E_{f}$ and the change in volume $\Delta V$ is B$_{52}$ with two $4c$-site atoms in the out-of-plane configuration. In the second column ’-2b’ indicates a removal of a 2b atom. $2\times 4c$ should be read as two atoms at $4c$ sites. All the data are obtained by GGA, while LDA results are added for $\Delta E_{f}$ in parentheses. []{data-label="tbl:HfofB50"}

The energy ordering in $\Delta E_{f}$ can be understood by inspecting the bonding environment of these IS as shown in Fig. \[fig:POS\]. As seen, the $2b$ site is most perfectly tetrahedrally coordinated, which is the best configuration for a $sp^{3}$-type covalent bond. The short bond length of 1.63 Å also supports its strong covalent character. Hence, it is reasonable to obtain the perfect occupancy for this site. The $2a$ site also has perfect tetrahedral coordination, too. But, the nearest neighbor distance of 2.15 Å is too long. Furthermore, none of the atoms of the neighboring B$_{12}$ icosahedron are oriented in the direction linking the $2a$ site to B$_{12}$. Therefore, the bonding of $2a$-site atoms is weak. The site $4c$ is also close to tetrahedral coordination with a bond length of 1.79 Å, however the bond angles are largely distorted in the 