C60的原子坐标：旋转矩阵

C60的对称性，可以用120个完全不同的矩阵来表示。

120个矩阵两两（包括自己和自己）相乘，共有14400次。

因为这120个矩阵构成了一个点群（Ih），因此14400个结果（矩阵）完全跑不出原来120个矩阵的范围。

俺折腾了好长时间，终于让俺的计算机把120个矩阵完全计算出来了。

程序里那（几个）数组的维数必须至少是14400，再大点俺的计算机就不给干了（内存溢出）。

折腾这120个矩阵有什么用呢？

当然有用了，用这些矩阵操作化学键很容易求出60个原子的原子坐标。

C60点群转动操作表明，六角形/六角形组成的边和五角星/六角形组成的边是否相等没有什么要求。因此，这两种边（键长）可以在一定的范围内变化，是两个独立的变量。

C60的原子坐标，文献中好像已经有报道了。他们是如何计算的，俺不知道，反正俺的是转出来的。

但是各种各样的Z@C60的原子坐标，有时找不到现成的数据。

（Z表示周期表中的原子）

C60的三个参数：

键长d1=边1（五边形/六边形）

键长d2=边2（六边形/六边形）

C60球半径R（格点到球心的距离）

C60的这三个参数，只有2个是独立的。

只要已知其中的2个，就可以用点群转动出所有的原子坐标。

C60点群的乘法表（示意图）

如果你有Z@C60结构，而且已知两个参数（d1，d2，R），那么，你就用电子邮件把这两个参数发过来，我可以给你计算出原子坐标，很简单的事情，120个矩阵一旋转，60个坐标就出来了。

附录1 C60的原子坐标

"d5/6=",1.45

"d6/6=",1.39

R为各原子到球心的距离。

1.233,0,3.318,"    R=",3.5397

.381,1.173,3.318,"    R=",3.5398

.381,-1.173,3.318,"    R=",3.5398

-.998,.725,3.318,"    R=",3.5399

-.998,-.725,3.318,"    R=",3.5399

2.416,0,2.587,"    R=",3.5397

.747,2.298,2.587,"    R=",3.54

.747,-2.298,2.587,"    R=",3.54

-1.955,1.42,2.587,"    R=",3.5399

-1.955,-1.42,2.587,"    R=",3.5399

2.797,1.173,1.825,"    R=",3.5397

2.797,-1.173,1.825,"    R=",3.5397

1.98,2.298,1.825,"    R=",3.54

1.98,-2.298,1.825,"    R=",3.54

-.251,3.023,1.825,"    R=",3.5401

-.251,-3.023,1.825,"    R=",3.5401

-1.573,2.593,1.825,"    R=",3.5396

-1.573,-2.593,1.825,"    R=",3.5396

-2.952,.695,1.825,"    R=",3.5395

-2.952,-.695,1.825,"    R=",3.5395

3.414,.725,.591,"    R=",3.5398

3.414,-.725,.591,"    R=",3.5398

1.744,3.023,.591,"    R=",3.5397

1.744,-3.023,.591,"    R=",3.5397

.365,3.471,.591,"    R=",3.5398

.365,-3.471,.591,"    R=",3.5398

-2.336,2.593,.591,"    R=",3.5397

-2.336,-2.593,.591,"    R=",3.5397

-3.188,1.42,.591,"    R=",3.5396

-3.188,-1.42,.591,"    R=",3.5396

3.188,1.42,-.591,"    R=",3.5396

3.188,-1.42,-.591,"    R=",3.5396

2.336,2.593,-.591,"    R=",3.5397

2.336,-2.593,-.591,"    R=",3.5397

-.365,3.471,-.591,"    R=",3.5398

-.365,-3.471,-.591,"    R=",3.5398

-1.744,3.023,-.591,"    R=",3.5397

-1.744,-3.023,-.591,"    R=",3.5397

-3.414,.725,-.591,"    R=",3.5398

-3.414,-.725,-.591,"    R=",3.5398

2.952,.695,-1.825,"    R=",3.5395

2.952,-.695,-1.825,"    R=",3.5395

1.573,2.593,-1.825,"    R=",3.5396

1.573,-2.593,-1.825,"    R=",3.5396

.251,3.023,-1.825,"    R=",3.5401

.251,-3.023,-1.825,"    R=",3.5401

-1.98,2.298,-1.825,"    R=",3.54

-1.98,-2.298,-1.825,"    R=",3.54

-2.797,1.173,-1.825,"    R=",3.5397

-2.797,-1.173,-1.825,"    R=",3.5397

1.955,1.42,-2.587,"    R=",3.5399

1.955,-1.42,-2.587,"    R=",3.5399

-.747,2.298,-2.587,"    R=",3.54

-.747,-2.298,-2.587,"    R=",3.54

-2.416,0,-2.587,"    R=",3.5397

.998,.725,-3.318,"    R=",3.5399

.998,-.725,-3.318,"    R=",3.5399

-.381,1.173,-3.318,"    R=",3.5398

-.381,-1.173,-3.318,"    R=",3.5398

-1.233,0,-3.318,"    R=",3.5397

附录2 键距d1和d2及其两头的原子坐标

1,"d1=",1.44977

"ATOM1=",1.233,0,3.318

"ATOM2=",.381,1.173,3.318

2,"d1=",1.44977

"ATOM1=",1.233,0,3.318

"ATOM2=",.381,-1.173,3.318

3,"d1=",1.449947

"ATOM1=",.381,1.173,3.318

"ATOM2=",-.998,.725,3.318

4,"d1=",1.449947

"ATOM1=",.381,-1.173,3.318

"ATOM2=",-.998,-.725,3.318

5,"d1=",1.45

"ATOM1=",-.998,.725,3.318

"ATOM2=",-.998,-.725,3.318

6,"d1=",1.449736

"ATOM1=",2.416,0,2.587

"ATOM2=",2.797,1.173,1.825

7,"d1=",1.449736

"ATOM1=",2.416,0,2.587

"ATOM2=",2.797,-1.173,1.825

8,"d1=",1.44946

"ATOM1=",.747,2.298,2.587

"ATOM2=",1.98,2.298,1.825

9,"d1=",1.449922

"ATOM1=",.747,2.298,2.587

"ATOM2=",-.251,3.023,1.825

10,"d1=",1.44946

"ATOM1=",.747,-2.298,2.587

"ATOM2=",1.98,-2.298,1.825

11,"d1=",1.449922

"ATOM1=",.747,-2.298,2.587

"ATOM2=",-.251,-3.023,1.825

12,"d1=",1.449999

"ATOM1=",-1.955,1.42,2.587

"ATOM2=",-1.573,2.593,1.825

13,"d1=",1.449233

"ATOM1=",-1.955,1.42,2.587

"ATOM2=",-2.952,.695,1.825

14,"d1=",1.449999

"ATOM1=",-1.955,-1.42,2.587

"ATOM2=",-1.573,-2.593,1.825

15,"d1=",1.449233

"ATOM1=",-1.955,-1.42,2.587

"ATOM2=",-2.952,-.695,1.825

16,"d1=",1.450569

"ATOM1=",2.797,1.173,1.825

"ATOM2=",3.414,.725,.591

17,"d1=",1.450569

"ATOM1=",2.797,-1.173,1.825

"ATOM2=",3.414,-.725,.591

18,"d1=",1.450544

"ATOM1=",1.98,2.298,1.825

"ATOM2=",1.744,3.023,.591

19,"d1=",1.450544

"ATOM1=",1.98,-2.298,1.825

"ATOM2=",1.744,-3.023,.591

20,"d1=",1.450143

"ATOM1=",-.251,3.023,1.825

"ATOM2=",.365,3.471,.591

21,"d1=",1.450143

"ATOM1=",-.251,-3.023,1.825

"ATOM2=",.365,-3.471,.591

22,"d1=",1.450836

"ATOM1=",-1.573,2.593,1.825

"ATOM2=",-2.336,2.593,.591

23,"d1=",1.450836

"ATOM1=",-1.573,-2.593,1.825

"ATOM2=",-2.336,-2.593,.591

24,"d1=",1.450544

"ATOM1=",-2.952,.695,1.825

"ATOM2=",-3.188,1.42,.591

25,"d1=",1.450544

"ATOM1=",-2.952,-.695,1.825

"ATOM2=",-3.188,-1.42,.591

26,"d1=",1.45

"ATOM1=",3.414,.725,.591

"ATOM2=",3.414,-.725,.591

27,"d1=",1.449947

"ATOM1=",1.744,3.023,.591

"ATOM2=",.365,3.471,.591

28,"d1=",1.449947

"ATOM1=",1.744,-3.023,.591

"ATOM2=",.365,-3.471,.591

29,"d1=",1.44977

"ATOM1=",-2.336,2.593,.591

"ATOM2=",-3.188,1.42,.591

30,"d1=",1.44977

"ATOM1=",-2.336,-2.593,.591

"ATOM2=",-3.188,-1.42,.591

31,"d1=",1.44977

"ATOM1=",3.188,1.42,-.591

"ATOM2=",2.336,2.593,-.591

32,"d1=",1.450544

"ATOM1=",3.188,1.42,-.591

"ATOM2=",2.952,.695,-1.825

33,"d1=",1.44977

"ATOM1=",3.188,-1.42,-.591

"ATOM2=",2.336,-2.593,-.591

34,"d1=",1.450544

"ATOM1=",3.188,-1.42,-.591

"ATOM2=",2.952,-.695,-1.825

35,"d1=",1.450836

"ATOM1=",2.336,2.593,-.591

"ATOM2=",1.573,2.593,-1.825

36,"d1=",1.450836

"ATOM1=",2.336,-2.593,-.591

"ATOM2=",1.573,-2.593,-1.825

37,"d1=",1.449947

"ATOM1=",-.365,3.471,-.591

"ATOM2=",-1.744,3.023,-.591

38,"d1=",1.450143

"ATOM1=",-.365,3.471,-.591

"ATOM2=",.251,3.023,-1.825

39,"d1=",1.449947

"ATOM1=",-.365,-3.471,-.591

"ATOM2=",-1.744,-3.023,-.591

40,"d1=",1.450143

"ATOM1=",-.365,-3.471,-.591

"ATOM2=",.251,-3.023,-1.825

41,"d1=",1.450544

"ATOM1=",-1.744,3.023,-.591

"ATOM2=",-1.98,2.298,-1.825

42,"d1=",1.450544

"ATOM1=",-1.744,-3.023,-.591

"ATOM2=",-1.98,-2.298,-1.825

43,"d1=",1.45

"ATOM1=",-3.414,.725,-.591

"ATOM2=",-3.414,-.725,-.591

44,"d1=",1.450569

"ATOM1=",-3.414,.725,-.591

"ATOM2=",-2.797,1.173,-1.825

45,"d1=",1.450569

"ATOM1=",-3.414,-.725,-.591

"ATOM2=",-2.797,-1.173,-1.825

46,"d1=",1.449233

"ATOM1=",2.952,.695,-1.825

"ATOM2=",1.955,1.42,-2.587

47,"d1=",1.449233

"ATOM1=",2.952,-.695,-1.825

"ATOM2=",1.955,-1.42,-2.587

48,"d1=",1.449999

"ATOM1=",1.573,2.593,-1.825

"ATOM2=",1.955,1.42,-2.587

49,"d1=",1.449999

"ATOM1=",1.573,-2.593,-1.825

"ATOM2=",1.955,-1.42,-2.587

50,"d1=",1.449922

"ATOM1=",.251,3.023,-1.825

"ATOM2=",-.747,2.298,-2.587

51,"d1=",1.449922

"ATOM1=",.251,-3.023,-1.825

"ATOM2=",-.747,-2.298,-2.587

52,"d1=",1.44946

"ATOM1=",-1.98,2.298,-1.825

"ATOM2=",-.747,2.298,-2.587

53,"d1=",1.44946

"ATOM1=",-1.98,-2.298,-1.825

"ATOM2=",-.747,-2.298,-2.587

54,"d1=",1.449736

"ATOM1=",-2.797,1.173,-1.825

"ATOM2=",-2.416,0,-2.587

55,"d1=",1.449736

"ATOM1=",-2.797,-1.173,-1.825

"ATOM2=",-2.416,0,-2.587

56,"d1=",1.45

"ATOM1=",.998,.725,-3.318

"ATOM2=",.998,-.725,-3.318

57,"d1=",1.449947

"ATOM1=",.998,.725,-3.318

"ATOM2=",-.381,1.173,-3.318

58,"d1=",1.449947

"ATOM1=",.998,-.725,-3.318

"ATOM2=",-.381,-1.173,-3.318

59,"d1=",1.44977

"ATOM1=",-.381,1.173,-3.318

"ATOM2=",-1.233,0,-3.318

60,"d1=",1.44977

"ATOM1=",-.381,-1.173,-3.318

"ATOM2=",-1.233,0,-3.318

1,"d2=",1.390629

"ATOM1=",1.233,0,3.318

"ATOM2=",2.416,0,2.587

2,"d2=",1.390663

"ATOM1=",.381,1.173,3.318

"ATOM2=",.747,2.298,2.587

3,"d2=",1.390663

"ATOM1=",.381,-1.173,3.318

"ATOM2=",.747,-2.298,2.587

4,"d2=",1.390408

"ATOM1=",-.998,.725,3.318

"ATOM2=",-1.955,1.42,2.587

5,"d2=",1.390408

"ATOM1=",-.998,-.725,3.318

"ATOM2=",-1.955,-1.42,2.587

6,"d2=",1.390365

"ATOM1=",2.797,1.173,1.825

"ATOM2=",1.98,2.298,1.825

7,"d2=",1.390365

"ATOM1=",2.797,-1.173,1.825

"ATOM2=",1.98,-2.298,1.825

8,"d2=",1.390174

"ATOM1=",-.251,3.023,1.825

"ATOM2=",-1.573,2.593,1.825

9,"d2=",1.390174

"ATOM1=",-.251,-3.023,1.825

"ATOM2=",-1.573,-2.593,1.825

10,"d2=",1.39

"ATOM1=",-2.952,.695,1.825

"ATOM2=",-2.952,-.695,1.825

11,"d2=",1.389685

"ATOM1=",3.414,.725,.591

"ATOM2=",3.188,1.42,-.591

12,"d2=",1.389685

"ATOM1=",3.414,-.725,.591

"ATOM2=",3.188,-1.42,-.591

13,"d2=",1.39014

"ATOM1=",1.744,3.023,.591

"ATOM2=",2.336,2.593,-.591

14,"d2=",1.39014

"ATOM1=",1.744,-3.023,.591

"ATOM2=",2.336,-2.593,-.591

15,"d2=",1.389253

"ATOM1=",.365,3.471,.591

"ATOM2=",-.365,3.471,-.591

16,"d2=",1.389253

"ATOM1=",.365,-3.471,.591

"ATOM2=",-.365,-3.471,-.591

17,"d2=",1.39014

"ATOM1=",-2.336,2.593,.591

"ATOM2=",-1.744,3.023,-.591

18,"d2=",1.39014

"ATOM1=",-2.336,-2.593,.591

"ATOM2=",-1.744,-3.023,-.591

19,"d2=",1.389685

"ATOM1=",-3.188,1.42,.591

"ATOM2=",-3.414,.725,-.591

20,"d2=",1.389685

"ATOM1=",-3.188,-1.42,.591

"ATOM2=",-3.414,-.725,-.591

21,"d2=",1.39

"ATOM1=",2.952,.695,-1.825

"ATOM2=",2.952,-.695,-1.825

22,"d2=",1.390174

"ATOM1=",1.573,2.593,-1.825

"ATOM2=",.251,3.023,-1.825

23,"d2=",1.390174

"ATOM1=",1.573,-2.593,-1.825

"ATOM2=",.251,-3.023,-1.825

24,"d2=",1.390365

"ATOM1=",-1.98,2.298,-1.825

"ATOM2=",-2.797,1.173,-1.825

25,"d2=",1.390365

"ATOM1=",-1.98,-2.298,-1.825

"ATOM2=",-2.797,-1.173,-1.825

26,"d2=",1.390408

"ATOM1=",1.955,1.42,-2.587

"ATOM2=",.998,.725,-3.318

27,"d2=",1.390408

"ATOM1=",1.955,-1.42,-2.587

"ATOM2=",.998,-.725,-3.318

28,"d2=",1.390663

"ATOM1=",-.747,2.298,-2.587

"ATOM2=",-.381,1.173,-3.318

29,"d2=",1.390663

"ATOM1=",-.747,-2.298,-2.587

"ATOM2=",-.381,-1.173,-3.318

30,"d2=",1.390629

"ATOM1=",-2.416,0,-2.587

"ATOM2=",-1.233,0,-3.318