{"id":1586,"date":"2021-12-12T00:00:00","date_gmt":"2021-12-11T15:00:00","guid":{"rendered":"https:\/\/www.insilico.jp\/blog\/?p=1586"},"modified":"2022-02-22T14:30:21","modified_gmt":"2022-02-22T05:30:21","slug":"superpose_molecules_1","status":"publish","type":"post","link":"https:\/\/www.insilico.jp\/blog\/2021\/12\/12\/superpose_molecules_1\/","title":{"rendered":"\u5206\u5b50\u306e\u6bcd\u6838\u69cb\u9020\u306e\u91cd\u306d\u5408\u308f\u305b\u3000\u524d\u7de8"},"content":{"rendered":"\n

\u4eca\u56de\u306e\u8a71\u984c\u306f\u5206\u5b50\u306e\u56de\u8ee2\u306b\u3064\u3044\u3066\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u3057\u3066\u307f\u307e\u3059\u3002\u8a71\u304c\u9577\u304f\u306a\u308a\u305d\u3046\u306a\u306e\u3067\u3001\u30d6\u30ed\u30b0\u3092\u5e7e\u3064\u304b\u306b\u5206\u5272\u3057\u3066\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u306f\u3058\u3081\u306b<\/h2>\n\n\n\n

\u524d\u56de\u306f\u6bcd\u6838\u69cb\u9020\u3092\u56fa\u5b9a\u3057\u3066\u30b3\u30f3\u30db\u30e1\u30fc\u30b7\u30e7\u30f3\u30b5\u30fc\u30c1\u3092\u3057\u3066\u307f\u307e\u3057\u305f\u3002\u8272\u3005\u306a\u30b3\u30f3\u30db\u30e1\u30fc\u30b7\u30e7\u30f3\u3092\u767a\u751f\u3055\u305b\u305f\u5f8c\u306b\u3084\u3063\u3066\u307f\u305f\u304f\u306a\u308b\u306e\u306f\u6bcd\u6838\u69cb\u9020\u306e\u91cd\u306d\u5408\u308f\u305b\u3067\u3057\u3087\u3046\u3002\u4f8b\u3048\u3070\u30bf\u30f3\u30d1\u30af\u8cea\u7d50\u6676\u69cb\u9020\u4e2d\u306e\u30ea\u30ac\u30f3\u30c9\u306b\u3001\u767a\u751f\u3055\u305b\u305f\u5206\u5b50\u306e\u6bcd\u6838\u69cb\u9020\u3092\u91cd\u306d\u3066\u307f\u308b\u3068\u3044\u3046\u3053\u3068\u306a\u3069\u7c21\u5358\u306b\u601d\u3044\u3064\u304d\u307e\u3059\u3002\u601d\u3044\u3064\u304f\u306e\u306f\u7c21\u5358\u3067\u3059\u304c\u3001\u5206\u5b50\u3092\u3046\u307e\u304f\u4e26\u9032\u30fb\u56de\u8ee2\u3055\u305b\u306a\u3044\u3068\u76ee\u7684\u306e\u6bcd\u6838\u69cb\u9020\u3068\u7dba\u9e97\u306b\u91cd\u306a\u308a\u307e\u305b\u3093\u3002\u5206\u5b50\u306e\u4e26\u9032\u306f\u5b9a\u30d9\u30af\u30c8\u30eb\u3092\u5168\u3066\u306e\u539f\u5b50\u5ea7\u6a19\u306b\u8db3\u3057\u7b97\u3059\u308c\u3070\u826f\u3044\u306e\u3067\u7c21\u5358\u3067\u3059\u3002\u3057\u304b\u3057\u306a\u304c\u3089\u5206\u5b50\u306e\u56de\u8ee2\u306f\u305d\u3093\u306a\u306b\u7c21\u5358\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u307e\u3057\u3066\u3084\u6bcd\u6838\u69cb\u9020\u304c\u30d4\u30c3\u30bf\u30ea\u91cd\u306a\u308b\u3088\u3046\u306b\u56de\u8ee2\u3059\u308b\u306e\u3069\u3046\u3059\u308c\u3070\u3044\u3044\u306e\u3067\u3057\u3087\u3046\u304b\uff1f<\/p>\n\n\n\n

\u5b9f\u3092\u8a00\u3046\u3068\u3053\u306e\u6bcd\u6838\u69cb\u9020\u306e\u91cd\u306d\u5408\u308f\u305b\u306b\u3064\u3044\u3066\u306f\u3001OpenBabel\u306e\u4e2d\u306b\u3082\u305d\u308c\u3089\u3057\u3044\u95a2\u6570\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u306f\u79c1\u3082\u78ba\u8a8d\u3057\u3066\u3044\u308b\u306e\u3067\u3059\u304c\u3001API\u30c9\u30ad\u30e5\u30e1\u30f3\u30c8\u306b\u4e00\u5207\u8aac\u660e\u304c\u3042\u308a\u307e\u305b\u3093\u3002\u4f7f\u3044\u65b9\u304c\u826f\u304f\u308f\u304b\u3089\u306a\u3044\u3082\u306e\u3092\u7121\u7406\u306b\u4f7f\u3046\u3088\u308a\u306f\u81ea\u5206\u3067\u4f5c\u308d\u3046\u304b\u306a\u3068\u601d\u3044\u307e\u3057\u305f\u3002\u3068\u3044\u3046\u3053\u3068\u3067\u3001\u5148\u305a\u306f\u524d\u7de8\u3068\u3057\u3066\u5206\u5b50\u306e\u91cd\u306d\u5408\u308f\u305b\u8a08\u7b97\u306e\u305f\u3081\u306e\u7406\u8ad6\u3092\u7d39\u4ecb\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u5206\u5b50\u306e\u56de\u8ee2<\/h2>\n\n\n\n

\u7269\u4f53\u306e3\u6b21\u5143\u7684\u306a\u56de\u8ee2\u3092\u6271\u3046\u65b9\u6cd5\u3068\u3057\u3066\u30aa\u30a4\u30e9\u30fc\u89d2\u3068\u56db\u5143\u6570\u306e2\u3064\u304c\u3042\u308a\u307e\u3059\u3002\u30aa\u30a4\u30e9\u30fc\u89d2\u306e\u65b9\u306f\u5927\u62b5\u306e\u529b\u5b66\u306e\u6559\u79d1\u66f8\u306b\u51fa\u3066\u304f\u308b\u306e\u3067\u3001\u7269\u7406\u5b66\u3092\u5b66\u3093\u3060\u4e8b\u304c\u3042\u308b\u65b9\u306f\u826f\u304f\u3054\u5b58\u3058\u304b\u3068\u601d\u3044\u307e\u3059\u3002\u3053\u308c\u306f\u9ad8\u6821\u6570\u5b66\u3067\u52c9\u5f37\u3059\u308b2\u6b21\u5143\u5e73\u9762\u3067\u306e\u56de\u8ee2\u884c\u5217$$ \\begin{pmatrix} \\cos \\theta & -\\sin \\theta \\\\ \\sin \\theta & \\cos \\theta \\end{pmatrix} $$\u3092\u7d44\u307f\u3042\u308f\u305b\u30663\u6b21\u5143\u306e\u56de\u8ee2\u3092\u8868\u73fe\u3059\u308b\u306e\u3067\u3001\u4e00\u898b\u3059\u308b\u3068\u76f4\u89b3\u7684\u306a\u8868\u73fe\u306a\u306e\u3067\u3059\u304c\u3001\u56de\u8ee2\u884c\u5217\u306b\u306f\\( \\sin \\theta \\)\u3068\u304b\\( \\cos \\theta \\)\u304c\u8907\u96d1\u306b\u51fa\u3066\u304d\u307e\u3059\u3002\u5206\u5b50\u306e\u91cd\u306d\u3042\u308f\u305b\u8a08\u7b97\u306b\u306f\u3053\u308c\u304c\u5927\u5909\u9762\u5012\u3067\u3001\u9811\u5f35\u308c\u3070\u4f55\u3068\u304b\u306a\u308b\u3068\u306f\u601d\u3046\u3051\u3069\u6b63\u76f4\u3042\u307e\u308a\u3084\u308a\u305f\u304f\u306a\u3044\u3067\u3059\u306d\u3002\u4e00\u65b9\u3001\u56db\u5143\u6570\u306e\u65b9\u306f\u6982\u5ff5\u306f\u96e3\u3057\u3044\u3067\u3059\u304c\u5b9f\u969b\u306e\u8a08\u7b97\u306e\u65b9\u306f\u610f\u5916\u3068\u5358\u7d14\u3067\u3059\u3002\u305d\u3046\u3044\u3046\u3053\u3068\u3067\u3001\u3053\u3053\u3067\u306f\u56db\u5143\u6570\u3092\u4f7f\u3063\u3066\u5206\u5b50\u306e\u91cd\u306d\u5408\u308f\u305b\u3092\u8003\u3048\u308b\u3053\u3068\u3068\u3057\u307e\u3059\u3002<\/p>\n\n\n\n

\u56db\u5143\u6570<\/h2>\n\n\n\n

\u9ad8\u6821\u6570\u5b66\u3067\u52c9\u5f37\u3059\u308b\u8907\u7d20\u6570\u306f\u5b9f\u6570\u90e8\u5206\u3092\\(a\\)\u3001\u865a\u6570\u90e8\u5206\u3092\\(b\\)\u3001\u865a\u6570\u5358\u4f4d\u3092\\(i\\)\u3068\u3057\u3066\\(a+bi\\)\u3068\u8868\u3055\u308c\u3066\u3001\u865a\u6570\u5358\u4f4d\u306b\u306f\\(i^2=-1\\)\u3068\u3044\u3046\u95a2\u4fc2\u304c\u3042\u308a\u307e\u3059\u3002\u4ef6\u306e\u56db\u5143\u6570\u306f\u3053\u306e\u8907\u7d20\u6570\u3092\u62e1\u5f35\u3057\u3066\u3001\u865a\u6570\u5358\u4f4d\u3068\u3057\u3066\\(i, j, k\\)\u306e3\u7a2e\u985e\u3092\u7528\u3044\u307e\u3059\u3002\u5b9f\u6570\u90e8\u5206\u3092\\(a\\)\u3001\u865a\u6570\u90e8\u5206\u3092\\(a, b, c\\)\u3068\u3059\u308b\u3068\u56db\u5143\u6570\u306f
$$ a+bi+cj+dk $$\u3068\u8868\u3055\u308c\u307e\u3059\u3002\u305d\u3057\u3066\u865a\u6570\u5358\u4f4d\\(i, j, k\\)\u306e\u95a2\u4fc2\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002
$$
\\begin{align}
i^2&=-1, & j^2&=-1, & k^2&=-1 \\\\
ij&=k, & jk&=i, & ki&=j \\\\
ji&=-k, & kj&=-i, & ik&=-j
\\end{align}
$$
\u3082\u3046\u30e4\u30d0\u3044\u611f\u3058\u3067\u3059\u3088\u306d\uff08\u7b11\uff09\u3002\u3067\u3082\u3001\u9762\u5012\u3067\u3082\u843d\u3061\u7740\u3044\u3066\u8a08\u7b97\u3059\u308c\u3070\u5927\u4e08\u592b\u3060\u3068\u8a00\u3046\u3053\u3068\u304c\u5206\u304b\u308b\u306e\u3067\u662f\u975e\u624b\u3092\u52d5\u304b\u3057\u3066\u8a08\u7b97\u3057\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u56db\u5143\u6570\u306e\u6027\u8cea<\/h2>\n\n\n\n

\u3053\u3053\u3067\u306f\u56db\u5143\u6570\u306e\u6027\u8cea\u3092\u7c21\u5358\u306b\u7d39\u4ecb\u3057\u307e\u3059\u3002\u8a73\u3057\u304f\u306fWikipedia<\/a>\u306a\u3069\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044\u3002\u307e\u305a2\u3064\u306e\u56db\u5143\u6570\\(p=a+bi+cj+dk\\)\u3068\\(q=w+xi+yj+zk\\)\u306e\u8db3\u3057\u7b97\u3067\u3059\u3002
$$
p+q=(a+bi+cj+dk)+(w+xi+yj+zk)=(a+w)+(b+x)i+(c+y)j+(d+z)k
$$
\u3068\u306a\u308a\u307e\u3059\u3002\u6b21\u306b\u639b\u3051\u7b97\u3067\u3059\u3002
$$
\\begin{align}
pq=(a+bi+cj+dk)(w+xi+yj+zk)=aw&+axi+ayj+azk \\\\
&+bwi+bxi^2+byij+bzik \\\\
&+cwj+cxji+cyj^2+czjk \\\\
&+dwk+dxki+dykj+dzk^2
\\end{align}
$$
\u3068\u306a\u308a\u56db\u5143\u6570\u306e\u865a\u6570\u5358\u4f4d\u306e\u95a2\u4fc2\u5f0f\u3088\u308a\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
$$
\\begin{align}
pq=&aw-bx-cy-dz \\\\
&+(ax+bw+cz-dy)i \\\\
&+(ay-bz+cw+dx)j \\\\
&+(az+by-cx+dw)k
\\end{align}
$$
\u3053\u3053\u3067\u56db\u5143\u6570\u306e\u7a4d\u306f\u53ef\u63db\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u3064\u307e\u308a\\(pq \\neq qp\\)\u3067\u3059\u3002
\u3042\u3068\u3001\u7d50\u5408\u6cd5\u5247\u3068\u5206\u914d\u6cd5\u5247\u306f\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u3064\u307e\u308a\u884c\u5217\u3068\u540c\u3058\u4ee3\u6570\u306e\u6cd5\u5247\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u56db\u5143\u6570\u3092\u30b9\u30ab\u30e9\u30fc\u90e8\u3068\u30d9\u30af\u30c8\u30eb\u90e8\u306b\u5206\u3051\u3066\u8868\u73fe\u3059\u308b\u65b9\u6cd5\u304c\u3042\u308a\u307e\u3059\u3002\u865a\u6570\u5358\u4f4d\\(i,j,k\\)\u3092\u305d\u308c\u305e\u308c\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\\(\\mathbf{i},\\mathbf{j},\\mathbf{k}\\)\u3068\u307f\u306a\u3057\u307e\u3059\u3002\u4f8b\u3048\u3070\u4e0a\u8a18\u306e\u56db\u5143\u6570\\(p,q\\)\u306f
$$
\\begin{align}
p&=a+bi+cj+dk\\\\
&=(a,\\mathbf{p}), \\qquad \\mathbf{p}=(b,c,d)\\\\
q&=w+xi+yj+zk\\\\
&=(w,\\mathbf{q}), \\qquad \\mathbf{q}=(x,y,z)\\\\
\\end{align}
$$
\u3068\u8868\u73fe\u3055\u308c\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u56db\u5143\u6570\u306e\u7a4d\u306f\u30d9\u30af\u30c8\u30eb\u306e\u5185\u7a4d\u3068\u5916\u7a4d\u3092\u4f7f\u3063\u3066
$$
pq=(aw-\\mathbf{p}\\cdot\\mathbf{q}, a\\mathbf{q}+w\\mathbf{p}+\\mathbf{p}\\times\\mathbf{q})
$$
\u3068\u306a\u308a\u307e\u3059\u3002\u82e5\u5e72\u30b7\u30f3\u30d7\u30eb\u306b\u307e\u3068\u307e\u308a\u307e\u3057\u305f\u3088\u306d\u3002\u3053\u3053\u3067\u3001\u5916\u7a4d\u3092\u4f7f\u3063\u3066\u3044\u308b\u3053\u3068\u304b\u3089\u56db\u5143\u6570\u306e\u7a4d\u306e\u9806\u756a\u306b\u3088\u3063\u3066\u7d50\u679c\u304c\u7570\u306a\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n

\u5171\u5f79\u56db\u5143\u6570\u3068\u304b<\/h2>\n\n\n\n

\u8907\u7d20\u6570\u306b\u8907\u7d20\u5171\u5f79\u304c\u3042\u308b\u3088\u3046\u306b\u3001\u56db\u5143\u6570\u306b\u3082\u5171\u5f79\u56db\u5143\u6570\u304c\u3042\u308a\u307e\u3059\u3002\u5171\u5f79\u56db\u5143\u6570\u306f$$ \\bar{q}=w-xi-yj-zk $$\u3068\u306a\u308a\u307e\u3059\u3002\u56db\u5143\u6570\u306e\u639b\u3051\u7b97\\(pq\\)\u306e\u5171\u5f79\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u9806\u756a\u304c\u9006\u306b\u5165\u308c\u66ff\u308f\u308a\u307e\u3059\u3002
$$ \\overline{pq}=\\bar{q}\\bar{p} $$
\u56db\u5143\u6570\u306e\u30ce\u30eb\u30e0\u306f\u81ea\u8eab\u306e\u5171\u5f79\u3068\u306e\u639b\u3051\u7b97\u3068\u306a\u308a\u307e\u3059\u3002
$$ ||q||^2 = q\\bar{q} = \\bar{q}q = a^2+b^2+c^2+d^2 $$
\u305d\u3057\u3066\u56db\u5143\u6570\u306e\u639b\u3051\u7b97\u306e\u30ce\u30eb\u30e0\u306f\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002
$$ ||pq||^2 = \\overline{pq}pq = \\bar{q}\\bar{p}pq = ||p||^2\\bar{q}q = ||p||^2||q||^2 $$
\u6700\u5f8c\u306b\u56db\u5143\u6570\u306e\u9006\u5143\u306f
$$ q^{-1}=\\frac{\\bar{q}}{||q||^2} $$
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u30d9\u30af\u30c8\u30eb\u306e\u56de\u8ee2<\/h2>\n\n\n\n

\u30ce\u30eb\u30e0\u304c1\u306e\u56db\u5143\u6570\u3092\u4f7f\u3046\u3068\u30d9\u30af\u30c8\u30eb\u306e\u56de\u8ee2\u3092\u7c21\u5358\u306b\u8868\u73fe\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3042\u308b\u30d9\u30af\u30c8\u30eb\\(\\mathbf{r}\\)\u3092\u5b9f\u6570\u90e8\u5206\u304c0\u306e\u56db\u5143\u6570\u3068\u3057\u3066
$$ r=(0, \\mathbf{r}) $$
\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u305d\u3057\u3066\u30ce\u30eb\u30e0\u304c1\u306e\u56db\u5143\u6570\\(q\\)\u3092\u30d9\u30af\u30c8\u30eb\\(r\\)\u306e\u56de\u8ee2\u306f
$$ r^{\\prime}=qr\\bar{q} $$
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u56db\u5143\u6570\\(q\\)\u306f\u4e09\u89d2\u95a2\u6570\u3092\u4f7f\u3063\u3066\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u8868\u73fe\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u3066\u3001
$$ q=(\\cos\\frac{\\theta}{2}, \\mathbf{u}\\sin\\frac{\\theta}{2}), \\quad ||\\mathbf{u}||=1 $$
\u3053\u3053\u3067\u30d9\u30af\u30c8\u30eb\\(\\mathbf{u}\\)\u306f\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u3067\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u56db\u5143\u6570\u3092\u8868\u73fe\u3057\u305f\u5834\u5408\u306b\u4e0a\u8a18\u306e\u56de\u8ee2\u306e\u5b9a\u7fa9\u5f0f\u306f\u3001\u30d9\u30af\u30c8\u30eb\\(\\mathbf{u}\\)\u3092\u56de\u8ee2\u8ef8\u3068\u3057\u3066\\(\\theta\\)\u3060\u3051\u53f3\u306d\u3058\u306e\u65b9\u5411\u306b\u56de\u8ee2\u3055\u305b\u308b\u3068\u3044\u3046\u8a08\u7b97\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n

\u6700\u5f8c\u306b3\u6b21\u5143\u306e\u56de\u8ee2\u884c\u5217\\(R\\)\u3068\u56db\u5143\u6570\\(q=a+bi+cj+dk\\)\u3068\u306e\u95a2\u4fc2\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002\\(qr\\bar{q}\\)\u3092\u9811\u5f35\u3063\u3066\u8a08\u7b97\u3057\u3066\u66f8\u304d\u4e0b\u3057\u3066\u307f\u308b\u3068\u4ee5\u4e0b\u306e\u5f0f\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002\u3082\u3061\u308d\u3093\u56db\u5143\u6570\u306e\u30eb\u30fc\u30eb\u3067\u3042\u308b\\(i^2=j^2=k^2=ijk=-1\\)\u3092\u9069\u7528\u3057\u3066\u6574\u7406\u3057\u307e\u3059\u3002
$$
\\begin{align}
r^{\\prime}=qr\\bar{q}&=(a+bi+cj+dk)(xi+yj+zk)(a-bi-cj-dk) \\\\
&=\\left\\{(a^2+b^2-c^2-d^2)x+2(bc-ad)y+2(ac+bd)z\\right\\}i \\\\
&\\quad+\\left\\{2(ad+bc)x+(a^2-b^2+c^2-d^2)y+2(-ab+cd)z\\right\\}j \\\\
&\\quad+\\left\\{2(bd-ac)x+2(cd+ab)y+(a^2-b^2-c^2+d^2)z\\right\\}k \\\\
\\end{align}
$$
\u3053\u3053\u304b\u3089\u884c\u5217\u306e\u5f62\u306b\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
$$
\\begin{align}
\\mathbf{r}^{\\prime}&=R\\mathbf{r} \\\\
R&=
\\begin{pmatrix}
a^2+b^2-c^2-d^2 & 2(bc-ad) & 2(ac+bd) \\\\
2(ad+bc) & a^2-b^2+c^2-d^2 & 2(-ab+cd) \\\\
2(bd-ac) & 2(cd+ab) & a^2-b^2-c^2+d^2
\\end{pmatrix}
\\end{align}
$$<\/p>\n\n\n\n

\u3068\u308a\u3042\u3048\u305a\u4eca\u56de\u306f\u3053\u3053\u307e\u3067\u3068\u3057\u307e\u3059\u3002
\u6b21\u56de\u306f\u5206\u5b50\u3092\u91cd\u306d\u5408\u308f\u305b\u308b\u305f\u3081\u306e\u6570\u5f0f\u3092\u5c0e\u51fa\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

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