Chemical Physics Letters 480 (2009) 278–283

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Chemical Physics Letters

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Third-order nonlinear optical properties of trigonal, rhombic and bow-tie graphenenanoflakes with strong structural dependence of diradical character

Kyohei Yoneda a, Masayoshi Nakano a,*, Ryohei Kishi a, Hideaki Takahashi a, Akihiro Shimizu b,Takashi Kubo b, Kenji Kamada c, Koji Ohta c, Benoît Champagne d, Edith Botek d

a Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560 8531, Japanb Department of Chemistry, Graduate School of Science, Osaka University, Toyonaka, Osaka 560 0043, Japanc Photonics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Ikeda, Osaka 563 8577, Japand Laboratoire de Chimie Théorique Appliquée Facultés Universitaires Notre-Dame de la Paix (FUNDP), rue de Bruxelles 61, 5000 Namur, Belgium

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 August 2009In final form 13 September 2009Available online 17 September 2009

0009-2614/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.cplett.2009.09.047

* Corresponding author.E-mail address: mnaka@cheng.es.osaka-u.ac.jp (M

Using broken-symmetry hybrid density functional theory, we investigate the relationship between thediradical character and the second hyperpolarizabilities, c, for a trigonal graphene nanoflake (trigonalGNF) and two types of GNFs with rhombic and bow-tie structures composed of two trigonal GNFs. It isshown that the GNF architecture has a large impact on the diradical character and subsequently on c.In particular, c of rhombic GNF with intermediate diradical character is found to be more than twiceas large as that of bow-tie GNF with nearly pure diradical character. Then, switching from the singletto triplet state rhombic GNF exhibits a significant reduction of c.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Over the past few years, a huge number of experimental andtheoretical studies on graphenes have been performed from chem-ical and physical viewpoints because they exhibit many uniqueelectronic and spin properties, and thus are expected to be usedin a wide range of applications in future electronics, spintronicsand photonics [1–16]. These systems are constructed from benzenerings and thus have two types of edges, armchair and zigzag edges.It has already been elucidated that nanographenes with low-dimensional structures, i.e., one-dimensional graphene ribbons[7–16], and finite-size graphene nanoflakes (GNFs) with variousshapes [17–22], exhibit unique edge states with localized p-electron distribution at the zigzag edges. For example, the localizedelectrons at the both-end zigzag edges of rectangular GNFs arefound to form up and down spin densities, respectively, which im-plies that such GNFs are predicted to be antiferromagnetic, i.e., topresent an open-shell singlet ground state [9–16]. Then, such un-ique edge states of GNFs are closely related to their magneticand electronic properties, and strongly depend on their sizes andshapes [9–27].

On the other hand, based on theoretical and computationalstudies, we have recently found a novel class of nonlinear optical(NLO) systems, i.e., open-shell singlet molecular systems, [28–38]. In such molecular systems, the second hyperpolarizability(c) – the microscopic origin of the third-order NLO properties –

ll rights reserved.

. Nakano).

exhibits a remarkable dependence on the diradical character[39,40]: the systems with intermediate y value tend to exhibit alarge enhancement of c as compared to the closed-shell and purediradical systems. This dependence has been theoretically clarifiedby the valence configuration interaction (VCI) method for two-sitediradical model [36], and has been exemplified by the spin-unrestricted post Hartree–Fock (UHF) and density functionaltheory (DFT) calculations on various real and model systems[28–38] including square and hexagonal GNFs [37,38]. Experimen-tally, polyaromatic diphenalenyl diradicals have been found toexhibit huge two-photon absorption cross section (a typicalthird-order NLO property) [41], which is in agreement with ourstructure–property relationship of c in diradical systems. More-over, the systems with intermediate diradical character exhibitstrong spin state dependence of c: a significant reduction of camplitude by changing singlet to triplet state [32,37].

In our previous study [38], we have examined the spin polariza-tion, diradical character and c values of hexagonal GNFs (HGNFs)with armchair and zigzag edges, and have found that zigzag-edgedHGNFs possess intermediate diradical character even in the rela-tively large size as compared to square GNFs, causing a furtherenhancement of c values in the direction of the spin polarization.In this study, therefore, we investigate the structure–property rela-tionships in c for a trigonal GNF and two types of GNFs with rhom-bic and bow-tie structures composed of two trigonal GNFs, all ofwhich possess zigzag edges. On the basis of the present results,we discuss the possibility of structural control of diradical charac-ter and thus of c values for GNFs composed of trigonal GNF build-ing blocks as well as spin state control of the c values.

K. Yoneda et al. / Chemical Physics Letters 480 (2009) 278–283 279

2. Methodology

2.1. Graphene nanoflakes (GNFs) and diradical characters

Fig. 1 shows the structures of trigonal (a), rhombic (b) and bow-tie (c) GNFs, which are optimized using the spin-unrestricted (U)B3LYP/6-31G* method in the singlet state. The optimized geometryof singlet trigonal GNF belongs to D3h symmetry, while for singletrhombic and bow-tie GNFs, their geometries belong to D2h symme-try. It is found that only the optimized geometry of singlet rhombicGNF coincides with that using the spin-restricted (R) B3LYP/6-31G*method. For singlet trigonal GNF, the bond-length alternation(BLA) of carbon(C)–carbon(C) bond, the increase of which repre-sents a decrease of aromaticity, is enhanced toward the end ofthe edges [0.029 Å (a–b, which indicates the bond length differencebetween a and b shown in Fig. 1) vs. 0.012 Å (b–c)], and that in thecenter [0.0 Å (e–e)] disappears (by symmetry). For singlet rhombicand bow-tie GNFs, the BLA is smaller in the center regions of con-stituent trigonal GNF blocks than in other regions: 0.011 Å (g0–i0)and 0.0 Å (k0 0–l0 0), whereas the BLA is significantly increased inthe linked regions of trigonal GNF blocks: 0.090 Å (e0–f0) and

Fig. 1. Structures of trigonal (a), rhombic (b) and bow-tie (c) GNFs optimized by the UB3represented by dark and white circles, respectively.

0.093 Å (d0 0–e0 0). In order to investigate the spin state dependencesof c, we also examine these systems in the triplet states with thesame geometries as those of their singlet states.

The diradical character yi related to the HOMO�i and LUMO+i isdefined by twice the weight of the doubly-excited configuration inthe multi-configurational (MC)-SCF theory [39,40]. In this study,the yi is calculated using spin-unrestricted Hartree–Fock (UHF) cal-culations. Within the spin-projected UHF (PUHF) theory, this is for-mally expressed as [39,40]

yi ¼ 1� 2T

1þ T2i

; ð1Þ

where Ti, the overlap between the corresponding orbital pairs, canbe expressed in terms of the occupation numbers (ni) of the UHFnatural orbitals (UNOs) [39,40]:

Ti ¼nHOMO�i � nLUMOþi

2: ð2Þ

The diradical character yi takes a value between 0 and 1, which cor-responds to closed-shell and pure diradical states, respectively. Thepresent scheme using the UNOs is the simplest but it can well

LYP/6-31G* method as well as the coordinate axis. Carbon and hydrogen atoms are

280 K. Yoneda et al. / Chemical Physics Letters 480 (2009) 278–283

reproduce the diradical characters calculated by other methodssuch as the ab initio configuration interaction (CI) method [42].

2.2. Calculation and analysis methods of c

It was found in our previous studies that the UBHandHLYPmethod [43] provides reliable c values of diradical systems withintermediate and large diradical characters [31]. The UBHandHLYPmethod is, therefore, employed to calculate the cxxxx diagonal ten-sor components, where x-axis is the primary spin polarizationdirection for singlet rhombic and bow-tie GNFs (see Figs. 2 and3). For trigonal GNF, the D3h symmetry ensures that cxxxx = cyyyy

Since c strongly depends on the size of systems (i.e., in p-conju-gated systems c increases nonlinearly with increasing the p-conju-gation lengths until saturation is reached [44]), rhombic and bow-tie GNFs have been chosen with similar dimension in the x direc-tion, so that one could highlight the spin polarization effect oncxxxx, i.e., diradical character dependence of cxxxx.We employ the fi-nite-field (FF) approach to calculate the static cxxxx values. This ap-proach consists in the fourth-order differentiation of the energy orthe third-order differentiation of the x-component of dipole mo-ment with respect to different amplitudes of the applied externalelectric field [28,32]. The employed differentiation scheme andfield amplitudes are respectively chosen so as to reduce thenumerical error to within 1% of the cxxxx value. All calculationsare performed using the GAUSSIAN 03 program package [43]. In orderto elucidate the spatial contributions of the electrons to cxxxx, weperform c density [qð3ÞxxxðrÞ] analysis [44], which utilizes the third-order derivative of the electron density with respect to the appliedfield:

qð3ÞxxxðrÞ ¼@3q

@Fx@Fx@Fx

�����F¼0

: ð3Þ

Using this density, c is represented by

cxxx ¼ �13!

Zrxqð3ÞxxxðrÞdr; ð4Þ

where rx denotes the x component of the position vector r. As a sim-ple example, we consider a pair of localized cxxxx densities with po-sitive and negative values. The contribution of electrons to cxxxx ispositive when the direction of an arrow drawn from positive to neg-ative cxxxx density coincides with the positive direction of the x axis.

Fig. 2. Clar’s sextets (circles) in resonance forms of trigonal (a), rhombic (b) an

The amplitude of the contribution associated with this pair ofqð3ÞxxxðrÞ is also proportional to the distance between them.

3. Results and discussion

3.1. Diradical character and spin density distribution

The UBHandHLYP/6-31G* singlet–triplet energy gaps[DES-T � ETðtripletÞ � ETðsingletÞ] of these systems are given in Ta-ble 1. For trigonal GNF, DES-T amounts to �0.664 eV, which meansthe ground state is triplet, while DES-T for both rhombic and bow-tie GNFs are positive and the ground states are singlets [DES-

T = 1.041 (rhombic) and 0.188 eV (bow-tie)]. This difference inthe ground-state spin state for these systems can be explainedby Lieb’s theorem [45]: when all C atoms in a system are alter-nately marked and are separated into marked and unmarked sites,the ground state has spin S = |n* � n|/2, in which n* and n representthe numbers of marked and unmarked C atoms, respectively. Theseresults on the spin multiplicity of the GNFs ground-state are ingood agreement with this theorem.

Table 1 lists the diradical characters y0 [Eqs. (1) and (2)] of sin-glet trigonal (y0 = 0.996), rhombic (y0 = 0.418) and bow-tie(y0 = 0.970) GNFs. These systems are regarded as diradical systemsbecause other yi values (i P 1) are sufficiently small comparedwith y0 values: y1 = 0.006 (trigonal), 0.086 (rhombic) and 0.204(bow-tie). Trigonal and bow-tie GNFs are nearly pure diradical sys-tems, whereas rhombic GNF has an intermediate diradical charac-ter. The small DES-T for bow-tie GNF relative to that of rhombicGNF corresponds well to the nearly pure diradical character ofbow-tie GNF. Because rhombic and bow-tie GNFs, composed oftwo zigzag-edged trigonal GNFs, possess similar p-conjugationlengths, the significant difference in y0 between them is predictedto be caused by other structural effects, as it can be understood bycomparing their resonance structures (Fig. 2) based on the Clar’ssextet rule [46]. Indeed, rhombic GNF displays both closed-shelland diradical resonance, while there are no closed-shell forms fortrigonal and bow-tie GNFs. The number of Clar’s sextets in thediradical form of rhombic GNF (3) is the same as that in theclosed-shell form and is also less than that of diradical bow-tieGNF (4). This suggests that the thermal stabilities of closed-shelland diradical forms are similar for rhombic GNF.

The features of BLAs clarified in Section 2.1 are also found to berelated to these differences in diradical characters. For example,

d bow-tie (c) GNFs. The number of Clar’s sextets are given in parentheses.

Fig. 3. Spin density distributions of singlet trigonal (a), rhombic (b) and bow-tie (c) GNFs as well as triplet trigonal (d), rhombic (e) and bow-tie (f) GNFs calculated by theUBHandHLYP/6-31G* method. The yellow and blue surfaces represent a and b spin densities with ±0.02 a.u. iso-surfaces, respectively. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)

Table 1Singlet–triplet energy gaps (DES-T)a, diradical characters (y0)b and second hyperpo-larizabilities (cxxxx)c for trigonal, rhombic and bow-tie GNFs in the singlet and tripletstates.

DES-T

[eV]Spin state y0 [�] cxxxx

[�103 a.u.]

Trigonal �0.664 (�0.662)d Singlet 0.996 21.4– Triplet – 21.0 (21.0)d

Rhombic 1.041 (0.809)d Singlet 0.418 452– Triplet – 115 (101)d

Bow-tie 0.188 (0.183)d Singlet 0.970 199– Triplet – 191 (189)d

a DES�T (� ETðtripletÞ � ETðsingletÞ) are calculated using the UBHandHLYP/6-31G* method.

b y0 are obtained from the occupation numbers of UNOs/6-31G*.c c is calculated using the UBHandHLYP/6-31G* method.d Values in parentheses are obtained by optimizing the geometries in the triplet

states.

K. Yoneda et al. / Chemical Physics Letters 480 (2009) 278–283 281

the significant increases in BLAs in linked regions of trigonal GNFblocks, 0.090 Å (e0–f0) and 0.093 Å (d0 0–e0 0) (see Fig. 1b and c), inrhombic and bow-tie GNFs, respectively, represent significant con-tributions to closed-shell configurations in rhombic and diradicalform in bow-tie GNFs (see Fig. 2b and c). The larger diradical char-acter for bow-tie GNF than for rhombic GNF is well exemplified bythese resonance forms characterized by their BLAs.

Fig. 3 shows the spin density distributions of the singlet (a, b, c)and triplet (d, e, f) GNFs. For all systems, the spin density distribu-tions is primarily localized at the zigzag edges, the amplitudes ofwhich are significantly larger than in the center regions. For singletstates, the dominant spin polarization is observed between right-and left-hand side edges (in the x-direction) for rhombic andbow-tie GNFs, while it is observed in the y-direction for trigonalGNF, though this system can also exhibit spin polarization in otherdirections owing to its D3h symmetry. The amplitudes of spin den-sities of singlet trigonal and bow-tie GNFs are also larger than thatof singlet rhombic GNFs, exemplifying the relative amplitudes of y0

for these systems.

In the triplet state, the spin density amplitude of bow-tie GNF issimilar to that in the singlet state, while for rhombic GNFs, the spindensity is larger than that in the singlet state. This difference orig-inates in the intermediate diradical character in singlet rhombicGNF in contrast to the nearly pure diradical character in singletbow-tie GNF. These comparisons between the triplet states arebased on singlet optimized geometries. As listed in Table 1, how-ever, additional calculations have shown that when using the trip-let optimized geometries, c changes by less than one percent forthe trigonal and bow-tie GNFs, while c decreases by about 12%for the rhombic GNF. Similar changes are also observed in the sin-glet–triplet gap energies. Such large changes in rhombic GNF orig-inate from its intermediate diradical character in contrast to othershaving nearly pure diradical characters, which provide similar spa-tial distributions of spin density amplitudes and charges to the cor-responding triplet states. Consequently, using the singletgeometries has a limited impact on the analysis of the c results.

3.2. Diradical character and spin state dependences of c

The cxxxx values of these GNFs in their singlet and triplet statesare listed in Table 1. cxxxx of the singlet rhombic GNF (y0 = 0.418,cxxxx = 452 � 103 a.u.) is more than twice that of singlet bow-tieGNF (y0 = 0.970, cxxxx = 199 � 103 a.u.). Changing from singlet totriplet for rhombic GNF, the cxxxx value is significantly reduced byabout a factor of 4 (cxxxx = 115 � 103 a.u.). In contrast, trigonaland bow-tie GNFs hardly exhibit spin state dependences of cxxxx.These results are in conformity with our previous results on thediradical NLO systems [28,32].

Since the switching of the spin polarization between the twosides of the bow-tie GNF hardly changes the amplitudes (seeFig. 3c and f), the dependence of cxxxx on the degree of spin polar-ization is negligible in bow-tie GNF. Then, one can predict that thedifference in p-conjugation length is the origin of the fact thatbow-tie GNF presents cxxxx values one order of magnitude largerthan trigonal GNF. A similar argument can also account for the dif-ference in cxxxx value between the triplet rhombic and bow-tie

Fig. 4. cxxxx density distributions of singlet trigonal (a), rhombic (b) and bow-tie (c) GNFs as well as triplet trigonal (d), rhombic (e) and bow-tie (f) GNFs calculated by theUBHandHLYP/6-31G* method. The yellow and blue meshes represent positive and negative cxxxx densities with ±500 a.u. iso-surfaces, respectively. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)

282 K. Yoneda et al. / Chemical Physics Letters 480 (2009) 278–283

GNFs, which further substantiates the strong diradical effect on thecxxxx value of the corresponding singlets.

The cxxxx density [qð3ÞxxxðrÞ] for these GNFs in the singlet and trip-let states are shown in Fig. 4 and are described by p-electron con-tributions. The singlet trigonal GNF (Fig. 4a) shows negligibleamplitudes of cxxxx densities as compared to other systems. Thecxxxx density amplitudes for singlet rhombic GNF (Fig. 4b) are obvi-ously the largest and dominant positive and negative cxxxx densitiesare shown to be distributed on the left- and right-hand side trigo-nal blocks (particularly on the zigzag edges), respectively, thoughalternate change in sign of cxxxx densities are observed, causingsome cancellations between positive and negative contributionsto cxxxx. This exemplifies the enhanced cxxxx value for singlet rhom-bic GNF. For singlet bow-tie GNF (Fig. 4c), the positive and negativep-electron cxxxx densities distributed well separately on the left-and right-hand side trigonal blocks, respectively, give the positivecontribution to the cxxxx values though the amplitudes of cxxxx den-sities on the zigzag edges rapidly decrease toward the central re-gions in contrast to singlet rhombic GNF. It is also found that achange from singlet to triplet state remarkably reduces the ampli-tudes of cxxxx densities in rhombic GNF, while provides negligibleeffect on the cxxxx densities for other systems. In general, these cxxxx

density distributions support coherently the structural and spinstate dependences of cxxxx.

4. Summary

In this Letter, we have theoretically elucidated the relationshipsbetween the third-order NLO properties of trigonal, rhombic andbow-tie graphene nanoflakes (GNFs), their diradical characters aswell as their spin multiplicities. First, the rhombic GNF presentsintermediate diradical character, contrary to trigonal and bow-tieGNFs having nearly pure diradical characters. This difference hasbeen explained in terms of their resonance forms and Clar’s sextetrule. Therefore, the diradical character for GNFs depends not onlyon the edge shapes, i.e., zigzag or armchair, and their p-conjuga-tion lengths, but also on their geometry or shape. Then, the longi-tudinal second hyperpolarizability (c) of singlet rhombic GNF hasbeen calculated to be very large, and about twice larger than in sin-

glet bow-tie GNF having similar p-conjugation length. Moreover,switching from the singlet to the triplet, c of the bow-tie structurehardly changes whereas it decreases by a factor of 4 for rhombicGNF. These results are in agreement with our structure-c relation-ship for diradical systems. According to the present results, we canfurther conclude that GNFs constitute a promising class of open-shell NLO materials because of the possibility to control c byadjusting their shape and the nature of the edge in addition tothe p-conjugation size. The investigation of larger size GNFs com-posed of plural and/or larger-size trigonal building blocks are inprogress.

Acknowledgements

This work is supported by Grant-in-Aid for Scientific Research(Nos. 21350011 and 20655003) and ‘Japan-Belgium CooperativeProgram’ from Japan Society for the Promotion of Science (JSPS),Grant-in-Aid for Scientific Research on Priority Areas from the Min-istry of Education, Culture, Sports, Science and Technology (MEXT)(No. 18066010), and the global COE (center of excellence) program‘Global Education and Research Center for Bio-EnvironmentalChemistry’ of Osaka University. E.B. thanks the IUAP program N�P6-27 for her postdoctoral grant. B.C. thanks the Fund for ScientificResearch – FNRS for his Research Director position.

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