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J. Phys. Chem. C 2007, 111, 7862-7867
ARTICLES Which Configuration is More Stable for La2@C80, D3d or D2h? Recomputation with ZORA Methods within ADF Junfeng Zhang,†,‡ Ce Hao,*,†,§ Shenmin Li,| Weihong Mi,† and Peng Jin† State Key Laboratory of Fine Chemicals, Dalian UniVersity of Technology, Dalian, 116024, People’s Republic of China, School of Physics and Information Engineering, Shanxi Normal UniVersity, Linfen, 041004, People’s Republic of China, Virtual Laboratory of Computational Chemistry, CNIC, Chinese Academy of Sciences, Beijing, People’s Republic of China, and Key Laboratory of Bio-organic Chemistry, Dalian UniVersity, Dalian, 116622, People’s Republic of China ReceiVed: December 17, 2006; In Final Form: March 28, 2007
Geometry optimization and frequency analysis were performed for a La2@C80 molecule using the Amsterdam density functional (ADF) package. The relativistic effects are taken into account by the zero-order regular approximation (ZORA) basis sets. Interestingly, we found that the D2h configuration is the global minimum in total energy, being about 1 kcal/mol lower than the D3d configuration, which has been recently considered to be the most stable by Shimotani et al. On the basis of calculations, a new La2 pair motion scenario is proposed, in which the two La ions passing through saddle-point C2h, travel between equivalent D2h configurations. The motion of these La ions forms a pentagonal-dodecahedral path just like the pattern of La2 MEM charge density in the Ih C80 cage. In addition, the calculated Raman spectrum on low-frequency metalcage vibrations is also in good agreement with the experimental results.
1. Introduction As a representative and abundant endohedral metallofullerene, La2@C80, has been extensively investigated from both experimental and theoretical points of view since the first observation in 1991.1,2 Among these investigations, of particular interest has been characterizing the structural features, such as the cage structure and symmetry, as well as the location, motion, and electronic states of the encapsulated La ions. Important progress has been marked by some successful experiments. The 139La nuclear magnetic resonance (139La NMR)3 reported there is only one peak indicating the two La ions are equivalent in the icosahedral cage symmetry (abbreviated as Ih) C806- cage. The K-edge XAFS4 experiments explored the nearest La-cage distance is 2.42 Å at 40 K and 2.44 Å at 295 K, and the LaLa distance is 3.90 Å at 40 K and 3.88 Å at 295 K. X-ray observation4 and Raman analysis5 on La2@C80 verified the ionization of each La is 3+; thus, the electronic structure of La2@C80 can be described as (La3+)2C806-. An experiment study6 showed that the encapsulated La atoms can participate in the electrical conduction. The dynamic behavior of endocluster has be found in C80-based endohullerenes.7-10 In addition, similardynamicbehavioralsocanbefoundinotherendofullerenes.11-14 Recently, Yamada et al.15 successfully controlled the motion of two La ions from their random circulation in La2@C80 by adding the extra adduct. Interestingly, by synchrotron radiation (SR) with MEM/Rietveld analysis, the structure of an endohedral metallofullerene was visualized through the charge densities of * Corresponding author. E-mail:
[email protected]. † Dalian University of Technology. ‡ Shanxi Normal University. § Chinese Academy of Sciences. | Dalian University.
the La2 pair, which is a perfect pentagonal-dodecahedron in an Ih-C806- cage.16 However, some other works have been reported that the widely used MEM/Rietveld method analysis is not reliable enough to determine of the structures of endofullerenes.7,17,18 In addition to the success made in experiments, theoretical research has also made great progress. Kobayashi et al.19,20 carried out quantum chemistry calculations on a C80 hollow cage and on La2@C80. The key geometries were optimized at the Hartree-Fock (HF) level with the effective core potential basis sets developed by Hay and Wadt,21 using (5s5p3d)/[4s4p3d] for La and using the split-valence 3-21G basis set for C. The nonlocal DFT single-point energy calculations were used to improve the accuracy of the HF energy calculations. The calculations predicted that the most stable configuration of La2@C80 is with D2h symmetry. The two La ions are uniformly located on the C2 axis, which passes through the centers of two hexagonal rings. The distance between the La ions is 3.655 Å, and each La ion is 2.568 Å away from the hexagons. In addition, Kobayashi et al.22 indicated that the La2 pair could circulate inside the fullerene cage with a small barrier of about 5 kcal/ mol. Recently, Shimotani et al.23,24 also investigated the stable configurations and the molecular vibration modes of La2@C80 by the use of quantum chemistry calculations. The optimized geometries were obtained by the Hartree-Fock method with the double ζ quality basis sets both for C atoms (3-21G) and La atoms (LanL2DZ). It should be noted that the basis sets used by Kobayashi et al. and Shimotani et al. are almost the same. The total numbers of the Gaussian functions are 782 and 764, respectively. Some similar results were reached, such as (La3+)2@C806- electronic structure and the instability of the D5d configuration. However, Shimotani et al. ’s results for the La-
10.1021/jp0686590 CCC: $37.00 © 2007 American Chemical Society Published on Web 05/15/2007
D3d or D2h Configuration for La2@C80 La distance and the nearest La-cage distance are more similar to the XAFS4 experimental values than Kobayashi et al. ’s were. To Shimotani et al., the most stable configuration is not D2h symmetry but D3d symmetry. D3d symmetry is where two La ions are uniformly located on the C3 axis which passes through the carbons shared by three hexagonal rings. Shimotani et al. ’s frequency analysis shows there is one imaginary frequency for a D2h configuration, which indicates that it is not a local minimum but a saddle point. On the contrary, there is no imaginary frequency for a D3d configuration indicating it is a local minimum. Furthermore, on basis of the potential energy surface calculations, Shimotani et al. proposed a La2 ions traveling scenario inside C806- cage. The traveling scenario path consists of ten equivalent D3d positions, which are the energy valleys in the potential energy surface. This traveling scenario gives an explanation fitting with the results of the MEM charge densities experiment.16 Which configuration of La2@C80 is more stable, D3d or D2h? Obviously, Kobayashi et al. and Shimotani et al. gave quite different answers from each other. To answer this question, in this paper we carry out 3 new investigations: on the geometry structure, on the vibrational spectrum of La2@C80, and on the La2 pair motion in the C806- cage. Quantum chemistry calculations are employed by using the density function theory method within the Amsterdam density functional (ADF2005.01) package. The relativistic effects are taken into account by using the zero-order regular approximation (ZORA) basis sets, which are worked with the Dirac equation with a two-component Hamiltonian form. It should be noted that for precise calculations, relativistic effects need to be taken into account even for light systems.25 Therefore, the ZORA basis sets were used not only for lanthanum atoms but also for carbon atoms in the computations. Interestingly, the results indicate that the most stable configuration is with D2h symmetry, in contrast to a D3d configuration proposed by Shimotani et al. However, both theoretical calculations show satisfactory agreement with some experimental evidence, such as the dodecahedral La2 MEM charge density and the Raman spectrum in the low-frequency part. The remainder of the paper is structured as follows. Section 2 describes the details of the computational methods. The stable configuration of a La2@C80 molecule and the motion of the La2 pair in the Ih C806- cage, as well as the calculated Raman spectroscopy are reported in Section 3. Finally, a brief summary is concluded in Section 4. 2. Computational Methods By means of the ADF2005.01 program,26-28 the relativistic density functional theory calculations were carried out for a La2@C80 molecule. The geometry optimization and vibrational frequency analysis29-32 of the configurations were calculated by using the zero-order regular approximation (ZORA) basis sets,32 in which the relativistic effects are considered. All calculations employed the local density approximation (LDA) and generalized gradient approximation (GGA),33-35 with the DZP basis sets used for C atoms and the TZP basis sets for La atoms. In addition the frozen-core approximation up to the 1s orbital for C atoms and the 4d orbital for La atoms was used. This method is referred to as ZRDTZP in this paper. In addition, some other relativistic basis sets and nonrelativistic basis sets for geometry optimization were also used for comparisons. 3. Results and Discussion 3.1. Geometry Optimization. To describe the symmetry of a C80 (Ih) cage more simply, we took a right angle patch under
J. Phys. Chem. C, Vol. 111, No. 22, 2007 7863
Figure 1. Representative patches (shaded) C5, C2, and C3 represent a five-fold axis, a two-fold axis, and a threefold axis, respectively; σ represents a reflection plane of the C80 (Ih) cage (left). Ih geometry structure (right).
a circum-spherical surface as a representative patch of the C80 (Ih) cage (as shown in Figure 1). The area of the patch is equal to 1/120 of the total surface. There are six key points on the patch: point A is at the center of five-membered rings with D5d local symmetry; point B is at the center of six-membered rings with D2h local symmetry; point C at the carbon atom shared with three six-membered rings with D3d local symmetry; point D at the center of the 6-6 bond with C2h local symmetry; point E at the carbon atom with C2h local symmetry; and point F at the center of the 6-5 bond with Ci local symmetry. The boundary of the patch is characterized by a fivefold axis OA (point O is the center of the cage), a twofold axis OB and a threefold axis OC. It should be noted that all independent symmetrical elements belonging to the Ih point group can be found in the representative patch. When the elements of the Ih group perform an operation on the patch, the result will encompass the entire surface of the polyhedron. Therefore, we can consider the patch ABC (shaded) as the smallest structural unit rather than the whole cage. The point group symmetry of La2@C80 is reduced from Ih due to the encapsulation of two lanthanum atoms. The symmetry varies depending on the relative positions of the two La ions inside the cage. Six configurations, or energy stationary points, referred to as D5d, D2h, D3d, C2h, C′2h, Ci, were found, which are named according to the geometric symmetry. Table 1 shows the La-La distances (RLa-La) and the shortest La-C distances (RLa-C) as well as the relative energies for the six configurations of La2@C80. It also D2h the methods of calculation used and who performed them. We can see that for the six configurations there is no significant difference for the calculated RLa-La and RLa-C respectively, and the results are in good agreement with the corresponding experimental values of MEM/Rietveld and XAFS. Note that with the ZRDTZP basis sets, all calculated values of RLa-La are shorter than the experimental ones but with the errors of less than 0.1 Å. Also the RLa-C values of the D3d, C2h, and Ci configurations are much closer to the experimental results. The results of geometry optimization without the relativity effects are also interesting. As shown in Table 1, without the relativity effects, the RLa-La and RLa-C are similar to the results of Shimotani et al. The differences between the calculated and experimental results are larger than those obtained by the relativistic basis sets. Interestingly, with the ZRDTZP basis sets, the most stable La2@C80 configuration is a D2h configuration, which is about 1 kcal/mol lower than that of a D3d configuration. The D5d configuration is the most unstable with a relative energy 8.86 kcal/mol higher than the D2h configuration. These results agree well with those of Kobayashi et al., but not with Shimotani et al.’s. Shimotani et al. proposed that the most stable configuration is with D3d symmetry, with
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Zhang et a.
TABLE 1: Geometric Parameters and Relative Energies for Six Configurations of La2@C80 structure
method
RLa-La (Å)
RLa-C (Å)
D3d D2h D3d D2h D3d (C) D2h (B) C2h (D) C2h (E) Ci (F) D5d (A) D2h (B) D3d (C) C2h (D) C2h (E) Ci (F) D5d (A)
MEM/Rietveld XAFS XAFS HF/DZ HF/DZ B3LYP/DZP B3LYP/DZP DZP/TZP DZP/TZP DZP/TZP DZP/TZP DZP/TZP DZP/TZP ZRDTZP ZRDTZP ZRDTZP ZRDTZP ZRDTZP ZRDTZP
3.84 3.90(40K) 3.88(295K) 3.645 3.652 3.731 3.743 3.731 3.744 3.729 3.718 3.725 3.693 3.828 3.823 3.818 3.80 3.799 3.775
2.39 2.42 (40 K) 2.44 (295 K) 2.467 2.573 2.429 2.533 2.426 2.538 2.452 2.423 2.452 2.509 2.513 2.398 2.418 2.399 2.427 2.487
energy (kcal/mol)
ref
0 1.8 0 0.1 0 0.02 0.43 2.18 3.82 7.60 0 1.08 1.40 2.38 3.41 8.86
16 4 4 23 23 23 23 this work this work this work this work this work this work this work this work this work this work this work this work
a RLa-La is the La-La distance; RLa-C is the shortest La-C distance. C2h configuration represents two La ions located on the C2 axis which passes through two equivalent key point Ds; C′2h represents two La ions located on the C2 axis which passes through two equivalent key point Es. b The capital letters in brackets represent the key points in Figure 1.
TABLE 2: La-La Distances and the Shortest La-C Distances for D2h, D3d and C2h(1) Configurations by Some ZORA Basis Sets ZORA sets DZP-C TZP-La DZP-C TZP-La.4d DZP-C.1s TZP-La
basic structure
RLa-La (Å)
RLa-C (Å)
relative energy (kcal/mol)
D2h (B) D3d (C) C2h (D) D2h (B) D3d (C) C2h (D) D2h (B) D3d (C) C2h (D)
3.822 3.823 3.821 3.820 3.821 3.791 3.828 3.823 3.827
2.519 2.397 2.419 2.520 2.40 2.428 2.511 2.398 2.412
0 1.13 1.61 0 1.34 1.40 0 1.20 1.54
a (DZP-C TZP-La) represents all electrons ZORA DZP basis sets for C atoms and all electrons ZORA TZP basis sets for La atoms. b (DZP-C TZP-La.4d) represents all electrons ZORA DZP basis sets for C atoms and ZORA TZP basis sets for La atoms with the frozencore approximation up to the 4d orbital. c (DZP-C.1s TZP-La) represents the ZORA DZP basis sets for C atoms with the frozen-core approximation up to the 1s orbital and all electrons ZORA TZP basis sets for La atoms.
the D2h configuration only being a saddle point. However, in our nonrelativistic calculations, the D3d configuration showed to be the most stable total energy structure, about 0.025 kcal/ mol higher than that of the D2h configuration. This implies that the relativistic effects play an important role in determining the structure and energy of La2@C80. In order to justify the basis sets that were used, especially in their effective core approximations, additional optimizations for the D2h, D3d, and C2h configurations of La2@C80 were also computed, using all electron basis sets with the DZP basis sets for C atoms and the TZP basis sets for La atoms. The results are listed in Table 2. From Table 2 we can see that the RLa-La and RLa-C values for the D2h, D3d and C2h configurations in all three electrons basis sets have almost no change, and the biggest differences are within 0.01 Å. The relative energy for the D2h, D3d and C2h configurations is from low to high, and keeps the same order in the three basis sets. The energy of a D2h configuration is always the minimum, being consistently about 1 kcal/mol lower than that of a D3d configuration. This suggests that the effective core approximation is good enough to describe the geometry
structure and obtain reasonable relative energy for La2@C80. In addition, for comparison, geometry optimizations for the D2h, D3d, and C2h configurations were also computed by using the maximum available ZORA TZ2P basis sets, which include a double polarized function, with the frozen-core approximation up to the 4d orbital for La atoms. Very similar results with the ZRDTZP calculations were obtained. Despite the above discussions, which suggest a D2h configuration is the most stable configuration in contrast to Shimotani et al.’s D3d configuration, we find that at these two calculated levels, the energy differences between the two configurations are very similar, within about 1 kcal/mol. Therefore, these calculations suggest that thermal motion may help the two La ions pass small energy barriers and transfer between the key configurations inside the C806- cage. This is supported by the MEM/Rietveld experiments and Shimotani et al.’s calculations. Furthermore, our calculations discovered that the greatest energy difference among the six configurations is 8.86 kcal/mol. Shimotani et al. verified the D3d configuration is an energy minimum point and the D2h is a saddle point, and suggested the trajectory of La traveling is from a D3d position to another D3d position through a D2h saddle point, forming a pentagonal dodecahedron of La when averaged in a long time scale. It should be noted that the trajectory of La atoms reported by Shimotani et al. is in good agreement with the experimental MEM charge density of La2@C80. On the other hand, our calculations at ZRDTZP level suggest the D2h configuration is the energy minimum point, and the D3d configuration is the second saddle point. We believe the trajectory of La traveling is from a D2h point to a D2h point through an energy saddle point. To justify this guess, we scanned the potential profile between the point C and the point D, as shown in Figure 2. Next, using the energy minimum point (G) of the parabola of energy as an initial configuration of the saddle point, we performed a geometry optimization of the transition state. In that configuration, the La ions are located under a point G. This configuration is referred to as a G configuration, which also has C2h symmetry. At a G point, the La-La distance (3.806 Å) and the nearest La-C distances (2.414 Å) are closer to that of the XAFS experiments. As we expected, by frequency analysis the G point
D3d or D2h Configuration for La2@C80
J. Phys. Chem. C, Vol. 111, No. 22, 2007 7865
Figure 2. The energy of single-point between point C and point D.
Figure 4. Two views of visual angle for La ions traveling inside Ih symmetry cage. (a) Along the C2 axes; (b) along S10 axes.
Figure 3. The path of the La ions traveling in the representative patch.
3.2. Frequency Analyses. For Ih cage of C80, the vibrational modes correspond to the symmetry types: 3Ag + 4T1g + 5T2g + 8Gg + 11Hg + 1Au + 7T1u + 8Gu + 9Hu
shows to be the saddle point since it has an imaginary frequency with a value of 29.64i cm-1. However, the energy of the G configuration is only 1.056 kcal/mol higher than that of the D2h configuration. The path of the La2 ions is represented in Figure 3, where the La ions travel along the dashed lines, passing through the saddle point G. Obviously, this is quite a different La2 pair motion scenario (as shown in Figure 4) from Shimotani et al.’s. In this new scenario, the La ions would quickly pass through a D2h point compared with a point G (C2h). That is because when they move downhill from a saddle point G (C2h), they could gain more kinetic energy at the minimum point (D2h). Furthermore, there are three pathways near a D3d point, whereas there are only two pathways passing through a D2h point. Since a point G is very close to a D3d point (only about 0.16 Å), a conclusion seems to be made that the probability of the La ions at a D2h point would be small compared with that of the G point. However, considering the Boltzmann distribution using energy difference of 1.056 kcal/mol and room temperature, the existing probability of La ions at a point D2h is more than three times of that at a point G. This implies that although this La2 motion scenario is similar with MEM/Rietveld analysis results, but the La’s most existing probability is at a D2h point, which is not in consistent with the results by Shimotani et al.23 and MEM/ Rievteld analysis.16
When the two La ions are encapsulated in the cage to form the most stable configuration (D2h), the system symmetry degenerates from point group Ih to D2h, splitting the degenerated vibrational modes of Ih -symmetry C80 as follows:
A g f Ag T1g f B1g + B2g + B3g T2g f B1g + B2g + B3g Gg f Ag + B1g + B2g + B3g Hg f 2Ag + B1g + B2g + B3g A u f Au T1u f B1u + B2u + B3u T2u f B1u + B2u + B3u Gu f Au + B1u + B2u + B3u Hu f 2Au + B1u + B2u + B3u It is known that the coupling of the pair of La2 ions with the cage results in additional vibrational modes. Here, the Hg mode (222 cm-1) of C806- with the lowest computed wavenumber23
7866 J. Phys. Chem. C, Vol. 111, No. 22, 2007
Zhang et a. point by Shimotani et al. Interestingly, we draw the same conclusions as Shimotani et al. when using the nonrelativistic basis sets of same size within the DFT frame, that is, the D3d configuration is more stable in energy than that of the D2h configuration, although in our calculations the energy difference is only 0.025 kcal/mol. The calculations imply that the relativistic effects definitely play an important role at least for this endohedral metallofullerene. On the basis of the calculations of the six energy stationary points of the representative patch, a new La2 pair motion scenario is proposed, in which two La ions, passing through saddle point C2h, travel between equivalent D2h configurations. The trajectory of this La2 motion forms a pentagonal-dodecahedron, which partly agrees with the pattern of La2 charge density inside the Ih C80 cage as reported by Nishibori et al.16 The calculated Raman spectrum on low-frequency vibrations shows that the cage vibration mode (Hg) for Ih cage splits into five nondegenerate modes when two La atoms are encapsulated. Among them, the Ag mode with Raman peak at 158.08 cm-1 is assigned as the coupling of the metal ions with the carbon cage, which is also in good agreement with the experimental results.5
Figure 5. Low-wavenumber region of the Raman spectra of (a) C806-,23 (b) D2h-symmetric La2@C80 (computed) and (c) La2@C80 (experimental). (The intensities of La2@C80 were taken from experimental data.5)
is such a cage vibration. From the analysis above, the Hg mode will split into five modes: two Ag modes and B1g, B2g, and B3g modes, with wave numbers of 158.08, 215.69, 217.09, 231.35, and 232.22 cm-1, respectively. The calculated peak of 158.08 cm-1, which corresponds to 161 cm-1 in the Raman experiment,5 is assigned as the in-phase synchronously coupled mode of the C806- cage elongation with the La-La stretching. Since this peak is only found in endohedral metallofullerenes, and relates to the motion of encapsulated metal ions in carbon cages, it has been taken as the fingerprint vibration of endohedral metallofullerenes. Figure 5 shows the splitting of degenerated Hg mode of C806at 222 cm-1 and the comparison of computational and experimental spectra. It can be seen that the calculated values are in good agreement with the experimental data. The average relativistic error is only about 1.67%. The modes at 215.69 cm-1 (Ag) and 217.09 cm-1 (B2g) could be assigned to the peaks at 219 cm-1 and 224 cm-1 of the Raman experiment5 respectively. Notice that among the five split vibrational modes, the difference of wavenumber of mode B1g (231.35 cm-1) and B3g (232.22 cm-1) is within 1 cm-1. Therefore, these two modes could be assigned as one peak (234 cm-1 in the experiment5). Interestingly, in Shimotani et al. ’s calculations, where the D3d configuration is more stable, the Hg mode of C806- is split into only three modes, one nondegenerate A1g mode and two double Eg modes. Those three modes also have fairly good agreement with the experimental Raman spectrum.5 4. Conclusions Geometry optimization and vibrational frequency analysis for La2@C80 are carried out with the ADF2005.01 program. The relativistic effects for lanthanum as well as carbon are considered by using the zero-order regular approximation (ZORA) basis sets. It is found that the most stable configuration for La2@C80 is of D2h symmetry, which was just considered to be a saddle
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