14892
J. Phys. Chem. 1996, 100, 14892-14898
Electronic Structures and Geometries of C60 Anions via Density Functional Calculations William H. Green, Jr.* Corporate Research Laboratories, Exxon Research & Engineering Co., Route 22 East, Annandale, New Jersey 08801
Sergiu M. Gorun*,† Corporate Research Laboratories, Exxon Research & Engineering Co., Route 22 East, Annandale, New Jersey 08801, and School of Chemical Sciences, UniVersity of East Anglia, Norwich NR4 7TJ, U.K.
George Fitzgerald‡ Cray Research, Inc., 655 E. Lone Oak Dr., Eagan, Minnesota 55121
Patrick W. Fowler Department of Chemistry, UniVersity of Exeter, Stocker Rd., Exeter EX4 4QD, UK
Arnout Ceulemans and Bruno C. Titeca Department of Chemistry, Catholic UniVersity of LeuVen, Celestijnenlaan 200F, 3001 HeVerlee, (LeuVen) Belgium ReceiVed: March 6, 1996; In Final Form: June 11, 1996X
The geometries and electronic structures of C60 and its mono- through hexaanions, all of which have been prepared in macroscopic quantities, are calculated using modern density functional techniques. Clear assignments of the electronic states, symmetries, and Jahn-Teller distorted geometries of the ions, which are difficult to determine experimentally, are obtained. The results are compared with available experimental data, and the agreement is excellent. Our calculations predict the geometries and electronic structures which have not yet been measured. Comparison with previous theoretical work indicates that density functional theory is the method of choice for the fullerene anions.
Introduction There has been considerable interest in the anions of C60 for a number of years, based in part on the predicted1,2 and observed3,4 high electron affinity of this novel form of carbon. The preparation of macroscopic amounts of C60-, and the measurement of its visible and near-IR spectrum opened new synthetic opportunities.5 The beautiful cyclic voltammogram of C60, which shows six distinct reduction potentials on the way to C606-, many of them reversible, attracted further interest.6-8 Reports that certain C603- salts are superconductors, with the highest Tc’s of any known molecule-based material, have led to even more activity in this field.9-10 The electronic structure of the C60 anions has been something of a puzzle. Addition of electrons to the 3-fold degenerate LUMO of C60 will cause most of the anions to be Jahn-Teller distorted away from the icosahedral symmetry of neutral C60. These Jahn-Teller distortions are thought to be responsible for the observed splittings of some of the Raman bands of C60 upon electrochemical reduction.11 However, the extent of this geometric distortion and the size of the electronic splittings between the previously degenerate orbitals are not known. As with transition metal complexes, many of the anions can exist in “high-spin” or “low-spin” states, and it is not always clear † Temporary address: Department of Chemistry, Princeton University, Princeton, NJ 08544. ‡ Present address: Oxford Molecular Group, Inc., 201 W. Burnsville Parkway, Suite 118, Burnsville, MN 55337. X Abstract published in AdVance ACS Abstracts, August 15, 1996.
S0022-3654(96)00689-2 CCC: $12.00
which state will be more stable. The Jahn-Teller distortions found in the present work (in agreement with other theoretical and experimental studies) are generally fairly small, leading to orbital splittings of less than 0.1 eV (800 cm-1), but they may be important in a number of ways. First, these small splittings, and the related couplings between electronic and nuclear motion, are thought to be critical for the observed conductivity of certain fulleride salts and important in controlling a variety of magnetoelectric and magnetic resonance phenomena observed in these systems. Second, and perhaps of more interest to chemists, the Jahn-Teller distortions break the symmetry of C60 anions, causing partial localization of the diffuse charge and spin of the anions, which may result in enhanced reactivity. The tremendous interest in the fulleride anions has prompted a number of theoretical studies. (No attempt is made here to provide a comprehensive review of the many published studies of fullerenes and fullerides; for reviews of relevant calculations, see refs 1 and 12.) The most comprehensive published theoretical studies of the anions were done by Negri, Orlandi, and Zerbetto13-15 using the QCFF/PI semiempirical method. Wastberg and Rosen16 used CNDO/S-CI to estimate the optical spectra of C60-. Saito, Dresselhaus, and Dresselhaus17 used another semiempirical method, MOPAC, to calculate all six anions; this study was restricted to icosahedrally symmetric geometries (no Jahn-Teller distortions). Although the semiempirical calculations explain some of the experimental data, one would not expect their predictions to be quantitative since these open-shell anions are very different than the molecules used in © 1996 American Chemical Society
Structures and Geometries of C60 Anions developing the semiempirical parameters. Instead, good ab initio calculations are needed. Hartree-Fock calculations have been performed on some of the anions,18-20 but these studies have serious deficiencies because they did not optimize the geometry (constraining it to be icosahedral in order to take advantage of the symmetry) and/or because they did not include necessary polarization and diffuse functions in their basis sets. Koga and Morokuma20 allowed geometrical relaxation of the monoanion but used only a very small 3-21G basis set. In general, density functional calculations are more accurate than Hartree-Fock calculations because they include some effects of electron correlation. Density functional calculations on open-shell carbon-centered radicals, using methodology similar to that presented here, have been demonstrated to give bond strengths within a few kcal/mol of experiment for many different systems.21-25 Density functional calculations are found to be as good as very expensive correlated methods for predicting electron affinities.26 Furthermore, density functional calculations, unlike Hartree-Fock calculations, are also known to give reliable results for open-shell transition metal complexes27 whose electronic structures resemble the C60 anions. A number of density functional studies of neutral (gas phase and solid) C60 have been published, using various local density functionals and different types of basis sets.28-31 Dunlap et al. also studied the monoanion using a local density functional.29 The monoanion and dianion were studied using the local density approximation (LDA) and the generalized gradient approximation (GGA) by Pederson and Quong, who also used large basis sets to compute the polarizabilities.30 Yannouleas and Landman studied all the C60n- anions (up to n ) 12), using a less detailed jellium/LDA model.31 All of these previous density functional studies constrained the anions to be icosahedrally symmetric and thus prevented the necessary Jahn-Teller distortion. We report here results for the first six anions allowing unconstrained geometry optimization, including modern “nonlocal” (gradient) corrections to the exchange-correlation density functional, and using the larger basis sets known to be required to achieve “chemical accuracy”. Computational Methodology Calculations were performed using the DGAUSS32 density functional code included in the UniChem33 package. This program iteratively solves the unrestricted Kohn-Sham (UKS) equations expanded in a basis set of atom-centered Gaussians. (For closed-shell molecules such as C60 neutral, the restricted Kohn-Sham (RKS) formalism was used.) Integer occupation numbers were used for all the orbitals, i.e., the standard KS method. The geometry was optimized without symmetry restrictions, using analytical gradients of the energy. The local density functional of Vosko, Wilk, and Nusair34 (which is built on Slater’s35 expression for the exchange functional) was used to optimize the electron density, but the energy was calculated including the gradient-corrected exchange functional of Becke36 and the gradient-corrected correlation functional of Perdew.37 There is usually little difference between energies computed using the gradient corrections self-consistently and those computed using the less computationally expensive two-step procedure employed here.38,39 The Becke-Perdew functional used here is known to give results very similar to those computed with other popular functionals. No corrections were made for zero-point energy effects, nor was any attempt made to bring the calculations (formally done in Vacuo at 0 K) up to the temperatures where the experimental measurements were made. These corrections would tend to cancel out in the present case and in any event are smaller than the precision of our
J. Phys. Chem., Vol. 100, No. 36, 1996 14893 calculations. Similarly, no attempt was made to precisely compare the calculated equilibrium geometries with the vibrationally averaged geometries measured experimentally, nor to make any corrections for the solvent and crystal packing effects which undoubtedly influence many of the experimental data. The DZVP basis set used (double-zeta plus polarization) consists of three contracted s functions and two contracted sets of p functions on each carbon plus one set of d functions, a total of 15 contracted basis functions (from 30 primitives) on each atom (9s5p1d/3s2p1d). This basis set, which was optimized for density functional calculations,40 has been shown to give excellent results for a variety of organic molecules.21-22,41 (Density functional techniques apparently do not need the large basis sets required to get converged results using correlated postHartree-Fock methods like MP2.42) DGAUSS reduces the number of integrals computed by fitting the electron density to an auxiliary basis set. We used the recommended40 auxiliary basis set for the DZVP calculations. When we added diffuse functions to the basis set, we also added them to the auxiliary basis set. It should be noted that no basis set could really be adequate for the computations on the trianions and higher anions addressed in the present study, since these species are thought to be unstable in a vacuum. If formed, they would spontaneously emit one or more electrons. (The trianions and higher anions are only stable in a condensed phase which can stabilize the negative charge.) By using a finite basis centered on the carbons, we are forcing the electrons to remain on the C60 so we will never find the true ground state energy. Our assumption is that the geometrical and electronic structures of these anions in solution are not too different from the structures we calculate using our finite basis in a vacuum. (Similar approximations were made by Negri et al.13 and others.) These calculations are rather big and time-consuming. Essentially, we are performing open-shell self-consistent and analytic gradient calculations involving 900 contracted basis functions from 1800 primitives at each geometry as we optimized 180 Cartesian coordinates without the benefit of symmetry. (Indeed, one of the purposes of this work was to test and improve the DGAUSS algorithms for handling very large systems.) The supercomputer CPU time required prevented us from doing many of the checks (e.g., extended basis set studies, second-derivative calculations, searches for other local minima on the PES) ordinarily done with ab initio calculations on smaller species. We do not believe we have reached the basis set limit, either for the ordinary basis or for the auxiliary basis set. Although the techniques used here have met with considerable success, one should be aware of several assumptions and approximations made. The computed energies of the anions are reported relative to the energy of neutral C60. The energies reported are the simple unrestricted Kohn-Sham energies for the states of various 〈Sz〉; it is not known for certain how to constrain the value of 〈S2〉 within density functional theory, nor is it clear when fractional orbital occupancies, or the multiconfigurational approach advocated by Ziegler and Baerends43-45 for symmetrical transition metal systems, should be used for molecular systems of lower symmetry such as the Jahn-Teller distorted C60 anions. Hence, there is some uncertainty about the reliability of the computed splittings between states of different multiplicity discussed below. The calculated energies of the Kohn-Sham orbitals are used to compute UV-vis transitions; this is certainly an approximation. (The correct way to compute electronically excited states within density functional theory is still a matter of some controversy.)
14894 J. Phys. Chem., Vol. 100, No. 36, 1996
Green et al.
Figure 1. Calculated Kohn-Sham orbital energy patterns of the C60 anions.
Despite all these caveats, the calculations reported here are undoubtedly the highest-level calculations performed so far on the C60 anions. A number of results of the computations are compared below with experimental observables. Results A summary of the calculated electronic structures of the C60 anions is presented in Figure 1. The anions are formed by partially filling the unoccupied triply degenerate t1u orbitals of neutral C60. Since the C60 structure is fairly rigid and the added electrons are distributed (more or less equally) over 60 carbon atoms, it is not surprising that the gross geometries and electronic structures of all the anions are found to be fairly similar and not very different from what one would expect based on the structure of neutral C60. For example, the HOMOLUMO gaps of all the anions are found to be very similar, lying around 1.1 eV, a value similar to the t1u-t1g gap in neutral C60. (By “HOMO” we mean the three (nearly) degenerate orbitals which are partially occupied; the “LUMO” also consists of three (nearly) degenerate orbitals.) As electrons are added to C60, the short (double) bonds connecting the pentagons increase in length, from a calculated 1.395 Å for C60 neutral to 1.429 A for C606-, presumably due to partial occupancy of the antibonding π* orbitals. For C60- to C605- the three (originally degenerate) orbitals are only partially filled, and these orbitals extend over the entire surface of the buckyball, so these anions have a number of nearly degenerate low-lying electronic configurations, each with a distinct (often Jahn-Teller distorted) equilibrium geometry. The electromagnetic and chemical properties are expected to be sensitive to the specifics of the ground state electronic configuration, the energy required to access the excited electronic states, and the molecular geometry for each anion.
Specific comparisons with experimental data (or predictions where the experimental data are lacking) for each anion are discussed below. Finally, an attempt is made to correlate the calculated energies of the isolated anions with the experimental redox potentials of their solvated counterparts. Assignments of Symmetries. The Jahn-Teller distortions of the anions are fairly subtle, and the analysis is complicated by numerical uncertainties in the atom positions which is limited by the precision of the DFT calculations. (The problem is analogous to those found in analyzing the experimental crystal structures of fulleride salts.) The point groups of the various distorted anion geometries were assigned and the distortions analyzed as follows. First, the molecule is oriented in a fixed Cartesian frame, coinciding with the Boyle-Parker setting of the icosahedron in a cube46 as used in earlier work on the force field of C60,47 i.e., with x, y, and z directions coinciding with C2 symmetry axes. In this way rotations and translations are removed, leaving only vibrational distortions of the originally icosahedral carbon framework. For each atom, the Cartesian displacements from the icosahedral position (i.e., from the position of that atom in an Ih C60 cage with the same average single and double bond lengths as the anion) were calculated and collected in a distortion vector of 180 entries. The norm of this vector gives a measure of the extent of the Jahn-Teller distortion. Approximate normal modes and frequencies were produced for displacements from the icosahedral position using a parametrized six-parameter force field from previous work.47 The distortion vector was then expressed as a linear combination of these normal modes. The choice of reference configuration removes the totally symmetric Ag components of the distortion. The normal modes and frequencies derived in this procedure vary slightly with anionic charge as a result of the variation in average bond lengths. Only the Hg modes are found to have
Structures and Geometries of C60 Anions
J. Phys. Chem., Vol. 100, No. 36, 1996 14895
TABLE 1: Point Group Symmetries, Average Bond Lengths (Å), Cage Radii, and Jahn-Teller Distortions of C60 Anionsa point group
CsC CdC
C60 (doublet)
D3d
C602- (singlet)
D2h
C602- (triplet)
D3d
C603- (doublet)
Ci
C604- (triplet)
D5d
C605- (doublet)
D2h
1.4435 1.3997 1.4431 1.4046 1.4432 1.4047 1.4430 1.4106 1.4436 1.4160 1.4442 1.4223
-
overlap (%) (Hg modes only) 727 1091 1180 1477 cm-1 cm-1 cm-1 cm-1
1617 cm-1
sum
norm distortion vector
2.2
10.1
99.5
0.051
1.1
2.4
8.0
99.3
0.117
0.4
1.8
2.0
9.2
99.4
0.053
5.9
0.7
1.3
2.6
9.7
99.3
0.092
12.6
9.5
0.4
1.6
1.9
9.1
99.6
0.053
8.6
5.4
0.5
1.0
2.2
9.6
99.9
0.051
R
214 cm-1
387 cm-1
516 cm-1
3.5379
13.5
51.7
9.4
9.8
0.8
2.1
3.5417
26.8
54.5
0.9
5.0
0.7
3.5419
16.7
48.9
11.8
8.7
3.5469
24.6
51.8
2.8
3.5526
15.0
49.6
3.5592
20.3
52.3
a For all Hg modes the frequency and percentage overlap with the distortion vector are calculated as described in the text; the sum of the overlaps with the Hg modes is always almost 100%, indicating that no other symmetry type is significantly Jahn-Teller active.
significant Jahn-Teller activity, as expected for a partially occupied t1u orbital. The first, second, and eighth Hg modes carry the bulk of the activity, as Table 1 shows. Point groups of the various distorted geometries were determined by a stepwise process. Bond lengths were tabulated and grouped in orbits; tentative conclusions based on these patterns were checked by inducing splitting of the Hg modes by selective isotopic substitution in the vibrational calculation. The Jahn-Teller activity was then traced to vibrational modes that were totally symmetric in the suspected point group. The results are listed in Table 1. One trend apparent from Table 1 is the gradual increase of the radius of the ball with total charge, an effect already well documented in the literature48,49 and readily understood in terms of progressive occupation of the LUMO set: orbitals that are antibonding along the hexagon-hexagon edges. Some contradictory predictions exist in the literature regarding the symmetry of C60-. Using the UHF version of the MINDO/3 semiempirical model, a D3d distorted cage was identified.50 Koga and Morokuma20 performed RHF and UHF calculations with a 3-21G basis set and found structures of the three epikernel symmetries D5d, D3d, and D2h differing by less than 0.1 kJ/mol. Clearly, the potential surface is almost flat. The present prediction that C60- has D3d symmetry relies on a more sophisticated electronic structure calculation, but the energies and distortions involved are still small. Neutral C60. The truncated icosahedral structure of C60 is described by two bond lengths. The experimental values from neutron diffraction studies of solid C60 are 1.45 and 1.39 Å, similar to the 1.45 and 1.40 Å inferred from nuclear magnetic resonance.51,52 Our corresponding calculated values lie within the experimental error bars at 1.444 and 1.395 Å. Our values are close to the 1.45 and 1.39 Å calculated by Dunlap et al.29 using a similar approach to ours, with a larger basis and a slightly different local density functional. Another local density functional study, using soft pseudopotentials and a basis set of plane waves rather than atom-centered Gaussians, found 1.444 and 1.382 Å.28 For comparison, the Hartree-Fock values, using a large basis set, are 1.448 and 1.370 Å.53 The ultraviolet spectrum of C60 has been reported by several workers.54 The hu f t1u transition is formally forbidden and is observed as a weak band at about 1.95 eV. It is likely that this is actually a vibronic transition lying above the electronic origin, so it really gives only an upper bound on the hu f t1u gap. Measurements on thin films of C60 suggest that the gap is 1.6 eV.55 A lower bound is given by the energy of the lowest-lying triplet state, which has been measured to be 1.56 eV (36.0 kcal/ mol).56 The calculated gap of 1.64 eV is bracketed by these
experimental data. The allowed hu f t1g transition is observed at 3.07 eV; the density functional calculation gives 2.74 eV, about 0.3 eV too low. C60-. The monoanion can be readily prepared in solution or in vacuum. The measured electron affinity (EA) of C60 is 2.666 eV;3 our calculated value, using the gradient-corrected density functional, is 2.88 eV, too high by 0.21 eV. (The published Hartree-Fock results lie around 0.9 eV, 1.7 eV too low; presumably larger basis sets and better geometries would increase this value.) Single-point density functional calculations done at the same geometry, but adding a diffuse s and p function to each carbon atom, give an EA of 2.81 eV, 0.14 eV larger than the experimental value. (Surprisingly, the addition of diffuse functions lowers the energy of the neutral C60 more than it lowers the energy of the anion.) The local density approximation also gives a good EA: with the Vosko-WilkNusair functional and our DZVP basis set, we get 3.3 eV; with a larger basis and the Perdew-Zunger57 density functional, Dunlap et al.29 find 2.8 eV. Using the generalized gradient approximation, Pederson and Quong30 calculate an EA of 2.75 eV. The Jahn-Teller distortion of C60- is rather complex and is biggest around the “equator” of C60- (with the C3 axis going through the “poles”). Because the C60 framework is very stiff and the extra electron is spread over many carbon atoms, the magnitude of the distortion is very small, with no dimensionless normal mode coordinate qi > 0.5. (The dimensionless normal coordinates are defined so that V ) 1/2hcΣωiqi2; i.e., qi ) 1 at the classical turning point for the vibrational ground state.) Complete geometries are available from the authors. The normal modes were computed using the C60 neutral force field computed using the AM1 semiempirical technique. These force constants and normal modes are presumed to be adequate to describe the anions for present purposes, though it should be noted that there is considerable disagreement in the literature about the details of the C60 force field, and it would not be surprising if the AM1 frequencies were 20% off. The lowestlying icosahedrally symmetric geometry is computed to lie 1.2 kcal/mol above the equilibrium geometry. Because the distortion is so small, the molecule is predicted to vibrate through the icosahedrally symmetric geometry even at the zero-point level. Presumably, this provides an extremely rapid mechanism to switch from one distorted geometry to another; i.e., an atom that is on the equator at one instant may find itself at the north pole a few picoseconds later. However, Wan et al.58 have recently measured the crystal structure of [Ni(C5Me5)2+][C60-]‚CS2, which shows an oblate distortion to a C2h or D2h geometry, which they suggest may be due to the Jahn-Teller
14896 J. Phys. Chem., Vol. 100, No. 36, 1996 distortion. Of course, many other factors may contribute to the distortion in this crystal; for example, Wan et al. note that the conductivity of this material is probably due to significant interactions between the adjacent C60 anions in the crystal. Because of the Jahn-Teller distortion, the 60 carbon atoms are no longer equivalent; for example, the Mulliken charges vary from 0.008e to 0.037e, more than twice the charge each atom in the anion would have at icosahedrally symmetric geometries. Similarly, the spin on each carbon atom, calculated from the Mulliken populations, varies from 0 to 0.059 of an electron spin, 3.5 times the value expected for the symmetric geometry. The large (and rapidly changing, as the molecule vibrates) differences in spin densities on each atom are expected to lead to significant broadening in the magnetic resonance spectrum, due to variable coupling between the spin of the electron and the 13C nuclei in C60 anion. These charge and spin localizations on particular carbon atoms may also lead to enhanced chemical reactivity. The calculated “HOMO f LUMO gap” (i.e., from the singly occupied orbital in the lower triad to the centroid of the next highest triad) is 1.17 eV, in precise agreement with the observed transitions centered at 1.17 eV.5 C602-. There are two low-lying electronic states of C602-, a triplet and a singlet. To the accuracy of our calculations, the two are essentially degenerate. Magnetic susceptibility measurements on salts of C602- indicate that the ground state is a singlet but that the triplet state is so low lying that it is significantly thermally populated, even at 6 K.59 The simple interpretation of the lack of any significant splitting between the singlet and triplet states is that both (1) the Jahn-Teller splitting is small because the atoms are held stiffly in place by the C-C bonds and (2) the two unpaired electrons do not interact very strongly because they are spatially separated. This is consistent with the interpretation of the measured ESR and ENDOR spectra. For example, from the ESR spectra, Bhyrappa et al. infer that the two unpaired electrons in the triplet are separated by about 9.7 Å on average. The lowest singlet and triplet electronic states are Jahn-Teller distorted away from icosahedral symmetry to D2h and D3d geometries, respectively. Paul et al.60 have measured the crystal structure of the bis(triphenylphosphine)iminium salt of C602-; they report that the fulleride has only Ci symmetry. They attribute the prolate geometrical distortion from icosahedral as due prinicipally to the Jahn-Teller distortion, though crystal packing forces and the presence of the cations may play a role. As for C60-, Mulliken population analysis indicates that the charge (and for the triplet, the spin) is distributed very unevenly at the distorted geometries. For the singlet state the Mulliken charges vary from 0.011e to 0.064e; i.e., the most reactive carbon atoms have about twice the average charge. For the triplet, the spins range from -0.008 to +0.055 of an electron spin on each carbon. The localization of charge and spin will lead to enhanced reactivity. If the singlet and triplet are in rapid equilibrium, as suggested by the experiments of Bhyrappa et al., it may be difficult to drive the chemistry selectivity along free radical vs nucleophilic pathways. The dianion singlet is predicted to be more Jahn-Teller distorted than any of the other anions studied, which is not surprising since in the singlet the two extra electrons occupy the same orbital, while the other spatial orbitals derived from the t1u triplet in C60 are empty. The distortion is so large that the icosahedrally symmetric geometry is only marginally accessible by zero-point motion (the largest qi ) 1.0). The triplet is less distorted and freely vibrates through an icosahedrally symmetric geometry. We calculate that C602- lies 2.7 eV below neutral C60; this
Green et al. value becomes 2.62 eV with the addition of the diffuse functions to the basis set. C602- is apparently stable in gas phase,61,62 indicating that its energy is less than or comparable to that of C60-, i.e., about 2.67 eV below neutral C60, in close agreement with our calculations. The disproportionation reaction C602+ C60 f 2 C60- is predicted to be quite exothermic; i.e., C602and C60 do not coexist in the gas phase. We calculate the electron affinity of C60- to be -0.2 eV (negative meaning that it takes energy to add an electron to the monoanion); by comparison, Pederson and Quong30 compute -0.3 eV, and Yannouleas and Landman arrive at -0.1 eV. The stability of the dianion suggests that the electron affinity is probably positive, though Yannouleas and Landman31 argue that there is a large centrifugal barrier (because in their model the valence electrons originate from l ) 5 orbitals) to emitting an electron from C602-, which explains its long lifetime. Normally, the electronic angular momentum is effectively quenched in molecular systems, so no large centrifugal barrier to electron emission is observed; the Jahn-Teller distortion of the dianion should increase the coupling between the nuclei and the electrons, quenching the electronic angular momentum even more effectively. Another possible barrier which would slow emission from metastable C602- could be the geometric distortion between C60- and C602-. However, our calculations indicate that the zero-point energy of the dianion will be enough to overcome the effects of this Franck-Condon barrier. These issues have been discussed in detail recently by Scheller, Compton, and Cederbaum.62 As they point out, it is likely that there is also a significant “monopole-induced dipole” interaction stabilizing the dianion due to the high polarizability of C60-; it is unlikely that the local density functional calculations reported here accurately compute this effect. The “HOMO f LUMO” transition of C602- is predicted to lie at 1.1 eV, very close to the analogous transition in C60-. Experimentally, this transition has been reported at 1.4 eV.59,63 C603-. Hund’s rules would suggest that the high-spin quartet state should be the most stable, but magnetic susceptibility measurements indicate that, at least in certain salts, the ground state is actually a doublet.59 In our calculations, the ground state is an icosahedrally symmetric quartet, but the Jahn-Teller distorted doublet state lies less than 1 kcal/mol higher in energy. ESR measurements suggest59 that a second doublet state, 2A, lies less than a kcal/mol above the ground (doublet) state. Since this second state has the same orbital occupancy as the quartet state, differing only by a spin flip, it is reasonable to infer that the quartet lies close indeed to the ground state doublet. As for C602-, both the orbital splitting induced by the Jahn-Teller distortion and the interactions between the unpaired electrons are quite small, so the unpaired electrons must be well-separated. These small splittings are thought to be important in the observed superconductivity of salts of C603-. It is possible that the trianion will show very different reactivity depending on its spin state, even though the two are not so different in energy; the classic example is the wildly different reactivities of singlet and triplet CH2. However, it seems likely that the normal concept of an electron “pair” may not be applicable to partially filled “t1u” orbitals in the C60 anions, since the electrons may be so widely separated that the “pair” may act more like a diradical. The highest spin densities of the doublet are calculated to be similar to the 0.05 electon spins/atom found on the symmetric quartet, so one would expect that they could react by a free-radical mechanism with similar substrates, albeit with rates that should be about 3 times smaller if the paired electrons in the doublet are not reactive as radicals. The doublet should be more reactive as a nucleophile than the quartet state,
Structures and Geometries of C60 Anions
J. Phys. Chem., Vol. 100, No. 36, 1996 14897
TABLE 2: Calculated vs Experimental Electron Affinities and Reduction Potentials of C60 and Its Anionsa calc vs (expt) C60 C60C602C603C604C605-
EA
abs red pot
CH2Cl2/SCE5
CH3CN + toluene/Fc7
C6H6/SCE6
THF/SCE64
2.81 (2.67) -0.2 (>0) -3.14 -6.16 -9.2 -12.1
-2.81 -3.15 -3.56 -3.89 -4.2 -4.65
-0.58 (-0.44) -0.92 (-0.82) -1.33 (-1.25) -1.66 (-1.72) -1.97 -2.42
-1.12 (-0.98) -1.46 (-1.37) -1.87 (-1.87) -2.20 (-2.35) -2.51 (-2.85) -2.96 (-3.26)
-0.50 (-0.36) -0.84 (-0.83) -1.25 (-1.42) -1.72 (-2.01) -1.89 (-2.60) -2.34
-0.47 (-0.33) -0.81 (-0.92) -1.22 (-1.49) -1.55 (-1.99) -2.06 -2.56
a Experimental data from refs 6-8 and 64. The zeroes of the calculated relative potentials were shifted so that the experimental EA of C 60 corresponds to the experimental first reduction potential. Solvent corrections were applied to the higher reduction potentials as described in the text.
since its distorted geometry localizes the charge, with a maximum computed Mulliken charge of 0.08e. The icosahedrally symmetric quartet state is predicted to have bond lengths of 1.443 and 1.411 Å. The density functional calculation predicts that C603- lies 0.44 eV above the energy of neutral C60 and more than 3 eV higher than C602-. It is therefore probably not possible to form significant amounts of the trianion in the gas phase, since it will rapidly emit an electron to form the dianion. The HOMOLUMO gap of the trianion is predicted to be 1.1 eV, very similar to that of the monoanion and the dianion. Higher Anions. The electronic structure of C604- resembles that of C602-, with two holes rather than two electrons partially filling the three low-lying orbitals. Calculations were only performed on the triplet state, though we also expect that two singlet states have similar energies. In the gas phase, the tetraanion is calculated to lie 6.6 eV above neutral C60. The pentaanion is a doublet like the monoanion. In the gas phase it is calculated to lie 15.8 eV above C60 neutral. The hexaanion has a completely full t1u shell and maintains icosahedral symmetry. The calculated bond lengths are 1.445 and 1.429 Å. Its gas phase energy is calculated to be 27.9 eV greater than that of the C60 neutral. Comparison with Measured Redox Potentials. The trianion and the higher anions have only been produced in condensed phases, where the dielectric properties of the medium and other solvation effects stabilize the localization of charge on the C60 core. In the simplest picture, a dielectric reduces the energy required to bring two chargessin our case, an electron and a C60 anionstogether by
∆E ) (qe2/4π0R)(1 - 1/κe)
(1)
where q is the charge of the anion (e.g., -1, -2, etc.), R is the effective radius of the solvent cavity surrounding the anion, and κe is the dielectric of the medium. This correction is added to the calculated electron affinities to get estimates of the absolute reduction potentials in solution. (Note that this zero-order correction vanishes for the first reduction of neutral C60.) The precise value of the dielectric constant for the solutions used by electrochemists is not usually reported, but the values of κe are typically large enough that this term causes only a few percent uncertainty. (It explains some of the differences between the reduction potentials vs SCE measured in the different solvents, Table 2.) Taking R to be 4.3 Å, as suggested by Negri et al.,13 and assuming that κe is large enough that the second term is essentially unity, one gets the corrections of 3.35, 6.7, 10.05, 13.4, and 16.75 V for respectively the second, third, fourth, fifth, and sixth reduction potentials of C60. A 1% change in the assumed R (i.e., 4.34 vs 4.30 Å) would cause a 1% change (i.e., up to 0.17 V) in these correction factors. Since there are uncertainties of 0.2 eV or more in the solvent corrections, we do not expect quantitative agreement between the experi-
mental6-8,64 and calculated values (even disregarding errors in the electronic structure calculations). There are further complications in comparing the isolatedanion energies with the measured redox potentials. One of the most troublesome is the well-known difficulty in converting from absolute potentials to the relative potentials used by electrochemists (and the related problem of dealing with all the higher-order solvation energy corrections). We have shifted the zero of all of the computed absolute reduction potentials to bring the measured electron affinity of C60 into line with the measured first reduction potential in solution. The computed absolute reduction potentials in the gas phase and in a high dielectric solvent and the shifted values are compared with the experimental redox potentials in Table 2. The predictions for the higher anions are in reasonable accord with experiment, though in light of the approximations made to connect the gas phase values with the solution phase experiment, this agreement is certainly not conclusive. The experimental half-cell reduction potentials are found to vary by as much as 0.6 V, depending on the solvent/electrolyte combination used; some of this can be seen in Table 2. For more examples and detailed discussions of the origins of these effects, see refs 64 and 65. Equation 1 provides an estimate of the poorest dielectrics which could possibly stabilize the anions C60n- (n > 2), which are unstable in the gas phase, as isolated species. For C603-, a dielectric with κe as small as 1.9 could be sufficient, while C606cannot be stabilized unless κe > 3.7. (Solutions of these anions may still be unstable for other reasons, e.g., if the solvent can accept or effectively solvate free electrons.) This calculation suggests that it might be possible to prepare stable solutions containing “free” C603- anions in hydrocarbon solvents such as benzene, while the higher anions will require polar solvents. Conclusions Good quality density functional calculations, with moderately large basis sets, have been performed to determine the geometries and electronic structures of the C60 anions. These calculations are the best source of information available on the geometrical structures of the isolated anions. The electronic transition energies and electron affinities are within a few tenths of an electronvolt of the available experimental data and provide predictions for the many states which have not been measured. Comparisons with available condensed phase data, e.g., electrochemical reduction potentials and crystal structures, are reasonably good; the remaining errors are likely due as much to uncertainties in the solvent corrections and crystal packing forces as to errors in the density functional calculations. The di- and trianions have nearly degenerate electronic states of different multiplicity; also, the Jahn-Teller distorted anions typically have several low-lying geometrical structures. It is clear that these near degeneracies strongly affect many of the measured physical properties of the anions, and they may also affect their chemical reactivity.
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