Comparing the QMC results with those of the various DFT functionals,
several important trends in the relative performance of the different
functionals are highlighted. The overall quality of a functional for
,
, and
clusters
is best judged by the agreement with the DMC data for the overall
shapes of the relative energy data of figures 8.7
and 8.9. The best agreement is given by the PBE and
B3LYP functionals, with the LDA being slightly inferior and the BLYP
functional being worst. The tendency of the BLYP functional to favour
structures with lower average coordination number and the tendency of
the LDA to favour structures with higher average coordination numbers
is consistent with the results on
reported by
Grossman et al. [43]
The LDA, perhaps surprisingly, performs at least as well as the
gradient corrected functionals. The LDA appears to favour systems with
high average densities, therefore placing the rings anomalously high
in energy. The gradient corrected functionals, which normally would
be expected to improve the ordering do not consistently reduce
the differences between the DFT and DMC results; for
the rms deviation for the PBE functional is larger than for the LDA
and the overall spread in energies is increased from 3.04 to 3.41 eV.
The BLYP functional performs poorly, consistently placing the
structures of low average coordination substantially lower in energy
relative to DMC and other DFT values. The B3LYP functional, which
includes an exact exchange component, performs most consistently, and
also gives the same increase in stability of
as DMC
when spin-polarization is included. The PBE and B3LYP density
functionals give the best description of the relative energies of the
isomers, while the BLYP functional gives the poorest.
The DMC binding energies show a clear increase in stability with
cluster size for both ring and fullerene isomers. The cumulenic
ring is found to be more weakly bound per
atom than the adjacent polyacetylenic
and
rings, in agreement with early semi-empirical
calculations of
rings. [165]
However, the difference in binding energies of the fullerene and rings
(either cumulenic or polyacetylenic) increases nearly linearly with
cluster size.
The final test of the DMC predictions must lie with experiment. It
is clear that the actual abundances of different clusters depend
sensitively on the precise experimental conditions. The stability of
clusters against fragmentation, growth and other chemical reactions is
clearly a very complicated subject. One issue is that the clusters are
formed at temperatures of order 10 K and therefore the vibrational
contributions to the free energy may be significant. A relatively
simple picture emerges from computations of vibrational
properties [151,152,156]).
Fullerenes are relatively rigid and have smaller vibrational free
energies than rings, which have many low-lying vibrational modes.
Consequently, the ring isomers are favoured relative to fullerenes at high
temperatures. If thermodynamic stability alone were to determine
which cluster sizes were observed then only the larger
fullerenes would ever be observed, but in a recent experiment the
abundance of the
fullerene was found to be
greater than
.[143] There is more
evidence that thermodynamic stability to rearrangements of clusters of
a particular size are important in determining which isomers are
observed. For example, in the experimental study of
Ref. [143], fullerenes were mostly observed for
clusters containing more than about 30 carbon atoms, while for smaller
clusters mostly rings were formed. This behaviour closely matches the
critical size for fullerene stability predicted by our DMC calculations.
There have been several proposals that cluster solids could be
synthesized by surface deposition of
fullerenes. [162,163] Our DMC results support
these proposals: the
fullerene is predicted to be
stable, yet it is likely to readily polymerise due to its geometry and
spin-polarised ground-state. However, unless
sufficiently high abundances of the fullerene are produced, its high
reactivity could also hinder production of such solids, as the
fullerene is likely to react with other clusters of different size or
with any impurities present. The
fullerene, although
predicted to be stable, appears less likely to polymerise
due to its strained structure and less reactive ground state.