原子雕刻技术：从晶体到团簇

有一种晶体Al₁₂Mo，立方体晶系（a=7.5815），空间群为204，Mo原子占2a位置，Al原子占24g。

在晶体空间，如果以Mo为参考原子，环看四周，就会看到12个Al原子围绕着Mo原子，这12个Al原子距离Mo原子的距离都是2.724埃。

围绕Mo的12个Al原子在空间中是怎么分布的呢？

这12个Al原子组成一个正20面体，每个面由3条边组成，是等边三角形。

在正20面体的表面上，12个Al原子可以连接出30条键，键长为2.88埃。

如果用刀（特殊刀）把这13个原子（1Mo＋12Al）雕刻出来，就是1个Al₁₂Mo团簇。距离中心Mo原子4.7109埃的较外层空间，分布着24个Al原子，这24个Al原子组成了1个14面体【6（4）＋8（6）】，6个正4边形和8个正6变形；24个Al原子可组成36条棱边（键），键长也为2.88埃。如果用刀（特殊刀）把这37（＝13＋24）个原子雕刻出来，就是一个团簇（Al₃₆Mo）。

围绕Mo原子的Al原子及其坐标数据（计算机输出格式）：

"2a-Mo"," is a reference atom"

"      Cartesian =  ",-3.7908,-3.7908,-3.7908

"      SP coordinates=",0,0,0

"      d of neighbors from the reference atom =",2.724

"      The number of neighbor atoms"," N=",12

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-6.1259,-3.7908,-5.1933

"      SP coordinates=  ",-.308,0,-.185

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-6.1259,-3.7908,-2.3882

"      SP coordinates=  ",-.308,0,.185

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-5.1933,-6.1259,-3.7908

"      SP coordinates=  ",-.185,-.308,0

"     Neighbor atom:","24g-Al"

"      Cartesian =  ",-5.1933,-1.4556,-3.7908

"      SP coordinates=  ",-.185,.308,0

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-3.7908,-5.1933,-6.1259

"      SP coordinates=  ",0,-.185,-.308

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-3.7908,-5.1933,-1.4556

"      SP coordinates=  ",0,-.185,.308

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-3.7908,-2.3882,-6.1259

"      SP coordinates=  ",0,.185,-.308

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-3.7908,-2.3882,-1.4556

"      SP coordinates=  ",0,.185,.308

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-2.3882,-6.1259,-3.7908

"      SP coordinates=  ",.185,-.308,0

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-2.3882,-1.4556,-3.7908

"      SP coordinates=  ",.185,.308,0

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-1.4556,-3.7908,-5.1933

"      SP coordinates=  ",.308,0,-.185

"      Neighbor atom:","24g-Al"

"      Cartesian =  ",-1.4556,-3.7908,-2.3882

"      SP coordinates=  ",.308,0,.185

12个Al原子组成1个正20面体，Al原子位于顶点（Vertices），30条键（Edges）的数据如下：

1,"  The distance of","24g-Al","and","24g-Al","d==",2.81

"       Coordinates of these two atoms:"

"       Cartesian=",-6.126,-3.791,-5.193,"---",-6.126,-3.791,-2.388

"       SP coordinates=",-.308,0,-.185,"---",-.308,0,.185

2,"  The distance of","24g-Al","and","24g-Al","d==",2.81

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-6.126,-3.791,"---",-2.388,-6.126,-3.791

"       SP coordinates=",-.185,-.308,0,"---",.185,-.308,0

3,"  The distance of","24g-Al","and","24g-Al","d==",2.81

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-1.456,-3.791,"---",-2.388,-1.456,-3.791

"       SP coordinates=",-.185,.308,0,"---",.185,.308,0

4,"  The distance of","24g-Al","and","24g-Al","d==",2.81

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-6.126,"---",-3.791,-2.388,-6.126

"       SP coordinates=",0,-.185,-.308,"---",0,.185,-.308

5,"  The distance of","24g-Al","and","24g-Al","d==",2.81

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-1.456,"---",-3.791,-2.388,-1.456

"       SP coordinates=",0,-.185,.308,"---",0,.185,.308

6,"  The distance of","24g-Al","and","24g-Al","d==",2.81

"       Coordinates of these two atoms:"

"       Cartesian=",-1.456,-3.791,-5.193,"---",-1.456,-3.791,-2.388

"       SP coordinates=",.308,0,-.185,"---",.308,0,.185

7,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-6.126,-3.791,-2.388,"---",-5.193,-1.456,-3.791

"       SP coordinates=",-.308,0,.185,"---",-.185,.308,0

8,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-6.126,-3.791,-2.388,"---",-3.791,-5.193,-1.456

"       SP coordinates=",-.308,0,.185,"---",0,-.185,.308

9,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-6.126,-3.791,-2.388,"---",-3.791,-2.388,-1.456

"       SP coordinates=",-.308,0,.185,"---",0,.185,.308

10,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-6.126,-3.791,"---",-3.791,-5.193,-6.126

"       SP coordinates=",-.185,-.308,0,"---",0,-.185,-.308

11,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-6.126,-3.791,"---",-3.791,-5.193,-1.456

"       SP coordinates=",-.185,-.308,0,"---",0,-.185,.308

12,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-6.126,-3.791,"---",-2.388,-6.126,-3.791

"       SP coordinates=",-.185,-.308,0,"---",.185,-.308,0

13,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-1.456,-3.791,"---",-3.791,-2.388,-6.126

"       SP coordinates=",-.185,.308,0,"---",0,.185,-.308

14,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-1.456,-3.791,"---",-3.791,-2.388,-1.456

"       SP coordinates=",-.185,.308,0,"---",0,.185,.308

15,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-5.193,-1.456,-3.791,"---",-2.388,-1.456,-3.791

"       SP coordinates=",-.185,.308,0,"---",.185,.308,0

16,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-6.126,"---",-3.791,-2.388,-6.126

"       SP coordinates=",0,-.185,-.308,"---",0,.185,-.308

17,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-6.126,"---",-2.388,-6.126,-3.791

"       SP coordinates=",0,-.185,-.308,"---",.185,-.308,0

18,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-6.126,"---",-1.456,-3.791,-5.193

"       SP coordinates=",0,-.185,-.308,"---",.308,0,-.185

19,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-1.456,"---",-3.791,-2.388,-1.456

"       SP coordinates=",0,-.185,.308,"---",0,.185,.308

20,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-1.456,"---",-2.388,-6.126,-3.791

"       SP coordinates=",0,-.185,.308,"---",.185,-.308,0

21,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-5.193,-1.456,"---",-1.456,-3.791,-2.388

"       SP coordinates=",0,-.185,.308,"---",.308,0,.185

22,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-2.388,-6.126,"---",-2.388,-1.456,-3.791

"       SP coordinates=",0,.185,-.308,"---",.185,.308,0

23,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-2.388,-6.126,"---",-1.456,-3.791,-5.193

"       SP coordinates=",0,.185,-.308,"---",.308,0,-.185

24,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-2.388,-1.456,"---",-2.388,-1.456,-3.791

"       SP coordinates=",0,.185,.308,"---",.185,.308,0

25,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-3.791,-2.388,-1.456,"---",-1.456,-3.791,-2.388

"       SP coordinates=",0,.185,.308,"---",.308,0,.185

26,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-2.388,-6.126,-3.791,"---",-1.456,-3.791,-5.193

"       SP coordinates=",.185,-.308,0,"---",.308,0,-.185

27,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-2.388,-6.126,-3.791,"---",-1.456,-3.791,-2.388

"       SP coordinates=",.185,-.308,0,"---",.308,0,.185

28,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-2.388,-1.456,-3.791,"---",-1.456,-3.791,-5.193

"       SP coordinates=",.185,.308,0,"---",.308,0,-.185

29,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-2.388,-1.456,-3.791,"---",-1.456,-3.791,-2.388

"       SP coordinates=",.185,.308,0,"---",.308,0,.185

30,"  The distance of","24g-Al","and","24g-Al","d==",2.88

"       Coordinates of these two atoms:"

"       Cartesian=",-1.456,-3.791,-5.193,"---",-1.456,-3.791,-2.388

"       SP coordinates=",.308,0,-.185,"---",.308,0,.185

注意：只有前6条Edges的长度是2.81，略不同于2.88，这可能是由于原子坐标（24g：y=0.1854，z=0.3083）的实验误差引起的。