- Email: [email protected]
Polarization-dependent optical properties of open (N,N) carbon tubules
Chemical Physics Letters 391 (2004) 212–215 www.elsevier.com/locate/cplett
Polarization-dependent optical properties of open (N ; N ) carbon tubules Yasushi Nomura *, Hiroya Fujita, Susumu Narita, Tai-ichi Shibuya Department of Chemistry, Faculty of Textile Science and Technology, Shinshu University, Tokida 3-15-1, Ueda, Nagano-ken 386-8567, Japan Received 20 February 2004 Available online 25 May 2004
Abstract Polarization-dependent optical properties of the open (N; N ) carbon tubules C4Nþ2Nn H4N (N ¼ 5–8 and n ¼ 0–11) are studied in low energy region with the CNDO/S-TDA calculations. It is shown that the low-lying excitation energies periodically decrease with n but not with N, irrespective of polarization directions. The N - and n-dependences of the oscillator strengths are also discussed. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction The discovery of carbon tubules or nanotubes [1] has greatly stimulated research work on single-wall nanotubes (SWNTs) [2–13]. A SWNT is characterized by a chiral vector ðn; mÞ n~ a þ m~ b with integers n and m, where ~ a and ~ b denote the unit vectors of the hexagonal honeycomb lattice [5]. Metallic and semiconducting characters have been discussed on various infinitely long (n,m) SWNTs [2–4]. The optical properties of SWNTs are also an interesting subject [7,13]. Liang et al. [7] made the semiempirical PM3 calculations on several types of SWNTs and analyzed their optical transitions. They categorized the transitions into the end modes and the tube modes, and showed that the end modes were sensitive to the tube length, radius and chirality. They also showed that the p–p transitions played important roles in the end modes in the low energy region. Previously [12] we theoretically examined the optical properties of the capped (5,5) SWNTs C60þ10n in the low energy region, and showed that their lowest optical transitions were due to the two types of single-electron excitations (SEs): [HOMO ! LUMO + 1] and [HOMO)1 ! LUMO]. These
*
Corresponding author. Fax: +81-268-21-5398. E-mail address: [email protected] (Y. Nomura).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.04.106
SEs were also responsible to low-lying prominent absorption bands. In this Letter, we consider polarization-dependent optical properties of the open (N ; N ) SWNTs C4Nþ2Nn H4N (N ¼ 5–8; n ¼ 0–11) in the low energy region. These SWNTs have the CN -rotational axes around the tubular axes and have the DNh or DNd symmetry depending upon odd or even n. In this Letter, transitions induced by lights polarized along the cylindrical and transversal axes are called z- and (x; y)-transitions, respectively. To calculate the excitation energies and the oscillator strengths, we employ the Tamm–Dancoff approximation (TDA) scheme in the CNDO/S approximation [14,15], as was used in our previous calculations [12,13]. We analyze the optical transitions of the SWNTs for the longest absorption edges and the prominent peaks. We also discuss their dependences on both N and n.
2. Results and discussion According to the geometry optimization with GA U S S I A N 98 at the AM1 level, the open (N ; N ) SWNTs C4Nþ2Nn H4N (designated as TðN ; nÞ, hereafter) with N ¼ 5–8 and n ¼ 0–11 have the DNh symmetry for odd n and the DNd for even n. CNDO/S calculations with these optimized geometries show that the HOMOs and LUMOs of TðN ; nÞ are all nondegenerate. Table 1
Y. Nomura et al. / Chemical Physics Letters 391 (2004) 212–215 Table 1 HOMO–LUMO gap energies (in eV) of TðN ; nÞ obtained by CNDO/S calculations n
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
0 1 2 3 4 5 6 7 8 9 10 11
7.24 4.92 5.30 5.16 3.67 4.21 4.41 3.03 3.57 3.94 2.16 3.17
6.69 5.20 5.59 4.73 3.88 4.46 4.04 3.19 3.77 3.64 2.79 3.33
6.34 5.40 5.80 4.45 4.02 4.63 3.75 3.30 3.91 3.36 2.88 3.44
6.09 5.53 5.94 4.26 4.12 4.75 3.55 3.38 4.01 3.16 2.94 3.52
shows the HOMO–LUMO gap energies of TðN ; nÞ. We see that the energies periodically decrease with n and take minimum values at n ¼ 1; 4; . . ., as was previously noted for various armchair SWNTs [6,8,9,11,12]. To obtain the transition energies and amplitudes for the low-lying excited states, we made TDA calculations in truncated active MO-spaces around the HOMO– LUMO gaps. Active MO-spaces used in our TDA calculations are constructed with about 20 occupied and about 20 unoccupied MOs. The TDA calculations with these truncated MO-spaces are expected reasonable because only the MOs nearby the HOMO–LUMO gaps make dominant contributions to the low-lying excited states. The values of HOMO–LUMO gap energies obtained from our CNDO/S calculations may be compared with those obtained from ab initio calculations. Previously, we compared the CNDO/S value 5.4 eV of the gap energy of C60 with the ab initio (STO-3G) value 8.7 eV. In the case of the gap energy of the capped (5,5) SWNTs C100 , for instance, our CNDO/S calculation gave the value 4.6 eV [12] while with the ab initio (B3LYP/6-311G) gave the value 2.21 eV [9].
213
2.1. Low-lying z-transitions Table 2 shows the excitation energies of TðN ; nÞ for the lowest z- and (x; y)-transitions. We see that the longest absorption edges of TðN ; nÞ with large n are of the lowest z-transitions. Analyzing the CNDO/S-TDA data for TðN ; n > 1Þ, we notice that four nondegenerate MOs including both the HOMO and LUMO participate in each of the lowest z-transitions. Fig. 1 shows our CNDO/ S-MO energy-levels nearby the HOMO–LUMO gap of Tð7; 8Þ as a typical example for a large n. For Tð7; 8Þ, [HOMO ! LUMO + 1] and [HOMO)1 ! LUMO] are the two SEs dominantly contributing to the lowest ztransition (indicated by solid arrows). Clearly these SEs have smaller excitation energies than those dominantly contributing to the lowest (x; y)-transition (indicated by dotted arrows). In general, two SEs make dominant contributions to the lowest z-transition of TðN ; n > 1Þ. Considering the symmetric property of TðN ; nÞ with respect to the operation rh or rv depending upon whether the molecular symmetry is DNh or DNd , the nondegenerate MOs are classified into two types: Symmetric (called a-type) and antisymmetric ðb-type) MOs. Then, the two SEs can be generally expressed as [highest occupied a-type MO (HO a-MO) ! lowest unoccupied a-type MO (LU a-MO)] and [HO b-MO ! LU b-MO]. The excitation energies of TðN ; nÞ in the lowest ztransitions decrease with n, as is seen in Table 2. The periodic oscillations of the energies vs. n are rather weakened, when compared with the HOMO–LUMO gaps, due to the participation of other MOs than HOMOs and LUMOs. These energies are, however, almost insensitive to N . According to our CNDO/S data, the HOMOs, LUMOs and neighboring MOs are well described in terms of the pseudo p-orbitals that are normal to the tubular surfaces, i.e., the pertinent transitions are practically p–p transitions. Then, the simple H€ uckel
Table 2 Excitation energies (in eV) of TðN ; nÞ in the lowest optically allowed z- and (x; y)-transitions n
0 1 2 3 4 5 6 7 8 9 10 11
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
z
x; y
z
x; y
z
x; y
z
x; y
7.04 3.78 2.87 2.30 2.61 2.23 1.91 2.05 1.86 1.67 1.02 1.64
4.52 3.29 3.47 3.18 2.75 2.99 2.89 2.47 2.73 2.73 2.01 2.57
7.29 3.96 3.03 2.43 2.79 2.37 2.02 2.17 1.96 1.74 1.82 1.70
4.16 3.38 3.55 3.02 2.82 3.00 2.72 2.49 2.72 2.57 2.35 2.55
7.48 4.08 3.14 2.47 2.90 2.46 2.09 2.28 2.05 1.81 1.89 1.77
3.89 3.40 3.59 2.84 2.80 2.97 2.55 2.47 2.66 2.41 2.32 2.47
7.61 4.17 3.20 2.53 2.96 2.55 2.16 2.34 2.07 1.87 1.93 1.81
3.67 3.39 3.61 2.69 2.76 2.99 2.41 2.43 2.57 2.27 2.24 2.38
214
Y. Nomura et al. / Chemical Physics Letters 391 (2004) 212–215
E (eV) 0
-2
e1g e1u
a2g
a2u
-4
-6
a1u
-8
a1g
e1u e1g
-10
Fig. 1. MO energy levels of Tð7; 8Þ in the vicinity of the HOMO– LUMO gap, computed by CNDO/S. The central column shows the nondegenerate levels, and the two columns on the both sides show the doubly degenerate levels. Dominant SEs for the lowest z- and (x; y)transitions are denoted by the solid and dotted arrows, respectively.
MO model for TðN ; nÞ shows that the energies of these nondegenerate MOs strongly depend on n but not on N . Our results on the HOMO–LUMO gap and the excitation energies for the lowest z-transitions reflect the H€ uckel MO-like properties of the HOMOs and LUMOs. However, Tð5; 10Þ has a much smaller value of the excitation energy than other TðN ; 10Þ. Its origin is currently under study. The SEs [HO a-MO ! LU a-MO] and [HO bMO ! LU b-MO] contributing to the lowest z-transition of each TðN ; n > 1Þ also make dominant contributions to the second lowest z-transition. The second transition gives an isolated prominent peak in the low energy
region of the absorption spectrum due to the z-polarized light. Table 3 shows the excitation energies and the oscillator strengths of TðN ; n > 1Þ for the second lowest z-transitions. Note that the N - and n-dependences of the excitation energies are similar to those for the lowest z-transitions. The oscillator strengths, however, almost monotonously increase with both N and n. This is because the number of electrons participating in the z-transition increases with the number of carbon atoms. The increase of the effective integral space for the transition dipole moment of the z-transition may also contribute to the increase of the oscillator strength with n. For TðN ; n ¼ 0Þ, on the other hand, the [HO a-MO ! LU a-MO] and [HO b-MO ! LU b-MO] make only minor contributions to the low-lying z-transitions. In this case, transitions between the doubly degenerate MOs make major contributions. The energy separations between these MO levels are larger, so that the lowest z-transitions of TðN ; n ¼ 0Þ lie exceptionally high, as is seen in Table 2. In the case of TðN ; n ¼ 1Þ, the low-lying z-transitions are mainly described by the [HO a-MO ! LU a-MO] and [HO b-MO ! LU b-MO], but contributions from the doubly degenerate MOs are also significant. 2.2. Low-lying (x,y)-transitions Our CNDO/S-TDA data show that the doubly degenerate MOs (e-MOs) are the ones which participate in the lowest (x; y)-transitions. For TðN ; 3mÞ and TðN ; 3m þ 1Þ with m ¼ 0–3, the major SEs are [HOMO ! lowest unoccupied e-MO (LU e-MO)] and [highest occupied e-MO (HO e-MO) ! LUMO]. These SEs also contribute to the second lowest (x; y)-transitions with small oscillator strength values. For TðN ; 3m þ 2Þ, the dominant SEs are [HOMO ! second LU e-MO (2nd-LU e-MO)] and [HOMO-1 ! LU e-MO], but their neighboring SEs [2nd-HO e-MO ! LUMO] and [HO e-MO ! LUMO+1] are also significant (see Fig. 1). These SEs contribute not only to the second lowest (x; y)-transitions but also to their neigh-
Table 3 Excitation energies DE (in eV) and oscillator strengths f of TðN ; n > 1Þ for the second lowest z-transitions n
2 3 4 5 6 7 8 9 10 11
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
DE
f
DE
f
DE
f
DE
f
4.24 3.46 3.80 3.29 2.89 3.02 2.76 2.53 1.99 2.45
0.902 0.904 1.41 1.76 1.91 2.54 2.91 3.16 3.41 4.23
4.39 3.51 3.89 3.35 2.90 3.07 2.79 2.53 2.62 2.46
1.20 1.04 1.74 1.98 2.10 2.64 3.14 3.41 4.27 4.66
4.50 3.57 3.98 3.42 2.95 3.14 2.84 2.55 2.66 2.49
1.56 1.22 2.15 2.26 2.41 3.00 3.50 3.65 4.54 4.99
4.59 3.63 4.06 3.51 3.00 3.20 2.90 2.58 2.70 2.51
1.97 1.44 2.63 2.93 2.87 3.46 4.10 4.06 4.98 5.29
Y. Nomura et al. / Chemical Physics Letters 391 (2004) 212–215 Table 4 Oscillator strengths of TðN ; nÞ for the lowest (x; y)-transitions n
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
0 1 2 3 4 5 6 7 8 9 10 11
1.45 0.87 0.29 0.26 0.55 0.14 0.15 0.40 0.13 0.06 0.21 0.15
2.00 1.36 0.27 0.78 1.09 0.05 0.59 0.78 0.05 0.41 0.66 0.08
2.50 1.76 0.21 1.33 1.51 0.01 1.11 1.26 0.00 0.92 1.06 0.01
2.97 2.08 0.13 1.94 1.86 0.00 1.72 1.63 0.01 1.52 1.43 0.00
boring transitions. Owing to the participation of HOMOs and LUMOs in the lowest (x; y)-transitions of TðN ; nÞ, the N - and n-dependences of their excitation energies are similar to those for the lowest z-transitions, as is seen in Table 2. Table 4 shows the oscillator strengths for the lowest (x; y)-transitions. It is clear that the oscillator strengths of TðN ; nÞ with given n increase monotonously with N , except for the case of TðN ; 3m þ 2Þ. For TðN ; 3m þ 2Þ, the second lowest (x; y)-transitions give rather large values of the oscillator strengths, which increase monotonously with N , but tend to decrease as n increases. That is, the expansion of the system along the cylindrical axis does not increase the oscillator strength for the lowlying (x; y)-transitions, but the expansion in the transversal directions increases it.
3. Conclusion CNDO/S-TDA calculations were made on the open (N ; N ) SWNTs, TðN ; nÞðN ¼ 5–8; n ¼ 0–11Þ to investigate their polarization-dependent optical properties in the low energy region. According to the CNDO/ S-TDA data, the low-lying optical transitions are practically the p–p transitions including the HOMOs and LUMOs (and their neighboring MOs). It is shown that excitation energies of the lowest z- and (x; y)-transitions
215
almost periodically decrease with n but not with N . Oscillator strengths for the second lowest z-transitions increase with both N and n. On the other hand, for the lowest (x; y)-transitions, they increase with N but not with n. The expansion of the TðN ; nÞ system in the transversal directions increases the oscillator strength for both the low-lying z- and (x; y)-transitions, though the expansion along the cylindrical axis increases it for the z-transitions but not for the (x; y)-transitions.
Acknowledgements This work was supported by Grant-in-Aid for 21st Century COE Program by Ministry of Education, Culture, Sports, Science, and Technology.
References [1] S. Iijima, Nature 354 (1991) 56. [2] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 631. [3] N. Hamada, S. Sawada, A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1579. [4] R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhause, Phys. Rev. B 46 (1992) 1804. [5] M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, CA, 1996. [6] A. Rochefort, D.R. Salahub, P. Avouris, J. Phys. Chem. B 103 (1999) 641. [7] W. Liang, X.J. Wang, S. Yokojima, G. Chen, J. Am. Chem. Soc. 122 (2000) 11129. [8] T. Yamabe, M. Imade, M. Tanaka, T. Sato, Synth. Met. 117 (2001) 61. [9] J. Cioslowski, N. Rao, D. Moncrieff, J. Am. Chem. Soc. 124 (2002) 8485. [10] J. Li, Y. Zhang, M. Zhang, Chem. Phys. Lett. 364 (2002) 328. [11] J. Li, Y. Zhang, M. Zhang, Chem. Phys. Lett. 364 (2002) 338. [12] Y. Nomura, H. Fujita, S. Narita, T. Shibuya, Chem. Phys. Lett. 375 (2003) 72. [13] Y. Nomura, H. Fujita, S. Narita, T. Shibuya, Int. Electron. J. Mol. Des. 3 (2004) 29. [14] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 50 (1969) 1126. [15] R.L. Ellis, G. Kuehnlenz, H.H. Jaffe, Theor. Chim. Acta 26 (1972) 131.
Polarization-dependent optical properties of open (N ; N ) carbon tubules Yasushi Nomura *, Hiroya Fujita, Susumu Narita, Tai-ichi Shibuya Department of Chemistry, Faculty of Textile Science and Technology, Shinshu University, Tokida 3-15-1, Ueda, Nagano-ken 386-8567, Japan Received 20 February 2004 Available online 25 May 2004
Abstract Polarization-dependent optical properties of the open (N; N ) carbon tubules C4Nþ2Nn H4N (N ¼ 5–8 and n ¼ 0–11) are studied in low energy region with the CNDO/S-TDA calculations. It is shown that the low-lying excitation energies periodically decrease with n but not with N, irrespective of polarization directions. The N - and n-dependences of the oscillator strengths are also discussed. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction The discovery of carbon tubules or nanotubes [1] has greatly stimulated research work on single-wall nanotubes (SWNTs) [2–13]. A SWNT is characterized by a chiral vector ðn; mÞ n~ a þ m~ b with integers n and m, where ~ a and ~ b denote the unit vectors of the hexagonal honeycomb lattice [5]. Metallic and semiconducting characters have been discussed on various infinitely long (n,m) SWNTs [2–4]. The optical properties of SWNTs are also an interesting subject [7,13]. Liang et al. [7] made the semiempirical PM3 calculations on several types of SWNTs and analyzed their optical transitions. They categorized the transitions into the end modes and the tube modes, and showed that the end modes were sensitive to the tube length, radius and chirality. They also showed that the p–p transitions played important roles in the end modes in the low energy region. Previously [12] we theoretically examined the optical properties of the capped (5,5) SWNTs C60þ10n in the low energy region, and showed that their lowest optical transitions were due to the two types of single-electron excitations (SEs): [HOMO ! LUMO + 1] and [HOMO)1 ! LUMO]. These
*
Corresponding author. Fax: +81-268-21-5398. E-mail address: [email protected] (Y. Nomura).
0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.04.106
SEs were also responsible to low-lying prominent absorption bands. In this Letter, we consider polarization-dependent optical properties of the open (N ; N ) SWNTs C4Nþ2Nn H4N (N ¼ 5–8; n ¼ 0–11) in the low energy region. These SWNTs have the CN -rotational axes around the tubular axes and have the DNh or DNd symmetry depending upon odd or even n. In this Letter, transitions induced by lights polarized along the cylindrical and transversal axes are called z- and (x; y)-transitions, respectively. To calculate the excitation energies and the oscillator strengths, we employ the Tamm–Dancoff approximation (TDA) scheme in the CNDO/S approximation [14,15], as was used in our previous calculations [12,13]. We analyze the optical transitions of the SWNTs for the longest absorption edges and the prominent peaks. We also discuss their dependences on both N and n.
2. Results and discussion According to the geometry optimization with GA U S S I A N 98 at the AM1 level, the open (N ; N ) SWNTs C4Nþ2Nn H4N (designated as TðN ; nÞ, hereafter) with N ¼ 5–8 and n ¼ 0–11 have the DNh symmetry for odd n and the DNd for even n. CNDO/S calculations with these optimized geometries show that the HOMOs and LUMOs of TðN ; nÞ are all nondegenerate. Table 1
Y. Nomura et al. / Chemical Physics Letters 391 (2004) 212–215 Table 1 HOMO–LUMO gap energies (in eV) of TðN ; nÞ obtained by CNDO/S calculations n
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
0 1 2 3 4 5 6 7 8 9 10 11
7.24 4.92 5.30 5.16 3.67 4.21 4.41 3.03 3.57 3.94 2.16 3.17
6.69 5.20 5.59 4.73 3.88 4.46 4.04 3.19 3.77 3.64 2.79 3.33
6.34 5.40 5.80 4.45 4.02 4.63 3.75 3.30 3.91 3.36 2.88 3.44
6.09 5.53 5.94 4.26 4.12 4.75 3.55 3.38 4.01 3.16 2.94 3.52
shows the HOMO–LUMO gap energies of TðN ; nÞ. We see that the energies periodically decrease with n and take minimum values at n ¼ 1; 4; . . ., as was previously noted for various armchair SWNTs [6,8,9,11,12]. To obtain the transition energies and amplitudes for the low-lying excited states, we made TDA calculations in truncated active MO-spaces around the HOMO– LUMO gaps. Active MO-spaces used in our TDA calculations are constructed with about 20 occupied and about 20 unoccupied MOs. The TDA calculations with these truncated MO-spaces are expected reasonable because only the MOs nearby the HOMO–LUMO gaps make dominant contributions to the low-lying excited states. The values of HOMO–LUMO gap energies obtained from our CNDO/S calculations may be compared with those obtained from ab initio calculations. Previously, we compared the CNDO/S value 5.4 eV of the gap energy of C60 with the ab initio (STO-3G) value 8.7 eV. In the case of the gap energy of the capped (5,5) SWNTs C100 , for instance, our CNDO/S calculation gave the value 4.6 eV [12] while with the ab initio (B3LYP/6-311G) gave the value 2.21 eV [9].
213
2.1. Low-lying z-transitions Table 2 shows the excitation energies of TðN ; nÞ for the lowest z- and (x; y)-transitions. We see that the longest absorption edges of TðN ; nÞ with large n are of the lowest z-transitions. Analyzing the CNDO/S-TDA data for TðN ; n > 1Þ, we notice that four nondegenerate MOs including both the HOMO and LUMO participate in each of the lowest z-transitions. Fig. 1 shows our CNDO/ S-MO energy-levels nearby the HOMO–LUMO gap of Tð7; 8Þ as a typical example for a large n. For Tð7; 8Þ, [HOMO ! LUMO + 1] and [HOMO)1 ! LUMO] are the two SEs dominantly contributing to the lowest ztransition (indicated by solid arrows). Clearly these SEs have smaller excitation energies than those dominantly contributing to the lowest (x; y)-transition (indicated by dotted arrows). In general, two SEs make dominant contributions to the lowest z-transition of TðN ; n > 1Þ. Considering the symmetric property of TðN ; nÞ with respect to the operation rh or rv depending upon whether the molecular symmetry is DNh or DNd , the nondegenerate MOs are classified into two types: Symmetric (called a-type) and antisymmetric ðb-type) MOs. Then, the two SEs can be generally expressed as [highest occupied a-type MO (HO a-MO) ! lowest unoccupied a-type MO (LU a-MO)] and [HO b-MO ! LU b-MO]. The excitation energies of TðN ; nÞ in the lowest ztransitions decrease with n, as is seen in Table 2. The periodic oscillations of the energies vs. n are rather weakened, when compared with the HOMO–LUMO gaps, due to the participation of other MOs than HOMOs and LUMOs. These energies are, however, almost insensitive to N . According to our CNDO/S data, the HOMOs, LUMOs and neighboring MOs are well described in terms of the pseudo p-orbitals that are normal to the tubular surfaces, i.e., the pertinent transitions are practically p–p transitions. Then, the simple H€ uckel
Table 2 Excitation energies (in eV) of TðN ; nÞ in the lowest optically allowed z- and (x; y)-transitions n
0 1 2 3 4 5 6 7 8 9 10 11
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
z
x; y
z
x; y
z
x; y
z
x; y
7.04 3.78 2.87 2.30 2.61 2.23 1.91 2.05 1.86 1.67 1.02 1.64
4.52 3.29 3.47 3.18 2.75 2.99 2.89 2.47 2.73 2.73 2.01 2.57
7.29 3.96 3.03 2.43 2.79 2.37 2.02 2.17 1.96 1.74 1.82 1.70
4.16 3.38 3.55 3.02 2.82 3.00 2.72 2.49 2.72 2.57 2.35 2.55
7.48 4.08 3.14 2.47 2.90 2.46 2.09 2.28 2.05 1.81 1.89 1.77
3.89 3.40 3.59 2.84 2.80 2.97 2.55 2.47 2.66 2.41 2.32 2.47
7.61 4.17 3.20 2.53 2.96 2.55 2.16 2.34 2.07 1.87 1.93 1.81
3.67 3.39 3.61 2.69 2.76 2.99 2.41 2.43 2.57 2.27 2.24 2.38
214
Y. Nomura et al. / Chemical Physics Letters 391 (2004) 212–215
E (eV) 0
-2
e1g e1u
a2g
a2u
-4
-6
a1u
-8
a1g
e1u e1g
-10
Fig. 1. MO energy levels of Tð7; 8Þ in the vicinity of the HOMO– LUMO gap, computed by CNDO/S. The central column shows the nondegenerate levels, and the two columns on the both sides show the doubly degenerate levels. Dominant SEs for the lowest z- and (x; y)transitions are denoted by the solid and dotted arrows, respectively.
MO model for TðN ; nÞ shows that the energies of these nondegenerate MOs strongly depend on n but not on N . Our results on the HOMO–LUMO gap and the excitation energies for the lowest z-transitions reflect the H€ uckel MO-like properties of the HOMOs and LUMOs. However, Tð5; 10Þ has a much smaller value of the excitation energy than other TðN ; 10Þ. Its origin is currently under study. The SEs [HO a-MO ! LU a-MO] and [HO bMO ! LU b-MO] contributing to the lowest z-transition of each TðN ; n > 1Þ also make dominant contributions to the second lowest z-transition. The second transition gives an isolated prominent peak in the low energy
region of the absorption spectrum due to the z-polarized light. Table 3 shows the excitation energies and the oscillator strengths of TðN ; n > 1Þ for the second lowest z-transitions. Note that the N - and n-dependences of the excitation energies are similar to those for the lowest z-transitions. The oscillator strengths, however, almost monotonously increase with both N and n. This is because the number of electrons participating in the z-transition increases with the number of carbon atoms. The increase of the effective integral space for the transition dipole moment of the z-transition may also contribute to the increase of the oscillator strength with n. For TðN ; n ¼ 0Þ, on the other hand, the [HO a-MO ! LU a-MO] and [HO b-MO ! LU b-MO] make only minor contributions to the low-lying z-transitions. In this case, transitions between the doubly degenerate MOs make major contributions. The energy separations between these MO levels are larger, so that the lowest z-transitions of TðN ; n ¼ 0Þ lie exceptionally high, as is seen in Table 2. In the case of TðN ; n ¼ 1Þ, the low-lying z-transitions are mainly described by the [HO a-MO ! LU a-MO] and [HO b-MO ! LU b-MO], but contributions from the doubly degenerate MOs are also significant. 2.2. Low-lying (x,y)-transitions Our CNDO/S-TDA data show that the doubly degenerate MOs (e-MOs) are the ones which participate in the lowest (x; y)-transitions. For TðN ; 3mÞ and TðN ; 3m þ 1Þ with m ¼ 0–3, the major SEs are [HOMO ! lowest unoccupied e-MO (LU e-MO)] and [highest occupied e-MO (HO e-MO) ! LUMO]. These SEs also contribute to the second lowest (x; y)-transitions with small oscillator strength values. For TðN ; 3m þ 2Þ, the dominant SEs are [HOMO ! second LU e-MO (2nd-LU e-MO)] and [HOMO-1 ! LU e-MO], but their neighboring SEs [2nd-HO e-MO ! LUMO] and [HO e-MO ! LUMO+1] are also significant (see Fig. 1). These SEs contribute not only to the second lowest (x; y)-transitions but also to their neigh-
Table 3 Excitation energies DE (in eV) and oscillator strengths f of TðN ; n > 1Þ for the second lowest z-transitions n
2 3 4 5 6 7 8 9 10 11
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
DE
f
DE
f
DE
f
DE
f
4.24 3.46 3.80 3.29 2.89 3.02 2.76 2.53 1.99 2.45
0.902 0.904 1.41 1.76 1.91 2.54 2.91 3.16 3.41 4.23
4.39 3.51 3.89 3.35 2.90 3.07 2.79 2.53 2.62 2.46
1.20 1.04 1.74 1.98 2.10 2.64 3.14 3.41 4.27 4.66
4.50 3.57 3.98 3.42 2.95 3.14 2.84 2.55 2.66 2.49
1.56 1.22 2.15 2.26 2.41 3.00 3.50 3.65 4.54 4.99
4.59 3.63 4.06 3.51 3.00 3.20 2.90 2.58 2.70 2.51
1.97 1.44 2.63 2.93 2.87 3.46 4.10 4.06 4.98 5.29
Y. Nomura et al. / Chemical Physics Letters 391 (2004) 212–215 Table 4 Oscillator strengths of TðN ; nÞ for the lowest (x; y)-transitions n
Tð5; nÞ
Tð6; nÞ
Tð7; nÞ
Tð8; nÞ
0 1 2 3 4 5 6 7 8 9 10 11
1.45 0.87 0.29 0.26 0.55 0.14 0.15 0.40 0.13 0.06 0.21 0.15
2.00 1.36 0.27 0.78 1.09 0.05 0.59 0.78 0.05 0.41 0.66 0.08
2.50 1.76 0.21 1.33 1.51 0.01 1.11 1.26 0.00 0.92 1.06 0.01
2.97 2.08 0.13 1.94 1.86 0.00 1.72 1.63 0.01 1.52 1.43 0.00
boring transitions. Owing to the participation of HOMOs and LUMOs in the lowest (x; y)-transitions of TðN ; nÞ, the N - and n-dependences of their excitation energies are similar to those for the lowest z-transitions, as is seen in Table 2. Table 4 shows the oscillator strengths for the lowest (x; y)-transitions. It is clear that the oscillator strengths of TðN ; nÞ with given n increase monotonously with N , except for the case of TðN ; 3m þ 2Þ. For TðN ; 3m þ 2Þ, the second lowest (x; y)-transitions give rather large values of the oscillator strengths, which increase monotonously with N , but tend to decrease as n increases. That is, the expansion of the system along the cylindrical axis does not increase the oscillator strength for the lowlying (x; y)-transitions, but the expansion in the transversal directions increases it.
3. Conclusion CNDO/S-TDA calculations were made on the open (N ; N ) SWNTs, TðN ; nÞðN ¼ 5–8; n ¼ 0–11Þ to investigate their polarization-dependent optical properties in the low energy region. According to the CNDO/ S-TDA data, the low-lying optical transitions are practically the p–p transitions including the HOMOs and LUMOs (and their neighboring MOs). It is shown that excitation energies of the lowest z- and (x; y)-transitions
215
almost periodically decrease with n but not with N . Oscillator strengths for the second lowest z-transitions increase with both N and n. On the other hand, for the lowest (x; y)-transitions, they increase with N but not with n. The expansion of the TðN ; nÞ system in the transversal directions increases the oscillator strength for both the low-lying z- and (x; y)-transitions, though the expansion along the cylindrical axis increases it for the z-transitions but not for the (x; y)-transitions.
Acknowledgements This work was supported by Grant-in-Aid for 21st Century COE Program by Ministry of Education, Culture, Sports, Science, and Technology.
References [1] S. Iijima, Nature 354 (1991) 56. [2] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 631. [3] N. Hamada, S. Sawada, A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1579. [4] R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhause, Phys. Rev. B 46 (1992) 1804. [5] M.S. Dresselhaus, G. Dresselhaus, P.C. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, CA, 1996. [6] A. Rochefort, D.R. Salahub, P. Avouris, J. Phys. Chem. B 103 (1999) 641. [7] W. Liang, X.J. Wang, S. Yokojima, G. Chen, J. Am. Chem. Soc. 122 (2000) 11129. [8] T. Yamabe, M. Imade, M. Tanaka, T. Sato, Synth. Met. 117 (2001) 61. [9] J. Cioslowski, N. Rao, D. Moncrieff, J. Am. Chem. Soc. 124 (2002) 8485. [10] J. Li, Y. Zhang, M. Zhang, Chem. Phys. Lett. 364 (2002) 328. [11] J. Li, Y. Zhang, M. Zhang, Chem. Phys. Lett. 364 (2002) 338. [12] Y. Nomura, H. Fujita, S. Narita, T. Shibuya, Chem. Phys. Lett. 375 (2003) 72. [13] Y. Nomura, H. Fujita, S. Narita, T. Shibuya, Int. Electron. J. Mol. Des. 3 (2004) 29. [14] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 50 (1969) 1126. [15] R.L. Ellis, G. Kuehnlenz, H.H. Jaffe, Theor. Chim. Acta 26 (1972) 131.