Electronic excitations of small clusters of C60

Download 2.2 Mb.

Page	4/4
Date	02.05.2018
Size	2.2 Mb.
	#47242

1 2 3 4

Larger cluster models of C₆₀ ~~reveal~~ hint more aspects of ~~the~~ clustering influence on the absorption spectra. ~~The~~ All electronic transitions of the whole studied region were obtained for ~~all the~~ every clusters model~~s except~~ but ~~for~~ (C₆₀)₁₃. ~~The~~ This largest model was used to study the effect of the cluster size over the spectral edge. ~~In these calculations, the~~ CIS matrices involve 30105, 48000 and 86400 SECs for (C₆₀)_N ~~where~~ for N = 3, 4 and 6, respectively while the low energy region of the ~~of the~~ (C₆₀)₁₃ model spectrum was appropriately described using 27208 SECs. ~~The~~ Resulting calculated absorption spectra of all clusters are shown in Figure 4. The lowest energy regions were ~~described~~ simulated with a half bandwidth of 0.03 eV, whereas the other bands were simulated with a half bandwidth of 0.15 eV. The inset shows the relation between ~~the~~ different selected half-width of the c, e and g bands with the number of molecules in the cluster ~~models~~. The widths of ~~the~~ bands e and g tend to increase progressively when the cluster models become larger. As in the case of the dimer, the major spectral differences appear in the low energy region. In general, an increase of the cluster size implies more allowed electronic excitations in the low energy region of the spectrum.

Noteworthy, the peak at around 2.85 eV for (C₆₀)₄ and (C₆₀)₁₃ resulted with a reduced intensity as compared to with the cases of ~~the~~ dimer and ~~the~~ trimer. This is must be another symmetry or shape effect, as the monomer, tetramer and tridecamer are equant objects, compared with the prolate dimer and the oblate trimer. Moreover, neither the (C₆₀)₄ ~~and~~ nor the (C₆₀)₁₃ respective spectrum ~~are~~ ~~not~~ appear polarized.

There are no experimental data on small fullerene clusters to compare with our simulation. Our results suggest the presence of small clusters in experiments, provided the fine structure of the absorption spectrum as measured in the range 2.5-3.2 eV. It has been suggested that the aggregates of C₆₀ in water could be small spherical particles containing at least four C₆₀ molecules.¹⁷ The optical absorption spectra of them above 3.0 eV show broad bands that are slightly red-shifted (around 0.1 eV) in comparison with those of the n-hexane solution.^{17, 50} In addition, a low intensity and very broad band covers the spectral range between 2.3 and 3.0 eV. This spectral feature appears when the formation of C₆₀ clusters is evident by electron microscopy studies,^{17, 50, 51} as well as in the absorption spectra of the solid film.¹⁵ This broad band reveals an activation of dipole-forbidden states by a number of possible causes, e.g., clustering effects, solvent interaction, or dynamic geometry fluctuations. Nevertheless, our calculations suggest that optically active states in clusters could be localized in a narrow spectral range and cannot fully account for this broad band.

FIG. 4. (Color online) CNDOL theoretical absorption spectra of (C₆₀)_N models. The low energy region of the spectrum has been inset amplified 30. The inside graphic shows half-widths of the c, e and g bands for the (C₆₀)_N models where N = 2, 3, 4 and 6. All of (C₆₀)_N modeled absorption spectrum intensities has been divided by the N units. Absorption is shown in arbitrary units (a.u.) to allow comparisons.

Figure 5 shows the very interesting landscape of densities of singlet CIS states (DOS-CIS) for all of the studied clusters. DOS values are divided by the number of C₆₀ units to normalize comparisons. These DOS-CIS are broad functions shown as a “band – like” representation in the whole range from 2.0 to 6.5 eV, illustrating the possibility of electron transitions assisted by some of the above mentioned mechanisms. Two clear behaviors are distinguished for the normalized DOS: i) bands seem similar between 2 and about 3.5 eV; ii) bands broad at higher energies with the size of clusters. It can be ~~This is~~ easily understood as and excitonic process where every CIS excitation involves an electron and a hole, which can be located at any (or shared among several) of the C₆₀ units. ~~and~~ Thus, the total number of excited states is proportional to the square of the number of C₆₀ units. ~~The~~ Intermolecular excitations are present in ~~the~~ DOS for energies higher than 3.5 eV.

FIG. 5. (Color Online) CNDOL density of CIS states (DOS-CIS) obtained for the study systems. Each DOS is divided by the number of C₆₀ units.

The CNDO/S method has been previously used to study the nature of electronic excitations of a van der Waals dimer of C₆₀.³¹ In that study, the CIS basis comprised 2000 SECs, and showed qualitatively the same results. The lowest states appear at around the same energies as the monomer transitions, although the oscillator strengths were not reported and the double peak structure at 2.8-3.0 eV was not revealed. On the other hand, the similarity of our absorption spectra of all the clusters above 3.5 eV reveals that intermolecular excitations are not dipole allowed. However, their presence must influence the spectrum by acting as additional relaxation channels for the allowed excitations. Hence, one expects an increment of the line broadening as a consequence of clustering effects. They may be already masked by the large bandwidth of the high-energy transitions, and it would be desirable to reveal these states by a direct method.

Let us consider the c band, that is fitted by two Gaussians centered at 3.69 and 3.76 eV (see Figure 7 in Appendix B) and their oscillator strength amounts up to 0.70. If the Gaussian centered at 3.69 eV is not attributed to an electronic transition, the remaining contribution to c band at 3.76 eV will be the only one transition with experimental oscillator strength smaller than the values from TDDFT and CNDOL calculations. Hence, we think that the broad band centered at 3.69 eV is also due to the 2¹T_1u level. A splitting of the 2¹T_1u level in 0.07 eV is incompatible with the symmetry of an isolated C₆₀. Our CNDOL calculations of the (C₆₀)_N clusters predict splits of this magnitude, i.e., 0.05 eV. The fitted strengths have a ratio of 0.46/0.24=1.9, while the CNDOL calculations predict the ratios 1.0, 1.6, 1.8, and 1.4, for N = 2, 3, 4 and 6, respectively. Bandwidths of this doublet ( = 0.28 and 0.08 eV) are remarkably different and we cannot offer a satisfactory explanation yet. For the sake of consistency, if clusters are present in n-hexane solution, a doublet fine structure should be observed at the low energy, as discussed above. It is possible that the solvent may affect differently the states of the doublet and they merge in a single peak, or that the oscillator strength may be transferred to one of the components. Evidently, additional simulations accounting for the solvent and clusters are necessary to clarify this issue, as well as experiments focused on this part of the spectrum. A plausible alternative is the Jahn-Teller effect, considering the triple degeneracy of the the 2¹T_1u level, and the existence of multiple vibrational modes. Jahn-Teller effects are invoked to explain the absorption spectra in the range 2-3 eV, where shifts of this magnitude are observed, as well as asymmetry in the bandwidths.⁵² This effect may be present in a smaller measure for the bands e and g, and could be responsible of the shoulders d and f.

CONCLUSIONS

The CNDOL method predicts transition strengths for C₆₀ in agreement with previous TDDFT calculations, although we attained better transition energy agreements with experimental results. Being less demanding in computer resources, CNDOL can be used to simulate vibronic transitions together with an appropriate force field, as well as Orlandi and Negri used CNDO/S []. u and to simulate the molecule with its environment at atomic level, in order to probe the mechanisms for activation of the silent states. We have advanced in the route of simulating collective phenomena observed in light absorption bands by small clusters of C₆₀, showing that the density of states is enhanced by charge-transfer excitations in the region above 3.5 eV. However, the optical absorption due to dipole allowed transitions presents only minor changes due to clustering at the highers energies. Only the fine structure at the red edge of the spectrum appears modified by explicit collective effects, showing the possibility of new optically active states in clusters that could be significant for possible applications. ~~The intense absorption bands are affected by small splittings of nearly 0.1 eV, which are too small compared to the bandwidths and are hard to detect.~~

ACKNOWLEDGEMENTS

This work was supported by the Spanish Agency for International Cooperation for Development (AECID) and the Grant SB2010-0119 from the Ministry of Education of Spain. E. M.-P. thanks R. Gebauer for many useful comments on TDDFT and optical properties.

TABLE I. Transition energies E and strengths f obtained by different levels of theory.

CNDO/S^a E; f	CNDO/S^b E; f	TDDFT^c E; f	CNDOL^d (cutoff) E; f	CNDOL^d (Full CIS) E; f
3.4;0.08	3.4;0.12	2.82;0.006	3.00;0.003	2.98;0.003

4.06;0.41	4.02;0.54	3.51;0.417	3.82;0.429	3.74;0.447
4.38;2.37	4.28;2.70	4.48;1.107	4.54;0.081	4.52;0.141
	4.64;0.54	5.02;0.000	4.54;0.081	4.52;0.141
4.70 5.07; 0.3	5.03;2.94	5.10;0.009	4.81;1.02	4.71;0.75
5.24;7.88	5.20;5.79	5.47;2.295	4.93;0.021 5.13;0.060	4.92;0.012 5.12;0.033
5.54;1.18	5.51;0.81	5.72;0.024	5.49;0.015 5.68;0.030	5.47;0.033 5.66;0.045
5.78;10.74	5.62;3.21	5.98;2.438	6.27;2.10	6.09;1.71
	6.31;0.42

^a Ref. 53.

^b Ref. 6.

^c Ref.21. The f are multiplied by 3 to account for degeneration (see text).

^d This work. The cutoff is 13.2 eV.

TABLE II. Transition energies E, oscillator strengths f, and bandwidth parameters σ determined from the absorption spectrum of C₆₀ in n-hexane.⁷ The third column contains the results of a Lorentzian fit with solvent effects filtered with the Onsager correction.⁴⁹  is the Lorentzian full width at half maximum. The units of E and σ are eV.

Ref. 7 E; f	Gaussian fit (this work) E; f; σ	Corrected fit⁴⁹ E; f;
3.04;0.015	3.04;0.002;0.02
3.30; ---- 3.78;0.37	3.69;0.46;0.28 3.76;0.24;0.08	3.78;0.26;0.09
4.06- 4.35 ; 0.10 4.84;2.27	4.17;0.096;0.11 4.47;0.49;0.15 4.84;2.27;0.15	4.87;1.41;0.13
5.46;0.22	5.35;0.25;0.10
5.88;3.09	5.90;4.90;0.34	6.04;5.32;0.36
6.36; ----	6.46;0.44;0.12

APPENDIX A: Effect of the CIS basis size

Figure 6 shows the effect of the size of the CIS basis, expressed in terms of the number of SECs. The 1200 SECs basis is roughly the size of the former CNDO/S calculations, and it is evident a bad description of the region above 6 eV. The Full CIS (with the atomic minimal basis set) provides a much better qualitative and quantitative agreement with the experiments. The 4621 SECs basis includes all the SECs with energy smaller than 13.2 eV, plus a few SECs added to complete the excited states shells present in the basis. This basis is qualitatively similar to the Full CIS basis, for a much lower computational cost, which is needed for the study of C₆₀ clusters.

FIG. 6. Effect of the size of the CIS basis over the CNDOL calculated spectrum of C₆₀.

APPENDIX B: Fit of the spectrum

Figure 7 shows the experimental data and fits of light absorption by C₆₀ in n-hexane solution. The thick lines show the fitted spectrum and the thin lines show the contribution of each transition. The dotted lines is the curve with the parameters estimated by Leach et al.⁷ The oscillator strength fitted for the allowed transitions at 3.76, 4.84 and 5.90 eV are 0.24, 2.27 and 4.90 (proportion 1:9:20), respectively. Note the broad band b centered at 3.69 eV. Its presence decreases the strength of the transition at 3.76 eV. If it is adjusted with a simple Gaussian, the height of peak c is well reproduced, but there is a considerable amount of oscillator strength unaccounted, which is associated to the feature b. Also note that the peak e, at 4.84 eV, can be described by a single Gaussian and is little affected by the transition d. Hence, peak e can be used to gauge the fitting procedure, and we have fixed its oscillator strength at 2.27 to agree with Ref. 7.

FIG. 7. Experimental data and fits of absorption spectrum by C₆₀ in n-hexane solution. In the fit we have fixed the 2.27 oscillator strength. Thin solid lines show the Gaussian components of the fitted function.

REFERENCES

1. T. M. Clarke and J. R. Durrant, Chem. Rev. 110 (11), 6736-6767 (2010).

2. M. Ohtani and S. Fukuzumi, Fuller Nanotub Car N 18 (3), 251-260 (2010).

3. P. Chawla, V. Chawla, R. Maheshwari, S. A. Saraf and S. K. Saraf, Mini-Rev. Med. Chem. 10 (8), 662-677 (2010).

4. D. V. Schur, S. Y. Zaginaichenko, A. F. Savenko, V. A. Bogolepov, N. S. Anikina, A. D. Zolotarenko, Z. A. Matysina, T. N. Veziroglu and N. E. Skryabina, Int. J. Hydrogen Energy 36 (1), 1143-1151 (2011).

5. K. R. S. Chandrakumar and S. K. Ghosh, Nano Lett. 8 (1), 13-19 (2007).

6. G. Orlandi and F. Negri, Photoch Photobio Sci 1 (5), 289-308 (2002).

7. S. Leach, M. Vervloet, A. Desprès, E. Breheret, J. P. Hare, T. J. Dennis, H. W. Kroto, R. Taylor and D. R. M. Walton, Chem. Phys. 160 (451-466) (1992).

8. R. E. Haufler, Y. Chai, L. P. F. Chibante, M. R. Fraelich, R. B. Weisman, R. F. Curl and R. E. Smalley, J. Chem. Phys. 95 (3), 2197 (1991).

9. Q. Gong, Y. Sun, Z. Huang, X. Zhou, Z. Gu and D. Qiang, J. Phys. B: At., Mol. Opt. Phys. 29 (21), 4981 (1996).

10. A. Sassara, G. Zerza, M. Chergui and S. Leach, ApJS 135 (2), 263-273 (2001).

11. J. Catalán and P. Pérez, Fuller Nanotub Car N 10 (2), 171 (2002).

12. J. D. Close, F. Federmann, K. Hoffmann and N. Quaas, CPL 276 (5-6), 393-398 (1997).

13. N. Sogoshi, Y. Kato, T. Wakabayashi, T. Momose, S. Tam, M. E. DeRose and M. E. Fajardo, J. Phys. Chem. A 104 (16), 3733-3742 (2000).

14. W. Kratschmer, L. D. Lamb, K. Fostiropoulos and D. R. Huffman, Nature 347 (6291), 354-358 (1990).

15. Y. Wang, J. M. Holden, A. M. Rao, P. C. Eklund, U. D. Venkateswaran, D. Eastwood, R. L. Lidberg, G. Dresselhaus and M. S. Dresselhaus, PhRvB 51 (7), 4547-4556 (1995).

16. A. L. Smith, J. Phys. B: At., Mol. Opt. Phys. 29 (21), 4975 (1996).

17. L. Bulavin, I. Adamenko, Y. Prylutskyy, S. Durov, A. Graja, A. Bogucki and P. Scharff, PCCP 2 (1627-1629) (2000).

18. E. Westin, A. Rosen, G. T. Velde and E. J. Baerend, J. Phys. B: At. Mol. Opt. Phys. 29, 5087–5113 (1996).

19. S. Iglesias-Groth, A. Ruiz, J. Breton and J. M. G. Llorente, J. Chem. Phys. 116 (24), 10648-10655 (2002).

20. Miguel A.L. Marques, Carsten A. Ullrich, Fernando Nogueira, Angel Rubio, Kieron Burke and E. K. U. Gross, Time-Dependent Density Functional Theory. (Springer Berlin / Heidelberg, 2006).

21. R. Bauernschmitt, R. Ahlrichs, F. H. Hennrich and M. M. Kappes, J. Am. Chem. Soc. 120 (20), 5052-5059 (1998).

22. A. Tsolakidis, D. Sánchez-Portal and R. M. Martin, PhRvB 66 (23), 235416 (2002).

23. J. Del Bene and H. H. Jaffe, J. Chem. Phys. 48 (4), 1807-1813 (1968).

24. T. Hara, S. Narita and T.-i. Shibuya, FST 3 (4), 459-467 (1995).

25. D. Rocca, R. Gebauer, Y. Saad and S. Baroni, J. Chem. Phys. 128, 154105 (2008).

26. P. Koval, D. Foerster and O. Coulaud, J. Chem. Theory Comput. 6 (9), 2654–2668 (2010).

27. L. Rincon, A. Hasmy, C. A. Gonzalez and R. Almeida, J. Chem. Phys. 129, 044107 (2008).

28. A. V. Nikolaev, I. V. Bodrenko and E. V. Tkalya, PhRvA 77, 012503 (2008).

29. D. M. Guldi, G. M. A. Rahman, V. Sgobba and C. Ehli, ChSRv 35, 471-487 (2006).

30. A. A. Voityuk and M. Duran, J. Phys. Chem. C 112 (5), 1672-1678 (2008).

31. P. R. Surjan, L. Udvardi and K. Nemeth, Synth. Met. 77, 107-110 (1996).

32. L. A. Montero, L. Alfonso, J. R. Alvarez and E. Perez, Int. J. Quantum Chem 37 (4), 465-483 (1990).

33. L. A. Montero-Cabrera, U. Röhrig, J. A. Padron-García, R. Crespo-Otero, A. L. Montero-Alejo, J. M. García de la Vega, M. Chergui and U. Röthlisberger, J. Chem. Phys. 127 (14), 145102 (2007).

34. M. E. Fuentes, B. Peña, C. Contreras, A. L. Montero, R. Chianelli, M. Alvarado, R. Olivas, L. M. Rodríguez, H. Camacho and L. A. Montero-Cabrera, Int. J. Quantum Chem 108, 1664–1673 (2008).

35. A. L. Montero-Alejo, M. E. Fuentes, E. Menéndez-Proupin, W. Orellana, C. F. Bunge, L. A. Montero and J. M. G. d. l. Vega, PhRvB 81, 235409 (2010).

36. D. L. Dorset and M. P. McCourt, AcCrA 50 (3), 344-351 (1994).

37. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. Dal Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj, M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri, R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto, C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen, A. Smogunov, P. Umari and R. Wentzcovitch, J. Phys.: Condens. Matter 21, 395502 (2009).

38. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).

39. N. Marzari, D. Vanderbilt, A. De Vita and M. C. Payne, Phys. Rev. Lett. 82, 3296 (1999).

40. K. Hedberg, L. Hedberg, D. S. Bethune, C. A. Brown, H. C. Dorn, R. D. Johnson and M. De Vries, Science 254 (5030), 410-412 (1991).

41. A. K. Soper, W. I. F. David, D. S. Sivia, T. J. S. Dennis, J. P. Hare and K. Prassides, J. Phys.: Condens. Matter 4, 6087-6094 (1992).

42. W. Branz, N. Malinowski, A. Enders and T. P. Martin, PhRvB 66 (9), 094107 (2002).

43. J. A. Pople and G. A. Segal, J. Chem. Phys. 44 (9), 3289-3296 (1966).

44. N. Mataga and K. Nishimoto, Z. Phys. Chem. 13, 140-157 (1957).

45. K. Nishimoto and N. Mataga, Z. Phys. Chem. 12, 335-338 (1957).

46. R. Pariser, J. Chem. Phys. 21, 568-569 (1953).

47. A. Szabo and N. S. Ostlung, Modern quantum chemistry: introduction to advanced electronic structure theory. (Dover Publications Inc., New York, 1996).

48. F. Negri, G. Orlandi and F. Zerbetto, J. Chem. Phys. 97 (9) (1992).

49. J. U. Andersen and E. Bonderup, EPJD 11 (3), 435-448 (2000).

50. P. Scharff, K. Risch, L. Carta-Abelmann, I. M. Dmytruk, M. M. Bilyi, O. A. Golub, A. V. Khavryuchenko, E. V. Buzaneva, V. L. Aksenov, M. V. Avdeev, Y. I. Prylutskyy and S. S. Durov, Carbon 42 (5-6), 1203-1206 (2004).

51. G. V. Andrievsky, V. K. Klochkov, E. L. Karyakina and N. O. McHedlov-Petrossyan, CPL 300 (3–4), 392-396 (1999).

52. I. D. Hands, J. L. Dunn, C. A. Bates, M. J. Hope, S. R. Meech and D. L. Andrews, PhRvB 77 (11), 115445 (2008).

53. M. Braga, S. Larsson, A. Rosen and A. Volosov, A&A 245, 232-238 (1991).

Download 2.2 Mb.

Share with your friends:

1 2 3 4