{"id":7994,"date":"2026-10-05T17:12:45","date_gmt":"2026-10-05T08:12:45","guid":{"rendered":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/?p=7994"},"modified":"2026-10-05T21:08:57","modified_gmt":"2026-10-05T12:08:57","slug":"%e9%ab%98%e6%a0%a1%e3%81%be%e3%81%a7%e3%81%97%e3%81%8b%e6%95%b0%e5%ad%a6%e3%82%92%e3%82%84%e3%81%a3%e3%81%a6%e3%81%84%e3%81%aa%e3%81%84%e4%ba%ba%e3%81%a7%e3%82%82%e5%88%86%e3%81%8b%e3%82%8b%e3%83%8f","status":"publish","type":"post","link":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/2026\/10\/05\/%e9%ab%98%e6%a0%a1%e3%81%be%e3%81%a7%e3%81%97%e3%81%8b%e6%95%b0%e5%ad%a6%e3%82%92%e3%82%84%e3%81%a3%e3%81%a6%e3%81%84%e3%81%aa%e3%81%84%e4%ba%ba%e3%81%a7%e3%82%82%e5%88%86%e3%81%8b%e3%82%8b%e3%83%8f\/","title":{"rendered":"\u9ad8\u6821\u307e\u3067\u3057\u304b\u6570\u5b66\u3092\u3084\u3063\u3066\u3044\u306a\u3044\u4eba\u3067\u3082\u5206\u304b\u308b\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">\n\n\n\n<p class=\"wp-block-paragraph\">\u672c\u8a18\u4e8b\u306e\u4f5c\u6210\u306b\u3042\u305f\u308a\u3001\u8aac\u660e\u30fb\u6570\u5f0f\u306e\u78ba\u8a8d\u3068\u4fee\u6b63\u306bChatGPT\u3092\u6d3b\u7528\u3057\u307e\u3057\u305f\u3002\u5185\u5bb9\u306e\u6700\u7d42\u7684\u306a\u8cac\u4efb\u306f\u8457\u8005\u306b\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8a08\u7b97\u5316\u5b66\u30d7\u30ed\u30b0\u30e9\u30e0\u3067\u5206\u5b50\u8ecc\u9053\u3092\u6c42\u3081\u308b\u8a08\u7b97\u306f\u4f55\u3092\u3084\u3063\u3066\u3044\u308b\u304b\u77e5\u3063\u3066\u304a\u304d\u305f\u3044\u3002\u5206\u5b50\u8ecc\u9053\u3068\u96fb\u5b50\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u6c42\u3081\u308b\u5358\u7d14\u30d2\u30e5\u30c3\u30b1\u30eb\u6cd5\u306e\u6b21\u306e\u6bb5\u968e\u3068\u3057\u3066\uff08DFT\u306e\u524d\u306b\uff09\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u3001\u3067\u304d\u308b\u3060\u3051\u5168\u90e8\u306e\u6570\u5b57\u3092\u66f8\u304d\u306a\u304c\u3089\u8a08\u7b97\u3092\u9032\u3081\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u6559\u79d1\u66f8\u306f\u3001<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u4e2d\u4e95\u6d69\u5df3\u300c<a href=\"https:\/\/www.maruzen-publishing.co.jp\/book\/b10112338.html\">\u624b\u3067\u89e3\u304f\u91cf\u5b50\u5316\u5b66I \u57fa\u790e\u91cf\u5b50\u5316\u5b66\u30fbHartree-Fock\u7de8<\/a>\u300d2022\u5e74\u3001\u4e38\u5584\u51fa\u7248.<\/li>\n\n\n\n<li>\u85e4\u6c38 \u8302\u300c<a href=\"https:\/\/www.kspub.co.jp\/book\/detail\/1533257.html\">\u5165\u9580\u5206\u5b50\u8ecc\u9053\u6cd5 \u5206\u5b50\u8a08\u7b97\u3092\u624b\u304c\u3051\u308b\u524d\u306b<\/a>\u300d1990\u5e74\u3001\u8b1b\u8ac7\u793e.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u3067\u3059\u3002<br><br>\u4ee5\u4e0b\u3001\u30a6\u30a7\u30d6\u3067\u95b2\u89a7\u3067\u304d\u308b\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u306e\u89e3\u8aac\u3001<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5fd7\u8cc0 \u57fa\u4e4b,&nbsp;\u96fb\u5b50\u72b6\u614b\u7406\u8ad6\u306e\u521d\u6b69I,&nbsp;\u30a2\u30f3\u30b5\u30f3\u30d6\u30eb,&nbsp;2012,&nbsp;14 \u5dfb,&nbsp;2 \u53f7,&nbsp;p. 102-105.&nbsp;<a href=\"https:\/\/doi.org\/10.11436\/mssj.14.102\">https:\/\/doi.org\/10.11436\/mssj.14.102<\/a><\/li>\n\n\n\n<li>\u5fd7\u8cc0 \u57fa\u4e4b,&nbsp;\u96fb\u5b50\u72b6\u614b\u7406\u8ad6\u306e\u521d\u6b69II,&nbsp;\u30a2\u30f3\u30b5\u30f3\u30d6\u30eb,&nbsp;2012,&nbsp;14 \u5dfb,&nbsp;3 \u53f7,&nbsp;p. 134-137.&nbsp;<a href=\"https:\/\/doi.org\/10.11436\/mssj.14.134\">https:\/\/doi.org\/10.11436\/mssj.14.134<\/a><\/li>\n\n\n\n<li>\u5fd7\u8cc0 \u57fa\u4e4b,&nbsp;\u96fb\u5b50\u72b6\u614b\u7406\u8ad6\u306e\u521d\u6b69III,&nbsp;\u30a2\u30f3\u30b5\u30f3\u30d6\u30eb,&nbsp;2012,&nbsp;14 \u5dfb,&nbsp;4 \u53f7,&nbsp;p. 178-181.&nbsp;<a href=\"https:\/\/doi.org\/10.11436\/mssj.14.178\">https:\/\/doi.org\/10.11436\/mssj.14.178<\/a><\/li>\n\n\n\n<li>\u5fd7\u8cc0 \u57fa\u4e4b,&nbsp;\u96fb\u5b50\u72b6\u614b\u7406\u8ad6\u306e\u521d\u6b69IV,&nbsp;\u30a2\u30f3\u30b5\u30f3\u30d6\u30eb,&nbsp;2013,&nbsp;15 \u5dfb,&nbsp;1 \u53f7,&nbsp;p. 53-56.&nbsp;<a href=\"https:\/\/doi.org\/10.11436\/mssj.15.53\">https:\/\/doi.org\/10.11436\/mssj.15.53<\/a><\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u8a08\u7b97\u30b3\u30fc\u30c9<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>dc1394(Hiroyuki Kawai) (2025-08-21) \u3067\u304d\u308b\u3060\u3051\u7c21\u5358\u306bHartree-Fock\u6cd5\u3092\u89e3\u8aac\u3057\u3066\u307f\u308b\uff08C++17\u3068Julia\u306e\u30b3\u30fc\u30c9\u4ed8\u304d\uff09\u3001<a href=\"https:\/\/qiita.com\/dc1394\/items\/2333bf7444d8e67cfe6f\">https:\/\/qiita.com\/dc1394\/items\/2333bf7444d8e67cfe6f<\/a> .<\/li>\n\n\n\n<li>dc1394(Hiroyuki Kawai) (2020-07-07) \u3067\u304d\u308b\u3060\u3051\u7c21\u5358\u306bHartree-Fock\u6cd5\u3092\u89e3\u8aac\u3057\u3066\u307f\u308bpart2\u3001<a href=\"https:\/\/qiita.com\/dc1394\/items\/054748db9e1ee0a73a10\">https:\/\/qiita.com\/dc1394\/items\/054748db9e1ee0a73a10<\/a> .<\/li>\n\n\n\n<li>dc1394(Hiroyuki Kawai) (2023-01-02) \u6709\u9650\u8981\u7d20\u6cd5\u3067\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306b\u5bfe\u3059\u308bHartree-Fock\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u307f\u308b\uff08Julia\u3067\u3084\u3063\u3066\u307f\u305f\uff09\u3001<a href=\"https:\/\/qiita.com\/dc1394\/items\/9444ca19937545b30ea1\">https:\/\/qiita.com\/dc1394\/items\/9444ca19937545b30ea1<\/a> .<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">\u8a08\u7b97\u306e\u6e96\u5099<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u9ad8\u6821\u6570\u5b66\u306e\u5fa9\u7fd2<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">\u6307\u6570\u95a2\u6570\u306e\u5fae\u5206<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u203b \u5fae\u5206\u306f \\((f(x))&#8217;\\) \u3067\u306f\u306a\u304f \\(\\frac{d f(x)}{dx}\\) \u3084\\(\\frac{d}{dx} f(x)\\)\u3068\u66f8\u304d\u307e\u3059\u3002 \\((f(x))'{}&#8217;\\) \u306f \\(\\frac{d^2 f(x)}{dx^2}\\) \u3068\u66f8\u304d\u307e\u3059\u3002\\[\\frac{d}{dx}e^{kx} = k e^{kx} \\]\n\n\n\n<h4 class=\"wp-block-heading\">\u6307\u6570\u95a2\u6570\u306e\u7a4d\u5206<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\[\\int_a^b e^{kx}dx = \\left[\\frac{1}{k} e^{kx} \\right]_a^b = \\frac{1}{k}\\left(e^{kb} &#8211; e^{ka}\\right)\\]\n\n\n\n<h4 class=\"wp-block-heading\">\u95a2\u6570\u306e\u7a4d\u306e\u5fae\u5206<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\frac{d}{dx} (f(x)g(x)) = \\frac{d f(x)}{dx}g(x) + f(x)\\frac{d g(x)}{dx} \\]\n\n\n\n<h4 class=\"wp-block-heading\">\u90e8\u5206\u7a4d\u5206<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\"> \\[ \\int \\left(\\frac{d}{dx}f(x)\\right)g(x)\\, dx = f(x)g(x) &#8211; \\int f(x)\\left(\\frac{d}{dx}g(x)\\right) dx  \\]\n\n\n\n<h4 class=\"wp-block-heading\">\u884c\u5217\u306e\u8db3\u3057\u7b97<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\begin{pmatrix} a &amp; b \\\\ c &amp; d \\end{pmatrix} +  \\begin{pmatrix} p &amp; q \\\\ r &amp; s \\end{pmatrix}  =  \\begin{pmatrix} a+p &amp; b+q \\\\ c+r &amp; d+s \\end{pmatrix}  \\]\n\n\n\n<h4 class=\"wp-block-heading\">\u884c\u5217\u306e\u639b\u3051\u7b97<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\begin{pmatrix} a &amp; b \\\\ c &amp; d \\end{pmatrix} \\times \\begin{pmatrix} p &amp; q \\\\ r &amp; s \\end{pmatrix} = \\begin{pmatrix} ap+br &amp; aq+bs \\\\  cp+dr &amp; cq+ds \\end{pmatrix} \\]\n\n\n\n<h3 class=\"wp-block-heading\">\u30cf\u30fc\u30c8\u30ea\u30fc\u539f\u5b50\u5358\u4f4d\u7cfb<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b50\u306e\u4f4d\u7f6e\u3092\u539f\u5b50\u6838\u304b\u3089\u306e\u8ddd\u96e2\\(r\\)\u3001\u6975\u89d2\\(\\theta\\)\u3001\u65b9\u4f4d\u89d2\\(\\phi\\)\u3067\u8868\u308f\u3057\u305f\u6642\u3001\u6c34\u7d20\u539f\u5b50\u306e1s\u539f\u5b50\u8ecc\u9053\u306f\\[ \\psi_{1s}(r,\\theta,\\phi) = \\frac{1}{\\sqrt{\\pi a_{0}^3}}\\exp\\left(-\\frac{r}{a_0}\\right) \\]\u3067\u3059\u3002\u7403\u5bfe\u79f0\u306a\u306e\u3067\u6975\u89d2\uff08\\(\\theta\\)\uff09\u3068\u65b9\u4f4d\u89d2\uff08\\(\\phi\\)\uff09\u3092\u542b\u307e\u306a\u3044\u8ddd\u96e2\\(r\\)\u3060\u3051\u306e\u95a2\u6570\u3067\u3059\u3002\u3053\u3053\u3067\u3001 \\(a_0 = \\frac{4\\pi\\varepsilon_0\\hbar^2}{m_e e^2}\\) \u306f\u30dc\u30fc\u30a2\u534a\u5f84\uff08\u7d040.529 \u00c5\uff09\u3001\\(m_e\\)\u306f\u96fb\u5b50\u306e\u8cea\u91cf\u3001\\(\\hbar \\)\u306f\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\\(h\\)\u3092\\(2\\pi\\)\u3067\u5272\u3063\u305f\u5b9a\u6570\u3001\\(e\\)\u306f\u96fb\u6c17\u7d20\u91cf\u3001\\(\\varepsilon_0\\)\u306f\u771f\u7a7a\u306e\u8a98\u96fb\u7387\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u539f\u5b50\u5358\u4f4d\u7cfb\u3067\u306f\u3001\u96fb\u8377\u306e\u57fa\u672c\u5358\u4f4d\u3092\\(e\\)\u3001\u8cea\u91cf\u306e\u57fa\u672c\u5358\u4f4d\u3092\\(m_e\\)\u3001\u4f5c\u7528\u306e\u57fa\u672c\u5358\u4f4d\u3092\\(\\hbar\\)\u3001\u9577\u3055\u306e\u57fa\u672c\u5358\u4f4d\u3092\\(a_0\\)\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u57fa\u672c\u5358\u4f4d\u3092\\(E_h\\)\uff08\\(\\approx 4.36\\times 10^{-18} \\mathrm{J}\\)\uff09\u3068\u3057\u307e\u3059\u3002\u8a98\u96fb\u7387\u306e\u7d44\u7acb\u5358\u4f4d\u306f\\(\\frac{e^2}{a_0 E_h} = 4\\pi \\varepsilon_0\\) \u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u539f\u5b50\u5358\u4f4d\u7cfb\u3067\u306f1s\u539f\u5b50\u8ecc\u9053\u306f\\[ \\psi_{1s}(r) = \\frac{1}{\\sqrt{\\pi}}\\exp(-r) \\]\u3068\u3044\u3046\u5358\u7d14\u306a\u5f0f\u306b\u306a\u308a\u307e\u3059\u3057\u3001\u6c34\u7d20\u539f\u5b50\u6838\u3068\u96fb\u5b50\u306e\u9759\u96fb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\\(V(r) = -\\frac{1}{4\\pi \\varepsilon_0}\\frac{e^2}{r}\\)\u3082\\(V(r) = -\\frac{1}{r} \\) \u3068\u66f8\u3051\u307e\u3059\u3002\u4ee5\u4e0b\u3067\u306f\u8ddd\u96e2\u3082\u30dc\u30fc\u30a2\u534a\u5f84&nbsp;\\(a_0\\)&nbsp;\u3092\u5358\u4f4d\u3068\u3057\u3066\u8868\u3059\u305f\u3081\u3001\u4f8b\u3048\u3070&nbsp;\\(r=1\\)&nbsp;\u306f\u300c\u539f\u5b50\u6838\u304b\u3089&nbsp;\\(1a_0\\)&nbsp;\u96e2\u308c\u305f\u4f4d\u7f6e\u300d\u3092\u610f\u5473\u3057\u307e\u3059\u3002<br><br>\\(\\exp(-r)\\)\u306f\u6307\u6570\u95a2\u6570\u3067\u3001\\(e^{-r}\\)\uff08\u30cd\u30a4\u30d4\u30a2\u6570\\(e \\approx 2.72\\) \u306e\\(-r\\)\u4e57\uff09\u306e\u5225\u306e\u66f8\u304d\u65b9\u3067\u3059\uff08\u30cd\u30a4\u30d4\u30a2\u6570\u3068\u96fb\u6c17\u7d20\u91cf\u306f\u3069\u3061\u3089\u3082\\(e\\)\u3067\u8868\u308f\u3059\u306e\u3067\u7d1b\u3089\u308f\u3057\u3044\u306e\u3067\u3059\u304c\u3001\u539f\u5b50\u5358\u4f4d\u3067\u306f\u4ee5\u5f8c\u96fb\u6c17\u7d20\u91cf\\(e\\)\u306f\u51fa\u3066\u304d\u307e\u305b\u3093\uff09\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u6c34\u7d20\u539f\u5b50\u306e1s\u8ecc\u9053\u306f\u3001\u6709\u540d\u306a\u30b7\u30e5\u30ec\u30fc\u30c7\u30a3\u30f3\u30ac\u30fc\u65b9\u7a0b\u5f0f\\[ \\hat{H}\\psi = E\\psi \\]\u3092\u89e3\u3044\u3066\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u65b9\u7a0b\u5f0f\u306e\u610f\u5473\u306f\u3001\u95a2\u6570\\(\\psi\\) \u306b\\(\\hat{H}\\)\u3068\u3044\u3046\u6f14\u7b97\uff08\u639b\u3051\u7b97\u3084\u5fae\u5206\u306a\u3069\uff09\u3092\u3059\u308b\u3068\u3001\u5143\u306e\u95a2\u6570\u306b\u4fc2\u6570\\(E\\)\u304c\u639b\u304b\u308b\u3088\u3046\u306a\u95a2\u6570\\(\\psi\\)\u3092\u6c42\u3081\u306a\u3055\u3044\u3001\u3068\u3044\u3046\u3082\u306e\u3067\u3059\u3002\\(f(x) = e^{ax}\\)\u3068\u3044\u3046\u95a2\u6570\u306b\u5fae\u5206\u3068\u3044\u3046\u6f14\u7b97\u3092\u3059\u308b\u3068\\(f'(x)=a e^{ax}\\)\u3068\u306a\u3063\u3066\u3001\u5143\u306e\u95a2\u6570\u306b\u4fc2\u6570\\(a\\)\u304c\u639b\u304b\u308a\u307e\u3059\u306d\u3001\u307f\u305f\u3044\u306a\u3053\u3068\u3067\u3059\u3002\\(E\\)\u306f\u30a8\u30cd\u30eb\u30ae\u30fc\u3067\u3042\u308a\u3001\u96fb\u5b50\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u6b63\u306e\u5024\uff09\u3068\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u8ca0\u306e\u5024\uff09\u304b\u3089\u306a\u308a\u307e\u3059\u3002\\(\\hat{H}\\)\u306f\u7c92\u5b50\u306e\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u548c\uff09\u3092\u73fe\u308f\u3059\u6f14\u7b97\u5b50\u3067\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\( \\psi_{1s}(r) = \\frac{1}{\\sqrt{\\pi}}e^{-r} \\) \u306b\u3001\\(\\hat{H}\\)\u3068\u3044\u3046\u6f14\u7b97\u3092\u3057\u3066\u4fc2\u6570\u304c\u639b\u304b\u308b\u304b\uff08\\(\\frac{1}{\\sqrt{\\pi}}e^{-r}\\)\u304c\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u56fa\u6709\u95a2\u6570\u306b\u306a\u3063\u3066\u3044\u308b\u304b\u3069\u3046\u304b\uff09\u78ba\u304b\u3081\u3066\u307f\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\\(x,y,z\\)\u8ef8\u304c\u3042\u308b\u76f4\u4ea4\u5ea7\u6a19\u3067\u306f\u95a2\u6570\u3092\\(x,y,z\\)\u3067\u305d\u308c\u305e\u308c2\u56de\u5fae\u5206\uff08\u504f\u5fae\u5206\uff09\u3057\u3066\u8db3\u3057\u5408\u308f\u305b\u308b\u8a08\u7b97\u3092\u3057\u307e\u3059\u3002\u9ad8\u6821\u7269\u7406\u3067\u306f\u30c9\u30fb\u30d6\u30ed\u30a4\u6ce2\uff08\u7269\u8cea\u6ce2\uff09\u3092\u5b66\u3073\u307e\u3059\u304c\u3001\u7c92\u5b50\u306e\u904b\u52d5\u91cf\u3092\\(p\\)\u3068\u7f6e\u3044\u305f\u306e\u4e00\u6b21\u5143\u30c9\u30fb\u30d6\u30ed\u30a4\u6ce2\u5f0f\u306f\\( f(x) = e^{i\\frac{p}{\\hbar}x}\\)\u3067\u3059\uff08\\(\\hbar\\)\u306f\u30d7\u30e9\u30f3\u30af\u5b9a\u6570\\(h\\)\u3092\\(2\\pi\\)\u3067\u5272\u3063\u305f\u5b9a\u6570\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9ad8\u6821\u7269\u7406\u3067\u306f\\(\u904b\u52d5\u91cf p = \u8cea\u91cf \u00d7 \u901f\u3055 = mv\\)\u3001\\(\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc T = 1\/2 \u00d7 \u8cea\u91cf \u00d7 \u901f\u3055\u306e\u4e8c\u4e57 = \\frac{1}{2}mv^2\\)\u3068\u7fd2\u3044\u307e\u3059\u3002\u3053\u306e2\u3064\u306e\u5f0f\u304b\u3089\\(T = \\frac{p^2}{2m}\\) \u304c\u5c0e\u304b\u308c\u307e\u3059\u3002\u30c9\u30fb\u30d6\u30ed\u30a4\u6ce2\u306e\u5f0f\u30921\u56de\u5fae\u5206\u3059\u308b\u3068\u5143\u306e\u6ce2\u306b\\(i\\frac{p}{\\hbar} \\)\u304c\u639b\u304b\u308a\u30012\u56de\u5fae\u5206\u3059\u308b\u3068\\(-\\frac{p^2}{\\hbar^2}\\)\u304c\u639b\u304b\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\\(-\\frac{\\hbar^2}{2m}\\frac{d^2}{dx^2}\\)\u3092\u4f5c\u7528\u3055\u305b\u308b\u3068\u3001\u9ad8\u6821\u7269\u7406\u3067\u7fd2\u3046\u53e4\u5178\u529b\u5b66\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\\(\\frac{p^2}{2m}\\) \u304c\u4fc2\u6570\u3068\u3057\u3066\u73fe\u308f\u308c\u307e\u3059\u3002\u539f\u5b50\u5358\u4f4d\u7cfb\u3092\u4f7f\u3046\u306e\u3067\u3001\u4ee5\u4e0b\u306e\u5f0f\u306b\u306f\\(\\hbar\\)\u3068\\(m_e\\)\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6c34\u7d20\u539f\u5b50\u306e1s\u539f\u5b50\u8ecc\u9053\u3092\\(x,y,z\\)\u3067\u306f\u306a\u304f\u539f\u5b50\u6838\u304b\u3089\u306e\u8ddd\u96e2\\(r\\)\u3060\u3051\u306e\u95a2\u6570\u3067\u8868\u308f\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u5c11\u3057\u9055\u3046\u8a08\u7b97\uff08\u6f14\u7b97\uff09\u3092\u3057\u307e\u3059\u3002\u6f14\u7b97\u5b50\u306f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\hat{T} = -\\frac{1}{2}\\left[\\frac{1}{r^2}\\frac{d}{dr}\\left(r^2 \\frac{d}{dr}\\right)\\right] \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\hat{T}\\psi_{1s}(r) = -\\frac{1}{2}\\left[\\frac{1}{r^2}\\frac{d}{dr}\\left(r^2 \\frac{d}{dr}\\right)\\right] \\psi_{1s}(r) \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3068\u3044\u3046\u5f0f\u3067\u884c\u3046\u3053\u3068\u306f\u3001\\(\\psi_{1s}(r)\\)\u3092\u5fae\u5206\u3057\u3066\u3001\\(r^2\\)\u3092\u639b\u3051\u3066\u3001\u307e\u305f\u5fae\u5206\u3057\u305f\u5f8c\u306b\u3001\\(\\frac{1}{r^2}\\)\u3092\u639b\u3051\u3066\u3001\u6700\u5f8c\u306b\\(-\\frac{1}{2}\\)\u3092\u639b\u3051\u308b\u3001\u3067\u3059\u3002\u5b9f\u969b\u306b\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002\u9014\u4e2d\u3067\u3001\u95a2\u6570\u306e\u7a4d\u306e\u5fae\u5206 \\((r^2 \\times e^{-r})&#8217; = (r^2)&#8217; \\times e^{-r} + r^2 \\times (e^{-r})&#8217; = (2r &#8211; r^2) e^{-r} \\) \u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> \\[ -\\frac{1}{2}\\left[\\frac{1}{r^2}\\frac{d}{dr}\\left(r^2 \\frac{d}{dr}\\right)\\right] \\frac{1}{\\sqrt{\\pi}}e^{-r} \\]\n\n\n\n<p class=\"wp-block-paragraph\">\\[ =&nbsp; -\\frac{1}{2}\\left[\\frac{1}{r^2}\\frac{d}{dr}\\right] r^2\\left(-1\\right)&nbsp; \\frac{1}{\\sqrt{\\pi}}e^{-r}\\]\n\n\n\n<p class=\"wp-block-paragraph\">\\[ =  -\\frac{1}{2}\\left[\\frac{1}{r^2}\\right] (-1)(2r &#8211; r^2) &nbsp; \\frac{1}{\\sqrt{\\pi}}e^{-r}   \\]\n\n\n\n<p class=\"wp-block-paragraph\">\\[ = \\left(\\frac{1}{r} &#8211; \\frac{1}{2}\\right) &nbsp; \\frac{1}{\\sqrt{\\pi}}e^{-r} \\]\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\frac{1}{r}\\)\u3068\u3044\u3046\u9805\u304c\u6b8b\u3063\u3066\u3044\u307e\u3059\u304c\u3001\u3053\u306e\u9805\u306f\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50\u306b\u3088\u3063\u3066\u6253\u3061\u6d88\u3055\u308c\u307e\u3059\u3002\u96fb\u5b50\u306e\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8868\u308f\u3059\u6f14\u7b97\u5b50\uff08\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\uff09\u306f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[\\hat{H} = \\hat{T} + \\hat{V}\\] <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3067\u3001\u6c34\u7d20\u539f\u5b50\u306e\u5834\u5408\u306f\\(\\hat{V}= -\\frac{1}{r}\\)\uff08\u539f\u5b50\u6838\u3068\u96fb\u5b50\u306e\u30af\u30fc\u30ed\u30f3\u76f8\u4e92\u4f5c\u7528\uff09\u306a\u306e\u3067\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\hat{H}\\psi_{1s}(r) = (\\hat{T} + \\hat{V})\\psi_{1s}(r) = \\left[\\left(\\frac{1}{r} &#8211; \\frac{1}{2}\\right) + \\left(-\\frac{1}{r}\\right)\\right]  \\psi_{1s}(r)  = -\\frac{1}{2} \\psi_{1s}(r) \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3067\u3001\u6c34\u7d20\u539f\u5b50\u306e\u57fa\u5e95\u72b6\u614b\u306e\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\\(E =-0.5\\;E_{\\mathrm h}\\) \u304c\u4fc2\u6570\u3068\u3057\u3066\u51fa\u3066\u304f\u308b\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\\(\\langle{}T\\rangle\\)\u3068\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\\(\\langle{}V\\rangle\\)\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\\(\\hat{H}\\psi = E\\psi\\)\u306e\u4e21\u8fba\u306b\u5de6\u304b\u3089\\(\\psi^*\\)\uff08\\(\\psi\\)\u306e\u8907\u7d20\u5171\u5f79\u3002\u4f8b\u3048\u3070\\(f(x) = \\cos x + i\\sin x\\)\u306e\u8907\u7d20\u5171\u5f79\u306f\\(f^*(x) = \\cos x &#8211; i\\sin x\\)\u3002\u6c34\u7d20\u539f\u5b50\u306e1s\u8ecc\u9053\u306f\u5b9f\u95a2\u6570\u306a\u306e\u3067\u3053\u3053\u3067\u306f\\(\\psi^{*} = \\psi\\)\uff09\u3092\u639b\u3051\u3066\u3001\u5168\u7a7a\u9593\u306b\u6e21\u3063\u3066\u7a4d\u5206\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4efb\u610f\u306e\u6ce2\u52d5\u95a2\u6570\u306b\u5bfe\u3059\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u671f\u5f85\u5024\u306f\u3001<br>\\[\\langle{}H\\rangle_\\psi = \\frac{\\int\\psi^* \\hat{H}\\psi d\\tau}{\\int|\\psi|^2 d\\tau} \\]\u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\\(\\psi\\)&nbsp;\u304c\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308c\u3070\u5206\u6bcd\u306f1\u3067\u3059\u3002\u3055\u3089\u306b&nbsp;\\(\\psi\\)&nbsp;\u304c\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306e\u56fa\u6709\u95a2\u6570\u306a\u3089\u3001\\(\\langle H\\rangle=E\\)&nbsp;\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5206\u6bcd\u306f1\u3068\u3057\u3066\u3001\u7a4d\u5206\u3092\u5168\u90e8\u66f8\u304f\u3068\u3001\\[ \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\psi^* \\hat{H} \\psi\\ r^2 \\sin\\theta dr d\\theta d\\phi = E \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\psi^* \\psi\\ r^2 \\sin\\theta dr d\\theta d\\phi. \\]<br><br>\\(E\\)\u306f\u5358\u306a\u308b\u6570\u5024\u306a\u306e\u3067\u3001\u7a4d\u5206\u8a18\u53f7\u306e\u5916\u306b\u51fa\u305b\u307e\u3059\u30021s\u539f\u5b50\u8ecc\u9053\u306f\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308b\uff08\u7d76\u5bfe\u5024\u306e\u4e8c\u4e57\u3092\u7a4d\u5206\u3059\u308b\u30681\u306b\u306a\u308b\u3088\u3046\u306b\u4fc2\u6570\u304c\u8abf\u6574\u3055\u308c\u3066\u3044\u308b\uff09\u306e\u3067\u3001\\(\\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\psi^* \\psi\\ r^2 \\sin\\theta dr d\\theta d\\phi =1\\)\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ E = \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\psi^* \\hat{H} \\psi\\ r^2 \\sin\\theta dr d\\theta d\\phi \\]\n\n\n\n<p class=\"wp-block-paragraph\">\\[ = \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\psi^* \\hat{T} \\psi r^2 \\sin\\theta dr d\\theta d\\phi + \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\psi^* \\hat{V} \\psi\\cdot{} r^2 \\sin\\theta dr d\\theta d\\phi \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u306f\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\langle{}T\\rangle = \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\frac{1}{\\sqrt{\\pi}}e^{-r} \\left(\\frac{1}{r} &#8211; \\frac{1}{2}\\right) \\frac{1}{\\sqrt{\\pi}}e^{-r} \\cdot{} r^2 \\sin\\theta dr d\\theta d\\phi \\]\n\n\n\n<p class=\"wp-block-paragraph\">\\[ = \\frac{1}{\\pi} \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\left(r &#8211; \\frac{r^2}{2}\\right) e^{-2r} \\sin\\theta dr d\\theta d\\phi .\\]\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\theta\\)\u3068\\(\\phi\\)\u306e\u7a4d\u5206\u3092\u5148\u306b\u6e08\u307e\u305b\u307e\u3059\u3002\\(\\int_0^{\\pi} \\sin\\theta d\\theta = [-\\cos\\theta]_0^{\\pi} = -(-1 &#8211; 1) = 2 \\)\u3001\\(\\int_0^{2\\pi}  d\\phi  = [\\phi]_0^{2\\pi} = 2\\pi \\)\u3002<br>\\[ \\langle{}T\\rangle =4\\int_0^{\\infty} \\left(r &#8211; \\frac{r^2}{2}\\right)  e^{-2r}  dr \\]<br>\u90e8\u5206\u7a4d\u5206\u306e\u516c\u5f0f \\(\\int f(r) g'(r) dr = f(r)g(r) &#8211; \\int f'(r)g(r) dr \\)\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\int_0^{\\infty} r e^{-2r} dr = \\left[r\\left(\u2013\\frac{1}{2}\\right)e^{-2r}\\right]_0^{\\infty} &#8211; \\int_0^{\\infty} \\left(\u2013\\frac{1}{2}\\right)e^{-2r} dr \\] <br>\\[= 0 &#8211; \\left[\\frac{1}{4} e^{-2r}\\right]_0^{\\infty} = \\frac{1}{4} .\\]<br>\u3053\u306e\u7a4d\u5206\u306b\u306f\u516c\u5f0f\\(\\int_0^{\\infty} r^n e^{-ar} dr = \\frac{n!}{a^{n+1}}\\) \u304c\u3042\u3063\u3066\u3001\u3053\u308c\u3092\u4f7f\u3046\u3068\u3001\\[ \\int_0^{\\infty} r^2 e^{-2r} dr = \\frac{2!}{2^3} = \\frac{1}{4}.\\] <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001\\[ \\langle{}T\\rangle =4\\int_0^{\\infty} \\left(r &#8211; \\frac{r^2}{2}\\right) e^{-2r} dr = 4\\times \\left(\\frac{1}{4} &#8211; \\frac{1}{2}\\times \\frac{1}{4}\\right) = \\frac{1}{2}\\,E_{\\mathrm h}.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u6b21\u306b\u3001\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u306f\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\langle{}V\\rangle = 4\\int_0^{\\infty} e^{-r} \\left(-\\frac{1}{r}\\right) e^{-r} \\times r^2 dr = -4\\int_0^{\\infty} r e^{-2r} dr = -4 \\times \\frac{1}{4} = -1\\,E_{\\mathrm h}.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\\(V\\propto-1\/r\\)&nbsp;\u578b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306b\u5bfe\u3059\u308b\u5b9a\u5e38\u675f\u7e1b\u72b6\u614b\u3067\u306f\u3001\u30d3\u30ea\u30a2\u30eb\u5b9a\u7406\u304b\u3089\u3001\\[ \\langle{}V\\rangle = -2\\langle{}T\\rangle \\] \u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002\u540c\u3058 \\(1\/r\\) \u578b\u306e\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092\u3082\u3064\u91cd\u529b\u7cfb\u3067\u3082\u30d3\u30ea\u30a2\u30eb\u5b9a\u7406\u304c\u6210\u308a\u7acb\u3064\u306e\u3067\u3001\u91cd\u529b\u7684\u306b\u5341\u5206\u5e73\u8861\u5316\u3055\u308c\u305f\uff08\u30d3\u30ea\u30a2\u30eb\u5316\u3057\u305f\uff09\u9280\u6cb3\u3084\u9280\u6cb3\u56e3\u306e\u8cea\u91cf\u306e\u63a8\u5b9a\u306b\u3082\u5229\u7528\u3067\u304d\u308b\u305d\u3046\u3067\u3059\u3002\u30d3\u30ea\u30a2\u30eb\u5b9a\u7406\u304b\u3089\u63a8\u5b9a\u3057\u305f\u9280\u6cb3\u56e3\u306e\u7dcf\u8cea\u91cf\u304c\u5149\u304b\u3089\u898b\u7a4d\u3089\u308c\u308b\u7269\u8cea\u306e\u8cea\u91cf\u3088\u308a\u306f\u308b\u304b\u306b\u5927\u304d\u3044\u3053\u3068\u306f\u3001\u5b87\u5b99\u306e\u6697\u9ed2\u7269\u8cea\uff08\u30c0\u30fc\u30af\u30de\u30bf\u30fc\uff09\u306e\u5b58\u5728\u3092\u793a\u3059\u8a3c\u62e0\u306e\u4e00\u3064\u3068\u3055\u308c\u3066\u3044\u308b\u305d\u3046\u3067\u3059\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u8a08\u7b97<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u6ce2\u52d5\u95a2\u6570<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306f\u96fb\u5b50\u304c2\u3064\u3042\u308a\u307e\u3059\u3002\u53b3\u5bc6\u306a\u7a7a\u9593\u6ce2\u52d5\u95a2\u6570\u306f\u3001\u96fb\u5b501\u306e\u539f\u5b50\u6838\u304b\u3089\u306e\u8ddd\u96e2\\(r_1\\)\u3001\u96fb\u5b502\u306e\u539f\u5b50\u6838\u304b\u3089\u306e\u8ddd\u96e2\\(r_2\\)\u3060\u3051\u3067\u306a\u304f\u3001\u96fb\u5b50\u9593\u8ddd\u96e2\\(r_{12}\\)\u306b\u3082\u4f9d\u5b58\u3059\u308b\\[ \\psi(r_1,r_2,r_{12}) \\]\u306e\u3088\u3046\u306a\u95a2\u6570\u306b\u306a\u308a\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u6c34\u7d20\u539f\u5b50\u306e\u3088\u3046\u306a\u9589\u3058\u305f\u5f62\u306e\u53b3\u5bc6\u306a\u89e3\u6790\u89e3\u306f\u77e5\u3089\u308c\u3066\u3044\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001\u591a\u96fb\u5b50\u6ce2\u52d5\u95a2\u6570\u3092\u4e00\u96fb\u5b50\u8ecc\u9053\u304b\u3089\u4f5c\u3063\u305f1\u500b\u306e\u30b9\u30ec\u30fc\u30bf\u30fc\uff08Slater\uff09\u884c\u5217\u5f0f\u3067\u8fd1\u4f3c\u3057\u307e\u3059\u3002\u305d\u308c\u304c\u4f55\u304b\u306f\u5f8c\u304b\u3089\u8aac\u660e\u3057\u307e\u3059\u3002\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u57fa\u5e95\u72b6\u614b\u3067\u306f\u30012\u3064\u306e\u96fb\u5b50\u304c\u540c\u3058\u7a7a\u9593\u8ecc\u9053\u3092\u5360\u6709\u3059\u308b\u305f\u3081\u3001\u305d\u306e\u7a7a\u9593\u90e8\u5206\u306f\u307e\u305a\\[ \\Psi(\\mathbf{r_1},\\mathbf{r_2}) \\approx \\phi(\\mathbf r_1)\\phi(\\mathbf r_2)\\]\u306e\u3088\u3046\u306a\u95a2\u6570\u306e\u7a4d\u3067\u8868\u308f\u3059\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u6ce2\u52d5\u95a2\u6570\u306e\u53cd\u5bfe\u79f0\u5316<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b50\u306e\u4f4d\u7f6e\u306b\u4f9d\u5b58\u3059\u308b\u7a7a\u9593\u8ecc\u9053\u3092\\(\\phi_{1s}(\\mathbf r)\\)\u3068\u66f8\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u3001(\\(\\mathbf r\\))\u306f\u96fb\u5b50\u306e\u4f4d\u7f6e\u30d9\u30af\u30c8\u30eb\u3067\u3059\u3002 \u4e0a\u5411\u304d\u3068\u4e0b\u5411\u304d\u306e\u30b9\u30d4\u30f3\u72b6\u614b\u3092\u8868\u3059\u95a2\u6570\u3092\u3001\u305d\u308c\u305e\u308c \\(\\alpha(\\sigma)\\)\u3001\\(\\beta(\\sigma)\\) \u3068\u3057\u307e\u3059\u3002\u7a7a\u9593\u8ecc\u9053\u3068\u30b9\u30d4\u30f3\u95a2\u6570\u306e\u7a4d\u3092<strong>\u30b9\u30d4\u30f3\u8ecc\u9053<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002\u4f4d\u7f6e\u3068\u30b9\u30d4\u30f3\u306e\u5909\u6570\u3092\u307e\u3068\u3081\u3066 \\(x=(\\mathbf r,\\sigma)\\) \u3068\u66f8\u304f\u3068\u3001\u4eca\u56de\u7528\u3044\u308b2\u3064\u306e\u30b9\u30d4\u30f3\u8ecc\u9053\u306f\\[ \\chi_{1s\\alpha}(x)=\\phi_{1s}(\\mathbf r)\\alpha(\\sigma),<br>\\qquad<br>\\chi_{1s\\beta}(x)=\\phi_{1s}(\\mathbf r)\\beta(\\sigma) \\]\u3067\u3059\u3002\u3053\u306e2\u3064\u306f\u7a7a\u9593\u90e8\u5206\u304c\u540c\u3058\u3067\u3082\u3001\u30b9\u30d4\u30f3\u90e8\u5206\u304c\u7570\u306a\u308b\u305f\u3081\u3001\u7570\u306a\u308b\u4e00\u96fb\u5b50\u72b6\u614b\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305a\u3001\u96fb\u5b501\u306b \\(\\alpha\\)\u3001\u96fb\u5b502\u306b \\(\\beta\\) \u306e\u30b9\u30d4\u30f3\u8ecc\u9053\u3092\u5272\u308a\u5f53\u3066\u305f\u7a4d\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\phi_{1s}(\\mathbf r_1)\\phi_{1s}(\\mathbf r_2)\\alpha(\\sigma_1)\\beta(\\sigma_2). \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u304b\u3057\u3001\u96fb\u5b50\u306f\u4e92\u3044\u306b\u533a\u5225\u3067\u304d\u306a\u3044\u540c\u7a2e\u306e\u30d5\u30a7\u30eb\u30df\u7c92\u5b50\u3067\u3059\u3002\u96fb\u5b50\u3092\u8868\u3059\u30e9\u30d9\u30eb1\u30682\u3092\u4ea4\u63db\u3057\u305f\u3068\u304d\u306b\u3001\u96fb\u5b50\u306e\u5168\u6ce2\u52d5\u95a2\u6570\u306f\u7b26\u53f7\u304c\u53cd\u8ee2\u3057\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001\u30e9\u30d9\u30eb\u3092\u4ea4\u63db\u3057\u305f\uff08\\(\\phi_{1s}(\\mathbf r_2)\\phi_{1s}(\\mathbf r_1)<br>\\alpha(\\sigma_2)\\beta(\\sigma_1)\\)\uff09\u3068\u3044\u3046\u9805\u3082\u542b\u3081\u3066\u8003\u3048\u307e\u3059\u3002\u3053\u3053\u3067\u300c\u30e9\u30d9\u30eb\u3092\u4ea4\u63db\u3059\u308b\u300d\u3068\u306f\u3001\u96fb\u5b50\u3092\u8868\u3059\u30e9\u30d9\u30eb1\u30682\u3092\u3001\u4f4d\u7f6e\u3068\u30b9\u30d4\u30f3\u306e\u4e21\u65b9\u3067\u5165\u308c\u63db\u3048\u308b\u3053\u3068\u3067\u3059\u3002\u9078\u629e\u80a2\u3068\u3057\u3066\u306f\u5272\u308a\u5f53\u3066\u3092\u5165\u308c\u63db\u3048\u305f\u9805\u3092\u5143\u306e\u5f0f\u304b\u3089\u8db3\u3059\u304b\u5f15\u304f\u304b\u3067\u3059\u30022\u3064\u306e\u7a4d\u3092\u8db3\u3057\u305f\u6ce2\u52d5\u95a2\u6570\u306f\u3001\\[ \\Psi_{+}(x_1,x_2) = \\frac{1}{\\sqrt{2}} \\left\\{\\phi_{1s}(\\mathbf r_1)\\phi_{1s}(\\mathbf r_2) \\alpha(\\sigma_1)\\beta(\\sigma_2)+\\phi_{1s}(\\mathbf r_2)\\phi_{1s}(\\mathbf r_1) \\alpha(\\sigma_2)\\beta(\\sigma_1)\\right\\} \\]\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001\\(x_i=(\\mathbf r_i,\\sigma_i)\\) \u306f\u3001\u96fb\u5b50 \\(i\\) \u306e\u4f4d\u7f6e\u3068\u30b9\u30d4\u30f3\u306e\u5909\u6570\u3092\u307e\u3068\u3081\u305f\u3082\u306e\u3067\u3059\u3002\u3053\u306e\u6ce2\u52d5\u95a2\u6570\u306f\u3001\u96fb\u5b501\u3068\u96fb\u5b502\u306e\u30e9\u30d9\u30eb\u3092\u5165\u308c\u63db\u3048\u3066\u3082\u3001\\[\\Psi_{+}(x_2,x_1)=\\Psi_{+}(x_1,x_2) \\]\u3068\u306a\u308a\u3001\u7b26\u53f7\u304c\u5909\u308f\u308a\u307e\u305b\u3093\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u96fb\u5b50\u306e\u6ce2\u52d5\u95a2\u6570\u306b\u5fc5\u8981\u306a\u53cd\u5bfe\u79f0\u6027\u3092\u6e80\u305f\u3057\u307e\u305b\u3093\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e00\u65b9\u30012\u3064\u306e\u7a4d\u3092\u5f15\u3044\u305f\u6ce2\u52d5\u95a2\u6570\u306f\u3001\\[\\Psi_{-}(x_1,x_2) = \\frac{1}{\\sqrt{2}} \\left\\{\\phi_{1s}(\\mathbf r_1)\\phi_{1s}(\\mathbf r_2)\\alpha(\\sigma_1)\\beta(\\sigma_2) -\\phi_{1s}(\\mathbf r_2)\\phi_{1s}(\\mathbf r_1)\\alpha(\\sigma_2)\\beta(\\sigma_1)\\right\\} \\]\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u306f\u3001\\[ \\Psi_{-}(x_2,x_1)=-\\Psi_{-}(x_1,x_2) \\]\u3068\u306a\u308b\u305f\u3081\u3001\u5fc5\u8981\u306a\u53cd\u5bfe\u79f0\u6027\u3092\u6e80\u305f\u3057\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u5f0f\u306b\u73fe\u308f\u308c\u308b\uff08\\(A\\times B &#8211; C\\times D\\)\uff09\u3068\u3044\u3046\u5f62\u306f\\(2\\times 2\\)\u306e\u884c\u5217\u5f0f\\[ \\begin{vmatrix} A &amp; C \\\\ D &amp;B\\end{vmatrix} \\] \u3068\u540c\u3058\u5f62\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u4e0a\u306e\u6ce2\u52d5\u95a2\u6570\u306f\\[ \\Psi_{-}(x_1,x_2) = \\frac{1}{\\sqrt{2}} \\begin{vmatrix} \\phi_{1s}(\\mathbf r_1)\\alpha(\\sigma_1) &amp; \\phi_{1s}(\\mathbf r_1)\\beta(\\sigma_1) \\\\ \\phi_{1s}(\\mathbf r_2)\\alpha(\\sigma_2) &amp; \\phi_{1s}(\\mathbf r_2)\\beta(\\sigma_2) \\end{vmatrix} \\]\u3068\u66f8\u3051\u307e\u3059\u3002\u3053\u306e\u884c\u5217\u5f0f\u304c\u300c\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u300d\u3068\u547c\u3070\u308c\u307e\u3059\u3002\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u3088\u3046\u306a2\u96fb\u5b50\u7cfb\u3067\u306f\u3001\u308f\u3056\u308f\u3056\u884c\u5217\u5f0f\u3067\u66f8\u304f\u3068\u304b\u3048\u3063\u3066\u5f0f\u304c\u5927\u304d\u304f\u898b\u3048\u307e\u3059\u304c\u3001\u96fb\u5b50\u6570\u304c\u5897\u3048\u308b\u3068\u3001\u53cd\u5bfe\u79f0\u306a\u6ce2\u52d5\u95a2\u6570\u3092\u7c21\u6f54\u306b\u8868\u305b\u308b\u306e\u3067\u975e\u5e38\u306b\u4fbf\u5229\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d1\u30a6\u30ea\u306e\u6392\u4ed6\u539f\u7406\u3068\u306e\u95a2\u4fc2\u3092\u78ba\u8a8d\u3059\u308b\u305f\u3081\u306b\u30012\u3064\u306e\u96fb\u5b50\u3092\u540c\u3058\u30b9\u30d4\u30f3\u8ecc\u9053 \\(\\chi_a\\) \u306b\u5165\u308c\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u5834\u5408\u3001\u53cd\u5bfe\u79f0\u5316\u3057\u305f\u5f0f\u306f\u3001\\[ \\frac{1}{\\sqrt{2}} \\left\\{ \\chi_a(x_1)\\chi_a(x_2) &#8211; \\chi_a(x_2)\\chi_a(x_1) \\right\\} =0 \\]\u3068\u306a\u308a\u3001\u6ce2\u52d5\u95a2\u6570\u304c\u6d88\u3048\u3066\u3057\u307e\u3044\u307e\u3059\u3002\u3064\u307e\u308a\u30012\u3064\u306e\u96fb\u5b50\u304c\u540c\u3058\u30b9\u30d4\u30f3\u8ecc\u9053\u3092\u5360\u3081\u308b\u72b6\u614b\u306f\u4f5c\u308c\u307e\u305b\u3093\u3002\u3053\u308c\u304c\u30d1\u30a6\u30ea\u306e\u6392\u4ed6\u539f\u7406\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002\u5404\u30b9\u30d4\u30f3\u8ecc\u9053\u304c\u898f\u683c\u5316\u3055\u308c\u3001\u4e92\u3044\u306b\u76f4\u4ea4\u3057\u3066\u3044\u308b\u3068\u304d\u3001\u4fc2\u6570 \\(1\/\\sqrt{2}\\) \u306b\u3088\u3063\u3066\u5168\u6ce2\u52d5\u95a2\u6570\u3082\u898f\u683c\u5316\u3055\u308c\u307e\u3059\u3002\u30b9\u30d4\u30f3\u8ecc\u9053\u306e\u5b9a\u7fa9\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\u5171\u901a\u306e\u7a7a\u9593\u90e8\u5206\u3092\u304f\u304f\u308a\u51fa\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><munder><munder><mfrac><mrow><mi>\u03b1<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>\u03c3<\/mi><mn>1<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>\u03b2<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>\u03c3<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mi>\u03b2<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>\u03c3<\/mi><mn>1<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mi>\u03b1<\/mi><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msub><mi>\u03c3<\/mi><mn>2<\/mn><\/msub><mo form=\"postfix\" stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">)<\/mo><\/mrow><msqrt><mn>2<\/mn><\/msqrt><\/mfrac><mo stretchy=\"true\" style=\"math-depth:0;\">\u23df<\/mo><\/munder><mtext>\u53cd\u5bfe\u79f0\u306a\u30b9\u30d4\u30f3\u90e8\u5206<\/mtext><\/munder><\/mrow><annotation encoding=\"application\/x-tex\">\\underbrace{\n\\frac{\n\\alpha(\\sigma_1)\\beta(\\sigma_2)\n&#8211;\n\\beta(\\sigma_1)\\alpha(\\sigma_2)\n}{\\sqrt{2}}\n}_{\\text{\u53cd\u5bfe\u79f0\u306a\u30b9\u30d4\u30f3\u90e8\u5206}}<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30b9\u30d4\u30f3\u90e8\u5206\u306f\u3001\u5168\u30b9\u30d4\u30f3\u304c0\u306e<strong>\u30b9\u30d4\u30f3\u4e00\u91cd\u9805<\/strong>\u3092\u8868\u3057\u307e\u3059\u3002\u7a7a\u9593\u90e8\u5206\u306f\u96fb\u5b50\u306e\u4ea4\u63db\u306b\u5bfe\u3057\u3066\u5bfe\u79f0\u3067\u3059\u304c\u3001\u30b9\u30d4\u30f3\u90e8\u5206\u304c\u53cd\u5bfe\u79f0\u306a\u306e\u3067\u3001\u5168\u4f53\u306f\u53cd\u5bfe\u79f0\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306f+2\u306e\u96fb\u8377\u3092\u6301\u30641\u3064\u306e\u539f\u5b50\u6838\u30682\u3064\u306e\u96fb\u5b50\u304b\u3089\u69cb\u6210\u3055\u308c\u307e\u3059\u3002\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306b\u5bfe\u3059\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306f\u3001\\[ \\hat{H} = \\left[-\\frac{1}{2}\\nabla_1^2 &#8211; \\frac{2}{r_1}\\right] + \\left[-\\frac{1}{2}\\nabla_2^2 &#8211; \\frac{2}{r_2}\\right] + \\frac{1}{r_{12}} \\]\u3068\u306a\u308a\u307e\u3059\u3002\\({r_1}\\)\u306f\u96fb\u5b501\u306e\u539f\u5b50\u6838\u304b\u3089\u306e\u8ddd\u96e2\u3001\\({r_2}\\)\u306f\u96fb\u5b502\u306e\u539f\u5b50\u6838\u304b\u3089\u306e\u8ddd\u96e2\u3001\\({r_{12} = |\\mathbf{r_1} &#8211; \\mathbf{r_2} |}\\)\u306f\u96fb\u5b501\u3068\u96fb\u5b502\u306e\u8ddd\u96e2\u3067\u3059\u3002\\(-\\frac{2}{r_1}\\) \u306e\u5206\u5b50\u304c2\u306b\u306a\u3063\u3066\u3044\u308b\u306e\u306f\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u6838\u306e\u96fb\u8377\u304c+2\u3067\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u304c2\u500d\u306b\u306a\u308b\u304b\u3089\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306f\u6ce2\u52d5\u95a2\u6570\u306e\u30b9\u30d4\u30f3\u90e8\u5206\u306b\u306f\u4f5c\u7528\u3057\u306a\u3044\u306e\u3067\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8a08\u7b97\u3059\u308b\u3068\u304d\u306b\u306f\u3001\u898f\u683c\u5316\u3055\u308c\u305f\u30b9\u30d4\u30f3\u90e8\u5206\u3092\u7a4d\u5206\u3059\u308b\u30681\u306b\u306a\u308a\u3001\u7a7a\u9593\u90e8\u5206\\(\\phi_{1s}(r_1)\\phi_{1s}(r_2)\\) \u3060\u3051\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3059\u308b\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u3002\u3057\u304b\u3057\u3001\u5f8c\u3067\u4ea4\u63db\u9805\u304c\u3069\u3053\u304b\u3089\u73fe\u308f\u308c\u308b\u304b\u3092\u7406\u89e3\u3059\u308b\u305f\u3081\u306b\u3001\u3053\u3053\u3067\u306f\u30b9\u30d4\u30f3\u90e8\u5206\u3092\u542b\u3081\u305f\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u306e\u307e\u307e\u8a08\u7b97\u3092\u9032\u3081\u307e\u3059\u3002\u4f4d\u7f6e\u3068\u30b9\u30d4\u30f3\u3092\u5408\u308f\u305b\u305f\u5909\u6570\u3092\\[x=(\\mathbf r,\\sigma)\\]\u3068\u3059\u308b\u3068\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u57fa\u5e95\u72b6\u614b\u306e\u6ce2\u52d5\u95a2\u6570\u306f \\[ \\psi(x_1,x_2)= \\frac{1}{\\sqrt{2}}\\begin{vmatrix} \\chi_{1s\\alpha}(x_1) &amp; \\chi_{1s\\beta}(x_1)\\\\ \\chi_{1s\\alpha}(x_2) &amp; \\chi_{1s\\beta}(x_2)\\end{vmatrix} = \\frac{1}{\\sqrt{2}}(\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2) &#8211; \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2))\\]\u306e\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305f\u3001\u30b9\u30d4\u30f3\u8ecc\u9053\u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3068\u304d\u306b\u306f\u3001\u7a7a\u9593\u5ea7\u6a19\u3068\u30b9\u30d4\u30f3\u5909\u6570\u306e\u4e21\u65b9\u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u306e\u3067\u3001\\[<br>dx=d\\tau\\,d\\sigma\\]\u3068\u66f8\u304f\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Slater\uff08\u30b9\u30ec\u30fc\u30bf\u30fc\uff09\u578b\u8ecc\u9053\uff08STO\uff09\u3092\u4f7f\u3063\u3066\u30a8\u30cd\u30eb\u30ae\u30fc\u8a55\u4fa1\u3059\u308b<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">He\u306e1s\u7a7a\u9593\u8ecc\u9053\u3092\\[ \\phi_{1s}(r) = \\sqrt{\\frac{\\zeta^3}{\\pi}}e^{-\\zeta r} \\]\u3068\u3057\u307e\u3059\u3002\\(\\zeta\\)\u306f\u5f8c\u3067\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u6700\u5c0f\u306b\u306a\u308b\u3088\u3046\u306b\u8abf\u6574\u3059\u308b\u305f\u3081\u306e\u30d1\u30e9\u30e1\u30fc\u30bf\u3067\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305a\u3001\u96fb\u5b501\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\\[\\langle{}T\\rangle_1 = \\iint\\Psi^*(x_1,x_2)\\left(-\\frac{1}{2}\\nabla_1^2\\right)\\Psi(x_1,x_2)\\,dx_1dx_2\\]\u3092\u6c42\u3081\u307e\u3059\u3002\u4eca\u56de\u7528\u3044\u308b\u8ecc\u9053\u3084\u30b9\u30d4\u30f3\u95a2\u6570\u306f\u5b9f\u95a2\u6570\u3068\u3057\u3066\u6271\u3048\u308b\u306e\u3067\u3001\u4ee5\u4e0b\u3067\u306f\u8907\u7d20\u5171\u5f79\u306e\u8a18\u53f7 \\(^*\\) \u3092\u7701\u7565\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\\[ \\langle{}T\\rangle_1 = \\iint \\left(\\frac{1}{\\sqrt{2}}(\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2) &#8211; \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2))\\right)\\left(-\\frac{1}{2}\\nabla_1^2\\right)\\left(\\frac{1}{\\sqrt{2}}(\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2) &#8211; \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2))\\right)  dx_1 dx_2 \\]<br><br>\u3053\u308c\u3092\u5c55\u958b\u3059\u308b\u3068\u30014\u3064\u306e\u9805\u304c\u73fe\u308f\u308c\u307e\u3059\u3002\\[ \\langle{}T\\rangle_1 =    \\frac{1}{2}\\iint \\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2)\\left(-\\frac{1}{2}\\nabla_1^2\\right)\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2)dx_1 dx_2 \\] \\[+ \\frac{1}{2}\\iint \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2)\\left(-\\frac{1}{2}\\nabla_1^2\\right)\\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2)dx_1 dx_2\\]\\[-   \\frac{1}{2}\\iint \\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2)\\left(-\\frac{1}{2}\\nabla_1^2\\right)\\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2)dx_1 dx_2 \\]\\[-   \\frac{1}{2}\\iint \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2)\\left(-\\frac{1}{2}\\nabla_1^2\\right)\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2)dx_1 dx_2 \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u771f\u3093\u4e2d\u306e\u6f14\u7b97\u5b50\u306e\u90e8\u5206\u306f\u96fb\u5b501\u306b\u3060\u3051\u4f5c\u7528\u3059\u308b\u306e\u3067\u3001\u96fb\u5b502\u306b\u95a2\u3059\u308b\u90e8\u5206\u306f\u5225\u306b\u7a4d\u5206\u3067\u304d\u3066\u3001\u8a66\u884c1s\u8ecc\u9053\u95a2\u6570\u306f\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u03b1\u30b9\u30d4\u30f3\u540c\u58eb\u307e\u305f\u306f\u03b2\u30b9\u30d4\u30f3\u540c\u58eb\u3060\u3068\u7a4d\u5206\u306f1\u3001\u03b1\u30b9\u30d4\u30f3\u3068\u03b2\u30b9\u30d4\u30f3\u306e\u7d44\u307f\u5408\u308f\u305b\u3060\u3068\u7a4d\u5206=0\u306b\u306a\u308a\u307e\u3059\u3002<br>\\[ \\int \\chi_{1s\\alpha}(x_2)\\chi_{1s\\alpha}(x_2) dx_2= \\int \\chi_{1s\\beta}(x_2)\\chi_{1s\\beta}(x_2) dx_2= 1\\]\\[ \\int \\chi_{1s\\alpha}(x_2)\\chi_{1s\\beta}(x_2) dx_2= \\int \\chi_{1s\\beta}(x_2)\\chi_{1s\\alpha}(x_2) dx_2= 0\\]<br>\u3068\u306a\u3063\u3066\u7b2c3\u9805\u3068\u7b2c4\u9805\u306f\u6d88\u305b\u307e\u3059\u3002\u6b8b\u3063\u305f\u90e8\u5206\u306f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\langle{}T\\rangle_1 = \\frac{1}{2}\\int \\chi_{1s\\alpha}(x_1) \\left(-\\frac{1}{2}\\nabla_1^2\\right)  \\chi_{1s\\alpha}(x_1)   dx_1 +  \\frac{1}{2}\\int \\chi_{1s\\beta}(x_1) \\left(-\\frac{1}{2}\\nabla_1^2\\right)  \\chi_{1s\\beta}(x_1)   dx_1.  \\] <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b9\u30d4\u30f3\u90e8\u5206\u306e\u7a4d\u5206\u306f\u3069\u3061\u3089\u30821\u3068\u306a\u308a\u3001\u7b2c1\u9805\u3068\u7b2c2\u9805\u306f\u540c\u3058\u5024\u306b\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\\[ \\langle{}T\\rangle_1 =\\int \\phi_{1s}(r_1) \\left(-\\frac{1}{2}\\nabla_1^2\\right) \\phi_{1s}(\\mathbf r_1) d\\tau_1. \\] <br>\u4eca\u56de\u306e 1s \u8ecc\u9053\u306f\u7403\u5bfe\u79f0\u306a\u306e\u3067\u3001\\[ \\nabla_1^2 = \\frac{1}{r_1^2}\\frac{d}{dr_1}\\left(r_1^2 \\frac{d}{dr_1}\\right) \\]\u3092\u4f7f\u3046\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br>\u3057\u305f\u304c\u3063\u3066\u3001<br>\\[ \\langle{}T\\rangle_1= \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\sqrt{\\frac{\\zeta^3}{\\pi}}e^{-\\zeta r_1} \\left[-\\frac{1}{2} \\frac{1}{r_1^2}\\frac{d}{dr_1}\\left(r_1^2 \\frac{d}{dr_1}\\right) \\right] \\sqrt{\\frac{\\zeta^3}{\\pi}}e^{-\\zeta r_1} r_1^2 \\sin\\theta dr_1 d\\theta d\\phi \\] \\[= 4\\zeta^3 \\int_0^{\\infty} \\left(\\zeta{}r_1 &#8211; \\frac{\\zeta^2 r_1^2}{2}\\right) e^{-2\\zeta{}r_1} dr_1 \\]\\[ = 4\\zeta^3\\left(\\frac{\\zeta}{4\\zeta^2} &#8211; \\frac{\\zeta^2}{8\\zeta^3}\\right) = \\frac{\\zeta^2}{2}.\\]<br><br>\u96fb\u5b502\u306b\u3064\u3044\u3066\u3082\u307e\u3063\u305f\u304f\u540c\u3058\u5024\u306a\u306e\u3067\u3001\\[\\langle{}T\\rangle_2 = \\frac{\\zeta^2}{2}\\]\u3068\u306a\u308a\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u5168\u4f53\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\\[\\langle{}T\\rangle_1 + \\langle{}T\\rangle_2 = \\zeta^2\\]\u3067\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50\\(-\\frac{2}{r_1}\\)\u3082\u96fb\u5b501\u306b\u3057\u304b\u4f5c\u7528\u3057\u306a\u3044\u4e00\u96fb\u5b50\u6f14\u7b97\u5b50\u306a\u306e\u3067\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u8a08\u7b97\u3068\u540c\u69d8\u306b\u3001\u30b9\u30d4\u30f3\u90e8\u5206\u306e\u7a4d\u5206\u3068\u96fb\u5b502\u306b\u3064\u3044\u3066\u306e\u7a4d\u5206\u3092\u5148\u306b\u3057\u3066\u304b\u3089\u307e\u3068\u3081\u308b\u3068\u3001\u4ee5\u4e0b\u306e\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\langle{}V\\rangle_1 = 4\\zeta^3 \\int_0^{\\infty} e^{-\\zeta r_1} \\left(-\\frac{2}{r_1}\\right) e^{-\\zeta r_1} \\times r_1^2 dr_1 \\]<br>\\[ = -8\\zeta^3 \\int_0^{\\infty} {r_1} e^{-2\\zeta r_1} dr_1 = -8\\zeta^3 \\times \\frac{1}{4\\zeta^2} = -2\\zeta \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b502\u306b\u3064\u3044\u3066\u3082\u307e\u3063\u305f\u304f\u540c\u3058\u5024\u306a\u306e\u3067\u3001\\[\\langle{}V\\rangle_2 = -2\\zeta\\]\u3068\u306a\u308a\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u5168\u4f53\u306e\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\\[\\langle{}V\\rangle_1 + \\langle{}V\\rangle_2 = -4\\zeta\\]\u3067\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u96fb\u5b50-\u96fb\u5b50\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u3092\u53d6\u308b\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[\\langle V_{ee}\\rangle = \\iint \\left(\\frac{1}{\\sqrt{2}}(\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2) &#8211; \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2))\\right)\\frac{1}{r_{12}} \\left(\\frac{1}{\\sqrt{2}}(\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2) &#8211; \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2))\\right) dx_1 dx_2 \\]<br>\u3067\u3059\u3002\u3053\u308c\u3092\u5c55\u958b\u3059\u308b\u30684\u3064\u306e\u9805\u304c\u73fe\u308f\u308c\u307e\u3059\u3002<br>\\[\\langle V_{ee}\\rangle = \\frac{1}{2}\\iint \\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2)\\frac{1}{r_{12}}\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2) dx_1dx_2 \\]\\[+ \\frac{1}{2}\\iint \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2)\\frac{1}{r_{12}}\\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2) dx_1dx_2 \\]\\[- \\frac{1}{2}\\iint \\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2)\\frac{1}{r_{12}}\\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2) dx_1dx_2 \\]\\[- \\frac{1}{2}\\iint \\chi_{1s\\beta}(x_1)\\chi_{1s\\alpha}(x_2)\\frac{1}{r_{12}}\\chi_{1s\\alpha}(x_1)\\chi_{1s\\beta}(x_2) dx_1dx_2 \\] <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6700\u521d\u306e2\u9805\u306f\u3001\u96fb\u5b50\u304c\u305d\u308c\u305e\u308c\u306e\u8ecc\u9053\u306b\u5206\u5e03\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u3088\u308b\u901a\u5e38\u306e\u30af\u30fc\u30ed\u30f3\u53cd\u767a\u3092\u8868\u3059<strong>\u30af\u30fc\u30ed\u30f3\u9805<\/strong>\u3067\u3059\u3002\u4e00\u65b9\u3001\u5f8c\u308d\u306e2\u9805\u306f\u3001\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u3092\u53cd\u5bfe\u79f0\u5316\u3057\u305f\u3053\u3068\u306b\u3088\u3063\u3066\u751f\u3058\u305f<strong>\u4ea4\u63db\u9805<\/strong>\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7b2c3\u9805\u3068\u7b2c4\u9805\u3067\u306f\u3001\u30b9\u30d4\u30f3\u90e8\u5206\u306e\u7a4d\u5206\u304c0\\[ \\int\\alpha^*(\\sigma)\\beta(\\sigma)\\,d\\sigma=0\\]\u306b\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u7b2c3\u9805\u3068\u7b2c4\u9805\u30820\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[K_{\\alpha\\beta}=0.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u4e00\u65b9\u3001\u7b2c\u4e00\u9805\u3068\u7b2c\u4e8c\u9805\u3067\u306f\u3001\u30b9\u30d4\u30f3\u90e8\u5206\u306e\u7a4d\u5206\u304c1\\[ \\int\\alpha^*(\\sigma)\\alpha(\\sigma)\\,d\\sigma=\\int\\beta^*(\\sigma)\\beta(\\sigma)\\,d\\sigma=1\\]\u306b\u306a\u308a\u3001\u3055\u3089\u306b\u96fb\u5b501\u3068\u96fb\u5b502\u306f\u540c\u3058\u7a7a\u9593\u8ecc\u9053 \\(\\phi_{1s}\\) \u3092\u5360\u3081\u3066\u3044\u308b\u306e\u3067\u30012\u3064\u306e\u30af\u30fc\u30ed\u30f3\u9805\u306f\u7b49\u3057\u304f\u306a\u308a\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u3001\\[\\langle V_{ee}\\rangle = \\frac{1}{2}J_{12}+\\frac{1}{2} J_{21} = J\\]\u3068\u306a\u308a\u3001<br>\\[\\langle V_{ee}\\rangle = \\iint\\phi^*_{1s}(r_1)\\phi^*_{1s}(r_2)\\frac{1}{r_{12}} \\phi_{1s}(r_1)\\phi_{1s}(r_2) d\\tau_1 d\\tau_2 \\]\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u3088\u3046\u306b\u3001\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u304b\u3089\u51fa\u767a\u3059\u308b\u3068\u4e00\u822c\u306b\u306f\u300c\\(\u30af\u30fc\u30ed\u30f3\u9805 &#8211; \u4ea4\u63db\u9805\\)\u300d\u3068\u3044\u3046\u69cb\u9020\u304c\u81ea\u7136\u306b\u73fe\u308f\u308c\u307e\u3059\u304c\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u57fa\u5e95\u72b6\u614b\u3067\u306f2\u96fb\u5b50\u306e\u30b9\u30d4\u30f3\u304c\u7570\u306a\u308b\u305f\u3081\u3001\u4e21\u96fb\u5b50\u9593\u306e\u4ea4\u63db\u9805\u306f0\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u304b\u3089\u5148\u306f\u3001\u7a7a\u9593\u90e8\u5206\u3060\u3051\u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b502\u306b\u3064\u3044\u3066\u5148\u306b\u7a4d\u5206\u3059\u308b\u5f62\u306b\u66f8\u304d\u76f4\u3059\u3068\u3001\\[\\langle V_{ee}\\rangle = \\int\\phi^*_{1s}(\\mathbf r_1)\\left[\\int\\frac{{|\\phi_{1s}(\\mathbf r_2)|^2}}{r_{12}}d\\tau_2\\right] \\phi_{1s}(\\mathbf r_1) d\\tau_1 . \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u771f\u3093\u4e2d\u306e<br>\\[ J(\\mathbf r_1) = \\int \\frac{|\\phi_{1s}(\\mathbf r_2)|^2}{r_{12}} d\\tau_2 \\]<br>\u306e\u90e8\u5206\u306f\u3001\\(r_1,r_2\\)\u306e\u95a2\u6570\u3092\\(r_2\\)\u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u306e\u3067\u3001\\(r_1\\)\u3060\u3051\u306e\u95a2\u6570\u306b\u306a\u308a\u3001\u96fb\u5b501\u304c\u3001\u8a66\u884c\u8ecc\u9053\\(\\phi_{1s}\\)\u306b\u5206\u5e03\u3057\u3066\u3044\u308b\u96fb\u5b502\u304b\u3089\u611f\u3058\u308b\u5e73\u5747\u7684\u306a\u30af\u30fc\u30ed\u30f3\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3068\u89e3\u91c8\u3067\u304d\u307e\u3059\u3002\\(\\sqrt{\\frac{\\zeta^3}{\\pi}}e^{-\\zeta r_2}\\)\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001<br>\\[ J(\\mathbf r_1) = \\frac{\\zeta^3}{\\pi} \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\frac{e^{-2\\zeta r_2}}{r_{12}} r_2^2 \\sin\\theta dr_2 d\\theta d\\phi . \\] <br>\u3053\u3053\u3067\\(r_1\\)\u306e\u65b9\u5411\u3092\\(z\\)\u8ef8\u306b\u53d6\u308c\u3070\u3001\\(\\theta\\)\u306f\u30d9\u30af\u30c8\u30eb\\(\\mathbf r_1\\)\u3068\\(\\mathbf r_2\\)\u304c\u306a\u3059\u89d2\u3067\u3042\u308a\u3001\\[ r_{12} = \\sqrt{r_1^2+r_2^2-2r_1r_2\\cos\\theta} \\]\u3068\u306a\u308a\u307e\u3059\u3002<br>\u3053\u306e\u7a4d\u5206\u306f\u9ad8\u6821\u6570\u5b66\u3067\u3082\u304c\u3093\u3070\u308c\u3070\u89e3\u304f\u3053\u3068\u304c\u3067\u304d\u3066\uff08<a href=\"https:\/\/faculty.washington.edu\/seattle\/physics441\/online\/Webber.pdf?utm_source=chatgpt.com\">\u30ef\u30b7\u30f3\u30c8\u30f3\u5927\u5b66\u306e\u8b1b\u7fa9\u8cc7\u6599\u306epp.139-140<\/a>\u3092\u53c2\u7167\uff09\u3001<br>\\[ J(\\mathbf r_1) = \\frac{1-e^{-2\\zeta r_1}}{r_1} &#8211; \\zeta e^{-2\\zeta r_1} \\]<br>\u3068\u3001\\(r_1\\)\u3060\u3051\u306e\u95a2\u6570\u306b\u306a\u308a\u307e\u3057\u305f\u3002\u3057\u305f\u304c\u3063\u3066\u3001<br>\\[\\langle V_{ee}\\rangle = \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} \\sqrt{\\frac{\\zeta^3}{\\pi}}e^{-\\zeta r_1} \\left(\\frac{1-e^{-2\\zeta r_1}}{r_1} &#8211; \\zeta e^{-2\\zeta r_1}\\right) \\sqrt{\\frac{\\zeta^3}{\\pi}}e^{-\\zeta r_1} r_1^2 \\sin\\theta dr_1 d\\theta d\\phi \\]<br>\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3092\u3059\u308b\u3068\\[ \\langle V_{ee}\\rangle = \\frac{5}{8}\\zeta \\]\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u306f<br>\\[ E = 2\\langle{}T\\rangle + 2 \\langle{}V\\rangle + \\langle V_{ee}\\rangle = \\zeta^2 &#8211; 4\\zeta + \\frac{5}{8}\\zeta = \\zeta^2 &#8211; \\frac{27}{8}\\zeta\\]<br>\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u95a2\u6570\u306f\u4e0a\u306b\u958b\u3044\u305f\u4e8c\u6b21\u95a2\u6570\u306a\u306e\u3067\u3001\u6700\u3082\u4f4e\u3044\\(E\\)\u3092\u4e0e\u3048\u308b\u306e\u306f\u3001\\(E\\)\u306e\u50be\u304d\u304c0\u306b\u306a\u308b<br>\\[ \\frac{dE}{d\\zeta} = 0\\]<br>\\(\\zeta\\)\u3067\u3059\u3002\\(E\\)\u3092\u5fae\u5206\u3059\u308b\u3068<br>\\[ 2\\zeta- \\frac{27}{8}= 0 \\]<br>\\[\\zeta = \\frac{27}{16} \\approx 1.6875.\\]<br>\u3053\u306e\u5024\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ E = -\\frac{729}{256} \\approx -2.848 E_{\\mathrm h}\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u9ad8\u7cbe\u5ea6\u8a08\u7b97\u3067\u5f97\u3089\u308c\u305f\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\\(E \\approx -2.9037 E_{\\mathrm h}\\) \u306a\u306e\u3067\u3001\u6bd4\u8f03\u7684\u7cbe\u5ea6\u306e\u3088\u3044\u5024\u3067\u3059\u3002\u4ee5\u4e0a\u306e\u65b9\u6cd5\u306f1s\u8ecc\u9053\u3092&nbsp;\\[ \\phi_{1s}(\\mathbf r)= N(\\zeta)e^{-\\zeta r}\\]&nbsp;\u3068\u3044\u3046\u5f62\u306b\u9650\u5b9a\u3057\u305f\u30011\u30d1\u30e9\u30e1\u30fc\u30bf\u306e\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u578b\u5909\u5206\u8a08\u7b97\u3067\u3057\u305f\u3002\u3053\u3067\u3001\\(N(\\zeta)\\) \u306f\u898f\u683c\u5316\u5b9a\u6570\u3067\u3059\u3002\u3053\u308c\u306b\u5bfe\u3057\u3066\u8ecc\u9053\u95a2\u6570 \\(\\phi\\) \u306e\u5f62\u305d\u306e\u3082\u306e\u3092\u5909\u5316\u3055\u305b\u308b\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u3067\u306f\u3001\u901a\u5e38\u306e\u5fae\u5206\u306e\u4ee3\u308f\u308a\u306b\u6c4e\u95a2\u6570\u5fae\u5206\u3092\u4f7f\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\uff08Hartree\u2013Fock\uff09\u6cd5<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u672c\u6765\u306e\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u3067\u306f\u3001\u8ecc\u9053\u3092\u7279\u5b9a\u306e\u95a2\u6570\u5f62\u306b\u9650\u5b9a\u305b\u305a\u3001\u8ecc\u9053\\(\\phi(\\mathbf r)\\)\u306e\u5f62\u305d\u306e\u3082\u306e\u3092\u5909\u5316\u3055\u305b\u3066\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u505c\u7559\u3059\u308b\u8ecc\u9053\u3092\u63a2\u3057\u307e\u3059\u3002\u3053\u306e\u3068\u304d\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\\(E\\)\u3092\u6570\u5024\\(x\\)\u306e\u95a2\u6570\u3067\u306f\u306a\u304f\u3001\u95a2\u6570\\(\\phi(\\mathbf r)\\)\u3092\u5165\u529b\u3059\u308b\u3068\u6570\u5024\u3092\u8fd4\u3059\u3082\u306e\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u300c\u95a2\u6570\u3092\u5165\u529b\u3068\u3057\u3066\u6570\u5024\u3092\u8fd4\u3059\u3082\u306e\u300d\u3092\u6c4e\u95a2\u6570\u3068\u547c\u3073\u3001\\[ E[\\phi] \\] \u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\"><br>\u898f\u683c\u5316\u6761\u4ef6\u3068\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570<br><\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u8ecc\u9053\u3092\u6700\u9069\u5316\u3059\u308b\u6642\u306b\u306f\u3001\u8ecc\u9053\u304c\u898f\u683c\u5316 \\(\\int |\\phi(\\mathbf r)|^2\\,d\\tau= 1\\)\uff08\u4e00\u96fb\u5b50\u8ecc\u9053\u306e\u5b58\u5728\u78ba\u7387\u3092\u5168\u7a7a\u9593\u3067\u7a4d\u5206\u3059\u308b\u30681\u306b\u306a\u308b\uff09\u3055\u308c\u3066\u3044\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002\u52a0\u3048\u3066\u3001\u7570\u306a\u308b\u8ecc\u9053\u304c\u76f4\u4ea4\u3057\u3066\u3044\u308b\u3053\u3068\\(\\int \\chi_i^*(x)\\chi_j(x) dx = 0\\)\uff08\u7570\u306a\u308b\u8ecc\u9053\\(i\\neq j\\)\u306b\u3064\u3044\u3066\u8ecc\u9053\u3092\u639b\u3051\u5408\u308f\u305b\u3066\u7a4d\u5206\u3059\u308b\u30680\u306b\u306a\u308b\uff09\u3068\u3044\u3046\u898f\u683c\u76f4\u4ea4\u6761\u4ef6\u3092\u6e80\u305f\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002<br>\u3053\u306e\u898f\u683c\u76f4\u4ea4\u6761\u4ef6\u3092\u4fdd\u3061\u306a\u304c\u3089\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u6700\u9069\u5316\u3059\u308b\u305f\u3081\u306b\u3001\\(\\varepsilon_{ij}\\)\u3068\u3044\u3046\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u3092\u5c0e\u5165\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u57fa\u5e95\u72b6\u614b\u3067\u306f\u03b1\u30b9\u30d4\u30f3\u8ecc\u9053\u3068\u03b2\u30b9\u30d4\u30f3\u8ecc\u9053\u306e\u76f4\u4ea4\u6027\u304c\u81ea\u52d5\u7684\u306b\u6e80\u305f\u3055\u308c\u308b\u306e\u3067\u3001\\[\\int |\\phi(\\mathbf r)|^2\\,d\\tau= 1\\]\u3068\u3044\u30461\u3064\u306e\u898f\u683c\u5316\u6761\u4ef6\u3060\u3051\u3092\u8003\u3048\u308c\u3070\u5341\u5206\u3067\u3059\u3002<\/p>\n\n\n\n<div class=\"wp-block-group has-border-color has-base-border-color has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-width:1px\">\n<div class=\"wp-block-group has-border-color has-base-border-color has-global-padding is-layout-constrained wp-block-group-is-layout-constrained\" style=\"border-width:1px;border-top-left-radius:1px;border-top-right-radius:1px;border-bottom-left-radius:1px;border-bottom-right-radius:1px\">\n<p class=\"has-body-text-color has-text-color has-link-color wp-elements-1 wp-block-paragraph\">\u300c\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u6cd5\u300d\u306e\u8aac\u660e\u3092\u3057\u307e\u3059\u3002\u4f8b\u3048\u3070 \\(f(x,y) = x + 2y\\)\u3092\\(x^2+y^2 = 1\\)\uff08\u534a\u5f841\u306e\u5186\uff09\u3068\u3044\u3046\u5236\u7d04\u3092\u8ab2\u3057\u3066\u6700\u5927\u6700\u5c0f\u5316\u3059\u308b\u6642\u306f\u3001\\[ L(x,y,\\lambda) = x + 2y -\\lambda(x^2 + y^2 -1) \\]\u3068\u3044\u3046\u91cf\u3092\u4f5c\u3063\u3066\u3001<br>\\[ \\frac{\\partial L(x,y,\\lambda)}{\\partial x}=0, \\frac{\\partial L(x,y,\\lambda)}{\\partial y}=0, \\frac{\\partial L(x,y,\\lambda)}{\\partial \\lambda}=0 \\] \u304b\u3089\u5f97\u3089\u308c\u308b\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3053\u3068\u3067\u7b54\u3048\u304c\u5f97\u3089\u308c\u307e\u3059\uff08\u6700\u5927\u5024\u306f\\(\\sqrt{5}\\)\u3001\u6700\u5c0f\u5024\u306f\\(-\\sqrt{5}\\)\uff09\u3002<\/p>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u3067\u306f\u3001\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u00a0\\(\\varepsilon\\)\u00a0\u3092\u5c0e\u5165\u3057\u3066\u3001\\[L[\\phi] = E[\\phi] &#8211; 2\\varepsilon\\left(\\int|\\phi(\\mathbf r)|^2\\,d\\tau &#8211; 1 \\right) \\]\u3068\u3044\u3046\u65b0\u3057\u3044\u6c4e\u95a2\u6570\u3092\u4f5c\u308a\u307e\u3059\u3002\\(\\varepsilon\\)\u306e\u4fc2\u65702\u306f\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u3067\u306f\u540c\u3058\u7a7a\u9593\u8ecc\u9053\u30922\u96fb\u5b50\u304c\u5360\u6709\u3057\u3066\u3044\u308b\u3053\u3068\u306b\u5bfe\u5fdc\u3055\u305b\u305f\u3082\u306e\u3067\u3059\u3002\u3053\u306e\u4fc2\u6570\u3092\u4ed8\u3051\u3066\u304a\u304f\u3068\u3001\u5f8c\u3067\u73fe\u308c\u308b\u00a0\\(\\varepsilon\\)\u00a0\u3092\u8ecc\u9053\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u3057\u3066\u305d\u306e\u307e\u307e\u89e3\u91c8\u3057\u3084\u3059\u304f\u306a\u308a\u307e\u3059\u3002<br><br>\\(E[\\phi]\\)\u306f\u30a8\u30cd\u30eb\u30ae\u30fc\u3001\\(\\varepsilon\\) \u306e\u5f8c\u308d\u306e\u62ec\u5f27\u306e\u4e2d\u306f\u7121\u6b21\u5143\u306a\u306e\u3067\u3001\\(\\varepsilon\\) \u306f\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u540c\u3058\u6b21\u5143\u3092\u6301\u3061\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u300c\u8ecc\u9053\u3092\u5909\u5206\u3059\u308b\u300d\u3068\u306f\u3069\u3046\u3044\u3046\u3053\u3068\u304b<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\\(\\phi^*\\)\u3067\u5909\u5206\u3059\u308b\u306e\u306f\u3001\u95a2\u6570\\(f(x)\\)\u3092\u6700\u5c0f\u5316\u3059\u308b\u6642\u306b\u5909\u6570\\(x\\)\u3092\\(x \\longrightarrow x + \\delta x\\)\u3068\u5c11\u3057\u3060\u3051\u5909\u5316\u3055\u305b\u3066\u3001\u305d\u306e\u6642\u306e \\(f\\)\u306e\u5909\u5316\u3092\u8003\u3048\u308b\u306e\u3068\u8003\u3048\u65b9\u306f\u540c\u3058\u3067\u3059\u3002\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u3067\u306f\u95a2\u6570\u3092\u5c11\u3057\u3060\u3051\u5909\u5316\u3055\u305b\u307e\u3059\u3002<br>\\[\\phi(\\mathbf r) \\longrightarrow \\phi(\\mathbf r) + \\delta\\phi(\\mathbf r)\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5909\u5316\u3092\u3001\u3053\u306e\u5c0f\u3055\u306a\u5909\u5316\u91cf\u306b\u3064\u3044\u3066\u5c55\u958b\u3059\u308b\u3068\u3001<br>\\[ E[\\phi+\\delta\\phi]-E[\\phi] = \\delta E+\\text{\u4e8c\u6b21\u4ee5\u4e0a\u306e\u5c0f\u3055\u3044\u9805} \\]<br>\u3068\u66f8\u3051\u307e\u3059\u3002\\(\\delta E\\) \u306f\u3001\\(\\delta\\phi\\) \u307e\u305f\u306f \\(\\delta\\phi^*\\) \u306b\u3064\u3044\u3066\u4e00\u6b21\u306e\u9805\u3092\u96c6\u3081\u305f\u300c\u4e00\u6b21\u5909\u5206\u300d\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6c4e\u95a2\u6570\u5fae\u5206\u306f\u3001\u3053\u306e\u4e00\u6b21\u5909\u5206\u3092<br>\\[ \\delta E =\\int \\frac{\\delta E}{\\delta\\phi(\\mathbf r)}\\,\\delta\\phi(\\mathbf r)\\,d\\tau + \\int \\frac{\\delta E}{\\delta\\phi^*(\\mathbf r)}\\,\\delta\\phi^*(\\mathbf r)\\,d\\tau \\]<br>\u3068\u66f8\u3044\u305f\u3068\u304d\u306e\u4fc2\u6570\u306b\u76f8\u5f53\u3057\u307e\u3059\u3002\u5909\u5206\u8a08\u7b97\u4e0a\u306f\\(\\phi\\)\u3068\\(\\phi^*\\)\u3092\u72ec\u7acb\u306a\u5909\u6570\u3068\u3057\u3066\u6271\u3063\u305f\u3046\u3048\u3067\u3001\\(\\phi\\)\u306f\u56fa\u5b9a\u3057\u3066\u3001\\(\\phi^* \\longrightarrow \\phi^* + \\delta\\phi^*\\)&nbsp;\u3068\u3059\u308c\u3070\u6c4e\u95a2\u6570\u5fae\u5206\\(\\frac{\\delta L}{\\delta\\phi^*}\\) \u3092\u5c0e\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u6c4e\u95a2\u6570<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u3067\u5b9f\u969b\u306b\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u30021\u3064\u306e\u96fb\u5b50\u306b\u3060\u3051\u4f5c\u7528\u3059\u308b\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50\u3068\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u6f14\u7b97\u5b50\u3092\u307e\u3068\u3081\u3066<br>\\[ \\hat{h} = -\\frac{1}{2}\\nabla^2 &#8211; \\frac{2}{r} \\]<br>\u3068\u66f8\u304d\u307e\u3059\u3002\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u3067\u306f\u3001\u540c\u3058\u7a7a\u9593\u8ecc\u9053\\(\\phi\\)\u30922\u96fb\u5b50\u304c\u5360\u6709\u3059\u308b\u306e\u3067\u3001\u4e00\u96fb\u5b50\u90e8\u5206\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u3001\\[ E_{1e} = 2\\int \\phi^*(\\mathbf r)\\hat{h}\\phi(\\mathbf r)\\,d\\tau \\]\u3068\u66f8\u3051\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b50\u9593\u53cd\u767a\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u4ee5\u524d\u6c42\u3081\u305f\u901a\u308a\u3001\\[E_{ee} = \\iint\\frac{|\\phi(\\mathbf r_1)|^2 |\\phi(\\mathbf r_2)|^2}{r_{12}} d\\tau_1 d\\tau_2\\]\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u6c4e\u95a2\u6570\u306f\\[ E[\\phi] = 2\\int \\phi^*(\\mathbf r)\\hat{h}\\phi(\\mathbf r)\\,d\\tau +\\iint\\frac{|\\phi(\\mathbf r_1)|^2 |\\phi(\\mathbf r_2)|^2}{r_{12}} d\\tau_1 d\\tau_2 \\]<br>\u3067\u3059\u3002\u3053\u308c\u306b\u898f\u683c\u5316\u6761\u4ef6\u3092\u52a0\u3048\u305f\\[L = 2\\int \\phi^*(\\mathbf r)\\hat{h}\\phi(\\mathbf r)\\,d\\tau +\\iint\\frac{|\\phi(\\mathbf r_1)|^2 |\\phi(\\mathbf r_2)|^2}{r_{12}} d\\tau_1 d\\tau_2 &#8211; 2\\varepsilon\\left(\\int|\\phi(\\mathbf r)|^2\\,d\\tau &#8211; 1 \\right)  \\]\u3092\u5909\u5206\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e00\u96fb\u5b50\u9805<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\phi^* \\longrightarrow \\phi^* + \\delta\\phi^*\\)&nbsp;\u3068\u5909\u5316\u3055\u305b\u305f\\[ 2\\int(\\phi^* + \\delta\\phi^*)\\hat{h}\\phi d\\tau \\]\u3092\u5c55\u958b\u3059\u308b\u3068\u3001\\[ = 2\\int\\left[\\phi^*\\hat{h}\\phi + \\delta\\phi^*\\hat{h}\\phi\\right]d\\tau . \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u4e00\u6b21\u5909\u5206\u3068\u3057\u3066\u6b8b\u308b\u306e\u306f\u3001\\[\\delta E_{1e} = 2\\int\\delta\\phi^*\\hat{h}\\phi d\\tau \\]\u3067\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u96fb\u5b50\u9593\u53cd\u767a\u9805<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ (\\phi^*(\\mathbf r_1)+\\delta\\phi^*(\\mathbf r_1))\\phi(\\mathbf r_1)(\\phi^*(\\mathbf r_2) +\\delta\\phi^*(\\mathbf r_2))\\phi(\\mathbf r_2) \\]\u306e\u90e8\u5206\u3092\u5148\u306b\u5c55\u958b\u3059\u308b\u3068\u3001\\[ |\\phi_1|^2|\\phi_2|^2 + (\\delta\\phi_1^*\\phi_1)|\\phi_2|^2 + |\\phi_1|^2(\\delta\\phi_2^*\\phi_2) + (\\delta\\phi_1^*\\phi_1) (\\delta\\phi_2^*\\phi_2).\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u6b21\u306e\u90e8\u5206\\(\\delta E_{ee}\\) \u306f\u2028\\[ \\delta E_{ee} = \\iint \\frac{\\delta\\phi_1^*\\phi_1|\\phi_2|^2}{r_{12}} d\\tau_1d\\tau_2 + \\iint \\frac{|\\phi_1|^2\\delta\\phi_2^*\\phi_2}{r_{12}} d\\tau_1d\\tau_2 .\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b501\u3068\u96fb\u5b502\u306f\u540c\u3058\u7a7a\u9593\u8ecc\u9053\\(\\phi\\)\u3092\u5360\u3081\u3066\u3044\u307e\u3059\u3002\u307e\u305f\u3001\\(r_{12}=r_{21}\\) \u3067\u3042\u308a\u3001\u7a4d\u5206\u5909\u6570 \\(\\mathbf r_1\\) \u3068 \\(\\mathbf r_2\\) \u306f\u5165\u308c\u63db\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u2028\\[ \\delta E_{ee} =2\\iint \\frac{\\delta\\phi_1^*\\phi_1|\\phi_2|^2}{r_{12}}d\\tau_1 d\\tau_2 \\] \\[ = 2\\int\\delta\\phi^*(\\mathbf r_1)\\left[\\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right]\\phi(\\mathbf r_1)d\\tau_1. \\] <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u3001\\[ J(\\mathbf r_1) = \\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2 \\]\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u96fb\u5b501\u304c\u4f4d\u7f6e \\(\\mathbf r_1\\) \u306b\u3044\u308b\u3068\u304d\u3001\u8ecc\u9053\\(\\phi\\)\u306b\u5206\u5e03\u3057\u3066\u3044\u308b\u3082\u30461\u3064\u306e\u96fb\u5b50\u304b\u3089\u611f\u3058\u308b\u5e73\u5747\u7684\u306a\u30af\u30fc\u30ed\u30f3\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3067\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u898f\u683c\u5316\u6761\u4ef6<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ &#8211; 2\\varepsilon\\left(\\int (\\phi^*+\\delta\\phi^*)\\phi\\, d\\tau &#8211; 1\\right)\\] \u3092\u5c55\u958b\u3059\u308b\u3068\u3001\u2028\\[ -2\\varepsilon\\left[\\int \\left(|\\phi|^2 + \\delta\\phi^*\\phi \\right) d\\tau &#8211; 1\\right]. \\] \u3057\u305f\u304c\u3063\u3066\u3001\u4e00\u6b21\u306e\u9805\u306f \\[ \\delta L_{\u898f\u683c\u5316} = -2\\varepsilon\\int \\delta\\phi^*\\phi d\\tau.\\]\n\n\n\n<h4 class=\"wp-block-heading\">\u4e00\u96fb\u5b50\u9805\u3001\u96fb\u5b50\u9593\u53cd\u767a\u9805\u3001\u898f\u683c\u5316\u6761\u4ef6\u3092\u307e\u3068\u3081\u308b<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\\(\\delta\\phi^*\\)\u306b\u3064\u3044\u3066\u306e\u4e00\u6b21\u306e\u9805\u3001\u3064\u307e\u308a\u7b2c\u4e00\u5909\u5206\u3092\u307e\u3068\u3081\u308b\u3068\u3001\u2028\\[ \\delta L =2\\int\\delta\\phi^*\\hat{h}\\phi d\\tau  + 2\\int\\delta\\phi^*(\\mathbf r_1)\\left[\\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right]\\phi(\\mathbf r_1)d\\tau_1\\]\\[ -2\\varepsilon\\int \\delta\\phi^*\\phi d\\tau. \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u666e\u901a\u306e\u95a2\u6570\\(f(x)\\) \u3067\u306f\u3001\\(x\\)\u3092\u308f\u305a\u304b\u306b\u5909\u5316\u3055\u305b\u308b\uff08\\(x \\longrightarrow x + \\delta x\\)\uff09\u3068\u4e00\u6b21\u306e\u5909\u5316\u306f\\(\\delta f = \\frac{df}{dx}\\delta x\\)\u3068\u306a\u308a\u3001\\(\\delta x\\)\u306b\u639b\u304b\u3063\u3066\u3044\u308b\u4fc2\u6570\u304c\u5fae\u5206\\(\\frac{df}{dx}\\)\u3067\u3059\u3002\u3053\u308c\u3068\u540c\u3058\u3088\u3046\u306b\\(\\delta L =\\int \\delta\\phi^*(\\mathbf r)\\frac{\\delta L}{\\delta \\phi^*(\\mathbf r)}\\,d\\tau\\)\u3068\u66f8\u3044\u305f\u6642\u3001\\(\\delta\\phi^*\\)\u00a0\u306b\u639b\u304b\u3063\u3066\u3044\u308b\u4fc2\u6570\u304c\u3001\u6c4e\u95a2\u6570\u5fae\u5206\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\\(\\int\\delta\\phi^*\\hat{h}\\phi d\\tau \\)\u3067\u306f\u3001\\(\\delta\\phi^*\\)\u306b\u639b\u304b\u3063\u3066\u3044\u308b \\(\\hat{h}\\phi\\)\u3092\u53d6\u308a\u51fa\u3057\u3066\u3001\\(\\frac{\\delta}{\\delta \\phi^*}\\int\\phi^*\\hat{h}\\phi\\, d\\tau = \\hat{h}\\phi\\)\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ \\frac{\\delta L}{\\delta\\phi^*} = 2\\left[\\hat{h}\\phi + \\left(\\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right)\\phi -\\varepsilon\\phi\\right] \\] \u505c\u7559\u6761\u4ef6\\(\\frac{\\delta L}{\\delta\\phi^*} = 0 \\) \u3092\u8ab2\u3059\u3068\\(\\frac{\\delta L}{\\delta\\phi^*}\\) \u5168\u4f53\u30922\u3067\u5272\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u3055\u3089\u306b\u79fb\u9805\u3059\u308b\u3068\u3001 \\[ \\left( \\hat{h}+ \\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right)\\phi = \\varepsilon\\phi \\] \u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u95a2\u6570\\(\\phi\\)\u306b \\(\\left( \\hat{h}+ \\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right)\\) \u3068\u3044\u3046\u6f14\u7b97\u3092\u3057\u305f\u3089\u3001\u6f14\u7b97\u3057\u305f\u7d50\u679c\u304c\u5143\u306e\u95a2\u6570\\(\\phi\\)\u306e\\(\\varepsilon\\)\u500d\u306b\u306a\u308b\u3068\u3044\u3046\u56fa\u6709\u5024\u65b9\u7a0b\u5f0f\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002\u666e\u901a\u3068\u9055\u3046\u306e\u306f\u6f14\u7b97\u5b50\u306e\u4e2d\u306b\u7b54\u3048\u306e\\(\\phi\\)\u304c\u542b\u307e\u308c\u3066\u3044\u308b\u306e\u3067\u3001\u77db\u76fe\u304c\u306a\u304f\u306a\u308b\u307e\u3067\u53cd\u5fa9\u7684\u306b\u89e3\u304b\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002\u305d\u3057\u3066\u3001\u898f\u683c\u5316\u6761\u4ef6 \\(\\int \\phi^*\\phi d\\tau = 1\\) \u3068\u9023\u7acb\u3055\u305b\u308b\u3053\u3068\u3067\u3001\u898f\u683c\u5316\u3057\u305f\u56fa\u6709\u95a2\u6570\u3092\u6c42\u3081\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002 \\[\\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2 = J(\\mathbf r_1) \\] \u3068\u7f6e\u304d\u63db\u3048\u3066\u3001\u3055\u3089\u306b\\( \\hat J\\phi(\\mathbf r_1) = J(\\mathbf r_1)\\phi(\\mathbf r_1) \\)\u3068\u5b9a\u7fa9\u3059\u308c\u3070\u3001\u56fa\u6709\u5024\u65b9\u7a0b\u5f0f\u306f \\[ \\left( \\hat{h}+\\hat{J} \\right)\\phi = \\varepsilon\\phi \\] \u3068\u8868\u308f\u3055\u308c\u307e\u3059\u3002\u3053\u3053\u3067\u5f97\u3089\u308c\u305f\u00a0\\((\\hat h+\\hat J_1)\\phi_1=\\epsilon_1\\phi_1\\)\u00a0\u306f\u3001He\u57fa\u5e95\u72b6\u614b\u30672\u96fb\u5b50\u304c\u5171\u6709\u3059\u308b\u5360\u6709\u7a7a\u9593\u8ecc\u9053\u3092\u6c42\u3081\u308b\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u3067\u3059\u3002He\u306e\u3088\u3046\u306b\u5360\u6709\u7a7a\u9593\u8ecc\u9053\u304c1\u3064\u3060\u3051\u306e\u9589\u6bbbRHF\u3067\u306f\u3001\u6a19\u6e96\u7684\u306a\u30d5\u30a9\u30c3\u30af\u6f14\u7b97\u5b50\u306f\u00a0\\(\\hat f=\\hat h+2\\hat J_1-\\hat K_1\\)\u00a0\u3067\u3059\u3002\u5360\u6709\u8ecc\u9053\uff08\u88ab\u5360\u8ecc\u9053\uff09\u00a0\\(\\phi_1\\)\u00a0\u306b\u5bfe\u3057\u3066\u306f\u00a0\\(\\hat K_1\\phi_1=\\hat J_1\\phi_1\\)\u00a0\u306a\u306e\u3067\u3001\u4e0a\u306e\u65b9\u7a0b\u5f0f\u306b\u7c21\u7d04\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304f<br><\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\uff08\u975e\u76f8\u5bfe\u8ad6\u7684\u3067\u539f\u5b50\u6838\u3092\u56fa\u5b9a\u3057\u305f\u5834\u5408\u306e\uff09\u53b3\u5bc6\u306a\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306f \\(E_{exact} = -2.9037\\, E_{\\mathrm h}\\)\u3067\u3059\u3002\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u3067\u306f\u3001\u591a\u96fb\u5b50\u6ce2\u52d5\u95a2\u6570\u3092\u5358\u4e00\u306e\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u3067\u8fd1\u4f3c\u3057\u3066\u3044\u308b\u306e\u3067\u3001\u5909\u5206\u539f\u7406\u304b\u3089HF\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u3053\u306e\u5024\u3088\u308a\u9ad8\u304f\u306a\u308b\u306f\u305a\u3067\u3059\u3002\u3057\u304b\u3057\u3001HF\u9650\u754c\u306b\u304a\u3051\u308b\u8ecc\u9053\\(\\phi (r)\\)\u304c\u3069\u306e\u3088\u3046\u306a\u5358\u7d14\u306a\u89e3\u6790\u95a2\u6570\u3067\u8868\u308f\u305b\u308b\u304b\u306f\u3001\u3042\u3089\u304b\u3058\u3081\u5206\u304b\u308a\u307e\u305b\u3093\u3002\u5c11\u306a\u304f\u3068\u3082\u3001\u8ecc\u9053\u3092\u5358\u4e00\u306e\u6307\u6570\u95a2\u6570 \\(\\phi(r)\\propto e^{-\\zeta r}\\)&nbsp; \u306b\u9650\u5b9a\u3057\u305f\u5909\u5206\u8a08\u7b97\u3067\u306f\u3001\\(\\zeta=27\/16\\) \u306e\u3068\u304d\\(E =-2.8477\\, E_{\\mathrm h}\\)\u304c\u6700\u4f4e\u5024\u3067\u3057\u305f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u305d\u3053\u3067\u3001\u8ddd\u96e2\\(r\\)\u3092\\(0\\,a_0\\)\u304b\u3089\\(10 \\,a_0\\)\u307e\u3067\\(0.01\\)\u523b\u307f\u306b\u533a\u5207\u3063\u3066\u3001\u5168\u90e8\u30671001\u500b\u306e\u683c\u5b50\u70b9\u3092\u4f7f\u3063\u3066HF\u65b9\u7a0b\u5f0f\u3092\u6570\u5024\u7684\u306b\u89e3\u304fPython\u30b3\u30fc\u30c9\u3092ChatGPT\u306b\u751f\u6210\u3057\u3066\u3082\u3089\u3044\u307e\u3057\u305f\u3002\u8a08\u7b97\u3092\u7c21\u5358\u306b\u3059\u308b\u305f\u3081\u306b\u52d5\u5f84\u6ce2\u52d5\u95a2\u6570\\(\\phi(r)\\) \u305d\u306e\u3082\u306e\u3067\u306f\u306a\u304f\u3001\u3053\u308c\u306b\\(r\\)\u3092\u639b\u3051\u305f\u63db\u7b97\u52d5\u5f84\u6ce2\u52d5\u95a2\u6570\\(u(r) = r \\phi(r)\\)\u3092\u6700\u9069\u5316\u3057\u3066\u3044\u308b\u305d\u3046\u3067\u3059\u3002\\(u(r)\\)\u306e\u4e00\u968e\u5fae\u5206\uff08\u50be\u304d\uff09\u306f\u300c\\((u_{i+1} -u_{i-1})\/(2 \\Delta r\\))\u300d\u3068\u4e21\u96a3\u308a\u306e\u5024\u304b\u3089\u6c42\u3081\u3066\u3001\\(u(r)\\)\u306e\u4e8c\u968e\u5fae\u5206\uff08\u66f2\u7387\uff09\u306f\u300e\u53f3\u5074\u306e\u50be\u304d\u3068\u5de6\u5074\u306e\u50be\u304d\u306e\u5dee\u3092\u3001\u3055\u3089\u306b\u00a0\\(\\Delta r\\)\u00a0\u3067\u5272\u308b\u300f\\(\u00a0[(u_{i+1}-u_i) &#8211; (u_{i}-u_{i-1})]\/\\Delta r^2\\)\u00a0\u3067\u6c42\u3081\u307e\u3059\u3002\u521d\u671f\u5024\u306f\\(e^{-\\zeta r}\\)\uff08\\(\\zeta = \\frac{27}{16}\\)\uff09\u304b\u3089\u5272\u308a\u5f53\u3066\u3089\u308c\u307e\u3059\u304c\u3001\u8ecc\u9053\u306e\u5024\u306f\u53cd\u5fa9\u8a08\u7b97\u306e\u9593\u306b\u66f4\u65b0\u3055\u308c\u308b\u306e\u3067\u3001\u6700\u7d42\u7684\u306a\u8ecc\u9053\u5f62\u306f\u6307\u6570\u95a2\u6570\u306b\u306f\u5236\u9650\u3055\u308c\u307e\u305b\u3093\u3002<br>\u8a08\u7b97\u306e\u7d50\u679c\u3001\\[\\Delta r = 0.01\\, a_0: E_{HF} \\approx -2.86135\\, E_{\\mathrm h}\\]\u3068\u306a\u308a\u307e\u3057\u305f\u3002\u3055\u3089\u306b\u683c\u5b50\u3092\u7d30\u304b\u304f\u3059\u308b\u3068\u3001\\[\\Delta r = 0.001\\, a_0: E_{HF} \\approx -2.86168\\, E_{\\mathrm h}\\]\u3068\u65e2\u77e5\u306e\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u9650\u754c\\(E_{\\mathrm{HF}}^{\\mathrm{limit}} = -2.861679995612\\,E_h\\)\u3068\u307b\u307c\u4e00\u81f4\u3057\u307e\u3057\u305f\u3002\\(E_{exact}\\)\u3068\\(E_{\\mathrm{HF}}^{\\mathrm{limit}}\\)\u306e\u5dee\u304c\u300c\u96fb\u5b50\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u300d\\(E_{\\mathrm{corr}}\\)\\[E_{\\mathrm{corr}} = E_{\\mathrm{exact}}- E_{\\mathrm{HF}}^{\\mathrm{limit}}\\]<br>\u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u304c\u3001\u305d\u308c\u306f\u307e\u305f\u5225\u306e\u304a\u8a71\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u3092\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f\u306b\u3057\u3066\u89e3\u304f<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">\u6570\u5024\u7684\u306b\u683c\u5b50\u70b9\u4e0a\u3067\u89e3\u304f\u306e\u3067\u306f\u306a\u304f\u3001\u3084\u3063\u3071\u308a\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u8ecc\u9053\u3092\u6c42\u3081\u305f\u3044\u3068\u3053\u308d\u3067\u3059\u3002\u3057\u304b\u3057\u3001\u300c\u8ecc\u9053\u306e\u5f62\u3092\u5909\u3048\u3066\u3001\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u6700\u5c0f\u306b\u306a\u308b\u5f62\u3092\u63a2\u3059\u300d\u3068\u8a00\u308f\u308c\u3066\u3082\u3001\u95a2\u6570\u305d\u306e\u3082\u306e\u3092\u3069\u3046\u5909\u5316\u3055\u305b\u3066\u63a2\u305b\u3070\u3044\u3044\u306e\u304b\u3001\u7c21\u5358\u306b\u306f\u5206\u304b\u308a\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001\u5b9f\u969b\u306e\u91cf\u5b50\u5316\u5b66\u8a08\u7b97\u3067\u306f\u3001\u8ecc\u9053\u3092\u3042\u3089\u304b\u3058\u3081\u7528\u610f\u3057\u305f\u8907\u6570\u306e\u95a2\u6570\u306e\u8db3\u3057\u5408\u308f\u305b\u3068\u3057\u3066\u8868\u73fe\u3057\u307e\u3059\u3002<br>\\[ \\phi =C_1 \\chi_1 + C_2 \\chi_2 \\cdots \\] \u3064\u307e\u308a\u3001\u300c\u8ecc\u9053\u306e\u5f62\u3092\u5909\u3048\u308b\u300d\u3068\u3044\u3046\u554f\u984c\u3092\u300c\u4fc2\u6570\\(C_1, C_2, \\cdots\\)\u3092\u5909\u3048\u308b\u300d\u3068\u3044\u3046\u554f\u984c\u306b\u7f6e\u304d\u63db\u3048\u307e\u3059\u3002\u3053\u3046\u3059\u308b\u3068\u3001\u3082\u3068\u3082\u3068\u306f<strong>\u95a2\u6570\u3092\u672a\u77e5\u6570\u3068\u3059\u308b\u5fae\u5206\u65b9\u7a0b\u5f0f\u3060\u3063\u305f\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u3092\u3001\u4fc2\u6570\\(C_1, C_2, \\cdots\\)\u3092\u672a\u77e5\u6570\u3068\u3059\u308b\u9023\u7acb\u65b9\u7a0b\u5f0f\uff08\u884c\u5217\u306e\u56fa\u6709\u5024\u554f\u984c\uff09\u3068\u3057\u3066\u8868\u308f\u3059<\/strong>\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u884c\u5217\u306f\u4ee5\u524d\u306f\u9ad8\u6821\u306e\u6570C\u306e\u5185\u5bb9\u306b\u5165\u3063\u3066\u3044\u305f\u306e\u3067\u3059\u304c\u3001\u4eca\u306f\u3042\u307e\u308a\u3084\u3089\u306a\u3044\u3088\u3046\u3067\u3059\u3002\u3053\u306e\u5f62\u306b\u3057\u305f\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u3092<strong>\u30ed\u30fc\u30bf\u30fc\u30f3\uff08Roothaan\uff09\u65b9\u7a0b\u5f0f<\/strong>\u307e\u305f\u306f<strong>\u30ed\u30fc\u30bf\u30fc\u30f3\u30fb\u30db\u30fc\u30eb\uff08Roothaan\u2013Hall\uff09\u65b9\u7a0b\u5f0f<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8db3\u3057\u5408\u308f\u305b\u308b\u95a2\u6570\u306e\u5f62\u3068\u3057\u3066\u306f\u3001\u30ac\u30a6\u30b9\u95a2\u6570\\[ g_i(r) = \\left(\\frac{2\\alpha_i}{\\pi}\\right)^{3\/4} e^{-\\alpha_i r^2}\\]\u304c\u3088\u304f\u4f7f\u308f\u308c\u307e\u3059\u3002\u30ac\u30a6\u30b9\u95a2\u6570\u3092\u4f7f\u3046\u5927\u304d\u306a\u7406\u7531\u306f\u3001\u30ac\u30a6\u30b9\u95a2\u6570\u3092\u4f7f\u3046\u3053\u3068\u3067\u7570\u306a\u308b\u539f\u5b50\u6838\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u95a2\u6570\u306e\u7a4d\u5206\uff08\u591a\u4e2d\u5fc3\u7a4d\u5206\uff09\u3092\u89e3\u6790\u7684\u306b\u52b9\u7387\u3088\u304f\u8a08\u7b97\u3067\u304d\u308b\u305f\u3081\u3067\u3059\u3002\u8db3\u3057\u5408\u308f\u305b\u308b\u305f\u3081\u306b\u3042\u3089\u304b\u3058\u3081\u7528\u610f\u3057\u305f\u95a2\u6570\u306e\u30bb\u30c3\u30c8\u3092<strong>\u57fa\u5e95\u95a2\u6570\u7cfb<\/strong>\u3068\u547c\u3073\u307e\u3059\u3002\u901a\u5e38\u3001\u305d\u308c\u305e\u308c\u306e\u57fa\u5e95\u95a2\u6570\u306f\u539f\u5b50\u6838\u306e\u4f4d\u7f6e\u3092\u4e2d\u5fc3\u3068\u3057\u3066\u914d\u7f6e\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u30b8\u30e7\u30f3\u30fb\u30dd\u30fc\u30d7\u30eb\uff08John Pople\uff09\u306b\u3088\u308b3-21G\u57fa\u5e95\u95a2\u6570\u7cfb\u3092\u4f7f\u7528\u3057\u307e\u3059\u3002He\u306e3-21G\u57fa\u5e95\u3067\u306f\u3001s\u578b\u57fa\u5e95\u95a2\u6570\u30922\u672c\u7528\u3044\u307e\u3059\u30021\u672c\u76ee\u306f2\u500b\u306e\u539f\u59cb\u30ac\u30a6\u30b9\u95a2\u6570\u3092\u7e2e\u7d04\u3057\u305f\u95a2\u6570\u30012\u672c\u76ee\u306f1\u500b\u306e\u539f\u59cb\u30ac\u30a6\u30b9\u95a2\u6570\u304b\u3089\u306a\u308b\u95a2\u6570\u3067\u3059\u3002HF\u8ecc\u9053\u306f\u3001\u3053\u306e2\u672c\u306e\u57fa\u5e95\u95a2\u6570\u306e\u7dda\u5f62\u7d50\u5408&nbsp;\\[ C_1\\chi_1(r)+C_2\\chi_2(r) \\]\u3068\u3057\u3066\u8868\u308f\u3057\u307e\u3059\u3002\u5177\u4f53\u7684\u306b\u6570\u5b57\u3092\u66f8\u304f\u3068\u3001\\[= C_1 \\left[ 0.885750e^{-13.6267r^2} + 1.070687e^{-1.99935r^2} \\right] + C_2 \\left[ 0.346979e^{-0.382993r^2} \\right]&nbsp;\\]\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u539f\u59cb\u30ac\u30a6\u30b9\u95a2\u6570\u306f\u5168\u90e8\u30673\u3064\u3067\u3059\u304c\u3001\u3053\u306e\u8a08\u7b97\u3067\u5909\u5316\u3055\u305b\u308b\u306e\u306f\u3001\u305d\u308c\u305e\u308c\u306e\u57fa\u5e95\u95a2\u6570\\(\\chi_1,\\chi_2\\)\u3092\u3069\u308c\u3060\u3051\u6df7\u305c\u308b\u304b\u3092\u8868\u308f\u30592\u3064\u306e\u4fc2\u6570\\(C_1,C_2\\)\u3067\u3059\u30022\u3064\u524d\u306e\u7bc0\u3067\u306f\\[\\phi(r)=Ne^{-\\zeta r}\\]\u306e\u6307\u6570 \\(\\zeta\\) \u3092\u5909\u5316\u3055\u305b\u3066\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u6700\u4f4e\u306b\u306a\u308b\u8ecc\u9053\u3092\u63a2\u3057\u307e\u3057\u305f\u3002\u305d\u308c\u306b\u5bfe\u3057\u3066\u30013-21G\u57fa\u5e95\u3092\u7528\u3044\u308b\u30ed\u30fc\u30bf\u30fc\u30f3\u6cd5\u3067\u306f\u3001\u30ac\u30a6\u30b9\u95a2\u6570\u306e\u6307\u6570 \\(\\alpha_i\\) \u3084\u7e2e\u7d04\u4fc2\u6570\u306f\u3042\u3089\u304b\u3058\u3081\u6c7a\u3081\u3066\u304a\u304d\u3001\u5206\u5b50\u8ecc\u9053\u3092\u4f5c\u308b\u4fc2\u6570 \\(C_1,C_2\\) \u3092\u5909\u5316\u3055\u305b\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u898f\u683c\u5316<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u8ecc\u9053 \\(\\phi_{1s}^{\\mathrm{HF\/3\\text{-}21G}}(r)\\) \u306b\u3064\u3044\u3066\u3001\u4e00\u96fb\u5b50\u306e\u78ba\u7387\u5bc6\u5ea6 \\(|\\phi_{1s}(r)|^2\\) \u3092\u5168\u7a7a\u9593\u3067\u7a4d\u5206\u3059\u308b\u30681\u306b\u306a\u308b\u3088\u3046\u306b\u898f\u683c\u5316\u3057\u307e\u3059\u3002<br>\\[ \\int|\\phi_{1s}|^2 d\\tau = 1\\]<br>\\[\\int (C_1\\chi_1(r)+C_2\\chi_2(r))^2\\,d\\tau = 1.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\\((A + B)^2 = A^2 + 2AB+B^2\\)\u306a\u306e\u3067\u3001\\[ C_1^2\\int\\chi_1^2d\\tau + 2C_1C_2\\int\\chi_1\\chi_2d\\tau + C_2^2\\int\\chi_2^2d\\tau =1. \\]<br>\u3053\u3053\u3067\u3001\u57fa\u5e95\u95a2\u6570\u540c\u58eb\u306e\u91cd\u306a\u308a\u7a4d\u5206\u3092\\(S_{ij} = \\int \\chi_i\\chi_j d\\tau \\)\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u4eca\u56de\u4f7f\u3063\u3066\u3044\u308b\u57fa\u5e95\u95a2\u6570\\(\\chi_1,\\chi_2\\)\u306f\u305d\u308c\u305e\u308c\u898f\u683c\u5316\u3055\u308c\u3066\u3044\u308b\u306e\u3067\u3001\\[S_{11} =S_{22} = 1\\]\u3067\u3059\u3002\u554f\u984c\u306f2\u3064\u306e\u7570\u306a\u308b\u57fa\u5e95\u95a2\u6570\u306e\u91cd\u306a\u308a<br>\\[\\int\\chi_1(\\mathbf r)\\chi_2(\\mathbf r)\\,d\\tau\\]\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ d\\tau = r^2\\sin\\theta dr d\\theta d\\phi \\]\u306a\u306e\u3067\u3001\\[ S_{12} =  \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty}  \\left[ 0.885750e^{-13.6267r^2} + 1.070687e^{-1.99935r^2} \\right] \\times \\]\\[ \\left[ 0.346979e^{-0.382993r^2} \\right] r^2\\sin\\theta dr d\\theta d\\phi \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u89d2\u5ea6\u90e8\u5206\u3092\u5148\u306b\u7a4d\u5206\u3059\u308b\u3068\u3001\\[ S_{12} = 4\\pi \\left[ (0.885750 \\times 0.346979) \\int_0^{\\infty} r^2 e^{-(13.6267+0.382993)r^2} dr + (1.070687 \\times 0.346979) \\int_0^{\\infty} r^2 e^{-(1.99935+0.382993)r^2}dr\\right]. \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u7a4d\u5206\u516c\u5f0f\\[ \\int_0^{\\infty} r^2 e^{-ar^2} dr = \\frac{\\sqrt{\\pi}}{4a^{3\/2}} \\]<br>\u3092\u4f7f\u3046\u3068\u3001\\[ S_{12} = 4\\pi(0.885750)(0.346979) \\frac{\\sqrt{\\pi}} {4(14.009693)^{3\/2}} + 4\\pi(1.070687)(0.346979) \\frac{\\sqrt{\\pi}} {4(2.382343)^{3\/2}} \\]<br>\\[ = 0.03264 + 0.56258 \\approx 0.5952. \\] <br><br>\u6570\u5024\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\\[ C_1^2 + 2(0.5952)C_1C_2 + C_2^2 =1. \\]<br><br>\u3053\u308c\u3092\u884c\u5217\u306e\u5f62\u3067\u66f8\u304f\u3068\\[ \\mathbf C^{\\mathrm T}\\mathbf S\\mathbf C =1\\]\u3067\u3059\u3002 \\(\\mathbf C^{\\mathrm T}\\)\u306f\u8ee2\u7f6e\u884c\u5217\u3067\u3001\u884c\u5217\u306e\u884c\u3068\u5217\u3092\u5f15\u3063\u304f\u308a\u8fd4\u3057\u3066\u4f5c\u308a\u307e\u3059\u3002\u5177\u4f53\u7684\u306b\u306f\u3001\\[ \\begin{pmatrix} C_1 &amp; C_2\\end{pmatrix}  \\begin{pmatrix} S_{11} &amp; S_{12} \\\\ S_{21} &amp; S_{22} \\end{pmatrix} \\begin{pmatrix} C_{1} \\\\ C_{2} \\end{pmatrix}=1. \\]\u3059\u306a\u308f\u3061\u3001\\[ \\begin{pmatrix} C_1 &amp; C_2\\end{pmatrix} \\begin{pmatrix} 1 &amp; 0.5952 \\\\ 0.5952 &amp; 1 \\end{pmatrix} \\begin{pmatrix} C_{1} \\\\ C_{2} \\end{pmatrix} = 1 \\]<br>\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u771f\u3093\u4e2d\u306b\u631f\u307e\u308c\u305f\u91cd\u306a\u308a\u884c\u5217 \\(\\mathbf S\\) \u304c\u3001\u5f8c\u3067\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f\\[\\mathbf F\\mathbf C = \\mathbf S\\mathbf C\\boldsymbol{\\varepsilon}\\]\u306b\u73fe\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f\u3092\u4f5c\u308b<\/h4>\n\n\n\n<p class=\"wp-block-paragraph\">\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u306f\\[ \\left( \\hat{h}+ \\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right)\\phi = \\varepsilon\\phi \\]\u3067\u3057\u305f\u3002\u30d5\u30a9\u30c3\u30af\u6f14\u7b97\u5b50\u3092\u4f7f\u3048\u3070\u3001 \\[\\hat{F}\\phi = \\varepsilon\\phi \\] \u3068\u7c21\u6f54\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\\[C_1\\chi_1(r)+C_2\\chi_2(r)\\] \u3092\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001 \\[ \\hat{F}(C_1\\chi_1(r)+C_2\\chi_2(r)) = \\varepsilon(C_1\\chi_1(r)+C_2\\chi_2(r)) \\]\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u5f0f\u306e\u4e21\u8fba\u306b\u5de6\u304b\u3089\\(\\chi_1(r)\\)\u3092\u639b\u3051\u3066\u5168\u7a7a\u9593\u3067\u7a4d\u5206\u3057\u307e\u3059\u3002\\[ \\int\\chi_1(r)\\hat{F} C_1\\chi_1(r) d\\tau+\\int \\chi_1(r)\\hat{F} C_2\\chi_2(r) d\\tau = \\int \\chi_1(r)\\varepsilon C_1\\chi_1(r) d\\tau+\\int\\chi_1(r)\\varepsilon C_2\\chi_2(r) d\\tau.\\] \u4fc2\u6570 \\(C_1,C_2\\)\u3068\u56fa\u6709\u5024\\(\\varepsilon\\) \u306f\u4f4d\u7f6e\u306b\u4f9d\u5b58\u3057\u306a\u3044\u6570\u306a\u306e\u3067\u3001\u7a4d\u5206\u306e\u5916\u306b\u51fa\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\\[ C_1\\int\\chi_1(r)\\hat{F}\\chi_1(r) d\\tau+C_2\\int \\chi_1(r)\\hat{F}\\chi_2(r)) d\\tau = \\varepsilon C_1 \\int \\chi_1(r)\\chi_1(r) d\\tau+\\varepsilon C_2\\int\\chi_1(r)\\chi_2(r)) d\\tau.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u5c11\u3057\u3084\u3084\u3053\u3057\u3044\u306e\u3067\u3059\u304c\u3001\u3053\u308c\u307e\u3067\\(1,2\\)\u3068\u3044\u3046\u6570\u5b57\u306f\u96fb\u5b501\u3068\u96fb\u5b502\u3092\u6307\u3059\u30e9\u30d9\u30eb\u3068\u3057\u3066\u4f7f\u3063\u3066\u304d\u307e\u3057\u305f\u3002\u3057\u304b\u3057\u3001\u3053\u3053\u304b\u3089\u306f\u57fa\u5e95\u95a2\u6570\u3092\u6307\u3059\u30e9\u30d9\u30eb\u3068\u3057\u3066\u4f7f\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u3001\\( F_{11} = \\int\\chi_1(r)\\hat{F}\\chi_1(r) d\\tau \\) \u3092\u30d5\u30a9\u30c3\u30af\u884c\u5217\u8981\u7d20\u3001\\( S_{11} = \\int \\chi_1(r)\\chi_1(r) d\\tau \\)\u3092\u91cd\u306a\u308a\u884c\u5217\u8981\u7d20\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001\u4e0a\u306e\u5f0f\u306f\u3001\\[ C_1 F_{11} + C_2 F_{12} = \\varepsilon\\left( C_1 S_{11} + C_2 S_{12} \\right)\\]\u3068\u7c21\u6f54\u306b\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u524d\u306e\u7bc0\u3067\u65e2\u306b\u8a08\u7b97\u3057\u305f\u3088\u3046\u306b\u3001\\(S_{11} = 1,\\, S_{12} = 0.5952\\)\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u540c\u3058\u3088\u3046\u306b\u3001\u4eca\u5ea6\u306f\u3082\u3068\u306e\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u306b\u5de6\u304b\u3089 \\(\\chi_2\\) \u3092\u639b\u3051\u3066\u7a4d\u5206\u3057\u307e\u3059\u3002\u9014\u4e2d\u3092\u7701\u7565\u3057\u307e\u3059\u304c\u3001\u540c\u69d8\u306b\u3001\\[ C_1 F_{21} + C_2 F_{22} = \\varepsilon\\left( C_1 S_{21} + C_2 S_{22} \\right)\\] \u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u3053\u308c\u306b\u3088\u3063\u3066\u672a\u77e5\u306e\u95a2\u6570\\(\\phi\\)\u3092\u6c42\u3081\u308b\u5fae\u5206\u65b9\u7a0b\u5f0f\u304c\u3001\u57fa\u5e95\u95a2\u6570\u306e\u4fc2\u6570 \\(C_1\\)\u3068\\(C_2\\)\u3092\u6c42\u3081\u308b\u9023\u7acb\u65b9\u7a0b\u5f0f\u306b\u7f6e\u304d\u63db\u308f\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ C_1 F_{11} + C_2 F_{12} = \\varepsilon\\left( C_1 S_{11} + C_2 S_{12} \\right)\\]\\[ C_1 F_{21} + C_2 F_{22} = \\varepsilon\\left( C_1 S_{21} + C_2 S_{22} \\right)\\] <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9023\u7acb\u4e00\u6b21\u65b9\u7a0b\u5f0f\u306f\u884c\u5217\u306e\u5f62\u3067\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u884c\u5217\u306e\u5f62\u3067\u66f8\u304f\u3068\u3001\\[ \\begin{pmatrix} F_{11} &amp; F_{12} \\\\ F_{21} &amp; F_{22} \\end{pmatrix} \\begin{pmatrix} C_1 \\\\ C_2 \\end{pmatrix} = \\varepsilon \\begin{pmatrix} S_{11} &amp; S_{12} \\\\ S_{21} &amp; S_{22} \\end{pmatrix} \\begin{pmatrix} C_1 \\\\ C_2 \\end{pmatrix} \\]\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u304c\u4e00\u3064\u306e\u8ecc\u9053\u306b\u3064\u3044\u3066\u66f8\u3044\u305f<strong>\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f<\/strong>\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b21\u306b\u3001\u3059\u3079\u3066\u306e\u9805\u3092\u5de6\u8fba\u306b\u79fb\u3057\u3066 \\(=0\\) \u306e\u5f62\u306b\u3057\u307e\u3059\u3002\\[ C_1 (F_{11} &#8211; \\varepsilon S_{11}) + C_2(F_{12} &#8211; \\varepsilon S_{12}) = 0\\]\\[ C_1 (F_{21} &#8211; \\varepsilon S_{21}) + C_2(F_{22} &#8211; \\varepsilon S_{22}) = 0\\]\u884c\u5217\u3067\u66f8\u3051\u3070\u3001 \\[ \\begin{pmatrix} F_{11} &#8211; \\varepsilon S_{11} &amp; F_{12} &#8211; \\varepsilon S_{12} \\\\ F_{21} &#8211; \\varepsilon S_{21} &amp; F_{22} &#8211; \\varepsilon S_{22} \\end{pmatrix} \\begin{pmatrix} C_1 \\\\ C_2 \\end{pmatrix} = 0. \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u9023\u7acb\u65b9\u7a0b\u5f0f\u306b\u306f\\(C_1 = C_2 = 0\\)\u3068\u3044\u3046\u89e3\u304c\u5fc5\u305a\u3042\u308a\u307e\u3059\u3002\u3057\u304b\u3057\u305d\u308c\u3067\u306f\u3001\\(\\psi = 0\\)\u3068\u306a\u3063\u3066\u610f\u5473\u304c\u3042\u308a\u307e\u305b\u3093\u3002\u305d\u3053\u3067\u3001\\(C_1,C_2\\)\u304c\u540c\u6642\u306b0\u3067\u306a\u3044\u89e3\u3092\u6301\u3064\u305f\u3081\u306e\u6761\u4ef6\u3092\u8003\u3048\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u306b\u306f\u3001\u4fc2\u6570\u884c\u5217\u306e<strong>\u884c\u5217\u5f0f\u304c0\u3067\u306a\u3051\u308c\u3070\u306a\u308a\u307e\u305b\u3093<\/strong>\u3002\\[\\begin{vmatrix}F_{11}-\\varepsilon &amp; F_{12}-0.5952\\varepsilon \\\\ F_{21}-0.5952\\varepsilon &amp; F_{22}-\\varepsilon \\end{vmatrix}=[(F_{11}-\\varepsilon) \\times ( F_{22}-\\varepsilon) ]-[(F_{12}-0.5952\\varepsilon)\\times ( F_{21}-0.5952\\varepsilon)] = 0\\]\u304c\u6761\u4ef6\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u307e\u305f\u3001\u884c\u5217\u5f0f\u304c\u51fa\u3066\u304d\u307e\u3057\u305f\u3002\u4ee5\u524d\u51fa\u3066\u304d\u305f\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u306f\u3001\u96fb\u5b50\u4ea4\u63db\u306b\u5bfe\u3057\u3066\u53cd\u5bfe\u79f0\u306a\u591a\u96fb\u5b50\u6ce2\u52d5\u95a2\u6570\u3092\u7c21\u6f54\u306b\u8868\u3059\u305f\u3081\u306e\u3082\u306e\u3067\u3057\u305f\u3002\u4e00\u65b9\u3001\u3053\u3053\u3067\u51fa\u3066\u304d\u305f\u884c\u5217\u5f0f\u306f\u3001\u9023\u7acb\u65b9\u7a0b\u5f0f\u304c\u300c\\(C_1=C_2= 0\\)\u300d\u4ee5\u5916\u306e\u89e3\u3092\u6301\u3064\u305f\u3081\u306e\u6761\u4ef6\u3067\u3059\u3002\u540c\u3058\u300c\u884c\u5217\u5f0f\u300d\u3068\u3044\u3046\u6570\u5b66\u7684\u306a\u9053\u5177\u3067\u3059\u304c\u3001\u5f79\u5272\u306f\u307e\u3063\u305f\u304f\u7570\u306a\u308a\u307e\u3059\u3002\u5f0f\u3068\u3044\u3046\u540d\u524d\u304c\u4ed8\u3044\u3066\u3044\u307e\u3059\u304c\u884c\u5217\u5f0f\u306f\u6570\u5024\u3067\u3059\u3002\u884c\u5217\u5f0f\u306f\u9ad8\u6821\u6570\u5b66\u306b\u767b\u5834\u3057\u307e\u305b\u3093\u304c\u30012&#215;2\u306e\u6b63\u65b9\u884c\u5217\\(\\begin{pmatrix}A &amp; B \\\\ C &amp; D \\end{pmatrix}\\)\u306b\u5bfe\u3059\u308b\u884c\u5217\u5f0f\u306f\\(A\\times D &#8211; B\\times C\\)\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u3001\u9069\u5f53\u306a\\(C_1,C_2\\)\u3092\u4eee\u5b9a\u3057\u3066\u3042\u3089\u304b\u3058\u3081\u8a08\u7b97\u3057\u3066\u304a\u3044\u3066\\( F_{11} , F_{12} ,F_{21} , F_{22}\\)\u306e\u6570\u5024\u3092\u4ee3\u5165\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001\u4e0a\u306e\u5f0f\u306f\\(\\varepsilon\\)\u306b\u95a2\u3059\u308b\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u3092\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u306e\u89e3\u306e\u516c\u5f0f\\[x = \\frac{-b \\pm\\sqrt{b^2 &#8211; 4ac}}{2a}\\]\u3092\u4f7f\u3063\u3066\u89e3\u304f\u3068\u30012\u500b\u306e\u57fa\u5e95\u95a2\u6570\u3092\u4f7f\u3063\u3066\u3044\u308b\u306e\u3067\u3001\u901a\u5e38\u306f2\u3064\u306e\u56fa\u6709\u5024\\(\\varepsilon_1, \\varepsilon_2\\)\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u3055\u3089\u306b\u3001\u305d\u308c\u305e\u308c\u306e \\(\\varepsilon\\) \u3092\u9023\u7acb\u65b9\u7a0b\u5f0f\u306b\u623b\u305b\u3070\u3001\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066\\(C_1\\)\u3068\\(C_2\\)\u306e\u6bd4\\[\\frac{C_1}{C_2}\\]\u304c\u6c7a\u307e\u308a\u307e\u3059\u3002\u6700\u5f8c\u306b\u3001\u898f\u683c\u5316\u6761\u4ef6\\[C_1^2+2(0.5952)C_1C_2+C_2^2=1\\]\u3092\u4f7f\u3046\u3068\u3001\\(C_1\\) \u3068 \\(C_2\\) \u306e\u5024\u305d\u306e\u3082\u306e\u304c\u6c7a\u307e\u308a\u307e\u3059\u3002<br><br>\u3053\u306e\u3088\u3046\u306a\u624b\u9806\u30672\u3064\u306e\u56fa\u6709\u5024\\(\\varepsilon_1,\\varepsilon_2\\)\u306b\u5bfe\u5fdc\u3057\u3066\u3001\\[\\phi_{\\mathrm{occupied}} = C_{11}\\chi_1+C_{21}\\chi_2\\]\\[\\phi_{\\mathrm{unoccupied}} = C_{12}\\chi_1+C_{22}\\chi_2\\]\u3068\u3044\u30462\u3064\u306e\u8ecc\u9053\u304c\u5f97\u3089\u308c\u307e\u3059\u3002\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u57fa\u5e95\u72b6\u614b\u3067\u306f\u3001\u3053\u306e\u3046\u3061\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u4f4e\u3044\u65b9\u306e\u8ecc\u9053 \\(\\phi_{\\mathrm{occupied}}\\) \u306b2\u96fb\u5b50\u3092\u5165\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u307e\u3067\u306e\u3001\\[\\text{\u884c\u5217\u5f0f}=0\\]\u304b\u3089\u56fa\u6709\u5024\u3092\u6c42\u3081\u3001\u305d\u306e\u56fa\u6709\u5024\u304b\u3089\u4fc2\u6570\u3092\u6c42\u3081\u308b\u3068\u3044\u3046\u624b\u9806\u306f\u3001\u5358\u7d14\u30d2\u30e5\u30c3\u30b1\u30eb\u6cd5\u3068\u57fa\u672c\u7684\u306b\u540c\u3058\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5358\u7d14\u30d2\u30e5\u30c3\u30b1\u30eb\u6cd5\u3068\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u306e\u5927\u304d\u306a\u9055\u3044\u306f\u3001\\(F_{ij}\\)\u3092\u8a08\u7b97\u3059\u308b\u30d5\u30a9\u30c3\u30af\u6f14\u7b97\u5b50\u304c<br> \\[ \\hat{F} = \\hat{h} + \\int\\frac{|\\phi(\\mathbf r_2)|^2}{r_{12}}d\\tau = \\hat{h} + \\int\\frac{|C_1\\chi_1(r_2)+C_2\\chi_2(r_2)|^2}{r_{12}}d\\tau \\]\u3068\u3001\u89e3\\(C_1,C_2\\)\u3092\u542b\u3093\u3067\u3044\u308b\u70b9\u306b\u3042\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u4fc2\u6570\\(C_{11},C_{21}\\)\u306e\u5024\u304c\u66f4\u65b0\u3055\u308c\u308b\u2192\\(F_{ij}\\)\u306e\u5024\u3082\u66f4\u65b0\u3055\u308c\u308b\u2192\u9023\u7acb\u65b9\u7a0b\u5f0f\u3092\u89e3\u3044\u3066\u4fc2\u6570\\(C_{11},C_{21}\\)\u306e\u5024\u3092\u66f4\u65b0\u2192\\(F_{ij}\\)\u306e\u5024\u3092\u66f4\u65b0\u30fb\u30fb\u30fb\u3001\u3068\\[(C_{11},C_{21}){\\mathrm{\u65b0}}\\approx(C_{11},C_{21}){\\mathrm{\u524d\u56de}}\\]\u3068\u306a\u308b\u307e\u3067\u53cd\u5fa9\u8a08\u7b97\u3057\u306a\u3051\u308c\u3070\u3044\u3051\u307e\u305b\u3093\u3002\u3053\u308c\u304c<strong>\u81ea\u5df1\u7121\u649e\u7740\u5834\uff08self-consistent field, SCF\uff09\u8a08\u7b97<\/strong>\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u203b \\(\\phi_{\\mathrm{unoccupied}}\\)\u306f\u975e\u5360\u6709\u8ecc\u9053\uff08\u7a7a\u8ecc\u9053\uff09\u3067\u3059\u304c\u3001HOMO\/LUMO\u3092\u8b70\u8ad6\u3059\u308b\u6642\u306b\u4f7f\u308f\u308c\u308b\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u3067\u3044\u3046\u6b63\u6e96\u4eee\u60f3\u8ecc\u9053\uff08canonical virtual orbital\uff09\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u30d8\u30ea\u30a6\u30e0\u306e\u6a19\u6e96\u7684\u306a\u9589\u6bbbHF\u30d5\u30a9\u30c3\u30af\u6f14\u7b97\u5b50\u306f\\(\\hat{f} = \\hat{h} + 2\\hat{J}_1 &#8211; \\hat{K}_1\\)\u3067\u3059\u3002\u5360\u6709\u8ecc\u9053\\(\\phi_1\\)\u306b\u5bfe\u3057\u3066\u306f\\(\\hat{J}\\phi_1 = \\hat{K}\\phi_1\\)\u306a\u306e\u306b\u5bfe\u3057\u3066\u3001\u5225\u306e\u8ecc\u9053\\(\\phi_a\\)\u306b\u5bfe\u3057\u3066\u306f\u4e00\u822c\u306b\\(\\hat{J}\\phi_a \\neq \\hat{K}\\phi_a\\)\u3067\u3059\u3002\u305d\u306e\u305f\u3081\u3001\u6b63\u6e96\u4eee\u60f3\u8ecc\u9053\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\u53ce\u675f\u3057\u305f\u5360\u6709\u8ecc\u9053\u304b\u3089\u6a19\u6e96\u9589\u6bbb\u30d5\u30a9\u30c3\u30af\u884c\u5217&nbsp;\\(\\hat h+2\\hat J_1-\\hat K_1\\)&nbsp;\u3092\u65b0\u305f\u306b\u7d44\u307f\u7acb\u3066\u307e\u3059\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">SCF\u8a08\u7b97\u30011\u56de\u76ee\uff08\\(C_1 = 1, C_2= 0\\)\uff09<\/h5>\n\n\n\n<h6 class=\"wp-block-heading\">\u30d5\u30a9\u30c3\u30af\u884c\u5217\u3092\u8a08\u7b97\u3059\u308b<\/h6>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u3067\u306f\u3001\u524d\u7bc0\u3067\u5c0e\u3044\u305fHe\u306e\u5360\u6709\u7a7a\u9593\u8ecc\u9053\u306b\u5bfe\u3059\u308bHF\u65b9\u7a0b\u5f0f&nbsp;\\((\\hat h+\\hat J)\\phi=\\epsilon\\phi\\)&nbsp;\u3092\u81ea\u5df1\u7121\u649e\u7740\u5834\uff08SCF\uff09\u6cd5\u3067\u89e3\u304d\u307e\u3059\u3002\u3053\u306e\u7bc0\u3067\u7528\u3044\u308b\u884c\u5217\u3092&nbsp;\\(F_{ij}\\)&nbsp;\u3068\u66f8\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[ F_{ij}^{(n)} =\\int\\chi_i(\\mathbf r_1) \\left[ \u2212\\frac{1}{2}\\nabla_1^2 \u2013\\frac{2}{r_1}+ \\int\\frac{| C_1^{(n)}\\chi_1(\\mathbf r_2)+C_2^{(n)}\\chi_2(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right]\\chi_j(\\mathbf r_1) d\\tau_1 \\]\n\n\n\n<p class=\"wp-block-paragraph\">\u4eca\u56de\u306f\u3001\\(C_1 = 1, C_2= 0\\)\u3068\u3044\u3046\u6975\u7aef\u306a\u521d\u671f\u5024\u3092\u4f7f\u3063\u3066\u307f\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\\[\\phi^{(0)} = 1\\times \\chi_1(\\mathbf r) + 0 \\times \\chi_2(\\mathbf r) = 0.885750e^{\u221213.6267r^2}+1.070687e^{\u22121.99935r^2}\\]<br>\\[F_{ij}^{(0)}= \\int\\chi_i^*(\\mathbf r_1) \\left( \u2212\\frac{1}{2}\\nabla_1^2 \u2013\\frac{2}{r_1}+ \\int\\frac{| \\phi^{(0)}(\\mathbf r_2)|^2}{r_{12}}d\\tau_2\\right)\\chi_j(\\mathbf r_1) d\\tau_1\\] \u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30ac\u30a6\u30b9\u95a2\u6570\u306e\u7a4d\u5206\u516c\u5f0f\u3092\u4f7f\u3063\u3066\u7a4d\u5206\u3059\u308b\u3068\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u9805\u3001\u6838\u5f15\u529b\u9805\u3001\u96fb\u5b50\u9593\u53cd\u767a\u9805\u306e\u9806\u306b\u3001<br>\\[F_{11}^{(0)} = 3.91612-5.48968+1.80325=0.22969 \\]<br>\\[F_{12}^{(0)} = 0.57895-2.23529+0.87651=-0.77984\\]<br>\\[F_{21}^{(0)} = 0.57895-2.23529+0.87651=-0.77984 \\]<br>\\[F_{22}^{(0)} = 0.57449-1.97513+0.91701=-0.48363 \\]<br>\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u30011\u56de\u76ee\u306e\u884c\u5217\u306f\u3001\\[\\mathbf{F}^{(0)} = \\begin{pmatrix} 0.22969 &amp; -0.77984 \\\\ -0.77984 &amp; -0.48363\\end{pmatrix}\\]\u3068\u306a\u308a\u307e\u3059\u3002<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u89e3\u304f\u3079\u304d\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f\u306f\u3001\\[ \\begin{pmatrix} 0.22969 &amp; -0.77984 \\\\ -0.77984 &amp; -0.48363\\end{pmatrix}\\begin{pmatrix} C_1^{(1)}\\\\ C_2^{(1)} \\end{pmatrix} =\\begin{pmatrix}1 &amp; 0.5952 \\\\ 0.5952 &amp; 1\\end{pmatrix}\\begin{pmatrix}C_1^{(1)}\\\\ C_2^{(1)} \\end{pmatrix}\\varepsilon \\]\u3067\u3059\u3002\u5de6\u8fba\u306b\u307e\u3068\u3081\u3066\u3001\u884c\u5217\u5f0f=0\u306e\u6761\u4ef6\u3092\u7acb\u3066\u307e\u3059\u3002<br>\\[ \\begin{vmatrix} 0.22969 &#8211; \\varepsilon &amp; -0.77984 -0.5952\\varepsilon\\\\ -0.77984-0.5952\\varepsilon &amp; -0.48363-\\varepsilon\\end{vmatrix}= 0\\]<br>\u5c55\u958b\u3059\u308b\u3068\u3001<br>\\[(0.22969 &#8211; \\varepsilon)( -0.48363 &#8211; \\varepsilon) &#8211; (-0.77984 &#8211; 0.5952\\varepsilon)^2 = 0 \\]\\[ 0.64573696\\varepsilon^2 -0.674381536\\varepsilon -0.7192354003=0 \\]\u3068\u306a\u308a\u3001\u3053\u306e\\(\\varepsilon\\)\u306e\u4e8c\u6b21\u65b9\u7a0b\u5f0f\u3092\u89e3\u304f\u3068\u3001\\[\\varepsilon_1 \\approx -0.65531\\,E_h\\]\\[\\varepsilon_2 \\approx 1.69965\\,E_h\\]\u304c\u6c42\u307e\u308a\u307e\u3059\u3002\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u4f4e\u3044\u65b9\u306e\\(\\varepsilon_1 \\approx -0.65531\\,E_h\\)\u3092\u96fb\u5b50\u304c\u5360\u6709\u3059\u308b\u306e\u3067\u3001\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f\u306b\u623b\u3059\u3068\u3001\\[ \\begin{pmatrix} 0.885 &amp; -0.38980 \\\\ -0.38980 &amp; 0.17168\\end{pmatrix}\\begin{pmatrix} C_1^{(1)}\\\\ C_2^{(1)} \\end{pmatrix} =0\\]<br><\/p>\n\n\n\n<h6 class=\"wp-block-heading\">\u65b0\u3057\u3044\u4fc2\u6570\u3092\u6c42\u3081\u308b<\/h6>\n\n\n\n<p class=\"wp-block-paragraph\">\\(C_1^{(1)}\u3068C_2^{(1)}\\)\u306e\u6bd4\u306f\u3001\\[\\frac{C_2^{(1)}}{C_1^{(1)}}= 2.2705.\\]<br>\u898f\u683c\u5316\u6761\u4ef6\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001\\[\\left(C_1^{(1)}\\right)^2+2(0.5952)C_1^{(1)}(2.2705)C_1^{(1)}+\\left(2.2705 C_1^{(1)}\\right)^2=1\\]\\[C_1^{(1)} = \\pm 0.33600.\\]<br>\u6b63\u306e\u5024\u3092\u53d6\u3063\u3066\u3001\\[\\left\\{C_1^{(1)},C_2^{(1)}\\right\\} = \\left\\{0.33600,0.76289\\right\\} .\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u521d\u671f\u5024\u3068\u3057\u3066\\(\\left\\{C_1^{(0)},C_2^{(0)}\\right\\} = \\left\\{1,0\\right\\}\\)\u3092\u5165\u529b\u3057\u305f\u306e\u306b\u3001\u65b0\u3057\u3044\u4fc2\u6570\u3068\u3057\u3066\\(\\left\\{C_1^{(1)},C_2^{(1)}\\right\\} = \\left\\{0.33600,0.76289\\right\\}\\)\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u3053\u308c\u3092\\(C_1,C_2\\)\u306e\u5024\u304c\u5909\u5316\u3057\u306a\u304f\u306a\u308b\u307e\u3067\u7e70\u66ff\u3048\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">SCF\u8a08\u7b97\u306e\u7d9a\u304d<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\u7d9a\u304d\u306fPython\u306b\u4efb\u305b\u307e\u3057\u305f\u3002<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes has-small-font-size\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-right\" data-align=\"right\">iteration<\/th><th class=\"has-text-align-left\" data-align=\"left\">\\(C_1\\)<\/th><th class=\"has-text-align-left\" data-align=\"left\">\\(C_2\\)<\/th><th class=\"has-text-align-left\" data-align=\"left\">\\(\\varepsilon_1\\) \/ \\(E_h\\)<\/th><th class=\"has-text-align-left\" data-align=\"left\">\\(E\\) \/ \\(E_h\\)<\/th><th class=\"has-text-align-left\" data-align=\"left\">max\\(|\\Delta C|\\)<\/th><\/tr><\/thead><tbody><tr><td>1<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.335997<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.762871<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.655308<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.776430<\/td><td class=\"has-text-align-left\" data-align=\"left\">7.628708e-01<\/td><\/tr><tr><td>2<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.485176<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.632086<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.970905<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.832555<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.491784e-01<\/td><\/tr><tr><td>3<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.451576<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.663054<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.888560<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835513<\/td><td class=\"has-text-align-left\" data-align=\"left\">3.359954e-02<\/td><\/tr><tr><td>4<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.459384<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.655939<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.907073<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835671<\/td><td class=\"has-text-align-left\" data-align=\"left\">7.808075e-03<\/td><\/tr><tr><td>5<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457580<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657587<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.902763<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835679<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.804063e-03<\/td><\/tr><tr><td>6<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457997<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657206<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903759<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">4.174173e-04<\/td><\/tr><tr><td>7<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457901<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657294<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903528<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">9.654941e-05<\/td><\/tr><tr><td>8<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457923<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657274<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903582<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">2.233373e-05<\/td><\/tr><tr><td>9<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457918<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657279<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903569<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">5.166129e-06<\/td><\/tr><tr><td>10<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903572<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.195009e-06<\/td><\/tr><tr><td>11<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903571<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">2.764245e-07<\/td><\/tr><tr><td>12<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903572<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">6.394137e-08<\/td><\/tr><tr><td>13<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903572<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.479066e-08<\/td><\/tr><tr><td>14<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903572<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">3.421314e-09<\/td><\/tr><tr><td>15<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903572<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">7.914042e-10<\/td><\/tr><tr><td>16<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903572<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">1.830645e-10<\/td><\/tr><tr><td>17<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.457919<\/td><td class=\"has-text-align-left\" data-align=\"left\">0.657278<\/td><td class=\"has-text-align-left\" data-align=\"left\">-0.903572<\/td><td class=\"has-text-align-left\" data-align=\"left\">-2.835680<\/td><td class=\"has-text-align-left\" data-align=\"left\">4.234574e-11<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u6700\u7d42\u7684\u306b\u53ce\u675f\u3057\u305f\u5024\u306f\u3001\\[C_1 = 0.457919\\]\\[C_2 = 0.657278\\]\\[\\varepsilon = -0.903572\\, E_h\\]\\[E = -2.835680\\, E_h\\]\u3067\u3057\u305f\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306e\u4e00\u96fb\u5b50\u6ce2\u52d5\u95a2\u6570\u306f\u3001\\[\\phi_{\\mathrm{3-21G}} = (0.457919)\\left[ 0.885750e^{-13.6267r^2} + 1.070687e^{-1.99935r^2} \\right] + (0.657278)\\left[ 0.346979e^{-0.382993r^2} \\right]\\]\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"318\" class=\"wp-image-8145\" style=\"width: 600px;\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G.png\" alt=\"\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G.png 1635w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G-300x159.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G-1024x542.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G-768x407.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G-1536x814.png 1536w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u6b63\u6e96\u4eee\u60f3\u8ecc\u9053\u3092\u6c42\u3081\u308b<\/h5>\n\n\n\n<h6 class=\"wp-block-heading\">\u53ce\u675f\u3057\u305f\u4fc2\u6570\u304b\u3089\\(F = h + 2 J -K\\)\u3092\u4f5c\u308b<\/h6>\n\n\n\n<p class=\"wp-block-paragraph\">\u53ce\u675f\u3057\u305f\u4fc2\u6570\u3092\u4f7f\u3063\u3066\u6b63\u6e96\u30d5\u30a9\u30c3\u30af\u884c\u5217\\(\\mathbf F\\)\u3092\u4f5c\u308a\u307e\u3059\u3002\u5360\u6709\u8ecc\u9053\u306e\u4fc2\u6570\u884c\u5217\\(\\begin{pmatrix}C_{11}\\\\C_{21}\\end{pmatrix}\\)\u306b\u4f5c\u7528\u3055\u305b\u308b\u3068\u3001\\[\\mathbf{J}\\begin{pmatrix}C_{11}\\\\C_{21}\\end{pmatrix} = \\mathbf{K}\\begin{pmatrix}C_{11}\\\\C_{21}\\end{pmatrix}\\]\u306b\u306a\u3063\u3066\u3001\u6253\u3061\u6d88\u3055\u308c\u308b\u306e\u3067\u7c21\u7565\u5316\u3057\u305f\u30d5\u30a9\u30c3\u30af\u884c\u5217\u3092\u4f7f\u3063\u3066\u3044\u307e\u3057\u305f\u3002\u3057\u304b\u3057\u3001\u975e\u5360\u6709\u8ecc\u9053\u3067\u306f\u3001 \\[\\mathbf{J}\\begin{pmatrix}C_{12}\\\\C_{22}\\end{pmatrix} \\neq \\mathbf{K}\\begin{pmatrix}C_{12}\\\\C_{22}\\end{pmatrix}\\] \u306b\u306a\u308b\u306e\u3067\u3001\\(K\\)\u306e\u6570\u5024\u3082\u8a08\u7b97\u3057\u3066\u5165\u308c\u305f\u6b63\u6e96\u30d5\u30a9\u30c3\u30af\u884c\u5217\\(\\mathbf F\\)\u3092\u4f5c\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53ce\u675f\u4fc2\u6570\u306f\u3001\\[\\begin{pmatrix}C_{11}\\\\C_{21}\\end{pmatrix} = \\begin{pmatrix}0.4579188741\\\\0.6572778604&nbsp;\\end{pmatrix} , \\]\u91cd\u306a\u308a\u884c\u5217\u306f\\[S = \\begin{pmatrix}1 &amp; 0.5952159497 \\\\0.5952159497 &amp; 1\\end{pmatrix}.\\]<br><br>\u6b63\u6e96HF\u306e\u884c\u5217\u8981\u7d20\\[F_{11} = h_{11} + 2J_{11} &#8211; K_{11}\\]\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\\[h_{11} = \\int \\left(0.885750e^{-13.6267r^2} + 1.070687e^{-1.99935r^2}\\right)\\left[-\\frac{1}{2}\\nabla_1^2 &#8211; \\frac{2}{r_1}\\right]\\left(0.885750e^{-13.6267r^2} + 1.070687e^{-1.99935r^2}\\right)\\,d\\tau_1 \\]\\[ = -1.5735646753.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u57fa\u5e95\u95a2\u6570\\[\\chi_1 = 0.885750e^{-13.6267r^2} + 1.070687e^{-1.99935r^2} \\]\\[ \\chi_2 =  0.346979e^{-0.382993r^2}\\]\u540c\u58eb\u306e\u96fb\u5b50\u9593\u53cd\u767a\u7a4d\u5206\u3092\u306f\u3058\u3081\u306b\u8a08\u7b97\u3057\u3066\u304a\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u96fb\u5b50\u9593\u53cd\u767a\u7a4d\u5206\u3092 \\[&nbsp;(ij|kl)= \\iint \\chi_i(\\mathbf r_1)\\chi_j(\\mathbf r_1) \\frac{1}{r_{12}} \\chi_k(\\mathbf r_2)\\chi_l(\\mathbf r_2) \\,d\\tau_1d\\tau_2\\]\u3068\u5b9a\u7fa9\u3059\u308b\u3068\u3001\\[(11|11)=1.8032545920,\\]\\[ (11|12)=0.8765068190,\\]\\[ (11|22)=0.9170082837,\\]\\[ (12|12)=0.4520772565.&nbsp;\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30fc\u30ed\u30f3\u9805\\(J_{11}\\)\u306f\u3001\\[\\iint \\chi_1(\\mathbf r_1)^2\\frac{1}{r_{12}}\\phi(\\mathbf r_2)^2\\,d\\tau_1 d\\tau_2.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u3053\u306b&nbsp;\\(\\phi=C_{11}\\chi_1+C_{21}\\chi_2\\)&nbsp;\u3092\u4ee3\u5165\u3057\u3066\u5c55\u958b\u3059\u308b\u3068\u3001\\[J_{11} = C_{11}^2(11|11) +2C_{11}C_{21}(11|12) +C_{21}^2(11|22)\\]\u3067\u3059\u3002\u6570\u5024\u3092\u5165\u308c\u308b\u3068\u3001\\[&nbsp;J_{11} =(0.2096896952)(1.8032545920)+(0.6019598756)(0.8765068190)+(0.4320141858)(0.9170082837)\\]\\[=0.3781239058+0.5276219357+0.3961605870\\]\\[=1.3019064285\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u4ea4\u63db\u9805\\(K_{11}\\)\u306f\u3001\\[\\iint \\chi_1(\\mathbf r_1)\\phi(\\mathbf r_1)\\frac{1}{r_{12}}\\phi(\\mathbf r_2)\\chi_1(\\mathbf r_2)\\,d\\tau_1 d\\tau_2.\\]<br>&nbsp;\\(\\phi=C_{11}\\chi_1+C_{21}\\chi_2\\)&nbsp;\u3092\u4ee3\u5165\u3057\u3066\u5c55\u958b\u3059\u308b\u3068\u3001\\[K_{11} =C_{11}^2(11|11) +C_{11}C_{21}(11|12) +C_{11}C_{21}(12|11) +C_{21}^2(12|12).&nbsp;\\]\\[ = C_{11}^2(11|11) +2C_{11}C_{21}(11|12) +C_{21}^2(12|12).&nbsp;\\]<br>\u6570\u5024\u3092\u4ee3\u5165\u3059\u308b\u3068\u3001\\[ K_{11} =(0.2096896952)(1.8032545920)+(0.6019598756)(0.8765068190)+(0.4320141858)(0.4520772565)\\]\\[ =0.3781239058+0.5276219357+0.1953037879\\]\\[ =1.1010496294. \\]<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u308c\u3089\u3092\u8db3\u3057\u5408\u308f\u305b\u308b\u3068\u3001\\[F_{11} =h_{11}+2J_{11}-K_{11}=-1.5735646753 +2(1.3019064285) -1.1010496294\\]\\[ =-1.5735646753 +2.6038128570 -1.1010496294\\]\\[=-0.0708014476\\ E_h.\\]<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6b8b\u308a\u3082\u5168\u90e8\u540c\u69d8\u306b\u8a08\u7b97\u3059\u308b\u3068\u3001\\[\\mathbf{F} = \\begin{pmatrix}-0.0708014476&amp;-1.1180026983\\\\-1.1180026983&amp;-0.4993641791\\end{pmatrix}.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\\[\\begin{pmatrix}C_{11} &amp; C_{12}\\\\ C_{21} &amp; C_{22} \\end{pmatrix}^{T}\\mathbf{F}\\begin{pmatrix}C_{11} &amp; C_{12}\\\\ C_{21} &amp;C_{22} \\end{pmatrix}= \\begin{pmatrix}C_{11} &amp; C_{12} \\\\ C_{21} &amp; C_{22} \\end{pmatrix}^{T}\\mathbf{S}\\begin{pmatrix}C_{11} &amp; C_{12}\\\\ C_{21} &amp;C_{22} \\end{pmatrix}\\begin{pmatrix}\\varepsilon_1 &amp; 0 \\\\ 0 &amp; \\varepsilon_2\\end{pmatrix} = \\begin{pmatrix}\\varepsilon_1 &amp; 0 \\\\ 0 &amp; \\varepsilon_2\\end{pmatrix} \\]\u306a\u306e\u3067\u3001\\[\\begin{pmatrix}0.4579188741 &amp; 0.6572778604\\\\ C_{12} &amp; C_{22} \\end{pmatrix}<br>\\begin{pmatrix}-0.0708014476&amp;-1.1180026983\\\\-1.1180026983&amp;-0.4993641791\\end{pmatrix}\\begin{pmatrix}0.4579188741 &amp; C_{12}\\\\ 0.6572778604 &amp;C_{22} \\end{pmatrix}=\\begin{pmatrix}\\varepsilon_1 &amp; 0 \\\\ 0 &amp; \\varepsilon_2\\end{pmatrix} .\\]<br>\u884c\u5217\u306e\u639b\u3051\u7b97\u3092\u3059\u308b\u3068\u3001\\[ \\begin{pmatrix}-0.9035715084 &amp;-0.76725974\\,C_{12} -0.84017556\\,C_{22}&nbsp; \\\\ -0.76725974\\,C_{12} -0.84017556\\,C_{22}&nbsp; &amp;-0.0708014476\\,C_{12}^2 +2(-1.1180026983)\\,C_{12}C_{22} +(-0.49936417)C_{22}^2 \\end{pmatrix}= \\begin{pmatrix}\\varepsilon_1 &amp; 0 \\\\ 0 &amp; \\varepsilon_2\\end{pmatrix} .\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u975e\u5bfe\u89d2\u6210\u5206\u30920\u3068\u7f6e\u3044\u305f\u5f0f\\[-0.76725974\\,C_{12} -0.84017556\\,C_{22}&nbsp; = 0\\]\u304b\u3089\u3001\\[ \\frac{C_{22}}{C_{12}} = -\\frac{0.76725974}{0.84017556}= -0.9132136.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u898f\u683c\u5316\u6761\u4ef6\u3001\\[C_{12}^2 + 2(0.5952159497)C_{12}C_{22}+ C_{22}^2 =1 \\]<br>\u3092\u89e3\u3044\u3066\u3001\\(C_{12} &gt; 0\\)\u306e\u89e3\u3092\u9078\u3076\u3068\u3001\\[C_{12} = 1.1571404526,\\]\\[C_{22} = -1.0567163936&nbsp;.\\]\n\n\n\n<p class=\"wp-block-paragraph\">\u6700\u5f8c\u306b\\(\\epsilon_2\\)\u3092\u8a08\u7b97\u3059\u308b\u3068\u3001\\[\\varepsilon_2 = -0.0708014476C_{12}^2 + 2(-1.1180026983)C_{12}C_{22} + (-0.4993641791)C_{22}^2\\]\\[= 2.0817026437\\ E_h\\]\n\n\n\n<p class=\"wp-block-paragraph\">3-21G\u57fa\u5e95\u95a2\u6570\u7cfb\u3092\u4f7f\u3063\u305f\u6642\u306e\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u306eHF\u6b63\u6e96\u4eee\u60f3\u8ecc\u9053\u306e\u6ce2\u52d5\u95a2\u6570\u306f\u3001\\[\\phi_{virtual} = (1.1571404526)\\left[ 0.885750e^{-13.6267r^2} + 1.070687e^{-1.99935r^2} \\right] + (-1.0567163936)\\left[ 0.346979e^{-0.382993r^2} \\right]\\]\u3067\u3059\u3002\u56f3\u793a\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" width=\"600\" height=\"192\" class=\"wp-image-8231\" style=\"width: 600px;\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G_virtual.png\" alt=\"\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G_virtual.png 1525w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G_virtual-300x96.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G_virtual-1024x328.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G_virtual-768x246.png 768w\" sizes=\"auto, (max-width: 600px) 100vw, 600px\" \/><\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u4eee\u60f3\u8ecc\u9053\u306f\u5360\u6709\u8ecc\u9053\u3068\u76f4\u4ea4\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u305f\u3081\u30012\u672c\u306e\u6b63\u5024\u306es\u578b\u57fa\u5e95\u95a2\u6570\u3092\u53cd\u5bfe\u7b26\u53f7\u3067\u7d44\u307f\u5408\u308f\u305b\u308b\u3053\u3068\u306b\u306a\u308a\u30011\u3064\u306e\u52d5\u5f84\u7bc0\uff08\\(0.86796613\\,a_0&nbsp;\\)\uff09\u304c\u751f\u3058\u307e\u3059\u3002\u305d\u306e\u305f\u3081\u6c34\u7d20\u69d82s\u8ecc\u9053\u306b\u4f3c\u305f\u5f62\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u30d8\u30ea\u30a6\u30e0\u306e2s\u539f\u5b50\u8ecc\u9053\u305d\u306e\u3082\u306e\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<br><br>(\u3053\u3053\u307e\u3067)<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u8a18\u4e8b\u306e\u4f5c\u6210\u306b\u3042\u305f\u308a\u3001\u8aac\u660e\u30fb\u6570\u5f0f\u306e\u78ba\u8a8d\u3068\u4fee\u6b63\u306bChatGPT &#8230;<\/p>\n","protected":false},"author":1,"featured_media":8145,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"_locale":"ja","_original_post":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/?p=7994","footnotes":""},"categories":[5],"tags":[],"class_list":["post-7994","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-5","ja"],"aioseo_notices":[],"aioseo_head":"\n\t\t<!-- All in One SEO 5.0.2.1 - aioseo.com -->\n\t<meta name=\"description\" content=\"\u672c\u8a18\u4e8b\u306e\u4f5c\u6210\u306b\u3042\u305f\u308a\u3001\u8aac\u660e\u30fb\u6570\u5f0f\u306e\u78ba\u8a8d\u3068\u4fee\u6b63\u306bChatGPT\u3092\u6d3b\u7528\u3057\u307e\u3057\u305f\u3002\u5185\u5bb9\u306e\u6700\u7d42\u7684\u306a\u8cac\u4efb\u306f\u8457\u8005\u306b\u3042\u308a\u307e\u3059\" \/>\n\t<meta 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