Distinguishing the Effects of Bond-Length ... - ACS Publications

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Distinguishing the Effects of Bond-Length Alternation versus BondOrder Alternation on the Nonlinear Optical Properties of π‑Conjugated Chromophores Rebecca L. Gieseking,† Chad Risko,*,‡ and Jean-Luc Brédas*,†,§ †

School of Chemistry and Biochemistry and Center for Organic Materials for All-Optical Switching, Georgia Institute of Technology, Atlanta, Georgia 30332-0400, United States ‡ Department of Chemistry and Center for Applied Energy Research, University of Kentucky, Lexington, Kentucky 40506-0055, United States § Division of Physical Sciences and Engineering and Solar & Photovoltaics Engineering Research Center, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia S Supporting Information *

ABSTRACT: Understanding the relationships between the molecular nonlinear optical (NLO) properties and the bondlength alternation (BLA) or π-bond-order alternation (BOA) along the molecular backbone of linear π-conjugated systems has proven widely useful in the development of NLO organic chromophores and materials. Here, we examine model polymethines to elucidate the reliability of these relationships. While BLA is solely a measure of molecular geometric structure, BOA includes information pertaining to the electronic structure. As a result, BLA is found to be a good predictor of NLO properties only when optimized geometries are considered, whereas BOA is more broadly applicable. Proper understanding of the distinction between BLA and BOA is critical when designing computational studies of NLO properties, especially for molecules in complex environments or in nonequilibrium geometries. he concepts of bond-length alternation (BLA) and πbond-order alternation (BOA) are key physicochemical parameters used in the design of π-conjugated molecular and polymer active materials for a wide range of electronic1−3 and photonic4−8 devices. While the evolutions of linear and nonlinear optical properties with BLA have been associated experimentally through comparisons with molecular structures determined from X-ray crystallography,9,10 both BLA and BOA are tools much more widely used in computational studies.1−4,6−8,11,12 BLA and BOA have proven particularly useful in achieving materials with the nonlinear optical (NLO) properties required for device applications, as even small changes can lead to major changes in the magnitude or even reverse the sign of the molecular second-order polarizability β or the third-order polarizability γ.11,12 The average BLA is usually defined as the difference between the average lengths of the nominally single bonds and the nominally double bonds along the linear π-conjugated backbone, while the average BOA is defined as the difference between the average π-bond orders of the same two sets of bonds. While BOA directly assesses the molecular electronic structure, BLA presents a measure of the molecular geometric structure that is often but not always a good reflector of the electronic structure; the consequences are important, as we show below.

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Although the relationships between BLA and the molecular NLO properties have proven widely useful, the extent of the applicability of these relationships in molecular design has been questioned by a recent investigation, which suggested that molecular structures extracted from molecular dynamics simulations in explicit solvent present only a very weak correlation between BLA and β.13 To provide a comprehensive understanding of this question, we elucidate here the limits of when the commonly used relationships between BLA and the molecular NLO properties are applicable. In particular, we demonstrate that BLA and BOA are well correlated only when the molecular geometry is optimized in the environment of interest. As BOA probes the molecular electronic structure, BOA is a good predictor of the molecular linear optical and NLO properties regardless of whether BOA reflects the molecular geometry. These results have important implications for the computational approaches used to study molecular NLO properties. To separately elucidate the roles of BLA and BOA on the description of molecular properties, we consider a prototypical 5-carbon streptocyanine (Figure 1) in three series of calculations: Received: April 20, 2015 Accepted: May 26, 2015

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In Series 1 (blue diamonds, Figure 1), as the magnitude of the electric field increases, both BLA and BOA increase in magnitude from the cyanine limit (BLA = 0 Å, BOA = 0) to the polyene limit (BLA = −0.1 Å, BOA = +0.65; and BLA = +0.1 Å, BOA = −0.65), showing the characteristic range of geometric and electronic structures that have been previously observed for linear conjugated systems.11 (We note that our choice of convention leads to BLA and BOA having opposite signs.) In Series 2 (red circles, Figure 1), BLA remains fixed at 0 Å. BOA, however, varies between −0.5 and +0.5 within the range of electric-field strengths examined. Even though the geometric structure is fixed, the BOA at each electric-field strength is roughly 70% of its value in the corresponding structure from Series 1. In Series 3 (green triangles, Figure 1), BLA varies from −0.1 Å to 0.1 Å; however, BOA for each geometry is only 30−50% of its value when the electric field is present. Thus, in the absence of an electric field, the geometric change induces only a relatively small change in the electronic structure. The consideration of these three series of results allows us to distinguish between changes in the geometric and electronic structures of the polymethines to highlight the relative effects of these two parameters on the NLO properties. With the distinction between the geometric and electronic structures in mind, we analyze the evolution of the ground-state dipole moment μg with BLA and BOA in these three series; we recall that μg is the key molecular parameter as its first-order [second-order/third-order] derivative with respect to external electric field corresponds to the first-order (linear) polarizability α [second-order polarizability β/third-order polarizability γ]. In Series 1, μg is zero along the long molecular axis at the cyanine limit23 and is large in magnitude at large values of both BLA and BOA (Figure 2; we note that the direction of the BLA axis has been reversed to ease comparison of the BLA and BOA plots). The magnitude of μg near the polyene limit indicates that the positive charge is becoming localized primarily on one of the two terminal nitrogen atoms. In contrast, the other two series show very different relationships between BLA and μg. In Series 2, there is substantial variation in μg despite the fixed molecular geometry; in Series 3, the variation in μg is relatively small despite the large change in geometry. The comparison among the three series underlines that the geometric structure alone is not suf f icient to predict the molecular properties. In marked contrast, when bond orders are instead considered, all three series show nearly indistinguishable relationships between BOA and μg. Since BOA is a measure of the molecular electronic structure, BOA can be used to consistently predict the molecular properties. Importantly, when the geometry is far from the energetic minimum in the specific environment, BOA is a much more reliable predictor of the molecular properties than is BLA. The molecular NLO properties are likewise much more strongly correlated with BOA than with BLA across the three series. We focus here on the second-order polarizability component βx along the long molecular axis (Figure 3); the first-order and third-order molecular polarizabilities display very similar correlations with BOA (see SI). Series 1 shows the sinusoidal evolution of βx characteristic of linear conjugated systems with respect to either BLA or BOA.10 In Series 2 and 3, a similar evolution of βx is found only with respect to BOA. Thus, without knowledge of how the molecular structure relates to the environment, BLA cannot be used to predict whether βx is small or large. As βx (computed using a sum-over-

Figure 1. Chemical structure (top) and correlation between BLA and BOA (bottom) for the 5-carbon streptocyanine. The three sets of colored symbols represent the different geometric and electric-field approximations employed in the study.

In Series 1, an electric field is applied along the long molecular axis, and the molecular geometry is fully optimized for each value of the electric field, as was done in the original computational studies11 demonstrating the relationships between BLA and the NLO properties. In Series 2, an electric field is applied along the long molecular axis as in series 1 but the molecular geometry is constrained (and unrelaxed) to the C2v-symmetric molecular geometry. In Series 3, the geometries derived in series 1 are considered in the absence of an external electric field. We note that in both Series 2 and Series 3, the molecular geometries are displaced from the energetic minima in their respective environments; as will be discussed later, these imposed distortions allow us to distinguish clearly between changes in the molecular geometric and electronic structures. As linear conjugated systems are highly polarizable, the electric field has a very large effect on the charge distribution along the long molecular axis.11 Geometry optimizations were performed using density functional theory (DFT) at the ωB97X/cc-pVDZ level14−16 with applied electric fields ranging from −7.5 to +7.5 × 107 V/cm, which correspond to usual values10 in such calculations;17 the achievement of geometry minima was confirmed by the absence of imaginary frequencies. The excited-state properties and Mulliken bond orders were computed using the semiempirical INDO Hamiltonian with applied electric fields of the same magnitudes as in the DFT calculations, as detailed in the descriptions of each Series above. The excited-state calculations were performed using a configuration interaction approach incorporating single and double excitations (SDCI), with an active space of 15 HOMOs and 15 LUMOs for single excitations and 4 HOMOs and 4 LUMOs for double excitations; this approach has been widely used for evaluating the NLO properties of π-conjugated systems and shown to provide results consistent with experiment.11,18−21 The NLO properties were computed at the static (zero-frequency) limit in the presence of the static electric field described earlier using a sum-over-states approach22 that sums over 350 excited states, which provides for fully converged NLO properties. We first examine the evolution of BLA and BOA across each of the three series, which allow us to distinguish between the effects of BLA and BOA on the molecular properties. 2159

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Figure 2. Correlation of μg with BLA and BOA for the 5-carbon streptocyanine. The three sets of colored symbols represent the different geometric and electric-field approximations employed in the study.

Figure 3. Correlation of βx with BLA and BOA for the 5-carbon streptocyanine. The three sets of colored symbols represent the different geometric and electric-field approximations employed in the study.

states approach) depends on excited-state energies, state dipole moments, and transition dipole moments,22 it should come as no surprise that a measure of the electronic structure is better at predicting the molecular NLO properties than is a geometric measure. To demonstrate that this correlation can be generalized to other, larger systems, we turn to analogous computations for the longer 9-carbon streptocyanine (Figure 4). The electric field ranged here from −4.5 to +4.5 × 107 V/cm and the active space for single excitations in the INDO/SDCI calculations was increased to 25 HOMOs and 25 LUMOs; otherwise, the computational methodology and definitions of the three series are identical to those described previously. For this molecule, it is again clear that BLA is a good predictor of the molecular NLO properties only when the geometry is optimized in its electric-field environment; in contrast, BOA can be consistently used to predict the molecular NLO properties regardless of the geometric structure. Even though the specific trends described here center on the molecular NLO properties of linear π-conjugated molecules, the primary message of this paper is much more broadly applicable: Although molecular geometric and electronic structures are often considered interchangeable, it is in fact important in the case of nonequilibrium geometries to carefully distinguish between the two. Geometric descriptors such as BLA can be and have been successfully used to characterize a wide variety of properties relevant to materials derived from πconjugated molecules and polymers, such as electronic and optical gaps,1,2,24 ordering of the one-photon and two-photon allowed states,25−27 soliton delocalization,28 and two-photon

absorption.18,29 In each case, however, it is important to bear in mind the limits of when geometric descriptors can be applied, i.e., at equilibrium conditions within the constraints of the environment. Computational studies used to separate the geometric and electronic structures often involve constraints that are not accessible experimentally, and hence great care must be taken in selecting an approach that adequately addresses the questions of interest. Several considerations must be taken into account: I. To understand the relationships between the molecular geometry and NLO properties, it is important to consider a geometry that is an energetic minimum in the environment (electric field, solvent, counterion, etc.) in which the NLO properties are computed. II. When considering molecules in complex environments, removal of the molecule from its environment may substantially modify the electronic structure even if the molecular geometry is retained. Maintaining the key features of the environment is essential to accurately assess the NLO properties. III. When considering molecular geometries that are displaced from their energetic minimum, the electronic structure will not change as much as the geometric changes may seem to imply. This is consistent with the previously observed weak correlation between BLA and NLO properties in geometries extracted from molecular dynamics simulations.13 In this case, the focus should be on BOA and not BLA since changes in BOA will be a 2160

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University of Kentucky. J.-L.B. acknowledges generous support from King Abdullah University of Science and Technology.

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Figure 4. Chemical structure (top) and correlation βx with BLA and BOA (bottom) for the 9-carbon streptocyanine. The three sets of colored symbols represent the different geometric and electric-field approximations employed in the study.

much more accurate predictor of variations in the NLO properties than changes in BLA. Computational studies have thus to be carefully designed to account for the distinction between BLA and BOA and analyze properly the optical and NLO properties of π-conjugated molecules in complex environments.

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ASSOCIATED CONTENT

S Supporting Information *

Additional computational results are given for the 5-C and 9-C streptocyanines. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.jpclett.5b00812.

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AUTHOR INFORMATION

Corresponding Authors

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The work at Georgia Tech was supported by the AFOSR MURI program (FA9550-10-1-0558) within the Center for Organic Materials for All-Optical Switching (COMAS). C.R. thanks the University of Kentucky Vice President for Research for start-up research funds to support the work at the 2161

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