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Electric-dipole forbidden transitions in C60: oscillator strengths induced by the spin–orbit coupling
Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145 www.elsevier.com/locate/theochem
Electric-dipole forbidden transitions in C60: oscillator strengths induced by the spin –orbit coupling Toshiki Haraa,1, Yasushi Nomuraa,*, Susumu Naritaa, Hirotoshi Itob, Tai-ichi Shibuyaa a
Department of Chemistry, Faculty of Textile Science and Technology, Shinshu University, Tokida 3-15-1, Ueda, Nagano-ken 386-8567, Japan Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofugaoka 1-5-1, Chofu, Tokyo 182-8585, Japan
b
Received 28 January 2002; accepted 3 April 2002
Abstract The electric-dipole forbidden transitions of the C60 molecule are studied. The oscillator strengths induced by the spin – orbit (SO) coupling have been estimated, using wavefunctions of the singlet and triplet excited states and their excited energies previously obtained by the Tamm– Dancoff approximation including all the 14,400 particle-hole pairs in the CNDO/S approximation. Effects of the SO coupling are treated in terms of the first-order perturbation theory. Singlet – triplet (ST) transitions from the ground state 11 Ag are allowed only to the 3 Au ; 3 T1u and 3 Hu states. The 11 Ag ! 13 Au transition gives the largest oscillator strength among them, and its transition process via virtual intermediate states is analyzed. It is shown that the large oscillator strength is due to a small energy difference between the final 13 Au and the intermediate 31 T1u states. Compared with the oscillator strengths of transitions induced by the vibronic coupling, the magnitudes of the oscillator strengths of the ST transitions are very small. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Fullerene C60; Spin–orbit coupling; Singlet–triplet transition; Oscillator strength
1. Introduction The feature of the visible– UV absorption spectrum [1 – 13] of C60 is characterized with three prominent peaks in the UV region [1,3,4 – 6,11], which are ascribed to the electric-dipole allowed transitions from the ground state to the 1 T1u excited states. There are also very weak bands observed in the visible range of 420– 700 nm [3,10 – 13]. Leach et al. [3] assigned the lowest 1 T1u state to a relatively weak but sharp peak near 410 nm. Theoretically, the excitation energy of the lowest 1 T1u state was estimated by * Corresponding author. Tel.: þ 81-268-21-5398. E-mail address: [email protected] (Y. Nomura). 1 Present address: Technology Platform Research Center, SEIKO EPSON Corporation, 281 Fujimi, Nagano-ken 399-0293, Japan.
several groups [14 – 18] to be around 3.3 eV. Negri et al. [16,19] attempted to assign the very weak bands to some vibronic states, based on the CNDO/S approximation [20 – 23]. On the other hand, some of the weak bands were ascribed to another origins, i.e. transitions from the singlet ground state to low-lying triplet states (ST transitions) induced by spin – orbit (SO) coupling [3, 11]. The SO coupling of C60 was expected to be large because of the spherical p-conjugated system with a rather large radius [3,11]. Despite this expectation, no theoretical calculation has been performed to evaluate the intensities of the ST transitions of C60. In this paper, we estimate the oscillator strengths of the ST transitions from the ground state to the lowlying triplet states of the C60 molecule. Previously
0166-1280/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 1 2 8 0 ( 0 2 ) 0 0 2 5 4 - 3
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[18], we carried out the complete single-excitation mixing calculations on the electronic transitions of the C60 molecule using the Tamm– Dancoff approximation (TDA) and the random-phase approximation (RPA) schemes in the CNDO/S [20 – 23] and the INDO/S [24,25] approximations. The results of the energies and wavefunctions obtained in the CNDO/STDA scheme are adopted in the present calculations. Treating the SO coupling within the first-order perturbation theory, we can show group theoretically that ST transitions from the ground state 11 Ag are allowed only to the 3 Au ; 3 T1u and 3 Hu states. We can then calculate the oscillator strengths for the ST transitions to these triplet states lying in a low energy region. We show that the transition 11 Ag ! 13 Au gives the largest oscillator strength, and its transition process via virtual intermediate states is analyzed. According to the analysis, we show that the largest contribution is due to a transition process via the intermediate 31 T1u state, which is almost resonant with the final 13 Au state. We will then show that the calculated oscillator strengths for the ST transitions are very small compared with those for the vibronic transition.
excited states l~tl ðMÞl ðM ¼ 21; 0; þ1Þ are expanded in terms of the unperturbed wavefunctions as [25] l~s0 l ¼ ls0 l þ
h
2.1. Electronic excited states We use the singlet and the triplet electronic excited states of the C60 molecule previously obtained [18] in the CNDO/S approximation with the TDA scheme, that is equal to the configuration interaction, singles (CIS) scheme. The TDA calculation was performed on the 14400-dimensional p – h space constructed by 120 occupied and 120 unoccupied MOs. In the SCF MO calculation [18], we assumed the icosahedral ˚ geometry with CC bond lengths of 1.397 and 1.449 A [26], that were obtained from the INDO calculation and proved to be very close to the neutron diffraction ˚ [27]. values 1.391 and 1.455 A 2.2. Oscillator strengths of the ST transitions Treating the SO coupling in the first-order perturbation theory, the spin-contaminated wavefunctions for the singlet ground state l~s0 l and the lth triplet
lth ðMÞl
M
^ SO ls0 l kth ðMÞlH Eðs0 Þ 2 Eðth Þ
ð1Þ
^ SO ltl ðMÞl ksh lH ; Eðtl Þ 2 Eðsh Þ
ð2Þ
and l~tl ðMÞl ¼ ltl ðMÞl þ
X h
lsh l
respectively, where H^ SO is the SO coupling and Eðs0 Þ the energy of the unperturbed state ls0 l: The ST transition moment from lsh l to lth ðMÞl is defined as the transition dipole moment between the contaminated states l~s0 l and l~tl ðMÞl ^ ~tl ðMÞl Dðs0 ! tl ðMÞÞ ; k~s0 lDl ¼
X h
þ 2. Computational method
XX
^ hl ks0 lDls
ksh lH^ SO ltl ðMÞl Eðtl Þ 2 Eðsh Þ
^ SO lth ðMÞl X ks0 lH ^ l ðMÞl; kth ðMÞlDlt h Eðs0 Þ 2 Eðth Þ
ð3Þ
^ is the electric dipole operator. In this paper, where D we neglect the spin-other-orbit coupling in the SO coupling [25,28], and H^ SO is given by 2 elec: X ZN X nucl: ~ Ni ·s~i ; ^ SO ¼ a H ‘ 3 2 i N rNi
ð4Þ
~ where a ¼ 1=137 is the fine structure constant, and ‘ and s~ denote the orbital and spin angular momenta, respectively. Now consider the ST transitions in the C60 ^ and H^ SO belong molecule. Note that the operators D to the irreducible representations (irreps) T1u and T1g of Ih ; respectively. Then, noting that the ground state of the molecule belongs to the irrep Ag ; we see that only the 1 T1g and 3 T1u states can participate in the ST transitions as the intermediate states lsh l and lth ðMÞl in Eq. (3), respectively. Using this selection rule for the intermediate states, the ST transition moment from the ground state to the nth triplet state belonging
T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145 Table 1 Oscillator strengths for the ST transitions to the low-lying triplet states of C60 3
3
Au
3
T1u
Hu
n
f £ 104
n
f £ 104
n
f £ 104
1
0.383
1 2 3 4
0.001 0.002 0.065 0.149
1 2 3 4
0.000 0.023 0.005 0.001
141
moment is written as 1 T1u ; n3 L Dg 11 Ag ! n3 Lb ðiÞ ¼ cðLðiÞ; g; bÞDtot I 3 þcðLðiÞ; b; gÞDtot T1g ; n3 L ; II
ð7Þ
where cðLðiÞ; g; bÞ
to an irrep L is written as D 11 Ag ! n3 Lb ðiÞ ¼ X
dimðT X1u Þ D
m
þ
D
^ 1 T1u ðjÞ 11 Ag lDlm
j¼1
X
dimðT X1u Þ
m
j¼1
D
D
^ SO ln3 Lb ðiÞ E m1 T1u ðjÞlH Eðn3 LÞ 2 E m1 T1u
E
E ^ SO lm3 T b ðjÞ 11 A g l H 1g
b ^ 3 Lb ðiÞ ; £ m3 T1g ðjÞlDln
E m1 T kH kn3 L
1u SO ¼ 1 Ag kDkm T1u Eðn3 LÞ 2 Eðm1 T1u Þ m X 1 DI m T1u ; n3 L ; ; XD
1
ð8Þ
1
ð9aÞ
m
and
Eð11 Ag Þ 2 Eðm3 T1g Þ E
dimðT X1 Þ
kAð1ÞlT1 ðjÞT1 ðgÞlkT1 ðjÞlLðiÞT1 ðbÞl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; dimðAÞdimðT1 Þ j¼1 1 3 T ; n L Dtot 1u I ¼
ð5Þ
where b denotes the spin component of the triplet state and i ¼ 1; 2; …; dimðLÞ in which dimðLÞ means the dimensionality of the irrep L. The ST transitions from the ground state are allowed only to the triplet states belonging to irreps Au ; T1u and Hu ; because T1u £ T1g ¼ Au þ T1u þ Hu : According to the Wigner –Eckart theorem, the electric-dipole and the SO coupling integrals in Eq. (5) are decomposed into the products of their reduced integrals and the CG coefficients. For example, the SO coupling integral of the first term in Eq. (5) is written as D E ^ SO ln3 Lb ðiÞ m1 T1u ðjÞlH D E 1 ¼ pffiffiffiffiffiffiffiffiffiffi kT1 ðjÞlLðiÞT1 ðbÞl m1 T1u kHSO kn3 L ; dimðT1 Þ ð6Þ where kT1 ðjÞlLðiÞT1 ðbÞl is the CG coefficient of I, a subgroup of Ih ; and km1 T1u ðjÞlH^ SO ln3 Lb ðiÞl the reduced integral of the original one. With the electric-dipole and the SO coupling integrals decomposed in Eq. (5), the g-component of the ST transition
3 Dtot T1g ; n3 L II D E E X 11 Ag kHSO km3 T1g D 3 m T1g kDkn3 L ¼ 1 3 m E 1 Ag 2 E m T1g X 3 ; DII m T1g ; n3 L :
ð9bÞ
m
The two types of transition processes 11 Ag ! m1 T1u ! n3 L in Eq. (9a) and 11 Ag ! m3 T1g ! n3 L in Eq. (9b) are, hereafter, called type I and type II processes, respectively. We can calculate Eq. (8) with the CG coefficients of the point group I that were previously obtained [18]. The oscillator strength for the ST transition 11 Ag ! n3 L is then given by dimð XLÞ 2 f 11 Ag ! n3 L ¼ v 1 1 Ag ! n 3 L 3 i¼1 ð10Þ dimðT 2 X 1 Þ 1 £ D 1 Ag ! n3 Lb ðiÞ : b¼1
3. Results and discussion As mentioned above, the ST transitions from the singlet ground state of the C60 molecule are allowed only to the triplet states of Au ; T1u and Hu : Table 1
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T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145
Table 2 Values of kDlm ; k11 Ag kDkm1 T1u l; kHSO lm ; km1 T1u kHSO k13 Au l; DEm21 ; ½Eð13 Au Þ 2 Eðm1 T1u Þ21 and DI ðmÞ m
kDlm
kHSO lm
DEm21
DI ðmÞ
3 2 4 .. . 242 .. . 130 .. .
22.64 24.14 ..3.06 . 0.42 .. . p 24.26 .. .
28.69 £ 1026 21.54 £ 1025 26 ..26.57 £ 10 . 22.99 £ 1025 p .. . 1.31 £ 1027 .. .
22080.00p 167.00 2105.00 .. . 21.24 .. . 22.14 .. .
24.78 £ 1022 p 1.07 £ 1022 2.11 £ 1023 .. . 1.54 £ 1025 .. . 1.19 £ 1026 .. . 23.48 £ 1022
Dtot I The total ST transition moment for the type I process, Dtot I ¼ column is marked by an asterisk.
P
m
shows calculated values of the oscillator strengths for the ST transitions from the ground state to the low-lying excited states 13 Au ; n3 T1u ðn ¼ 1; 2; 3; 4Þ and n3 Hu ðn ¼ 1; 2; 3; 4Þ: Note that the oscillator strengths for the ST transitions to 13 Au and 43 T1u states are prominently large among those shown in Table 1. For the ST transition to the 13 Au state that gives the largest oscillator strength, we introduce quantities defined by
DI ðmÞ ; DI m1 T1u ; 13 Au
E m1 T kH k13 A
¼ 1 Ag kDkm T1u 3 1u SO 1 u ð11aÞ E 1 Au 2 E m T1u D
1
1
Table 3 Values of kDlm ; km3 T1g kDk13 Au l; kHSO lm ; k11 Ag kHSO km3 T1g l; DEm21 ; ½Eð11 Ag Þ 2 Eðm3 T1g Þ21 and DII ðmÞ m
kDlm
kHSO lm
DEm21
DII ðmÞ
8 3 10 1 9 .. .
4.18 4.87p 22.00 21.30 23.19 .. .
22.24 £ 1024 27.92 £ 1025 22.81 £ 1024 p 28.56 £ 1025 27.23 £ 1025 .. .
24.60 26.65 24.29 213.7p .. 24.41 .
4.29 £ 1023* 2.56 £ 1023 22.42 £ 1023 21.53 £ 1023 21.02 £ 1023 .. .
Dtot II
1.41 £ 1023
The total ST transition moment for the type II process, Dtot II ¼ m DII ðmÞ; is shown at the bottom. All values in a.u. The largest value in each column is marked by an asterisk.
P
DI ðmÞ; is shown at the bottom. All values in a.u. The largest value in each
and DII ðmÞ ; DII m3 T1g ; 13 Au D
E E 11 Ag kHSO km3 T1g D m3 T1g kDk13 Au : ¼ E 11 Ag 2 E m3 T1g ð11bÞ The DI ðmÞ and DII ðmÞ are the ST transition moments reflecting the transition processes 11 Ag ! m1 T1u ! 13 Au of type I and 11 Ag ! m3 T1g ! 13 Au of type II, respectively. With these quantities, we can analyze which factors influence the oscillator strength for the ST transition. Table 2 shows values of the DI ðmÞ for the type I process, in the decreasing order of magnitude, together with values of k11 Ag kDkm1 T1u l; km1 T1u kHSO k13 Au l and ½Eð13 Au Þ 2 Eðm1 T1u Þ21 : The total ST P transition moment for the type I process, Dtot m DI ðmÞ; is I ¼ shown at the bottom of the table. Similarly, Table 3 shows values, DII ðmÞ; km3 T1g kDk13 Au l; k11 Ag kHPSO km3 T1g l; ½Eð11 Ag Þ 2 Eðm3 T1g Þ21 and Dtot m DII ðmÞ for the type II process. Note II ¼ that the magnitudes of the SO coupling integrals kHSO lm in Tables 2 and 3 are comparable to those estimated for benzene (1025 a.u.) [29] and azulene (1026 – 1025) [30] with the CNDO/S-CIS. Note that the total ST transition moment Dtot 1 for the type I process is about ten times larger than Dtot 2 for the type II process. In the type I process, the process via 31 T1u state dominantly contributes to the ST transition moment as shown in Table 2. This is
T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145 Table 4 Oscillator strengths for the vibronic transitions to the low-lying singlet excited states of C60
f £ 104
11 T1g
11 T2g
11 Gg
1 1 Hg
11 T2u
11 Gu
11 Hu
14.84
30.16
143.67
32.32
0.06
0.08
0.58
143
the contribution of the ST transition to the weak bands is negligibly small compared with that of the vibronic transition.
4. Conclusion
because the energy difference between the 31 T1u and the 13 Au states is very small [18],2 i.e. these states are nearly degenerate. Therefore, it is expected that a resonance of the intermediate state m1 T1u with the final triplet state n3 L ðL ¼ Au ; T1u ; Hu Þ enhances the magnitude of the ST transition moment for the transition 11 Ag ! n3 L: In fact, the oscillator strength is large for the ST transition to 43 T1u that is nearly degenerate with 31 T1u : The calculated values of the energy difference and the SO coupling between the m1 T1u and the n3 L states satisfy the perturbative condition, i.e. E D 1 m T1u kHSO kn3 L p Eðn3 LÞ 2 Eðm1 T1u Þ :
The oscillator strengths of the C60 molecule for the ST transitions from the ground state to the low-lying triplet states were calculated by the CNDO/S-TDA scheme on the complete 14 400 p– h space. Analyzing the transition process of the most intensive ST transition to the 13 Au state, we have found that a resonance with the intermediate state 31 T1u enhances the oscillator strength. To examine the possibility that the ST transition contributes to the weak bands in the absorption spectrum of C60, the oscillator strengths for the ST transitions were compared with those for the vibronic transitions. We have found that the former are much smaller than the latter.
Thus, the perturbative estimation of the oscillator strengths for the ST transitions seems to be adequate in this calculation. If the perturbative condition fails, i.e. E D 1 m T1u kHSO kn3 L $ Eðn3 LÞ 2 E m1 T1u ;
Appendix
the states strongly would couple with each other and become compound eigenstates of the singlet and triplet states. Next, we examine the possibility that the ST transition contributes to the weak bands in the visible region of the absorption spectrum of C60. There exist seven singlet excited states in the region lower than the energy of the 11 T1u state [18]. Vibronic transitions of the ground state to the low-lying singlet states are expected to contribute to the weak bands. According to a Herzberg –Teller type approach (see Appendix), we have estimated oscillator strengths for the vibronic transitions as shown in Table 4. Compared with Table 1, the oscillator strengths for the vibronic transitions are much larger than those for the ST transitions, except the vibronic transitions to the ungerade states 11 T2u ; 11 Gu and 11 Hu : Therefore, we can expect that 2
In Ref. [18], the excitation energies of 31 T1u ; 13 Au and 43 T1u are 4.30, 4.29 and 4.28 eV, respectively.
According to the first-order perturbation theory of the Herzberg –Teller expression [31], the wavefunction of the lth singlet electronic state is written as lsl ðr; QÞl ¼ lsl ðr; 0Þl þ
X X h–l a
Qa lsh ðr; 0Þl
^ ›Qa Þ0 lsl ðr; 0Þl ksh ðr; 0Þlð›H= ; El ð0Þ 2 Eh ð0Þ ðA1Þ
where lsl ðr; 0Þl and El ð0Þ are the wavefunction and eigenenergy of the lth singlet electronic state at the equilibrium geometry and Qa denotes the ath normal coordinate. For simplicity, lsl ðr; 0Þl and El ð0Þ are hereafter abbreviated to lsl l and El : Here, the ^ ›Qa Þ0 lsl l is replaced vibronic coupling term ksh lð›H= by
› ^ 0l ðr; QÞl ; ksh lH ^ vib ðaÞlsl l; ðA2Þ ks0h ðr; QÞlHls ›Q a 0 where ls0l ðr; QÞl is a modified wavefunction of lsl l ¼ lsl ðr; 0Þl by taking into account the dependence of the AOs upon the nuclear motion. Under the Born – Oppenheimer approximation, the transition moment
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T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145
between the vibronic ground state and the onequantum excited level of the ath normal mode in the electronically excited state is written as Dðs0 ! sl ; aÞ ¼ kx00 ðQa ÞlQa lxl1 ðQa Þl " ^ X ^ h l ksh lHvib ðaÞlsl l £ ks0 lDls El 2 Eh h–l
£
where x00 ðQa Þ and xl1 ðQa Þ are vibrational wavefunctions on the electronic states s0 and sl ; respectively. Assuming that the vibrational frequency is va ; the same value in both the s0 and sl states, we can set that qffiffiffiffiffiffiffiffiffiffi kx00 ðQa ÞlQa lxl1 ðQa Þl ¼ 1=ð2va Þ: Because the electric dipole transitions from the ground state 1 Ag of the C60 molecule are allowed only to the 1 T1u states, the transition moment for the vibronic transition 11 Ag ! n1 LðiÞ; induced by the ath normal mode belonging to an irrep GðkÞ; is written as D 11 Ag ! n1 LðiÞ; a [ GðkÞ ^ 1 T1u ðjÞ 11 Ag lDlm
E
j¼1
D
E ^ vib ðaÞln1 LðiÞ m1 T1u ðjÞlH Eðn1 LÞ 2 E m1 T1u D E ^ vib ðaÞlm1 GðjÞ XGÞ 11 Ag lH X dimð þ E 11 Ag 2 Eðm1 GÞ m j¼1
ðA4Þ
# ^ LðiÞl : km GðjÞlDln 1
i¼1
k¼1
ðA5Þ
ðA3Þ
dimðT X1u Þ D
2 X dimð XLÞ dimð XGÞ 1 D 1 Ag ! n1 LðiÞ; a [ GðkÞ ; a
# ^ vib ðaÞlsh l X ks0 lH ^ ll ; þ ksh lDls E0 2 Eh h–0
sffiffiffiffiffiffiffi" X 1 ¼ 2v a m
the vibronic coupling is written as f 11 Ag ! n1 L ¼ 23 v 11 Ag ! n1 L
1
Using this transition moment, the oscillator strength for the transition 11 Ag ! n1 L induced by
1
1
where vð1 Ag ! n LÞ is the excitation energy.
References [1] W. Kra¨tschmer, L.D. Lamb, K. Fostiropoulos, D.R. Huffman, Nature 347 (1990) 354. [2] C. Reber, L. Yee, J. McKiernan, J.I. Zink, R.S. Williams, W.M. Tong, D.A.A. Ohlberg, R.L. Whetten, F. Diederich, J. Phys. Chem. 95 (1991) 2127. [3] S. Leach, M. Vervloet, A. Despre`s, E. Bre´heret, J.P. Hare, T.J. Denis, H.W. Kroto, R. Taylor, D.R.M. Walton, Chem. Phys. 160 (1992) 451. [4] H. Ajie, M.M. Alvarez, S.J. Anz, R.D. Beck, F. Diederich, K. Fostiropoulos, D.R. Huffman, W. Kra¨tchmer, Y. Rubin, K.E. Schriver, D. Sensharma, R.L. Whetten, J. Chem. Phys. 94 (1990) 8630. [5] J.P. Hare, H.W. Kroto, R. Taylor, Chem. Phys. Lett. 177 (1991) 394. [6] J. Cataln, Chem. Phys. Lett. 223 (1994) 159. [7] B.B. Brady, E.J. Beiting, J. Chem. Phys. 97 (1992) 3855. [8] S. Dai, L. Mac Toth, G.D. Del Cul, D.H. Metcalf, J. Chem. Phys. 101 (1994) 4470. [9] H. Yasumatsu, T. Kondow, H. Kitagawa, K. Tabayashi, K. Shobatake, J. Chem. Phys. 104 (1996) 899. [10] R.E. Haufler, Y. Chai, J.P.F. Chibante, M.R. Fraelich, R.B. Weisman, R.F. Curl, R.E. Smalley, J. Chem. Phys. 95 (1991) 2197. [11] Z. Gasyna, P.N. Schatz, J.P. Hare, T.J. Dennis, H.W. Kroto, R. Taylor, D.R.M. Walton, Chem. Phys. Lett. 183 (1991) 283. [12] A. Sasara, G. Zerza, M. Chergui, J. Phys. B 329 (1996) 4997. [13] J.D. Close, F. Federmann, K. Hoffmann, N. Quaas, Chem. Phys. Lett. 276 (1997) 393. [14] F. Negri, G. Orlandi, F. Zerbetto, Chem. Phys. Lett. 144 (1988) 31. [15] I. La´szlo´, L. Udvardi, J. Mol. Struct. 183 (1989) 271. [16] F. Negri, G. Orlandi, F. Zerbetto, J. Chem. Phys. 97 (1992) 6496. [17] T. Hara, S. Narita, T. Shibuya, Fullerene Sci. Technol. 3 (1995) 459. [18] T. Hara, S. Narita, S. Kumei, T. Shibuya, Int. J. Quant. Chem. 85 (2001) 136. [19] A. Sassara, G. Zerza, M. Chergui, F. Negri, G. Orlandi, J. Chem. Phys. 107 (1997) 8731. [20] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 48 (1968) 1807. [21] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 49 (1968) 1221. [22] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 50 (1969) 1126.
T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145 [23] R.L. Ellis, G. Kuehnlenz, H.H. Jaffe, Theor. Chim. Acta 26 (1972) 131. [24] J.A. Pople, D.L. Beveridge, P.A. Dobosh, J. Chem. Phys. 47 (1967) 2026. [25] H. Itoh, H. Kameshita, T. Shibuya, J. Chem. Phys. 104 (1996) 1309. [26] T. Shibuya, M. Yoshitani, Chem. Phys. Lett. 137 (1987) 13. [27] W.I.F. David, R.M. Ibberson, J.C. Mathewman, K. Prassides,
[28] [29] [30] [31]
145
T.J.S. Dennis, J.P. Hare, H.W. Kroto, R. Taylor, D.R.M. Walton, Nature 353 (1991) 147. H. Itoh, Y.J. I’Haya, Mol. Phys. 24 (1972) 1103. ˚ gren, O. Vahtras, Theor. Chim. S. Knuts, B.F. Minaev, H. A Acta 87 (1994) 343. R.L. Ellis, R. Squire, H.H. Jaffe, J. Chem. Phys. 55 (1971) 3499. G. Herzberg, E. Teller, Z. Phys. Chem. 21 (1933) 410.
Electric-dipole forbidden transitions in C60: oscillator strengths induced by the spin –orbit coupling Toshiki Haraa,1, Yasushi Nomuraa,*, Susumu Naritaa, Hirotoshi Itob, Tai-ichi Shibuyaa a
Department of Chemistry, Faculty of Textile Science and Technology, Shinshu University, Tokida 3-15-1, Ueda, Nagano-ken 386-8567, Japan Department of Applied Physics and Chemistry, The University of Electro-Communications, Chofugaoka 1-5-1, Chofu, Tokyo 182-8585, Japan
b
Received 28 January 2002; accepted 3 April 2002
Abstract The electric-dipole forbidden transitions of the C60 molecule are studied. The oscillator strengths induced by the spin – orbit (SO) coupling have been estimated, using wavefunctions of the singlet and triplet excited states and their excited energies previously obtained by the Tamm– Dancoff approximation including all the 14,400 particle-hole pairs in the CNDO/S approximation. Effects of the SO coupling are treated in terms of the first-order perturbation theory. Singlet – triplet (ST) transitions from the ground state 11 Ag are allowed only to the 3 Au ; 3 T1u and 3 Hu states. The 11 Ag ! 13 Au transition gives the largest oscillator strength among them, and its transition process via virtual intermediate states is analyzed. It is shown that the large oscillator strength is due to a small energy difference between the final 13 Au and the intermediate 31 T1u states. Compared with the oscillator strengths of transitions induced by the vibronic coupling, the magnitudes of the oscillator strengths of the ST transitions are very small. q 2002 Elsevier Science B.V. All rights reserved. Keywords: Fullerene C60; Spin–orbit coupling; Singlet–triplet transition; Oscillator strength
1. Introduction The feature of the visible– UV absorption spectrum [1 – 13] of C60 is characterized with three prominent peaks in the UV region [1,3,4 – 6,11], which are ascribed to the electric-dipole allowed transitions from the ground state to the 1 T1u excited states. There are also very weak bands observed in the visible range of 420– 700 nm [3,10 – 13]. Leach et al. [3] assigned the lowest 1 T1u state to a relatively weak but sharp peak near 410 nm. Theoretically, the excitation energy of the lowest 1 T1u state was estimated by * Corresponding author. Tel.: þ 81-268-21-5398. E-mail address: [email protected] (Y. Nomura). 1 Present address: Technology Platform Research Center, SEIKO EPSON Corporation, 281 Fujimi, Nagano-ken 399-0293, Japan.
several groups [14 – 18] to be around 3.3 eV. Negri et al. [16,19] attempted to assign the very weak bands to some vibronic states, based on the CNDO/S approximation [20 – 23]. On the other hand, some of the weak bands were ascribed to another origins, i.e. transitions from the singlet ground state to low-lying triplet states (ST transitions) induced by spin – orbit (SO) coupling [3, 11]. The SO coupling of C60 was expected to be large because of the spherical p-conjugated system with a rather large radius [3,11]. Despite this expectation, no theoretical calculation has been performed to evaluate the intensities of the ST transitions of C60. In this paper, we estimate the oscillator strengths of the ST transitions from the ground state to the lowlying triplet states of the C60 molecule. Previously
0166-1280/02/$ - see front matter q 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 1 2 8 0 ( 0 2 ) 0 0 2 5 4 - 3
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[18], we carried out the complete single-excitation mixing calculations on the electronic transitions of the C60 molecule using the Tamm– Dancoff approximation (TDA) and the random-phase approximation (RPA) schemes in the CNDO/S [20 – 23] and the INDO/S [24,25] approximations. The results of the energies and wavefunctions obtained in the CNDO/STDA scheme are adopted in the present calculations. Treating the SO coupling within the first-order perturbation theory, we can show group theoretically that ST transitions from the ground state 11 Ag are allowed only to the 3 Au ; 3 T1u and 3 Hu states. We can then calculate the oscillator strengths for the ST transitions to these triplet states lying in a low energy region. We show that the transition 11 Ag ! 13 Au gives the largest oscillator strength, and its transition process via virtual intermediate states is analyzed. According to the analysis, we show that the largest contribution is due to a transition process via the intermediate 31 T1u state, which is almost resonant with the final 13 Au state. We will then show that the calculated oscillator strengths for the ST transitions are very small compared with those for the vibronic transition.
excited states l~tl ðMÞl ðM ¼ 21; 0; þ1Þ are expanded in terms of the unperturbed wavefunctions as [25] l~s0 l ¼ ls0 l þ
h
2.1. Electronic excited states We use the singlet and the triplet electronic excited states of the C60 molecule previously obtained [18] in the CNDO/S approximation with the TDA scheme, that is equal to the configuration interaction, singles (CIS) scheme. The TDA calculation was performed on the 14400-dimensional p – h space constructed by 120 occupied and 120 unoccupied MOs. In the SCF MO calculation [18], we assumed the icosahedral ˚ geometry with CC bond lengths of 1.397 and 1.449 A [26], that were obtained from the INDO calculation and proved to be very close to the neutron diffraction ˚ [27]. values 1.391 and 1.455 A 2.2. Oscillator strengths of the ST transitions Treating the SO coupling in the first-order perturbation theory, the spin-contaminated wavefunctions for the singlet ground state l~s0 l and the lth triplet
lth ðMÞl
M
^ SO ls0 l kth ðMÞlH Eðs0 Þ 2 Eðth Þ
ð1Þ
^ SO ltl ðMÞl ksh lH ; Eðtl Þ 2 Eðsh Þ
ð2Þ
and l~tl ðMÞl ¼ ltl ðMÞl þ
X h
lsh l
respectively, where H^ SO is the SO coupling and Eðs0 Þ the energy of the unperturbed state ls0 l: The ST transition moment from lsh l to lth ðMÞl is defined as the transition dipole moment between the contaminated states l~s0 l and l~tl ðMÞl ^ ~tl ðMÞl Dðs0 ! tl ðMÞÞ ; k~s0 lDl ¼
X h
þ 2. Computational method
XX
^ hl ks0 lDls
ksh lH^ SO ltl ðMÞl Eðtl Þ 2 Eðsh Þ
^ SO lth ðMÞl X ks0 lH ^ l ðMÞl; kth ðMÞlDlt h Eðs0 Þ 2 Eðth Þ
ð3Þ
^ is the electric dipole operator. In this paper, where D we neglect the spin-other-orbit coupling in the SO coupling [25,28], and H^ SO is given by 2 elec: X ZN X nucl: ~ Ni ·s~i ; ^ SO ¼ a H ‘ 3 2 i N rNi
ð4Þ
~ where a ¼ 1=137 is the fine structure constant, and ‘ and s~ denote the orbital and spin angular momenta, respectively. Now consider the ST transitions in the C60 ^ and H^ SO belong molecule. Note that the operators D to the irreducible representations (irreps) T1u and T1g of Ih ; respectively. Then, noting that the ground state of the molecule belongs to the irrep Ag ; we see that only the 1 T1g and 3 T1u states can participate in the ST transitions as the intermediate states lsh l and lth ðMÞl in Eq. (3), respectively. Using this selection rule for the intermediate states, the ST transition moment from the ground state to the nth triplet state belonging
T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145 Table 1 Oscillator strengths for the ST transitions to the low-lying triplet states of C60 3
3
Au
3
T1u
Hu
n
f £ 104
n
f £ 104
n
f £ 104
1
0.383
1 2 3 4
0.001 0.002 0.065 0.149
1 2 3 4
0.000 0.023 0.005 0.001
141
moment is written as 1 T1u ; n3 L Dg 11 Ag ! n3 Lb ðiÞ ¼ cðLðiÞ; g; bÞDtot I 3 þcðLðiÞ; b; gÞDtot T1g ; n3 L ; II
ð7Þ
where cðLðiÞ; g; bÞ
to an irrep L is written as D 11 Ag ! n3 Lb ðiÞ ¼ X
dimðT X1u Þ D
m
þ
D
^ 1 T1u ðjÞ 11 Ag lDlm
j¼1
X
dimðT X1u Þ
m
j¼1
D
D
^ SO ln3 Lb ðiÞ E m1 T1u ðjÞlH Eðn3 LÞ 2 E m1 T1u
E
E ^ SO lm3 T b ðjÞ 11 A g l H 1g
b ^ 3 Lb ðiÞ ; £ m3 T1g ðjÞlDln
E m1 T kH kn3 L
1u SO ¼ 1 Ag kDkm T1u Eðn3 LÞ 2 Eðm1 T1u Þ m X 1 DI m T1u ; n3 L ; ; XD
1
ð8Þ
1
ð9aÞ
m
and
Eð11 Ag Þ 2 Eðm3 T1g Þ E
dimðT X1 Þ
kAð1ÞlT1 ðjÞT1 ðgÞlkT1 ðjÞlLðiÞT1 ðbÞl pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; dimðAÞdimðT1 Þ j¼1 1 3 T ; n L Dtot 1u I ¼
ð5Þ
where b denotes the spin component of the triplet state and i ¼ 1; 2; …; dimðLÞ in which dimðLÞ means the dimensionality of the irrep L. The ST transitions from the ground state are allowed only to the triplet states belonging to irreps Au ; T1u and Hu ; because T1u £ T1g ¼ Au þ T1u þ Hu : According to the Wigner –Eckart theorem, the electric-dipole and the SO coupling integrals in Eq. (5) are decomposed into the products of their reduced integrals and the CG coefficients. For example, the SO coupling integral of the first term in Eq. (5) is written as D E ^ SO ln3 Lb ðiÞ m1 T1u ðjÞlH D E 1 ¼ pffiffiffiffiffiffiffiffiffiffi kT1 ðjÞlLðiÞT1 ðbÞl m1 T1u kHSO kn3 L ; dimðT1 Þ ð6Þ where kT1 ðjÞlLðiÞT1 ðbÞl is the CG coefficient of I, a subgroup of Ih ; and km1 T1u ðjÞlH^ SO ln3 Lb ðiÞl the reduced integral of the original one. With the electric-dipole and the SO coupling integrals decomposed in Eq. (5), the g-component of the ST transition
3 Dtot T1g ; n3 L II D E E X 11 Ag kHSO km3 T1g D 3 m T1g kDkn3 L ¼ 1 3 m E 1 Ag 2 E m T1g X 3 ; DII m T1g ; n3 L :
ð9bÞ
m
The two types of transition processes 11 Ag ! m1 T1u ! n3 L in Eq. (9a) and 11 Ag ! m3 T1g ! n3 L in Eq. (9b) are, hereafter, called type I and type II processes, respectively. We can calculate Eq. (8) with the CG coefficients of the point group I that were previously obtained [18]. The oscillator strength for the ST transition 11 Ag ! n3 L is then given by dimð XLÞ 2 f 11 Ag ! n3 L ¼ v 1 1 Ag ! n 3 L 3 i¼1 ð10Þ dimðT 2 X 1 Þ 1 £ D 1 Ag ! n3 Lb ðiÞ : b¼1
3. Results and discussion As mentioned above, the ST transitions from the singlet ground state of the C60 molecule are allowed only to the triplet states of Au ; T1u and Hu : Table 1
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T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145
Table 2 Values of kDlm ; k11 Ag kDkm1 T1u l; kHSO lm ; km1 T1u kHSO k13 Au l; DEm21 ; ½Eð13 Au Þ 2 Eðm1 T1u Þ21 and DI ðmÞ m
kDlm
kHSO lm
DEm21
DI ðmÞ
3 2 4 .. . 242 .. . 130 .. .
22.64 24.14 ..3.06 . 0.42 .. . p 24.26 .. .
28.69 £ 1026 21.54 £ 1025 26 ..26.57 £ 10 . 22.99 £ 1025 p .. . 1.31 £ 1027 .. .
22080.00p 167.00 2105.00 .. . 21.24 .. . 22.14 .. .
24.78 £ 1022 p 1.07 £ 1022 2.11 £ 1023 .. . 1.54 £ 1025 .. . 1.19 £ 1026 .. . 23.48 £ 1022
Dtot I The total ST transition moment for the type I process, Dtot I ¼ column is marked by an asterisk.
P
m
shows calculated values of the oscillator strengths for the ST transitions from the ground state to the low-lying excited states 13 Au ; n3 T1u ðn ¼ 1; 2; 3; 4Þ and n3 Hu ðn ¼ 1; 2; 3; 4Þ: Note that the oscillator strengths for the ST transitions to 13 Au and 43 T1u states are prominently large among those shown in Table 1. For the ST transition to the 13 Au state that gives the largest oscillator strength, we introduce quantities defined by
DI ðmÞ ; DI m1 T1u ; 13 Au
E m1 T kH k13 A
¼ 1 Ag kDkm T1u 3 1u SO 1 u ð11aÞ E 1 Au 2 E m T1u D
1
1
Table 3 Values of kDlm ; km3 T1g kDk13 Au l; kHSO lm ; k11 Ag kHSO km3 T1g l; DEm21 ; ½Eð11 Ag Þ 2 Eðm3 T1g Þ21 and DII ðmÞ m
kDlm
kHSO lm
DEm21
DII ðmÞ
8 3 10 1 9 .. .
4.18 4.87p 22.00 21.30 23.19 .. .
22.24 £ 1024 27.92 £ 1025 22.81 £ 1024 p 28.56 £ 1025 27.23 £ 1025 .. .
24.60 26.65 24.29 213.7p .. 24.41 .
4.29 £ 1023* 2.56 £ 1023 22.42 £ 1023 21.53 £ 1023 21.02 £ 1023 .. .
Dtot II
1.41 £ 1023
The total ST transition moment for the type II process, Dtot II ¼ m DII ðmÞ; is shown at the bottom. All values in a.u. The largest value in each column is marked by an asterisk.
P
DI ðmÞ; is shown at the bottom. All values in a.u. The largest value in each
and DII ðmÞ ; DII m3 T1g ; 13 Au D
E E 11 Ag kHSO km3 T1g D m3 T1g kDk13 Au : ¼ E 11 Ag 2 E m3 T1g ð11bÞ The DI ðmÞ and DII ðmÞ are the ST transition moments reflecting the transition processes 11 Ag ! m1 T1u ! 13 Au of type I and 11 Ag ! m3 T1g ! 13 Au of type II, respectively. With these quantities, we can analyze which factors influence the oscillator strength for the ST transition. Table 2 shows values of the DI ðmÞ for the type I process, in the decreasing order of magnitude, together with values of k11 Ag kDkm1 T1u l; km1 T1u kHSO k13 Au l and ½Eð13 Au Þ 2 Eðm1 T1u Þ21 : The total ST P transition moment for the type I process, Dtot m DI ðmÞ; is I ¼ shown at the bottom of the table. Similarly, Table 3 shows values, DII ðmÞ; km3 T1g kDk13 Au l; k11 Ag kHPSO km3 T1g l; ½Eð11 Ag Þ 2 Eðm3 T1g Þ21 and Dtot m DII ðmÞ for the type II process. Note II ¼ that the magnitudes of the SO coupling integrals kHSO lm in Tables 2 and 3 are comparable to those estimated for benzene (1025 a.u.) [29] and azulene (1026 – 1025) [30] with the CNDO/S-CIS. Note that the total ST transition moment Dtot 1 for the type I process is about ten times larger than Dtot 2 for the type II process. In the type I process, the process via 31 T1u state dominantly contributes to the ST transition moment as shown in Table 2. This is
T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145 Table 4 Oscillator strengths for the vibronic transitions to the low-lying singlet excited states of C60
f £ 104
11 T1g
11 T2g
11 Gg
1 1 Hg
11 T2u
11 Gu
11 Hu
14.84
30.16
143.67
32.32
0.06
0.08
0.58
143
the contribution of the ST transition to the weak bands is negligibly small compared with that of the vibronic transition.
4. Conclusion
because the energy difference between the 31 T1u and the 13 Au states is very small [18],2 i.e. these states are nearly degenerate. Therefore, it is expected that a resonance of the intermediate state m1 T1u with the final triplet state n3 L ðL ¼ Au ; T1u ; Hu Þ enhances the magnitude of the ST transition moment for the transition 11 Ag ! n3 L: In fact, the oscillator strength is large for the ST transition to 43 T1u that is nearly degenerate with 31 T1u : The calculated values of the energy difference and the SO coupling between the m1 T1u and the n3 L states satisfy the perturbative condition, i.e. E D 1 m T1u kHSO kn3 L p Eðn3 LÞ 2 Eðm1 T1u Þ :
The oscillator strengths of the C60 molecule for the ST transitions from the ground state to the low-lying triplet states were calculated by the CNDO/S-TDA scheme on the complete 14 400 p– h space. Analyzing the transition process of the most intensive ST transition to the 13 Au state, we have found that a resonance with the intermediate state 31 T1u enhances the oscillator strength. To examine the possibility that the ST transition contributes to the weak bands in the absorption spectrum of C60, the oscillator strengths for the ST transitions were compared with those for the vibronic transitions. We have found that the former are much smaller than the latter.
Thus, the perturbative estimation of the oscillator strengths for the ST transitions seems to be adequate in this calculation. If the perturbative condition fails, i.e. E D 1 m T1u kHSO kn3 L $ Eðn3 LÞ 2 E m1 T1u ;
Appendix
the states strongly would couple with each other and become compound eigenstates of the singlet and triplet states. Next, we examine the possibility that the ST transition contributes to the weak bands in the visible region of the absorption spectrum of C60. There exist seven singlet excited states in the region lower than the energy of the 11 T1u state [18]. Vibronic transitions of the ground state to the low-lying singlet states are expected to contribute to the weak bands. According to a Herzberg –Teller type approach (see Appendix), we have estimated oscillator strengths for the vibronic transitions as shown in Table 4. Compared with Table 1, the oscillator strengths for the vibronic transitions are much larger than those for the ST transitions, except the vibronic transitions to the ungerade states 11 T2u ; 11 Gu and 11 Hu : Therefore, we can expect that 2
In Ref. [18], the excitation energies of 31 T1u ; 13 Au and 43 T1u are 4.30, 4.29 and 4.28 eV, respectively.
According to the first-order perturbation theory of the Herzberg –Teller expression [31], the wavefunction of the lth singlet electronic state is written as lsl ðr; QÞl ¼ lsl ðr; 0Þl þ
X X h–l a
Qa lsh ðr; 0Þl
^ ›Qa Þ0 lsl ðr; 0Þl ksh ðr; 0Þlð›H= ; El ð0Þ 2 Eh ð0Þ ðA1Þ
where lsl ðr; 0Þl and El ð0Þ are the wavefunction and eigenenergy of the lth singlet electronic state at the equilibrium geometry and Qa denotes the ath normal coordinate. For simplicity, lsl ðr; 0Þl and El ð0Þ are hereafter abbreviated to lsl l and El : Here, the ^ ›Qa Þ0 lsl l is replaced vibronic coupling term ksh lð›H= by
› ^ 0l ðr; QÞl ; ksh lH ^ vib ðaÞlsl l; ðA2Þ ks0h ðr; QÞlHls ›Q a 0 where ls0l ðr; QÞl is a modified wavefunction of lsl l ¼ lsl ðr; 0Þl by taking into account the dependence of the AOs upon the nuclear motion. Under the Born – Oppenheimer approximation, the transition moment
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T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145
between the vibronic ground state and the onequantum excited level of the ath normal mode in the electronically excited state is written as Dðs0 ! sl ; aÞ ¼ kx00 ðQa ÞlQa lxl1 ðQa Þl " ^ X ^ h l ksh lHvib ðaÞlsl l £ ks0 lDls El 2 Eh h–l
£
where x00 ðQa Þ and xl1 ðQa Þ are vibrational wavefunctions on the electronic states s0 and sl ; respectively. Assuming that the vibrational frequency is va ; the same value in both the s0 and sl states, we can set that qffiffiffiffiffiffiffiffiffiffi kx00 ðQa ÞlQa lxl1 ðQa Þl ¼ 1=ð2va Þ: Because the electric dipole transitions from the ground state 1 Ag of the C60 molecule are allowed only to the 1 T1u states, the transition moment for the vibronic transition 11 Ag ! n1 LðiÞ; induced by the ath normal mode belonging to an irrep GðkÞ; is written as D 11 Ag ! n1 LðiÞ; a [ GðkÞ ^ 1 T1u ðjÞ 11 Ag lDlm
E
j¼1
D
E ^ vib ðaÞln1 LðiÞ m1 T1u ðjÞlH Eðn1 LÞ 2 E m1 T1u D E ^ vib ðaÞlm1 GðjÞ XGÞ 11 Ag lH X dimð þ E 11 Ag 2 Eðm1 GÞ m j¼1
ðA4Þ
# ^ LðiÞl : km GðjÞlDln 1
i¼1
k¼1
ðA5Þ
ðA3Þ
dimðT X1u Þ D
2 X dimð XLÞ dimð XGÞ 1 D 1 Ag ! n1 LðiÞ; a [ GðkÞ ; a
# ^ vib ðaÞlsh l X ks0 lH ^ ll ; þ ksh lDls E0 2 Eh h–0
sffiffiffiffiffiffiffi" X 1 ¼ 2v a m
the vibronic coupling is written as f 11 Ag ! n1 L ¼ 23 v 11 Ag ! n1 L
1
Using this transition moment, the oscillator strength for the transition 11 Ag ! n1 L induced by
1
1
where vð1 Ag ! n LÞ is the excitation energy.
References [1] W. Kra¨tschmer, L.D. Lamb, K. Fostiropoulos, D.R. Huffman, Nature 347 (1990) 354. [2] C. Reber, L. Yee, J. McKiernan, J.I. Zink, R.S. Williams, W.M. Tong, D.A.A. Ohlberg, R.L. Whetten, F. Diederich, J. Phys. Chem. 95 (1991) 2127. [3] S. Leach, M. Vervloet, A. Despre`s, E. Bre´heret, J.P. Hare, T.J. Denis, H.W. Kroto, R. Taylor, D.R.M. Walton, Chem. Phys. 160 (1992) 451. [4] H. Ajie, M.M. Alvarez, S.J. Anz, R.D. Beck, F. Diederich, K. Fostiropoulos, D.R. Huffman, W. Kra¨tchmer, Y. Rubin, K.E. Schriver, D. Sensharma, R.L. Whetten, J. Chem. Phys. 94 (1990) 8630. [5] J.P. Hare, H.W. Kroto, R. Taylor, Chem. Phys. Lett. 177 (1991) 394. [6] J. Cataln, Chem. Phys. Lett. 223 (1994) 159. [7] B.B. Brady, E.J. Beiting, J. Chem. Phys. 97 (1992) 3855. [8] S. Dai, L. Mac Toth, G.D. Del Cul, D.H. Metcalf, J. Chem. Phys. 101 (1994) 4470. [9] H. Yasumatsu, T. Kondow, H. Kitagawa, K. Tabayashi, K. Shobatake, J. Chem. Phys. 104 (1996) 899. [10] R.E. Haufler, Y. Chai, J.P.F. Chibante, M.R. Fraelich, R.B. Weisman, R.F. Curl, R.E. Smalley, J. Chem. Phys. 95 (1991) 2197. [11] Z. Gasyna, P.N. Schatz, J.P. Hare, T.J. Dennis, H.W. Kroto, R. Taylor, D.R.M. Walton, Chem. Phys. Lett. 183 (1991) 283. [12] A. Sasara, G. Zerza, M. Chergui, J. Phys. B 329 (1996) 4997. [13] J.D. Close, F. Federmann, K. Hoffmann, N. Quaas, Chem. Phys. Lett. 276 (1997) 393. [14] F. Negri, G. Orlandi, F. Zerbetto, Chem. Phys. Lett. 144 (1988) 31. [15] I. La´szlo´, L. Udvardi, J. Mol. Struct. 183 (1989) 271. [16] F. Negri, G. Orlandi, F. Zerbetto, J. Chem. Phys. 97 (1992) 6496. [17] T. Hara, S. Narita, T. Shibuya, Fullerene Sci. Technol. 3 (1995) 459. [18] T. Hara, S. Narita, S. Kumei, T. Shibuya, Int. J. Quant. Chem. 85 (2001) 136. [19] A. Sassara, G. Zerza, M. Chergui, F. Negri, G. Orlandi, J. Chem. Phys. 107 (1997) 8731. [20] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 48 (1968) 1807. [21] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 49 (1968) 1221. [22] J. Del Bene, H.H. Jaffe, J. Chem. Phys. 50 (1969) 1126.
T. Hara et al. / Journal of Molecular Structure (Theochem) 589–590 (2002) 139–145 [23] R.L. Ellis, G. Kuehnlenz, H.H. Jaffe, Theor. Chim. Acta 26 (1972) 131. [24] J.A. Pople, D.L. Beveridge, P.A. Dobosh, J. Chem. Phys. 47 (1967) 2026. [25] H. Itoh, H. Kameshita, T. Shibuya, J. Chem. Phys. 104 (1996) 1309. [26] T. Shibuya, M. Yoshitani, Chem. Phys. Lett. 137 (1987) 13. [27] W.I.F. David, R.M. Ibberson, J.C. Mathewman, K. Prassides,
[28] [29] [30] [31]
145
T.J.S. Dennis, J.P. Hare, H.W. Kroto, R. Taylor, D.R.M. Walton, Nature 353 (1991) 147. H. Itoh, Y.J. I’Haya, Mol. Phys. 24 (1972) 1103. ˚ gren, O. Vahtras, Theor. Chim. S. Knuts, B.F. Minaev, H. A Acta 87 (1994) 343. R.L. Ellis, R. Squire, H.H. Jaffe, J. Chem. Phys. 55 (1971) 3499. G. Herzberg, E. Teller, Z. Phys. Chem. 21 (1933) 410.