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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
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