博文
20年了，魔方转出了原子坐标
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20年了，魔方转出了原子坐标
俺用魔方玩原子坐标，今天终于有了结果。
（注意，俺说的“结果”是转出了一些正确的结果，而非通过什么专家“鉴定”或“验收”，俺会把俺的结果帖在科学网上，希望有人能够用上这些结果，而不是通过那种“鉴定”帖个什么标签）
1985年研究生时学了一门课《群论》，后来用群论来玩魔方（1991年），出了两本书：《魔方及其应用》（业余爱好）和《魔方的科学和计算机表现》（国家科普基金资助），俺始终思考着如何用魔方转动出原子坐标。
余瑞璜先生有个“固体与分子的经验电子理论”，计算价键结构时，首先需要知道原子坐标，对于简单的结构，还可以比划出来，对于复杂结构就难了。
需要原子坐标的地方还很多，铁道警察各管一段，俺只管转出原子坐标，其他的事情别人可以接着干。
实际上，做学问和玩魔方是一样的，管它有没有结果，要一直转下去。当然了，在双手转动魔方的同时，大脑也要跟着魔方一起转动。
用魔方转出（晶体的）原子坐标，需要（科学地）具体干些什么呢？
1）要学（用）点群论
群论的内容很多，但是玩魔方需要点群，这和晶体学里的点群是一样的。点群解决转动的问题，加上平移就是空间群问题，求解原子坐标需要考虑两者。实质上，群论是一种数学变换工具，同时可以解决坐标的变换和函数的变换。坐标变换对应原子坐标，函数变换对应结构的性质（特征表）。
2）要学（用）点晶体学
晶体学的基础内容是关于原子在空间如何排列，还有组合问题。排列组合的本质可以归纳到群论，又回到了群论。
3）要学（用）点线性代数
没有线性代数，是解决不了实际的群论问题的。矩阵就是一个压扁了的魔方，对于一个转动或者平移操作，矩阵的两边联系着两组原子坐标。
4）要学（用）点计算机编程
如果没有计算机帮忙，俺玩魔方会很累的。同样，要解决原子坐标问题，也需要计算机的帮忙。编程的时候，如果马马虎虎，不是你玩计算机，而是计算机在玩你啊！
计算机编程工作是真正需要经验的，否则，计算机会绝对地拒绝运行你的程序，除了改错，一点办法也没有的。
5）要学（用）点X射线衍射理论和方法
X射线衍射确定晶体结构技术，透露了很多原子坐标的信息。不但要知道理论，还要知道实验结果及其指标化等技巧。
6）要学（用）点 International Tables for Crystallography（好几大本书）
7个晶系，14种点阵，32点群，230空间群，各种各样的不同的符号体系，不是为了原子坐标，俺才不看那么细呢。
7）要学（用）点固体物理
物理学把晶体学的晶胞切割得更小，一会儿正空间，一会儿倒空间。硬要抽出晶体结构的那根筋，还好，俺是学物理的。
8）还要学（用）点化学
因为分子也有对称性，而且化学特别喜欢点群的特征表。实际上，化学里化学键的理论就是关于原子坐标的理论，只不过化学家优先选择了极坐标体系（键长和键角），而俺这里选用的是笛卡尔坐标系（X，Y，Z）。
《数学手册》里，绝大多数的公式和函数都是用（X，Y，Z）表达的，因此，俺更喜欢笛卡尔坐标系（X，Y，Z）。
9）最后还要学（用）点“慢学”
20多年解决不了一个问题，这就是慢学。
着急有什么用呢？
“别着急，慢慢来”。
“一慢，二看，三通过”。
这都是关于“慢学”的经典名言！
俺才不着急呢？为什么要着急呢？
下面的数据是关于Cu晶体，很简单的结构，一大堆数据。
这次是计算机选键操作。
参考原子（0,0,0）
"4a","原子","配位数=",1
"和参考原子的键距＝",0
0,0,0,
"4a","原子","配位数=",12
"和参考原子的键距＝",.707106781186548
-.5,-.5,0,
-.5,0,-.5,
-.5,0,.5,
-.5,.5,0,
0,-.5,-.5,
0,-.5,.5,
0,.5,-.5,
0,.5,.5,
.5,-.5,0,
.5,0,-.5,
.5,0,.5,
.5,.5,0,
"4a","原子","配位数=",6
"和参考原子的键距＝",1
-1,0,0,
0,-1,0,
0,0,-1,
0,0,1,
0,1,0,
1,0,0,
"4a","原子","配位数=",24
"和参考原子的键距＝",1.22474487139159
-1,-.5,-.5,
-1,-.5,.5,
-1,.5,-.5,
-1,.5,.5,
-.5,-1,-.5,
-.5,-1,.5,
-.5,-.5,-1,
-.5,-.5,1,
-.5,.5,-1,
-.5,.5,1,
-.5,1,-.5,
-.5,1,.5,
.5,-1,-.5,
.5,-1,.5,
.5,-.5,-1,
.5,-.5,1,
.5,.5,-1,
.5,.5,1,
.5,1,-.5,
.5,1,.5,
1,-.5,-.5,
1,-.5,.5,
1,.5,-.5,
1,.5,.5,
"4a","原子","配位数=",12
"和参考原子的键距＝",1.4142135623731
-1,-1,0,
-1,0,-1,
-1,0,1,
-1,1,0,
0,-1,-1,
0,-1,1,
0,1,-1,
0,1,1,
1,-1,0,
1,0,-1,
1,0,1,
1,1,0,
"4a","原子","配位数=",8
"和参考原子的键距＝",1.73205080756888
-1,-1,-1,
-1,-1,1,
-1,1,-1,
-1,1,1,
1,-1,-1,
1,-1,1,
1,1,-1,
1,1,1,
"点群操作后，输出组成晶胞的原子坐标（x,y,z）=",14
"4a","组成晶胞的原子=",1
0,0,0,
"4a","组成晶胞的原子=",3
0,.5,.5,
.5,0,.5,
.5,.5,0,
"4a","组成晶胞的原子=",3
0,0,1,
0,1,0,
1,0,0,
"4a","组成晶胞的原子=",3
.5,.5,1,
.5,1,.5,
1,.5,.5,
"4a","组成晶胞的原子=",3
0,1,1,
1,0,1,
1,1,0,
"4a","组成晶胞的原子=",1
1,1,1,
"编号＝",1,"键距＝",.7071068
"原子-",1
0,0,0," ","在晶胞内的配位数===",3
"原子-",2
0,0,1," ","在晶胞内的配位数===",3
"原子-",3
0,.5,.5," ","在晶胞内的配位数===",8
"原子-",4
0,1,0," ","在晶胞内的配位数===",3
"原子-",5
0,1,1," ","在晶胞内的配位数===",3
"原子-",6
.5,0,.5," ","在晶胞内的配位数===",8
"原子-",7
.5,.5,0," ","在晶胞内的配位数===",8
"原子-",8
.5,.5,1," ","在晶胞内的配位数===",8
"原子-",9
.5,1,.5," ","在晶胞内的配位数===",8
"原子-",10
1,0,0," ","在晶胞内的配位数===",3
"原子-",11
1,0,1," ","在晶胞内的配位数===",3
"原子-",12
1,.5,.5," ","在晶胞内的配位数===",8
"原子-",13
1,1,0," ","在晶胞内的配位数===",3
"原子-",14
1,1,1," ","在晶胞内的配位数===",3
"编号＝",2,"键距＝",1
"原子-",1
0,0,0," ","在晶胞内的配位数===",3
"原子-",2
0,0,1," ","在晶胞内的配位数===",3
"原子-",3
0,.5,.5," ","在晶胞内的配位数===",1
"原子-",4
0,1,0," ","在晶胞内的配位数===",3
"原子-",5
0,1,1," ","在晶胞内的配位数===",3
"原子-",6
.5,0,.5," ","在晶胞内的配位数===",1
"原子-",7
.5,.5,0," ","在晶胞内的配位数===",1
"原子-",8
.5,.5,1," ","在晶胞内的配位数===",1
"原子-",9
.5,1,.5," ","在晶胞内的配位数===",1
"原子-",10
1,0,0," ","在晶胞内的配位数===",3
"原子-",11
1,0,1," ","在晶胞内的配位数===",3
"原子-",12
1,.5,.5," ","在晶胞内的配位数===",1
"原子-",13
1,1,0," ","在晶胞内的配位数===",3
"原子-",14
1,1,1," ","在晶胞内的配位数===",3
"编号＝",3,"键距＝",1.224745
"原子-",1
0,0,0," ","在晶胞内的配位数===",3
"原子-",2
0,0,1," ","在晶胞内的配位数===",3
"原子-",3
0,.5,.5," ","在晶胞内的配位数===",4
"原子-",4
0,1,0," ","在晶胞内的配位数===",3
"原子-",5
0,1,1," ","在晶胞内的配位数===",3
"原子-",6
.5,0,.5," ","在晶胞内的配位数===",4
"原子-",7
.5,.5,0," ","在晶胞内的配位数===",4
"原子-",8
.5,.5,1," ","在晶胞内的配位数===",4
"原子-",9
.5,1,.5," ","在晶胞内的配位数===",4
"原子-",10
1,0,0," ","在晶胞内的配位数===",3
"原子-",11
1,0,1," ","在晶胞内的配位数===",3
"原子-",12
1,.5,.5," ","在晶胞内的配位数===",4
"原子-",13
1,1,0," ","在晶胞内的配位数===",3
"原子-",14
1,1,1," ","在晶胞内的配位数===",3
"编号＝",4,"键距＝",1.414214
"原子-",1
0,0,0," ","在晶胞内的配位数===",3
"原子-",2
0,0,1," ","在晶胞内的配位数===",3
"原子-",3
0,1,0," ","在晶胞内的配位数===",3
"原子-",4
0,1,1," ","在晶胞内的配位数===",3
"原子-",5
1,0,0," ","在晶胞内的配位数===",3
"原子-",6
1,0,1," ","在晶胞内的配位数===",3
"原子-",7
1,1,0," ","在晶胞内的配位数===",3
"原子-",8
1,1,1," ","在晶胞内的配位数===",3
"编号＝",5,"键距＝",1.732051
"原子-",1
0,0,0," ","在晶胞内的配位数===",1
"原子-",2
0,0,1," ","在晶胞内的配位数===",1
"原子-",3
0,1,0," ","在晶胞内的配位数===",1
"原子-",4
0,1,1," ","在晶胞内的配位数===",1
"原子-",5
1,0,0," ","在晶胞内的配位数===",1
"原子-",6
1,0,1," ","在晶胞内的配位数===",1
"原子-",7
1,1,0," ","在晶胞内的配位数===",1
"原子-",8
1,1,1," ","在晶胞内的配位数===",1
Cu晶胞中原子和原子的作用：d表示键距，左边的单个数字是原子编号，右边的数字表示键的编号，三个数字一组的就是原子坐标：
"---------------------------------------",1
1,"--",2
0,0,0," ",0,.5,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",2
1,"--",3
0,0,0," ",.5,0,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",3
1,"--",4
0,0,0," ",.5,.5,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",4
2,"--",3
0,.5,.5," ",.5,0,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",5
2,"--",4
0,.5,.5," ",.5,.5,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",6
2,"--",5
0,.5,.5," ",0,0,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",7
2,"--",6
0,.5,.5," ",0,1,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",8
2,"--",8
0,.5,.5," ",.5,.5,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",9
2,"--",9
0,.5,.5," ",.5,1,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",10
2,"--",11
0,.5,.5," ",0,1,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",11
3,"--",4
.5,0,.5," ",.5,.5,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",12
3,"--",5
.5,0,.5," ",0,0,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",13
3,"--",7
.5,0,.5," ",1,0,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",14
3,"--",8
.5,0,.5," ",.5,.5,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",15
3,"--",10
.5,0,.5," ",1,.5,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",16
3,"--",12
.5,0,.5," ",1,0,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",17
4,"--",6
.5,.5,0," ",0,1,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",18
4,"--",7
.5,.5,0," ",1,0,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",19
4,"--",9
.5,.5,0," ",.5,1,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",20
4,"--",10
.5,.5,0," ",1,.5,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",21
4,"--",13
.5,.5,0," ",1,1,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",22
5,"--",8
0,0,1," ",.5,.5,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",23
6,"--",9
0,1,0," ",.5,1,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",24
7,"--",10
1,0,0," ",1,.5,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",25
8,"--",9
.5,.5,1," ",.5,1,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",26
8,"--",10
.5,.5,1," ",1,.5,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",27
8,"--",11
.5,.5,1," ",0,1,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",28
8,"--",12
.5,.5,1," ",1,0,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",29
8,"--",14
.5,.5,1," ",1,1,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",30
9,"--",10
.5,1,.5," ",1,.5,.5,
"d======================",.7071068
"4a-4a"
"---------------------------------------",31
9,"--",11
.5,1,.5," ",0,1,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",32
9,"--",13
.5,1,.5," ",1,1,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",33
9,"--",14
.5,1,.5," ",1,1,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",34
10,"--",12
1,.5,.5," ",1,0,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",35
10,"--",13
1,.5,.5," ",1,1,0,
"d======================",.7071068
"4a-4a"
"---------------------------------------",36
10,"--",14
1,.5,.5," ",1,1,1,
"d======================",.7071068
"4a-4a"
"---------------------------------------",37
3,"--",9
.5,0,.5," ",.5,1,.5,
"d======================",1
"4a-4a"
"---------------------------------------",38
1,"--",6
0,0,0," ",0,1,0,
"d======================",1
"4a-4a"
"---------------------------------------",39
1,"--",7
0,0,0," ",1,0,0,
"d======================",1
"4a-4a"
"---------------------------------------",40
2,"--",10
0,.5,.5," ",1,.5,.5,
"d======================",1
"4a-4a"
"---------------------------------------",41
5,"--",11
0,0,1," ",0,1,1,
"d======================",1
"4a-4a"
"---------------------------------------",42
5,"--",12
0,0,1," ",1,0,1,
"d======================",1
"4a-4a"
"---------------------------------------",43
1,"--",5
0,0,0," ",0,0,1,
"d======================",1
"4a-4a"
"---------------------------------------",44
6,"--",11
0,1,0," ",0,1,1,
"d======================",1
"4a-4a"
"---------------------------------------",45
6,"--",13
0,1,0," ",1,1,0,
"d======================",1
"4a-4a"
"---------------------------------------",46
4,"--",8
.5,.5,0," ",.5,.5,1,
"d======================",1
"4a-4a"
"---------------------------------------",47
7,"--",12
1,0,0," ",1,0,1,
"d======================",1
"4a-4a"
"---------------------------------------",48
7,"--",13
1,0,0," ",1,1,0,
"d======================",1
"4a-4a"
"---------------------------------------",49
11,"--",14
0,1,1," ",1,1,1,
"d======================",1
"4a-4a"
"---------------------------------------",50
12,"--",14
1,0,1," ",1,1,1,
"d======================",1
"4a-4a"
"---------------------------------------",51
13,"--",14
1,1,0," ",1,1,1,
"d======================",1
"4a-4a"
"---------------------------------------",52
4,"--",5
.5,.5,0," ",0,0,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",53
7,"--",8
1,0,0," ",.5,.5,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",54
7,"--",9
1,0,0," ",.5,1,.5,
"d======================",1.224745
"4a-4a"
"---------------------------------------",55
1,"--",9
0,0,0," ",.5,1,.5,
"d======================",1.224745
"4a-4a"
"---------------------------------------",56
1,"--",10
0,0,0," ",1,.5,.5,
"d======================",1.224745
"4a-4a"
"---------------------------------------",57
3,"--",6
.5,0,.5," ",0,1,0,
"d======================",1.224745
"4a-4a"
"---------------------------------------",58
1,"--",8
0,0,0," ",.5,.5,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",59
2,"--",7
0,.5,.5," ",1,0,0,
"d======================",1.224745
"4a-4a"
"---------------------------------------",60
4,"--",11
.5,.5,0," ",0,1,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",61
4,"--",12
.5,.5,0," ",1,0,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",62
8,"--",13
.5,.5,1," ",1,1,0,
"d======================",1.224745
"4a-4a"
"---------------------------------------",63
2,"--",12
0,.5,.5," ",1,0,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",64
4,"--",14
.5,.5,0," ",1,1,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",65
2,"--",13
0,.5,.5," ",1,1,0,
"d======================",1.224745
"4a-4a"
"---------------------------------------",66
9,"--",12
.5,1,.5," ",1,0,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",67
5,"--",9
0,0,1," ",.5,1,.5,
"d======================",1.224745
"4a-4a"
"---------------------------------------",68
5,"--",10
0,0,1," ",1,.5,.5,
"d======================",1.224745
"4a-4a"
"---------------------------------------",69
10,"--",11
1,.5,.5," ",0,1,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",70
3,"--",11
.5,0,.5," ",0,1,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",71
2,"--",14
0,.5,.5," ",1,1,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",72
6,"--",8
0,1,0," ",.5,.5,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",73
3,"--",13
.5,0,.5," ",1,1,0,
"d======================",1.224745
"4a-4a"
"---------------------------------------",74
6,"--",10
0,1,0," ",1,.5,.5,
"d======================",1.224745
"4a-4a"
"---------------------------------------",75
3,"--",14
.5,0,.5," ",1,1,1,
"d======================",1.224745
"4a-4a"
"---------------------------------------",76
5,"--",6
0,0,1," ",0,1,0,
"d======================",1.414214
"4a-4a"
"---------------------------------------",77
5,"--",7
0,0,1," ",1,0,0,
"d======================",1.414214
"4a-4a"
"---------------------------------------",78
6,"--",14
0,1,0," ",1,1,1,
"d======================",1.414214
"4a-4a"
"---------------------------------------",79
5,"--",14
0,0,1," ",1,1,1,
"d======================",1.414214
"4a-4a"
"---------------------------------------",80
6,"--",7
0,1,0," ",1,0,0,
"d======================",1.414214
"4a-4a"
"---------------------------------------",81
1,"--",11
0,0,0," ",0,1,1,
"d======================",1.414214
"4a-4a"
"---------------------------------------",82
11,"--",12
0,1,1," ",1,0,1,
"d======================",1.414214
"4a-4a"
"---------------------------------------",83
11,"--",13
0,1,1," ",1,1,0,
"d======================",1.414214
"4a-4a"
"---------------------------------------",84
1,"--",12
0,0,0," ",1,0,1,
"d======================",1.414214
"4a-4a"
"---------------------------------------",85
12,"--",13
1,0,1," ",1,1,0,
"d======================",1.414214
"4a-4a"
"---------------------------------------",86
1,"--",13
0,0,0," ",1,1,0,
"d======================",1.414214
"4a-4a"
"---------------------------------------",87
7,"--",14
1,0,0," ",1,1,1,
"d======================",1.414214
"4a-4a"
"---------------------------------------",88
6,"--",12
0,1,0," ",1,0,1,
"d======================",1.732051
"4a-4a"
"---------------------------------------",89
1,"--",14
0,0,0," ",1,1,1,
"d======================",1.732051
"4a-4a"
"---------------------------------------",90
5,"--",13
0,0,1," ",1,1,0,
"d======================",1.732051
"4a-4a"
"---------------------------------------",91
7,"--",11
1,0,0," ",0,1,1,
"d======================",1.732051
"4a-4a"
https://blog.sciencenet.cn/blog-2321-42926.html
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