  Since fullerene formation occurs under conditions where direct observation of
atomic-scale reactions is not possible, modeling is the only way to reveal
atomistic mechanisms which can lead to selection of abundant fullerene isomers
(like C60-Ih). In the present paper we review the results obtained for
different atomistic mechanisms by various modeling techniques. Although it
seems that atomic-scale processes related to odd fullerenes (such as growth by
consecutive insertions of single carbon atoms and rearrangements of the sp$^2$
structure promoted by extra sp atoms) provide the main contribution to
selection of abundant isomers, at the moment there is no conclusive evidence in
favor of any particular atomistic mechanism. Thus, the following multiscale
modeling strategy to solve the mystery of the high yield of abundant fullerene
isomers is suggested. On the one hand, sets of reactions between fullerene
isomers can be described using theoretical graph techniques. On the other hand,
reaction schemes can be revealed by classical molecular dynamics simulations
with subsequent refinement of the activation barriers by ab initio
calculations. Based on the reaction sets with the reaction probabilities
derived in this way, the different atomistic mechanisms of abundant fullerene
isomer selection can be compared by kinetic models.
