{"id":361,"date":"2019-10-11T21:57:40","date_gmt":"2019-10-11T13:57:40","guid":{"rendered":"http:\/\/132.145.126.25\/?p=361"},"modified":"2022-10-21T17:03:03","modified_gmt":"2022-10-21T09:03:03","slug":"coordinates-transformation-qt-zh","status":"publish","type":"post","link":"https:\/\/www.tsihyoung.com\/index.php\/2019\/10\/11\/coordinates-transformation-qt-zh\/","title":{"rendered":"\u91cf\u5b50\u529b\u5b78\u4e2d\u7684\u5169\u985e\u5ea7\u6a19\u8b8a\u63db"},"content":{"rendered":"\n<p>\u4e0a\u56de\u6211\u5011\u7c21\u8981\u4ecb\u7d39\u4e86<a href=\"http:\/\/www.tsihyoung.com\/index.php\/en\/2019\/04\/12\/molecular-orbitals-from-roothaan-equation\" data-type=\"URL\" data-id=\"http:\/\/www.tsihyoung.com\/index.php\/en\/2019\/04\/12\/molecular-orbitals-from-roothaan-equation\">Roothan\u65b9\u7a0b<\/a>\u7684\u89e3\u6cd5\u3002\u8a72\u89e3\u6cd5\u5148\u5c0d\u539f\u5b50\u8ecc\u9053\u57fa\u7d44\u7684\u91cd\u758a\u77e9\u9663\\(\\mathbf{S}\\)\u4f5c\u4e86Cholesky\u5206\u89e3\uff0c\u5f9e\u800c\u5c07\u539f\u672c\u7684Roothan\u65b9\u7a0b\u8f49\u5316\u6210\u7232\u6b63\u4ea4\u57fa\u7d44\u4e0b\u7684\u7279\u5fb5\u65b9\u7a0b\uff0c\u6700\u5f8c\u901a\u904e\u6c42\u89e3\u7279\u5fb5\u65b9\u7a0b\u5f97\u5230\u5206\u5b50\u8ecc\u9053\u7684\u80fd\u91cf\\(\\boldsymbol{\\epsilon}\\)<\/p>\n\n\n\n<blockquote style=\"text-align:center\" class=\"wp-block-quote\"><p>\\(\\mathbf{H}\\mathbf{C}=\\mathbf{S}\\mathbf{C}\\boldsymbol{\\epsilon}\\)<\/p><cite><font size=\"2\">Roothan\u65b9\u7a0b<\/font><\/cite><\/blockquote>\n\n\n\n<p>\u6ce8\u610f\u5230\u4e0a\u5f0f\u4e2d\\(\\mathbf{H}\\)\u7232\u539f\u5b50\u57fa\u7d44\u4e0b\u7684\u54c8\u5bc6\u9813\u91cf\uff0c\u800c\\(\\boldsymbol{\\epsilon}\\)\u53ef\u7406\u89e3\u7232\u5206\u5b50\u8ecc\u9053\u4e0b\u7684\u54c8\u5bc6\u9813\u91cf\uff08\u4ee5\u4e0b\u5206\u5225\u8a18\u7232\\(\\mathbf{H_{AO}}\\)\u548c\\(\\mathbf{H_{MO}}\\)\uff09\uff0c\u5247\u4e0a\u5f0f\u5176\u5be6\u8868\u9054\u4e86\u54c8\u5bc6\u9813\u91cf\u5728\u539f\u5b50\u8ecc\u9053\u57fa\u7d44\u548c\u5206\u5b50\u8ecc\u9053\u57fa\u7d44\u4e0b\u7684\u5ea7\u6a19\u8b8a\u63db\uff0c\u5373\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathbf{H_{MO}} = \\left(\\mathbf{S}\\mathbf{C}\\right)^{-1}\\mathbf{H_{AO}}\\mathbf{C}\\). (1)<\/p>\n\n\n\n<p>\u53e6\u5916\uff0c\u7531\u65bc\u5206\u5b50\u8ecc\u9053\u662f\u6b63\u4ea4\u6b78\u4e00\u7684<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\langle \\varphi_{i} | \\varphi_{j} \\rangle = \\mathbf{C}^{\\dagger}\\mathbf{S}\\mathbf{C} = \\mathbf{I}\\),<\/p>\n\n\n\n<p>\u5f0f(1)\u53c8\u53ef\u5beb\u4f5c<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathbf{H_{MO}} = \\mathbf{C}^{\\dagger}\\mathbf{H_{AO}}\\mathbf{C}\\).<\/p>\n\n\n\n<p>\u9019\u719f\u6089\u7684\u5473\u9053\uff0c\u662f\u6a19\u6e96\u7684\u8868\u8c61\u8b8a\u63db\u516c\u5f0f\uff01<\/p>\n\n\n\n<p>\u53e6\u4e00\u65b9\u9762\uff0c\u76f4\u63a5\u8003\u5bdf\\(\\hat{H}\\)\u5728\u5206\u5b50\u8ecc\u9053\u4e0b\u7684\u5143\u7d20\u4ea6\u53ef\u5f97\u5230\u540c\u6a23\u7d50\u679c\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\left(\\mathbf{H_{MO}}\\right)_{ij} = \\langle\\varphi_{i} | \\hat{H} | \\varphi_{j} \\rangle = \\sum_{\\mu\\nu}{C_{i{\\mu}}^{\\ast} \\langle\\chi_{\\mu}|\\hat{H}|\\chi_{\\nu}\\rangle C_{{\\nu j}}} = \\left( \\mathbf{C}^{\\dagger}\\mathbf{H_{AO}}\\mathbf{C} \\right)_{ij}\\).<\/p>\n\n\n\n<p>\u4e00\u822c\u800c\u8a00\uff0c\u5c0d\u4efb\u610f\u529b\u5b78\u91cf\\(\\hat{A}\\)\uff0c\u7d66\u5b9a\u5169\u5957\u6eff\u8db3\u5982\u4e0b\u8b8a\u63db\u7684\u57fa\u7d44\\(\\mathbf{X_{1}}\\)\u548c\\(\\mathbf{X_{2}}\\)\uff0c<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathbf{X_{2}} = \\mathbf{C}\\mathbf{X_{1}}\\), (2)<\/p>\n\n\n\n<p>\u5247\\(\\hat{A}\\)\u5728\u5169\u57fa\u7d44\u4e0b\u7684\u77e9\u9663\u8868\u793a\u6eff\u8db3<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathbf{A_{2}} = \\mathbf{C}^{\\dagger}\\mathbf{A_{1}}\\mathbf{C}\\). (3)<\/p>\n\n\n\n<p>\u63a5\u4e0b\u4f86\u6211\u5011\u8003\u5bdf\u53e6\u4e00\u985e\u4ee5\u5bc6\u5ea6\u77e9\u9663\u7232\u4ee3\u8868\u7684\u7b97\u7b26\u3002\u5bc6\u5ea6\u77e9\u9663\u662f\u5426\u4e5f\u6eff\u8db3(2)(3)\u7684\u5ea7\u6a19\u8b8a\u63db\u95dc\u4fc2\u5462\uff1f\u6211\u5011\u9996\u5148\u7d66\u51fa\u5bc6\u5ea6\u77e9\u9663\u7b97\u7b26\u5728\u80fd\u91cf\u8868\u8c61\u4e0b\u7684\u5b9a\u7fa9\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\hat{\\sigma} = \\sum_{i} f_{i}|\\varphi_{i}\\rangle\\langle\\varphi_{i}|\\),<\/p>\n\n\n\n<p>\u5176\u4e2d\\(f_{i}\\)\u662f\u614b\\(|\\varphi_{i}\\rangle\\)\u4e0a\u7684\u4f54\u64da\u6578\u3002\u4ee5\u539f\u5b50\u8ecc\u9053\u57fa\u7d44\u7232\u4f8b\uff0c\u4e0b\u6587\u5c07\u8b49\u660e\\(\\hat{\\sigma}\\)\u7684\u77e9\u9663\u8868\u793a\u7232\\(\\boldsymbol{\\sigma}^{\\mathbf{AO}} = \\mathbf{C}\\mathbf{f}\\mathbf{C}^{\\dagger}\\)\uff0c\u5176\u4e2d\\(\\mathbf{f}\\)\u662f\\(f_{i}\\)\u6392\u6210\u7684\u5c0d\u89d2\u77e9\u9663\u3002\u6211\u5011\u5148\u5617\u8a66\u6c42\u7b97<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathrm{tr}{\\left(\\boldsymbol{\\sigma}^{\\mathbf{AO}}\\mathbf{S_{AO}}\\right)} = \\sum_{i\\mu\\nu} f_{i}C_{i\\nu}^{\\ast}\\langle\\chi_{\\nu}|\\chi_{\\mu}\\rangle C_{\\mu i} = \\sum_{i} f_{i}\\). (4)<\/p>\n\n\n\n<p>\u9019\u8aaa\u660e(4)\u5f0f\u5de6\u908a\u8868\u793a\u4e86\u96fb\u5b50\u6578\u4e4b\u548c\uff0c\u56e0\u800c\u4e0d\u61c9\u96a8\u5ea7\u6a19\u8b8a\u63db\u800c\u8b8a\u5316\u3002\u5728\u5206\u5b50\u8ecc\u9053\u57fa\u7d44\u4e0b\uff0c\u7531\u65bc\u57fa\u7d44\u6b63\u4ea4\u6b78\u4e00\uff0c\u6211\u5011\u6709<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathrm{tr}{\\left(\\boldsymbol{\\sigma}^{\\mathbf{MO}}\\mathbf{S_{MO}}\\right)} = \\mathrm{tr}{\\left(\\boldsymbol{\\sigma}^{\\mathbf{MO}}\\right)}\\), (5)<\/p>\n\n\n\n<p>\u7232\u4f7f(5)\u5f0f\u5de6\u908a\u4ecd\u7136\u7b49\u65bc\\(\\sum_{i} f_{i}\\)\uff0c\u6700\u81ea\u7136\u7684\u69cb\u9020\u662f\\(\\boldsymbol{\\sigma}^{\\mathbf{MO}} = \\mathbf{f}\\)\uff0c\u5373<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\boldsymbol{\\sigma}^{\\mathbf{MO}} = \\mathbf{C}^{-1} \\boldsymbol{\\sigma}^{\\mathbf{AO}} \\left(\\mathbf{C}^{\\dagger}\\right)^{-1}\\). (6)<\/p>\n\n\n\n<p>\u53ef\u898b(6)\u4e4b\u8b8a\u63db\u8207(3)\u7684\u4e0d\u540c\u3002\u53e6\u5916\uff0c\u529b\u5b78\u91cf\\(\\hat{A}\\)\u7684\u671f\u671b\u503c\u4e5f\u53ef\u901a\u904e\u4e0b\u5f0f\u6c42\u5f97<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\langle\\hat{A}\\rangle = \\mathrm{tr}\\left(\\boldsymbol{\\sigma}\\mathbf{A}\\right)\\),<\/p>\n\n\n\n<p>\u800c<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathrm{tr}\\left(\\boldsymbol{\\sigma}^{\\mathbf{MO}}\\mathbf{A_{MO}}\\right) = \\mathrm{tr}\\left(\\mathbf{C}^{-1}\\boldsymbol{\\sigma}^{\\mathbf{AO}}\\mathbf{A_{AO}}\\mathbf{C}\\right) = \\mathrm{tr}\\left(\\boldsymbol{\\sigma}^{\\mathbf{AO}}\\mathbf{A_{AO}}\\right)\\),<\/p>\n\n\n\n<p>\u53ef\u898b\u8b8a\u63db(6)\u4fdd\u8b49\u4e86\u96fb\u5b50\u7e3d\u6578\u8207\u529b\u5b78\u91cf\u671f\u671b\u503c\u4e0d\u8b8a\uff0c\u56e0\u6b64\u662f\u7b26\u5408\u7269\u7406\u8981\u6c42\u7684\u3002<\/p>\n\n\n\n<p>\u90a3\u9ebc\u7232\u4ec0\u9ebc\u540c\u6a23\u662f\u7b97\u7b26\uff0c\u5bc6\u5ea6\u77e9\u9663\u8207\u529b\u5b78\u91cf\u7684\u8b8a\u63db\u95dc\u4fc2\u537b\u4e0d\u4e00\u81f4\u5462\uff1f\u6ce8\u610f\u5230\u529b\u5b78\u91cf\u7684\u6307\u6a19\u662f\u5beb\u5728\u4e0b\u6a19\u7684\uff0c\u800c\u5bc6\u5ea6\u77e9\u9663\u5247\u662f\u4e0a\u6a19\uff0c\u9019\u8aaa\u660e\u5169\u8005\u7684\u5f35\u91cf\u985e\u578b\u5176\u5be6\u662f\u8fe5\u7570\u7684\uff0c\u4e5f\u6b63\u5982\u6b64\uff0c\u5169\u8005\u624d\u9075\u5faa\u4e0d\u540c\u7684\u8b8a\u63db\u95dc\u4fc2\u3002\u4e0b\u6587\u5c07\u7c21\u8981\u4ecb\u7d39\u9019\u7a2e\u4e0d\u540c\u3002<\/p>\n\n\n\n<p>\u5c0d\u65bc\u975e\u6b63\u4ea4\u57fa\u7d44\\(\\left\\{|\\chi_{\\mu}\\rangle\\right\\}\\)\uff0c\u5176\u5ea6\u898f\u5f35\u91cf\uff0c\u5373\u91cd\u758a\u77e9\u9663\u7232<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\mathbf{S}_{\\mu\\nu} = \\langle\\chi_{\\mu}|\\chi_{\\nu}\\rangle\\).<\/p>\n\n\n\n<p>\u901a\u904e\u5ea6\u898f\u53ef\u5c07\u8a72\u57fa\u7d44\u8f49\u7232\u5c0d\u5076\u57fa\u7d44<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(|\\chi^{\\mu}\\rangle = \\sum_{\\nu} \\left(\\mathbf{S}^{-1}\\right)^{\\nu\\mu}|\\chi_{\\nu}\\rangle\\).<\/p>\n\n\n\n<p class=\"has-cyan-bluish-gray-color has-text-color\" style=\"font-size:12px\">\u8a3b\uff1a\u5be6\u969b\u4e0a\\(|\\chi^{\\mu}\\rangle\\)\u4ecd\u7136\u4f4d\u65bc\u77e2\u91cf\u7a7a\u9593\u800c\u975e\u5c0d\u5076\u7a7a\u9593\uff0c\u5176\u81ea\u7136\u8a8d\u540c\u7684\u5de6\u77e2\\(\\langle\\chi^{\\mu}|\\)\u624d\u662f\u771f\u6b63\u7684\u5c0d\u5076\u77e2\u91cf\u3002<\/p>\n\n\n\n<p>\u5206\u5225\u7a31\\(|\\chi_{\\mu}\\rangle\\)\u548c\\(|\\chi^{\\mu}\\rangle\\)\u7232\u9006\u8b8a\u548c\u5354\u8b8a\u77e2\u91cf\u3002<\/p>\n\n\n\n<p>\u986f\u7136\u9019\u5c0d\u57fa\u7d44\u6eff\u8db3\u96d9\u6b63\u4ea4\u95dc\u4fc2<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\langle\\chi_{\\mu}|\\chi^{\\nu}\\rangle = \\delta_{\\mu}^{\\hphantom{\\mu}\\nu}, \\langle\\chi^{\\mu}|\\chi_{\\nu}\\rangle = \\delta_{\\hphantom{\\mu}\\nu}^{\\mu}\\).<\/p>\n\n\n\n<p>\u57fa\u7d44\u5b8c\u5099\u6027\u53ef\u8868\u793a\u7232<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\sum_{\\mu}|\\chi_{\\mu}\\rangle\\langle\\chi^{\\mu}| = 1\\).<\/p>\n\n\n\n<p>\u5c0d\u65bc\u529b\u5b78\u91cf\u7b97\u7b26\\(\\hat{A}\\)\uff0c\u5176\u7269\u7406\u610f\u7fa9\u7522\u751f\u65bc\u5b83\u4f5c\u7528\u5728\u9006\u8b8a\u77e2\u91cf\u7684\u904e\u7a0b\u4e2d\uff0c\u56e0\u800c\u5b83\u672c\u8eab\u662f\u5354\u8b8a\u5f35\u91cf\uff0c\u6709\u610f\u7fa9\u7684\u662f\u5176\u9006\u8b8a\u5206\u91cf\\(\\mathbf{A}_{\\mu\\nu}\\)\u3002\u800c\u5c0d\u65bc\u5bc6\u5ea6\u77e9\u9663\\(\\hat{\\sigma}\\)\uff0c\u5b83\u7684\u7269\u7406\u610f\u7fa9\u7522\u751f\u65bc\u5b83\u4f5c\u7528\u5728\u529b\u5b78\u91cf\u7b97\u7b26\u7684\u904e\u7a0b\u4e2d\uff0c\u56e0\u800c\u5b83\u672c\u8eab\u662f\u9006\u8b8a\u5f35\u91cf\uff0c\u6709\u610f\u7fa9\u7684\u662f\u5176\u5354\u8b8a\u5206\u91cf\\(\\boldsymbol{\\sigma}^{\\mu\\nu}\\)\u3002<\/p>\n\n\n\n<p>\u8003\u616e\\(\\hat{\\sigma}\\)\u5728\u539f\u5b50\u8ecc\u9053\u4e0b\u7684\u5354\u8b8a\u5206\u91cf<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\boldsymbol{\\sigma}^{\\mu\\nu} = \\sum_{i} f_{i}\\langle\\chi^{\\mu}|\\varphi_{i}\\rangle\\langle\\varphi_{i}|\\chi^{\\nu}\\rangle = \\left(\\mathbf{C}\\mathbf{f}\\mathbf{C}^{\\dagger}\\right)^{\\mu\\nu}\\), (7)<\/p>\n\n\n\n<p>\u5373<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\boldsymbol{\\sigma}^{\\mathbf{AO}} = \\mathbf{C}\\mathbf{f}\\mathbf{C}^{\\dagger}\\).<\/p>\n\n\n\n<p>(7)\u5f0f\u7b2c\u4e8c\u500b\u7b49\u865f\u7528\u5230\u4e86\u96d9\u6b63\u4ea4\u95dc\u4fc2\u3002<\/p>\n\n\n\n<p>\u7232\u7372\u5f97\\(\\hat{\\sigma}\\)\u5728\u5206\u5b50\u8ecc\u9053\u4e0b\u7684\u5354\u8b8a\u5206\u91cf\uff0c\u6211\u5011\u9996\u5148\u8981\u5f97\u5230\u5206\u5b50\u8ecc\u9053\u7684\u5354\u8b8a\u77e2\u91cf<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(|\\boldsymbol{\\varphi}^{\\mathbf{MO}}\\rangle = \\mathbf{S_{MO}}^{-1} |\\boldsymbol{\\varphi}_{\\mathbf{MO}}\\rangle = \\left(\\mathbf{C}^{\\dagger}\\mathbf{SC}\\right)^{-1} |\\boldsymbol{\\varphi}_{\\mathbf{MO}}\\rangle\\),<\/p>\n\n\n\n<p>\u5f9e\u800c\u5f97\u5230\\(\\hat{\\sigma}\\)\u9006\u8b8a\u5206\u91cf\u8207\u5354\u8b8a\u5206\u91cf\u7684\u95dc\u4fc2<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\boldsymbol{\\sigma}^{\\mathbf{MO}} = \\left(\\mathbf{C}^{\\dagger}\\mathbf{SC}\\right)^{-1} \\boldsymbol{\\sigma}_{\\mathbf{MO}}\\left(\\mathbf{C}^{\\dagger}\\mathbf{SC}\\right)^{-1}\\) . (8)<\/p>\n\n\n\n<p>\u8f03\u6613\u6c42\u5f97<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\boldsymbol{\\sigma}_{ij} = \\sum_{k} f_{k}\\langle\\varphi_{i}|\\varphi_{k}\\rangle\\langle\\varphi_{k}|\\varphi_{j}\\rangle = \\sum_{k} \\left(\\mathbf{C}^{\\dagger}\\mathbf{SC}\\right)_{ik} f_{k} \\left(\\mathbf{C}^{\\dagger}\\mathbf{SC}\\right)_{kj}\\),<\/p>\n\n\n\n<p>\u5373<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\boldsymbol{\\sigma}_{\\mathbf{MO}} = \\mathbf{C}^{\\dagger}\\mathbf{S}\\boldsymbol{\\sigma}^{\\mathbf{AO}}\\mathbf{SC}\\). (9)<\/p>\n\n\n\n<p>\u5c07(9)\u4ee3\u5165(8)\u5373\u5f97(6)<\/p>\n\n\n\n<p class=\"has-text-align-center\">\\(\\boldsymbol{\\sigma}^{\\mathbf{MO}} = \\mathbf{C}^{-1} \\boldsymbol{\\sigma}^{\\mathbf{AO}} \\left(\\mathbf{C}^{\\dagger}\\right)^{-1}\\).<\/p>\n\n\n\n<p>\u7d9c\u4e0a\uff0c\u5bc6\u5ea6\u77e9\u9663\u5c6c\u65bc\u9006\u8b8a\u5f35\u91cf\u7684\u672c\u8cea\u5c0e\u81f4\u4e86\u5176\u8207\u666e\u901a\u529b\u5b78\u91cf\u5ea7\u6a19\u8b8a\u63db\u7684\u5dee\u7570\u3002\u5728\u5be6\u969b\u61c9\u7528\u4e2d\uff0c\u9084\u6709\u4e00\u985e\u77e9\u9663\u662f\u8207\u5bc6\u5ea6\u77e9\u9663\u985e\u4f3c\u7684\uff0c\u5728\u4f5c\u5ea7\u6a19\u8b8a\u63db\u6642\u4e00\u5b9a\u8981\u5c0f\u5fc3\u5224\u65b7\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4e0a\u56de\u6211\u5011\u7c21\u8981\u4ecb\u7d39\u4e86Roothan\u65b9\u7a0b\u7684\u89e3\u6cd5\u3002\u8a72\u89e3\u6cd5\u5148\u5c0d\u539f\u5b50\u8ecc\u9053\u57fa\u7d44\u7684\u91cd\u758a\u77e9\u9663\\(\\mathbf{S}\\)\u4f5c\u4e86Cholesky\u5206\u89e3\uff0c\u5f9e\u800c\u5c07\u539f\u672c\u7684Roothan\u65b9\u7a0b\u8f49\u5316\u6210\u7232\u6b63\u4ea4\u57fa\u7d44\u4e0b\u7684\u7279\u5fb5\u65b9\u7a0b\uff0c\u6700\u5f8c\u901a\u904e\u6c42\u89e3\u7279\u5fb5\u65b9\u7a0b\u5f97\u5230\u5206\u5b50\u8ecc\u9053\u7684\u80fd\u91cf\\(\\boldsymbol{\\epsilon}\\) \\(\\mathbf{H}\\mathbf{C}=\\mathbf{S}\\mathbf{C}\\boldsymbol{\\epsilon}\\) Roothan\u65b9\u7a0b \u6ce8\u610f\u5230\u4e0a\u5f0f\u4e2d\\(\\mathbf{H}\\)\u7232\u539f\u5b50\u57fa\u7d44\u4e0b\u7684\u54c8\u5bc6\u9813\u91cf\uff0c\u800c\\(\\boldsymbol{\\epsilon}\\)\u53ef\u7406\u89e3\u7232\u5206\u5b50\u8ecc\u9053\u4e0b\u7684\u54c8\u5bc6\u9813\u91cf\uff08\u4ee5\u4e0b\u5206\u5225\u8a18\u7232\\(\\mathbf{H_{AO}}\\)\u548c\\(\\mathbf{H_{MO}}\\)\uff09\uff0c\u5247\u4e0a\u5f0f\u5176\u5be6\u8868\u9054\u4e86\u54c8\u5bc6\u9813\u91cf\u5728\u539f\u5b50\u8ecc\u9053\u57fa\u7d44\u548c\u5206\u5b50\u8ecc\u9053\u57fa\u7d44\u4e0b\u7684\u5ea7\u6a19\u8b8a\u63db\uff0c\u5373\uff1a \\(\\mathbf{H_{MO}} = \\left(\\mathbf{S}\\mathbf{C}\\right)^{-1}\\mathbf{H_{AO}}\\mathbf{C}\\). (1) \u53e6\u5916\uff0c\u7531\u65bc\u5206\u5b50\u8ecc\u9053\u662f\u6b63\u4ea4\u6b78\u4e00\u7684 \\(\\langle \\varphi_{i} | \\varphi_{j} \\rangle = \\mathbf{C}^{\\dagger}\\mathbf{S}\\mathbf{C} = \\mathbf{I}\\), \u5f0f(1)\u53c8\u53ef\u5beb\u4f5c \\(\\mathbf{H_{MO}} = \\mathbf{C}^{\\dagger}\\mathbf{H_{AO}}\\mathbf{C}\\). \u9019\u719f\u6089\u7684\u5473\u9053\uff0c\u662f\u6a19\u6e96\u7684\u8868\u8c61\u8b8a\u63db\u516c\u5f0f\uff01 \u53e6\u4e00\u65b9\u9762\uff0c\u76f4\u63a5\u8003\u5bdf\\(\\hat{H}\\)\u5728\u5206\u5b50\u8ecc\u9053\u4e0b\u7684\u5143\u7d20\u4ea6\u53ef\u5f97\u5230\u540c\u6a23\u7d50\u679c\uff1a \\(\\left(\\mathbf{H_{MO}}\\right)_{ij} = \\langle\\varphi_{i} | \\hat{H} | \\varphi_{j} \\rangle = \\sum_{\\mu\\nu}{C_{i{\\mu}}^{\\ast} \\langle\\chi_{\\mu}|\\hat{H}|\\chi_{\\nu}\\rangle C_{{\\nu j}}} = \\left( \\mathbf{C}^{\\dagger}\\mathbf{H_{AO}}\\mathbf{C} \\right)_{ij}\\). \u4e00\u822c\u800c\u8a00\uff0c\u5c0d\u4efb\u610f\u529b\u5b78\u91cf\\(\\hat{A}\\)\uff0c\u7d66\u5b9a\u5169\u5957\u6eff\u8db3\u5982\u4e0b\u8b8a\u63db\u7684\u57fa\u7d44\\(\\mathbf{X_{1}}\\)\u548c\\(\\mathbf{X_{2}}\\)\uff0c \\(\\mathbf{X_{2}} = \\mathbf{C}\\mathbf{X_{1}}\\), (2) \u5247\\(\\hat{A}\\)\u5728\u5169\u57fa\u7d44\u4e0b\u7684\u77e9\u9663\u8868\u793a\u6eff\u8db3 \\(\\mathbf{A_{2}} = \\mathbf{C}^{\\dagger}\\mathbf{A_{1}}\\mathbf{C}\\). (3) \u63a5\u4e0b\u4f86\u6211\u5011\u8003\u5bdf\u53e6\u4e00\u985e\u4ee5\u5bc6\u5ea6\u77e9\u9663\u7232\u4ee3\u8868\u7684\u7b97\u7b26\u3002\u5bc6\u5ea6\u77e9\u9663\u662f\u5426\u4e5f\u6eff\u8db3(2)(3)\u7684\u5ea7\u6a19\u8b8a\u63db\u95dc\u4fc2\u5462\uff1f\u6211\u5011\u9996\u5148\u7d66\u51fa\u5bc6\u5ea6\u77e9\u9663\u7b97\u7b26\u5728\u80fd\u91cf\u8868\u8c61\u4e0b\u7684\u5b9a\u7fa9\uff1a \\(\\hat{\\sigma} = \\sum_{i} [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[72],"tags":[],"post_formats":[],"languages":[6],"post_translationss":[],"class_list":["post-361","post","type-post","status-publish","format-standard","hentry","category-computational-chemistry"],"guten_post_layout_featured_media_urls":{"full":false,"thumbnail":false,"medium":false,"medium_large":false,"large":false,"1536x1536":false,"2048x2048":false,"guten_post_layout_landscape_large":false,"guten_post_layout_portrait_large":false,"guten_post_layout_square_large":false,"guten_post_layout_landscape":false,"guten_post_layout_portrait":false,"guten_post_layout_square":false,"sciencexlite-blog-thumb":false},"_links":{"self":[{"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/posts\/361","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/comments?post=361"}],"version-history":[{"count":5,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/posts\/361\/revisions"}],"predecessor-version":[{"id":625,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/posts\/361\/revisions\/625"}],"wp:attachment":[{"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/media?parent=361"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/categories?post=361"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/tags?post=361"},{"taxonomy":"post_format","embeddable":true,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/post_formats?post=361"},{"taxonomy":"language","embeddable":true,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/languages?post=361"},{"taxonomy":"post_translations","embeddable":true,"href":"https:\/\/www.tsihyoung.com\/index.php\/wp-json\/wp\/v2\/post_translationss?post=361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}