  This work aims at exploring the potential energy surfaces of
C$_{24}$H$_n${n=0,6,12,18,24} up to 20-25\,eV using the genetic algorithm in
combination with the density functional based tight binding (DFTB) potential.
The structural diversity of the non fragmented structures was analysed using
order parameters which were chosen as the number of 5 or 6 carbon rings and the
asphericity constant $\beta$. The most abundant and lowest energy population
was found to correspond to a flakes population, constituted of isomers of
variable shapes possessing a large number of 5 or 6-carbon rings. This
population is characterized by a larger number of spherical isomers when
$n_H/n_C$ increases. Simultaneously, the fraction of the pretzels population
constituted of spherical isomers possessing fewer 5 or 6 carbon ring cycles
increases. For all hydrogenation rates, the fraction of cages population
remains extremely minor while the branched population is the highest energy
population for all $n_H/n_C$ ratios. For all C$_{24}$H$_n${n=0,6,12,18,24}
clusters, a detailed study of the evolution of the carbon ring size
distribution as a function of energy clearly shows that the stability is
correlated to the number of 6-carbon rings. A similar study for hybridization
$sp^n$ (n=1-3) shows that the number of $sp^1$ carbon atoms increases with
energy while globally the number of $sp^3$ carbon atoms increases with
$n_H/n_C$. The average values of the ionization potentials of all populations
were found to decrease when $n_H/n_C$ increases, ranging from 7.9\,eV down to
6.4\,eV that we correlated to carbon atoms hybridization $sp^n$ (n=1-3). These
results are of astrophysical interest as they should be taken into account in
astrophysical models especially regarding the role of carbonaceous species in
the gas ionization.
