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Oct 6, 2015 - Interplay between Open-Shell Character, Aromaticity, and Second. Hyperpolarizabilities in Indenofluorenes. Kotaro Fukuda, Takanori Nagam...
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Interplay between Open-Shell Character, Aromaticity, and Second Hyperpolarizabilities in Indenofluorenes Kotaro Fukuda, Takanori Nagami, Jun-ya Fujiyoshi, and Masayoshi Nakano* Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan S Supporting Information *

ABSTRACT: Focusing on the original and extended indenofluorene frameworks, we theoretically investigate the interplay between the open-shell character, the aromaticity, and the second hyperpolarizabilities γ. Interestingly, the oddelectron density distribution, which illustrates the spatial contribution of the open-shell character, is found to well correlate with the magnetic shielding tensor distribution, which indicates the magnetic criteria of the aromaticity. This can be explained with the partial destruction of the πdelocalization due to the emergence of odd (unpaired) electrons. Further investigation on the γ values, which are third-order nonlinear optical (NLO) properties at the molecular scale, reveals the correlation of strong enhancement of γ and its density distribution to the intermediate open-shell character and its odd-electron density distribution. These results will contribute not only to the detailed understanding of the structure−NLO property relationships in the indenofluorene frameworks but also to building new design guidelines for highly efficient NLO molecules based on the open-shell character− aromaticity correlation.

1. INTRODUCTION

between the diradical character and the traditional chemical concepts/indices that most chemists are familiar with. Among several chemical indices related to diradical character, aromaticity is one of the most essential and intuitive concepts for chemists regardless of their major fields.18−20 There have been lots of discussions on the basis of the fact that antiaromatic systems tend to exhibit a small energy gap between the HOMO (highest occupied molecular orbital) and the LUMO (lowest unoccupied molecular orbital),19,20 which leads to an increase in the diradical character,11,15 defined by twice the weight of double excitation configuration from the HOMO to LUMO. However, this correlation is still unclarified, in particular, on the relationship between the spatial contributions of the open-shell character and aromaticity. In the present paper, therefore, we investigate the spatial correlation between the open-shell character, aromaticity, and the second hyperpolarizabilitythe third-order NLO property at the molecular scaleby focusing on indenofluorene frameworks (Figure 1). A series of indenofluorenes are πconjugated fused-ring systems with alternating structures composed of three six-membered and two five-membered rings. They exhibit several structural isomers, and four of them have been synthesized by Haley’s and Tobe’s groups.21−24

Recently, open-shell singlet systems have attracted a great deal of attention from both scientific and engineering points of view due to their unique electronic structures and high functionalities.1−5 Since the proposition of the concept of “open-shell singlet” around 1970, many experimental and theoretical investigations have been performed primarily on the electronic structures of reactive intermediates.6 More recently, several stable open-shell singlet systems, e.g., organic π-conjugated fused-ring systems such as oligoacenes7,8 and graphene nanoflakes,1,3,9 have been synthesized and their structures, physicochemical properties, and reactivity have been extensively investigated.1−5,7−9 In general, the electronic structures of open-shell singlet systems are characterized by the quantumchemically defined index, “diradical character” y (0 (closedshell) ≤ y ≤ 1 (pure open-shell)).5,10−13 In this context, we have constructed new design principles for a new class of openshell singlet molecular systems exhibiting highly active nonlinear optical (NLO) properties5,14−16 or causing efficient singlet fission,17 which are superior to the traditional closedshell systems. In the application of these principles, the important point is the comprehensive understanding of structure−diradical character relationships and the finding of chemical and physical tuning schemes of the diradical character. Thus, to realize open-shell-character-based design for functional molecular systems, it is essential to clarify the relationships © XXXX American Chemical Society

Received: September 1, 2015 Revised: October 5, 2015

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Using the odd-electron density, the diradical character y is expressed by 1 2 1 = 2

y=

∫ dr [nLUNO(|ϕHONO(r)|2 + |ϕLUNO(r)|2 )]

(3)

In the triplet case, the orbitals included in the definition of the odd-electron density are assumed to be two singly occupied orbitals (nHONO = nLUNO = 1). The magnetic shielding tensor is calculated using the gauge invariant atomic orbital (GIAO) method.33 Since the systems except for the para-2 system exhibit completely planar structure because of the symmetric constraints, the molecular plane, where the magnetic shielding tensor is calculated 1 Å above the center of each ring, is uniquely defined. For the para-2 system, which exhibits a nonplanar structure because of the steric repulsion, the values calculated 1 Å above and below the center of the molecular plane differ from each other. In the present study, therefore, the average of the values above and below is adopted as the magnetic shielding tensor values 1 Å above the ring center for para-2. For the calculation of the longitudinal component of the static second hyperpolarizabilities γzzzz, we have adopted the fourth-order differentiation of the total energy according to the static electric field, finite field (FF) method in B-convention. The spatial distribution of the γzzzz, γzzzz density ρ(3) zzz(r), is defined as31

Figure 1. Molecular frameworks of synthesized indenofluorenes.

Since these systems possess 20π electrons, they have been investigated as antiaromatic analogues of pentacene.21 On the other hand, these systems exhibit proaromatic quinodimethane framework within the structures, and therefore these systems can also exhibit open-shell singlet character.22,24,25 These features enable us to perform detailed comparison between the open-shell character and the local aromaticity. Furthermore, in order to assess these impacts on the molecular functionality, we evaluate the second hyperpolarizabilities of these systems. These results would contribute not only to the clarification of the correlation between the open-shell character and the local aromatic features but also to the construction of new design guidelines for the highly efficient NLO species composed of the π-conjugated fused-ring systems involving antiaromatic rings.

(3) ρzzz (r) =

2. METHODOLOGY Geometry optimizations for all the systems are performed with the U(R)B3LYP/6-311+G** method under the symmetry constraints of C2h for para-1, C2 for para-2, and C2v for meta and ortho systems. The U(R)B3LYP method was employed in the previous studies, and was found to be sufficient for reproducing the X-ray crystallographic structures even though the applied basis sets were smaller in some previous cases.21−24 For each optimized geometry, we have carried out the Hessian calculations and have confirmed that all the structures are at the potential energy local minima. For the determination of the open-shell character (diradical character y and odd-electron density), the magnetic shielding tensor component −σyy and the second hyperpolarizabilities γ, we have employed the long-range corrected unrestricted density functional theory (DFT) method, LC-UBLYP, with the range separating parameter μ of 0.33 bohr−1 26 together with the basis set of 6-311+G**. These methods are found to provide semiquantitatively correct results on the optical excitation energies, transition moments, optical response properties, magnetic shielding tensors, and so on for openshell singlet condensed-ring π-conjugated systems.27−30 Within the single determinantal UDFT scheme, the diradical character is defined as the occupation number of the lowest unoccupied natural orbital (LUNO) of the unrestricted wave functions nLUNO:12,31,32 y = nLUNO = 2 − nHONO (1)

∂ 3ρ(r) ∂Fz 3

F=0

(4)

where ρ(r) is the total electron density at the position r, and Fz is the z component of the external electric field F. This γzzzz density relates to the γzzzz value as γzzzz = −

1 3!

∫ rzρzzz(3)(r) dr

(5)

According to the above relationship, a pair of positive and negative γzzzz densities with large amplitudes, separated by a large distance, contributes to the increase in γzzzz values. Considering the calculation of the triplet states, we apply the geometries optimized at singlet states because we focus on the difference of the electronic structures without the geometric change. All the calculations are performed with the Gaussian09 program package.34

3. RESULTS AND DISCUSSION 3.1. Open-Shell Character and Local Aromaticity of Indenofluorenes. At the first stage, we evaluate the open-shell singlet character for each indenofluorene system. Figure 2 shows the resonance structures together with the calculated diradical character y. It is found that the systems involving the para or ortho quinodimethane framework, referred to as para-1, para-2, and ortho, exhibit negligible diradical character (y < 0.1), and the system with the meta quinodimethane framework, referred to as meta, exhibits a larger value (y = 0.645). This implies that the para and ortho systems are categorized into nearly closed-shell systems, while the meta system is categorized into intermediate open-shell (diradical) singlet systems. Such a variation in the diradical character is consistent with the previous report,24,25 and is explained with the resonance structures (Figure 2). With their resonance

Odd-electron density ρodd at position r is calculated using the frontier natural orbitals ϕHONO and ϕLUNO as follows:12,31,32 ρodd (r) = nLUNO(|ϕHONO(r)|2 + |ϕLUNO(r)|2 )

∫ dr ρodd (r)

(2) B

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aromaticity in the six-membered rings can also be explained with the number of Clar’s sextets in the resonance structures (see Figure 2). The open-shell resonance structures of all the systems, as noted in the last paragraph, are shown to exhibit the Clar’s sextets at all the six-membered rings, which contribute to the aromaticity at all the six-membered rings. In contrast, in the closed-shell resonance structures, the para and ortho systems exhibit Clar’s sextets at both terminal rings, while the meta system exhibits Clar’s sextet at only one of the two terminal rings. As a result, the terminal six-membered rings of the nearly closed-shell systems (para-1, para-2, and ortho) exhibit fully Clar’s sextet aromatic nature for all the resonance structures, while those of the meta system exhibit less aromatic nature due to the contribution of the fully aromatic nature in the openshell resonance structure and the half in the closed-shell resonance structure. Similarly, significantly reduced antiaromatic or nonaromatic nature of the central six-membered ring of the meta system can be explained with the fact that the meta system has a larger contribution of the open-shell resonance structure, which exhibits a Clar’s sextet at the central benzene ring. These local aromaticity features are more clearly displayed with the magnetic shielding tensor maps, which visualize the spatial distributions of −σyy with color contours (Figure 3).

Figure 2. Resonance structure and diradical character y of each indenofluorene system calculated at the LC-UBLYP/6-311+G**// U(R)B3LYP/6-311+G** level of theory. Clar’s sextets are shown by the delocalized benzene-ring forms.

structures, it is easily found that the open-shell resonance structures exhibit larger numbers of Clar’s sextets, that is, all three benzene rings, than the closed-shell structures. This feature stems from the existence of the proaromatic quinodimethane structure in these molecules. More importantly, only the meta system exhibits a smaller number of Clar’s sextets in the closed-shell form than other systems (one for meta, and two for para and ortho systems). This difference explains the relatively larger stability of the open-shell resonance structures, resulting in the larger diradical character of meta system than those of para and ortho ones. In order to evaluate the local aromaticity of the present systems, we next investigate the magnetic shielding tensor component (−σyy) 1 Å above the center of each six- and fivemembered ring plane35,36 (Table S1). The negative and positive −σyy values indicate local aromatic and antiaromatic features of each ring, respectively. It is found that, for all the systems, the middle three rings, the six-membered ring together with the adjoining two five-membered rings, exhibit positive −σyy (antiaromatic nature), while the terminal benzene rings exhibit negative −σyy (aromatic nature). It should be however noted that the meta system with an intermediate y value exhibits considerably different amplitudes of −σyy as compared to those of the other systems. For the nearly closed-shell para and ortho systems, the −σyy values of the aromatic terminal benzene rings are around −19 ppm, and those of the antiaromatic central six-membered rings are around 7.0−9.5 ppm, while the meta system exhibits less aromatic terminal (−11.1 ppm) and significantly reduced antiaromatic or nonaromatic central (0.2 ppm) six-membered rings. These observations show that the aromaticity of the six-membered rings correlates with the open-shell character and, more specifically, with the feature that the difference in the local aromaticity between the central and the terminal rings is much smaller in the intermediate open-shell meta system than in the nearly closed-shell para and ortho systems. This variation of the

Figure 3. −σyy maps in the singlet states calculated at the LC-UBLYP/ 6-311+G** level of theory.

Here, the aromatic region (with negative −σyy) is shown with blue contours, while the antiaromatic region (with positive −σyy) is shown with yellow ones. As seen from these contour maps, the central benzene ring together with the adjoining two five-membered rings exhibits yellow contours, i.e., local antiaromatic nature, except for the meta system, whose central benzene ring exhibits almost white contours, i.e., local nonaromatic nature. Such spatial features of −σyy maps give more detailed spatial contribution features of the local aromaticities in the indenofluorene series discussed in the last paragraph. In order to clarify the spatial correlation between the local aromaticity and the open-shell singlet character, we also examine the maps of odd (unpaired) electron density distribution12,31,32 (Figure 4). It is found from these figures that large odd-electron densities are generally distributed around the zigzag-edge region of the five-membered rings. Judging from the fact that this feature is consistent with that observed in the open-shell resonance structures, the oddC

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Figure 5. −σyy maps (left) and odd-electron densities (right) of para-1 and meta systems in the triplet states calculated at the LC-UBLYP/6311+G** level of theory (contour value of 0.004 a.u. for the oddelectron density distributions).

Figure 4. Odd-electron densities in the singlet states calculated at the LC-UBLYP/6-311+G** level of theory (contour value of 0.0004 a.u. except for meta system (0.004 a.u.)).

benzene rings are stabilized with Clar’s sextets, while the singlet systems have contribution of closed-shell resonance structures, which in particular reduce the aromaticity or increase the antiaromaticity in the central six-membered ring (see Figure 2). It is furthermore found that the odd-electron density distribution feature of the para-1 system in the triplet state (see Figure 5) significantly differs from that in the singlet state (see Figure 4), and resembles that of the open-shell singlet meta system, where the odd-electron density distributions are also observed in the terminal benzene rings in addition to the central one. These observations of the spatial distributions of the local aromaticity and the odd-electron density support the present prediction that there exists a correlation between the difference in the local aromatic nature and that in the amplitudes of odd-electron densities of the six-membered rings. 3.2. Second Hyperpolarizaibilities of Indenofluorene Systems. In order to confirm the impact of the open-shell character and the aromaticity on the molecular functionality, we address the longitudinal components of second hyperpolarizabilities γ, γzzzz, which are the dominant components for all the indenofluorene systems. Although there is a report on the calculation of the γ values for these indenofluorene series,25 the interplay between the open-shell character, aromaticity, and γ values, as well as their spatial contributions in both the singlet and triplet states, has never been investigated. Calculated γzzzz values together with the γzzzz densities in singlet states are shown in Figure 6. It is found that, among these indenofluorene series, the intermediate open-shell meta system exhibits the largest γzzzz values, of which tendencies agree with the previous consideration25 of γzzzz of indenofluorene series and our previous results.14−16 On the other hand, among the almost closed-shell systems, the para-2 system exhibits almost half the value of those of para-1 and ortho systems. It is expected that this variation of the γzzzz values stems from the relatively smaller system size along the longitudinal axis and slightly nonplanar structure of the indenofluorene framework due to the steric repulsion. For the spatial distributions of γzzzz densities, it is found that the spatial distributions qualitatively vary according to the amplitudes of the open-shell character. It is found that, for the almost closed-shell para and ortho systems, the γzzzz densities are primarily localized in the quinodimethane frameworks, while that for the meta system, that spreads over the molecule, and is not localized in the quinodimethane region. These tendencies lead to the spatial-distribution

electron density maps also substantiate Clar’s sextet rule in these molecules. Focusing on the six-membered rings of the odd-electron density maps, it turns out that, for the para and ortho systems, odd-electron densities are more significantly distributed at the central benzene rings than at the terminal ones, while for the meta system, odd-electron densities are more delocalized over both the central and terminal benzene rings. This distribution difference is not straightforwardly understood from the resonance structures. Interestingly, the primary odd-electron density distribution region is found to well correspond to the local antiaromatic or weaker aromatic ones of the six-membered rings. As discussed before, the difference in the local aromaticity between the six-membered rings is more distinct in the para and ortho systems than in the meta system, the feature of which corresponds to the fact that, in the para and ortho systems, the odd-electron densities in the six-membered rings are shown to be distributed primarily in the central antiaromatic six-membered rings, while in the meta system, the odd-electron densities in the six-membered rings are shown to be shared in both the middle and terminal rings. This indicates that, for each indenofluorene system, the sixmembered rings with larger odd-electron densities exhibit relatively antiaromatic nature. This observed spatial correlation between the odd-electron density and local aromaticity is reasonable since the emergence of odd-electron density in the aromatic ring implies the partial destruction of the full πdelocalization over the ring, resulting in the reduction of aromaticity or in the emergence of antiaromaticity. In order to further confirm the present prediction, we examine the odd-electron density together with the −σyy maps of the triplet states, which correspond to the pure open-shell states (Figure 5 for para-1 and meta systems and Figure S1 for the other systems). Since the para-2 and ortho systems exhibit qualitatively the same result as the para-1 system, we discuss only the para-1 and meta systems. As seen from the −σyy maps and their values of these triplet states, unlike the corresponding singlet systems (which exhibit antiaromaticity in the central benzene rings (Figure 3)), both structures exhibit a similar spatial feature that all the benzene rings exhibit similar aromatic nature (Table S2, Figures 5 and S1). This is explained by the fact that the triplet states are described as pure open-shell resonance structures for both the systems, where all the D

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molecular frameworks, not localized at the central quinodimethane region. In this regard, the spatial distributions of the closed-shell para and ortho systems in the triplet states resemble that of the meta system in the singlet state. In the triplet states of all the systems, however, the γzzzz densities are not shown to have large distribution around the zigzag edge region of the five-membered rings, where the large odd-electrons are distributed. These features indicate that, in the triplet states, the spatial contribution to the γzzzz is not so correlated to the odd-electron density distributions unlike the intermediate open-shell singlet systems. This is predicted to be caused by the Pauli effects,37 where triplet diradical electrons are prohibited to be delocalized due to the Pauli principle and thus do not contribute to the enhancement of γzzzz. Thus, in the above two sections on the original indenofluorene frameworks, we have shown that there exists strong correlation between the absolute values and the spatial distributions of open-shell character, local aromaticity, and γ values. The emergence and the variation of the open-shell character, which give impact on the aromaticity and the γ values, can be explained with the relative stability of the openshell resonance structures, which varies with the number of Clar’s sextets in the closed-shell and the open-shell resonance structures. In this regard, within the indenofluorene frameworks, such a variation of the relative stability of the resonance structures is achieved with the geometrical difference around the antiaromatic five-membered rings. With these understandings, in section 3.3, we investigate the larger analogue of the para-1 system, which exhibits larger relative stability of the open-shell resonance structures even though the system exhibits para-1 type geometrical structure, and will show that the correlations obtained in the original indenofluorene series are qualitatively preserved even in the extended systems. 3.3. Larger Analogues of para-1 and meta Systems. Figure 7 shows the resonance structures of extended para-1 and meta systems, which exhibit naphthalene structure instead of the central benzene ring of the original systems. With the resonance structures, it is easily found that the present extended systems exhibit larger stabilization of the open-shell structure originating from the increase of the number of open-shell resonance structures, though the number of Clar’s sextets in

Figure 6. γzzzz density maps together with the γzzzz values in the singlet states calculated at the LC-UBLYP/6-311+G** level of theory (contour value of 1000 a.u.).

similarity of the γzzzz densities to the other two spatial distributions, i.e., odd-electron density and −σyy maps, especially around the six-membered ring regions. Considering the triplet states, it is found that, for nearly closed-shell para and ortho systems, the amplitudes of the γzzzz in the triplet states are comparable to those in the singlet states, triplet γsinglet zzzz /γzzzz = 1.21 for para-1, 0.991 for para-2, and 0.928 for ortho, while for the meta system the γzzzz value in the triplet state exhibits only a half amplitude of the singlet state γzzzz value; in other words, the γzzzz value in the singlet state is about triplet twice that in the triplet state, γsinglet zzzz /γzzzz = 2.18 for meta (the absolute values of γzzzz and the γzzzz density distributions in the triplet states are shown in Figure S2). These observations indicate that the intermediate open-shell character strongly enhances the γ value only in the meta system. On the other hand, for nearly closed-shell para and ortho systems, despite their similarities of γ zzzz amplitudes, the γ zzzz density distributions in the triplet states are shown to be considerably different from those in the singlet states. All the γzzzz density distributions in the triplet states are found to spread over all the

Figure 7. Resonance structures of extended para-1 and meta systems. E

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The γ values on these extended systems are also shown to well describe the characteristic feature of the intermediate open-shell singlet systems. The extended para-1 and meta systems exhibit much larger γzzzz values (832000 a.u. and 758000 au for extended para-1 and meta systems, respectively) than the original systems (277000 a.u. and 370000 a.u. for original para-1 and meta systems, respectively). Especially, the extended para-1 exhibits the γzzzz value of about three times that of the original para-1 system, while the γzzzz of the extended meta system is about twice the value of original one. This difference of the enhancement can be explained with the variation of the open-shell singlet character of the para-1 type systems. With the extension of the central benzene ring region of the original para-1 system, the intermediate open-shell singlet character emerges, while for the meta system, the original meta system already exhibits intermediate open-shell singlet character. The impact of the open-shell singlet character is confirmed not only with the similarity of the γzzzz density to the odd-electron density in the singlet states (Figures 8 and S4) but also with the comparison of the γzzzz values between the singlet and the triplet states. It is found for both the extended systems that the ratio of the γzzzz values of the singlet state to triplet that of the triplet state, γsinglet zzzz /γzzzz , is about 2.3 and this ratio is similar to that of the original meta system (2.18). Considering the fact that the ratio of the almost closed-shell original para-1 system is only 1.21, the variation of the ratio is predicted to be quite well correlated with the open-shell singlet character of each system. At the last part, we compare the γzzzz values with one of the real open-shell singlet NLO molecules, s-indaceno[1,2,3-cd;5,6,7-c′d′]diphenalene (IDPL).38 At the same level of theory applied in the present paper, it is found that IDPL exhibits a diradical character y of 0.700 and a longitudinal second hyperpolarizability γzzzz of 2080000 a.u., which is about 2.5 times as large as those of the extended systems. In this regard, the present systems themselves are predicted to be not so superior to IDPL, and while taking it into account that there exist several analogues of diradical and potentially tetraradical species,39−41 the indenofluorene frameworks are expected to be the potential building blocks of the open-shell singlet NLO molecules with larger π-conjugated fused-ring structures.42 Thus, it is concluded that the extension of the central benzene ring of the para-1 system to the naphthalene moiety, as expected, results in the emergence of the intermediate openshell singlet character. This suggests that the extension of the central benzene ring of the para-1 system impacts on the increase of the open-shell character similarly to the geometric change to the meta type structure, the fact of which can be expected from the resonance structures.

each structure is the same as the original ones. In fact, the extended systems exhibit larger open-shell character as compared with the original indenofluorene systems, resulting in intermediate open-shell singlet character for both systems (y = 0.434 for extended para-1 and 0.772 for extended meta systems) though the para-1 type system still exhibits smaller diradical character as compared to that of the meta type one. Considering the previously obtained correlation between the open-shell character and the local aromaticity, it is expected that such an increase of the open-shell character affects the value and the spatial contributions of the odd-electron density and the local aromaticity. Figure 8 shows the −σyy maps and the

Figure 8. −σyy maps (left) and odd-electron densities (right) of extended para-1 system in the singlet state calculated at the LCUBLYP/6-311+G** level of theory (contour value of 0.004 a.u. for the odd-electron density distributions).

odd-electron density distributions of the extended systems in the singlet states. As seen from the −σyy maps, the extended para-1 system exhibits weak aromatic character at the middle naphthalene region, the fact of which is considerably different from that of the original para-1 system, whose central benzene ring exhibits antiaromatic character (see also Tables S1 and S3). In this regard, the qualitative variation of the spatial feature of the −σyy map of the present extended para-1 system is similar to that of the larger open-shell indenofluorene systems, meta and the triplet systems, though direct comparisons between the absolute values are difficult because of the structural difference. Furthermore, the extended meta system exhibits larger aromatic character at the middle naphthalene region even in the singlet state, which is almost the same value as that in the triplet state (Table S4). This tendency is different from the other two intermediate open-shell singlet systems, extended para-1 and original meta systems, and reflects the relatively larger openshell singlet character of the extended meta one. Namely, the extension of the middle ring structure of the indenofluorene series tends to increase the open-shell singlet character, and to make the middle benzene rings relatively aromatic, resulting in reducing the gap of the aromaticity between the middle and terminal benzene rings. The odd-electron density distributions of extended systems are also found to differ from the original systems. The extended systems exhibit less density distribution at the middle naphthalene region than that of the central benzene ring region of the original ones. This observation agrees with the correlation obtained in section 3.1 between the open-shell character and the local aromaticity, in the point that the middle naphthalene region exhibits more aromatic character together with the less density of the odd-electron density distribution.

4. CONCLUSION In this study, we have investigated the spatial correlation between the open-shell character, aromaticity, and second hyperpolarizabilities by focusing on the synthesized indenofluorene frameworks. It has been found that the spatial distributions of open-shell character and the magnetic criteria of the local aromaticity vary as a function of the amplitude of open-shell character. Especially, the emergence of local antiaromatic nature in the six-membered rings is found to be related to the large odd-electron density distribution. From these results, we conclude that the distribution of the magnetic criteria of the local aromaticity, −σyy, correlates with the oddelectron density distribution. Further investigation of the second hyperpolarizabilities γzzzz has shown that there also F

DOI: 10.1021/acs.jpca.5b08520 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

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exist spatial correlations in the primary distributions between the γzzzz densities and the odd-electron densities in the small and intermediate open-shell singlet region. In order to confirm these observations, we also consider the extended indenofluorene systems. It is revealed that the replacement of the central benzene ring with the larger naphthalene moiety increases the open-shell character, resulting in the intermediate open-shell singlet character for both the extended para-1 and meta systems. The correlation between the open-shell character and the local aromaticity obtained in the original indenofluorene series is shown to be fulfilled even in the extended indenofluorene frameworks. The variation of the open-shell singlet character is also found to give a large effect on the γ values. The correlation between the open-shell singlet character and the local aromaticity of indenofluorene series is demonstrated to be achieved by the existence of the various proaromatic moieties with antiaromatic five-membered rings within the molecular structure. However, only the existence of antiaromatic rings is found to be insufficient for obtaining the intermediate open-shell singlet character, which enhances the second hyperpolarizabilities. In this regard, with the help of the resonance structures, it turns out that the meta-type geometrical framework or the extension of the middle benzene ring region stabilizes the open-shell resonance structure, resulting in the intermediate open-shell singlet character. These findings would contribute to a deeper understanding of the controllability of the open-shell singlet character of the π-conjugated fused-ring frameworks involving antiaromatic rings, which is expected to be useful for designing high functional molecules for efficient NLO and singlet fission.17,43

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.5b08520. Calculation details and the other detailed data (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Fax: +81-6-6850-6268. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by JSPS Research Fellowship for Young Scientists (No. 15J04949), Grant-in-Aid for Scientific Research (A) (No. 25248007) from Japan Society for the Promotion of Science (JSPS), a Grant-in-Aid for Scientific Research on Innovative Areas “Stimuli-Responsive Chemical Species” (No. A24109002a), “π-System Figuration” (15H00999), “Photosynergetics” (A26107004a), MEXT, the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan.

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REFERENCES

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DOI: 10.1021/acs.jpca.5b08520 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.jpca.5b08520 J. Phys. Chem. A XXXX, XXX, XXX−XXX