqs 19 and 20. Besides, the effect of the dielectric constant of the solvent in the solution is taken into account via the intensity and energy of the peaks of the UV absorption spectrum in the solvent, as explained above. On the other hand, we have taken into account that σχ(3) for m-NA is 1.99 × 1014 molecules/cm2 in the solution (typically, in NLO-determinations are used concentrated solutions) as it was calculated in β-carotene,6 which is associated to a single and double bond in a linear chain. This differs from the value for the solid crystalline state which is 5.69 × 1014 molecules/cm2, considering a density of m-NA of 1.3003 g/cm3.40b Importantly, the interaction of the laser beam is with the m-NA molecule as a linear ring in which the electronic ring current runs. Indeed, the high dipole moment present in the m-NA implies that the solvent molecules, in this case acetone,24 surround the m-NA molecule and diminish its surface density σχ(3). Thus, the value of σχ(3) for m-NA lies between the values of the PTS polyenyne (low σχ(3) = 1.36 × 10 14 molecules/cm2) and the polyacetylene polymer (high σχ(3) = 3.2 × 10 14 molecules/cm2).6

The degeneracy factors gc and gv and number of transition moments fvc and fcv for the LUMO and HOMO in eqs 19 and 20 were determined following the FGR54

graphic file with name ao9b03063_m025.jpg 24

Wvc is the rate transition from the HOMO to the LUMO, a is the length of a bond, fcv is the number of transition moments from the HOMO to the LUMO, and gc is the degeneracy of the LUMO level.

The final equations were obtained from a dimensional analysis of the FGR in relation to the cofactor χ0(3) in eq 22 and to the variable Sij in eq 23. We have also taken into account the polarization with its respective susceptibilities, which give eqs 14 and 15 from Genkin and Mednis’ analysis.57 Thus, a higher transition moment (square)58 and higher degeneracy of the energy levels increase the polarization through the increment of the susceptibilities, proportional to the square of the transition moment as expressed by the FGR.

For ω ≠ 0, in accordance to Agrawal and Flitzanis,4 and again taking into account the energy degeneracy54,57,58

graphic file with name ao9b03063_m026.jpg 25
graphic file with name ao9b03063_m027.jpg 26

“P” indicates the total permutations which are eight for a one-dimensional chain and result from the permutations of ω1, ω2, and ω3 between the values +ω and −ω (2 × 2 × 2). Besides, because we have considered that m-NA is a linear chain, the fourth rank tensor χxxxx(3) becomes a zero-rank tensor, χxxxx = χ(3), which is a complex number.

Crucially, it was realized that the simulation of the linear and NLO properties of m-NA, o-NA, and p-NA in solution has more physicochemical coherence without the application of the Fermi’s golden rule and then gc and fvc are both equal to 1. In this way, both m-NA and p-NA have similar NLO properties and the fourth part corresponds to that of o-NA.

Acknowledgments

We are thankful for the commentaries on the work of Dr. José Luis Hernández-Pozos and Dr. Luis Guillermo Mendoza-Luna from the UAM Iztapalapa. The authors would like to express their gratitude to Dr. Iraís Vera Robles from UAM (Área de Biofisicoquímica y Laboratorios de Química de Docencia, Departamento de Química-CBI-UAM-Iztapalapa) for the UV-equipment used. We are very grateful for the use of the supercomputer cluster Yoltla, which is maintained by the Laboratorio de Súpercomputo y Visualización en Paralelo (LSVD) at UAM-I. Dr. Cesar A. Guarin acknowledges CONACYT for grant-683-Catedra-CONACYT. The authors acknowledge grant SIIUAM-12301022 (Referencia i27, Programa Especial de Apoyo a la Investigación-2019) for financial support.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.9b03063.

  • Resonance structures m-NA, secular equations m-NA, Pople coefficients, linear optical properties, evaluation of βz, and second hyperpolarizability γ (PDF)

The authors declare no competing financial interest.

Footnotes

a

In a previous article, we have used the relation η = ℏ/β2/τR/2; then, the values in this work correspond to the common notation in the literature.

b

In a previous article,6 we had the same situation with the films of the semiamorphous polyenyne PHDAz, which was obtained experimentally and the value of the absorption coefficient α (cm–1) allowed to determine the value of σχ(1) = 1.2 × 1012 molecules/cm2. The semiamorphous film of this polyenine had crystalline regions and can be compared to a solution, because the amorphous regions can be considered as the solvent and the crystalline regions as the active sites. In this way, in the evaluation of χ(3)for PHDAz by Hückel–Agrawal’s methodology, it was necessary to take into account a value of σχ(3) of 1.36 × 1014 molecules/cm2, a value corresponding to the crystalline parts of the polyenine PHDAz.

Supplementary Material

ao9b03063_si_001.pdf (745.6KB, pdf)

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