First-Principles Investigation on Structural and Optical Properties of M

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First-principles Investigation on Structural and Optical Properties of M@C (where M = H, Li, Na, and K) +

60

Yoshifumi Noguchi, Osamu Sugino, Hiroshi Okada, and Yutaka Matsuo J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 28 Jun 2013 Downloaded from http://pubs.acs.org on June 28, 2013

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The Journal of Physical Chemistry

First-principles Investigation on Structural and Optical Properties of M+@C60 (where M = H, Li, Na, and K) Yoshifumi Noguchi,∗,† Osamu Sugino,† Hiroshi Okada,‡ and Yutaka Matsuo‡ Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan, and Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan E-mail: [email protected]

KEYWORDS: Green’s function, Fullerene, all-electron mixed basis approach, photoabsorption spectra, Bethe-Salpeter method Abstract Structural and optical properties of proton and alkali-metal atom cation encapsulated inside C60 molecule, M+ @C60 (where M=H, Li, Na, and K), are investigated in detail by first-principles. Two possible ground state atomic geometries where M+ can exist outside or inside C60 are proposed by the optimization using B3LYP hybrid ∗

To whom correspondence should be addressed Institute for Solid State Physics, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan ‡ Department of Chemistry, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan †

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functional based on the density functional theory. From the potential energy surface obtained in the same level calculation, we confirm that the energy barrier existing at the surface of C60 plays an important role to confine M+ inside. We also apply an all-electron GW +Bethe-Salpeter method based on the many-body perturbation theory and calculate the (inverse) photoemission spectra and the photoabsorption spectra. The calculated spectra are in reasonable agreement with available experimental results, while minor deviation exists suggesting non-negligible environment effect.

Introduction Lithium-encapsulated endohedral fullerene (Li@C60 ) has become an emerging nanocarbon material since its production was first reported in 1996. 1 A mixture of Li@C60 and empty C60 was obtained by ion implantation in a vacuum chamber. This mixture was chemically oxidized and was successfully separated to give Li+ @C60 salts. 2 Since then, Li+ @C60 has received considerable attention as a new kind of functional material 3,4 and intensive experimental investigations have been performed on its chemical modification, 5 photoinduced electron transfer, 6 and photovoltaic properties. 7 Structural and optical properties of Li+ @C60 have been investigated experimentally. 2,3 The C60 frame was found to maintain neutrality even after the insertion of Li+ . The positive charge of the encapsulated Li+ in the C60 cage is electronically compensated for by the negative charge of counteranions in both the solution phase and the solid phase. In + the crystalline phase, [Li+ @C60 ][SbCl− 6 ], Li is located at the off-center position near the

six-membered ring of C60 , according to X-ray diffraction studies. 3 In contrast, in solution, Li+ @C60 has Ih symmetry as does the empty C60 according to

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C NMR spectroscopy of

2 + [Li+ @C60 ][SbCl− 6 ]. The reason for the different Li distributions has not been explained in

detail. These experiments raised important questions on how the Li+ distribution is affected by the environment, such as the crystal field, solvent, salts, and temperature. This issue may 2 ACS Paragon Plus Environment

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be important also for controlling the functionality of this material. As a first step toward full understanding of this problem, it will be important to theoretically obtain the potential energy surface of an isolated Li+ @C60 ion at zero temperature. The electronic structures were also measured to find the role played by the inserted Li+ . 2 Electrochemical measurement done on [Li+ @C60 ][PF− 6 ] in solution indicted that the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are lowered by a similar amount (0.5–0.7 eV) when Li+ is inserted. This is considered to be the result of stabilization by the positive charge of Li+ . The experiment on [Li+ @C60 + ][PF− 6 ] in o-dichlorobenzene showed that the optical gaps are unchanged by the Li insertion.

Even when HOMO and LUMO are shifted by the same amount, this result may not be so trivial when the localized nature of the excitation is taken into account. The electron–hole pair generated by the optical process will be confined in the C60 frame and thus it is not surprising that the pair is affected by the existing Li+ . Environmental effects may be required to understand the experimental result. Toward a full understanding, the first step will be to study an isolated Li+ @C60 cation. The aim of this study is to investigate from first principles an isolated Li+ @C60 cation at zero temperature as well as protons or alkali-metal ions encapsulated in C60 , i.e,. M+ @C60 (where M = H, Li, Na, and K). We will provide the atomic structure and the potential energy surface (PES) of M+ @C60 using standard density functional calculations and compare the results in detail with those measured by X-ray diffraction and

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C NMR. We will also

calculate the HOMO-LUMO gap and the optical gap by using the all-electron GW -BetheSalpeter method and discuss how the insertion of M+ affects these excited-state properties.

Theoretical methods An all-electron GW +Bethe-Salpeter method is employed for calculating the optical properties of empty C60 and M+ @C60 in this study. This method consists of a three-step



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calculation. First is a conventional ground-state calculation made by using the local density approximation (LDA) based on the density functional theory (DFT). The LDA wave functions and Kohn-Sham orbital energies obtained here are used as input data for constructing the GW self-energy operator 8 in the second step of the calculation. Although our GW method is a one-shot version, 9 as discussed in next section, Blase et al.’s manner 10 is employed in this study to avoid strong overscreening caused by too small of a LDA gap. Promising single excited spectra related to (inverse) photoemission spectra are obtained in this stage. In the last step, the Bethe-Salpeter equation (BSE) is solved within GW approximation (GWA) to take the excitonic effect into account and the photoabsorption spectra comparable with the experimental spectra are calculated. 11–13 These calculations are carried out using an all-electron mixed basis approach, 14,15 in which the LDA one-electron wave function is expanded as a linear combination of plane waves (PWs) and atomic orbitals (AOs). This tactic is required because perturbation theories such as of GW and Bethe-Salpeter require all the information on the electronic states from the core electron states localized around the nucleus to free electron states above the vacuum level delocalizing the entire unit cell. Our all-electron method can fully respond to this high-level demand owing to the use of both PWs and AOs. This is an essential point of the present method. M+ @C60 is placed in a cubic FCC supercell with an edge length of about 31 ˚ A , which is large sufficiently to be regarded as almost an isolated system. Since the GW +Bethe-Salpeter calculation is sensitively affected by the long tail part of the Coulomb interactions with M+ @C60 in neighboring cells, we completely eliminate the Coulomb interactions among the unit cells by implementing a Coulomb cutoff. 16,17 We have checked the following parameters very carefully: PWs with cutoff energy of about 10.0 Ry, G vectors with cutoff energy of 20.6 Ry, G (G′ ) vectors with cutoff energy of 2.3 Ry, and also 9600 empty levels corresponding to about 65.6 eV are sufficient to converge LDA, GWA, and BSE energies. A total of 120 occupied levels and 50 empty levels are used to construct the BSE Hamiltonian.


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Results and discussion Structural properties We calculated the ground state of M+ @C60 (where M= H, Li, Na, and K) using B3LYP/ccpVTZ implemented in the Ganssian09 package. 18 We begin by mentioning the basis set effect on the structural properties. The effect was investigated on the line connecting the center of C60 toward the outside via the center of the six-membered ring. When M+ (where M = Li, Na, and K) was moved along this line, the total energy obtained using B3LYP/6-311G* showed a significant difference from the one obtained using B3LYP/cc-pVTZ, as shown in Fig. 1. The difference in the total energy is always more than 5.9 eV, which already indicates insufficient convergence of the total energy. The difference exhibits a maximum at the center of the six-membered ring and the maximum amounts to 7.1 eV for K+ @C60 , alerting us to the use of a smaller basis set in obtaining a reliable PES. In the following, we use cc-pVTZ, 19 the largest basis set allowed in our computational environment, though its convergence test is left to a future study. We note that the standard but smaller basis sets, such as 3-21G, 6-31G, and 6-311G, used in previous studies, 20–24 generate a serious error in the PES. All the cations energetically prefer to be located outside C60 rather than inside. The difference in energy is 2.03 eV for H+ , 0.56 eV for Li+ , 0.63 eV for Na+ , and 0.64 eV for K+ . We now show the most stable and metastable geometries separately. The optimized atomic geometries are shown in Fig. 2. In contrast to other metal cations, H+ is chemically bonded with a carbon atom of C60 , with a C–H distance of 1.10 ˚ A , thereby distorting the local frame of C60 . M+ has the lowest total energy when it is located away from the center of the six-membered ring outward by about 1.83 ˚ A for Li+ , 2.23 ˚ A for Na+ , and 2.58 ˚ A for K+ . These cations are just weakly bound to C60 . 25 Our interest is, however, rather in C60 encapsulating M+ . M+ has a metastable structure inside of C60 , as shown in Fig. 3. Again, H+ is chemically bonded to a carbon atom of C60 , whereas other cations are only weakly bound to C60 being located on the line connecting the center of C60 and the six-membered



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Figure 1: Basis set effect in eV, which is the difference between B3LYP/6-311G* and B3LYP/cc-pVTZ total energies. Here Li+ is moved from the center of C60 corresponding to the origin of horizontal axis to outside C60 via the center of six-membered ring.


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ring. The stable positions of Li+ , Na+ , and K+ are about 1.50, 0.78, and 0.00 ˚ A distant from the center of C60 , respectively. 26 A way to ascertain the validity of the present results is to compare with the results from experiment. According to the X-ray diffraction experiment on 3 + ˚ solid [Li+ @C60 ][SbCl− 6 ] in Ref. , Li is located away from the center of C60 by 1.34 A in the

direction toward the center of the six-membered ring. The present result is consistent with the experimental result, suggesting that the difference in the environment, in the isolated phase or in the condensed phase, plays only a minor role on the stable position.

Figure 2: Most stable atomic geometries of M+ @C60 (M = H, Li, Na, and K) optimized by B3LYP/cc-pVTZ. M+ is located outside C60 .

Now we discuss how and why M+ is confined inside of C60 . To investigate this, we calculated the total energy along the linear path of M+ going from the centers of C60 outward through the center of the six-membered ring. In this calculation no geometry optimization was performed. The results are shown in Fig. 4. Here the total energy is referred to that obtained by locating M+ at the center of C60 . 27 The energy rapidly increases near the sixmembered ring (marked by the dark yellow line) and exhibits a maximum at the center of



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Figure 3: Meta-stable atomic geometries of M+ @C60 (M = H, Li, Na, and K) optimized by B3LYP/cc-pVTZ.

the six-membered ring for Li+ and Na+ and at a position slightly outward from the center for H+ and K+ . The larger the ionic radius, the higher is the energy barrier. The large barrier height found for Li+ , Na+ and K+ indicates that those cations cannot escape from C60 once encapsulated. The situation is opposite for H+ where no barrier is found. The results can be intuitively understood by comparing the ionic radius, 0.90 ˚ A for Li+ , 1.16 ˚ A for Na+ , and 1.52 ˚ A for K+ , and the size of the six-membered ring whose incircle (circumcircle) radius is 1.23 ˚ A (1.42 ˚ A ). As depicted in Fig. 5, the ionic radius is small for Li+ and Na+ compared with the incircle radius, but it may not be sufficiently small to pass through the six-membered ring without touching the electrons responsible for the C–C σ bond. 28 The ionic radius is already larger than the incircle radius for K+ , where the energy barrier is maximum. In contrast, the ionic radius is significantly smaller for H+ , where no barrier is found. These PESs give us further insight into a related experiment. A recent


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C NMR exper-

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Figure 4: Total energies plotted with respect to the position of M+ . The inset is of energy range between −3 eV and +3 eV.



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2 + iment on [Li+ @C60 ][PF− 6 ] in o-dichlorobenzene suggests that Li @C60 has Ih symmetry

as does the empty C60 . The result is seemingly inconsistent with the calculated potential 3 energy surface and the experiment on solid [Li+ @C60 ][SbCl− 6 ] in Ref. . There are two pos-

sible explanations for this, as already mentioned in Ref. 2 . The first explanation is that the stable position of Li+ depends on the chemical environment, i.e., in the solvent and in the solid. The second explanation is that Li+ is hopping around the center of C60 much faster than the time scale of the NMR measurement, so that a time-averaged geometry with higher symmetry Ih is observed. Closely looking at the PES as shown in the inset in Fig. 4, we find that Li+ is lower in total energy by 0.6 eV when displaced by 1.50 ˚ A . It is unlikely that the environmental effect is sufficiently large to make the center of C60 the most stable. From this, we conclude that the second explanation most likely accounts for the present calculation.

Figure 5: Cartoon of a six-membered ring and M+ ions. Single bonds of the six-membered ring referred to as C1–C2, C3–C4, and C5–C6 are about 1.39 ˚ A and double bonds referred ˚ to as C2–C3, C4–C5, and C6–C1 are about 1.45 A . The ion radius is about 0.90 ˚ A for Li+ + + (pink circle), 1.16 ˚ A for Na (blue circle), and 1.52 ˚ A for K (green circle).


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Optical properties Table 1: LDA and GWA HOMO-LUMO gaps in electron volts. The experimental values 29,30 are also listed here for comparison.

C60 H @C60 Li+ @C60 Na+ @C60 K+ @C60 +

LDA GWA Expt. 29,30 1.68 4.78 4.91 1.29 4.39 — 1.64 4.89 — 1.68 4.84 — 1.70 4.85 —

Next, we study the optical properties of the empty C60 and M+ @C60 using the allelectron GW +Bethe-Salpeter method, which is a scheme to apply many-body corrections to the LDA. Considering that B3LYP/cc-pVTZ yields the most reliable structure, we use the geometries optimized using B3LYP/cc-pVTZ although the all-electron GW +Bethe-Salpeter method is based on a different scheme and basis functions. The HOMO-LUMO gaps are listed in Table 1 together with available experimental data. 29,30 As already well known, the LDA tends to underestimate the HOMO-LUMO gap by a few electron volts. This general tendency is seen in the C60 molecule as well and the difference from the experimental gap is about 3.2 eV, which is somewhat larger than that of other systems such as metal and semiconductor clusters. The one-shot GW scheme cannot sufficiently correct such an underestimated HOMO-LUMO gap, thereby causing the overscreening problem as reported by Blase et al. 10 They suggested replacing the Kohn-Sham eigenvalues with the HartreeFock levels in performing the GW calculation as a simple prescription. Following this, we employed the modified one-shot GW method, or the G0 W0 (HFdiag ). Then, the LDA HOMOLUMO gap of C60 is enlarged by about 3.0 eV in G0 W0 (HFdiag ) and the deviation from the experiment becomes only about 0.1 eV. Table 1 lists the results of the G0 W0 (HFdiag ) calculation for M+ @C60 . The HOMOLUMO gap of the empty C60 is enlarged only by 0.1 eV when M+ (where M = Li, Na, and K) is inserted into the empty C60 . This is the case not only for GW but also for LDA. 11 ACS Paragon Plus Environment


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Figure 6: GW quasiparticle energies involving some occupied levels (red bars) and unoccupied levels (blue bars) near the Fermi level (in electron volts). The LUMO level is set to be zero.

Note that both the HOMO and LUMO levels are shifted downward relative to the vacuum level by a few electron volts (not shown here) by the insertion, but the HOMO-LUMO gap is unchanged. The situation is different for H+ . Because H+ makes a chemical bond with the carbon atom, the electronic structure and the atomic geometry are affected more significantly than in other M+ @C60 cases. The LDA and GWA HOMO-LUMO gaps of H+ @C60 are about 0.4 eV smaller than those of others. These results are seen in Fig. 6 too. The GW quasiparticle energies shown here are referred to the corresponding LUMO level. We notice that the highly degenerated levels of the empty C60 are, in most cases, split off into some one- or two-hold degenerate levels after insertion of H+ , Li+ , or Na+ . The splitting is not so large for K+ . Considering that the level splitting is the smallest for K+ , which is located at the center of C60 , and the largest for H+ , which is chemically bonded with a carbon atom, we conclude the position of the cations rather than the atomic species plays a key role in removing the degeneracy. Note that the level positions referred to the vacuum level 12 ACS Paragon Plus Environment

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are almost unchanged for Li+ , Na+ , and K+ , whereas the positions are shifted upward by almost half an electron volt for H+ .

Figure 7: BSE photoabsorption spectra of empty C60 (black line) and Li+ @C60 (red line). The experimental spectra 3,31 are also shown here for comparison. The experimental spectra labeled as “Experiment (A)" 31 have been converted in Ref. 32 from wavelength in nanometers to photon energy in electron volts and the experimental spectra labeled as “Experiment (B)" is from Ref. 3 where the unit conversion from wavelength in nanometers to photon energy in electron volts has been done.

Using the information obtained in the LDA and GW calculations, we next construct the BSE Hamiltonian and calculate the photoabsorption spectra including the explicit excitonic effect. To see how well the GW +Bethe-Salpeter method works for the present systems, we first focus on C60 having rich experimental data. The minimum BSE optical gap of C60 is about 1.8 eV. This is the HOMO-LUMO gap and the transition is forbidden. Since the GW HOMO-LUMO gap is about 4.8 eV, the corresponding exciton binding energy is estimated to be about 3.0 eV. Such a large value emphasizes importance of the excitonic effect. There 13 ACS Paragon Plus Environment


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are four observable peaks in the BSE spectra up to 6 eV (Fig. 7). The first optically allowed BSE gap is about 3.4 eV and involves HOMO-1 to LUMO, HOMO to LUMO+1, HOMO to LUMO+2, and HOMO-2 to LUMO transitions. We compare in Fig. 7 the present result (black line) with the experimental spectra (blue line), labeled as “Experiment (A),” which were measured for C60 in hexanes at room temperature. 33 Experiment (A) has four peaks in this photon energy region too. The experimental peak positions, marked by blue arrows, are at about 3.10, 3.76, 4.82, and 5.85 eV, whereas the corresponding BSE peak positions are at about 3.4, 3.7, 4.9, and 5.5 eV, respectively (and note that the remaining error is only 0.35 eV even in the worst case). The BSE spectra (black line) and the experimental one (blue line) are consistent in both the number of observable peaks and the peak positions. Also the peak heights are consistent; although the first one is a little too high, others are consistent with experiment. Overall, the BSE spectra agree with experiment well. Next we compare C60 (black line) and Li+ @C60 (red line) in Fig. 7. The peak position and the profile are not so different between C60 and Li+ @C60 as expected from the electronic structure in Fig. 6. A blue shift of 0.2–0.4 eV is seen for the main peaks of Li+ @C60 (3.6, 3.9, 5.2, and 5.9 eV). The blue shift was, however, not observed by a recent UV–visible 3 experiment on C60 and [Li+ @C60 ][PF− 6 ] in o-dichlorobenzene done at room temperature,

which is labeled as “Experiment (B)” in Fig. 7. Indeed, the experimental spectra of Li+ @C60 (purple line) have a broad peak, which is centered at 3.70 eV and ranges from 2.8 to 4.0 eV; the peak position is almost exactly the same as that of the empty C60 (green line). Note that the small peak located at 3.1 eV for C60 is not seen in Li+ @C60 possibly because the corresponding peak in Li+ @C60 is involved in the broad peak. We emphasize that, for the C60 spectra, Experiment (B), done for the solid up to 4.1 eV, is strikingly consistent with Experiment (A) measured in hexanes. This indicates that the solvent effect is small at least for C60 . In addition, those spectra for C60 are consistent with the calculation obtained without incorporating the solvent. Therefore, it is not clear why the blue shift found by the theory is not observed in Experiment (B). It is possible that, when Li+ is encapsulated into


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C60 , the spectra are unexpectedly affected by the conditions, e.g., temperature, solvent, and the effect of existing salt. These factors should be investigated in future studies.

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Figure 8: BSE photoabsorption spectra of empty C60 and M+ @C60 .

We have applied the present method to other M+ @C60 as well. Fig. 8 shows the photoabsorption spectra of M+ @C60 together with that of empty C60 . The spectra are significantly affected when H+ is encapsulated, a result of the distorted local frame of C60 . (Note that, as mentioned above, GW quasiparticle energies of H+ @C60 are already very different.) Other spectra are, however, very similar to each other: The first two peaks below 4 eV are very similar among M+ @C60 (where M = Li, Na, and K) and the next two peaks around 5.3 eV are similar between Li+ @C60 and Na+ @C60 . Note that, similarly to Li+ @C60 , a blue shift appears also for Na+ @C60 and K+ @C60 .



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Conclusions We have determined the most stable and metastable atomic geometries of M+ @C60 (where M = H, Li, Na, and K) from first-principles B3LYP/cc-pVTZ calculation. In the most stable atomic geometries, all the M+ except to H+ are weakly bound to outside C60 without chemical bonding and H+ is bound to outside C60 by chemical bonding. In the metastable atomic geometries, M+ is located inside C60 and again only H+ makes a chemical bond with C60 . We calculated the PESs where M+ is moved on the line connecting the centers of C60 and a six-membered ring and confirmed that M+ @C60 cations (where M = Li, Na, and K) have a very high energy barrier around the surface of C60 and the energy barrier confines M+ inside C60 . A different situation occurs for H+ . H+ has no conspicuous energy barrier, at least on the line we checked; however, it remains inside C60 by forming a chemical bond with a carbon atom. In addition, we found another energy barrier around the center of C60 . Its existence suggests that the Ih symmetry of Li+ @C60 measured in a 13 C NMR experiment results from Li+ moving around the center of C60 much faster than the time scale of

13

C

NMR. We calculated the (inverse) photoemission spectra and the photoabsorption spectra of empty C60

and M+ @C60

by using an all-electron first-principles GW +Bethe-Salpeter

method. The GW HOMO-LUMO gap and the BSE photoabsorption spectra of empty C60 agree with experiment well. We compared the present results for empty C60 and Li+ @C60 with the experiment. For both cases of empty C60 and Li+ @C60 , the BSE main peak position is consistent with experiment; however, a blue shift was found in the BSE spectra of Li+ @C60 . The discrepancies between BSE and experimental spectra might be caused by some mismatches between theoretical and experimental conditions.


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Acknowledgement The authors thank the Institute for Solid State Physics and Information Technology Center, the University of Tokyo and Research Institute for Information Technology, Kyushu University, for the use of their supercomputers. One of authors, Y. N., has been supported by the Grant-in-Aid for Young Scientists (B) (No. 23740288) from Japan Society for the Promotion of Science (JSPS). Y. M. and H. O. acknowledge financial supports by the Funding Program for Next Generation World-Leading Researchers.

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(26) We did similar calculation for Li+ @C60 with CAM-B3LYP to see the effect of longrange interaction between Li+ and C60 . The optimized atomic geometry is same as that of B3LYP. So we continue to use B3LYP in this study. (27) For H+ @C60 , we set the total energy at 0.60 ˚ A distant from the center of C60 to be zero because, at a few positions of H+ especially around the center of C60 , the total energy calculations become unstable and fail to converge the electronic states in self-consistent cycle. (28) http://en.wikipedia.org/wiki/Ionic_radius. (29) Muigg, D.; Scheier, P.; Becker, K.; Mark, T. D. Measured Appearance Energies of C+ n (3 ≤ n ≤ 10) Fragment Ions Produced by Electron Impact on C60 . J. Phys. B 1996, 29, 5193–5198. (30) Wang, X. B.; Woo, H. K.; Wang, L. S. Vibrational Cooling in a Cold Ion Trap: Vibrationally Resolved Photoelectron Spectroscopy of Cold C− 60 Anions. J. Chem. Phys. 2005, 123, 051106. (31) Smith, A. L. Comparison of the Ultraviolet Absorption Cross Section of C60 Buckminsterfullerene in the Gas Phase and in Hexane Solution. J. Phys. B 1996, 29, 4975–4980. (32) Bertsch, G. F.; Bulgac, A.; Tom´ anek, D.; Wang, Y. Collective Plasmon Excitations in C60 Clusters. Phys. Rev. Lett. 1991, 67, 2690–2693. (33) Ajie, H.; Alvarez, M. M.; Anz, S. J.; Beck, R. D.; Diederich, F.; Fostiropoulos, K.; Huffman, D. R.; Kratschmer, W.; Rubin, Y.; Schriver, K. E.; Sensharma, D.; Whetten, R. L. Comparison of the Pseudospinodal to the Transition from Metastability to Instability in a Binary Liquid Mixture. J. Phys. Chem. 1990, 94, 8630–8631.


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First-Principles Investigation on Structural and Optical Properties of M

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