First-principles studies on the intrinsic stability of the magicFe13O8cluster

Complementary to our previous study [Wang et al., Phys. Rev. B $59,$ 12 672 (1999)], the intrinsic stability of the magic ${\mathrm{Fe}}_{13}{\mathrm{O}}_8$ cluster is confirmed by the calculations on the vibrational properties. For all the vibration modes, the frequency is positive, and therefore the structure is dynamically stable. $A_{1g},$ $B_{1g},$ and $B_{2g}$ modes are Raman active, $A_{2g},$ $A_{2u},$ and $E_u$ modes are infrared active, while the $E_g$ mode is active both in Raman and in infrared spectroscopy. The intrinsic stability combined with the relative stability make this cluster to be magic.