  The mechanism of ferromagnetism stability in sandwich clusters
V$_n$(C$_6$H$_6$)$_{n+1}$ has been studied by first-principles calculation and
model analysis. It is found that each of the three types of bonds between V and
benzene (Bz) plays different roles. V 3d$_{z^2}$ orbital, extending along the
molecular axis, is weakly hybridized with Bz's HOMO-1 orbital to form the
$\sigma$-bond. It is quite localized and singly occupied, which contributes
1$\mu_B$ to the magnetic moment but little to the magnetic coupling of
neighboring V magnetic moments. The in-plane d$_{x^2-y^2}$, d$_{xy}$ orbitals
are hybridized with the LUMO of Bz and constitute the $\delta$-bond. This
hybridization is medium and crucial to the magnetic coupling though the
$\delta$ states have no net contribution to the total magnetic moment.
d$_{xz}$, d$_{yz}$ and HOMO of Bz form a quite strong $\pi$-bond to hold the
molecular structure but they are inactive in magnetism because their energy
levels are far away from the Fermi level. Based on the results of
first-principles calculation, we point out that the ferromagnetism stability is
closely related with the mechanism proposed by Kanamori and Terakura [J.
Kanamori and K. Terakura, J. Phys. Soc. Jpn. 70, 1433 (2001)]. However, the
presence of edge Bz's in the cluster introduces an important modification. A
simple model is constructed to explain the essence of the physical picture.
