The transition metal center $\mathbf{A}$ located in the d-block or ds-block can form a complex structure of the form $\left[\mathbf{AB}_6\mathbf{C}_{12}\right]$, where the order of its point group is $24$, and there are no $C_4$ or $S_4$ axes present. Here, $\mathbf{A, B, C}$ are atoms and can be treated as points. All bond angles in this complex are non-acute, and the distances between symmetry-independent $\mathbf{C-C}$ atom pairs are pairwise distinct. Determine the number of faces of the convex polyhedron formed by the outermost atoms as vertices in this complex molecule.
位于d区或ds区的过渡金属中心$\mathbf{A}$可形成形如$\left[\mathbf{AB}_6\mathbf{C}_{12}\right]$的配合物结构，其点群阶数是$24$，且不存在任何$C_4$或$S_4$轴，其中$\mathbf{A,B,C}$是原子，可视为点。该配合物的所有键角中不存在锐角，且在对称性上无关的$\mathbf{C-C}$原子对的距离两两不同。求该配合物分子中，最外层原子为顶点构成的凸多面体的面数。
2
3
4
6
8
9
10
12
14
15
18
20
24
30
45
48
55
60
66
72
2
3
4
6
8
9
10
12
14
15
18
20
24
30
45
48
55
60
66
72
The point groups of order 24 include  

$C_{24}, C_{12\mathrm{v}}, C_{12\mathrm{h}}, D_{12}, D_{6\mathrm{h}}, D_{6\mathrm{d}}, S_{24}, T_\mathrm{h}, T_\mathrm{d}, O$  

and others. Among these, only the two point groups $D_{6\mathrm{h}}$ and $T_\mathrm{h}$ satisfy the condition of having no $C_4$ or $S_4$ axis.  

<Checkpoint>[Only the point groups $D_{6\mathrm{h}}$ and $T_\mathrm{h}$ satisfy the condition](1)</Checkpoint>  

If the point group is $D_{6\mathrm{h}}$, then with $\mathbf{A}$ at the center, six $\mathbf{B}$ atoms form a regular planar hexagon, with each $\mathbf{B}$ atom connected to two $\mathbf{C}$ atoms. The $\mathbf{C}$ atoms are symmetrically distributed above and below the plane, forming a convex polyhedron that is a hexagonal prism.  

<Checkpoint>[If the point group is $D_{6h}$, its structure is a hexagonal prism](1)</Checkpoint>  

<MultiModal>![The molecular structure of $D_{6h}$, featuring 1 silver sphere at the center, 6 blue spheres connected to the silver sphere forming a planar hexagon, with each blue sphere bonded to two red spheres symmetrically distributed on both sides of the plane](/media/uploads/cb458e8c-f2b8-4abf-9ce9-5d64574a975a.png)</MultiModal>  

However, since d-block and ds-block elements lack available f orbitals for bonding, they cannot form a regular planar hexagon coordination. Therefore, the point group of this complex is not $D_{6\mathrm{h}}$.  

<Checkpoint>[d-block and ds-block elements lack available f orbitals for bonding](1)</Checkpoint>  

On the other hand, if the point group is $T_\mathrm{h}$, then with $\mathbf{A}$ at the center, six $\mathbf{B}$ atoms form a regular octahedron, with each $\mathbf{B}$ atom connected to two $\mathbf{C}$ atoms. The $\mathbf{BC}_2$ units lie within the $\mathbf{AB}_4$ cross-section of the octahedron, and the planes of adjacent $\mathbf{BC}_2$ units are mutually perpendicular. The outermost 12 $\mathbf{C}$ atoms form the shape of a triangular icosahedron.  

<Checkpoint>[If the point group is $T_h$, its structure is a triangular icosahedron](2)</Checkpoint>  

<MultiModal>![The molecular structure of $T_h$, featuring 1 silver sphere at the center, 6 blue spheres forming a regular octahedron with red spheres, each blue sphere bonded to 2 red spheres. The red-blue-red units lie within the plane formed by 1 silver sphere and 4 blue spheres, with adjacent blue spheres having their red-blue-red planes mutually perpendicular](/media/uploads/acc77075-4dc6-444d-83df-b4d08ca618de.png)</MultiModal>  

The triangular icosahedron has 20 triangular faces.  

<Checkpoint>[There are 20 triangular faces](1)</Checkpoint>
阶数为24的点群有

$C_{24},C_{12\mathrm{v}},C_{12\mathrm{h}},D_{12},D_{6\mathrm{h}},D_{6\mathrm{d}},S_{24},T_\mathrm{h},T_\mathrm{d},O$

等。其中，只有$D_{6\mathrm{h}}$和$T_\mathrm{h}$两个点群满足不存在$C_4$或$S_4$轴的条件。

<Checkpoint>[满足条件的点群只有$D_{6\mathrm{h}}$和$T_\mathrm{h}$](1)</Checkpoint>

若为$D_{6\mathrm{h}}$点群，则以$\mathbf{A}$为中心，6个$\mathbf{B}$原子构成平面正六边形，每个$\mathbf{B}$原子连接两个$\mathbf{C}$原子，且$\mathbf{C}$原子对称地分布在平面的上下两侧，构成的凸多面体为六棱柱。

<Checkpoint>[若为$D_{6h}$点群，其结构为六棱柱](1)</Checkpoint>

<MultiModal>![D_{6h}的分子结构，其中1个银色球位于中心，6个蓝色球与银色球相连构成平面六边形，每个蓝色球连接两个红色球，红色球对称得分布在平面两侧](/media/uploads/cb458e8c-f2b8-4abf-9ce9-5d64574a975a.png)</MultiModal>

但是由于d区和ds区元素没有可用于成键的f轨道，无法形成平面正六边形配位，故该配合物的点群不是$D_{6\mathrm{h}}$。

<Checkpoint>[d区和ds区元素没有可用于成键的f轨道](1)</Checkpoint>

另一方面，若为$T_\mathrm{h}$点群，则以$\mathbf{A}$为中心，6个$\mathbf{B}$构成正八面体，每个$\mathbf{B}$上连接两个$\mathbf{C}$，且$\mathbf{BC}_2$位于八面体的截面$\mathbf{AB}_4$内，邻位的$\mathbf{BC}_2$平面相互垂直。最外层的12个$\mathbf{C}$原子构成三角二十面体的形状。

<Checkpoint>[若为$T_h$点群，其结构为三角二十面体](2)</Checkpoint>

<MultiModal>![T_h的分子结构，其中1个银色球位于中心，6个蓝色球与红色球相连构成正八面体，每个蓝色球上连接2个红色球，相连接的红-蓝-红组合位于由1个银色球和4个蓝色球构成的平面内，邻位的蓝色球所在的红-蓝-红组合的平面相互垂直](/media/uploads/acc77075-4dc6-444d-83df-b4d08ca618de.png)</MultiModal>

三角二十面体上有20个三角形面。

<Checkpoint>[有20个三角形面](1)</Checkpoint>