J. Phys. Chem. 1995,99, 11061-11066
11061
Continuous Symmetry Analysis of Hyperpolarizabilities. Characterization of the Second-Order Nonlinear Optical Response of Distorted Benzene David R. Kanis, Jason S. Wong, Tobin J. Marks,* and Mark A. Ratner* Department of Chemistry and the Materials Research Center, Northwestem University, Evanston, Illinois 60208-3113
Ha@t Zabrodsky Department of Computer Sciences, Bar-llan University, Ramat-Gan 52900, Israel
Shahar Keinan and David Avnir* Department of Organic Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel Received: February 20, 1995; In Final Form: May 9, I 9 9 P
A direct relation between the first hyperpolarizability, p, of noncentrosymmetric molecular structures and the metric centrosymmetricity content, S(i), of such structures is shown for the first time. For a series of systematic, in-plane distortions (stretch, pull, shift, and squish deformations) of the model NLO chromophore benzene, we find a convincing monotonic relationship between calculated values of /3 and S(i). These results suggest that the dominant variation in p for these structures arises from the change in oscillator strength. More striking, these comparisons demonstrate the utility of the S(i) metric in correlating observable behavior with symmetry content.
I. Background
polarizability, the quadratic hyperpolarizability, and the cubic hyperpolarizability. The second-order response, /3, the magniUnderstanding the origins of the second-order nonlinear tude of which increases as the molecule becomes less cenoptical (NLO) response of molecular chromophores is currently trosymmetric, is the quantity of interest in this work. In this an area of intense research interest.’-* Molecule-based NLO contribution we will explore how /Idepends (quantitatively) on materials offer many intriguing characteristics such as ultrafast the degree of centrosymmetricity. response times, superior processability characteristics, and lower The notion that symmetry content can be quantified as a dielectric constants relative to inorganic solids, the materials continuous structural property has been developed recently, and currently used in photonic technologies.2 Through extensive a scale that allows one to evaluate quantitatively the amount experimentalltheoretical studies, a number of chemically-based that any symmetry element is present in a molecular configudesign criteria have been suggested by which one correlates ration has been constructed (continuous symmetry measure nonlinear optical response with particular molecular structural a p p r ~ a c h ) . ~Using - ~ this method, we demonstrate here for the motifs (electronic asymmetry, charge-transfer character, inducfirst time, that a quantitative relationship exists between /3 and tive effects, and delocalization).2 the symmetry measure of inversion, S(i). We do so by observing Theoretical methodologies have played a crucial role in a monotonic correlation between the calculated /3 and S(i) for arriving at the current understanding of medium- to large-sized a series of distorted benzene geometries (Scheme 1). The ability second-order molecular NLO chromophores,’.2 and several comprehensive reviews have been published on this s ~ b j e c t . ~ . ~ to quantify the effects of molecular distortion on molecular hyperpolarizability has significant implications for understanding In general, one wishes to use theoretical techniques either to the properties of “real-world” NLO materials in which geometric design molecules with the largest possible responses (for device distortion may significantly influence the actual molecular applications) or to derive simple models for interpreting the symmetry. More importantly, the observed correlation between responses for large classes of structures. The aim here is quite S(i) and /3 clearly demonstrates the utility of the symmetry different: we wish to analyze the development of the nonlinear metric. response as molecules are distorted from point group symmetries that include an inversion center and, therefore by symmetry, 11. The Continuous Symmetry Approach have vanishing hyperpolarizabilities. Formally, hyperpolarizability terms enter into the expansion We have proposed to treat symmetry as a continuous of the overall polarization of a molecule of an applied electric structural property, rather than treating it in classical termss as field.’ The expansion can be expressed as “symmetric or not”. Our main motivation has been the view that defining a continuous measure is bound both to enrich the descriptive capabilities of the structures and to provide correlations between structure and physical or chemical properties. Physical intuition seems to agree with this proposition: Retumwhere pi, pi, Fi,ag,p i j k , and yijk/ are respectively the molecular ing for instance to distorted benzene, one may wonder whether polarization, the molecular permanent dipole moment, the it is justified to jump from a D6h description to, say, a C I electromagnetic field along the ith axis ( i = x, y, z), the description at any level of distortion or substitution. This Abstract published in Advance ACS Absrracts, June 15, 1995. question is intimately resolution dependent. The recognition @
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11062 J. Phys. Chem., Vol. 99, No. 28, 1995 SCHEME 1 The distorrion
Tt$:fird
The nearest
ernhosymmetne
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have a zero fvst hyperpolarizability and a zero permanent dipole m ~ m e n t .Since ~ the continuous symmetry metric provides a quantitative characterization of the extent to which the inversion center is present for a particular geometry, it is possible to compare computed quadratic hyperpolarizabilities with the extent to which the inversion center remains. Thus, this application provides the ideal testing ground for a continuous symmetry measure.
III. Computational Details
that distorted benzene is no longer D6h is determined by the spatial and temporal limitations of the instrumentation used. Furthermore, even for fully developed distortions only vaguely resembling an original reference structure, some symmetry is still present. The ideal symmetry fades as the distortion becomes more pronounced, but for many physical processes the forbiddeness or allowedness is determined by such diminished, residual symmetry. The evaluation of the symmetry content of a given structure is carried out by searching for the nearest configuration of vertices which has the desired symmetry: n
S(G) = (lOO/n)C I [Pi-
Pi[
i= I
Here Pi are the n vertices of the, original configuration (normalized to maximum radius l), Pi are the corresponding points in the nearest G symmetric configuration (where G is a symmetry element or a symmetry point group) and the factor of 100 expands the scale, for convenience, from 0 (the original configuration is already G symmetric) to 100. Equation 2 defines a metric on the space of all sets of n poinp. The primary practical problem is then to locate the set of Pi's which will minimize S(G),the continuous symmetry measure. A detailed explanation of how this problem was solved is given in detail elsewhere.5s6 Briefly, the idea is to fold the configuration of points into a dense cluster by applying the symmetry operations for the group in an ordered sequence to the vertices (the folding step), averaging the cluster, and unfolding the averaged point into the symmetry shape by applying in reverse order the group elements to it. Rigorous mathematical proof was provided that this approach indeed leads to minimization of S(G).5-7 For the analysis of the hyperpolarizability changes of benzene, we apply two Syllllnetry groups: ci {E,i}and c6 { E , c6, c62, cfj-1, c62 ) . The former is a case where the number of elements is smaller than the number of vertices. Here the foldingunfolding procedure is carried out for each possible pair within the hexagon and minimized (see section 2.4 in ref 6). As an example of the utility and applicability of the continuous symmetry measure approach, we examine here the molecular second-order response4 of distorted chromophore benzene structures. Molecules possessing a center of symmetry
A number of computational schemes are capable of computing the second-order NLO response for molecular chrom o p h o r e ~ . In ~ , ~general, computationally intensive ab initio procedures (contained in electronic structure packages such as GAUSSIAN? ACES II,Io GAMESS," or HOND0I2) are the methods of choice for smaller molecular structures, while larger, more computationallydemanding architectures generally require semiempirical schemes, such as CNDO-SOS,I3 PPP-S0S,l4 INDO-SOS,15316 or MNDO-FF.I7 The relatively modest structures (in size) examined in this contribution suggest that ab initio routines are appropriate here. l 8 Moreover, the relatively small responses for the computed distortions dictate that a reasonable degree of computational accuracy is required for this study to obtain quantitative trends in P versus degree of noncentrosymmetry. The semiempirical schemes with minimal basis sets and approximate Hamiltonians are not likely to detect changes in small second-order responses. In general, the electromagnetic fields in eq 1 are frequencydependent, as are the polarizabilities and hyperpolarizabilities. For simplicity, we restrict ourselves in this contribution to the hyperpolarizability at zero frequency. The static field (zero frequency) hyperpolarizability computations discussed in this contribution were performed using the GAUSSIAN 92 program? employing a coupled perturbed Hartree-Fock (CPHF) scheme.19 In this formulation based upon the HellmanFeynman theorem,I9 the molecular Hamiltonian explicitly includes a term (-rF)describing the interaction between an external static field (F) and the molecular electronic structure. The second-order response is then obtained by differentiating the total energy E(F), with respect to the applied field in the zero field limit (eq 3).
(3) Since the electric field along a given axis i (F'J must transform as i itself, a centrosymmetric molecular environment will yield a vanishing quadratic hyperpolarizability (expectation value of an odd function). By virtue of Kleinman symmetry,20the 27 components of the 3 x 3 x 3 tensor are reduced to 10 in the static field limit. One can then convert the tensor information into a vector via eq 4.
where i and j run over the molecular Cartesian directions x , y , and z. Two combinations of the vector components are important in NLO studies, PVecand Ptot.The former is the vector component along the ground state dipole moment @) and is the component of ,d sampled in EFISH experiments (eq 51, while the latter is the total intrinsic hyperpolarizability (eq 6 ) . Ptotis a positive quantity by definition and is the parameter of interest in this study.
J. Phys. Chem., Vol. 99, No. 28, 1995 11063
Continuous Symmetry Analysis of Hyperpolarizabilities
The approach used in this paper ignores correlation effects in the ground state. These effects have been shown to be important in smaller molecule^;^ however, here we make the physically reasonable assumption that whatever error arises from neglecting correlation effects in one distorted benzene structure should be nearly equal for all slightly distorted benzene structures. The quadratic hyperpolarizability was computed for all distorted structures in three basis sets: 3-21G, 6-31G*, and 6-31+G.21 The 3-21G basis set is a split-valence basis set of contracted Gaussian functions, the 6-3 1G* improves on the 3-21G by including more primitive functions and by adding a single set of d-type polarization functions on each carbon atom. The 6-31+G improves on the 6-31G set by adding a diffuse function on each carbon atom. For quantitative NLO response, the basis set must include diffuse functions, and the computed response is very dependent upon the functional form of the diffuse function. For the planar distortions examined herein, the qualitative trends were found to be largely independent of basis set for a given distortion. We also examined out-of-plane distortions; these perturbations required very large basis sets to correlate with centrosymmetricity. The modest basis sets (321G, 6-31G*, even 6-31fG) were found not to be appropriate for out-of-plane distortions. In one section of this contribution we also compare the ab initio results with those derived from a semiempirical ZINDOSOS treatmentsi5 The ZINDO-SOS approach has been shown to provide reasonably reliable responses for medium- to largesized high$ chromophore^.^^,^^ The details of this method have been described elsewhereI5and will be briefly summarized here. An INDOE minimal-basis semiempirical scheme devised by Zerner and co-workersZ2provides the necessary ground state and excited state information to compute p via a sum-overexcited states (SOS)perturbative treatment.23 In this formalism, the ground state is not correlated; however, the excited states include monoexcited configuration interaction (MECI). The parent centrosymmetric structure possesses C-C distances of 1.40 and C-H distances of 1.08 The C-H distances were left unchanged in these computations. The stretch distortion involves moving CI-HI along the x axis in increments of 0.1 A. In the shift distortion, the x and y coordinates of CI-HIare systematically altered, changing the C ~ - C ~ - C Iand c2-cIvc6 angles as well as c2-c1 and c1c 6 distances. In the pull distortion, C2 and C3 and associated hydrogen atoms are moved in unison away from the other four carbon atoms by modifying the y coordinates in increments of 0.1 A. Finally, for the squish distortion, C2 and C3 are systematically moved toward and away from one another by altering the x coordinates (Scheme 1).
A
A.
IV. Results and Discussion Figures 1-4 present a comparison of the GAUSSIAN-CPHF derived second-order susceptibilities vet; referred to as @ in the figures) against the continuous symmetry measures for the four in-plane distortions of the benzene molecule (stretch, pull, shift, and squish, respectively). The quadratic hyperpolarizability is reported in the standard units of cm5 esu-I. For each distortion type, the extent of distortion has been varied,
1.
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Figure 1. GAUSSIAN-derived second-order NLO response computed cm5 esu-I) versus symmetry contents at zero frequency (in units of of the inversion operator for squish distortions of benzene.
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11064 J. Phys. Chem., Vol. 99, No. 28, 1995 I
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S(iJ
Figure 4. GAUSSIAN-derivedsecond-order NLO response computed at zero frequency (in units of cm5esu-I) versus symmetry contents of the inversion operator for shift distortions of benzene.
Figure 6. Summary plot of GAUSSIAN-derived second-order NLO
cm5 esu-l) computed using a 3-21G basis response (in units of set versus symmetry contents of the inversion center for all four classes of distortions. Note a near linear correlation with each type of distortion, but with substantially different slopes.
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Figure 5. GAUSSIAN-derivedsecond-orderNLO response computed
at zero frequency (in units of cm5esu-I) versus symmetry contents of the C g rotation for squish distortions of benzene.
Figures 3 (stretch distortion) and 4 (shift distortion) also show a distinctive, monotonic relationship between the computed response and the symmetry measure. For the stretch distortion, the relationship is monotonic; however, it appears to somewhat more quadratic than linear in nature for both basis sets. Strict linear correlation between hyperpolarizability and symmetry measure is actually not expected. For example, if two carbon atoms in a distorted structure were to come too close to one another, there would be a strong repulsive effect, greatly altering the energetics and nature of the excitation patterns. This is not evident in the symmetry measure, but it would certainly affect the nonlinear response and would cause a breakdown of the near-linear relationships found in Figures 1 and 2. Another possible explanation for the apparent nonlinear, yet monotonic observed behavior is the inadequacy of the basis set for extreme distortions. For the most distorted structure displayed in Figure 3 (S(i)= 0.23), the two C-C bond distances become 1.78 A, far removed from the 1.40 A distances of the centrosymmetric benzene structure. At such extreme distortions, a much larger basis set is required to accurately treat the NLO response. The computations with the 6-31G* basis set (Figure 4) display a marked sinusoidal component, the origin of which is not evident. The presence of the inversion center causes the hyperpolarizability in benzene to vanish. On the other hand, the c6 rotation axis is related to the delocalization of charge in the benzene ring, and it was suspected that a quantification of the nonlinear response as a function of the c6 content might also prove informative. Figure 5, which should be compared with Figure 1, displays the GAUSSIAN-derived responses for the squish distortion as a function of s(c6).A roughly linear correlation in both basis sets between /3 and the symmetry measure S(c6) is again observed. Note that the s(c6)values are larger than
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Ccntrorymctriei?y, S ( i )
Figure 7. Summary plot of GAUSSIAN-derived second-order NLO cm5 response computed using a 6-31G* basis set (in units of esu-I) versus symmetry contents of the inversion center for all four classes of distortions. Note a near linear correlation with each type of distortion, but with substantially different slopes.
the S(i) values for comparable quadratic hyperpolarizabilities. This reflects the fact that the corrective motions of the carbon atoms of the, distorted benzene in order to attain the desired symmetry (Pi, eq 1) are larger for the higher of the two symmetry groups, c6 and i. The computational results for several distortions are compared in Figures 6 and 7. Note that for each of the four distortions monotonic relationships between the computed response and the symmetry content of inversion are observed. Also note that the different symmetry breakings display different slopes. For example, Figure 7 shows that the quadratic hyperpolarizability responds similarly to squish, stretch, and pull distortions, while the shift distortion is slightly less sensitive to the breaking of inversion symmetry. These observations, which hold for both the 3-21G (Figure 6) and the 6-31G* (Figure 7) basis sets show clearly that symmetry cannot be used as the sole structural parameter in exploring the cause/effect relationship between molecular geometry and hyperpolarizability. In general, conclusions for basis set 6-31-tG are identical to those for the 3-21G and 6-31G* basis sets, as shown for the squish distortion in Figure 8. However, this basis set leads to several inconsistencies in the computed response for highly distorted structures. It is likely that optimization of the diffuse functions is perhaps required if one wants to compare symmetry measures with computed /3 values in this basis set. The important point here is that the trends observed for the 3-21G and 6-31G* basis sets are again reproduced with the diffuse basis set.
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Continuous Symmetry Analysis of Hyperpolarizabilities
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J. Phys. Chem., Vol. 99, No. 28, 1995 11065
away from centrosymmetry; Ap vanishes when S(i) = 0, as does /?. Oscillator strengths can also vanish between states of the same symmetry. Transition energies, on the other hand, should be less sensitive to small distortions and therefore not be the two-level term dictating the change in /? with increase in centrosymmetricity.
6-31tG
A
V. Conclusions
A
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Figure 8. GAUSSIAN-derived second-order NLO response computed at zero frequency (in units of cm5 esu-I) versus symmetry contents of the inversion operator for squish distortions of benzene. This figure differs from Figure 1 in that GAUSSIAN computations employing a 6-31+G basis set and ZINDO-derived responses are included for purposes of comparison. Notice that all of the results from the ZINDO model disagree substantially with those employing the a b initio method.
Hyper-Raman spectra are proportional to the square of the derivative of the hyperpolarizability with respect to nuclear coordinate^.^^ Calculations of hyper-Raman spectra neglecting correlation using different basis sets have been reported.24 For an appropriate basis set (i.e., incorporating diffuse functions) the computed spectra compare very well with experiment. These calculations are very much related to the current calculations, where the change in the second-order response with molecular geometry is being examined. The fact that ab initiobased methods using reasonable basis sets provide hyper-Raman responses in excellent agreement with experiment gives some level of comfort to the current ab initio calculations, despite their ignoring correlation effects. For purposes of comparison, ZINDO-SOS-derived responses for the squish structures were also computed and are compared in Figure 8 with the ab initio results. The ZINDO method is clearly less satisfactory. For example, larger distortions lead to decreased responses, in stark contrast to the ab initio calculations. This is not a surprising result. Semiempirical models using minimum basis descriptions should provide reasonably accurate responses for high-/? chromophores where a few optical excitations dominate the second-order response. They cannot, however, describe the full polarization of a fairly symmetric delocalized charge cloud such as that found in distorted benzene structures. One can rationalize the different slopes observed for different distortions and the lack of a perfect linear relationship between the quantitative S(i) and the computed NLO response in terms of different contributions to the hyperpolarizability. While the full spectral representation of the second-order susceptibility contains both two-level and three-level terms, in many situations one can approximate the response using the two-level formula25 (the first level being ground state g and the second level being the crucial excited state n)
Heref,,, Apgn,and hug, are respectively the oscillator strength of the optical transition between the two states, the change in dipole moment, and the energy difference between the ground and excited states. The photon energy tiw also enters into the formula. Within the two level model, the change in dipole moment is clearly affected by a molecular distortion of benzene
Several previous investigations have examined the variation of the computed NLO responses, both second order and third order, with change in molecular g e ~ m e t r y . ' The ~.~~ important conclusion from the present contribution is that a rather complicated molecular response property, the quadratic hyperpolarizability, correlates rather well with the quantitative symmetry measure. A good correlation between the symmetry measure and /? is found, except with extreme nonphysical distortions; then the basis set used for the calculations becomes quantitatively inadequate, but a near linear correlation between beta and the symmetry distortion is obtained. In molecular second-order materials, constituent chromophores are only slightly deformed by environmental effects relative to their isolated geometries. This contribution suggests that one could use the centrosymmetricity metric to estimate the molecular response of chromophores in actual materials relative to computed or EFISH-derived responses of the isolated chromophores. The complex functionality of the NLO response is also clear in the differing slopes for the quasi-linear relationships observes for different classes of molecular distortions. Overall, the behavior noticed in these studies is the first demonstration of a direct correlation between the symmetry metric (S(i))for a particular symmetry element (the inversion center) and a molecular response property.
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