4616
J. Phys. Chem. 1993,97, 4616-4620
Exploring the Stability, Structure, and Electronic Properties of Zr, Ti, V, Fe, and Si Metallocarbohedrenes Robin W. Grimes' and Julian D. Gale The Royal Institution of Great Britain, 21 Albemarle St., London W1X 4BS, United Kingdom Received: November 2, 1992; In Final Form: February 8, 1993
The recent exciting discovery of dodecahedral metallocarbohedrene molecules with the formula M&, 2 points the way toward a new branch of cluster chemistry. In this study, we describe the results of quantum cluster local density approximation (LDA) calculations, whose purpose is to predict the characteristics of those clusters where M = Zr, Ti, V, Fe, or Si. In all cases, the molecules are stable and, although they exhibit deviations from an ideal dodecahedral geometry, they remain remarkably spherical. In the Zr, Ti, and V variants, the difference in Mulliken charge distribution between M and C shows that each C2 component assumes approximately a single negative charge. These results are used to explain why, in this geometry, the molecules are so stable.
Introduction Considerable effort has been invested in attempts to alter the propertiesof C60 and related fullerenes through the incorporation of dopant ions. This has resulted in the formation of molecules which contain at their centers a variety of metal ions including lanthanum,' iron,2 and uranium3 or the gas atom helium? In addition, it has been possible to substitutea number of heterovalent boron atoms for framework carbon to form BxC6sx,where x is in the range l-6.s A variety of theoretical techniques have been useful in this regard, not only in confirming the stability of known molecules but also in predicting new variants.6.' Most recently, cluster studies have been extended through the work of Guo et a1.8-11who observed stable gas-phase M& molecules (for M = Ti, V,Zr, or Hf). The molecules were formed by the plasma reaction of CI,C2,or C3 hydrocarbons with laserevaporated metals and their compositions determined using mass spectroscopy with isotopic variation of the hydrocarbon. The predominance of the mass peaks corresponding to a M& ratio was such that the latter were termed "super magic" peaks.8 Guo et al. interpreted their findings as indicating that a particularly stable molecule had been formed and suggested that this could correspond to a dodecahedral structure. In this, the M& moleculesare formed from edge-sharingpentagons, each pentagon being composed of two metal and three carbon atoms. Alternatively, the structure can be viewed as a cube of eight metal ions with C2 diatomic molecules adjacent to the six faces12(see Figure 1). The stability of the Ti& molecule was supported by theoretical calculations which predicted a slight distortion of the perfect dodecahedral structure.12 In addition, Ceulemans and Fowleri3have rationalized the stability of the Ti~C12cluster in the context of the 2(k 1)2 electron-counting rule for fully delocalized 1 bonding on the surface of a ~phere.1~Because titanium has the same number of valence electrons as carbon, albeit in different orbitals, this particular metallocarbohedrene can be considered to be analogous to the hypothetical dodecahedrene (C~O); however, both species have two excess electrons relative to the closed shell number of p electrons for k = 2. The metallocarbohedrenes (met-cars) are novel in comparison to conventional fullerenes in that they incorporate a much higher proportion of non-carbon atoms and are significantly smaller in size. Subsequent experimental work'' has shown that larger clusters of the zirconium compound can be formed, the formulas of which suggest that these larger clusters are comprised of multiple dodecahedral clusters which share cage faces in a similar manner to some zeolite materials.15 The purpose of this study is to compare and contrast the results of recent quantum cluster calculations performed on a variety of
+
Figure 1. MsC12 molecular geometry viewed from two perspectives: an array of interlocking pentagons (left) and an M8 cluster with associated Cz molecules (right).
met-cars. In addition to geometry and charge distributions, the cluster first ionization and electron affinities are discussed.
Methodology For this study, the ab initio DMol code was employed.I6 This solves the local density functional (LDF) equations variationally and self-con~istently.'~ The calculation begins with the generation of a trial wave function, $, which is an antisymmetrized product of the molecular orbitals, vi, which are themselves linear combinations of atomic basis functions, xi. The associated charge density, p, is simply a sum over the occupied orbitals
In this regard, the method is identical to most other quantum chemical techniques. If we now express the ground-state properties of the cluster as functions of the charge density, the total energy E,[p] may be written
where T [ p ] is the kinetic energy of the system, u[p] is due to Coulombic interactions, and E x c [ p ] is the exchange correlation component. In practice, the kinetic energy is calculated directly from the wave function while the Coulombic interactions, being essentially classical in origin, are determined from the charge density. The many-electron exchange correlation term is also determined from the charge density but is subject to two assumptions which constitute the local density approximation (LDA). The first is that we presume thechargedensityisvarying slowly on an atomic scale. This allows us to invoke the second, which is to express the exchange correlation energy as a simple function of the density, that is, 4 p ] . Together, these allow the total exchange correlation energy to be obtained by integrating
0022-3654/93/2097-4616$04.00/0 0 1993 American Chemical Society
Zr, Ti, V, Fe, and Si Metallocarbohedrenes the uniform electron gas result so thatl8, 1
where the explicitform of tXcis that derived by Barth and Hedin.19 Since the total energy is a function of the density, its minimum is obtained by optimizing E, variationally with respect to p. In a similar way to HartreeFock theory, this leads to a set of coupled equations with analogous Fock operators.20 Thus
The Journal of Physical Chemistry, Vol. 97, No. 18, 1993 4617
TABLE I: Calculated Structural Properties of M& Clusters Zr& TisClz VSCI~FesC12 SisC12 Distances in MSC12Molecules (A) M-M
3.29 1.42 2.12 0.94
c-c
M-C
center of C2 to center of cube plane
3.06 1.40 1.98 0.94
2.85 1.44 1.89 1.02
2.61 1.36 1.84 1.13
3.03 1.38 1.88 0.82
Deviation from Perfect Spherical Geometry, V (%)"
cluster charge where V2 is the kinetic energy operator; VNand V, are due to the nuclear-electron and electron-electron interactions; and pxc,the exchange correlation potential, is related to exc by
-1 0 +1
-5 -6
-2 -3
-7
-4
+6 +3 +1
+20 +12 +2
-10 -10 -9
As a guide, a +1% deviation corresponds to a center-to-C distance which is -0.05 A longer than the center-to-M distance. a
Each quantum cluster technique has its advantages and disadvantages, making each method more or less appropriate to a specific problem. In the present case, the disadvantage is inherent in the approximate way in which the exchangecorrelation energies are treated. However, this imparts the advantage that it is not necessary to calculate four center integrals as it is, for instance,in Hartree-Fock methods. Consequently, a much higher quality of basis set can be employed for a given system. In these calculations, a double numeric basis was used. This has been specifically designed for use in DMol and is subject to the orthonormality conditions. Each atomic orbital, xi,is a sum of two numerical exact LDA spherical atom orbitals, one derived for the neutral atom and the other for the 2+ ion.I7 Additional double numeric core and higher angular momentum number polarization functionswere also included in the basis. The richness of such a basis should guarantee negligiblebasis set superposition error. In this study, carbon was described by 1 X Is, 2 X 2s, 2 X 2p, and 1 X 3d double numeric functions. Similar extensive basis sets were used for the metal atoms. Inclusion of nonlocal corrections, as for example in the form proposed by Becke and Perdew,Z1 generally leads to some improvement in binding energyz2while only slightly perturbing the equilibrium geometry. The exception to this is in cases such as hydrogen bonding and weak intermolecular interactions. Thus in the present study, given the near metallic state of the metallocarbohedrenes, the calculations are performed within the local spin density (LSD) approximation. Lastly, as a testament to the suitability of LDF to the present task, the marked success that DMol has had in reproducing the experimental behavior of a large number of similarly sized carbonand nitrogen-containing transition-metal molecules is noted.17
Results and Discussion The cluster properties can be classified into two groups, structural and electronic, the results for which are presented in Tables I and 11,respectively. While all calculations were carried out within the point group D z ~no , distortion from the starting Th geometry was considered. Structural Properties. The characteristic cluster interatomic distances (Table I) show that despite the variation between the different molecules of 0.68 A in metal-metal (M-M) separation, the carbon-rbon (C-C) bond length remains remarkably constant. The metal-carbon (M-C) bond distances, which also vary, are in fact always considerably longer than the C-C bonds. Therefore, the pentagons which comprise the dodecahedrons are not regular but have one side which is between 24% and 33% shorter than the other four. In addition, although the M-C bond distances order in the same way as the M-M length, thevariation is not as great as in the M-M separation. This is achieved through a marked variation in the relative distance of the C2 units above the faces of the metal cube. This leads to deviations from an
ideal geometry in which all atoms lie on a sphere. In other words, the metal and carbon atoms are not equidistant from the center of the dodecahedron. This can be quantified by considering the percentage deviation from unity, V, of the ratio
v = [l-(D,/D*)]
x 100%
where D, and D, are the distances from the center of the cluster to carbon and metal atoms, respectively. This is a measure of how closely the CZdimer is bound to the M8 cube since the more negative the deviation, the more closely bound to a face of the M8 cluster is the C2 dimer. In Table I, the distance, in angstroms, from the center of a C2 to the center of its nearest metal cube face is also given. The values, detailing the variation from sphericity (V), are reported in Table I and show that both positive and negative deviations exist. Interestingly, it is evident that the deviation always becomes more positive in the negatively charged molecule (i.e., the carbon dimers move further away from metal faces when an electron is trapped by the molecule) and more negative in theionizedcluster (i.e., thecarbon dimers move inward when an electron is removed from the molecule). The possible exception from this trend is Si8C12, for which the distortion is essentially independent of charge state. Electronic hoperties. The charge distribution within the clusters (see Table 11) was analyzed using a Mulliken population analysis. It shows that the M8core assumes a positive charge to an extent proportional to the electronegativityz3of thecomponent M atom (see Figure 2). In particular, in the Zr8, Tis, and v8cl2 clusters, this results in the C2components becoming effectively C2-. This charge transfer creates an electrostatic attraction between the CZdimer and M8 cluster that decreases in strength from Zr to Fe. This is reflected in the corresponding decrease in the negative distortion from sphericity (see Table I). However, using this simple electrostatic argument to explain the trend in sphericity is only useful if the M ions are similar electronically: Si, not being a transition metal, is electronically sufficiently different that its distortionfrom sphericityresults fromdifferences in bonding. It is useful at this point to consider the electronic structure of the M&2!l molecule and how it relates to its Cz and M8 components. To this end, in Figure 3 we show the distribution of energy levels for Zr8Cl2 and the neutral Zr8 and C2 clusters. On formation of the molecular cluster, the C2 2s orbitals have increased in energy in response to their assumption of a negative charge. The positive shift of the valence 2s set is not as great due to stabilization as a result of mixing with the Zrs 4d band. The C 2p and Zr 4d orbitals have formed bonding/antibonding sets. Since the Zrs 4d orbital eigenvalues are higher than those of the C2 2p, the upper C 2p/Zr 4d set is more metallic in character. The important point to note is that despite the eight Zr atoms assuming a total charge of +6, the predominantly metal-like C 2p/Zr 4d eigenvalues are not stabilized to the extent expected.
-
The Journal of Physical Chemistry, Vol. 97, No. 18, 1993
4618
Grimes and Gale
TABLE II: Calculated Electronic Properties of M8Cl2Clusters Zr: 1.11+ C: 0.743
Mulliken charges distribution ground-state multiplicity orbital energies (eV) and their symmetries HOMO a LUMO a HOMO 0 LUMO p formation energies (eV) neutral atoms Mg and 6C2 metal and graphite metal carbide and graphite ionization energy (eV) distribution of hole in cluster Mg:
c2:
electron affinity (eV) distribution of electron in cluster Mg: Cz:
Charge 0 Ti: 0.75+ C: 0.503
V: 0.65+ C: 0.445
Fe: 0.29+ c : 0.2013
Si: 0.13+ C: 0.095
-3.31 -2.66 -3.17 -3.17
-3.57 -3.28 -3.44 -3.44
-4.20 -3.79 4.62 -3.87
-5.36 4.61 -5.23 -5.23
-5.49 4.61 -5.13 -5.13
-138.08 -58.27 0.77 17.54
-133.37 -58.62 -6.16 9.04
-1 36.10 -57.96 -5.26 -
-125.36 -5 1.42 -2.56 -
-120.51 -46.20 -5.38 10.71
4.89
Charge 1+ 5.33
5.92
7.49
6.92
0.97 0.03
0.60 0.40
0.65 0.35
0.59 0.41
0.95 0.05
1.59
Charge 11.71
2.18
4.24
3.40
1.05 -0.05
0.63 0.37
0.78 0.22
0.79 0.21
0.93 0.07
-0.1
-0.2
g
0.8-
c
-0.3
.-
Y
2
e
c,
0.6-
?
a
d -0.4
r:
0.4-
v
5
0.2-
5
r-u
0.04
1.3
P a
"
"
"
"
"
1.4
1.5
1.6
1.7
1.8
"
1.9
.'
2s
-0.5
-0.6 2.0
Pauling Electronegativity Figure 2. Relationship between the Mulliken populations in the M&12 molecules and the Pauling electronegativity of the metal atom.
As will be explained later, the reason for this is one of the keys to understanding the stability of the metallocarbohedrenes dodecahedron molecules. In Figure 4, the orbital energies of all five molecules are presented. In each case it is possible to see the same orbital energy distribution as described for Zr&: a pair of bonding/ antibonding M d/C 2p sets and a pair of C 2s levels. It is clear, however, that as a function of increasing electronegativity, the orbital energies are becoming more negative and the range in energy encompassed by the orbitals is expanding. This is a consequence of the reduction in the atomic orbital energies of the metal atoms of which the electronegativity trend is a manifestation (see Figure 5 ) . The molecular orbital explanation is that as the eigenvalues of the metal and carbon atoms become closer in energy, the interaction increases, a broader distribution of orbitals is formed, and the charge transfer decreases (see Table 11). This is most marked for Si, whose 3p orbitals also overlap especially well with the carbon 2p orbitals. Many of the properties of the MsC12 molecules which will determine whether they are technologically useful depend upon the values of the ionization energies or electron affinities. Calculated values are tabulated in Table I1 along with the distribution of the associated hole or excess electron in the cluster. It is clear that, in all cases, the majority of the hole or electron is delocalized over the eight metal ions. The extent to which the hole or electron is metal-like decreases as the metal atom becomes more electronegative. This reflects the character of the HOMO
-0.7
-0.8 c 2
Figure 3. Formation of the Zr&2 molecular orbitals from their Zr8 and C2 components.
and LUMO of these neutral molecules, whose metal component becomes less dominant with increasing electronegativity as discussed above. This distribution of the hole and electron on the metal ions (see Table 11) can be used to explain the change in sphericity noted in the last section. Since the C2 units are negatively charged, the increase in positive charge on the metal ions will attract the C2units and lead to a decrease in the value of the deviation from sphericity. Conversely, since the association of an electron to the molecule primarily leads to a decrease in the positive charge of the metal atoms, the attraction of the C2- molecules to the metal core decreases and the deviation from sphericity becomes more positive. Since in Si8C12 the C2 molecules are neutral, the association of an electron or a hole, albeit mostly on the metal ions, has no strong electrostatic effect on the binding of the C2 molecules. As such, there is no change in the deviation from sphericity in Si&2 (see Table 11). Stability. It is important to assess how stable the M& molecules are with respect to possible dissociation products including both other gaseous clusters and solid phases. For example, in Table 11, the stability of the molecules with respect to M8and C2clusters has been calculated. In all cases, the M8Cl2
Zr, Ti, V, Fe, and Si Metallocarbohedrenes
-
I
p t :-f P
The Journal of Physical Chemistry, Vol. 97, NO. 18, 1993 4619
-0.5
w
-
=
=
- -
-
-0.8
Fe Si Metal ion in MSClzcluster
Zr
Ti
V
Increasing electronegativity
Figure 4. Relationship between different MgC12 molecular orbital energies.
-1
-0.1I-
:
? v
p
-0.2
- -
-0.3
"I -0.4
Metabmetal
(A)
Zr
Ti
V
Fe
Si
2.73
2.44
2.26
2.24
2.37
Figure 5. Orbital energies for relaxed, neutral Mg clusters. cluster is much more stable, reflecting both the strong intramolecular bonding compared to these component clusters and also the benefit of the charge transfer. The stabilityof the dodecahedralmolecular clusterswith respect to the component elements, M,in their standard states can be estimated by use of thermodynamic data24to complete the BornHarber cycle BM(g)
+
M(s)
+
C(s)
tu*
C(S)
'WC)
WO)+
I
1WO)
7 Fed%)
Ofthefiveclustersconsidered,Ti,V,and Fe8C12possessanegative heat of formation while ZrsClz and Si&* are both metastable with respect to their component elements. For the three cases where solid metal carbides are formed, all of the clusters are unstable according to the reaction
However, such a comparison is unequal insofar as thedodecahedral molecules will gain energy by condensing into a crystalline phase. Certainly, in a condensed phase, the charge distribution around the molecule will lead to strong multipole-multipole intermolecular attractions. Alternatively, as has recently been suggested,"J dodecahedron face-sharing networks may form. This will also lead to greater stability. With this in mind, the stability of the Ti, V, and Fe variants with respect to metal and graphite testifies to the remarkable stability of this family of molecules. However, it is clear that further work is necessary before it is
possible to determine how stable an M~clz-typestoichiometric solid is compared to more conventional materials. The question as to why the M&]Z molecular form is so stable still remains. In their original paper, Guo et al. went some way toward an explanation, and, as described in the Introduction, Ceulemans and Fowler have expressed some interesting ideas. Nevertheless, a full explanation is still lacking. In this regard, we believe some insight may be gained as to why certain variants are more stable than others by noting that the most stable molecules are also those with the smallest deviations from a spherical geometry (see Table I). The importance of sphericity as a contributing factor toward stability can be explained on the basis of a simple electrostatic argument. First, it must be realized that the C2 molecule has an unusually high electron affinity: 3.54 eV.25 Thus, donating an electron to each CZcomponent of the M& molecule is energetically very advantageous. Of course, this requires the Mg unit to donate six electrons which, if it were an isolated species, would be energetically very unfavorable. Indeed, with the removal of every successive electron, the metal orbitals would become more negative, reflecting the increasing difficultyof progressively ionizing the cluster. This general effect is due to the increasing Coulombic field of the MS cluster as a whole. However, in Figures 3 and 4 it is clear that the metal eigenvalues are not stabilizedto theextent that would beexpected. This is because the atoms of the M& molecule lie approximately on a sphere. The charge transfer is, in fact, a symmetric charge rearrangement on the surface of that sphere. As such, the Coulombic field experienced by the metal atoms is zero, and the energy to remove electrons from the metal orbitals does not increase dramatically. Summary
The aim of this paper is to compare and contrast the properties of a series of experimentally observed and yet to be observed M8Cl2 dodecahedral molecules. We focus on the relationship between geometric and electronic properties and their effect on the stability of the molecules. In particular, a description of the evolution of the M&12 cluster molecular orbitals from M8 and C2 components is given. This is shown to be useful in understanding the reasons for the strong variation in charge distribution, electron affinity, and ionization energy between different molecules. The sphericity of the dodecahedron is also an important concept and relates to the relative stability of the molecules. Since this paper was submitted for publication, a number of other theoretical studies have been One of these2' predicts that a Ti8C12molecule with a small Jahn-Teller-driven D2h distortion to the dodecahedron molecule is more stable compared to the structure considered in the present work. Anotherz8 promotes an even more radical distortion to the dodecahedron structure in which the eight Ti ions form a tetracapped tetrahedron while the molecule retains an overall Td symmetry. Lastly, Bo et al.30have suggested that the CZunits are more stable when parallel to the diagonals of the M8 metal framework.
References and Notes (1) Chai, Y.;Guo, T.; Jin, C.; Haufler, R.E.; Chibante, L. P. F.;Fure, J.; Wang, L.; Alford, J. M.; Smalley, R. E. J . Phys. Chem. 1991,95,7564. (2) Pradeep, T.; Kulkarni, G. U.; Kannan, K. R.; Guru Row, T. N.; Rao, C. N.R. J . Am. Chem. SOC.1992, 114, 2212. (3) Guo, T.; Diener, M. D.; Chai, Y.;Alfard, M. J.; Haufler, R. E.; McClure, S. M.; Ohno, T.; Weaver, J. H.; Scuseria, G. E.; Smalley, R. E. Science 1992, 257, 1661. (4) Ross, M. M.; Callahan, J. H. J. Phys. Chem. 1991, 95, 5720. (5) Guo, T.; Changming, J.; Smalley, R. E. J . Phys. Chem. 1991, 95, 4948.
(6) Dunlap, B. I.; Brenner, D. W.; Mintmire, J. W.; Mowrey, R. C.; White, C. T. J. Phys. Chem. 1991, 95, 8737. (7) Andreoni, W.; Gygi, F.; Parrinello, M. Chem. Phys. Leff. 1992,190, 159.
4620 The Journal of Physical Chemistry, Vol. 97, No. 18, 1993 (8) Guo, B. C.; Kerns, K. P.; Castleman, A. W., Jr. Science 1992,255, 1411. (9) Guo, B. C.; Wei, S.; Purnell, J.; Buzza, S.; Castleman, A. W., Jr. Science 1992. 256. 5 15. (IO) Wei,'S.; Guo, B. C.; Purnell, J.; Buzza, S.; Castleman, A. W., Jr. Science. 1992. - -, 256. -. .. 818. . (1 1) Wei, S.;Guo, B. C.; Purnell, J.; Buzza, S.; Castleman, A. W., Jr. J. Phys. Chem. 1992, 96,4166. (12) Grimes, R. W.; Gale. J. D. J . Chem. Soc.. Chem. Commun. 1992. 1222. (13) Ceulemans, A.; Fowler, P. W. J. Chem. Soc.,Faraday Trans. 1992, 88,2797. (14) Fowler, P. W.; Woolrich, J. Chem. Phys. Lett. 1986, 127, 78. (15) Newsam, J. M. Solid State Chemistry: Compounds; Oxford University Press: New York, 1992;Chapter 7. (16) DMol version 2.1, BIOSYM Technologies, San Diego, CA, 1991. ( I 7) See, for example: Delley, B. J . Chem. Phys. 1990, 92,508. Sasa, C.; Andzelm, J.; Elkin, B. C.; Wimmer, E.; Dobbs, K. D.; Dixon, D. A. J . Chem. Phys. 1992, 96,6630.
Grimes and Gale Hohenberg, P.; Kohn, W. Phys. Rev. [Sect.] B 1964, 136,864. von Barth, U.;Hedin, L. J . Phys. C 1972, 5 , 1629. Kohn, W.; Sham, L. J. Phys. Rev. A 1965, 140, 1133. Perdew, J. P. Phys. Rev. B Condens. Matter 1986, 33, 8822. Andzelm, J.; Wimmer, E. J . Chem. Phys. 1992, 96, 1280. Allred. A. L.J. InorP. Nucl. Chem. 1961. 17. 215. (24j CRC Handbook of Ehemistry and Physics, 65th ed.; CRC Press: Boca Raton, FL; 1984;p D-51. (25) Huber, K. P.; Hertzberg, G. Molecular Spectra and Structure IV: ConstantsofDiatomicMolecules,vanNostrand Rheinhold: New York, 1979. (26) Reddy, B. V.; Khanna, S.N.; Jena, P. Science 1992, 258, 1640. (27) Rantaia, T.T.;Jelski, D. A,; Bowser, J. R.; Xai, X.; George, T.F. 2.Phys. D At., Mol., Clusters, in press. (28) Dance, I. J . Chem. Sot., Chem. Commun. 1992, 1779. (29) Rohmer, M.-M.; de Vaal, P.; BCnard, M. J . Am. Chem. Sot. 1992, 11 4,9696. (30) Bo, C.; Poblet, J.-M.; Henriet, C.; Rohmer, M.-M.; BCnard, M., manuscript in preparation.
