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Arè endohedral metal(IV)C28 compounds hypervalent?
Volume 200, number 4
CHEMICAL PHYSICS LETTERS
11 December 1992
Ark endohedral metal (IV) Cx8compounds hypervalent? Oliver D. Hiiberlen, Notker Riisch Lehntuhlfir Theorebsche Chemie, Technische UniversitiitMiinchen, W-8046 Garching, Germany
and Brett I. Dunlap Code 6179, Naval Research Laboratory, Washington,DC 20375-5000, USA Received 20 August 1992; in final form 21 September 1992
Nonrelativistic and, where appropriate, scalar-relativistic linear combination of Gaussian-type orbital (LCGTG) local density functional (LDF) calculations on Ti@C2s, Ce@C2s, Ti@&H,, Ce&sH,, the corresponding empty fullerene C2s, and the corresponding hydrogenated fullerene CznH.,are compared. The empty fullerene is tetravalent and strongly binds four hydrogen atoms on its exterior or a tetravalent atom inside. Combining a tetravalent endohedral atom with four exterior hydrogen atoms significantly weakens the two different sets of bounds and leads to an open-shell electronic structure.
1. Introduction The magic-number fullerenes [ 1,2] and the fullerenes that are now available in macroscopic quantities as a result of the Krtitschmer-Huffman process [ 31 are known to have or are believed to have a closed-shell electronic structure. These include CsO, CTO,Cs4 and higher-mass fullerenes [ 4 1. The endohedral fullerene complexes that are now available in quantity tend to have other numbers of carbon atoms, namely 82 [ 5-91 and 28 [ lo]. This is strong evidence that the guest metal atom inside the carbon cage stabilizes the fullerene shell by donating electrons to the shell [ 111 and/or by forming multiple covalent bonds to the shell [ 12]. Neither type of bonding, however, would be expected to completely pacify the carbon shell because its positive curvature requires that each carbon atom have all of its chemical bonds lying completely within a tangential plane. Thus, outside such a plane the electron distribution about each carbon atom must have some lone-pair Correspondence to: N. R&h, Lehrstuhl fi Theoretische Chemie, Technische Universitiit Milnchen, W-8046 Garching, Germany.
418
character, which may enable additional bonds to be formed. Phrased differently, the question is whether an endohedral complex could show some characteristics of “hypervalency” [ 13-l 5 1. Such effects should be largest when the surface curvature is largest, i.e. when the number of carbon atoms is smallest. The fullerene Czs has recently been shown to form endohedral compounds with the tetravalent metal atoms uranium, hafnium, zirconium, and titanium [ 10 1. In that work a Czs fullerene structure of tetrahedral symmetry has been shown to be tetravalent. The clearest way, among several others, to establish its tetravalency is to consider attaching four hydrogen atoms to the exterior of the structure. That case is drawn in fig. 1. The surface of the fullerene itself can be thought of as four sets of a hexagonal facet directly opposite a vertex shared by three pentagonal facets. These four vertices are formed by the “black” carbon atoms. Alternating “gray” and white” atoms form the hexagons. The “gray” set of symmetryequivalent atoms are nearest neighbors of the “black’ atoms, and the “white” atoms are next-nearest neighbors of the “black” atoms. The four hydrogen atoms are attached to the four “black” carbon atoms and these C-H moieties can be though of as being
0009-2614/92/S 05.00 Q 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 200, number 4
CHEMICAL PHYSICS LETTERS
I1 December 1992
to model the f-element uranium with which it shares some similarities in its organometallic chemistry [ 20,2 11. For the lanthanide cerium a relativistic description is important, which we have implemented at the level of a scalar-relativistic approximation [ 22,231 that allows a self-consistent two-component treatment, essentially equivalent to the inclusion of Darwin and mass-velocity terms.
2. Method
Fig. 1. Structure of the tetrahedral exohedral fullerene C28Hb.A single plane that contains representatives of each of the four types of symmetry-inequivalent atoms is indicated by darkened bonds and dotted lines. That plane also contains representatives of each of the two different types of symmetry axes. Threefold axeS go from the center to a “black” atom, twofold axes go from the center to the center of a bond between two neighboring “white” atoms.
the vertices of a regular tetrahedron. This is but one of many hydrogenation isomers of Cz8H4, but the pentagonal bond angle, 108’, is so much closer to the tetrahedral bond angle, 109.5”, than the hexagonal bond angle, 120°, that these may be assumed to be the preferred bonding sites. In principle, more than four hydrogen atoms could be attached to the outside of this Cz8 fullerene, but the strain associated with this substitution builds up rapidly and eventually forces some hydrogen atoms to be more stable if bound endohedrally [ 16,17 1. In the present work we study the competition between endohedral bonding of a tetravalent metal atom and exohedral bonding of four monovalent hydrogen atoms to the tetrahedral isomer of Czs using the first-principles linear combination of Gaussiantype orbitals (LCGTO) local density functional (LDF) method [ 18 1. The tetravalent endohedral atoms that we have considered are titanium, which is smaller than the interior volume [ 191 and cerium which just fits [ 12 1. The latter atom may be taken
The local density approximation used in this work is the Vosko-Wilk-Nusair [ 241 functional form for the exchange-correlation potential that interpolates between the essentially exact free-electron gas results [25] in the completely ferromagnetic and completely paramagnetic limits. The starting orbital basis set for carbon, a 9s/Spld basis [ 261, was augmented with a d exponent of 0.6 [27] (all exponents are given in au) and contracted to an atomic 5s/4p/ Id basis according to a spin-restricted calculation. The starting orbital basis set for titanium, a 14s/ 1Op/ 5d basis [28], was augmented with one s (0.205), two p (0.0611,0.156), and one d exponent (0.072) [29,30]. The titanium orbital basis set was contracted in analogous fashion to a 7s/6p/3d basis. The orbital basis set of cerium was a 2 1s/ 16p/ 1ld/9f basis [ 23 ] which was contracted similarly to a 11s/ lOp/8d/Sf basis. The fitting basis sets used in the LCGTO-LDF method were constructed by properly scaling the orbital exponents [ 18 1, but were left uncontracted; they have been described elsewhere [ 12,191. In a previous ab initio study the geometries, restricted to tetrahedral symmetry, of C&, Ti@&, Zr@?&, Cz8H4and Ti@Cz8H4were optimized at the Hartree-Fock (HF) level of theory using a doublezeta basis [ 10 1. When the hydrogen atoms are added or removed, the largest and most significant changes in the geometry of the carbon cage occur in the bond distance of the “black” to the “gray” atoms. For an empty cage, this bond distance dbgat the HF level is 1.440 A. With titanium or zirconium inside, the HF calculations [ lo] yield a slight expansion of the entire shell, and db, grows to 1.453 and 1.456 A, respectively. For CZBH4,however, this bond distance increases by 7Ohto 1.542 8, compared to its length in 419
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the empty fullerene. Similarly, this bond increases from 1.453 to 1.565 A (an 8Ohincrease) ingoing from Ti@& to Ti@CZ8H4. Taking these HF results into account, we have considered only partial geometry optimizations. In the absence of exohedral hydrogen atoms we use an empirical potential [ 3 1] geometry for CIBfor which the bond distance c& is 1.465 A [ 19 1. When hydrogen atoms are present we have only optimized d,, or equivalently the distance from the center to the “black” carbon atoms, rb, which is 2.5 14 A in the empty cage using the empirical geometry. All other distances were kept fixed, including the C-H bond distances which we chose as 1.09 A.
3. Results and discussion The results of the LCGTO-LDF geometry optimizations are displayed in table 1. We find a somewhat smaller elongation (by 4%) of the bond distance dbg in CZaH4than was found in the HF study. This distance, 1.527 A, is just about the length of C-C single bond. Again, placing a metal atom, either titanium or cerium, at the center of the fullerene leads to a slight expansion of the cage, increasing the distance dbgby about 0.015 A. For cerium, the equilibrium position endohedral to CZswas found to be essentially at the center [ 121, as this atom is just big enough to fill the cage. For the smaller titanium atom, an optimal distance to-
11 December 1992
ward the “black” atom (see fig. 1) was determined in LCGTO-LDF calculations [ 191 on Ti@C& to occur at a radial position rb of 0.52 A (solid curve in fig. 2), which lowers the molecular symmetry to C,,. Therefore we decided to partially optimize the location of titanium in and the cage structure of [email protected] from an LCGTO-LDF optimization of dbpin Td symmetry, the titanium atom was moved radially outward from the center along a threefold axis toward a “black” carbon atom (see fig_ 1). A more shallow minimum was found at a radial distance of 0.425 A (see fig. 2). The structure was then refined further by optimizing the shortest Ti-C (black) distance, resulting in a bond length dbg of 1.558 A (see table 1) for that corner of the tetrahedron. The level of optimization as described above is assumed to be sufficient for a comparison of the HF and LDF electronic structure of the bare fullerene and its various derivatives which are compared in table 2. In both methods the empty cage is an openshell (OS) system - having no HOMO-LUMO gap in its one-electron spectrum; thus it is expected to undergo a Jahn-Teller distortion. Due to the difference in the meaning of the one-electron eigenvalues in the HF and LDF methods [ 32 1, the values of the HOMO-LUMO gap are calculated to be different, but the ionization potentials, computed as total energy differences in both methods, show that the results are in reasonable agreement for the closed-shell
Table 1 Energy gain AE (in eV) from optimizing the radial position r, of the “black” carbon atoms in fig. 1 (or equivalently, the bond distance &) after adding four hydrogen atoms outside the Css fullerene, both without and with each of the two metal atoms, Ti and Ce, added inside (see text). Alldistancesin A Molecule
Symmetry
Cl8
Td
CzaH4 T@GSH4 T@G&L CeQCssH~
Td Td Gv Td
db
rb
Al!?
1.465 1.527 1.545 1.558 a’ 1.542
2.514 2.678 2.719 2.748 .) 2.712
1.37 2.76 3.42 2.55
‘) Optimizing the titanium-atom position along a threefold axis and then the bond distance to the nearest “black” carbon atom starting from the values of dbpand r, in the line immediately above.
420
-2.5 0.0 0.2 0.4 0.6 0.8 Radial distance [A] Fig. 2. The potential energy curves for an endohedral titanium atom moving radially outward from the center along a threefold axis towards a “black” carbon atom of fig. 1 in Css (solid curve) and in t&H4 (dashed curve).
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Table 2 Comparison of Hartree-Fock and LDF values for the HOMO-LUMO gap and the fmt ionization potentials (in eV) for various endoand exo-hedrally modified C2s fullerene derivatives. For open-shell systems (indicated by OS), the ionization potentials are given for the high-spin electronic structure Molecule
‘) Ref. [IO].
Symmetry
ClS
Td
C2&4
T*
T&G T@C28
Td C3”
ZrW28
Td
Ce@C28
Td
T@C2sH4 TGGsH, Ce@G&
f4 Csv Td
HOMO-LUMO gap
Ionization potential
HF”
LDF b,
HF”’
LDF ”
OS 9.1 8.1
OA 1.74 1.07 1.74
7.1 8.0 8.7
7.8 7.4 7.6 7.4
8.6
8.6 4.6
2.37 OS OS OS
4.9
8.3 6.1 6.4 6.6
b, This work.
systems in which the shell is tetravalent. An apparently significant difference between the HF and LDF descriptions of these fullerene derivatives occurs for the potentially hypervalent [ 13- 15 ] compound Ti@&H4, This is one of two possibly hypervalent compounds listed in the last three rows of table 2 in which the cage has a nominal valence of 8. In the HF investigation [ lo], the titanium compound was studied in a ‘Al state and a large HOMO-LUMO gap was found. In the present LDF work, this system has an OS electronic structure, with four electrons occupying the I7t2 HOMO when the titanium atom is at the center. From this configuration the following states arise: 3T,, ‘A,, ‘E, and *Tz. The triplet state is the LDF ground state. The geometry optimization for Ti@C2sH4 described above was performed using a spin-averaged description of the electronic structure to reduce the computational effort, The total energy difference toward the triplet ground state was 0.36 eV at equilibrium. The symmetry of these complexes is high enough that many effects can be described quite well by taking a spherical approximation for the fullerene or by considering only a special planar slice through the molecule containing the center and the centers of each of the four different symmetry-inequivalent atoms of C,,H4. That plane is indicated by bold lines in fig. 1. The 17t, HOMO of tetrahedral Ti@CZ8H4is plotted in this plane in fig. 3; it is spread out all over the molecule - even on the hydrogen atoms at the lower
Fig. 3. Contour plot of the HOMO ( 17t2) of Ti@CssH, when the titanium atom lies at the center, i.e. in tetrahedral symmetry. The plane shown is the one indicated in fig. 1. The positions of the carbon and hydrogen atoms are marked by black circles, the empty circle marks the middle of the bond between neighboring “white” atoms. Apart from having only a small amplitude on the “black” atoms, the orbital is quite uniformly spread out over the entire molecule. In particular, there is significant amplitude on the hydrogen atoms (see the lower left- and right-hand corners of the figure). The values of the contour lines are ho.01 5, CO.03, +0.06, and k 0.12 au, solid and dashed lines indicating values of opposite sign. 421
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left and right of the figure. Its amplitude on the “black” atoms is rather small and the only bonding contribution is between titanium and the “gray” atoms. The weakness of the hypervalency of this compound is also indicated by the dashed curve of fig, 2 where the depth of the well is reduced substantially from the tetravalent compound, which also suggests a significantly weakened Ti-Cz8H4 interaction. Another way to compare the LDF bonding of titanium and cerium inside Cz8 to that inside C18H, is via their binding energies. The former are almost twice as large as the latter. The binding energy of Ti@& with respect to Ti d’s2 and C28 ( ‘A2, which is the calculated LDF and HF [lo] ground state of the empty fullerene), 11.22 eV, is much larger than the binding energy of Ti inside Cz8H4,5.97 eV. The corresponding values for the cerium compounds (with respect to Ce f1d1s2,the experimental ground state), are 13.7 and 7.35 eV, respectively.
4. Conclusions All-electron total-energy LCGTO-LDF calculations were used to study the interaction between a tetrahedral C2*fullerene, an endohedral titanium or cerium atom, and four exohedral hydrogen atoms. Despite the fact that cerium fills up the endohedral volume while titanium is smaller and has an off-center equilibrium position, both give similar pictures of the potentially hypervalent M@Cz8H4 compounds. The presence of four exohedral hydrogen atoms substantially weakens the metal-fullerene bonds. This endohedral binding energy is reduced to almost half and OS systems are created in which the HOMO is delocalized throughout the molecule. Based on these results, it is tempting to speculate that if a tetrahedral lattice (or polymer) of M&B monomers connected via “black” atoms could be made, then this extended structure would be conducting. Of course, if the metal atoms are inside C28 then they cannot escape and a nominally hypervalent compound will be stable if our hydrogen atoms are stably attached outside. The binding energy per hydrogen atom in Ti@Cz8H4 is 2.25 eV and in Ce@&H4 is 2.00 eV. Thus the titanium compound, but not the cerium compound, is stable against loss of 2 Hz. This 422
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almost energy-neutral situation might allow a synthetic route to solid M@Cz8.
Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft (NR), by the Fonds der Chemischen Industrie (NR), and by the US Office of Naval Research through the Naval Research Laboratory (BID). The stay of BID at the Technische Universitit Milnchen, where this work was done, was made possible through a NATO travel grant (CRG 920132).
References [I] H.W. Kroto, JR. Heath, S.C. O’Brien, R.F. Curl and RE. Smalley, Nature 318 (1985) 162. [2] H.W. Kroto, Nature 329 (1987) 529. [3] W. Kriatschmer, L.D. Lamb, K. Fostiropolous and D.R. Huffman, Nature 347 (1990) 354. [4] K. Kikuchi, N. Nakahara, T. Wakabayashi, M. Honda, H. Matsumiya, T. Moriwaki, S. Suzuki, H. Shiromaru, K Saito, K. Yamauchi, I. lkemoto and Y. Achiba, Chem. Phys. Letters 188 (1992) 177. [ 5 ] Y. Chai, T. Guo, C. Jin, R.E. Haufler, L.P.E. Chibante, J. Furc, L. Wang, J.M. Alford and R.E. Smalley, I. Phys. Chem. 95 (1991) 7564. [6]M.M. Alvarez, E.G. Gillan, K. Holczer, R.B. Kaner, KS. Min and R.L. Whetten, J. Phys. Chem. 95 (1991) 10561. [ 71 R.D. Johnson, M.S. de V&s, J. Salem, D.S. Bethune and C.S. Yannoni, Nature 355 (1992) 239. [8] J.H. Weaver, Y. Chai, G.H. Kroll, C. Jin, T.R. Ohno, R.E. Haufler, T. Guo, J.M. Alford, J. Conceicao, L.P.F. Chibame, A. Jam, G. Palmer and R.E. Smalley, Chem. Phys. Letters 190 (1992) 460. [ 91 H. Shinohara, H. Sate, M. Ohkohchi, Y. Ando, T. Kodama, T. Shida, T. Kato andY. Saito, Nature 357 (1992) 52. [ lo] T. Guo, M.D. Diener, Y. Chai, M.J. Alford, R.E. Haufler, S.M. McClure, T. Ohno, J.H. Weaver, G.E. Scuseria and R.E. Smalley, to be published. [ 111D.E. Manolopoulos and P.W. Fowler, Chem. Phys. Letters 187 (1991) 1. [ 121N. Rlisch, 0. Hdberlen and B.I. Dunlap, to be published. [ 131 R.E. Rundle, Surv. Progr. Chem. 1 (1963) 81. [ 141J.I. Musher, Angew. Chem. 81 (1969) 68. [ 151A.E. Reed and P. von Schleyer, J. Am. Chem. Sot. 112 (1990) 1434. [ 161B.I. Dunlap, D.W. Brenner, J.W. Mintmire, R.C. Mowrey andC.T. White, J. Phys.Chem. 95 (1991) 5763. [ 171M. Saunders, Science 253 ( 1991) 330.
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[ 181B.I. Dunlap and N. R&h, Advan. Quantum Cbem. 21 (1990) 317. [ 191B.I. Dunlap, 0. Hliberlen and N. Riisch, to be published. [ 201 N. RaSch and J.A. Streitwieser, J. Am. Chem. Sot. 105 (1983) 7237. [ 2 1 ] J.A. Streitwieser, S.A. Rinsley, J.T. Rigsbee, I.L. Fragala, E. Ciliberto and N. Rtisch, J. Am. Chem. Sot. 107 (1985) 7786. [ 221 O.D. Hiiberlen and N. Riisch, Chem. Phys. Letters 199 (1992) 491. [ 231 P. Knappe and N. Rasch, J. Chem. Phys. 92 ( 1990) 1153. [24] S.H. Vosko, L. Wilkand M. Nusair, Can. J. Phys. 58 ( 1980) 1200. [25 ] D.M. Ceperley and B.J. Alder, Phys. Rev. Letters 45 ( 1980) 566.
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[26] F.B. van Duijneveldt, IBM Research Report No. IU 945 (1971). [27] S. Huzinaga, ed. Gaussian basis sets for molecular calculations (Elsevier, Amsterdam, 1984). [28] A.J.H. Wachters, I. Chem. Phys. 52 (1970) 1033. [29] P. Rnappe and N. Riisch, J. Organometal. Chem. 359 (1989) C5. [ 301 P. Rnappe, Dissertation, Technical University Munich (1990). [ 311 D. W. Brenner, Phys. Rev. B 42 ( 1990) 9458. [ 321 R.G. Parr and W. Yang, Density functional theory for atoms and molecules (Oxford Univ. Press, Oxford, 1989) .
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CHEMICAL PHYSICS LETTERS
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Ark endohedral metal (IV) Cx8compounds hypervalent? Oliver D. Hiiberlen, Notker Riisch Lehntuhlfir Theorebsche Chemie, Technische UniversitiitMiinchen, W-8046 Garching, Germany
and Brett I. Dunlap Code 6179, Naval Research Laboratory, Washington,DC 20375-5000, USA Received 20 August 1992; in final form 21 September 1992
Nonrelativistic and, where appropriate, scalar-relativistic linear combination of Gaussian-type orbital (LCGTG) local density functional (LDF) calculations on Ti@C2s, Ce@C2s, Ti@&H,, Ce&sH,, the corresponding empty fullerene C2s, and the corresponding hydrogenated fullerene CznH.,are compared. The empty fullerene is tetravalent and strongly binds four hydrogen atoms on its exterior or a tetravalent atom inside. Combining a tetravalent endohedral atom with four exterior hydrogen atoms significantly weakens the two different sets of bounds and leads to an open-shell electronic structure.
1. Introduction The magic-number fullerenes [ 1,2] and the fullerenes that are now available in macroscopic quantities as a result of the Krtitschmer-Huffman process [ 31 are known to have or are believed to have a closed-shell electronic structure. These include CsO, CTO,Cs4 and higher-mass fullerenes [ 4 1. The endohedral fullerene complexes that are now available in quantity tend to have other numbers of carbon atoms, namely 82 [ 5-91 and 28 [ lo]. This is strong evidence that the guest metal atom inside the carbon cage stabilizes the fullerene shell by donating electrons to the shell [ 111 and/or by forming multiple covalent bonds to the shell [ 12]. Neither type of bonding, however, would be expected to completely pacify the carbon shell because its positive curvature requires that each carbon atom have all of its chemical bonds lying completely within a tangential plane. Thus, outside such a plane the electron distribution about each carbon atom must have some lone-pair Correspondence to: N. R&h, Lehrstuhl fi Theoretische Chemie, Technische Universitiit Milnchen, W-8046 Garching, Germany.
418
character, which may enable additional bonds to be formed. Phrased differently, the question is whether an endohedral complex could show some characteristics of “hypervalency” [ 13-l 5 1. Such effects should be largest when the surface curvature is largest, i.e. when the number of carbon atoms is smallest. The fullerene Czs has recently been shown to form endohedral compounds with the tetravalent metal atoms uranium, hafnium, zirconium, and titanium [ 10 1. In that work a Czs fullerene structure of tetrahedral symmetry has been shown to be tetravalent. The clearest way, among several others, to establish its tetravalency is to consider attaching four hydrogen atoms to the exterior of the structure. That case is drawn in fig. 1. The surface of the fullerene itself can be thought of as four sets of a hexagonal facet directly opposite a vertex shared by three pentagonal facets. These four vertices are formed by the “black” carbon atoms. Alternating “gray” and white” atoms form the hexagons. The “gray” set of symmetryequivalent atoms are nearest neighbors of the “black’ atoms, and the “white” atoms are next-nearest neighbors of the “black” atoms. The four hydrogen atoms are attached to the four “black” carbon atoms and these C-H moieties can be though of as being
0009-2614/92/S 05.00 Q 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 200, number 4
CHEMICAL PHYSICS LETTERS
I1 December 1992
to model the f-element uranium with which it shares some similarities in its organometallic chemistry [ 20,2 11. For the lanthanide cerium a relativistic description is important, which we have implemented at the level of a scalar-relativistic approximation [ 22,231 that allows a self-consistent two-component treatment, essentially equivalent to the inclusion of Darwin and mass-velocity terms.
2. Method
Fig. 1. Structure of the tetrahedral exohedral fullerene C28Hb.A single plane that contains representatives of each of the four types of symmetry-inequivalent atoms is indicated by darkened bonds and dotted lines. That plane also contains representatives of each of the two different types of symmetry axes. Threefold axeS go from the center to a “black” atom, twofold axes go from the center to the center of a bond between two neighboring “white” atoms.
the vertices of a regular tetrahedron. This is but one of many hydrogenation isomers of Cz8H4, but the pentagonal bond angle, 108’, is so much closer to the tetrahedral bond angle, 109.5”, than the hexagonal bond angle, 120°, that these may be assumed to be the preferred bonding sites. In principle, more than four hydrogen atoms could be attached to the outside of this Cz8 fullerene, but the strain associated with this substitution builds up rapidly and eventually forces some hydrogen atoms to be more stable if bound endohedrally [ 16,17 1. In the present work we study the competition between endohedral bonding of a tetravalent metal atom and exohedral bonding of four monovalent hydrogen atoms to the tetrahedral isomer of Czs using the first-principles linear combination of Gaussiantype orbitals (LCGTO) local density functional (LDF) method [ 18 1. The tetravalent endohedral atoms that we have considered are titanium, which is smaller than the interior volume [ 191 and cerium which just fits [ 12 1. The latter atom may be taken
The local density approximation used in this work is the Vosko-Wilk-Nusair [ 241 functional form for the exchange-correlation potential that interpolates between the essentially exact free-electron gas results [25] in the completely ferromagnetic and completely paramagnetic limits. The starting orbital basis set for carbon, a 9s/Spld basis [ 261, was augmented with a d exponent of 0.6 [27] (all exponents are given in au) and contracted to an atomic 5s/4p/ Id basis according to a spin-restricted calculation. The starting orbital basis set for titanium, a 14s/ 1Op/ 5d basis [28], was augmented with one s (0.205), two p (0.0611,0.156), and one d exponent (0.072) [29,30]. The titanium orbital basis set was contracted in analogous fashion to a 7s/6p/3d basis. The orbital basis set of cerium was a 2 1s/ 16p/ 1ld/9f basis [ 23 ] which was contracted similarly to a 11s/ lOp/8d/Sf basis. The fitting basis sets used in the LCGTO-LDF method were constructed by properly scaling the orbital exponents [ 18 1, but were left uncontracted; they have been described elsewhere [ 12,191. In a previous ab initio study the geometries, restricted to tetrahedral symmetry, of C&, Ti@&, Zr@?&, Cz8H4and Ti@Cz8H4were optimized at the Hartree-Fock (HF) level of theory using a doublezeta basis [ 10 1. When the hydrogen atoms are added or removed, the largest and most significant changes in the geometry of the carbon cage occur in the bond distance of the “black” to the “gray” atoms. For an empty cage, this bond distance dbgat the HF level is 1.440 A. With titanium or zirconium inside, the HF calculations [ lo] yield a slight expansion of the entire shell, and db, grows to 1.453 and 1.456 A, respectively. For CZBH4,however, this bond distance increases by 7Ohto 1.542 8, compared to its length in 419
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the empty fullerene. Similarly, this bond increases from 1.453 to 1.565 A (an 8Ohincrease) ingoing from Ti@& to Ti@CZ8H4. Taking these HF results into account, we have considered only partial geometry optimizations. In the absence of exohedral hydrogen atoms we use an empirical potential [ 3 1] geometry for CIBfor which the bond distance c& is 1.465 A [ 19 1. When hydrogen atoms are present we have only optimized d,, or equivalently the distance from the center to the “black” carbon atoms, rb, which is 2.5 14 A in the empty cage using the empirical geometry. All other distances were kept fixed, including the C-H bond distances which we chose as 1.09 A.
3. Results and discussion The results of the LCGTO-LDF geometry optimizations are displayed in table 1. We find a somewhat smaller elongation (by 4%) of the bond distance dbg in CZaH4than was found in the HF study. This distance, 1.527 A, is just about the length of C-C single bond. Again, placing a metal atom, either titanium or cerium, at the center of the fullerene leads to a slight expansion of the cage, increasing the distance dbgby about 0.015 A. For cerium, the equilibrium position endohedral to CZswas found to be essentially at the center [ 121, as this atom is just big enough to fill the cage. For the smaller titanium atom, an optimal distance to-
11 December 1992
ward the “black” atom (see fig. 1) was determined in LCGTO-LDF calculations [ 191 on Ti@C& to occur at a radial position rb of 0.52 A (solid curve in fig. 2), which lowers the molecular symmetry to C,,. Therefore we decided to partially optimize the location of titanium in and the cage structure of [email protected] from an LCGTO-LDF optimization of dbpin Td symmetry, the titanium atom was moved radially outward from the center along a threefold axis toward a “black” carbon atom (see fig_ 1). A more shallow minimum was found at a radial distance of 0.425 A (see fig. 2). The structure was then refined further by optimizing the shortest Ti-C (black) distance, resulting in a bond length dbg of 1.558 A (see table 1) for that corner of the tetrahedron. The level of optimization as described above is assumed to be sufficient for a comparison of the HF and LDF electronic structure of the bare fullerene and its various derivatives which are compared in table 2. In both methods the empty cage is an openshell (OS) system - having no HOMO-LUMO gap in its one-electron spectrum; thus it is expected to undergo a Jahn-Teller distortion. Due to the difference in the meaning of the one-electron eigenvalues in the HF and LDF methods [ 32 1, the values of the HOMO-LUMO gap are calculated to be different, but the ionization potentials, computed as total energy differences in both methods, show that the results are in reasonable agreement for the closed-shell
Table 1 Energy gain AE (in eV) from optimizing the radial position r, of the “black” carbon atoms in fig. 1 (or equivalently, the bond distance &) after adding four hydrogen atoms outside the Css fullerene, both without and with each of the two metal atoms, Ti and Ce, added inside (see text). Alldistancesin A Molecule
Symmetry
Cl8
Td
CzaH4 T@GSH4 T@G&L CeQCssH~
Td Td Gv Td
db
rb
Al!?
1.465 1.527 1.545 1.558 a’ 1.542
2.514 2.678 2.719 2.748 .) 2.712
1.37 2.76 3.42 2.55
‘) Optimizing the titanium-atom position along a threefold axis and then the bond distance to the nearest “black” carbon atom starting from the values of dbpand r, in the line immediately above.
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-2.5 0.0 0.2 0.4 0.6 0.8 Radial distance [A] Fig. 2. The potential energy curves for an endohedral titanium atom moving radially outward from the center along a threefold axis towards a “black” carbon atom of fig. 1 in Css (solid curve) and in t&H4 (dashed curve).
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Table 2 Comparison of Hartree-Fock and LDF values for the HOMO-LUMO gap and the fmt ionization potentials (in eV) for various endoand exo-hedrally modified C2s fullerene derivatives. For open-shell systems (indicated by OS), the ionization potentials are given for the high-spin electronic structure Molecule
‘) Ref. [IO].
Symmetry
ClS
Td
C2&4
T*
T&G T@C28
Td C3”
ZrW28
Td
Ce@C28
Td
T@C2sH4 TGGsH, Ce@G&
f4 Csv Td
HOMO-LUMO gap
Ionization potential
HF”
LDF b,
HF”’
LDF ”
OS 9.1 8.1
OA 1.74 1.07 1.74
7.1 8.0 8.7
7.8 7.4 7.6 7.4
8.6
8.6 4.6
2.37 OS OS OS
4.9
8.3 6.1 6.4 6.6
b, This work.
systems in which the shell is tetravalent. An apparently significant difference between the HF and LDF descriptions of these fullerene derivatives occurs for the potentially hypervalent [ 13- 15 ] compound Ti@&H4, This is one of two possibly hypervalent compounds listed in the last three rows of table 2 in which the cage has a nominal valence of 8. In the HF investigation [ lo], the titanium compound was studied in a ‘Al state and a large HOMO-LUMO gap was found. In the present LDF work, this system has an OS electronic structure, with four electrons occupying the I7t2 HOMO when the titanium atom is at the center. From this configuration the following states arise: 3T,, ‘A,, ‘E, and *Tz. The triplet state is the LDF ground state. The geometry optimization for Ti@C2sH4 described above was performed using a spin-averaged description of the electronic structure to reduce the computational effort, The total energy difference toward the triplet ground state was 0.36 eV at equilibrium. The symmetry of these complexes is high enough that many effects can be described quite well by taking a spherical approximation for the fullerene or by considering only a special planar slice through the molecule containing the center and the centers of each of the four different symmetry-inequivalent atoms of C,,H4. That plane is indicated by bold lines in fig. 1. The 17t, HOMO of tetrahedral Ti@CZ8H4is plotted in this plane in fig. 3; it is spread out all over the molecule - even on the hydrogen atoms at the lower
Fig. 3. Contour plot of the HOMO ( 17t2) of Ti@CssH, when the titanium atom lies at the center, i.e. in tetrahedral symmetry. The plane shown is the one indicated in fig. 1. The positions of the carbon and hydrogen atoms are marked by black circles, the empty circle marks the middle of the bond between neighboring “white” atoms. Apart from having only a small amplitude on the “black” atoms, the orbital is quite uniformly spread out over the entire molecule. In particular, there is significant amplitude on the hydrogen atoms (see the lower left- and right-hand corners of the figure). The values of the contour lines are ho.01 5, CO.03, +0.06, and k 0.12 au, solid and dashed lines indicating values of opposite sign. 421
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left and right of the figure. Its amplitude on the “black” atoms is rather small and the only bonding contribution is between titanium and the “gray” atoms. The weakness of the hypervalency of this compound is also indicated by the dashed curve of fig, 2 where the depth of the well is reduced substantially from the tetravalent compound, which also suggests a significantly weakened Ti-Cz8H4 interaction. Another way to compare the LDF bonding of titanium and cerium inside Cz8 to that inside C18H, is via their binding energies. The former are almost twice as large as the latter. The binding energy of Ti@& with respect to Ti d’s2 and C28 ( ‘A2, which is the calculated LDF and HF [lo] ground state of the empty fullerene), 11.22 eV, is much larger than the binding energy of Ti inside Cz8H4,5.97 eV. The corresponding values for the cerium compounds (with respect to Ce f1d1s2,the experimental ground state), are 13.7 and 7.35 eV, respectively.
4. Conclusions All-electron total-energy LCGTO-LDF calculations were used to study the interaction between a tetrahedral C2*fullerene, an endohedral titanium or cerium atom, and four exohedral hydrogen atoms. Despite the fact that cerium fills up the endohedral volume while titanium is smaller and has an off-center equilibrium position, both give similar pictures of the potentially hypervalent M@Cz8H4 compounds. The presence of four exohedral hydrogen atoms substantially weakens the metal-fullerene bonds. This endohedral binding energy is reduced to almost half and OS systems are created in which the HOMO is delocalized throughout the molecule. Based on these results, it is tempting to speculate that if a tetrahedral lattice (or polymer) of M&B monomers connected via “black” atoms could be made, then this extended structure would be conducting. Of course, if the metal atoms are inside C28 then they cannot escape and a nominally hypervalent compound will be stable if our hydrogen atoms are stably attached outside. The binding energy per hydrogen atom in Ti@Cz8H4 is 2.25 eV and in Ce@&H4 is 2.00 eV. Thus the titanium compound, but not the cerium compound, is stable against loss of 2 Hz. This 422
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almost energy-neutral situation might allow a synthetic route to solid M@Cz8.
Acknowledgement This work was supported by the Deutsche Forschungsgemeinschaft (NR), by the Fonds der Chemischen Industrie (NR), and by the US Office of Naval Research through the Naval Research Laboratory (BID). The stay of BID at the Technische Universitit Milnchen, where this work was done, was made possible through a NATO travel grant (CRG 920132).
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