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Global minima of (C60)nCa2+, (C60)nF− and (C60)nI− clusters
Chemical Physics Letters 410 (2005) 404–409 www.elsevier.com/locate/cplett
Global minima of (C60)nCa2+, (C60)nF and (C60)nI clusters J. Herna´ndez-Rojas a, J. Breto´n a, J.M. Gomez Llorente a b
a,*
, D.J. Wales
b
Departamento de Fı´sica Fundamental II, Universidad de La Laguna, 38205 Tenerife, Spain University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom Received 5 April 2005; in final form 6 May 2005 Available online 21 June 2005
Abstract Likely candidates for the lowest potential energy minima of (C60)nCa2+, (C60)nF and (C60)nI clusters are located using basinhopping global optimisation. In each case, the potential energy surface is constructed using the Girifalco form for the C60 intermolecular interaction, an averaged Lennard–Jones C60–ion interaction, and a polarisation potential, which depends on the first few non-vanishing C60 multipole polarisabilities. We find that the ions generally occupy the interstitial sites of a (C60)n cluster, the coordination shell being tetrahedral for Ca2+ and F. The I ion has an octahedral coordination shell in the global minimum for (C60)6I, however for 12 P n P 8 the preferred coordination geometry is trigonal prismatic. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction C60, the most abundant single shell fullerene under various experimental conditions, has a remarkably high ˚ 3) [1,2]. This large electric-dipole polarisability (a1 80 A polarisability leads to what we have called a Ôpolarisation bondÕ [3] in C60X complexes, X being an alkali ion, with binding energies as high as 1 eV. In a previous publication [4], motivated by this property, we studied the effect of the high polarisation energies on the geometrical structure and energetics of (C60)nX clusters, where X was chosen to be either an alkali metal ion or Cl. We carried out searches for the likely global potential energy minima of those clusters using a potential energy function that incorporates the Girifalco potential between C60 molecules, an averaged Lennard–Jones ion–C60 interaction, and a polarisation potential depending on the first few fullerene multipole polarisabilities. The basin-hopping approach [5,6] was used to locate the likely global min*
Corresponding author. E-mail addresses: [email protected] (J. Herna´ndez-Rojas), jbreton@ ull.es (J. Breto´n), [email protected] (J.M. Gomez Llorente), dw34@ cam.ac.uk (D.J. Wales). 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.05.075
ima. We found that the ions were small enough to occupy an interstitial site of the fullerene (C60)n matrix, the coordination shell being triangular for Li+, tetrahedral for Na+ and K+, and octahedral for Cl, in agreement with standard radius ratio rules. When the required coordination site did not exist in the (C60)n global minimum, the global minimum of the doped cluster usually changed to provide such a site. For all the ions, we found significant binding energies, DEB, in the range 1.5 to 2.5 eV. These values are exceptionally high for a nonpolar matrix like C60, and agree quite well with the predictions of the simple Born model [7], DEB = (e 1)q2/2ek, when one uses the fullerite value (e = 4.4) [8] for the dielectric permittivity and the ion site radius k. The relative contribution to the association energy of the multipole polarisabilities beyond the dipole contribution was quite significant and increased from 8% for Cl to 34% for Li+. The corresponding contribution to the polarisation energy was larger, with values ranging from 30% for Cl to 45% for Li+. In the present contribution, we extend our previous study to the clusters (C60)nCa2+, (C60)nF and (C60)nI. In particular, we wish to explore the effect of both a larger ionic charge and a larger ionic radius on
J. Herna´ndez-Rojas et al. / Chemical Physics Letters 410 (2005) 404–409
the predicted structure. We will present likely candidates for the global potential energy minima in each case. We focus our study on clusters with n < 14, but some cases with n P 14 are also considered. For the (C60)nCa2+ complexes we have obtained a rather unexpected result. Namely, the arrangements þ ðC60 Þþ nm þ ðC60 Þm Ca ðm < nÞ can have lower energy than the cluster asymptotic dissociation limit (C60)n + Ca2+. For n = 1 the energy difference between the two possible dissociation limits is the sum of the C60 ionization potential and the Ca2+ electron affinity, which we predict to be somewhat lower than the binding energy of the corresponding C60Ca2+ complex. Therefore the C60Ca2+ complex would be unstable relative to the charge-transfer dissociation limit. However, for n P 2, we have found that the ion complex is stable with respect to all the charge-transfer dissociation channels. This stabilisation of Ca2+ by a neutral non-polar molecule is a consequence of the high polarisability of C60. As in our previous work [4], we will be interested in the optimal coordination environment of each ion and in its ability to change the geometry of the optimal (C60)n fullerene cluster to achieve this preferred coordination. These details depend upon the balance between the dispersion–repulsion and the polarisation energies. Our analysis will provide guidance for future molecular beam experiments. Various gas phase spectroscopy techniques could probably also be applied to probe the predicted structures. Furthermore, the ionic species could be investigated by drift tube arrival time measurements. This paper is organised as follows. In Section 2, we discuss the expression previously derived in [4], which provides the potential energy surface as a sum of dispersion–repulsion and polarisation contributions. In Section 3, we present likely candidates for the cluster global minima and analyse these results. Finally, Section 4 summarises our conclusions.
2. The potential energy The closed-shell electronic structure of C60, and of the chosen ions, makes semi-empirical methods particularly suitable for constructing the potential energy surface (PES) of the corresponding clusters. The expression providing this PES was presented in [4] as a sum of three contributions V ¼ V ff þ V if þ V pol ;
ð1Þ
where Vff is the sum of pairwise fullerene–fullerene dispersion–repulsion interactions, Vif is the sum of pairwise ion–fullerene dispersion–repulsion interactions, and Vpol is a term including the relevant polarisation-induced contributions. The form of each of these terms has been obtained by calculating the average interaction over all fullerene orientations; i.e., we have disregarded the finer
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details due to the non-spherical symmetry of the charge density in the C60 molecule. This is a rather good approximation, justified by the high symmetry of this molecule, whose lowest non-vanishing multipole moment and anisotropic polarisability correspond to spherical harmonic indexes l = 6 and l = 3, respectively. With these assumptions the fullerene–fullerene dispersion–repulsion interaction is given by GirifalcoÕs potential [9], which corresponds to the dominant isotropic contribution obtained from a sum of pairwise Lennard–Jones carbon–carbon interactions, whose parameters eCC and rCC are given standard values. Similarly, the ion–fullerene dispersion–repulsion energy is represented by an expression [10] that corresponds to the dominant isotropic contribution obtained from a sum of pairwise Lennard–Jones carbon–ion interactions, with parameters eXC and rXC. Values for these parameters were estimated from gas phase data using the Lorentz–Berthelot combination rules. There are several sources for the ion parameters [11–16]. For Ca2+, we have used the values provided in [16]. For the anions, we have chosen values adjusted to fit molecular properties of the alkali halides [17] (binding energy, bond distance and vibrational frequency) when the potential is written as the sum of Lennard–Jones, polarisation and Coulomb terms. Actually, rXC is the most important parameter because it fixes the ion size, which determines its coordination in the complex. The dispersion energy associated with the parameter eXC is less than 1% of the values corresponding to the polarisation and Girifalco contributions. The values chosen for all these parameters are given in Table 1. The polarisation term Vpol in Eq. (1) includes the energy associated with the polarisation of the fullerene molecules due to the electric field of the ion charge and the multipole moments induced in the other fullerene molecules. This contribution was evaluated self-consistently in terms of the first three C60 multipole polarisabilities (al, l = 1, 2, 3). The required al were obtained from a semiempirical treatment of the C60 electronic structure (see [4] and references therein), which gives an a1 value in agreement with experimental data. These values were later slightly adjusted to include the effect of higher order terms, which may be relevant at the smaller ion–fullerene distances. The three al values adjusted in this way are given in [4]. When our parameter set is used to represent the interaction in (C60)2 and ðC60 Þþ 2 , the agreement with the experimental values [18] is very good [4]. Table 1 Values of the parameters for the ion–carbon Lennard–Jones interaction ˚) X eXC (meV) rXC (A Ca2+ F I
3.527 2.373 5.020
3.18 3.46 4.33
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3. Global potential energy minima As in [4], the likely candidates for the global potential energy minima were located using the basin-hopping scheme [5], which corresponds to the ÔMonte Carlo plus energy minimizationÕ approach of Li and Scheraga [6]. This method has been used successfully in both neutral [5] and charged clusters [19,20], along with many other applications [21]. In the size range considered here the global optimization problem is relatively straightforward. The global minima are always found in less than 500 steps of each basin-hopping run, independent of the random starting geometry. For the (C60)nX complexes, association energies are defined for the process nC60 þ X ¼ ðC60 Þn X;
DEa .
ð2Þ
We also define the X ion binding energy as the difference between the association energy of the complex and that of the (C60)n cluster, i.e. ðC60 Þn þ X ¼ ðC60 Þn X;
DEb .
ð3Þ
Our (C60)n global minima and association energies coincide precisely with those obtained by Doye and Wales [22]. In Fig. 1, we plot the association and binding energies defined in (2) and (3) as a function of the number of fullerene molecules n. Both tend to asymptotic values as n increases. We also include in this figure the point group symmetry obtained for each complex. As in the complexes studied in [4], the contribution of the polarisation energy Vpol to the binding energies is the dominant one (more than 90% of the total).
Fig. 1. Association energies per fullerene molecule DEa/n (circles) and binding energies DEb (crosses) for (C60)nX complexes. The arrow indicates the size at which a complete coordination shell is achieved. Asterisks mark those complexes in which the structure of (C60)n matrix differs from the global minimum of the corresponding (C60)n cluster. We also include the point group for each complex.
As mentioned in Section 1, for the (C60)nCa2+ comþ plexes, the arrangements ðC60 Þnm þ ðC60 Þm Caþ ðm < nÞ can have energies lower than the cluster asymptotic dissociation limit (C60)n + Ca2+. For n = 1, the energy difference, DEct, between the two possible dissociation channels is the sum of the C60 ionization potential and the Ca2+ electron affinity, i.e., DEct = 4.27 eV, which is somewhat lower than the binding energy of the corresponding C60Ca2+ complex DEb = 4.06 eV. Therefore, the C60Ca2+ complex is unstable relative to the chargetransfer dissociation limit. However, using simple electrostatic arguments we have estimated that the crossing between the two electronic states correlating with the two asymptotic limits takes place at an ion–fullerene ˚ and an energy of 1 eV above centre distance of 6 A 2+ the C60Ca global minimum, which would make charge-transfer a very unlikely process in this compound under normal conditions. For n = 2, we estimate DEct DEb 2.3 eV for m = 0 and 1.8 eV for m = 1, which means that (C60)2Ca2+ is already stable against the charge-transfer dissociation channels. For increasing n the stabilisation effect increases sharply until the first coordination shell is formed. This stabilisation of the Ca2+ charge by C60 is a consequence of the high polarisability of this molecule. As we know from [4], the initial sharp decrease shown by the binding energies in Fig. 1 is related to the formation of the ion coordination shell. This process represents the largest contribution (more than 90%) to the asymptotic binding energies. The completion of this first shell is indicated by an arrow in Fig. 1 and the corresponding structures, which will be called coordination compounds or complexes, are illustrated in Fig. 2. Like the alkali ions Na+ and K+, the smaller F and Ca2+ ions favour tetrahedral coordination, while the larger I prefers an octahedral coordination environment, like Cl. These results agree with the predictions of the standard radius ratio rules if we use the geometrical ˚ ) and the Pauling radii for the radius for C60 (R0 = 3.55 A ions. In the coordination compounds of Fig. 2, the geometrical structure of the fullerene matrix is that of the corresponding (C60)n global minimum; in other words, the ion occupies an interstitial site of the fullerene cluster. However, this fullerene cluster undergoes a mean contraction of 1.6% in the Ca2+ complex and a mean expansion of 1.6% and 1.7% in the F and I complexes, respectively, in order to accommodate the corresponding ion. This distortion is significantly larger than that found in the corresponding alkali and chlorine ion complexes with the same geometrical structure (0.5% contraction for Na+, 1.1% expansion for K+, and no distorsion for Cl). The energetic contribution coming from the multipole polarisabilities al with l > 1 to the polarisation energies of the above complexes is just as significant as for the alkali and chloride ion complexes, with relative
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Fig. 2. Global minima for ion-doped fullerene clusters corresponding to a complete first coordination shell. All the figures in this work involving cluster structures were prepared using the program XCRYSDEN [24].
values of 29% for I, 34% for F, and 40% for Ca2+. Perhaps more interesting is the effect of these terms on the structure of the cluster, since if these multipole polarisabilities are neglected, the geometry changes for the I and F complexes to one in which the ions move outside the fullerene matrix. Furthermore, for the (C60)nF clusters with n P 6 the ion coordination changes from tetrahedral (see below) to octahedral, with the ion very close to the centre of an octahedral site in a modified (C60)n matrix. On the other hand, the structure of the (C60)nI complexes for n > 6, which will be presented below, is (except for n = 8) not affected by the l > 1 polarisabilities. All these structural effects of the al with l > 1, which were not observed in the alkali and chloride ion complexes, are a result of the significant geometrical distortion induced by the I and F ions in the (C60)n global minimum. The energetic penalty for this distortion is compensated by the favourable contribution from the multipole polarisabilities. In the Ca2+ coordination compounds, the (C60)4 matrix experiences a mean contraction; removal of the l > 1 polarisabilities reduces the stress in this case and does not have any other effect on the cluster structure. The magnitude of the calculated binding energies is consistent with the Born model. Using the ion–C60 distance, Rec, in the complex and setting k = RecR0 ˚ for F, 3.67 A ˚ for I, and 2.50 A ˚ for (k = 2.70 A 2+ Ca ), the Born model gives polarisation energies somewhat higher in magnitude than those obtained with our detailed model by 4% for Ca2+, 8% for F and 17% for I. The larger deviations found for the F and I complexes are consistent with the distortion of the (C60)n
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matrix induced by I and F ions, which produces a more open coordination shell and thus reduces the binding energy from the value predicted by the Born model. For clusters where the number of C60 molecules is different from the value required to complete the first coordination shell, the geometrical structure of the (C60)n global minimum may differ from that of the corresponding (C60)n cluster. We mark these situations with an asterisk in Fig. 1. For n P 4 and n 6¼ 6 all the (C60)n global minima include tetrahedral sites, and the Ca2+ and F ions occupy one of these sites in each case. The octahedral geometry of the (C60)6 cluster provides the only exception to this rule (as seen in Fig. 1). In the (C60)6Ca2+ complex this original structure changes to the one shown in Fig. 3, to provide the required tetrahedral site. However, in the (C60)6F complex this change does not occur and the F ion has an unsymmetrical location inside the octahedral matrix (see Fig. 3). The coordination number six required by the I ion is not compatible with global minima based on icosahedral packing, which are observed for (C60)n clusters for n < 14. Consequently, and as seen in Fig. 1, most of these original structures change to provide such a coordination site for the I ion in the corresponding (C60)nI cluster, as for (C60)nCl complexes. However, the coordination geometry of the I ion in these compounds is not octahedral, as it is for Cl complexes, but trigonal prismatic, as can be seen for the global minima presented in Fig. 3. The larger iodide ion seems to favour this particular geometry, in which the fullerene matrix would be subject to less stress. When the ion is not able to modify the icosahedral packing of the (C60)n matrix, as for n = 7,10, it moves to an external location, as seen in Fig. 3. The (C60)13I global minimum has an interesting structure: in this case the icosahedral global minimum for (C60)13 changes to an Ino decahedron in the corresponding iodide complex. Although less stable than the icosahedron, this decahedral geometry lies only 0.01 eV higher in energy. This small difference is compensated by the appearance of five-coordinate sites on some of the faces. By moving to one of these sites the iodide ion only lacks a single fullerene molecule to complete its preferred octahedral coordination. We encountered the same situation in the (C60)13Cl complex [4] (in [4] the fullerene matrix structure in this complex was incorrectly labelled icosahedral). For n P 14, the Girifalco (C60)n clusters have either decahedral or close-packed global minima; in the corresponding (C60)nI complexes these original structures are generally preserved, the ion being located, as in the (C60)13I cluster, at a five-coordinate site on the surface. This behaviour contrasts with that found in the corresponding chloride complexes [4], in which the smaller Cl ion occupies instead an inner octahedral site of the same (C60)n structure. The distortion of the original fullerene matrix that would take place in the
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Fig. 3. Global minima of selected (C60)nX complexes.
substitution of the chloride ion by iodide is the main cause of the different behaviour. Consistent with the contraction or expansion produced by the ions in the (C60)n global minima, the fullerene–fullerene distances tend to be smaller in the Ca2+ complexes and larger in the F and I complexes than in the corresponding (C60)n cluster. The changes decrease in magnitude with the distance from the ion. The lowest minima obtained in the present work will be made available for download from the Cambridge Cluster Database [23].
4. Conclusions In [4], we studied the global minima of fullerene–ion complexes (C60)nX, where X was either an alkali metal ion or Cl. In the present contribution, we have extended this study to the clusters (C60)nCa2+, (C60)nF and (C60)nI. We were particularly interested in the structural consequences of a larger ionic charge and ionic radius. We have used the potential energy surface introduced in [4], and the basin-hopping approach to locate the likely global minima. For all the ions we obtain significant binding energies, in good agreement with the predictions of the Born model. We have found that the relative contribution to the polarisation energy
of the multipole polarisabilities with l > 1 is quite significant (30–40%), and thus cannot be ignored. We find that the ions usually prefer to occupy interstitial sites of a fullerene (C60)n matrix, the coordination shell being tetrahedral for Ca2+ and F, as for Na+ and K+ clusters [4]. Hence these ions always occupy a tetrahedral site in the (C60)n global minima for n P 4 and n 6¼ 6. The octahedral n = 6 cluster would have to modify its structure to provide the tetrahedral site required by all these ions. This modification always occurs except for the fluoride ion. As for Cl [4], I prefers an octahedral coordination shell in the n = 6 complex. However, for 7 < n < 14 the preferred coordination geometry is not octahedral but trigonal prismatic. In these cases, the original (C60)n global minima, which are based on icosahedral packing, change to provide the preferred environment for I. The larger size of the iodide ion seems to favour this particular geometry, in which the fullerene matrix is subject to less strain. For complexes with n P 14 the iodide ion is located at a five-coordinate site on the surface of the fullerene matrix, which preserves the original structure. This surface coordination is a direct consequence of the larger ionic radius: the energetic penalty associated with the distortion of the original close-packed or icosahedral structure would not be compensated by the increased polarization energy. This exclusion effect was not observed for the
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smaller chloride ion [4], so that diffusion and solvation in fullerene matrices are predicted to be quite different for chloride and iodide ions. Finally, we have shown that the larger charge of the Ca2+ ion produces stability with respect to all chargetransfer dissociation processes for (C60)nCa2+ complexes with n P 2. This stabilization of the Ca2+ charge by a neutral non-polar C60 matrix is a remarkable consequence of the high polarizability of buckminsterfullerene. Therefore, it may also be possible to observe exchange of C60 and water molecules in hydrated alkaline earth metal ions, as observed for benzene [25]. Acknowledgement This work was supported by ÔMinisterio de Ciencia y Tecnologı´a (Spain)Õ and ÔFEDER fund (EU)Õ under contract No. BFM2001-3343. References [1] A. Ballard, K. Bonin, J. Louderback, J. Chem. Phys. 113 (2000) 5732. [2] R. Antoine, Ph. Dugourd, D. Rayane, E. Benichou, M. Broyer, F. Chandezon, C. Guet, J. Chem. Phys. 110 (1999) 9771. [3] A. Ruiz, J. Herna´ndez-Rojas, J. Breto´n, J.M. Gomez Llorente, J. Chem. Phys. 109 (1998) 3573.
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[4] J. Herna´ndez-Rojas, J. Breto´n, J.M. Gomez Llorente, D.J. Wales, J. Chem. Phys. 121 (2004) 12315. [5] D.J. Wales, J.P.K. Doye, J. Phys. Chem. A 101 (1997) 5111. [6] Z. Li, H.A. Scheraga, J. Mol. Struct. 179 (1988) 333. [7] M. Born, Z. Phys. 1 (1920) 45. [8] A.F. Hebard, R.C. Haddon, R.M. Fleming, A.R. Kortan, Appl. Phys. Lett. 59 (1991) 2109. [9] L.A. Girifalco, J. Phys. Chem. 96 (1992) 858. [10] J. Breto´n, J. Gonza´lez-Platas, C. Girardet, J. Chem. Phys. 99 (1993) 4036. [11] P. Bopp, W. Dietz, K. Heinzinger, Z. Naturforsch 34a (1979) 1424. [12] L.X. Dang, Chem. Phys. Lett. 200 (1992) 21. [13] L.X. Dang, D.E. Smith, J. Chem. Phys. 99 (1993) 6950. [14] D.E. Smith, L.X. Dang, J. Chem. Phys. 100 (1994) 3757. [15] S.H. Lee, J.C. Rasaiah, J. Chem. Phys. 101 (1994) 6964. [16] L.X. Dang, D.E. Smith, J. Chem. Phys. 102 (1995) 3483. [17] D.O. Welch, O.W. Lazareth, G.J. Dienes, R.D. Hatcher, J. Chem. Phys. 64 (1976) 835. [18] W. Branz, N. Malinowski, A. Enders, T.P. Martin, Phys. Rev. B 66 (2002) 094107. [19] J.P.K. Doye, D.J. Wales, Phys. Rev. B 59 (1999) 2292. [20] J. Herna´ndez-Rojas, D.J. Wales, J. Chem. Phys. 119 (2003) 7800. [21] D.J. Wales, Energy Landscapes, Cambridge University Press, Cambridge, 2003. [22] J.P.K. Doye, D.J. Wales, Chem. Phys. Lett. 262 (1996) 167. [23] D.J. Wales, et al., The Cambridge Cluster Database, Available from:. [24] A. Kokalj, J. Mol. Graph. Model. 17 (1999) 176. [25] S.E. Rodriguez-Cruz, E.R. Williams, J. Am. Soc. Mass. Spectrom. 12 (2001) 250.
Global minima of (C60)nCa2+, (C60)nF and (C60)nI clusters J. Herna´ndez-Rojas a, J. Breto´n a, J.M. Gomez Llorente a b
a,*
, D.J. Wales
b
Departamento de Fı´sica Fundamental II, Universidad de La Laguna, 38205 Tenerife, Spain University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom Received 5 April 2005; in final form 6 May 2005 Available online 21 June 2005
Abstract Likely candidates for the lowest potential energy minima of (C60)nCa2+, (C60)nF and (C60)nI clusters are located using basinhopping global optimisation. In each case, the potential energy surface is constructed using the Girifalco form for the C60 intermolecular interaction, an averaged Lennard–Jones C60–ion interaction, and a polarisation potential, which depends on the first few non-vanishing C60 multipole polarisabilities. We find that the ions generally occupy the interstitial sites of a (C60)n cluster, the coordination shell being tetrahedral for Ca2+ and F. The I ion has an octahedral coordination shell in the global minimum for (C60)6I, however for 12 P n P 8 the preferred coordination geometry is trigonal prismatic. Ó 2005 Elsevier B.V. All rights reserved.
1. Introduction C60, the most abundant single shell fullerene under various experimental conditions, has a remarkably high ˚ 3) [1,2]. This large electric-dipole polarisability (a1 80 A polarisability leads to what we have called a Ôpolarisation bondÕ [3] in C60X complexes, X being an alkali ion, with binding energies as high as 1 eV. In a previous publication [4], motivated by this property, we studied the effect of the high polarisation energies on the geometrical structure and energetics of (C60)nX clusters, where X was chosen to be either an alkali metal ion or Cl. We carried out searches for the likely global potential energy minima of those clusters using a potential energy function that incorporates the Girifalco potential between C60 molecules, an averaged Lennard–Jones ion–C60 interaction, and a polarisation potential depending on the first few fullerene multipole polarisabilities. The basin-hopping approach [5,6] was used to locate the likely global min*
Corresponding author. E-mail addresses: [email protected] (J. Herna´ndez-Rojas), jbreton@ ull.es (J. Breto´n), [email protected] (J.M. Gomez Llorente), dw34@ cam.ac.uk (D.J. Wales). 0009-2614/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.05.075
ima. We found that the ions were small enough to occupy an interstitial site of the fullerene (C60)n matrix, the coordination shell being triangular for Li+, tetrahedral for Na+ and K+, and octahedral for Cl, in agreement with standard radius ratio rules. When the required coordination site did not exist in the (C60)n global minimum, the global minimum of the doped cluster usually changed to provide such a site. For all the ions, we found significant binding energies, DEB, in the range 1.5 to 2.5 eV. These values are exceptionally high for a nonpolar matrix like C60, and agree quite well with the predictions of the simple Born model [7], DEB = (e 1)q2/2ek, when one uses the fullerite value (e = 4.4) [8] for the dielectric permittivity and the ion site radius k. The relative contribution to the association energy of the multipole polarisabilities beyond the dipole contribution was quite significant and increased from 8% for Cl to 34% for Li+. The corresponding contribution to the polarisation energy was larger, with values ranging from 30% for Cl to 45% for Li+. In the present contribution, we extend our previous study to the clusters (C60)nCa2+, (C60)nF and (C60)nI. In particular, we wish to explore the effect of both a larger ionic charge and a larger ionic radius on
J. Herna´ndez-Rojas et al. / Chemical Physics Letters 410 (2005) 404–409
the predicted structure. We will present likely candidates for the global potential energy minima in each case. We focus our study on clusters with n < 14, but some cases with n P 14 are also considered. For the (C60)nCa2+ complexes we have obtained a rather unexpected result. Namely, the arrangements þ ðC60 Þþ nm þ ðC60 Þm Ca ðm < nÞ can have lower energy than the cluster asymptotic dissociation limit (C60)n + Ca2+. For n = 1 the energy difference between the two possible dissociation limits is the sum of the C60 ionization potential and the Ca2+ electron affinity, which we predict to be somewhat lower than the binding energy of the corresponding C60Ca2+ complex. Therefore the C60Ca2+ complex would be unstable relative to the charge-transfer dissociation limit. However, for n P 2, we have found that the ion complex is stable with respect to all the charge-transfer dissociation channels. This stabilisation of Ca2+ by a neutral non-polar molecule is a consequence of the high polarisability of C60. As in our previous work [4], we will be interested in the optimal coordination environment of each ion and in its ability to change the geometry of the optimal (C60)n fullerene cluster to achieve this preferred coordination. These details depend upon the balance between the dispersion–repulsion and the polarisation energies. Our analysis will provide guidance for future molecular beam experiments. Various gas phase spectroscopy techniques could probably also be applied to probe the predicted structures. Furthermore, the ionic species could be investigated by drift tube arrival time measurements. This paper is organised as follows. In Section 2, we discuss the expression previously derived in [4], which provides the potential energy surface as a sum of dispersion–repulsion and polarisation contributions. In Section 3, we present likely candidates for the cluster global minima and analyse these results. Finally, Section 4 summarises our conclusions.
2. The potential energy The closed-shell electronic structure of C60, and of the chosen ions, makes semi-empirical methods particularly suitable for constructing the potential energy surface (PES) of the corresponding clusters. The expression providing this PES was presented in [4] as a sum of three contributions V ¼ V ff þ V if þ V pol ;
ð1Þ
where Vff is the sum of pairwise fullerene–fullerene dispersion–repulsion interactions, Vif is the sum of pairwise ion–fullerene dispersion–repulsion interactions, and Vpol is a term including the relevant polarisation-induced contributions. The form of each of these terms has been obtained by calculating the average interaction over all fullerene orientations; i.e., we have disregarded the finer
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details due to the non-spherical symmetry of the charge density in the C60 molecule. This is a rather good approximation, justified by the high symmetry of this molecule, whose lowest non-vanishing multipole moment and anisotropic polarisability correspond to spherical harmonic indexes l = 6 and l = 3, respectively. With these assumptions the fullerene–fullerene dispersion–repulsion interaction is given by GirifalcoÕs potential [9], which corresponds to the dominant isotropic contribution obtained from a sum of pairwise Lennard–Jones carbon–carbon interactions, whose parameters eCC and rCC are given standard values. Similarly, the ion–fullerene dispersion–repulsion energy is represented by an expression [10] that corresponds to the dominant isotropic contribution obtained from a sum of pairwise Lennard–Jones carbon–ion interactions, with parameters eXC and rXC. Values for these parameters were estimated from gas phase data using the Lorentz–Berthelot combination rules. There are several sources for the ion parameters [11–16]. For Ca2+, we have used the values provided in [16]. For the anions, we have chosen values adjusted to fit molecular properties of the alkali halides [17] (binding energy, bond distance and vibrational frequency) when the potential is written as the sum of Lennard–Jones, polarisation and Coulomb terms. Actually, rXC is the most important parameter because it fixes the ion size, which determines its coordination in the complex. The dispersion energy associated with the parameter eXC is less than 1% of the values corresponding to the polarisation and Girifalco contributions. The values chosen for all these parameters are given in Table 1. The polarisation term Vpol in Eq. (1) includes the energy associated with the polarisation of the fullerene molecules due to the electric field of the ion charge and the multipole moments induced in the other fullerene molecules. This contribution was evaluated self-consistently in terms of the first three C60 multipole polarisabilities (al, l = 1, 2, 3). The required al were obtained from a semiempirical treatment of the C60 electronic structure (see [4] and references therein), which gives an a1 value in agreement with experimental data. These values were later slightly adjusted to include the effect of higher order terms, which may be relevant at the smaller ion–fullerene distances. The three al values adjusted in this way are given in [4]. When our parameter set is used to represent the interaction in (C60)2 and ðC60 Þþ 2 , the agreement with the experimental values [18] is very good [4]. Table 1 Values of the parameters for the ion–carbon Lennard–Jones interaction ˚) X eXC (meV) rXC (A Ca2+ F I
3.527 2.373 5.020
3.18 3.46 4.33
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3. Global potential energy minima As in [4], the likely candidates for the global potential energy minima were located using the basin-hopping scheme [5], which corresponds to the ÔMonte Carlo plus energy minimizationÕ approach of Li and Scheraga [6]. This method has been used successfully in both neutral [5] and charged clusters [19,20], along with many other applications [21]. In the size range considered here the global optimization problem is relatively straightforward. The global minima are always found in less than 500 steps of each basin-hopping run, independent of the random starting geometry. For the (C60)nX complexes, association energies are defined for the process nC60 þ X ¼ ðC60 Þn X;
DEa .
ð2Þ
We also define the X ion binding energy as the difference between the association energy of the complex and that of the (C60)n cluster, i.e. ðC60 Þn þ X ¼ ðC60 Þn X;
DEb .
ð3Þ
Our (C60)n global minima and association energies coincide precisely with those obtained by Doye and Wales [22]. In Fig. 1, we plot the association and binding energies defined in (2) and (3) as a function of the number of fullerene molecules n. Both tend to asymptotic values as n increases. We also include in this figure the point group symmetry obtained for each complex. As in the complexes studied in [4], the contribution of the polarisation energy Vpol to the binding energies is the dominant one (more than 90% of the total).
Fig. 1. Association energies per fullerene molecule DEa/n (circles) and binding energies DEb (crosses) for (C60)nX complexes. The arrow indicates the size at which a complete coordination shell is achieved. Asterisks mark those complexes in which the structure of (C60)n matrix differs from the global minimum of the corresponding (C60)n cluster. We also include the point group for each complex.
As mentioned in Section 1, for the (C60)nCa2+ comþ plexes, the arrangements ðC60 Þnm þ ðC60 Þm Caþ ðm < nÞ can have energies lower than the cluster asymptotic dissociation limit (C60)n + Ca2+. For n = 1, the energy difference, DEct, between the two possible dissociation channels is the sum of the C60 ionization potential and the Ca2+ electron affinity, i.e., DEct = 4.27 eV, which is somewhat lower than the binding energy of the corresponding C60Ca2+ complex DEb = 4.06 eV. Therefore, the C60Ca2+ complex is unstable relative to the chargetransfer dissociation limit. However, using simple electrostatic arguments we have estimated that the crossing between the two electronic states correlating with the two asymptotic limits takes place at an ion–fullerene ˚ and an energy of 1 eV above centre distance of 6 A 2+ the C60Ca global minimum, which would make charge-transfer a very unlikely process in this compound under normal conditions. For n = 2, we estimate DEct DEb 2.3 eV for m = 0 and 1.8 eV for m = 1, which means that (C60)2Ca2+ is already stable against the charge-transfer dissociation channels. For increasing n the stabilisation effect increases sharply until the first coordination shell is formed. This stabilisation of the Ca2+ charge by C60 is a consequence of the high polarisability of this molecule. As we know from [4], the initial sharp decrease shown by the binding energies in Fig. 1 is related to the formation of the ion coordination shell. This process represents the largest contribution (more than 90%) to the asymptotic binding energies. The completion of this first shell is indicated by an arrow in Fig. 1 and the corresponding structures, which will be called coordination compounds or complexes, are illustrated in Fig. 2. Like the alkali ions Na+ and K+, the smaller F and Ca2+ ions favour tetrahedral coordination, while the larger I prefers an octahedral coordination environment, like Cl. These results agree with the predictions of the standard radius ratio rules if we use the geometrical ˚ ) and the Pauling radii for the radius for C60 (R0 = 3.55 A ions. In the coordination compounds of Fig. 2, the geometrical structure of the fullerene matrix is that of the corresponding (C60)n global minimum; in other words, the ion occupies an interstitial site of the fullerene cluster. However, this fullerene cluster undergoes a mean contraction of 1.6% in the Ca2+ complex and a mean expansion of 1.6% and 1.7% in the F and I complexes, respectively, in order to accommodate the corresponding ion. This distortion is significantly larger than that found in the corresponding alkali and chlorine ion complexes with the same geometrical structure (0.5% contraction for Na+, 1.1% expansion for K+, and no distorsion for Cl). The energetic contribution coming from the multipole polarisabilities al with l > 1 to the polarisation energies of the above complexes is just as significant as for the alkali and chloride ion complexes, with relative
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Fig. 2. Global minima for ion-doped fullerene clusters corresponding to a complete first coordination shell. All the figures in this work involving cluster structures were prepared using the program XCRYSDEN [24].
values of 29% for I, 34% for F, and 40% for Ca2+. Perhaps more interesting is the effect of these terms on the structure of the cluster, since if these multipole polarisabilities are neglected, the geometry changes for the I and F complexes to one in which the ions move outside the fullerene matrix. Furthermore, for the (C60)nF clusters with n P 6 the ion coordination changes from tetrahedral (see below) to octahedral, with the ion very close to the centre of an octahedral site in a modified (C60)n matrix. On the other hand, the structure of the (C60)nI complexes for n > 6, which will be presented below, is (except for n = 8) not affected by the l > 1 polarisabilities. All these structural effects of the al with l > 1, which were not observed in the alkali and chloride ion complexes, are a result of the significant geometrical distortion induced by the I and F ions in the (C60)n global minimum. The energetic penalty for this distortion is compensated by the favourable contribution from the multipole polarisabilities. In the Ca2+ coordination compounds, the (C60)4 matrix experiences a mean contraction; removal of the l > 1 polarisabilities reduces the stress in this case and does not have any other effect on the cluster structure. The magnitude of the calculated binding energies is consistent with the Born model. Using the ion–C60 distance, Rec, in the complex and setting k = RecR0 ˚ for F, 3.67 A ˚ for I, and 2.50 A ˚ for (k = 2.70 A 2+ Ca ), the Born model gives polarisation energies somewhat higher in magnitude than those obtained with our detailed model by 4% for Ca2+, 8% for F and 17% for I. The larger deviations found for the F and I complexes are consistent with the distortion of the (C60)n
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matrix induced by I and F ions, which produces a more open coordination shell and thus reduces the binding energy from the value predicted by the Born model. For clusters where the number of C60 molecules is different from the value required to complete the first coordination shell, the geometrical structure of the (C60)n global minimum may differ from that of the corresponding (C60)n cluster. We mark these situations with an asterisk in Fig. 1. For n P 4 and n 6¼ 6 all the (C60)n global minima include tetrahedral sites, and the Ca2+ and F ions occupy one of these sites in each case. The octahedral geometry of the (C60)6 cluster provides the only exception to this rule (as seen in Fig. 1). In the (C60)6Ca2+ complex this original structure changes to the one shown in Fig. 3, to provide the required tetrahedral site. However, in the (C60)6F complex this change does not occur and the F ion has an unsymmetrical location inside the octahedral matrix (see Fig. 3). The coordination number six required by the I ion is not compatible with global minima based on icosahedral packing, which are observed for (C60)n clusters for n < 14. Consequently, and as seen in Fig. 1, most of these original structures change to provide such a coordination site for the I ion in the corresponding (C60)nI cluster, as for (C60)nCl complexes. However, the coordination geometry of the I ion in these compounds is not octahedral, as it is for Cl complexes, but trigonal prismatic, as can be seen for the global minima presented in Fig. 3. The larger iodide ion seems to favour this particular geometry, in which the fullerene matrix would be subject to less stress. When the ion is not able to modify the icosahedral packing of the (C60)n matrix, as for n = 7,10, it moves to an external location, as seen in Fig. 3. The (C60)13I global minimum has an interesting structure: in this case the icosahedral global minimum for (C60)13 changes to an Ino decahedron in the corresponding iodide complex. Although less stable than the icosahedron, this decahedral geometry lies only 0.01 eV higher in energy. This small difference is compensated by the appearance of five-coordinate sites on some of the faces. By moving to one of these sites the iodide ion only lacks a single fullerene molecule to complete its preferred octahedral coordination. We encountered the same situation in the (C60)13Cl complex [4] (in [4] the fullerene matrix structure in this complex was incorrectly labelled icosahedral). For n P 14, the Girifalco (C60)n clusters have either decahedral or close-packed global minima; in the corresponding (C60)nI complexes these original structures are generally preserved, the ion being located, as in the (C60)13I cluster, at a five-coordinate site on the surface. This behaviour contrasts with that found in the corresponding chloride complexes [4], in which the smaller Cl ion occupies instead an inner octahedral site of the same (C60)n structure. The distortion of the original fullerene matrix that would take place in the
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Fig. 3. Global minima of selected (C60)nX complexes.
substitution of the chloride ion by iodide is the main cause of the different behaviour. Consistent with the contraction or expansion produced by the ions in the (C60)n global minima, the fullerene–fullerene distances tend to be smaller in the Ca2+ complexes and larger in the F and I complexes than in the corresponding (C60)n cluster. The changes decrease in magnitude with the distance from the ion. The lowest minima obtained in the present work will be made available for download from the Cambridge Cluster Database [23].
4. Conclusions In [4], we studied the global minima of fullerene–ion complexes (C60)nX, where X was either an alkali metal ion or Cl. In the present contribution, we have extended this study to the clusters (C60)nCa2+, (C60)nF and (C60)nI. We were particularly interested in the structural consequences of a larger ionic charge and ionic radius. We have used the potential energy surface introduced in [4], and the basin-hopping approach to locate the likely global minima. For all the ions we obtain significant binding energies, in good agreement with the predictions of the Born model. We have found that the relative contribution to the polarisation energy
of the multipole polarisabilities with l > 1 is quite significant (30–40%), and thus cannot be ignored. We find that the ions usually prefer to occupy interstitial sites of a fullerene (C60)n matrix, the coordination shell being tetrahedral for Ca2+ and F, as for Na+ and K+ clusters [4]. Hence these ions always occupy a tetrahedral site in the (C60)n global minima for n P 4 and n 6¼ 6. The octahedral n = 6 cluster would have to modify its structure to provide the tetrahedral site required by all these ions. This modification always occurs except for the fluoride ion. As for Cl [4], I prefers an octahedral coordination shell in the n = 6 complex. However, for 7 < n < 14 the preferred coordination geometry is not octahedral but trigonal prismatic. In these cases, the original (C60)n global minima, which are based on icosahedral packing, change to provide the preferred environment for I. The larger size of the iodide ion seems to favour this particular geometry, in which the fullerene matrix is subject to less strain. For complexes with n P 14 the iodide ion is located at a five-coordinate site on the surface of the fullerene matrix, which preserves the original structure. This surface coordination is a direct consequence of the larger ionic radius: the energetic penalty associated with the distortion of the original close-packed or icosahedral structure would not be compensated by the increased polarization energy. This exclusion effect was not observed for the
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smaller chloride ion [4], so that diffusion and solvation in fullerene matrices are predicted to be quite different for chloride and iodide ions. Finally, we have shown that the larger charge of the Ca2+ ion produces stability with respect to all chargetransfer dissociation processes for (C60)nCa2+ complexes with n P 2. This stabilization of the Ca2+ charge by a neutral non-polar C60 matrix is a remarkable consequence of the high polarizability of buckminsterfullerene. Therefore, it may also be possible to observe exchange of C60 and water molecules in hydrated alkaline earth metal ions, as observed for benzene [25]. Acknowledgement This work was supported by ÔMinisterio de Ciencia y Tecnologı´a (Spain)Õ and ÔFEDER fund (EU)Õ under contract No. BFM2001-3343. References [1] A. Ballard, K. Bonin, J. Louderback, J. Chem. Phys. 113 (2000) 5732. [2] R. Antoine, Ph. Dugourd, D. Rayane, E. Benichou, M. Broyer, F. Chandezon, C. Guet, J. Chem. Phys. 110 (1999) 9771. [3] A. Ruiz, J. Herna´ndez-Rojas, J. Breto´n, J.M. Gomez Llorente, J. Chem. Phys. 109 (1998) 3573.
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