Bond lengths in the beaded model

In the beaded model of fullerenes that contains only one type of beads, we don't really need to know the relationship between bond length and size of beads to construct a faithful model. But, in molecules such as EMACs, there are at least two types of bond lengths, central metal-metal bonds and ligand bonds, and two types of hybrizations, sp² and d²sp³. It is important to get the correct relation between bond lengths and diameter of beads, in order to get the right pitch angle of EMACs.

In the sp² hybridized carbon atom, we use three spherical beads to mimic carbon-carbon bonds as shown in the left of the following figure. These spherical beads are in contact with each other and the atom is located at the center of three spheres. The relation between bond length and diameter of bead is given by r=1.155a. Similarly, the metal-metal bond lengths for the d2sp3 hybridized atom, is given by r=1.414a.



In the previous post, I gave a simple formula for the pitch angle. But the a and b in the formula should stand for the central metal-metal bond length and the ligand bond length, respectively.