Time-dependent density functional study of the static secondhyperpolarizability of BB-, NN- and BN-substituted C60

Lasse Jensen a,*, Piet Th. van Duijnen a, Jaap G. Snijders a, Delano P. Chong b

a Theoretical Chemistry, Material Science Centre, Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AG Groningen, Netherlandsb Department of Chemistry, University of British Columbia, 2036 Main Mall, Vancouver, BC, Canada V6T 1Z1

Received 6 February 2002; in final form 3 May 2002

Abstract

In this work we have investigated the effects of substituting carbon atoms with B and N on the second hyperpo-

larizability of C60 using time-dependent density functional theory. We have calculated the second hyperpolarizability of

the double substitute-doped fullerenes C58NN, C58BB and C58BN. For C60 only small changes in the second hyper-

polarizability were found when doping with either 2B or 2N. However, by doping C60 with both B and N, creating an

donor-acceptor system, an increase in the second hyperpolarizability with about 50% was found. � 2002 Elsevier

Science B.V. All rights reserved.

1. Introduction

Molecules which exhibit large nonlinear optical(NLO) response properties are of great techno-logical importance. They define a new generationof compounds useful in information technology,where rapid communication processes are basedon optical methods rather than electronics [1,2].Conjugated organic molecules with delocalizedelectron systems are interesting because of theirpotentially large optical response properties.Fullerenes and carbon nanotubes have an ex-tended p-system and are therefore promising can-didates for new photonic materials.

Recently both experiments [3,4] and theory[5–7] have shown that the third-order nonlinearityof C60 is smaller than first assumed. For this rea-son the third-order nonlinearity of derivatives ofC60 have been experimentally investigated andfound to provide a large increase in the nonlin-earity compared with the pure C60, see e.g. [4,8].Theoretically, substitute-doping of the fullereneC60 with B or N has been suggested as another wayof increasing the nonlinearity of the fullerenes[9,10]. These studies have been done using an ex-tended Su–Schrieffer–Heeger (SSH) model [11] andshow that doping enlarges the second hyperpo-larizability by several orders of magnitude com-pared to the pure fullerenes, suggesting B-, N-, andBN-doped fullerenes as serious candidates forphotonic devices. Since the SSH model predictsresults [9] for the pure fullerene which are largerthan that predicted with first-principle methods

27 June 2002

Chemical Physics Letters 359 (2002) 524–529

www.elsevier.com/locate/cplett

* Corresponding author. Fax: +31-50-363-4441.

E-mail address: l.jensen@chem.rug.nl (L. Jensen).

0009-2614/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0009 -2614 (02 )00739 -X

such as DFT and ab initio SCF [5–7] it is inter-esting also to investigate the effects of substitute-doping on the second hyperpolarizability of thefullerenes using DFT.

In this work we will investigate the effects ofsubstitute-doped C60 with 2B, 2N and BN on thesecond hyperpolarizability component along thedoping axis. The calculation of the second hyper-polarizability will be done using time-dependentdensity functional theory (TD-DFT).

2. Computational details

Here we use TD-DFT for the calculations of thesecond hyperpolarizability, c, as described in[6,12,13]. First the first hyperpolarizability, b, iscalculated analytically in the presence of a smallelectric field. The second hyperpolarizability canthen be obtained by finite-field differentiation ofthe analytically calculated first hyperpolarizabilityas

cabcdð0; 0; 0; 0Þ ¼ limEd!0

babcð0; 0; 0ÞjEd

Ed: ð1Þ

For all the TD-DFT calculations we used theRESPONSE code [14,12] in the Amsterdam den-sity functional (ADF) program [15,16]. The ADFprogram uses basis sets of Slater functions wherein this work a triple zeta valence plus polarization(in ADF basis set IV) were chosen as basis. Thebasis set was then augmented with field-inducedpolarization (FIP) functions of Zeiss et al. [17].They chose the exponents based on results of anexact analysis of field induced changes in hydrogenatom orbitals. We use both first- and second-orderFIP functions since for the calculation of c thesecond-order FIP functions are needed in order toget good results [18]. This basis set will here bedenoted as IV++ ([6s4p2d1f] for C, N, B and[4s2p1d] for H). In order to test the quality of thisbasis set we also calculated c for benzene andborazine using a large even-tempered basis set withadditional diffuse functions ([8s6p4d4f] for C, N, Band [4s3p3d] for H), here called ET.

For the calculations of c for the fullerenes weused the local density approximation (LDA) andthe calculations were restricted to the static czzzz

component, which is the component along thedoping axis. This was done for two reasons. Thefirst is to keep the computational burden as low aspossible and the second is that the aim was to in-vestigate the effects on c by doping the fullerenenot obtaining highly accurate results. However, inorder to assess the validity of LDA we also testedthree different exchange-correlation (xc) poten-tials, the Becke–Lee–Yang–Parr (BLYP) [19,20],the Perdew–Wang (PW91) [21] and the van Leeu-wen–Baerends (LB94) [22] potentials for the cal-culations of c for the benzene molecule.

All molecular geometries were optimized usingthe PW91 xc-potential. The benzene and borazinemolecules are placed in the xy-plane. All geome-tries are available from the authors on request.The location of the doping atoms in the fullereneare illustrated in Fig. 1. For C60 the doping atomsare on opposite sides of the carbon cage and areidentical to the ð1; 60Þ doping configuration of Xuet al. [10]. The z-axis is taken along the directionfrom the pentagon in the bottom to the pentagonon the top. This configuration of the doping atomswere previously found to give the largest dopingeffect on the second hyperpolarizability [10] and istherefore the configuration used in this work.

3. Results

3.1. Second hyperpolarizability of benzene andborazine

The results for the static polarizability andsecond hyperpolarizability tensor for benzene and

Fig. 1. Position of substituted atoms in C60.

L. Jensen et al. / Chemical Physics Letters 359 (2002) 524–529 525

borazine are displayed in Tables 1 and 2, re-spectively. The errors introduced in adopting thecombination of single-side finite-field and ana-lytical techniques can be estimated from thedifferences between cxxxx and cyyyy which by sym-metry arguments should be identical for bothbenzene and borazine. It is found that the dif-ference between cxxxx and cyyyy is very small, es-pecially using LDA. With LDA we performedcalculations both with the IV, IV++ and the ETbasis sets. In general there is reasonable agree-ment between the results obtained with the IV++and the ET basis sets, especially for the polariz-ability. The IV basis set clearly underestimatesthe second hyperpolarizability most noticeable inthe direction perpendicular to the ring, i.e., czzzz,

which illustrates the importance of the FIP’s.Comparing the results obtained with IV++ andET the largest deviation, around 25%, is foundfor the cxxxx and cyyyy components of benzene.However, comparing the mean hyperpolarizabil-ity, �cc, for benzene with the recent results of Iwataet al. [23] we only find around an 8% deviation.In their work they use a real-space TD-DFTmethod which gives results for the second hy-perpolarizability in agreement with TD-DFT re-sults close to the basis set limit [23]. We thereforebelieve that the basis set (IV++) used here, cap-tures most features of the second hyperpolariz-ability and is adequate for the study on largersystems where the basis set effect are expected tobe smaller.

Table 1

Static polarizability and second hyperpolarizability for benzene (in a.u.)

LDA/ET LDA/IV LDA BLYP PW91 LB94 SCFa MP2a

axx 84.46 80.43 84.05 84.03 82.11 81.19 74.21 76.02

azz 45.46 39.68 45.42 46.37 44.46 36.76 39.47 41.28

cxxxx 21 878 9459 16 251 19 530 17 302 8540 14 190 19 332

cyyyy 21 875 9417 16 216 18 875 17 317 8550 14 190 19 332

cxxyy 7288 3142 5419 6899 5767 2851 4728 6402

czzzz 16 050 2381 14 863 18 190 16 323 3724 12 480 15 708

cyyzz 8506 2140 7369 9276 7974 2190 6456 8688

cxxzz 8506 2140 7369 9276 7974 2190 6456 8688

cb 21 681 7220 17 529 21 499 18 874 7055 15 228 20 386

cc 19 000 23 800 – 12 900 – –

a From [24].b c ¼ 1

15

Pi;j ðciijj þ cijij þ cijjiÞ.

c Real-space results from [23].

Table 2

Static polarizability and second hyperpolarizability for borazine (in a.u.)

LDA/ET LDA/IV LDA SCFa MP2a

axx 76.45 72.20 76.66 66.85 75.64

azz 48.51 40.83 46.53 40.39 44.31

cxxxx 21 814 12 273 17 745 – –

cyyyy 21 654 12 128 17 606 – –

cxxyy 7213 4042 5865 – –

czzzz 13 203 4221 13 901 – –

cyyzz 5874 1775 5565 – –

cxxzz 5874 1775 5565 – –

c 18 919 8761 16 648 – –

a From [33].

526 L. Jensen et al. / Chemical Physics Letters 359 (2002) 524–529

In Table 1 we also present the polarizability andsecond hyperpolarizability for benzene calculatedusing the basis set IV++ and various xc-potentials.First it is noted that the effect of using a GGAfunctional (PW91, BLYP) gives an increase in thesecond hyperpolarizability but the effect is verysmall. This is in good agreement with studies of thesecond hyperpolarizability for small molecules[13,18,23]. Secondly, the effect of using a xc-po-tential which has the correct asymptotic behavior(LB94) on the second hyperpolarizability is large.The results using the LB94 potential is betweentwo and four times smaller than the LDA results.This finding is in agreement with the results of vanGisbergen et al. [13] using the same functional.However, the LDA results for the second hyper-polarizability for benzene obtained here is in goodagreement with both the SCF and MP2 results ofPerrin et al. [24]. Also, the static polarizability forbenzene and borazine calculated using LDA arelarger than the MP2 and SCF results in goodagreement with previous findings [25]. Therefore,to improve the LDA results a functional whichprovides both correct inner and outer parts of thexc-potential is needed [26].

3.2. Second hyperpolarizability of C60

The static second hyperpolarizability compo-nents along the z-axis for the fullerenes C60 arepresented in Table 3. For C60 there is only oneindependent component and czzzz is therefore equalto the average second hyperpolarizability. Alsopresented in Table 3 are three recent theoreticalpredictions of the second hyperpolarizablility ofC60. A comparison with experiments are difficult

due to the large differences in the experimentalresults [5–7] and the comparison will thereforeonly be made with theoretical results. In this workwe have neglected vibrational, dispersion and sol-vent effect and these factors have to be consideredwhen comparing with experiments. Karna et al.[27] also discussed other problems in relatingcalculated values with experimental results forbenzene. Our results for the second hyperpolariz-ability of C60 is about 35% larger than the previ-ously reported LDA result [6] but is in reasonablegood agreement with the recent real-space LDAresult [23] and the SCF result [7]. This again sup-ports the notion that the basis set used here isadequate to study the second hyperpolarizabilityof large systems, especially, because of the agree-ment with our results and the real-space resultsboth for benzene and C60.

3.3. Second hyperpolarizability of substituted C60

In Table 3 the results for the first and secondhyperpolarizability of B,N-substituted C60 are dis-played. Only the BN-substituted fullerene has afirst hyperpolarizability which is different fromzero. We find only small differences in the secondhyperpolarizability between the 2N- and 2B-sub-stituted C60 and the pure C60 molecule. Substitutingwith 2N decreases the second hyperpolarizabilitywith about 5% and substituting with 2B increasesthe second hyperpolarizability with about 12%.This is in contrast to the results obtained with aSSH model [9,10,28] where increases by severalorders of magnitude were found both by substi-tuting with B and with N. This increase in thesecond hyperpolarizability were found both formono-substituted C60 [9] and double substitutedC60 [10] where the largest increase were found fortwo B-substituted C60. From populations analysisit was found that N acts as an electron acceptor andB as an electron donor in the doubly substitutedfullerene 1 which was also found in a previouslystudy [29]. Therefore, the effects of doping C60 with

Table 3

Second hyperpolarizability of pure and substituted fullerenes

(in a.u.)

C60 C58N2 C58B2 C58BN

bzzz – – – )602.01

czzzz 137 950 130 147 155 430 208 470

czzzza 124 000

czzzzb 87 438

czzzzc 113 765

a Real-space LDA result from [23].b LDA result from [6].c SCF results from [7].

1 The Mulliken charge on B in C58B2 was 0.97 and the charge

on N in C58N2 was )0.53. In C58BN the charge on B was 0.93

and on N )0.52.

L. Jensen et al. / Chemical Physics Letters 359 (2002) 524–529 527

either B or N will affect the p-electron density indifferent ways and therefore also the nonlinearoptical properties. For this reason we also calcu-lated the second hyperpolarizability of C60 substi-tuted with both B and N creating a donor–acceptorsystem with B as donor and N as acceptor. For thissystem it is found that the second hyperpolariz-ability is increased with around 50% comparedwith the undoped system. Also, the first hyperpo-larizability is of the same order as that of p-nitro-aniline calculated with SCF, see e.g. [30]. However,in general the doping effect on the hyperpolariz-abilties of C60 found here is very small comparedwith that found for fullerene derivatives [4,8]. Sincethe hyperpolarizabilities are expected to increasewith an increase in the distance between the donorand acceptor it would be interesting to investigatethe scaling behaviour of the hyperpolarizability ofdoped fullerenes as a function of the cage size.Also, in push–pull derivatives of C60 a relationshipbetween the conjugation path length between do-nor/acceptor groups and the first hyperpolariz-ability was found [31]. For pure carbon structures itis found that the carbon nanotubes have largersecond hyperpolarizabilities than the fullereneswith the same number of atoms [32] and the effectof substituting nanotubes instead of fullerenes as away of increasing the second hyperpolarizabilitywould also be interesting.

4. Conclusion

We have used TD-DFT to calculate the secondhyperpolarizability of the doubly substitute-dopedfullerenes C58NN, C58BB and C58BN after testingthe procedure on benzene and borazine. Reason-able good agreement is found with previously re-sults for benzene, borazine and C60, especiallyconsidering the small size of the basis set used inthis work. Only small changes in the second hy-perpolarizability were found when doping C60 witheither 2B or 2N. However, an increase with 50%was found when doping C60 with both B and N.We therefore propose BN-doped fullerenes as aninteresting starting point for further investigationsof the hyperpolarizabilities of doped fullerenes andcarbon nanotubes.

Acknowledgements

The Danish Research Training Council isgratefully acknowledged for financial support ofL.J.

References

[1] P.N. Prasad, D.J. Williams, Introduction to Nonlinear

Optical Effects in Molecules and Polymers, Wiley, New

York, 1991.

[2] D.P. Shelton, J.E. Ruce, Chem. Rev. 94 (1994) 3.

[3] L. Geng, J.C. Wright, Chem. Phys. Lett. 249 (1996)

105.

[4] J. Li, S. Wang, H. Yang, Q. Gong, X. An, H. Chen, D.

Qiang, Chem. Phys. Lett. 288 (1998) 175.

[5] P. Norman, Y. Luo, D. Jonsson, H. �AAgren, J. Chem. Phys.

106 (1997) 8788.

[6] S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, Phys.

Rev. Lett. 78 (1997) 3097.

[7] D. Jonsson, P. Norman, K. Ruud, H. �AAgren, T. Helgaker,

J. Chem. Phys. 109 (1998) 572.

[8] S. Wang, W. Huang, R. Liang, Q. Gong, H. Li, H. Chen,

D. Qiang, Phys. Rev. B 63 (2001) 153408.

[9] J. Dong, J. Jiang, J. Yu, Z. Wang, D. Xing, Phys. Rev. B

52 (1995) 9066.

[10] Q. Xu, J. Dong, J. Jinag, D.Y. Xing, J. Phys. B 29 (1996)

1563.

[11] W.P. Su, J.R. Schrieffer, A.J. Heeger, Phys. Rev. B 22

(1980) 2099.

[12] S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, J.

Chem. Phys. 109 (1998) 10644.

[13] S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, J.

Chem. Phys. 109 (1998) 10657.

[14] S.J.A. van Gisbergen, J.G. Snijders, E.J. Baerends, Com-

put. Phys. Commun. 118 (1999) 119.

[15] ADF 2000.02 Development Version., Theoretical Chemis-

try, Vrije Universiteit, Amsterdam, 2000.

[16] G. te Velde, F.M. Bickelhaupt, E.J. Baerends, C. Fonseca

Guerra, S.J.A. van Gisbergen, J.G. Snijders, T. Ziegler, J.

Comp. Chem. 22 (2001) 931.

[17] G.D. Zeiss, W.R. Scott, N. Suzuki, D.P. Chong, Mol.

Phys. 37 (1979) 1543.

[18] P. Calaminici, K. Jug, A.M. K€ooster, J. Chem. Phys. 109

(1998) 7756.

[19] A.D. Becke, Phys. Rev. A 38 (1988) 3098.

[20] C.L. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988)

785.

[21] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson,

M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46

(1991) 6671.

[22] R. van Leeuwen, E.J. Baerends, Phys. Rev. A 49 (1994)

2421.

[23] J.-I. Iwata, K. Yabana, G.F. Bertsch, J. Chem. Phys. 115

(2001) 8773.

528 L. Jensen et al. / Chemical Physics Letters 359 (2002) 524–529

[24] P.N. Perrin, E. amd Prasad, P. Mougenot, M. Dupuis, J.

Chem. Phys. 91 (1989) 4728.

[25] S.A.C. McDowell, R.D. Amos, N.C. Handy, Chem. Phys.

Lett. 235 (1995) 1.

[26] P.R.T. Schipper, O.V. Gritsenko, S.J.A. van Gisbergen,

E.J. Baerends, J. Chem. Phys. 112 (2000) 1344.

[27] S.P. Karna, G.B. Talapatra, P.N. Prasad, J. Chem. Phys.

95 (1991) 5873.

[28] R.H. Xie, Phys. Lett. A 258 (1999) 51.

[29] N. Kurita, K. Kobayashi, H. Kumahora, K. Tago, Phys.

Rev. B 48 (1993) 4850.

[30] S.P. Karna, P.N. Prasad, M. Dupuis, J. Chem. Phys. 94

(1991) 1171.

[31] M. Fanti, G. Orlandi, F. Zerbetto, J. Phys. Chem. A 101

(1997) 3015.

[32] J. Jiang, J. Dong, X. Wan, D.Y. Xing, J. Phys. B 31 (1998)

3079.

[33] E.F. Archibong, A.J. Thakkar, Mol. Phys. 81 (1994) 557.

L. Jensen et al. / Chemical Physics Letters 359 (2002) 524–529 529