Nonlinear Optical Properties of Stacked Conjugated Systems

ARTICLE pubs.acs.org/crystal

Nonlinear Optical Properties of Stacked Conjugated Systems Published as part of a virtual special issue on Structural Chemistry in India: Emerging Themes. Simil Thomas,† Y. A. Pati,† and S. Ramasesha*,† †

Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, India 560 012 ABSTRACT: We study linear and nonlinear optical properties of two pushpull polyenes stacked in head to head (HtH) and head to tail (HtT) configurations, at different stacking angles within the Pariser ParrPople model using exact diagonalization method. By varying the stacking angle between the polyenes, we find that the optical gap varies marginally, but transition dipoles show large variations. We find that the dominant first-order hyperpolarizability component βXXX for HtH arrangement and βYYY for HtT arrangement strongly depend on the distance of separation between molecules, while the other smaller component (βXYY for HtH arrangement and βXXY for HtT arrangement) does not show this variation with distance. We find that the βXXX for HtH configuration shows a maximum at an angle away from 0, in contrast with the oriented gas model. This angle varies with distance between the polyenes, and at large distance it falls to 0. The ratio of all components of β of a dimer to monomer is less than two for HtH configuration for all angles. But for HtT configurations the ratio of the dominant β component is greater than two at large angles. Our ZINDO study on two monomers (4hydroxy-40 -nitroazobenzene) connected in a nonconjugative fashion shows a linear increase in |βBav| without much red shift in optical gap. There is a linear increase in |βBav| with increase in number of monomers connected nonconjugatively without resulting in a red shift in optical gap.

’ INTRODUCTION Conjugated organic systems have been studied for their strong nonlinear optical (NLO) response for the last three decades.1 Large first and second order hyperpolarizability coefficients are of importance in several areas, such as telecommunication,2,3 photodynamic therapy,4 optical limiting,5 and deep UV conversion, to name a few. The important paradigm for achieving large second order polarizability is to substitute the π-conjugated system with a donor (D) and an acceptor (A) group at the two ends of the conjugation. However, such molecules possess a large dipole moment and tend to stack themselves in an antiparallel arrangement, resulting in an inversion center in the crystal. It is well-known that the macroscopic second order polarizability χ(2) vanishes in such crystal structures by symmetry. To overcome this stacking problem, several strategies have been followed, such as introducing chiral centers in the molecule.6 Theoretically, it is well-known that the second order polarizability tensor can be decomposed into irreducible tensors of rank-1, resembling dipole moment, and of rank-3, resembling octupolar moment. It is then possible to think of molecules which have no dipole moment but large octupole moment. Such molecules can stack in a crystal structure with no inversion center and thus can lead to a macroscopic system with a large χ(2) response.7 Although this is an interesting paradigm, crystals of octupolar molecules so far studied do not show large χ(2) responses. Another paradigm for achieving parallel alignment of the NLO phores is chemical and involves linking individual moieties by a spacer group such as CH2. Calix[n]arenes are one such system in which each chromophore consists of a π-system such as benzene r 2011 American Chemical Society

with a donor and an acceptor group at the para-position. The molecules are held fixed by the CH2 group, and the intermolecular angle can be varied by changing the donors and acceptors. Here the molecular second order polarizability β can be increased by increasing n. Kelderman et al. have synthesized calix[4]arenes with different donor and acceptor groups.8 The cone shaped calix[4]arenes with all the four donors as NO2 groups gave the maximum β values without much red shift in the absorption maximum, λmax. Furthermore, they also found that the β value is 2035% less than the vectorial sum of the β values of the four monomers.9 The dependence of β on the orientations of individual NLO phores in a calix[n]arene has been studied theoretically by Datta and Pati, within the DFT (density functional theory) method and the MollerPlesset (MP2) method.10 The orientation dependence on β has been experimentally studied on alkenyl and alkynyl expanded calix[4]arenes.11 It is well-known that the first order hyperpolarizability coefficient increases superlinearly with increase in conjugation of the system12 for system size less than the π coherence length. But this has the disadvantage of causing a red shift in the absorption band. For electro-optic applications in telecommunications, the material should exhibit a large β without a large red shift in the λmax. This is because a small λmax implies a significant tail in the absorption spectra in the telecommunication wavelength region 1.31.5 μm. The absorption in this region results in conversion Received: January 11, 2011 Revised: February 23, 2011 Published: March 28, 2011 1846

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Figure 1. Structure of monomer (4-hydroxy-40 -nitroazobenzene) M and its nonconjugated (1,8-dihydroxy-4,5-bis(4-nitrophenylazo)-9,10dihydroanthracene) S1 and conjugated (1,8-dihydroxy-4,5-bis(4-nitrophenylazo)anthracene) (P1) dimers, linked by bridges of varying length n.

of light into heat, which is detrimental to the device. So, various methods have been proposed to overcome the nonlinearity transparency trade-off. One of the systems which show the nonlinearitythermal stabilitytransparency trade-off is derivatives of dicyano methylene pyran molecules.13 In these molecules, the β is high while the red shift in absorption wavelength is small. High β is associated with two low-lying excited states in these molecules, as they have two donor groups and one acceptor group. In another system, direct conjugation between donor and acceptor molecules is avoided to reduce the red shift in energy gap.14 In these DA substituted 1,8-dihetero arylnapthalenes, conjugation through the π-system is reduced because of the 1,8substitution. It was observed that 1,8-aryl naphthalene derivatives have λmax less than 380 nm and higher β values than their conjugated counterparts. There also exist experimental studies on the nonconjugatively connected DA system. Mihailovic et al. reported a high β value for a nonconjugatively connected donoracceptor substituted triethylene diamine.15 Schuddeboom et al. synthesized NLO chromophores with nonconjugatively connected donor and acceptor moieties (saturated hydrocarbon bridge)16 to study the dependence of the length of the nonconjugated bridge between donor (D) and acceptor (A) groups on second order polarizability, β. The β values of these DσA compounds were comparable to that of the DπA compound, p-nitroaniline. Recently, a series of nonconjugated dimer molecules, two DA substituted azobenzenes connected by a nonconjugated linkage (via two CH2 groups), have been synthesized, and the β values of these compounds have been measured.17 The second order hyperpolarizability β is found to be twice that of the monomer in the case of D = OH and A = NO2, while the optical gap changed only marginally from that of the monomer. In this paper we model the NLO properties of a pair of stacked pushpull polyenes by the PariserParrPople (PPP) model. We compute the second order polarizability β as a function of distance between the polyene chains as well as angle between them and compare these to the result from the oriented gas model (OGM).18 We also study the second order polarizability β of 4-hydroxy-40 -nitro-azobenzene (M) and its nonconjugated 1,8-dihydroxy-4,5-bis(4-nitrophenylazo)-9,10-dihydroanthracene (DHAD, S1) and conjugated dimers (P1) (see Figure 1). These systems are studied using ZINDO at the SCI level. In the next section we give a brief introduction to the methods and models. This will be followed by the Results and Discussion.

’ METHODOLOGY Our studies consist of two parts, first is the computation of the effective β values of two pushpull polyenes chains in different relative orientations and at different distances. The second study is the ZINDO study of M and its conjugated (P1) and nonconjugated dimer (DHAD). The study of the pushpull systems is carried out in the PariserParrPople model, and model exact β values are obtained using the correction vector (CV) method.19,20 The PariserParrPople Hamiltonian is given by HPPP ¼

∑iσ εiâ†iσâiσ þ h∑ijiσ tij ðâ†iσâjσ þ h:c:Þ þ ∑i 2i^ni ð^ni 1Þ U

þ

∑ Vijð^ni zi Þð^nj zj Þ

ð1Þ

i>j

where the εis are the site energies and tijs are transfer integrals taken as 2.4 eV, nîs are the number operators, zi are chemical potentials, Uis are on-site electronelectron interaction (11.26 eV for carbon), and Vijs are the intersite electronelectron repulsions. The intersite interactions are parameterized using Ohno parameterization.21 In unsubstituted polyenes, it is a good approximation to consider that all carbon pz orbitals are in the same chemical environment and choose the site energy, εi = 0, at all sites. However, in the pushpull system (DA), we will have ε1 > 0 and εN < 0 to simulate substitution by an electron pushing group at the first site (1) and an electron pulling group at the other end of the chain (N).22 For simplicity, we have assumed that |ε1| = |εN| = 2.0 eV . While the above Hamiltonian is used for modeling an individual pushpull chain, only the intersite interaction term is used as the additional term in the Hamiltonian for a complete system of two pushpull chains. For both single and double chains, the intersite interaction is parametrized using Ohno parameterization, which extrapolates the intersite potential between U, when the distance between sites is zero, and e2/r, as the distance tends to infinity. The PPP Hamiltonian can be solved for ground state and lowlying excited states, exactly for up to 16 sites. So, while it is possible to obtain the low-lying states, properties that apparently depend on all the excited states seem to be difficult to compute. This is because for a 16 site conjugated system we have 34,763,300 singlet excited states with nonvanishing transition dipoles to the ground state. Restricting to only singlets on each of 1847

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the individual 8 carbon chains still leaves us with 3,111,696 singlets. Computing all these states and dipole matrix elements is impossible with the current computing capabilities of computer clusters that are commercially available. In order to circumvent this problem, the CV method for nonlinear optical coefficients was invented in 1988.20 The perturbation expression for the second harmonic generation coefficient of a system is given by βijk ð 2ω; ω, ωÞ ¼ þ

μign μjnm μkmg N P I p2 m, n ðωng 2ωÞðωmg ωÞ

∑

μjgn μinm μkmg ðωng þ ωÞðωmg ωÞ

þ

μjgn μknm μimg ðωng þ ωÞðωmg þ 2ωÞ

ð2Þ

where |gæ is the ground state, |næ and |mæ are the excited states of the Hamiltonian, and pωng and pωmg are the energy gaps from the ground state to the excited state |næ and |mæ, respectively. i, j, k are the Cartesian coordinates, μinm is the ith component of the transition dipole moment between eigenstates |næ and |mæ of the Hamiltonian, and PI is the intrinsic permutation operator which permutes (j,ω) and (k,ω) and leads to six terms, with each of the terms in eq 2 giving rise to two permuted terms. In the CV method,20,23,24 the correction vector |φ(1) i (ω)æ is computed by solving the set of linear algebraic equations defined by ð1Þ

ðHPPP Eg ( pωÞjφi ð ( ωÞæ ¼ μ˜ i jgæ

ð3Þ

where ω is the laser frequency at which β is measured and μi is the ith component of the dipole moment. First order CV, φi(1) is formally given by ð1Þ

jφi ð ( ωÞæ ¼

Ænj~ μ jgæ

∑n pωng (i pωjnæ

ð4Þ

where |næ is the excited state wave function. From φi(1)(ω), one can directly obtain components of β by the equation ð1Þ îjk ðÆφð1Þ μj jφk ð ωÞæÞ ð5Þ βijk ð 2ω; ω, ωÞ ¼ P i ð 2ωÞj^

where Pijk is the full permutation operator, which permutes (ω,i), (ω,j), and (2ω,k). Tumbling averaged β along the direction i is defined as βhi ¼

1 3

∑j βijj þ βjji þ βjij ∑i

such a theory of intermolecular interactions does not exist for nonlinear optic properties. In our calculations aimed at studying the effect of intermolecular interactions on the second harmonic generation (SHG) coefficient, we have computed model exact SHG coefficients for two pushpull polyene chains with chain axes oriented at different angles. We have compared the SHG coefficient with the OGM of Oudar and Zyss. This model assumes that the effective SHG coefficient of a collection of molecules with known β values is just a sum of the individual β values given by βtot IJK ¼

ð7Þ

Regarding linear optical properties such as optical gaps and transition dipole moments, the exciton theory treats the dipole dipole interaction term ! μ2 3 ! μ ! ð ! μ ! μ r12 Þð ! r12 Þ H 0 ¼ 1 33 2 3 1 3 ð8Þ 5 r12 r12 as a perturbation. The lowest excited state which is 2-fold degenerate (when we are dealing with two identical molecules) is split by the perturbation H0 , and depending upon the alignment of the transition dipoles, the lowest excited state is either weakly or strongly dipole coupled to the ground state. However,

1 V

∑ijk ∑s cos θi, s cos θj, s cos θk, s βsijk

ð9Þ

where βsijk is the (ijk)th component of the SHG coefficient of the sth molecule, and θi,s, θj,s, and θk,s are the angle made by the molecular frame of the sth molecule with the corresponding laboratory frame I, J, and K. V is the volume of the unit cell. Two pushpull chains stacked at an angle θ are as shown in Figure 2. According to this equation, the XXX and XYY components are given by βXXX ¼ 2 cos3 θ βxxx

ð10Þ

βXYY ¼ 2 sin2 θ cos θ βxxx

ð11Þ

and

ð6Þ

Spatially averaged β is given by

rffiffiffiffiffiffiffiffi ! j βav j ¼ βh2i

Figure 2. Schematic geometry of the two pushpull (DA) stacks studied in this paper. θ is the angle between the chains. (a) Head to head (HtH) stacking; (b) head to tail (HtT) stacking. The transfer integrals are taken to be uniform with a value of 2.4 eV, and all bond lengths are fixed at 1.4 Å.

Restricted CI Calculations. For all our calculations on DHAD, we have used the optimized geometry obtained from the ADF (Amsterdam Density Functional)25,26 package, using the double-ζ polarized basis set. The optimized geometries of the monomer, DazobenzeneA (M), and conjugated dimer (MπM (P1)) are planar. In the case of the nonconjugated (MσM, (S1)) dimer, two monomers make a dihedral angle of nearly 33 between them. The experimental geometry of the molecule, however, corresponds to a dihedral angle of 10.17 The optimized geometry is used as an input to obtain the Hamiltonian matrix and ground state wave function within the ZINDO method.27 The basis set for NLO calculations consists of only singly excited configurations over the space of the 40 highest occupied molecular orbitals and the 40 lowest unoccupied molecular orbitals, along with the HartreeFock ground state. We have also used the multireference singles and doubles 1848


Crystal Growth & Design configuration interaction (MRSDCI) method for nonlinear optical coefficient calculations. In this case, our reference slater determinants are, along with the HF ground state, three singly excited slater determinants wherein the excitations are from HOMO f LUMO, HOMO1 f LUMO, and HOMO f LUMOþ1, and one doubly excited configuration, where both excitations are from HOMO f LUMO. This includes quadruple excitations in the basis set. These calculations have been carried out mainly to check the excitation gaps. It has been shown earlier that, for the calculation of β, single configuration interaction (SCI) calculations are size consistent for optical gaps, and hence, we expect the NLO coefficient also to be size consistent.24,28,29 NLO coefficients are calculated using the correction vector method (CV) incorporated within the ZINDO package.30

’ RESULTS AND DISCUSSION We will first present and discuss the results for two pushpull polyene chains stacked at an angle (Figure 2) and will then move on to discussing the results on the DHAD system. Stack of Two PushPull Polyenes. In this study we focus on two pushpull polyenes, each of six carbon atoms, and we study the SHG coefficient at a laser excitation frequency of 0.65 eV. The Hilbert space spanned by the system consists of 226,512 singlets. In principle, it is sufficient to take the singlets at each chain, and the restricted Hilbert space will then correspond to 30,625. We have also carried out some representative calculations on two pushpull polyenes of eight carbons each. This system spans a restricted Hilbert space of 3,111,696 singlets while the full singlet space of the 16 carbon system at half-filling is 34,763,300. For simplicity, we have considered the pushpull system to be linear. The two orientations, head to head (HtH) and head to tail (HtT), are shown in Figure 2. Our earlier studies have shown that a zigzag pushpull system has very nearly the same βXXX coefficients as the linear system.31 For the chain lying along the x-axis, the SHG tensor β has only one nonzero component, βxxx. In the laboratory frame, with the chain lying in the XY plane, making an arbitrary angle to the X-axis, we will have nonzero βXXX and βXYY for the chain in the laboratory frame for HtH stacking and βYYY and βYYX for the HtT stacking. We present the optical gaps and transition dipoles of the stacked system (in Table 1) and compare these with that of the isolated chain. For this purpose, we consider the two low-energy excitations of the isolated chains and HtT stacking. We note that, in the noninteracting limit of the stacked chains, the excited state of the composite system is doubly degenerate, since the excitations can be on any one of the molecules. Interactions lift the degeneracy of the excitations, and the splitting of the degenerate states as well as the absolute shift relative to the excitation energy of the isolated chain are small. However, the transition dipoles show an interesting variation. The lower energy states in each pair have a vanishing transition dipole for the parallel orientation of the chain while the higher energy states have a large transition dipole. As the orientation of the two chains shifts to the perpendicular geometry, the transition dipoles shift to the lower energy state of the pair. This is due to the loss of inversion symmetry of the stack when the angle changes from θ = 0 to θ = 90. While the behavior of the optical gaps of a stacked system shows small variations, the transition dipoles show large variations as a function of stacking angle. Hence, it will be difficult to predict a priori the effect of stacking on the SHG coefficient. In

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Table 1. Variation in the Excitation Energy (in eV) from the Ground State, and Magnitude of the Transition Dipole |μB1g| (in D) as a Function of the Stacking Angle θ for Head to Tail Stacking. a 2θ (deg)

E1

|μB1g|

E2

|μ B2g|

E3

|μB3g|

E4

|μ B4g|

0

3.67

0.00

3.77

4.19

4.10

0.00

4.47

11.44

10

3.69

0.66

3.76

4.30

4.14

0.90

4.44

11.40

20

3.69

1.24

3.76

4.33

4.17

1.84

4.43

11.26

30

3.70

1.78

3.75

4.30

4.19

2.77

4.42

11.04

40

3.70

2.31

3.75

4.22

4.20

3.68

4.41

10.73

50

3.70

2.82

3.75

4.09

4.20

4.57

4.41

10.35

60 70

3.70 3.70

3.31 3.78

3.75 3.75

3.93 3.72

4.21 4.21

5.41 6.22

4.41 4.40

9.89 9.35

80

3.70

4.23

3.75

3.49

4.21

6.97

4.40

8.74

90

3.70

4.65

3.75

3.23

4.22

7.67

4.40

8.07

100

3.70

5.03

3.75

2.94

4.22

8.31

4.40

7.33

110

3.70

5.39

3.75

2.62

4.22

8.88

4.40

6.54

120

3.70

5.70

3.75

2.29

4.22

9.39

4.40

5.70

130

3.70

5.97

3.75

1.93

4.22

9.82

4.40

4.82

140 150

3.70 3.70

6.37 6.37

3.75 3.75

1.57 1.18

4.22 4.22

10.18 10.47

4.40 4.40

3.90 2.95

160

3.70

6.50

3.75

0.79

4.22

10.67

4.40

1.98

170

3.70

6.58

3.75

0.40

4.22

10.79

4.40

0.99

180

3.70

6.60

3.75

0.00

4.22

10.83

4.40

0.00

a

The monomer excitation energies are 3.80 eV and 4.30 eV, and the corresponding transition dipole moments are 5.55 D and 6.96 D. The distance of least separation between the molecules is 4 Å.

Figure 3 we show the dependence of βXXX and βXYY as a function of the stacking angle for various distances of separation for the two stacking geometries. We find that the dominant component shows a strong dependence on the distance of separation. At large distance of separation, our results are in agreement with the OGM. What is interesting is that the stacking angle θ at which the maximum in the βXXX component is observed is dependent upon the distance of separation R. In the OGM, θmax is independent of the distance, as the OGM assumes the two chains are noninteracting. We regain the OGM at large distances as the interaction becomes very weak. In Figure 4a, we show the dependence of the angle θ at which the large component is maximum, as a function of distance R between the chains. We find that this angle differs significantly from the OGM value at short distances and approaches the OGM value asymptotically at large separation. The θmax value of the small component, on the other hand, does not show this variation and is near the OGM value at all separations. What is more interesting is that both the small and large components do not show the variation in θmax in the HtT configurations (Figure 4b). In Figure 5, we see that the θmax is away from the OGM value even in the large pushpull chains of eight conjugated carbon atoms. For an interchain separation of 4 Å, the 2θmax value is 18 for the eight carbon system while that for the six carbon system it is 17. Since the longer system shows a θmax value larger than that for the six carbon system, we may conclude that in the polymer limit the βXXX component will show a maximum for a stacking angle away from zero. In Figure 6, we present the variation in the component of β for different pushpull strengths as a function of the stacking angle 1849



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Figure 3. Variation of βXXX and βXYY as a function of stacking angle 2θ, for different distances of minimum separation. The pushpull strength |ε| = 2.0 eV and the laser excitation frequency is 0.65 eV. βxxx of the monomer is 35.67 1030 esu. The geometry of stacking is HtH in parts a and b and HtT in parts c and d. In the inset of part a, we have shown βXXX vs 2θ for two distances of interchain separation of 4 Å and 8 Å, with an interval of 1.

Figure 4. Plot of the angle 2θmax at which the dominant component is maximum as a function of distance of minimum separation of the stacked dimers in the (a) HtH and (b) HtT configurations. Each pushpull polyene chain is six carbon atoms long. The pushpull strength |ε| = 2.0 eV, and the laser excitation frequency is 0.65 eV.

at a distance of separation R = 4 Å. Larger pushpull strength gives higher β values at all stacking angles, although θmax does not depend on the strength of the push and pull group. In order to compare the magnitude of the SHG coefficient in the stacked system with that of a single molecule, we have plotted the ratio of the SHG response of the stacked system to that of the monomer, as a function of stacking angle (Figure 7). We find

that, in the HtH stacking, for both components and all stacking angles, the stacked system always has less than twice the SHG response of the isolated molecule. However, in the HtT stacking, the large component shows a response which is more than twice the isolated molecule response at large stacking angle. In order to see the effect of medium on the above, we have considered the case in which the intermolecular electron repulsions Vij are 1850



Figure 5. Variation of βXXX and βXYY as a function of stacking angle 2θ, for distance of minimum separation R = 4 Å. The pushpull strength |ε| = 2.0 eV, and the laser excitation frequency is 0.65 eV. βxxx of the monomer is 97.13 1030 esu. Each pushpull polyene chain is eight carbon atoms long. The geometry of stacking is HtH.

screened by a dielectric medium, defined by the static dielectric constant κ. Vij values are scaled as Vij/κ, and κ is taken as 2 and 25. We find that weak screening shifts the θmax slightly and, for strong screening, the OGM result is regained as expected (see the Supporting Information). We have also calculated the β coefficients for the HtH and HtT stackings within the two state model (TSM) of Chemla and Oudar, obtained by restricting eq 2 to only the ground state and the optically allowed excited state. In the pushpull systems, there are several optically allowed excited states as a consequence of the breaking of inversion and alternancy symmetries. We have considered the state with the largest transition dipole to the ground state as the optically excited state in the TSM. While the TSM results show similar behavior in the dependence of the β components on the stacking angle, the actual value is almost an order of magnitude smaller than the exact value. This shows that the TSM is inadequate in dealing with pushpull systems with many optically allowed excited states. However, it appears that the qualitative features are reasonably well reproduced, possibly because, in the chains we consider, the transition dipole moment is always along the chain axis for all the states. Linear and Nonlinear Optical Properties of DHAD. We have studied the D (donor) and A (acceptor) substituted azobenzene monomers (M) and dimers connected by σ- (S1) and π-bridges (P1). We have varied the length of the π-conjugation bridge (CH) as well as the nonconjugated (CH2) bridge. We have also studied the NLO properties of the longer oligomers of nonconjugatively bridged systems. We have calculated the optical gap and |βBav| for these molecules in both the SCI and MRSDCI methods (Table 2). Within the SCI calculations, the optical gap of the monomer is 3.29 eV, which is in agreement with the experimental gap, 3.203.24 eV, in different solvents.17 The gap for the nonconjugated dimer (S1) is 3.38 eV, while the experimental gap is 3.123.16 eV. According to the exciton theory, when two transition dipole moments are in parallel orientation, we have an H-aggregate and the optical gap of the dimer is more than that of the monomer.32 Since the two monomers are bridged by a nonconjugated unit, we can treat the system as an H-aggregate. On the other hand, the optical gap for the conjugated molecule

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(P1) is 2.80 eV, which is 0.49 eV less than that of the nonconjugated dimer. This shows that the excited state is stabilized by large delocalization. We have also carried out MRSDCI calculations on these molecules. The optical gaps for M, S1, and P1 are 3.42 eV, 3.45 eV, and 2.76 eV, respectively, which are very similar to the SCI values. And in both the SCI and MRSDCI calculations, the optical gaps for the monomer and nonconjugated dimers are nearly the same, while there is a significant red shift in the case of the conjugated dimer. The optical gap, in the case of conjugated dimers, also decreases progressively with increasing number of carbon atoms in the conjugation bridge. This decrease is ≈0.15 eV per CH unit. Except in the case of the nonconjugated dimer with three interacting CH2 groups, the magnitude of the optical gaps and |βBav|, as well as the ratio of the |βBav| values, are all nearly the same in both calculations. The reduction in the |βBav| values of dimer in which monomers are separated by three CH2 groups may be because of nonplanarity of the molecule. In the rest of this section, we will report all our results based on the SCI method, as it has been reported earlier that, for β, SCI calculations tend to be reliable.28,29 In Table 3 we present the ground state dipole moment, the transition dipole moment, and difference in the dipole moment between the ground and the excited states. The dipole moment of the dimer is almost twice that of the monomer. The ground state dipole moment of the nonconjugated system is smaller than that of the corresponding conjugated system. On the other hand, the change in dipole moment is larger for the nonconjugated system compared to that of the conjugated system and in confirmity with the earlier observation by Schuddeboom et al.16 The transition dipole moment is marginally higher for nonconjugated systems than for the conjugated system. But the optical gap is significantly smaller for the conjugated dimers. According to the TSM, β is proportional to the change in dipole moment and the oscillator strength and is inversely proportional to the energy gap.33 The optical gap for the monomer and nonconjugated molecule are almost the same, while the oscillator strength of the nonconjugated dimer is almost twice that of the monomer (1.02 and 1.90). The change in dipole moment is slightly larger for the nonconjugated dimer compared to the conjugated dimer (see Table 3). Hence, β for the nonconjugated dimer is approximately twice that of the monomer (1.93), which is in agreement with the experimental results of Zhang et al.17 For a conjugated molecule, even though the oscillator strength and change in dipole moments are slightly smaller than those of the nonconjugated molecule, the optical gap is significantly smaller. Hence, we observe a nearly 3-fold increase in the β. With an increase in the number of bridging units, the optical gap decreases considerably for conjugated dimers whereas, for the nonconjugated dimers, the decrease is marginal. A nonconjugated molecule with three bridging units resembles two isolated monomer molecules. The ratio of |βBav| for a conjugated dimer with respect to a monomer increases by nearly 0.6 for every pair of additional CH groups to the conjugation bridge. So far we have described our studies on dimers with conjugated and nonconjugated bridges. We have extended our studies to oligomers with up to four monomers linked by σ bridges. In Table 4 we have given the values of optical gap and |βBav|. The optical gap increases slightly (≈0.08 ( 0.01) with the increase in the number of monomer units. Both oscillator strength and ground state dipole moment increase linearly with 1851



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Figure 6. (a) Variation of βXXX (filled symbols) and βXYY (unfilled symbols) as a function of stacking angle 2θ and (b) variation of βYYY (filled symbols) and βXXY (unfilled symbols) as a function of stacking angle 2θ, for a distance of minimum separation R = 4 Å. The laser excitation frequency is 0.65 eV.

Figure 7. Plot of the ratio of the largest component of the β of the stacked dimers to that of the single molecule, projected to the laboratory frame as a function of orientation (a) for HtH stacking and (b) HtT stacking. The pushpull strength |ε| is set to 2.0 eV, and each is six carbon atoms long with R = 4 Å. The components are labeled in the figure.

Table 2. Optical Gap (in eV) and |βBav| (in 1030 esu) from SCI and MRSDCI Calculations for Nonconjugated and Conjugated Dimers (ω = 0.65 eV and SCI with Configurations 1601)a SCI molecule

nonconjugated

conjugated

no. of CH2/CH units

energy gap

MRSDCI

|β Bav|

ratio

energy gap

|β Bav|

ratio

1

3.38

42.15

1.90

3.45

39.04

1.63

2

3.29

48.27

2.18

3.35

49.57

2.06

3

3.28

51.72

2.33

3.41

41.01

1.71

1

2.80

61.52

2.78

2.76

77.11

3.21

2

2.63

78.80

3.56

2.62

93.05

3.87

3

2.49

94.36

4.26

2.48

103.18

4.30

The optical gap for the monomer is 3.29 eV, and |βBav| is 22.15 1030 esu within SCI, and 3.42 eV and 24.02 1030 esu, respectively, within the MRSDCI calculation. The ratio in the table is the ratio of |βBav| of dimer to monomer molecules. a

the chain length (see Table 4). The change in dipole moment between the ground state and excited state varies marginally. Hence, |βBav| increases almost linearly with the increase in chain length. This is consistent with the OGM, wherein the β of the

of the supramolecule can be written in terms of the βmonomer zzz = n(cos (θ))3βmonomer , θ is the angle monomer as βoligomer zzz zzz between the monomers, and n is the number of monomers. In our case, θ is nearly zero. Indeed, this is also true for |βBav|. 1852



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Table 3. Ground State Dipole Moment (|μBG|) in Debye, Optical Gap (EG) in eV, Transition Dipole Moment (||) in Debye, Oscillator Strength (f), and Change in Dipole Moment !|) for the Monomer between the Ground and Excited States (|Δμ a and Dimer Molecules

’ ASSOCIATED CONTENT

bS

μG

EG

ÆG|μ|Eæ

f

Δμ

8.85

3.29

8.72 (y)

1.00

4.54

Supporting Information. Plot of the ratio of the largest component of the β of the stacked dimers to that of the single molecule, projected to the laboratory frame as a function of orientation for different dielectric constants, κ, of the medium: (a) for HtH stacking and (b) for HtT stacking. This material is available free of charge via the Internet at http://pubs.acs.org.

17.06

3.38

12.02 (y)

1.89

4.51

’ AUTHOR INFORMATION

n=2

17.54

3.29

11.10 (y)

1.55

3.50

Corresponding Author

n=3

17.91

3.28

8.87 (y)

0.99

1.32

*E-mail: [email protected], [email protected], [email protected].

n=1

18.73

2.80

11.41 (y)

1.42

3.32

n=2

19.52

2.63

11.33 (y)

1.29

2.57

n=3

9.99

2.97

6.35 (z)

0.45

4.85

molecule monomer nonconjugated dimer n=1

conjugated dimer

a

n gives the number of intervening CH2 (CH) groups in the case of nonconjugated (conjugated) dimers.

Table 4. Ground State Dipole Moment (μG) in Debye, Optical Gap (EG) in eV, Oscillator Strength (f), Change in Dipole Moment between the Ground and Excited States (Δμ), and |βBav| (in 1030 esu) as a Function of Number of Monomer Units in the Nonconjugated Oligomers μG

energy gap

f

Δμ

|β Bav|

1

8.79

3.27

1.02

4.90

22.32

2 3

17.03 25.08

3.34 3.43

1.90 2.82

5.04 4.69

42.99 59.23

4

32.94

3.51

3.78

4.77

76.45

no. of monomers

Hence, the β of the tetramer is almost 4 times that of the monomer. The slight increase in the optical gap with the oligomer size is in qualitative agreement with the experimental results of Clays et al.11 on alkynyl expanded donoracceptor substituted calix[4]arenes.

’ CONCLUSIONS In summary, we have studied the linear and nonlinear optical properties of the stacked pushpull polyenes in different orientations by the exact diagonalization method within the PPP model. We have varied the strength of the donor and acceptor groups. In the head to tail configuration, the dominant β component is more than twice that of the single molecule at large stacking angle, whereas, for head to head configuration, it is always less than two. We find that the βXXX for the head to head configuration shows a maximum at an angle θmax away from 0, contrary to the prediction of the OGM. This angle varies with distance between polyenes, and at large interpolyene distance it falls to 0. The angle θmax does not depend on the donor and acceptor strength of the pushpull system. We have also studied 0 4-hydroxy-4 -nitroazobenzene and its conjugated and nonconjugated dimers using the ZINDO SCI method. We find that a conjugated bridge between monomers enhances |βBav| beyond the OGM results, while the nonconjugated bridged system shows results closer to the prediction of OGM. Our studies show that one has to go beyond OGM for stacks, in which the intermolecular separation is within the van der Waals radii, or when the molecules are connected by conjugated bridges.

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Nonlinear Optical Properties of Stacked Conjugated Systems

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