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ARTICLE
on this topic which goes back to Herzfeld and Goeppert Mayer.
24
A
correct prediction of a phase transition requires an accurate calculation of
the free energy in the two phases, though stability criteria are still useful
and true transition pressures are usually not too distant from the calculated
values.
”
’ RESULTS AND DISCUSSION
Enthalpy of the Benzene Phases as a Function of Pressure.
In the gas phase benzene is a D
6h
molecule with C
ÀC = 1.40 Å
and C
ÀH = 1.09 Å. The melting point of benzene is 279 K; the
boiling point is 353 K. At normal pressures, one expects a variety
of polymorphs to be of roughly equal energy, a typical situation
for molecular crystals. Indeed, three phases are well-character-
ized experimentally: phases I (Pbca), II (P4
3
2
1
2), and III (P2
1
/c).
Our theoretical optimization reproduces the structures of these
phases well as far as C
ÀC and CÀH distances go, but less well
for the unit cell parameters. The details are in the Supporting
Information (SI). The calculated C
ÀC and CÀH distances
match the experimental ones. The unit cell parameters are within
5% of experiment.
Should we have done better or worse on these? The cell
parameters are determined essentially by dispersion forces and
electrostatic interactions (quadrupole
Àquadrupole being the
leading term for benzene). The DFT method we use is not good
at gauging dispersion forces, so the moderately poor agreement
with experimental force constants was not unexpected. We are
aware that it is possible to apply corrections to get a better
accounting of the van der Waals forces.
25
À27
We chose not do so
because our interest was not so much in the ambient pressure
structure as in that at elevated pressure, where such corrections
may not be necessary.
Figure 2.
Reoptimized benzene phases at 1 atm. Phases I (Pbca), II (P4
3
2
1
2), and III (P2
1
/c) were characterized experimentally by Thi
ery and Leger.
5
Phases I
0
(Cmca), III
0
(C
2
/c), IV (Pbam), and V (P2
1
) were theoretically predicted by Raiteri et al.
10
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J. Am. Chem. Soc. 2011, 133, 9023–9035
Journal of the American Chemical Society
ARTICLE
On elevating pressure, the error in the unit cell parameters is
reduced to within 1%, where structural parameters are available
(see SI). The lattice parameters of the higher pressure phases
remain a matter of controversy. Raiteri et al. predicted the
existence of four further phases, phases I
0
(Cmca), III
0
(C
2
/c),
IV (Pbam), and V (P2
1
).
10
We reoptimized the seven benzene
phases mentioned, since our methodology di
ffers somewhat
from that of Raiteri et al. Figure 2 shows these optimized benzene
phases at 1 atm. The lattice parameters and simulated powder
di
ffractions for these phases also at 1 atm are shown in the SI.
In addition, four more hypothetical benzene models were
considered: two known structures of C
6
F
6
and C
6
F
6
ÀC
6
H
6
and
two phases with parallel benzene molecules. These phases are not
competitive with the most stable phases III and V, as shown in the
enthalpy curves in the SI.
We proceeded to optimize the structures of these seven phases
over a wide range of pressures. Figure 3 shows the enthalpy
curves of the benzene phases as a function of pressure in the
ranges of 0
À20 and 0À80 GPa. At 1 atm, the enthalpy differences
among the seven benzene phases are tiny; these structures are
within 0.05 eV per benzene molecule of each other. Nothing
surprising here
—there clearly are many ways for anisotropic
dispersion forces to hold benzene molecules together in a molec-
ular crystal. The structures become di
fferentiated in enthalpy as
the pressure increases.
From Figure 3, we can see that, at least theoretically, the
phase I f phase II transition should occur at ∼4 GPa, phase II f
phase III at
∼7 GPa, and phase IIIf phase V at ∼40 GPa. Above
40 GPa, phase V comes into play and proceeds to become the
most stable structure at still higher pressures. Phases I
0
, III
0
, and
IV suggested by Raiteri et al. are not competitive with phases I, II,
III, and V by our calculations in the whole pressure range studied.
Experimentally, both the I
ÀII and IIÀIII phase transitions are
extremely sluggish, phase II existing between 1.4 and 4 GPa, and
phase III existing between 4 and 11 GPa. The current computa-
tion agrees reasonably well with the order of phase transitions
found in experimental studies in the low and moderate pressure
regime.
Might the benzene structures we calculate be rotational solids,
as solid H
2
,
28
solid CH
4
,
29
and the C
60
crystal structure
30
are? Let
us probe this with a numerical experiment. Consider phase III,
which has two benzene molecules in the unit. We rotate one of
the benzene molecules along its C
6
axis while another benzene
molecule is frozen. Other rotations encounter large barriers.
Figure 4 shows the energetic consequences of this rotational
motion as a function of pressure.
At 1 atm, this benzene rotation is free as it is in methane.
As expected, the rotational barrier is elevated with increasing
Figure 3.
Calculated relative enthalpy curves of benzene phases as a
function of pressure in the ranges of (a) 0
À20 and (b) 0À80 GPa. Here
phase I is taken as a reference.
Figure 4.
Energy barrier to benzene rotations in phase III structures at
various pressures. The
“perfect” structure at the corresponding pressure
is taken as a reference. Above: de
finition of the rotation studied.