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ARTICLE
Figure 14 shows the calculated total density of states (TDOS)
and Fermi surface for benzene phase III at 190 GPa, a pressure at
which it is metallic yet, at 0 K, still stable with respect to
rearrangement to a saturated polymer. The underlying band
structure is shown in the SI. Note the TDOS shows a character-
istic free-electron-like shape over a wide energy range. As we
noted earlier, above P = 210 GPa, phase III undergoes a pressure-
induced chemical transformation to polymer II. Once a saturated
molecule is achieved, the band gap opens up again
—polymer II
and the graphanes have large band gaps. Based on the band gaps
in phase III (Figure 13), we suggest that, upon increasing
pressure, benzene will turn from transparent to colored to black
and then colorless again as it polymerizes.
Could there exist a
“Kekule metal” in benzene in the (narrow)
pressure range where it is calculated to be metallic? The spec-
ulative notion here is of a Kekul
e state of the π systems for
each molecule, delocalized yet contained in a six-membered
carbon ring, that might then be phase-linked via the bridging
hydrogens, a Bloch-like state being the consequence. If so, then
this could be a realization of the resonant valence bond state
proposed by P. W. Anderson as a possible basis for high-
temperature superconductivity
59
(pairing via Kekul
e states in
a
“Kekule metal”, one might optimistically say).
Another way to follow the approach to metallicity of a material
is via its dielectric function. The probability of photon absorption
is also directly related to the imaginary part of the optical
dielectric function
ε(ω). In the SI we show the computed
dielectric functions of phase III at 0, 190, 200, and 210 GPa.
These calculations indicate
—through Drude behavior at low
frequencies
—that the structure at 0 GPa (molecular crystal) and
that at 210 GPa (CH saturated polymer) is an insulator, while at
190 and 200 GPa, the dielectric response is characteristic of a
metal, as we already concluded from the band gaps.
Figure 13.
Relationship among band gap (in eV), pressure, and Wigner
À
Seitz radius (r
s
).
Figure 14.
Calculated TDOS and Fermi surface of phase III at 190 GPa.
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ARTICLE
Some Model Structures.
Some lower-dimensional models
are also useful in getting an understanding of the factors
governing benzene metallization. Consider first
π stacking, as
shown in a one-dimensional array in Figure 15 (top). In the
process of compressing, the C
ÀC and CÀH distances in
benzene are frozen. Only the distance between benzene mole-
cules is allowed to vary. This one-dimensional array becomes
metallic at R < 2.5 Å, due to overlap of
π and π* bands. The
situation is similar to a previously studied stack of ethylenes.
60
Metallization can also be attained by
σ overlap of CÀH
bonding and antibonding orbitals, if the benzenes are so disposed
as to emphasize
σ interactions. The one-dimensional array
shown in the bottom of Figure 15 is one way to do this. This
becomes a metal at a center-to-center ring distance of 4.5 Å, at
which point the shortest H 3 3 3 H contact L = 1.25 Å.
We also calculated a two-dimensional array shown in Figure 16,
designed to create equalized H 3 3 3 H contact, L, in a net. When
this is compressed, it becomes metallic when the closest H---H
contact L = 1.35 Å.
We had an idea that in the region of metallization, there might
be some unusual dynamics, separating the scale of mobility of C
and H atoms. To be speci
fic, we wondered if the carbon rings
might move independently of the hydrogens or, alternatively, if
the hydrogens would move with relative ease if the carbon atom
positions were frozen. This was probed at geometries corre-
sponding to L values above and below the metallization threshold
(see SI). However, these motions proved very costly in energy.
Calculated Bulk and Shear Moduli.
The stiffness of materials
may be measured in a number of ways. Young
’s modulus (E)
describes the material
’s response to linear strain (like pulling on
the ends of a wire). The bulk modulus (K) describes the response
to uniform pressure, and the shear modulus (G) describes the
material
’s response
The calculated bulk modulus, shear modulus, and Young
’s
modulus of phase III at a series of pressures are listed in Table 1.
The Reuss de
finition is utilized to calculate the bulk modulus and
shear modulus.
61
The bulk and Young
’s moduli rise with
increasing pressure. At 150 GPa, the bulk modulus of phase III
is 450 GPa, which is lower than the value of diamond at that
pressure (computed as 850 GPa; as a calibration we calculated
the P = 1 atm bulk modulus of diamond as 450 GPa, experimental
value 443 GPa
62
). At 210 GPa, the computed bulk modulus (745
GPa) of polymer II is also lower than the theoretical value of
diamond at this pressure (1100 GPa). Therefore, it should be
possible to compress benzene in diamond anvil cells.
Primary Local Actions in Benzene under Pressure.
Theo-
reticians, attracted by symmetry, would like to see all solid-state
transformations happen in a concerted manner, preserving as
much symmetry as possible at every place in a macroscopic
crystal. In the real world, given unavoidable pressure and
temperature inhomogeneities, if there is a driving force, chemical
and physical transformations are likely to be nucleated, and
hence to occur locally. Given a multitude of such local reaction
nuclei, the result will likely be an amorphous polymer which is
difficult to handle theoretically.
We have seen very clearly that ordered all-saturated, four-
coordinate carbon polymers (graphanes and the related needles
and sheets) are favored thermodynamically over delocalized,
conjugated, three-coordinate polymers (graphite) and molecular
structures (benzene crystals) at all pressures. A simple argument
showed that converting a double bond to two single bonds is
favored. As an approach to the nucleated, local amorphous
polymerization problem, we have looked at the dimerization of
benzene in some detail. We need to mention here the prior work
of Engelke on this subject.
63,64
To jump-start the process of dimerization, we brought two
benzene molecules to an uncomfortably close contact and then
let a molecular (zero-dimensional) geometry optimization pro-
gram complete the optimization. The molecules either moved
apart or dimerized. This procedure generated many, but not all,
of the known dimers of benzene, as well as some dimers that were
really unexpected. We followed this brute-force process with a
detailed exploration, including missed known structures and
exploring the complete set of stereochemical possibilities for
(C
6
H
6
)
2
molecules.
A detailed account of this section of the work will be published
elsewhere,
65
as it is of substantive chemical interest
—we generated
Figure 15.
(Top) Stack of benzenes, a model for
π interaction.
(Bottom) One-dimensional benzene array, a model for
σ interaction.
Figure 16.
Model of 2D benzene.
Table 1. Calculated Bulk Modulus, Shear Modulus, and
Young
’s Modulus (All in GPa) for Phase III at a Series of
Pressures
pressure (GPa)
phase III
50
100
150
190
210
bulk modulus
182
375
450
523
745
shear modulus
68
163
146
99
338
Young
’s modulus
X
232
422
608
629
1052
Y
126
397
283
346
1081
Z
100
335
338
426
652