5763
J . Phys. Chem. 1991, 95, 5163-5168
+
found, indicating that the 1(NH3),-(L),)H+, n m = 5 , are
particularly stable. It is proposed that a central ammonium ion
NH4+ is bound to four constituents in any combination of ammonia and L molecules. Studies of metastable unimolecular
dissociation processes reveal that the ((NH3),.(L),)H+. n = 1, m
= 2-4, cluster ions lose one L molecule whereas the {(NH~),,.
(L),)H+, n = 2-1 1, m = 1-4, cluster ions lose one NH3 molecule
in a few tens of microseconds. The experimental results suggest
that the stable cluster ions ((NH3),,.(CH$N),IH+
and
I(NH3),.(CH3CHO),)H+, n m = 5 , have completed solvation
shell structures as expressed in structures I.
+
Acknowledgment. Support by the US.Department of Energy,
Grant NO. DE-FG02-88-ER60668, is gratefully acknowledged.
Registry No. NH3, 7664-41-7; NH4, 14798-03-9; CH,CN, 75-05-8;
CH,CHO, 75-07-0.
B. I. Dunlap,* D. W. Brenner, J. W. Mintmire, R. C. Mowrey, and C. T. White
Theoretical Chemistry Section, Code 61 19, Naval Research Laboratory, Washington, D. C. 20375-5000
(Received: February 8, 1991; In Final Form: March 13, 1991)
Local density functional methods and empirical hydrocarbon potentials are used to study the electronic and geometric structures
of icosahedral Cm C,Hm and C,FW Compared to isolated C,, the C-H and C-F bond strengths of these saturated species
are reduced by over 40% and over 15% from the C-H and C-F bond strengths in methane and in tetrafluoromethane,respectively.
Similar C-H bond strengths are found for C,Hw and CsoHso Other C&, species are optimized by using empirical potentials.
Outward bonding for all hydrogen atoms is not always energetically preferred.
1. Introduction
C, has been isolated in macroscopic quantities in several
laboratories.’” Carbon- 13 NMR of pure C, dissolved in various
solvents shows that this molecule has one symmetry inequivalent
atom*5 and therefore has the proposed buckminsterfullerene (BF)
structure with icosahedral ( I h ) symmetry.’ Haufler et ale6have
partially hydrogenated C, using Birch reduction to yield C60H36.
C60H36is only the first in a series of new molecules that probably
will be synthesized starting from c60. Perhaps methods will be
devised to fully hydrogenate C, to CWH, and other routes found
to yield CmFbO,which should be inert similar to poly(tetrafluoroethylene). These hypothetical fully saturated Ih molecules
have already been studied theoretically within the Hartree-Fock
(HF) approximation.* Stimulated by these results and possibilities, we have chosen to study the geometry and electronic
structure of these and related molecules using empirical potentialsg
and local density functional (LDF) calculations.’0
Of the many possible hydrogenated and fluorinated (260
structures, the fully saturated species, C,H,
depicted in Figure
1, and the corresponding C,F,
molecule, have the highest symmetry and hence are easiest to study by the all-electron, firstprinciples LDF methods herein. However, if intermediate steps
in the transformation of c 6 0 to C(,OHw are considered, the high
symmetry of the reactants and products is lost and a full geometry
optimization of these intermediate species rapidly becomes impractical. In addition, if the proposed CaH36structure$ depicted
in Figure 2, is made from a molecular modeling kit using flexible
tubes of roughly equal length to form nearest-neighbor bonds, then
the bonds connecting saturated carbon atoms bend slightly inward
while the bonds connecting unsaturated carbon atom pairs of their
four neighboring saturated atoms are quite planar. The inward
curvature of this model for these saturated C-C bonds is as great
as the outward curvature of a similarly constructed CWmolecule.
The inward curvature for a fully saturated Ih C6OH60 is even
greater. These results suggest that, rather than bonding all hydrogens to the exterior surface of the cage, lower energy isomers
might be obtained by bonding a subset of these hydrogens to the
interior surface of the molecule to better compensate local strains.
However, putting one or a few hydrogen atoms inside the BF cage
lowers both the tetrahedral (Th)symmetry” of the proposed
Corresponding author.
CsOH36structure6 depicted in Figure 2 and the I , symmetry of
the C&, molecule depicted in Figure 1, making a first-principles
study of these isomers difficult. We have optimized the geometries
of some of these lower symmetry structures using recently introduced empirical hydrocarbon potentials9 as a practical way to
address whether moving a subset of hydrogen atoms inside the
Cm cage lowers the total energy of these clusters. We also have
used these empirical potentials in a preliminary survey of the
energetics of hydrogen addition to the exterior surfaces of some
of these clusters.
In section 2 we describe our LDF approach and compare our
LDF results with experiment for CH4 and CF4. We also briefly
review in section 2 our recently developed empirical hydrocarbon
potentials. In section 3 we consider the icosahedral species, C,,
CWHmand C,F,
with all H and F atoms bonded to the outside
of the cage. Optimized LDF geometries allow us to compare
the C-H and C-F bond energies of C,H,
and CWFa and to
compare empirical potential and LDF geometries for CaHm In
section 4 we use our LDF approach to study the recently synthesized? partially hydrogenated species CWH36at the optimized
empirical potential geometry. In section 5 we consider a number
of other CWHnspecies using empirical potentials in a preliminary
(1) Kritschmer, W.; Fostiropoulos, K.; Huffman, D. R. Chem. Phys. Lett.
1990, 170, 167.
(2) Kratschmer, W.; Lamb, L. D.; Fostiropolous. K.; Huffman, D. R.
Nature 1990, 347, 354.
(3) Taylor, R.; Hare, J. P.; Abdul-Dada, A. K.; Kroto, H. W. J . Chem.
SOC.,Chem. Commun. 1990, 1423.
(4) Johnson, R. D.; Meijer, G.; Bethune, D. S. J . Am. Chem. Soc. 1990,
112, 8983.
( 5 ) Ajie, H.; Alvarez, M. M.; Anz, S. J.; Beck, R. D.; Dicderich, F.;
Fostiropoulos, K.; Huffman, D. R.; Kratschmer, W.; Rubin, Y.;Schrivcr, K.
E.; Sensharma, D.; Whetten, R. L. J . Phys. Chem. 1990, 94, 8630.
(6) Haufler, R. E.;Conceicao, J.; Chibente, L. P.F.; Chai, Y.;
Byrne, N.
E.;Flanagan, S.;Haley, M. M.; OBrien, S. C.; Pan, C.; Xiao, 2.;Billups,
W. E.;Ciufolini, M. A.; Hauge, R. H.; Margrave, J. L.; Wilson, L. J.; Curl,
R. F.; Smalley, R. E. J. Phys. Chem. 1990. 94, 8634.
(7) Kroto, H. W.; Heath, J. R.; OBrien, S.C.; Curl, R. F.; Smalley, R.
E. Nature 1985, 318, 163.
(8) Scuseria, G. E. Chem. Phys. Lett. 1991, 176,423.
(9) Brenner, D. W. Phys. Rev. B 1990, 42, 9458.
(IO) Dunlap, B. I.; Connolly, J. W. D.; Sabin, J. R. J. Chem. Phys. 1979,
71, 3396, 4993.
(11) Dunlap, B. 1.; Brenner, D. W.; Mowrey, R. C.; Mintmire, J. W.;
Robertson, D. H.; White, C. T. Mater. Res. Soc. Symp. Proc., in press.
This article not subject to US.Copyright. Published 1991 by the American Chemical Society
Dunlap et al.
5764 The Journal of Physical Chemistry, Vol. 95, No. 15, 1991
TABLE I: C m B-is Set SMk#
C-C bonds, A
basis
tvDe 1
tvw 2
91511
1.43
1.43
1.45
1.39
1.39
1.39
ll/6/l
ll/7/l
Eb,
kcal/mol
218
El,
hartrecs
201
197
-2.30
-1.72
-0.41
uSuccessivecolumns from the left are the primitive Gaussian basis
sets in s/p/d notation, the two optimized bond distances, the atomization energy per carbon atom, and El as computed using eq 1. Bonds of
type 1 are shared by a pentagon and hexagon while bonds of type 2 are
shared by two hexagons.
Figure 1. Icosahedral CmHm.
U
U
Figure 2. Tetrahedral CmH36 with the 36 hydrogen atoms pointing
outward.
survey of the energetics of hydrogenation and isomerization on
the Cso cage of these molecules.
2. Computational Methods
Computationally, LDF‘s differ from H F in that they have a
single one-electron (Fock) potential.I2 Thus, Abelian and nonAbelian groups are treated in the same fashion if one restricts
the Fock potential to be symmetric under the point group of the
nuclei. The trace of the Hamiltonian within each multidimensional
(for non-Abelian groups) irreducible representation of the group
is i n ~ a r i a n t ,reducing
’~
considerably the number of integrals involving this trace that must be actually computed.14 For example,
consider the 60-equivalent-atom case of BF with one basis function
on each atom. The 1830 matrix elements resulting from all unique
orbital pairs in the 60 X 60 orbital products reduce by symmetry
to 32 independent terms. The one-electron effective Hamiltonian
is real; thus, the real and complex components of one-dimensional
complex irreducible representations, which occur for the Thgroup
used herein, can be treated as if they comprise a two-dimensional
irreducible representation of a group. Maximal use of symmetry
has been incorporated into a computer codeI5 that extends the
practical a p roach of Sambe and FeltonI6 to variationally fitting
the density’ and that uses products of solid-spherical harmonics
and Gaussian functions as basis sets for both the one-electron
orbitals and the LDF potential that define the orbitals.” The
LDF used herein is the Perdew-Zunger’* (PZ) fit to the free-
!?
(12) Slater, J. C. Quanrum Theory of Molecules and Solids; McGrawHill: New York, 1974;Vol. 4.
(13) Pitzer, R. M. J . Chem. Phys. 1972,58, 31 11.
(14)Dunlap, B. 1. Adv. Chem. Phys. 1987, 69, 287.
(I 5) Dunlap, B. 1.; R k h , N. Advances in Quantum Chemistry; Trickey,
S.B., Ed.; Academic: New York, 1990,p 317.
(16)Sambe. H.; Felton, R. H. J . Chem. Phys. 1975.62, 1122.
(17)Dunlap, B. I. Phys. Rev. A 1990. 41, 1127.
(18)Perdew, J. P.;Zunger, A. Phys. Rev. B 1981, 23, 5948.
electron gas results of Ceperley and Alder.l9
Our previous LDF calculationsm on Cso used two orbital basis
sets for the carbon atom. The first two are the 9s/5p and 1ls/6p
orbital basis sets of van Duijneveldt2Iaugmented with a d function
of exponent 0.57 bohr2 and contracted to 2,3/1,3/0,1 and
2,4/1,4/0,1. In this orbital contraction notation each angular
momentum is separated by a slash, the first of the two numbers
being the number of lowest energy atomic orbitals used and the
second being the number of most diffuse Gaussian basis functions
used. Two additional totally symmetric Gaussian basis sets are
needed to fit the electron-electron LDF potential. Spherically
symmetric atom-centered functions were obtained by scaling the
orbital exponents appropriate for treating the direct Coulomb
potential and for treating the weaker exchange-correlation potential,1° discarding exponents smaller than 0.3 and 0.1, respectively, for the two basis sets.22 The p fitting functions were 0.3,
0.7, 2.0 on carbon and 1.0 on hydrogen, with an additional exponent 5.0 added for the larger basis set. Bond-centered fitting
functions of exponent 0.5 were added at the midpoint of each
nearest-neighbor bond.
The results of the previous study are included in Table I. The
two bond lengths are determined by fitting the computed total
energy on a grid of points and then computing the energy at the
optimal geometry predicted from the fit. This process is iterated
until the optimized bond distances are determined to within 0.01
A. Approximate A4 vibrational frequences can be determined
by fitting the potential energy surface to simple polynomials in
the various bond distances;1° thus, the greater the number of
degrees of freedom, the more sensitive the fitted frequencies are
to the exact nature of the fitting polynomial. Our previous
calculations suggested that the bond distances are insensitive to
further improvements in the orbital basis sets. For this work larger
fitting basis sets were used. The C orbital basis set, an 1ls/7p
basis21 augmented with a d exponent of 0.6 b ~ h r - * is, ~not
~ a
significant change. The orbital basis set was contracted 2,3/
1,2/0,1. The fitting basis sets were improved by attempting to
minimize the absolute value of El
El = Y2Jdr1 dr, i j ( r M r 2 ) - p(r2))/r12
(1)
which is the difference in the self-Coulomb repulsion of all the
electrons computed by using the fit to the density p interacting
with the true density p less the self-Coulomb repulsion computed
by using only the fitted charge density. This quantity is zero if
the charge density is variationally fit without using a constraint.I0
The constraint that the integral of the fitted charge density equals
the total number of electrons was used. Both constrained and
unconstrained fits, although different, yield total energies that
are identical within 10” hartrees. Using a constrained fit and
minimizing E , leads to the best potential energy surfaces for a
given orbital basis set.22 E l was reduced by a factor of 4 by
including all scaled s-orbital exponents, without cutoff. Use of
diffuse exchange-correlation fitting exponents required the use
(19)Ceperley, D. M.; Alder, B. J. Phys. Rev. Lerr. 1980, 45, 566.
(20)Dunlap, B. 1. Inr. J . Quanrum Chem., Quonrum Chem. Symp. 1988,
22, 257.
(21)van Duijneveldt, F. B. IBM Rcsearch Report RJ945. 1971.
(22)Dunlap, 8. 1.; Mei, W. N. J. Chem. Phys. 1983, 78,4997.
(23)Huzinaga, S.,Ed. Gaussian Basis Sers for Molecular Calcularionr;
Elsevier: Amsterdam, 1984.
Structures of CMHM,CMFM,and CMH36
The Journal of Physical Chemistry, Vol. 95, No. IS, 1991 5765
TABLE II: Comparison of LDF md ExperimentalSSC-H a d C-F
Avmge Bond Enet@e~(E,), Bond D ~ s ~ (R&
c ~ s 8 d Ai
Vibntionrl Frequencies (Y,) for CHI a d CF,
CHI
quantity
Rc-x, A
Eb, kcal/mol
v , , cm-I
exp
1.091
99
2916
CF,
LDF
1.10
116
2990
exp
1.320
102
909
LDF
1.32
146
920
of variational weights to ensure numerical stability.24 Both Sp
and 5d fitting exponents were used with exponents 0.25,0.37,0.7,
2.0, and 5.0 bohr-*. The bond-centered exponents were 1.0 and
0.33 bohr-* for the charge density and exchange-correlation basis
sets, respectively. The calculated binding energy per atom Ebof
CMdecreased slightly to 197 kcal/mol. The experimental enthalpy
of formation (0 K) of carbon from graphite is 170.0 kcal/m01.~~
This overbinding is consistent with the fact that recent L D F s in
general overbind most molecules.10 In going from the 11/6/1 basis
to the 1 1/7/ 1 basis, the two A, vibrational frequencies are reduced
from approximately 1820 and 710 to 1570 and 600 cm-I. Raman
spectroscopy suggests that they experimentally occur at 1469 and
497 cm-'.26 The LDF-optimized CMbond distances are within
0.02 A of bond distances obtained in Hartree-Fock8u27-29and
~ e m i e m p i r i c aoptimizations.
l~~~
The Hartree-Fock (HF) bond
distances obtained by using the most complete basis sets are 1.37
and 1.45 A for bonds shared by two hexagons and a hexagon and
pentagon, respectively.* A number of fixed-geometry LDF
electronic structure calculations for BF have also appeared.3638
The primitive basis sets for H and F were the 6s and 12s/7p
bases of ref 21. The polarization function exponents for H and
F were chosen as I .O and 1.3 bohr-*, respectively. The contraction
schemes for H and F were 1,4/0,1 and 1 ,S/ 1,4/0,1, respectively.
Three 9 fitting functions scaled from the second, fourth, and sixth
most diffuse p-orbital functions were used in the fitting bases.
These bases are tested in calculations on CH, and CF4 in Table
I1 and compared with experimental results for the JANAF tables.25
The average bond energies are computed by subtracting the
separated atomic energies from the total energies of the molecules
and dividing the result by 4. They are compared with numbers
derived from the experimental enthalpies of formation at 0 KaZ5
Thus, the experimental energies exclude one-quarter of the
zero-point vibrational energies of the molecules. Despite the fact
that fluorine is the most overbound atom in LDF methods, six
LDF reaction energies involving fluoromethanes agree with experiment to within 2 kcal/m01,~~
suggesting that relative energies
are reliable. The bond distances and vibrational frequencies are
also in excellent agreement with experiment.
The empirical potential energy function9 used herein is based
on a formalism orginally derived by Abel14 to describe universal
(24)Dunlap, B. 1.; Cook, M. In?. J . Quantum Chem. 1986. 29, 767.
(25)Chase, M. W.,Jr.; Davies, C. A.; Downey, J. R., Jr.; Frurip, D. J.;
McDonald, R. A.; Syvemd, A. N. J . Phys. Chem. Re5 Data 1985,14 (Suppl.
I).
- I .
(26)Bcthune, D. S.;Meijer, G.; Tang, W.C.; Rosen, H. J. Chem. Phys.
Lett. 1990, 174. 219.
(27)LUthi. H. P.;Almlaf. J. Chem. Phvs. Lett. 1987. 135. 357.
(28) Disch, R. L.; Schulman, J. M. Ckem. Phys. Lett. 1986, 125,465.
Schulman, J. M.; Disch, R. L.; Miller, M. A.; Peck, R. C. Ibid. 1987, 141,
45.
(29)Fowler, P.W.;
Lawretti, P.; Zansi, R. Chem. Phys. Lett. 1990, 165,
79.
(30)Marynick, D.S.;Estreicher, S. Chem. Phys. Lcrr. 1986, 132, 383.
(31)Newton, M. D.;Stanton, R. E. J. Am. Chem. Soc. 1986,108, 2469.
(32)Stanton, R. E.;Newton, M. D. J . Phys. Chem. 1988, 92, 2141.
(33)Haddon, R. C.; Brus, L. E.; Raghavachari, K. Chem. Phys. ha.
1986, 125,459.
(34)Kataoka, M.; Nakajima, T. Tetrahedron 1986, 42,6437.
(35) Lbzlb, I.; Udvardi, L. Chrm. Phys. Lett. 1987, 136,418.
(36)Hale, P.D. J . Am. Chem. Soc. 1986, 108.6087.
(37) Satpnthy, S. Chem. Phys. fer?. 1986, 130,545.
(38) Rostn, A.; Wistberg, B. J . Chem. Phys.' 1989. 90,2525.
(39)Dixon, D.A.; Andzelm, J.; Fitzgerald, G.; Wimmer, E.;Jasien, P.In
Dcyiry FunCrioMI Methods In Chemistry; Labanowski, J., Andzelm, J., Eds.;
Springer: New York, 1991;p 33.
TABLE III: HF (Ref 8) a d LDF C&, a d C&
POtellthl (I) c& ShYChvrl P 8 " e t c ~ "
molecule
CaH60 (1)
CaHa (HF)
CaHa (LDF)
C a F a (HF)
CaFa (LDF)
tYDe 1
1.55
1.57
1.55
1.60
1.61
tYDe 2
1.54
1.55
1.54
1.59
1.60
md Empirical
center%-X
anale. dea
1.09
1.08
1.10
1.34
1.35
178
180
180
"Bonds of type 1 are shared by a pentagon and hexagon while bonds
of type 2 are shared by two hexagons.
TABLE I V Occupied a d Selected Unoccupied LDF One-Electron
V a h e Levels of CJ&,
in eVa
8 ti,
-1.29
9 h,
-8.81
3 a,
-12.95
5 a,
-1.73
6 g,
-8.86
6 h,
-14.32
5 g,
-9.65
4 t2,
-14.68
-5.45
8 h,
-9.78
2 ti,
-14.97
7 h,
7 g,
-5.48
2 t2,
-9.91
4 ti,
-16.13
3 t2,
-5.69
6 t2,
-9.91
3 h,
-16.79
7 gu
-5.94
5 g,
-10.03
3 g,
-17.70
1 1 h,
-6.31
4 g,
-10.93
5 h,
-18.01
3 ti,
-6.74
6 ti,
-11.04
3 gu
-18.94
7 t2,
-6.88
5 t2,
-11.13
3 tiu -19.74
6 h,
-7.15
7 h,
-11.78
4 h,
-20.46
7 ti,
-7.36
4 a,
-12.25
3 tiu -21.20
IO h,
-7.43
5 ti,
-12.54
2 a,
-21.59
4 g,
-12.58
6 g,
-8.20
4 h,
-12.79
5 h,
-8.30
" A horizontal gap in the first column separates the occupied and
virtual levels.
features of chemical bonding and subsequently developed by
TersofP' to describe bonding in covalently bonded solids. The
potential energy is modeled by a short-range Morse-like pair
potential where the attractive terms of the pair potential are
coupled to bonding environments by a many-body empirical
function. This many-body function is closely related to a normalized Pauling bond order and depends on bond angles and local
atomic coordination and whether ?r bonds are part of a conjugated
system. The function is fit to the lattice constants and binding
energies for the diamond lattice, graphite sheets, hypothetical
simple cubic and face-centered cubic lattices of carbon, and bond
energies and lengths for molecular single, double, and triple
carbon-carbon bonds and carbon-hydrogen bonds. The function
provides a reasonable prediction of atomization energies for hydrocarbon molecules ranging from small ring systems to large
aromatic molecules as demonstrated by Table IV of ref 9. Further
properties and details of the potential form are found in ref 9.
3. Fully Saturated C a w and CwFw
Table I11 compares optimized CMHMand c6oF@geometries
in HF8 to those predicted by using empirical potential I of ref 9
and to our LDF geometries. The agreement is excellent between
our calculations. In fact, the LDF energies at the LDF and
empirical potential minima differ by only 0.02 hartree. The largest
bond distance between H F and the LDF geometries is 0.02 A.
The slight bending away from the radial direction of the originC-H angle in the empirical potential geometry is in the plane of
the origin and the C-C bond shared by two hexagons and shortens
the hexagonal H-H third-neighbor distance.
The LDF electronic structure of CmHM is closed-shell,
4a,~tl,7t2,3tl,l 1hg7g,3t2,7g,7h,, where these highest occupied
orbitals of each Ih symmetry are ordered according to increasing
one-electron eigenvalues. The electronic configuration agrees with
the H F result,8 and the one-electron energy levels are given in
Table IV.
C60F60 has L D F electronic structure
7a,14t1,14t2,1 5g,8tl,la,l 5g,8t2,22h,l 6h,, and the one-electron
levels are given in Table V. Cm has LDF electronic structure
(40)Abell, G.C. Phys. Reu. E 1985, 31,6184.
(41) Tersoff, J. Phys. Reu. Let?. 1986, 56,632;1988, 61,2879.
Dunlap et al.
5766 The Journal of Physical Chemistry, Vol. 95, No. 15, 1991
TABLE V: Occupied a d scketed
Valence Levels of COF, in eVo
16 tiu
-1.25
13 ti.
16 g8-1.69
12gu16 ti,
-1.79
19 h,
24 h,
-2.41
12g,
6 ti,
16 g,
-3.08
18 h,
15 ti,
-3.46
23 h,
-4.28
7 a,
12 ti,
15 ti,
-5.31
12 t2,
8 a,
-6.01
5 ti,
1 1 h,
16 h,
-9.50
17 h,
22 h,
-9.64
1 1 ti,
8 tz,
-9.93
I I g,
15 g,
-9.97
IO h,
1 a,
-10.05
1 1 tz,
14 g,
-10.08
5 tz,
8 ti,
-10.15
1 1 g,
15 g,
-10.24
IO gg
7 t2,
-10.26
16 h,
15 h,
-10.30
14g,
-10.31
9 h,
6 ti,
-10.55
log,
15 ha
21 h,
-10.61
14 h,
-10.64
IO t2,
9 g,
13 g,
-10.73
14 h,
7 ti,
-10.93
14 t2,
-10.97
9 g,
13 h,
-11.14
4 ti,
13 ha
13t2,
-11.18
20 h,
-11.19
IO ti,
8 g,
14 ti,
-11.21
13 g,
-11.27
6 a,
12 h,
-11.37
9 tz,,
Unoccupied LDF One-Electron
-11.38
-11.49
-11.52
-11.58
-11.72
-11.98
-12.14
-12.15
-12.17
-13.98
-14.12
-14.33
-14.33
-14.34
-14.40
-14.70
-14.70
-14.70
-14.83
-14.93
-15.34
-15.40
-15.44
-15.65
-15.88
-16.09
-16.20
-16.21
-16.41
-16.43
-16.59
-16.62
-16.71
9 tiu
8 h,.
8 g,
5a
12\,
4 ti,
8 tz,
8 ti,
7 h,
7g
1,
11
7 g,
7 t2,
IO h,
7 ti,
4 a,
3 tip
6 g,
6 g,
6 h,
6 ti,
9 h,
3 ti,
6 ti,
5 h,
5 g,
8 h,
5 g,
5 t2,
7 h,
5 ti,
3 a,
-17.27
-17.42
-17.57
-18.18
-18.38
-18.72
-18.80
-19.75
-19.89
-20.65
-21.03
-21.71
-22.56
-23.13
-23.77
-24.12
-31.26
-31.29
-31.59
-31.79
-31.89
-31.96
-32.40
-32.53
-32.87
-33.05
-33.07
-33.32
-33.50
-33.72
-33.93
-34.05
" A horizontal gap in the first column separates the occupied and
virtual levels.
?
I
v
x
F
2
w
" A horizontal gap in the first column separates the occupied and
virtual levels.
2t,,4aC6t,,2t2,6t2,6g,10h 6g 6h,, and the one-electron levels are
given in Table VI. The LDf: electronic configurations of C,F,
and C, also agree with the HF res~lts.**~'-~~
Figure 3 gives the
density-of-states Lorentz broadened by 0.2eV for these three
molecules. The levels of the fluorinated compound are shifted
down in energy relative to the other two because of the large
electronegativity of fluorine.
The one-electron LDF eigenvalue gaps between the highest
occupied molecular orbital (HOMO) and the lowest unoccupied
molecular orbital (LUMO) for Cbo,CaHm and C,F,
are 1.68,
3.72,and 3.48 eV, respectively. The LUMO's for Cbo,CboH,,
and CWFa are 7tl,, Sa,, and 8a,, respectively. Thus, HOMOLUMO excitations are electric-dipole forbidden in all three of
these molecules. The LDF energetics, bond energies, and A
vibrational frequencies are given in Table VII. The LDF C-I-!
and C-F bond strengths have been reduced by 44% in going from
CHI to CmHWand 16% in going from CF, to CboFbo.
If perfect tetrahedral bonds surrounded each carbon atom in
CboHbo,then our empirical potential would yield a C-H bond
i
P
-30
L
~
I
F w e 3. PZ one-electron occupied density of states for C,, C,H,, and
C,F, Lorentz broadened by 0.2 eV. Scales are chosen to yield uniform
maximum peak heights; all curves integrate to the total number of valence electrons. These molecules are all closed-shell and have a large
electric dipole forbidden HOMO-LUMO excitation.
TABLE VII: LDF C d F , and C a w and Average C-X Bond
Energies E , (Obtained by Subtracting the Total Energy of C, from
C& and Dividing by 60)and A. Vibrational Frequencies
molecule
kcal / mol
vibrational
frequencies, cm-I
C60H60
65
123
3020, 1390, 1210, 830
970, 750, 510, 360
Eb*
C
,
F
TABLE VI: Occupied and Selected Unoccupied LDF One-Electron
Valence Levels of Cu in eVo
1 1 h,
-2.07
2 ti,
-10.58
6 h,
-15.70
7 ti,
8 ha
-10.70
4 tz,
-16.79
-2.25
3 ti,
5 g,
-10.85
4 ti,
-17.78
-3.15
7t1,
7 h,
-11.26
3 h,
-17.89
-4.26
4 g,
-11.75
3 g,
-19.42
6 h,
-5.94
6 ti,
-1 1.84
5 h,
-19.90
6g
-7.12
4 a,
-12.44
3 gu
-21.18
3 tz,
-22.02
IO\,
5 tz,
-12.47
-7.24
4 ha
-23.04
5 ti,
-13.05
5 h,
-8.78
3 ti,
-24.02
4 h,
-13.25
6 g,
-8.83
2 a,
-24.53
4 g,
-14.07
9 h,
-9.09
6 12,
-9.38
3 a,
-15.34
5 gu
-10.10
2 ti,
-15.47
P'==
-10 h
TABLE VIII: Coordinates of the Symmetry Inequivalent C&%
Atoms
atom
x, A
Y,A
z, A
C
C
C
2.4879
3.2697
3.8962
4.0659
4.95 17
0.6636
1S280
0.7904
2.0222
1.0154
2.2277
1.2623
H
H
0.0000
1.7755
O.oo00
strength decrease of 34% relative to CH4instead of the more than
40% computed with either LDF or empirical potentials. Thus,
CmHm is destabilized because geometric constraints preclude
perfect tetrahedral bonding patterns around each carbon atom.
The strain introduced by closing the spherical carbon shell on itself
might be severe enough that lower energy configurations could
exist with a few of the hydrogen (and possibly even the larger and
more ionic fluorine) atoms bonded inward.
4. ThSymmetric
An isomer of C60H36with the 24 unsaturated carbon atoms
paired up along bonds shared by a hexagon and a pentagon to
form 12 double bonds that are as far apart as possible on the
surface yields one of five equivalent Th structures for the molecule,
depicted in Figure 2. In all five structures the double bonds, four
at a time, are bisected by three mutually orthogonal planes passing
through the center of the molecule, and each double bond is now
associated with a different pentagon of the original buckminsterfullerene (BF). The remaining 36 saturated carbon sites are
divided by the group into a set of 12 and a set of 24 C H groups.
The smaller, symmetry-equivalent set of 12 C H groups lie in the
planes that bisect the 12 double bonds and are second-nearest
neighbors to double-bond carbon atoms. Each carbon atom of
the larger, symmetry-equivalent set of 24 C H groups is a nearest
neighbor tocarbon atoms involved in two different double bonds.
Table VI11 gives the optimized coordinates of the symmetry-in-
The Journal of Physical Chemistry, Vol. 95, No. 15, 1991 5767
Structures of CmHm, CmFm,and CmH36
TABLE I X Total Energies in eV Relative to C, for Various C&,
Species Using Both Empirical Potentials
hydrogen atoms
total
inward
potential I,
kcal / mol
potential 11,
kcat/ mol
-4.03
-4.00
-80.51
-87.38
-87.10
-78.96
-52.12
-92.76
-143.96
-143.99
-148.73
-149.69
-1 48.24
-1 16.33
-4.19
-4.17
-83.20
-90. IO
-89.84
-80.47
-49.62
-95.43
-146.04
-146.08
-150.71
-151.76
-1 50.95
-1 08.8 1
~
20
26
34
36
36
36
36
38
58"
58'
60
60
60
60
0
0
0
0
I
12
24
0
0
0
0
1
12
24
Nearest neighbors connected by a bond shared by two hexagons.
'Nearest neighbors connected by a bond shared by a hexagon and a
pentagon.
h
ys
h
XP
Figure 4. Tetrahedral CmH3,with one symmetry-equivalentset of 24
hydrogen atoms pointing outward and one set of 12 symmetry-equivalent
hydrogen atoms, which are darkened, pointing inward.
equivalent atoms of C60H36with all the hydrogen atoms pointing
outward obtained by using potential I of ref 9. These coordinates
refer to a coordinate system in which the 3-fold axes of the tetrahedral group point in the (l,l,l), (1,-1,-l), etc., directions. In
contrast to the case for CmH60, comparison of our empirical
potential energies for CWH3,with the energies that would result
from assuming perfect planar bonding for the sp2 carbon atoms
and perfect tetrahedral bonding for the sp' carbon atoms suggest
little or no strain energy for this less saturated species. At this
empirical potential geometry, the LDF electronic structure is closed
shell, 6a,6eUl la,l le,22t,27t,.11 The CmH36LUMO is 7a,, and
like all BF derivatives studied herein the HOMO-LUMO excitation of 3.75 eV is electric dipole forbidden. Relative to BF and
separated hydrogen atoms the LDF average C-H bond energy
at this geometry is 72 kcal/mol. This average bond energy is larger
than that of CmHmand would only increase upon LDF geometry
optimization. Nevertheless, it is significantly closer to that of
CmHm than to that of CH4, and that result would be unlikely to
change upon optimization.
Using our empirical potentials, we have shown that putting any
of the 36 hydrogen atoms inside the BF cage raises the energy."
These results are included in Table IX. It costs a quite small
amount of energy, less than 0.3 eV, to point one of the hydrogen
atoms of C60H36inward. Pointing 12 hydrogen atoms attached
to carbon atoms that are second neighbors to the double-bonded
carbon atom pairs as shown in Figure 4. costs about 0.8 eV per
atom. This interesting molecule is shaped more like a cube than
the rather spherical isomer formed if all hydrogen atoms point
outward, which is shown in Figure 2. This cubic isomer should
be rather stable if synthesized, because 0.8 eV is much smaller
than the average C-H bond strength.
TABLE X Energy Gained by Adding the Two Hydrogen Atom To
Get C a m Where Available, and the Average C-H Bond Strengtb
(Calculated as in Table VII) Using Empirical Potential I
no. of
H atoms
2
34
36
38
58
60
E(CmHA -
Eb,
E ( C m H A eV
kcal / mol
-4.03
46.5
54.6
56.0
56.3
57.3
57.2
-6.86
-5.38
-4.73
5. Other BF Hydrides
Table IX contains further results concerning the energetics of
hydrogen addition and isomerization on the BF cage. These
calculations use both empirical potentials from ref 9. Potential
I generates bond distances more accurately and was used above,
while potential I1 generates stretching force constants more ac~ u r a t e l y .Potential
~
I gives -422.62 eV for the total energy of
Cm, which corresponds to a cohesive energy of 162.4 kcal/mol
per carbon atom. Potential I1 gives -419.21 eV for the total energy
of Cm, which corresponds to a cohesive energy of 161.1 kcal/mol
per carbon atom.
Table IX concerns relatively small changes in the hydrogen
atom number and bonding for stoichiometries near c m , CmH36,
and CmHbO. If the addition of two hydrogen atoms to Cm or
subtraction of two hydrogen atoms from CaHa is considered, then
the two processes can take place at any two of the 60 carbon atoms,
leading to 32 distinct molecules in both cases. We have only
considered altering the bonding of neighboring carbon atoms,
which leads to two cases. The first is indicated by a superscript
a in the table and corresponds to nearest neighbors connected by
a bond shared by two hexagons. The second is indicated by a
superscript b in the table and corresponds to nearest neighbors
connected by a bond shared between a pentagon and a hexagon.
The energy differences between these two cases is quite small,
being less than 0.04 eV using either potential.
We have also considered bonding hydrogen atoms to the inside
of the carbon cage. This widens the scope of possible molecules
greatly. To limit the scope of such studies, we have only considered
pointing 1, 12, and 24 atoms inward. For one atom pointing
inward we have included in Table IX only the results for the mast
stable isomer. It is energetically favorable to point one hydrogen
atom inward for CWHmbut not for CmH36. For more than one
atom pointing inward, we have chosen to preserve Th symmetry.
These molecules are significantly less stable than the corresponding
molecules with all hydrogen atoms pointing outward except for
the case of 12 atoms pointing inward in CmHm.
Table X considers only molecules with all hydrogen atoms
pointing outward. For the most stable isomer it lists the average
C-H bond energy and, for the calculations that have been done,
the energy gained by adding the last two hydrogen atoms. By
a substantial margin, adding hydrogen atoms to complete the
C60H36structure is energetically more favorable than going to
higher levels of saturation.
6. Conclusions
We have optimized the LDF Ih geometries of CpolCmHm, and
C60Fm. These are all closed-shell molecules, with substantial
HOMO-LUMO one-electron energy gaps, although the C-H and
C-F average bond energies are reduced by 44% and 16% from
those of methane and tetrafluoromethane. LDF bond energy
considerations and empirical potential strain energy considerations
suggest that a fully saturated CwH, molecule is somewhat less
stable than C60H36. Empirical potentials suggest that CmHm
would prefer to have at least one hydrogen atom point inward.
For the case of C60H36bonding one hydrogen atom inward is
almost neutral energetically. These results sugest that if appropriate chemical methods are developed, perhaps based on osmylation,42 adding substituents inside and outside BF could lead to
(42) Hawkins, J. M.; Lewis, T. A.; Loren, S.D.; Meyer, A,; Heath, J. R.;
Saykally, R. J. J . Org. Chem. 1990, 55, 6250.
5768
J . Phys. Chem. 1991, 95, 5768-5773
an incredibly rich set of new molecules of varying shape. Putting
fluorine atoms inside the cage could also dramatically alter the
response properties of C,Fm.
Acknowledgment. These calculations were made using a grant
of CRAY X-MP computer time from the US.Naval Research
Laboratory. This work was supported in part by the Office of
Naval Research (ONR) through the Naval Research Laboratory
and through the ONR Chemistry and Materials Science Divisions
Contracts NOOO14-91-WX-24150 and "14-91-WX-24154.
Structures and Vibrational Frequencies of Cbo,C,o, and C8,
Krishnan Raghavachari*
AT& T Bell Laboratories, Murray Hill, New Jersey 07974
and Celeste McMichael Roblfing*
Sandia National Laboratories. Livermore, California 94551 -0969 (Received: March 28, 1991)
The structures and vibrational frequencies of C, C7&and CMhave been determined with the semiempirical MNDO method.
Two highly stable minimum-energy structures have been identified for CE4. Of the two, a D6h form, which contains five
distinct carbon atoms, is found to be slightly more stable than a Td form with four distinct carbons. Both isomers are found
to be significantly more stable than either Csoor C7,,, consistent with the presence of a greater proportion of six-membered
rings in CE4.An additional structure containing seven-membered rings has also been considered for CM,but is significantly
less stable than structures containly only five- and six-membered rings. Finally, the geometries of C, C70, and the two
stable forms of CE4have been completely optimized by using ab initio Hartree-Fock theory with the double-f 3-21G basis
set.
Introduction
The recent macroscopic preparation' of spheroidal carbon
clusters* has stimulated a wide variety of experimental and theoretical studies on these novel systems. C, and CT0have now
been isolated by several different experimental groups and their
properties have been m e a ~ u r e d . ~Many
' ~ theoretical studies have
been reported on C, and a few on C70.'8-36Among the other
E. Nature 1985, 318, 162.
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0022-3654/91/2095-5768$02.50/0
TABLE I: Bond Leugtbs (A) rad Bond Aagles (deg) for Cm
param'
MNDO
HF/3-21G
CI-CI
1.473
1.452
c1-Cl
1.402
I.370
cl-c3
1.469
1.447
c3-c3
1.389
1.356
c3-c4
1.478
1.458
C4C4
1.442
1.414
c4-cs
1.430
1.403
cs-cs
1.484
1.475
cI-cI-cI
108.0
108.0
c,-cI-cl
119.7
119.7
c&-C3
120.3
120.2
c3c2-c3
106.9
106.9
c1c3-c3
120.1
120.1
CZ-C3<4
108.5
108.3
c3-c3-c4
119.8
119.9
c3-c4-c4
108.1
108.2
c3-c4-cs
121.4
121.5
c4-c4-c5
121.6
121.3
c4-cs-c5
118.7
118.4
c4-c5-c4
115.6
116.3
For the numbering system, see Figure 3.
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0 1991 American Chemical Society