\n
LDA\u4ea4\u63db\u30a8\u30cd\u30eb\u30ae\u30fc\u6c4e\u95a2\u6570\u306e\u5f0f\u306f\u3001\\[E_{x}[\\rho] = -\\frac{3}{4}\\left(\\frac{3}{\\pi}\\right)^{\\frac{1}{3}} \\int \\rho(\\mathbf r)^{\\frac{4}{3}}d\\mathbf{r}\\]\u3067\u3059\u30023\u5206\u306e4\u4e57\u306f\u300c4\u4e57\u306e3\u4e57\u6839 \\(\\sqrt[3]{x^4}\\)\u300d\u3092\u8868\u308f\u3057\u307e\u3059\u3002\u3053\u306e\u4ea4\u63db\u6c4e\u95a2\u6570\u306e\u6c4e\u95a2\u6570\u5fae\u5206\u3092\u5c0e\u304f\u305f\u3081\u306b\u3001\u96fb\u5b50\u5bc6\u5ea6\u3092\\(\\rho(\\mathbf{r}) \\to \\rho(\\mathbf{r}) + \\delta\\rho(\\mathbf{r})\\)\u3068\u5c11\u3057\u3060\u3051\u5909\u5316\u3055\u305b\u307e\u3059\u3002\\(x^{\\frac{4}{3}}\\) \u3092\u5fae\u5206\u3059\u308b\u3068\u3001\\[\\frac{d}{dx}x^{\\frac{4}{3}}=\\frac{4}{3}x^{\\frac{1}{3}}\\]\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\\(x\\)\u3092\u5fae\u5c0f\u306b\u5909\u5316\u3055\u305b\u305f\u6642\u306e\u4e00\u6b21\u5909\u5206\uff08\u4e8c\u6b21\u4ee5\u4e0a\u306e\u5909\u5316\u306f\u7121\u8996\uff09\u306f\\((x+\\delta x)^{4\/3}\\approx x^{4\/3}+\\frac43x^{1\/3}\\delta x\\)\u3068\u8fd1\u4f3c\u3067\u304d\u307e\u3059\u3002\u96fb\u5b50\u5bc6\u5ea6\u306b\u5f53\u3066\u306f\u3081\u308b\u3068\u3001\u96fb\u5b50\u5bc6\u5ea6\u3092\u5fae\u5c0f\u306b\u5909\u5316\u3055\u305b\u305f\u6642\u306e\u3001LDA\u4ea4\u63db\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5909\u5316\u306f\u3001\\[\\begin{aligned}
\\delta E_x&=E_x[\\rho+\\delta\\rho]-E_x[\\rho]\\\\
&\\approx-\\left(\\frac{3}{\\pi}\\right)^{\\frac{1}{3}}\\int\\rho(\\mathbf r)^{\\frac{1}{3}}\\delta\\rho(\\mathbf r)d\\mathbf r
\\end{aligned}\\]\u3068\u306a\u308a\u307e\u3059\u3002\\(\\delta\\rho(\\mathbf r)\\) \u306e\u4fc2\u6570\u304c\u6c4e\u95a2\u6570\u5fae\u5206\u306a\u306e\u3067\u3001\\[v_x(\\mathbf r) = \\frac{\\delta E_x[\\rho]}{\\delta\\rho(\\mathbf r)}=-\\left(\\frac{3}{\\pi}\\right)^{\\frac{1}{3}}\\rho(\\mathbf r)^{\\frac{1}{3}}\\]\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u306a\u305c\u3053\u308c\u304c\u4ea4\u63db\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u306a\u306e\u3067\u3057\u3087\u3046\u304b\u3002\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u65b9\u7a0b\u5f0f\u3092\u5c0e\u304f\u6642\u306b\u306f\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u6c4e\u95a2\u6570\u3092\u8ecc\u9053\u3067\u5909\u5206\u3059\u308b\u3068\u3001\\[\\delta\\rho(\\mathbf r) = 2\\phi_A(\\mathbf r)\\delta\\phi^*_A(\\mathbf r),\\]\\[\\delta E_x = \\int v_x(\\mathbf r)\\delta\\rho(\\mathbf r)\\,d^3 r = \\int 2v_x(\\mathbf r)\\phi_A(\\mathbf r)\\delta\\phi^*_A(\\mathbf r)\\,d^3 r\\]\\[\\frac{\\delta E_x}{\\delta\\phi^*_A(\\mathbf r)} = 2v_x(\\mathbf r)\\phi_A(\\mathbf r)\\]
\u3068\u3044\u3046\u4ea4\u63db\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3092\u8ecc\u9053\u306b\u639b\u3051\u305f\u5f62\u304c\u73fe\u308c\u307e\u3059\u3002\u3053\u308c\u306f\u4e00\u96fb\u5b50\u306b\u4f5c\u7528\u3059\u308b\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u540c\u3058\u5f62\u306a\u306e\u3067\u3001\u4ea4\u63db\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb<\/strong>\u3068\u547c\u3070\u308c\u307e\u3059\u3002<\/p>\n<\/div>\n\n\n\n\u6b21\u306b\u3001\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u306e\u4e00\u96fb\u5b50\u6f14\u7b97\u5b50\u3092\u5b9f\u95a2\u6570\\(\\chi_1\\)\u3067\u631f\u3093\u3067\u7a4d\u5206\u3057\u305f\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u884c\u5217\u8981\u7d20\\(H_{11}^{\\mathrm{KS,(0)}}\\)\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u884c\u5217\u8981\u7d20\\(H_{11}^{\\mathrm{KS,(0)}}\\)\u306e\u8a08\u7b97<\/h4>\n\n\n\n1) \u521d\u671f\u5bc6\u5ea6\u3092\u4f5c\u308b<\/h5>\n\n\n\n
KS-DFT\u306b\u304a\u3044\u3066\u96fb\u5b50\u5bc6\u5ea6\u306f\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u8ecc\u9053\u304b\u3089\u4f5c\u3089\u308c\u307e\u3059\u3002\u96fb\u5b50\u5bc6\u5ea6\u306f\\(\\rho = 2|\\phi_A|^2 + 2|\\phi_B|^2\\)\u306a\u306e\u3067\u3001\u521d\u671fKS\u8ecc\u9053\u3092\\[\\begin{aligned}
\\phi_A^{(0)}(r)&=\\chi_1(r),\\\\
\\phi_B^{(0)}(r)&=-0.141583535\\,\\chi_1(r)
+1.009973216\\,\\chi_2(r)
\\end{aligned}\\]\u3068\u3059\u308b\u3068\u3001\\[\\begin{aligned}\\rho^{(0)}(r)
&=2\\chi_1(r)^2
+2\\left[-0.141583535\\,\\chi_1(r)
+1.009973216\\,\\chi_2(r)\\right]^2\\\\ &= 2.040091795 \\chi_1(r)^2 + 2.040091794 \\chi_2(r)^2 – 0.5719823127063942 \\chi_1(r) \\chi_2(r) \\end{aligned}\\]\u3067\u3059\u3002\u3053\u3053\u3067\u3001\\[\\begin{aligned} \\chi_1(r)^2 =& (1.13361044)^2\\,e^{-2\\times 71.8876r^2} +(1.54675504)^2\\,e^{-2\\times 10.7289r^2} +(0.90270029)^2\\,e^{-2\\times 2.22205r^2}\\\\ &+ 2 (1.13361044)(1.54675504)\\,e^{-(71.8876+10.7289)r^2} + 2 (1.13361044)(0.90270029)\\,e^{-(71.8876+2.22205)r^2} \\\\ &+ 2 (1.54675504)(0.90270029)\\,e^{-(10.7289+2.22205)r^2}, \\\\ \\chi_2(r)^2 =& (0.36440247)^2\\,e^{-2\\times1.29548r^2}+(0.32575156)^2\\,e^{-2\\times 0.268881r^2}-2(0.36440247)(0.32575156)\\,e^{-(1.29548+0.268881)r^2}, \\\\ \\chi_1(r)\\chi_2(r) =& (1.13361044)(-0.36440247)\\,e^{-(71.8876+1.29548)r^2} + (1.13361044)(0.32575156)\\,e^{-(71.8876+0.268881)r^2} \\\\ &+ (1.54675504)(-0.36440247)\\,e^{-(10.7289+1.29548)r^2} + (1.54675504)(0.32575156)\\,e^{-(10.7289+0.268881)r^2} \\\\ &+ (0.90270029)(-0.36440247)\\,e^{-(2.22205+1.29548)r^2} + (0.90270029)(0.32575156)\\,e^{-(2.22205+0.268881)r^2} \\end{aligned} \\]\u3067\u3059\u3002
\u3053\u306e\u3088\u3046\u306b\u3001\u521d\u671f\u96fb\u5b50\u5bc6\u5ea6\u306f15\u500b\u306e\u30ac\u30a6\u30b9\u95a2\u6570\u306e\u548c\u3068\u3057\u3066\u8868\u305b\u307e\u3059\u3002<\/p>\n\n\n\n![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/initial_density-1024x393.png\")
\u56f32\uff0eBe\u539f\u5b50\u306e\u521d\u671f\u96fb\u5b50\u5bc6\u5ea6\u3068\u52d5\u5f84\u96fb\u5b50\u6570\u5206\u5e03\u3002<\/strong> \u5de6\u56f3\u306f\u521d\u671f\u96fb\u5b50\u5bc6\u5ea6 \\(\\rho^{(0)}(r)\\) \u3092\u5bfe\u6570\u76ee\u76db\u3067\u3001\u53f3\u56f3\u306f\u52d5\u5f84\u96fb\u5b50\u6570\u5206\u5e03 \\(4\\pi r^2\\rho^{(0)}(r)\\) \u3092\u793a\u3059\u3002\u5b9f\u7dda\u306f\u5408\u8a08\u3001\u7834\u7dda\u3068\u4e00\u70b9\u9396\u7dda\u306f\u305d\u308c\u305e\u308c\u521d\u671f\u8ecc\u9053 \\(\\phi_A^{(0)}\\)\u3001\\(\\phi_B^{(0)}\\) \u306b\u7531\u6765\u3059\u308b\u5bc4\u4e0e\u3092\u8868\u3059\u3002\u53f3\u56f3\u306e\u5206\u5e03\u3092\u534a\u5f84\u306b\u3064\u3044\u3066\u7a4d\u5206\u3059\u308b\u3068\u3001\u305d\u306e\u7bc4\u56f2\u306e\u7403\u6bbb\u5185\u306b\u542b\u307e\u308c\u308b\u96fb\u5b50\u6570\u304c\u5f97\u3089\u308c\u308b\u3002\u3053\u308c\u306fSCF\u8a08\u7b97\u3092\u958b\u59cb\u3059\u308b\u305f\u3081\u306e\u521d\u671f\u5206\u5e03\u3067\u3042\u308a\u3001\u53ce\u675f\u5f8c\u306e\u96fb\u5b50\u5bc6\u5ea6\u3067\u306f\u306a\u3044\u3002\\(a_0\\) \u306f\u30dc\u30fc\u30a2\u534a\u5f84\u3092\u8868\u3059\u3002<\/figcaption><\/figure>\n\n\n\n2) \u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc<\/h5>\n\n\n\n
\u3053\u306e\u90e8\u5206\u306f\u96fb\u5b50\u5bc6\u5ea6\u306b\u4f9d\u5b58\u3057\u307e\u305b\u3093\u3002\\[
T_{11}
=\\int\\chi_1\\left(-\\frac12\\nabla^2\\right)\\chi_1\\,d^3r,
\\]
\\[
V_{11}^{\\mathrm{ext}}
=\\int\\chi_1\\left(-\\frac4r\\right)\\chi_1\\,d^3r.
\\]\n\n\n\n
\\(\\chi_1\\)\u306f3\u3064\u306e\u539f\u59cb\u30ac\u30a6\u30b9\u95a2\u6570\u304b\u3089\u306a\u308b\\(\\chi_1(\\mathbf r) = A e^{-\\alpha r^2} + B e^{-\\beta r^2} + C e^{-\\gamma r^2}\\)\u306e\u3067\u3001\u3069\u3061\u3089\u3082\u5c55\u958b\u3059\u308b\u3068\\(3×3=9\\)\u9805\u3067\u3059\u304c\u3001\u5bfe\u79f0\u6027\u306b\u3088\u308a\u72ec\u7acb\u306a\u7a4d\u5206\u306f6\u7a2e\u985e\u3067\u3059\u3002<\/p>\n\n\n\n
\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u884c\u5217\u8981\u7d20\u306e\u8a08\u7b97<\/h6>\n\n\n\n
\\[\\begin{aligned}\\langle A e^{-\\alpha r^2}|\\hat{T}| B e^{-\\beta r^2}\\rangle &= AB \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} e^{-\\alpha r^2}\\left(-\\frac{1}{2}\\right) \\left(\\frac{d^2}{dr^2} + \\frac{2}{r}\\frac{d}{dr}\\right) e^{-\\beta r^2} \\cdot{} r^2 \\sin\\theta dr d\\theta d\\phi \\\\ &= 4\\pi AB \\int_0^{\\infty} e^{-\\alpha r^2} \\left(-\\frac{1}{2}\\right) \\left((-2\\beta+4\\beta^2 r^2) +\\frac{2}{r}(-2\\beta r )\\right) e^{-\\beta r^2}r^2\\,dr \\\\ & = 4\\pi AB\\left\\{
\\int_0^{\\infty}(3\\beta – 2\\beta^2 r^2) e^{-(\\alpha + \\beta) r^2} r^2\\,dr \\right\\} \\\\ &= 4\\pi AB\\left\\{ 3\\beta\\int_0^{\\infty} r^2 e^{-(\\alpha + \\beta) r^2} dr – 2\\beta^2 \\int_0^{\\infty} r^4 e^{-(\\alpha + \\beta) r^2} dr \\right\\} \\end{aligned} \\]\n\n\n\n
\n
\u7a4d\u5206 \\(\\int_0^{\\infty} r^n e^{-a r^2} dr\\) \uff08\\(n\\)\u304c0\u4ee5\u4e0a\u306e\u6574\u6570\u3067\\(a > 0\\)\u306e\u6642\uff09\u306f\u3001\u90e8\u5206\u7a4d\u5206\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u3067\u6c42\u3081\u3089\u308c\u307e\u3059\u3002\u516c\u5f0f\u306f\u3001\\(n\\)\u304c\u5076\u6570\uff08\\(n = 2m\\)\uff09\u306e\u6642\u3001\\[I_{2m} = \\frac{(2m)!}{2^{2m+1}m!}\\frac{\\sqrt{\\pi}}{a^{m+1\/2}},\\]\\(n\\)\u304c\u5947\u6570\uff08\\(n = 2m+1\\)\uff09\u306e\u6642\u3001 \\[I_{2m+1} = \\frac{m!}{2a^{m+1}}\\]\u3067\u3059\u3002<\/p>\n<\/div>\n\n\n\n
\u4eca\u56de\u5fc5\u8981\u306a\u306e\u306f\u3001\\[
I_2(\\alpha+\\beta)
=\\frac{\\sqrt{\\pi}}{4(\\alpha+\\beta)^{3\/2}},
\\qquad
I_4(\\alpha+\\beta)
=\\frac{3\\sqrt{\\pi}}{8(\\alpha+\\beta)^{5\/2}}
\\]\u306a\u306e\u3067\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u884c\u5217\u8981\u7d20\u306f\u3001\\[
\\begin{aligned}
\\langle Ae^{-\\alpha r^2}|\\hat T
|Be^{-\\beta r^2}\\rangle
&=4\\pi AB
\\left[
\\frac{3\\beta\\sqrt{\\pi}}{4(\\alpha+\\beta)^{3\/2}}
-\\frac{6\\beta^2\\sqrt{\\pi}}{8(\\alpha+\\beta)^{5\/2}}
\\right]\\\\
&=\\frac{3\\pi^{3\/2}AB
\\left[\\beta(\\alpha+\\beta)-\\beta^2\\right]}
{(\\alpha+\\beta)^{5\/2}}\\\\
&=
\\frac{3\\pi^{3\/2}AB\\,\\alpha\\beta}
{(\\alpha+\\beta)^{5\/2}}
\\end{aligned}
\\]\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u884c\u5217\u8981\u7d20\u306e\u8a08\u7b97<\/h6>\n\n\n\n
\\[\\begin{aligned}\\langle A e^{-\\alpha r^2}|\\hat{V}_{\\mathrm{ext}}| B e^{-\\beta r^2}\\rangle &= AB \\int_0^{2\\pi}\\!\\!\\!\\int_0^{\\pi}\\!\\!\\!\\int_0^{\\infty} e^{-\\alpha r^2}\\left(-\\frac{4}{r}\\right) e^{-\\beta r^2} \\cdot{} r^2 \\sin\\theta dr d\\theta d\\phi \\\\ & = -16\\pi AB\\int_0^{\\infty} r e^{-(\\alpha + \\beta) r^2} dr \\\\ &= -16\\pi AB\\int_0^{\\infty} -\\frac{1}{2(\\alpha + \\beta)} \\left(\\frac{d}{dr}e^{-(\\alpha + \\beta) r^2} \\right) dr \\\\ &= \\frac{8\\pi AB}{\\alpha + \\beta}\\left[ e^{-(\\alpha + \\beta) r^2} \\right]_0^{\\infty} \\\\ &= -\\frac{8\\pi AB}{\\alpha + \\beta}\\end{aligned} \\]\n\n\n\n
\u9014\u4e2d\u3067 \\(\\frac{d}{dr}e^{-(\\alpha +\\beta) r^2 } = -2(\\alpha + \\beta) r e^{-(\\alpha +\\beta) r^2} \\) \u3092\u4f7f\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n
\u5168\u3066\u306e\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u3066\u8db3\u3057\u5408\u308f\u305b\u308b\u3068\u3001\\[
T_{11}=6.905049356,\\qquad
V_{11}^{\\mathrm{ext}}=-14.815214657
\\]
\u306a\u306e\u3067\u3001
\\[
h_{11}=T_{11}+V_{11}^{\\mathrm{ext}}
=-7.910165301
\\]
\u3067\u3059\u3002<\/p>\n\n\n\n
3) \u30cf\u30fc\u30c8\u30ea\u30fc\u9805<\/h5>\n\n\n\n
\u5168\u96fb\u5b50\u5bc6\u5ea6\\(\\rho^{(0)}\\)\u304c\u4f5c\u308b\u30cf\u30fc\u30c8\u30ea\u30fc\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\\(v_{\\mathrm{H}}\\)\u3092\u3001\u57fa\u5e95\u95a2\u6570\\(\\chi_1\\)\u3067\u671f\u5f85\u5024\u3068\u3057\u3066\u8a55\u4fa1\u3057\u305f\u884c\u5217\u8981\u7d20\u3067\u3059\u3002<\/p>\n\n\n\n
\\[
V_{11}^{\\mathrm H}
=\\iint
\\frac{\\chi_1(\\mathbf r)^2\\rho^{(0)}(\\mathbf r’)}
{|\\mathbf r-\\mathbf r’|}
\\,d^3r\\,d^3r’.
\\]
\u4eca\u56de\u306f\u3001\\[V_{11}^{\\mathrm H} = 5.877695401\\]\u3067\u3059\u3002<\/p>\n\n\n\n
\n
3-21\u57fa\u5e95\u95a2\u6570\u7cfb\u306es\u578b\u90e8\u5206\u306b\u306f\u7e2e\u7d04\uff08contracted\uff09\u57fa\u5e95\u95a2\u6570\u304c3\u3064\uff08\\(\\chi_1, \\chi_2, \\chi_3\\)\uff09\u3057\u304b\u306a\u3044\u306e\u3067\u30014\u3064\u306e\u7e2e\u7d04\u95a2\u6570\u304c\u5168\u3066\u7570\u306a\u308b\u5834\u5408\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u305f\u3060\u3057\u3001\u539f\u59cb\u30ac\u30a6\u30b9\u95a2\u6570\u307e\u3067\u8003\u3048\u308b\u3068\u3001\u6700\u521d\u306b4\u3064\u306e\u7570\u306a\u308b\u30ac\u30a6\u30b9\u95a2\u6570\u304c\u542b\u307e\u308c\u308b\u7a4d\u5206\u3092\u8a08\u7b97\u3057\u3066\u304a\u304f\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\uff08\u4f8b\u3048\u3070\u3001\\(\\chi_1\\)\u306e1\u3064\u76ee\u306e\u539f\u59cb\u95a2\u6570\u3001\\(\\chi_1\\)\u306e2\u3064\u76ee\u306e\u539f\u59cb\u95a2\u6570\u3001\\(\\chi_2\\)\u306e1\u3064\u76ee\u306e\u539f\u59cb\u95a2\u6570\u3001\\(\\chi_3\\)\u306e\u7d44\u307f\u5408\u308f\u305b\uff09\u3002<\/p>\n\n\n\n
\u540c\u3058\u539f\u5b50\u4e0a\u306e4\u3064\u306e\u539f\u59cbs\u578b\u30ac\u30a6\u30b9\u95a2\u6570\u306e\u96fb\u5b50\u9593\u53cd\u767a\u7a4d\u5206\u3067\u306f\u3001\\[\\begin{aligned} (ij|kl) &= \\iint \\frac{N_i e^{-\\alpha_i r_1^2} N_j e^{-\\alpha_j r_1^2} N_k e^{-\\alpha_k r_2^2} N_l e^{-\\alpha_l r_2^2}}{|\\mathbf{r}_1 – \\mathbf{r}_2|} d\\mathbf{r}_1 d\\mathbf{r}_2 \\\\ &=N_i N_j N_k N_l \\iint \\frac{e^{-(\\alpha_i + \\alpha_j) r_1^2} e^{-(\\alpha_k + \\alpha_l) r_2^2}}{|\\mathbf{r}_1 – \\mathbf{r}_2|} d\\mathbf{r}_1 d\\mathbf{r}_2 \\\\ |\\mathbf{r}_1 – \\mathbf{r}_2| = \\sqrt{r_1^2 + r_2^2 – 2r_1r_2\\cos\\theta} &\\text{\u3068\u7f6e\u3044\u3066\u89d2\u5ea6\u90e8\u5206\u3092\u5148\u306b\u7a4d\u5206\u3059\u308b\u3068\u3001}\\\\ &= 16\\pi^2 N_i N_j N_k N_l \\left[ \\int_0^\\infty
r_1e^{-(\\alpha_i + \\alpha_j) r_1^2}
\\left(\\int_0^{r_1}r_2^2e^{-(\\alpha_k + \\alpha_l) r_2^2}dr_2\\right)dr_1 \\right. \\\\ &\\,\\,\\,\\left. +\\int_0^\\infty
r_2e^{-(\\alpha_k + \\alpha_l) r_2^2}
\\left(\\int_0^{r_2}r_1^2e^{-(\\alpha_i + \\alpha_j) r_1^2}dr_1\\right)dr_2 \\right]. \\end{aligned}\\] \u7a4d\u5206\u3092\u9032\u3081\u308b\u3068\u6700\u7d42\u7684\u306b\u306f\u3001\\[(ij|kl)=N_iN_jN_kN_l
\\frac{2\\pi^{5\/2}}
{(\\alpha_i+\\alpha_j)(\\alpha_k+\\alpha_l)
\\sqrt{\\alpha_i+\\alpha_j+\\alpha_k+\\alpha_l}}\\]\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n<\/div>\n\n\n\n
4) LDA\u4ea4\u63db\u9805<\/h5>\n\n\n\n
\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u6cd5\u3067\u306f\u4ea4\u63db\u9805\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u3001\u57fa\u5e95\u95a2\u6570\u306e\u6570\u306b\u4f9d\u5b58\u3057\u3066\u3001\u56db\u3064\u306e\u57fa\u5e95\u95a2\u6570\u3092\u542b\u3080\u4e8c\u96fb\u5b50\u7a4d\u5206\uff08\u4e00\u822c\u306e\u5206\u5b50\u3067\u306f\u56db\u4e2d\u5fc3\u7a4d\u5206\uff09\u3092\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u4e00\u65b9\u3001\u4eca\u56de\u4f7f\u3046LDA\u3067\u306f\u4ea4\u63db\u9805\u3092\u96fb\u5b50\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u8a55\u4fa1\u3067\u304d\u308b\u306e\u3067\u3001HF\u578b\u306e\u4ea4\u63db\u7a4d\u5206\u304c\u5fc5\u8981\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n
\u5404\u534a\u5f84\u3067\u5bc6\u5ea6\u3092\u6c42\u3081\u3001\u4ea4\u63db\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\\[ v_x(r)=-\\left(\\frac3\\pi\\right)^{1\/3}\\rho^{(0)}(r)^{1\/3} \\]\u3092\u8a55\u4fa1\u3057\u307e\u3059\u3002\u4eca\u56de\u306e\u96fb\u5b50\u5bc6\u5ea6\u3067\u306f\u3001\u5bc6\u5ea6\u306e\u5206\u6570\u4e57\u304c\u7a4d\u5206\u306b\u5165\u308b\u306e\u3067\u30ac\u30a6\u30b9\u7a4d\u5206\u306e\u516c\u5f0f\u304c\u4f7f\u3048\u305a\u3001\u6570\u5024\u7a4d\u5206\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u305f\u3081\u3001\u8a08\u7b97\u5316\u5b66\u30d7\u30ed\u30b0\u30e9\u30e0\u3067DFT\u8a08\u7b97\u3092\u3059\u308b\u6642\u306b\u306f\u3001\u5ea7\u6a19\u306e\u30b0\u30ea\u30c3\u30c9\u3092\u6307\u5b9a\u3059\u308b\u30aa\u30d7\u30b7\u30e7\u30f3\u304c\u3042\u308a\u307e\u3059\u3002\\[
V_{11}^{x}
=-\\left(\\frac{3}{\\pi}\\right)^{1\/3}
\\int\\chi_1^*(\\mathbf r)\\,
\\rho(\\mathbf r)^{1\/3}\\,
\\chi_1(\\mathbf r)\\,d^3r
\\]\u3092\u6570\u5024\u7a4d\u5206\u3059\u308b\u3068\u3001\\[{V_{11}^{x}=-1.338674832}\\]\u3067\u3059\u3002<\/p>\n\n\n\n
5) VWN5\u76f8\u95a2\u9805<\/h5>\n\n\n\n
\\[ E_c^{\\rm VWN5}[\\rho]=\\int\\rho(\\mathbf r)\\varepsilon_c^{\\rm VWN5}(\\rho(\\mathbf r))\\,d^3r,
\\qquad v_c(\\rho)=\\frac{d[\\rho\\varepsilon_c(\\rho)]}{d\\rho} \\]\n\n\n\n
\\[V_{11}^{c}
=4\\pi\\int_0^\\infty
r^2\\chi_1(r)^2v_c(r)\\,dr\\]\u3092\u6570\u5024\u7a4d\u5206\u3059\u308b\u3068\u3001\\[V_{11}^{c}=-0.085261629\\]\u3067\u3059\u3002<\/p>\n\n\n\n
6) 5\u3064\u3092\u8db3\u3057\u5408\u308f\u305b\u308b<\/h5>\n\n\n\n
\\[H_{11}^{\\mathrm{KS}}
=T_{11}+V_{11}^{\\mathrm{ext}}
+V_{11}^{\\mathrm H}+V_{11}^{x}+V_{11}^{c}\\]\u306a\u306e\u3067\u3001\\[H_{11}^{\\mathrm{KS,(0)}} = -3.456406361\\]\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u81ea\u5df1\u7121\u649e\u7740\u5834\uff08SCF\uff09\u8a08\u7b97<\/h4>\n\n\n\n
\u540c\u69d8\u306b\u884c\u5217\u8981\u7d20\u3092\u3059\u3079\u3066\u8a08\u7b97\u3059\u308b\u3068\u3001\\[H_s^{\\mathrm{KS},(0)}
=
\\begin{pmatrix}
-3.456406361&-0.765056014&-0.770064201\\\\
-0.765056014&-0.085156251&-0.263514314\\\\
-0.770064201&-0.263514314&-0.246804663
\\end{pmatrix}\\]\u3068\u306a\u308a\u307e\u3059\u3002HF\u6cd5\u306e\u6642\u306e\u30d5\u30a9\u30c3\u30af\u884c\u5217\u3068\u4f3c\u305f\u3088\u3046\u306a\u5024\u3067\u3059\u30021\u30b5\u30a4\u30af\u30eb\u76ee\u306b\u89e3\u304f\u3079\u304d\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u30fb\u30ed\u30fc\u30bf\u30fc\u30f3\u65b9\u7a0b\u5f0f\u306f\\[\\begin{pmatrix}
-3.456406361&-0.765056014&-0.770064201\\\\
-0.765056014&-0.085156251&-0.263514314\\\\
-0.770064201&-0.263514314&-0.246804663
\\end{pmatrix}\\begin{pmatrix}C_{11}&C_{12}&C_{13}\\\\
C_{21}&C_{22}&C_{23}\\\\
C_{31}&C_{32}&C_{33}
\\end{pmatrix} =\\begin{pmatrix}
1&0.140185435&0.177121893\\\\
0.140185435&1&0.798850479\\\\
0.177121893&0.798850479&1
\\end{pmatrix} \\begin{pmatrix}
C_{11}&C_{12}&C_{13}\\\\
C_{21}&C_{22}&C_{23}\\\\
C_{31}&C_{32}&C_{33}
\\end{pmatrix}\\begin{pmatrix}
\\varepsilon_1&0&0\\\\
0&\\varepsilon_2&0\\\\
0&0&\\varepsilon_3
\\end{pmatrix}\\]\u3067\u3059\u3002HF\u6cd5\u306e\u6642\u3068\u540c\u3058Python\u30b3\u30fc\u30c9\u3067\u89e3\u3044\u3066\u307f\u308b\u3068\u3001\u4fc2\u6570\u884c\u5217\\[\\mathbf{C}^{(1)}=\\begin{pmatrix}-0.988139323 &-0.233032528 &0.0408634 \\\\ -0.108402074 & 0.154002943 &-1.6517306 \\\\ 0.037472168 & 0.887127481 & 1.417294102\\end{pmatrix}\\]\u304c\u5f97\u3089\u308c\u307e\u3057\u305f\u3002\u5404\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306e\u7b26\u53f7\u306f\u72ec\u7acb\u306b\u9078\u3079\u307e\u3059\u3002\u4ee5\u4e0b\u3067\u306f\u3001\u56fa\u6709\u5024\uff08\u8ecc\u9053\u30a8\u30cd\u30eb\u30ae\u30fc\uff09\u304c\u4f4e\u3044\u65b92\u3064\u3092\u65b0\u3057\u3044KS\u5360\u6709\u8ecc\u9053\\[\\phi_{A}^{(1)} =0.988139323 \\chi_{1} + 0.108402074 \\chi_{2} -0.037472168 \\chi_{3},\\]\\[\\phi_{B}^{(1)} =-0.233032528 \\chi_{1} +0.154002943 \\chi_{2} +0.887127481 \\chi_{3}\\]\u3068\u3057\u3066\u3001\u65b0\u3057\u3044\u96fb\u5b50\u5bc6\u5ea6\u3092\u4f5c\u3063\u3066\u3001\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u884c\u5217\u3092\u66f4\u65b0\u3057\u3066\u3001\u4ee5\u5f8c\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u53ce\u675f\u3059\u308b\u307e\u3067\u7e70\u308a\u8fd4\u3057\u307e\u3059\u3002\u3053\u306e\u521d\u671f\u5024\u3068\u7403\u5bfe\u79f0\u30fb\u9589\u6bbb\u306e\u5236\u7d04\u306e\u4e0b\u3067\u306f\u3001\u4ee5\u4e0b\u306e\u89e3\u306b\u53ce\u675f\u3057\u307e\u3057\u305f\u3002<\/p>\n\n\n\n
\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u3001\\[E_{\\mathrm{KS}}=2\\langle\\phi_A|\\hat{ h}|\\phi_A\\rangle+2\\langle\\phi_B|\\hat{h}|\\phi_B\\rangle+E_{\\mathrm H}[\\rho]+E_{\\mathrm x}[\\rho]+E_{\\mathrm c}[\\rho].\\]\n\n\n\n| Iteration<\/th> | \\(E\\,\/\\,E_h\\)<\/th> | \\(C_{11}\\)<\/th> | \\(C_{21}\\)<\/th> | \\(C_{31}\\)<\/th> | \\(C_{12}\\)<\/th> | \\(C_{22}\\)<\/th> | \\(C_{32}\\)<\/th> | \\(\\varepsilon_{1}\\,\/\\,E_h\\)<\/th> | \\(\\varepsilon_{2}\\,\/\\,E_h\\)<\/th><\/tr><\/thead> |
|---|
| 0<\/td> | -14.037457<\/td> | 1<\/td> | 0<\/td> | 0<\/td> | -0.1415835<\/td> | 1.00997322<\/td> | 0<\/td> | <\/td> | <\/td><\/tr> |
| 1<\/td> | -14.354198<\/td> | 0.98813932<\/td> | 0.10840208<\/td> | -0.0374722<\/td> | -0.2330325<\/td> | 0.15400294<\/td> | 0.88712748<\/td> | -3.4809809<\/td> | -0.0826515<\/td><\/tr> |
| 2<\/td> | -14.357123<\/td> | 0.98952234<\/td> | 0.09953175<\/td> | -0.035148<\/td> | -0.229482<\/td> | 0.24107613<\/td> | 0.81129465<\/td> | -3.8380542<\/td> | -0.2150877<\/td><\/tr> |
| 3<\/td> | -14.357197<\/td> | 0.98934164<\/td> | 0.10071279<\/td> | -0.0354598<\/td> | -0.2299793<\/td> | 0.22711445<\/td> | 0.82365246<\/td> | -3.7940845<\/td> | -0.2012269<\/td><\/tr> |
| 4<\/td> | -14.357199<\/td> | 0.98936947<\/td> | 0.10053049<\/td> | -0.0354111<\/td> | -0.2299061<\/td> | 0.22934949<\/td> | 0.82167885<\/td> | -3.8010036<\/td> | -0.2034579<\/td><\/tr> |
| 5<\/td> | -14.357199<\/td> | 0.98936502<\/td> | 0.10055964<\/td> | -0.0354189<\/td> | -0.2299178<\/td> | 0.22899218<\/td> | 0.82199449<\/td> | -3.7998968<\/td> | -0.2031011<\/td><\/tr> |
| 6<\/td> | -14.357199<\/td> | 0.98936573<\/td> | 0.10055498<\/td> | -0.0354177<\/td> | -0.2299159<\/td> | 0.2290493<\/td> | 0.82194403<\/td> | -3.8000737<\/td> | -0.2031582<\/td><\/tr> |
| 7<\/td> | -14.357199<\/td> | 0.98936562<\/td> | 0.10055572<\/td> | -0.0354179<\/td> | -0.2299162<\/td> | 0.22904017<\/td> | 0.8219521<\/td> | -3.8000454<\/td> | -0.2031491<\/td><\/tr> |
| 8<\/td> | -14.357199<\/td> | 0.98936564<\/td> | 0.1005556<\/td> | -0.0354178<\/td> | -0.2299162<\/td> | 0.22904163<\/td> | 0.82195081<\/td> | -3.8000499<\/td> | -0.2031505<\/td><\/tr> |
| 9<\/td> | -14.357199<\/td> | 0.98936564<\/td> | 0.10055562<\/td> | -0.0354178<\/td> | -0.2299162<\/td> | 0.2290414<\/td> | 0.82195101<\/td> | -3.8000492<\/td> | -0.2031503<\/td><\/tr> |
| 10<\/td> | -14.357199<\/td> | 0.98936564<\/td> | 0.10055562<\/td> | -0.0354178<\/td> | -0.2299162<\/td> | 0.22904143<\/td> | 0.82195098<\/td> | -3.8000493<\/td> | -0.2031503<\/td><\/tr> |
| 11<\/td> | -14.357199<\/td> | 0.98936564<\/td> | 0.10055562<\/td> | -0.0354178<\/td> | -0.2299162<\/td> | 0.22904143<\/td> | 0.82195099<\/td> | -3.8000493<\/td> | -0.2031503<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n Be\u539f\u5b50\u306e\u53ce\u675f\u3057\u305fLDA\uff08Dirac\u4ea4\u63db\uff0bVWN5\u76f8\u95a2\uff09\uff0f3-21G \u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5185\u8a33\u306f\u3001\u4ee5\u4e0b\u306e\u3068\u304a\u308a\u3067\u3059\u3002\u5358\u4f4d\u306f\u3059\u3079\u3066hartree\uff08E<\/em>h<\/sub>\uff09\u3067\u3059\u3002<\/p>\n\n\n\n| \u6210\u5206<\/th> | \u8a18\u53f7<\/th> | \u30a8\u30cd\u30eb\u30ae\u30fc<\/th><\/tr><\/thead> |
|---|
| KS\u975e\u76f8\u4e92\u4f5c\u7528\u7cfb\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc<\/td> | \\(T_s\\)<\/td> | \\(+14.4229115261\\)<\/td><\/tr> | | \u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc<\/td> | \\(V_{\\mathrm{ne}}\\)<\/td> | \\(-33.3982887803\\)<\/td><\/tr> | | \u30cf\u30fc\u30c8\u30ea\u30fc\u30a8\u30cd\u30eb\u30ae\u30fc<\/td> | \\(J\\)<\/td> | \\(+7.1480571442\\)<\/td><\/tr> | | \u4ea4\u63db\u30a8\u30cd\u30eb\u30ae\u30fc<\/td> | \\(E_x\\)<\/td> | \\(-2.3050452405\\)<\/td><\/tr> | | \u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc<\/td> | \\(E_c\\)<\/td> | \\(-0.2248339062\\)<\/td><\/tr> | \u5168\u30a8\u30cd\u30eb\u30ae\u30fc<\/strong><\/td>\\(E\\)<\/strong><\/td>| \\(-14.3571992567\\)<\/strong><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n \u53b3\u5bc6\u306aKS\u7406\u8ad6\u3067\u306f\u3001\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u306b\u306f\u76f8\u95a2\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\\(T_c\\)\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u3059\u3002\u5b9a\u7fa9\u306f\u76f8\u4e92\u4f5c\u7528\u7cfb\u3068KS\u53c2\u7167\u7cfb\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5dee\\(T_c=T\u2212T_s\\)\u3067\u3059\u3002\u3053\u3053\u3067\u306fVWN5\u76f8\u95a2\u6c4e\u95a2\u6570\u306e\u4e00\u69d8\u5ea7\u6a19\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u304b\u3089\u3001\u8fd1\u4f3c\u7684\u306a\\(T_c\\)\u3068\\(U_c\\)\u3092\u8a55\u4fa1\u3057\u307e\u3057\u305f\uff08\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u5ea7\u6a19\u306e2\u968e\u5fae\u5206\u3001\u30af\u30fc\u30ed\u30f3\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u8ddd\u96e2\u306e\u9006\u6570\u3092\u542b\u3080\u306e\u3067\u3001\u6ce2\u52d5\u95a2\u6570\u3092\u4e00\u69d8\u306b\u5ea7\u6a19\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0 \\(\\psi_\\gamma(\\mathbf r_1,\\dots,\\mathbf r_N)=\\gamma^{3N\/2}\\psi(\\gamma\\mathbf r_1,\\dots,\\gamma\\mathbf r_N)\\) \u3068\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u3059\u308b\u3068\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u306f \\(\\gamma^2\\)\u500d\u3001\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u306f \\(\\gamma\\) \u500d\u306b\u306a\u308a\u307e\u3059\u3002\u53b3\u5bc6\u306a\u76f8\u95a2\u6c4e\u95a2\u6570\u3067\u306f\u3053\u306e\u30b9\u30b1\u30fc\u30ea\u30f3\u30b0\u5247\u3068\u5909\u5206\u539f\u7406\u304b\u3089\u3001\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u306b\u3064\u3044\u3066\\(\\left.\\frac{dE_{\\mathrm c}[\\rho_\\gamma]}{d\\gamma}\\right|_{\\gamma=1}
=2T_{\\mathrm c}+U_{\\mathrm c}\\)\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\uff09\u3002\u4ee5\u4e0b\u306e\u5024\u306fVWN5\u306e\u67a0\u5185\u3067\u306e\u5206\u89e3\u3067\u3042\u308a\u3001Be\u539f\u5b50\u306e\u53b3\u5bc6\u306a\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3084\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u76f4\u63a5\u6c42\u3081\u305f\u3082\u306e\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\\[
E_c=T_c+U_c
=\\underbrace{0.1381078096}_{T_c}
+\\underbrace{-0.3629417158}_{U_c}
=-0.2248339062\\]\u3067\u3059\u3002<\/p>\n\n\n\n \u6bd4\u8f03<\/h3>\n\n\n\nCCSD(T)\u306f\u3001\u7d50\u5408\u30af\u30e9\u30b9\u30bf\u30fc\u6cd5\u3068\u547c\u3070\u308c\u308bpost-HF\u6cd5\u306e\u9ad8\u7cbe\u5ea6\u306a\u8a08\u7b97\u30ec\u30d9\u30eb\u3067\u3059\u3002DFT\u6c4e\u95a2\u6570\u306e\u691c\u8a3c\u3084\u958b\u767a\u306b\u7528\u3044\u308b\u9ad8\u7cbe\u5ea6\u306a\u53c2\u7167\u8a08\u7b97\u306e\u4e00\u3064\u3067\u3059\u3002\u305f\u3060\u3057\u3001\u53b3\u5bc6\u89e3\u3067\u306f\u306a\u304f\u3001\u5f37\u3044\u9759\u7684\u76f8\u95a2\u3084\u57fa\u5e95\u95a2\u6570\u306e\u4e0d\u8db3\u306b\u3088\u308b\u8aa4\u5dee\u304c\u3042\u308a\u5f97\u307e\u3059\u3002<\/p>\n\n\n\n CCSD(T)\u5217\u306f\u3001HF\u306e\u5404\u6210\u5206\u3092\u305d\u306e\u307e\u307e\u6b8b\u3057\u3001\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5dee\u3092\u307e\u3068\u3081\u3066\u76f8\u95a2\u88dc\u6b63\u3068\u3057\u3066\u793a\u3057\u305f\u5e33\u7c3f\u4e0a\u306e\u5206\u89e3\u3067\u3059\u3002\u4e0a4\u884c\u306fCCSD(T)\u306e\u5bc6\u5ea6\u30fb\u5bc6\u5ea6\u884c\u5217\u304b\u3089\u8a55\u4fa1\u3057\u305f\u5404\u30a8\u30cd\u30eb\u30ae\u30fc\u6210\u5206\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n\n\n\n| \u6210\u5206<\/th> | HF\/3-21G<\/th> | LDA(VWN5)\/3-21G<\/th> | HF\/cc-pCVQZ-F12<\/th> | CCSD(T)\/cc-pCVQZ-F12
<HF\u57fa\u6e96><\/th><\/tr><\/thead> |
|---|
| \u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\\(T\\, \\text{or}\\, T_s\\)<\/td> | +14.478061<\/td> | +14.422912<\/td> | +14.572967<\/td> | +14.572967<\/td><\/tr> | | \u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\\(V_{\\mathrm{ne}}\\)<\/td> | \u221233.463645<\/td> | \u221233.398289<\/td> | \u221233.635078<\/td> | \u221233.635078<\/td><\/tr> | | \u30cf\u30fc\u30c8\u30ea\u30fc\u30a8\u30cd\u30eb\u30ae\u30fc\\(J\\)<\/td> | +7.166001<\/td> | +7.148057<\/td> | +7.156002<\/td> | +7.156002<\/td><\/tr> | | \u4ea4\u63db\u30a8\u30cd\u30eb\u30ae\u30fc\\(E_x^{\\mathrm{HF}}\\, \\text{or}\\, E_x^{\\mathrm{LDA}}\\)<\/td> | \u22122.667237<\/td> | \u22122.305045<\/td> | \u22122.666903<\/td> | \u22122.666903<\/td><\/tr> | | \u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\\(E_c^{\\mathrm{HF}\u57fa\u6e96} \\,\\text{or}\\, E_c^{\\mathrm{VWN5}}\\)<\/td> | 0<\/td> | \u22120.224834<\/td> | 0<\/td> | \u22120.090533<\/td><\/tr> | \u5168\u30a8\u30cd\u30eb\u30ae\u30fc<\/strong><\/td>\u221214.486820<\/strong><\/td>\u221214.357199<\/strong><\/td>\u221214.573013<\/strong><\/td>| \u221214.663545<\/strong><\/td><\/tr><\/tbody><\/table>i) HF\/cc-pCVQZ-F12\u3068CCSD(T)\/cc-pCVQZ-F12\u30ec\u30d9\u30eb\u306fORCA 6.1.1\u3067\u8a08\u7b97\u3057\u305f\u3002ii) \u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u9650\u754c\u306f-14.573023 Eh<\/sub>\u3001\u53b3\u5bc6\u306a\u975e\u76f8\u5bfe\u8ad6\u7684\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u221214.66735649269 Eh<\/sub>\u3002iii) CCSD(T)-F12\/cc-pCVQZ-F12\u30ec\u30d9\u30eb\u3067\u306e\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u221214.666660946795 Eh<\/sub>\u3002iV) CCSD(T)\/cc-pXVQZ-F12\u306eHF\u57fa\u6e96\u306e\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u3001CCSD(T)\u306e\u7de9\u548c\u5fdc\u7b54\u5bc6\u5ea6\u306b\u57fa\u3065\u304f\u5206\u89e3\u306b\u3088\u3063\u3066\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u6210\u5206\\(\\Delta T = +0.084797991 E_{h}\\) \u3068\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u30a8\u30cd\u30eb\u30ae\u30fc\u6210\u5206\\(\\Delta V = -0.175330519 E_h\\)\u306b\u5206\u3051\u3089\u308c\u308b\u3002<\/figcaption><\/figure>\n\n\n\n \u4eca\u56de\u3001LDA\u4ea4\u63db\u306e\u7d76\u5bfe\u5024\u306fHF\u4ea4\u63db\u3088\u308a\u5c0f\u3055\u304f\u3001VWN5\u76f8\u95a2\u306e\u7d76\u5bfe\u5024\u306fCCSD(T)\u3068HF\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5dee\u3088\u308a\u5927\u304d\u3044\u3068\u3044\u3046\u7d50\u679c\u3067\u3057\u305f\u3002\u305f\u3060\u3057\u3001\u5bc6\u5ea6\u3068\u5b9a\u7fa9\u304c\u7570\u306a\u308b\u305f\u3081\u3001\u5404\u6210\u5206\u306e\u8aa4\u5dee\u3092\u76f4\u63a5\u6bd4\u8f03\u306f\u3067\u304d\u307e\u305b\u3093\u3002<\/p>\n\n\n\n![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/orbital-1024x666.png\") \u56f33\uff0eBe\u539f\u5b50\u306e\u5360\u67091s\u30fb2s\u8ecc\u9053\u306e\u6bd4\u8f03\u3002<\/strong> HF\/3-21G\u3001KS-LDA\uff08Dirac\u4ea4\u63db\uff0bVWN5\u76f8\u95a2\uff09\/3-21G\u3001\u304a\u3088\u3073HF\/cc-pCVQZ-F12\u306b\u3088\u308b\u8ecc\u9053\u3092\u793a\u3059\u3002\u4e0a\u6bb5\u306f\u52d5\u5f84\u95a2\u6570 \\(R(r)\\)\u3001\u4e0b\u6bb5\u306f \\(u(r)=rR(r)\\) \u3067\u3001\u5de6\u5217\u304c1s\u8ecc\u9053\u3001\u53f3\u5217\u304c2s\u8ecc\u9053\u3067\u3042\u308b\u3002\u672c\u6587\u306es\u578b\u7a7a\u9593\u8ecc\u9053\u3068\u306e\u95a2\u4fc2\u306f \\(\\phi(\\mathbf r)=R(r)\/\\sqrt{4\\pi}\\) \u3067\u3042\u308a\u3001\\(\\int_0^\\infty |R(r)|^2r^2dr=\\int_0^\\infty |u(r)|^2dr=1\\) \u3068\u898f\u683c\u5316\u3055\u308c\u308b\u3002\u8ecc\u9053\u5168\u4f53\u306e\u7b26\u53f7\u306f\u4efb\u610f\u3067\u3042\u308b\u3002<\/figcaption><\/figure>\n\n\n\n![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/density-1024x384.png\") \u56f34\uff0eBe\u539f\u5b50\u306e\u96fb\u5b50\u5bc6\u5ea6\u3068\u52d5\u5f84\u96fb\u5b50\u6570\u5206\u5e03\u306e\u6bd4\u8f03\u3002<\/strong> HF\/3-21G\u3001KS-LDA\uff08Dirac\u4ea4\u63db\uff0bVWN5\u76f8\u95a2\uff09\/3-21G\u3001\u304a\u3088\u3073CCSD(T)\/cc-pCVQZ-F12\u306b\u3088\u308b\u7d50\u679c\u3092\u793a\u3059\u3002CCSD(T)\u306b\u306f\u7de9\u548c\u5bc6\u5ea6\uff08relaxed density\u3002\u5916\u90e8\u6442\u52d5\u306b\u5bfe\u3059\u308b\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5fdc\u7b54\u304b\u3089\u6c42\u3081\u3001\u8ecc\u9053\u306e\u5fdc\u7b54\u3082\u542b\u3081\u305f\u5bc6\u5ea6\uff09\u3092\u4f7f\u7528\u3057\u305f\u3002\u5de6\u56f3\u306f\u96fb\u5b50\u306e\u6570\u5bc6\u5ea6 \\(\\rho(r)\\) \u3092\u5bfe\u6570\u76ee\u76db\u3067\u3001\u53f3\u56f3\u306f\u52d5\u5f84\u96fb\u5b50\u6570\u5206\u5e03 \\(4\\pi r^2\\rho(r)\\) \u3092\u793a\u3059\u3002\\(4\\pi r^2\\rho(r)\\,dr\\) \u306f\u534a\u5f84 \\(r\\) \u304b\u3089 \\(r+dr\\) \u306e\u7403\u6bbb\u5185\u306b\u542b\u307e\u308c\u308b\u96fb\u5b50\u6570\u3092\u8868\u3057\u3001\u5168\u7a7a\u9593\u3067\u7a4d\u5206\u3059\u308b\u3068Be\u539f\u5b50\u306e\u96fb\u5b50\u65704\u306b\u306a\u308b\u3002<\/figcaption><\/figure>\n\n\n\n![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/density_difference-1024x341.png\") \u56f35\uff0eCCSD(T)\u3092\u57fa\u6e96\u3068\u3057\u305fBe\u539f\u5b50\u306e\u96fb\u5b50\u5bc6\u5ea6\u306e\u5dee\u3002<\/strong> CCSD(T)\/cc-pCVQZ-F12\u306e\u7de9\u548c\u5bc6\u5ea6\u3092\u57fa\u6e96\u3068\u3057\u3066\u3001HF\/3-21G\u304a\u3088\u3073KS-LDA\uff08Dirac\u4ea4\u63db\uff0bVWN5\u76f8\u95a2\uff09\/3-21G\u3068\u306e\u5dee\u3092\u6bd4\u8f03\u3057\u305f\u3002\u5de6\u56f3\u306f \\(\\Delta\\rho(r)=\\rho_{\\mathrm{\u5404\u65b9\u6cd5}}(r)-\\rho_{\\mathrm{CCSD(T)}}(r)\\)\u3001\u53f3\u56f3\u306f\u52d5\u5f84\u96fb\u5b50\u6570\u5206\u5e03\u306e\u5dee \\(4\\pi r^2\\Delta\\rho(r)\\) \u3092\u793a\u3059\u3002\u6b63\u306e\u5024\u306f\u57fa\u6e96\u3088\u308a\u96fb\u5b50\u5bc6\u5ea6\u304c\u5927\u304d\u304f\u3001\u8ca0\u306e\u5024\u306f\u5c0f\u3055\u3044\u3053\u3068\u3092\u8868\u3059\u3002\u8a08\u7b97\u65b9\u6cd5\u3068\u57fa\u5e95\u95a2\u6570\u7cfb\u306e\u4e21\u65b9\u304c\u7570\u306a\u308b\u305f\u3081\u3001\u3053\u306e\u5dee\u306b\u306f\u4e21\u8005\u306e\u5f71\u97ff\u304c\u542b\u307e\u308c\u308b\u3002<\/figcaption><\/figure>\n\n\n\n\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u5bc6\u5ea6\u6c4e\u95a2\u6570\u6cd5\u306e\u7406\u8ad6\u7684\u80cc\u666f\u306e\u52c9\u5f37<\/h2>\n\n\n\n\u7cfb\u306e\u6ce2\u52d5\u95a2\u6570\\(\\Psi\\)\u306e\u7d76\u5bfe\u5024\u306e\u4e8c\u4e57 \\(|\\Psi|^2\\) \u304b\u3089\u96fb\u5b50\u5bc6\u5ea6\\(\\rho\\)\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u96fb\u5b50\u5bc6\u5ea6\u306f\u3001\u307b\u304b\u306e\u96fb\u5b50\u306e\u5ea7\u6a19\u3092\u7a4d\u5206\u3057\u3001\u30b9\u30d4\u30f3\u306b\u3064\u3044\u3066\u548c\u3092\u53d6\u308a\u3001\u96fb\u5b50\u6570 \\(N\\) \u3092\u639b\u3051\u3066\u6c42\u3081\u307e\u3059\u3002\u6ce2\u52d5\u95a2\u6570\u7406\u8ad6\uff08wave function theory, WFT\uff09\u3067\u306f\u3001\u5916\u90e8\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\\(v_{\\mathrm{ext}}\\)\uff08\u539f\u5b50\u6838\u306e\u4f4d\u7f6e\u3068\u539f\u5b50\u756a\u53f7\uff09\u3068\u96fb\u5b50\u306e\u6570\\(N\\)\u304b\u3089\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u304c\u6c7a\u307e\u308b\u306e\u3067\u3001\\[(N,v_{\\mathrm{ext}}) \\rightarrow \\hat{H} \\rightarrow \\Psi_0 \\rightarrow \\rho_0\\]\u3067\u3059\u3002<\/p>\n\n\n\n \u4e00\u65b9\u3001DFT\u306e\u7406\u8ad6\u7684\u57fa\u790e\u3068\u306a\u308b\u306e\u306f\u3001\u57fa\u5e95\u72b6\u614b\u306b\u3064\u3044\u3066\u9006\u5411\u304d\u306e\u5bfe\u5fdc\u3082\u6210\u7acb\u3059\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u3067\u3059\u3002\u4ee5\u4e0b\u3067\u306f\u3001\u57fa\u5e95\u72b6\u614b\u304c\u7e2e\u9000\u3057\u3066\u3044\u306a\u3044\uff08\u5168\u4f53\u306e\u4f4d\u76f8\u306e\u9055\u3044\u3092\u9664\u3044\u3066\u57fa\u5e95\u72b6\u614b\u304c\u4e00\u610f\u306b\u6c7a\u307e\u308b\uff09\u5834\u5408\u3092\u8003\u3048\u307e\u3059\u3002\u57fa\u5e95\u72b6\u614b\u306e\u96fb\u5b50\u5bc6\u5ea6\u304b\u3089\u3001\u96fb\u5b50\u306e\u6570\u304c\u5206\u304b\u308a\u3001\u96fb\u5b50\u5bc6\u5ea6\u306b\u5bfe\u5fdc\u3059\u308b\u539f\u5b50\u6838\u306e\u4f4d\u7f6e\u3068\u539f\u5b50\u306e\u7a2e\u985e\uff08\u5916\u90e8\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\uff09\u304c\uff08\u52a0\u6cd5\u5b9a\u6570 \\(+ C\\) \u3092\u9664\u3044\u3066\uff09\u4e00\u610f\u306b\u6c7a\u307e\u308b\u306e\u3067\u3001\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u304c\u6c7a\u307e\u3063\u3066\u3001\u6ce2\u52d5\u95a2\u6570\u3082\u6c7a\u307e\u3063\u3066\u3001\uff08\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u96f6\u70b9\u3092\u56fa\u5b9a\u3059\u308c\u3070\uff09\u30a8\u30cd\u30eb\u30ae\u30fc\u3082\u6c7a\u307e\u308b\u3053\u3068\u304c\u8a3c\u660e\u3055\u308c\u3066\u3044\u307e\u3059\u3002\\[\\rho_0 \\rightarrow (N,v_{\\mathrm{ext}}) \\rightarrow \\hat{H} \\rightarrow \\Psi_0 \\rightarrow E_0 .\\] \u3055\u3089\u306b\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u96fb\u5b50\u5bc6\u5ea6\u306e\u6c4e\u95a2\u6570\u3068\u3057\u3066\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u3001\u5909\u5206\u539f\u7406\u304b\u3089\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u6c4e\u95a2\u6570\uff09\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3067\u57fa\u5e95\u72b6\u614b\u306e\u96fb\u5b50\u5bc6\u5ea6\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u3053\u3053\u3067\u3001\u57fa\u5e95\u72b6\u614b\u5bc6\u5ea6\u304b\u3089\u5916\u90e8\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u304c\u4e00\u610f\u306b\u6c7a\u307e\u308b\u3068\u3044\u3046\u8a71\u3068\u3001\u3042\u308b\u7cfb\u306e\u57fa\u5e95\u72b6\u614b\u5bc6\u5ea6\u3092\u5909\u5206\u306b\u3088\u3063\u3066\u6c42\u3081\u308b\u8b70\u8ad6\u3092\u533a\u5225\u3057\u3066\u304a\u304d\u307e\u3059\u3002\u5f8c\u8005\u3067\u306f\u3001\u539f\u5b50\u6838\u306e\u914d\u7f6e\u306a\u3069\u304c\u5b9a\u3081\u308b\u5916\u90e8\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3068\u96fb\u5b50\u6570\u3092\u56fa\u5b9a\u3057\u3066\u8a66\u884c\u5bc6\u5ea6\u3092\u5909\u5316\u3055\u305b\u307e\u3059\u3002<\/p>\n\n\n\n \u30dd\u30a4\u30f3\u30c8\u3092\u7b87\u6761\u66f8\u304d\u3067\u6574\u7406\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \n- \u539f\u5b50\u3084\u5206\u5b50\u306e\u7cfb\u306e\u975e\u76f8\u5bfe\u8ad6\u7684\u30b7\u30e5\u30ec\u30fc\u30c7\u30a3\u30f3\u30ac\u30fc\uff08Schr\u00f6dinger\uff09\u65b9\u7a0b\u5f0f\u3092\u89e3\u6790\u7684\u306b\u89e3\u304d\u3001\u6ce2\u52d5\u95a2\u6570\u3092\u6c42\u3081\u308b\u3053\u3068\u306f\u4e00\u822c\u7684\u306b\u56f0\u96e3\u3067\u3059\u3002<\/li>\n\n\n\n
- \u6ce2\u52d5\u95a2\u6570\u3092\u5358\u4e00\u306e\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f\u306b\u5236\u9650\u3057\u3066\u3001\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u3067\u5f97\u3089\u308c\u308b\u306e\u304c\u30cf\u30fc\u30c8\u30ea\u30fc\u30fb\u30d5\u30a9\u30c3\u30af\u65b9\u7a0b\u5f0f\u3067\u3059\u3002\u30b7\u30e5\u30ec\u30fc\u30c7\u30a3\u30f3\u30ac\u30fc\u65b9\u7a0b\u5f0f\u306e\u53b3\u5bc6\u89e3\u3092\u4e0e\u3048\u308b\u308f\u3051\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/li>\n\n\n\n
- DFT\u3067\u306f\u3001\u57fa\u5e95\u72b6\u614b\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u96fb\u5b50\u5bc6\u5ea6\u306e\u6c4e\u95a2\u6570\u3068\u3057\u3066\u8868\u308f\u305b\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u96fb\u5b50\u5bc6\u5ea6\u3068\u3044\u3046\u95a2\u6570\u3092\u5165\u529b\u3059\u308b\u3068\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u3044\u3046\u6570\u5024\u304c\u8fd4\u308b\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002\u305f\u3060\u3057\u3001\u8868\u308f\u305b\u308b\u3053\u3068\u304c\u5206\u304b\u3063\u3066\u3044\u308b\u3060\u3051\u3067\u3001\u53b3\u5bc6\u306a\u6c4e\u95a2\u6570\u306e\u5177\u4f53\u7684\u306a\u5f62\u306f\u4e00\u822c\u306b\u5206\u304b\u3063\u3066\u3044\u307e\u305b\u3093\u3002<\/li>\n\n\n\n
- \u7279\u306b\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3084\u4ea4\u63db\u30fb\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u3001\u96fb\u5b50\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u96e3\u984c\u3067\u3059\u3002\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u8aa4\u5dee\u306f\u3001\u7d50\u5408\u30a8\u30cd\u30eb\u30ae\u30fc\u306a\u3069\u306e\u5c0f\u3055\u306a\u30a8\u30cd\u30eb\u30ae\u30fc\u5dee\u306e\u8a55\u4fa1\u306b\u5927\u304d\u304f\u5f71\u97ff\u3057\u307e\u3059\u3002<\/li>\n\n\n\n
- \u305d\u3053\u3067\u3001\u8ecc\u9053\u304b\u3089\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u96fb\u5b50\u5bc6\u5ea6\u3092\u8a08\u7b97\u3057\u3066\u3001\u6b8b\u308a\u3092\u96fb\u5b50\u5bc6\u5ea6\u306e\u6c4e\u95a2\u6570\u3067\u6271\u3046\u67a0\u7d44\u307f\u3068\u3057\u3066\u3001\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\uff08Kohn\u2013Sham\uff09\u6cd5\u304c\u8003\u6848\u3055\u308c\u307e\u3057\u305f\u3002<\/li>\n\n\n\n
- \u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u65b9\u7a0b\u5f0f\u3092\u57fa\u5e95\u95a2\u6570\u3067\u5c55\u958b\u3057\u3066\u3001\u65b9\u7a0b\u5f0f\u3092\u884c\u5217\u8868\u793a\u3057\u305f\u3082\u306e\u304c\u3001\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u30fb\u30ed\u30fc\u30bf\u30fc\u30f3\uff08Kohn\u2013Sham\u2013Roothaan\uff09\u65b9\u7a0b\u5f0f\u3067\u3059\u3002<\/li>\n\n\n\n
- \u96fb\u5b50\u5bc6\u5ea6\u306b\u3088\u308b\u5909\u5206\u539f\u7406\u306f\u3001\u30ec\u30f4\u30a3\uff08Levy\uff09\u306e\u5236\u9650\u4ed8\u304d\u63a2\u7d22\u306b\u3088\u3063\u3066\u3001\u3088\u308a\u4e00\u822c\u7684\u306a\u5bc6\u5ea6\u3092\u6271\u3048\u308b\u5f62\u306b\u5b9a\u5f0f\u5316\u3055\u308c\u307e\u3057\u305f\u3002\u30db\u30fc\u30d8\u30f3\u30d0\u30fc\u30b0\u30fb\u30b3\u30fc\u30f3 (Hohenberg\u2013Kohn) \u306e\u7b2c2\u5b9a\u7406\u306f\u3053\u306e\u5236\u9650\u4ed8\u304d\u63a2\u7d22\u306b\u542b\u307e\u308c\u307e\u3059\u3002\u96fb\u5b50\u5bc6\u5ea6\u3068\u5916\u90e8\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\uff08\u539f\u5b50\u6838\u306e\u4f4d\u7f6e\u3068\u539f\u5b50\u306e\u7a2e\u985e\uff09\u306e\u95a2\u4fc2\u3092\u8aac\u660e\u3059\u308b\u305f\u3081\u306e\u91cd\u8981\u306a\u5b9a\u7406\u3068\u3057\u3066\u30db\u30fc\u30d8\u30f3\u30d0\u30fc\u30b0\u30fb\u30b3\u30fc\u30f3\u306e\u7b2c1\u5b9a\u7406\u304c\u3042\u308a\u307e\u3059\u3002<\/li>\n<\/ul>\n\n\n\n
\u4ee5\u4e0b\u3067\u306f\u539f\u5b50\u6838\u3092\u56fa\u5b9a\u3057\u3066\u3001\u6838\u9593\u53cd\u767a\u3092\u9664\u3044\u305f\u96fb\u5b50\u7cfb\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n\n\n\n \u6570\u5b66\u7684\u306a\u8a3c\u660e\u4e0a\u3001DFT\u306b\u304a\u3051\u308b\u5909\u5206\u8a08\u7b97\u306f\u3001\u300c\u5916\u90e8\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\uff08\u539f\u5b50\u6838\uff09\u3092\u8003\u3048\u305a\u306b\u3001\u4e0e\u3048\u3089\u308c\u305f\u96fb\u5b50\u5bc6\u5ea6\u306b\u5bfe\u3057\u3066\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u6700\u5c0f\u5316\u3059\u308b\u53cd\u5bfe\u79f0\u5316\u3055\u308c\u305f\u6ce2\u52d5\u95a2\u6570\u3092\u63a2\u7d22\u3059\u308b\u300d\u904e\u7a0b\u3068\u300c\u96fb\u5b50\u5bc6\u5ea6\u3092\u5909\u5316\u3055\u305b\u3066\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u6700\u5c0f\u5316\u3059\u308b\u300d\u904e\u7a0b\u306b\u5206\u3051\u3089\u308c\u307e\u3059\u3002\u524d\u8005\u306e\u63a2\u7d22\u3067\u5f97\u3089\u308c\u308b\u6700\u5c0f\u5024\u3092\u3001\u5bc6\u5ea6\u306e\u6c4e\u95a2\u6570\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u305f\u3082\u306e\u304c\u300c\u666e\u904d\u6c4e\u95a2\u6570\uff08universal functional\uff09\u300d\\[F[\\rho]=\\min_{\\Psi\\to\\rho} \\langle\\Psi|\\hat T+\\hat V_{ee}|\\Psi\\rangle\\]\u3067\u3059\u3002\u6c4e\u95a2\u6570\u306a\u306e\u3067\u3001\u96fb\u5b50\u5bc6\u5ea6\u95a2\u6570\u3092\u5165\u529b\u3059\u308b\u3068\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8fd4\u3057\u307e\u3059\u304c\u3001\u5185\u90e8\u7684\u306b\u306f\u300c\u4e0e\u3048\u3089\u308c\u305f\u96fb\u5b50\u5bc6\u5ea6\u306b\u5bfe\u3057\u3066\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u3068\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u6700\u5c0f\u5316\u3059\u308b\u53cd\u5bfe\u79f0\u5316\u3055\u308c\u305f\u6ce2\u52d5\u95a2\u6570\u3092\u63a2\u7d22\u300d\u3057\u3066\u3044\u307e\u3059\u30023\u7a2e\u985e\u306e\u96fb\u5b50\u5bc6\u5ea6\\(\\rho_A,\\rho_B,\\rho_C\\)\u3092\u8003\u3048\u3001\u3053\u306e\u4eee\u60f3\u4f8b\u3067\u306f\u3001\u3053\u306e3\u7a2e\u985e\u304c\u8a31\u3055\u308c\u308b\u5bc6\u5ea6\u306e\u3059\u3079\u3066\u3067\u3042\u308b\u3068\u3057\u307e\u3059\u3002\u305d\u308c\u305e\u308c\u306e\u96fb\u5b50\u5bc6\u5ea6\u3092\u751f\u6210\u3059\u308b\u6ce2\u52d5\u95a2\u6570\u306f\u7121\u6570\u306b\u5b58\u5728\u3057\u307e\u3059\u304c\u3001\u5404\u5bc6\u5ea6\u3092\u56fa\u5b9a\u3057\u3066\u3001\\(\\langle\\Psi|\\hat T+\\hat V_{ee}|\\Psi\\rangle\\)\u3092\u6700\u5c0f\u306b\u3059\u308b\\(\\Psi\\)\u3092\u63a2\u7d22\u3057\u3066\u304a\u304d\u307e\u3059\u3002\u305d\u3057\u3066\u3001\u5916\u90e8\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\u3082\u542b\u3081\u3066\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8a55\u4fa1\u3057\u307e\u3059\u3002\\[\\begin{aligned}&E[\\rho_A] = F[\\rho_A] + \\int \\rho_A(\\mathbf r)v_{\\mathrm{ext}}(\\mathbf r) d\\mathbf{r} = -0.4\\,\\,\\,\uff08\u63a2\u7d22\u3055\u308c\u305f\u6ce2\u52d5\u95a2\u6570\u306f\\Psi_{A}\uff09\\\\ &E[\\rho_B] = F[\\rho_B] + \\int \\rho_B(\\mathbf r)v_{\\mathrm{ext}}(\\mathbf r) d\\mathbf{r} = -0.5\\,\\,\\,\uff08\u63a2\u7d22\u3055\u308c\u305f\u6ce2\u52d5\u95a2\u6570\u306f\\Psi_{B}\uff09\\\\&E[\\rho_C] = F[\\rho_C] + \\int \\rho_C(\\mathbf r)v_{\\mathrm{ext}}(\\mathbf r) d\\mathbf{r} = -0.2\\,\\,\\,\uff08\u63a2\u7d22\u3055\u308c\u305f\u6ce2\u52d5\u95a2\u6570\u306f\\Psi_{C}\uff09\\end{aligned}\\] \u3053\u306e3\u3064\u306e\u4e2d\u3067\u306f\\(\\rho_B\\)\u304c\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u6700\u5c0f\u5024\u3092\u4e0e\u3048\u308b\u5bc6\u5ea6\\(\\rho_0\\)\u306a\u306e\u3067\u3001\\(\\Psi_{B}\\)\u304c\u771f\u306e\u57fa\u5e95\u72b6\u614b\u6ce2\u52d5\u95a2\u6570\\(\\Psi_{0}\\)\u3067\u3059\u3002\u3053\u308c\u306f\u6570\u5b66\u7684\u306a\u7406\u8ad6\u4e0a\u306e\u8a71\u3067\u3001\u5b9f\u969b\u306eDFT\u8a08\u7b97\u624b\u9806\u3068\u306f\u9055\u3044\u307e\u3059\u3002<\/p>\n\n\n\n \u7406\u8ad6\u4e0a\u3001\u57fa\u5e95\u72b6\u614b\u6ce2\u52d5\u95a2\u6570\\(\\Psi_{0}\\)\u306f\u65e2\u306b\u63a2\u7d22\u6e08\u307f\u306a\u306e\u3067\u3001\u666e\u904d\u6c4e\u95a2\u6570\u306e\u4e2d\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u671f\u5f85\u5024\u306f\u3053\u306e\u6ce2\u52d5\u95a2\u6570\u304b\u3089\u5b9a\u7fa9\u3067\u304d\u307e\u3059\u3002\\[T[\\rho_0] = \\int\\cdots\\int \\Psi^*_{0} \\hat{T} \\Psi_{0} dx_1\\cdots dx_N = \\langle\\Psi_0|\\hat{T}|\\Psi_0\\rangle\\] \u6b21\u306b\u3001\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\\(\\ \\int\\cdots\\int \\Psi^*_{0} \\hat{V}_{ee} \\Psi_{0} dx_1\\cdots dx_N = \\langle\\Psi_0|\\hat{V}_{ee}|\\Psi_0\\rangle\\)\u306e\u3046\u3061\u3001\u53e4\u5178\u7684\u306a\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\uff08\u30cf\u30fc\u30c8\u30ea\u30fc\u9805\uff09\u306f\u6ce2\u52d5\u95a2\u6570\u3067\u306f\u306a\u304f\u96fb\u5b50\u5bc6\u5ea6\u3060\u3051\u304b\u3089\u5b9a\u7fa9\u3067\u304d\u307e\u3059\u3002\\[J[\\rho_0] = \\frac{1}{2}\\iint\\frac{\\rho_0(\\mathbf r)\\rho_0(\\mathbf r’)}{|\\mathbf{r} – \\mathbf{r’}|}d^3 r d^3 r’ \\] \u305d\u3057\u3066\u3001\u666e\u904d\u6c4e\u95a2\u6570\u306e\u6b8b\u308a\u306e\u96fb\u5b50\u9593\u76f8\u4e92\u4f5c\u7528\u306e\u975e\u53e4\u5178\u7684\u90e8\u5206\u3092\\(W_{nc}\\)\u3068\u7f6e\u304d\u307e\u3059\u3002\\[W_{nc}[\\rho_0] = F[\\rho_0] – T[\\rho_0] – J[\\rho_0] \\] \u4e0a\u306e\u671f\u5f85\u5024\u306e\u5b9a\u7fa9\u304b\u3089\u975e\u53e4\u5178\u7684\u90e8\u5206\u3092\u76f4\u63a5\u8a55\u4fa1\u3059\u308b\u306b\u306f\u57fa\u5e95\u72b6\u614b\u6ce2\u52d5\u95a2\u6570\\(\\Psi_{0}\\)\u304c\u5fc5\u8981\u3067\u3059\u3002\u3055\u3089\u306b\u3001\u6838\u5f15\u529b\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u8a18\u53f7\u3067\u7f6e\u304d\u63db\u3048\u3066\u304a\u304d\u307e\u3059\u3002\\[ E_{\\mathrm{ext}}[\\rho_0] = \\int \\rho_0(\\mathbf r)v_{\\mathrm{ext}}(\\mathbf r) d^3 r \\] \u3057\u305f\u304c\u3063\u3066\u3001\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u5404\u8981\u7d20\u306b\u5206\u5272\u3059\u308b\u3068\u4ee5\u4e0b\u306e\u5f0f\u3068\u306a\u308a\u307e\u3059\u3002\\[E[\\rho_0] = T[\\rho_0] + E_{\\mathrm{ext}}[\\rho_0] + J[\\rho_0] + W_{nc}[\\rho_0] \\]\n\n\n\n \u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc \\(T[\\rho_0]\\) \u306f\u7406\u8ad6\u4e0a\u3001\u6ce2\u52d5\u95a2\u6570\\(\\Psi_{0}\\)\u304b\u3089\u8a08\u7b97\u3055\u308c\u3001\u96fb\u5b50\u5bc6\u5ea6\u304b\u3089\u76f4\u63a5\u8a08\u7b97\u3059\u308b\u3088\u3044\u8fd1\u4f3c\u6c4e\u95a2\u6570\u304c\u3042\u308a\u307e\u305b\u3093\u3067\u3057\u305f\u3002\u305d\u3053\u3067\u3001\u30b3\u30fc\u30f3\u3068\u30b7\u30e3\u30e0\u304c\u5c0e\u5165\u3057\u305f\u306e\u304c\u3001\u5b9f\u969b\u306e\u7cfb\u3068\u540c\u3058\u57fa\u5e95\u72b6\u614b\u96fb\u5b50\u5bc6\u5ea6\u3092\u6301\u3064\u4eee\u60f3\u7684\u306a\u300c\u76f8\u4e92\u4f5c\u7528\u306e\u306a\u3044\u53c2\u7167\u7cfb (non-interacting reference system)\u300d\u3067\u3059\u3002\u3053\u306e\u300c\u53c2\u7167\u7cfb\u300d\u306e\u305d\u308c\u305e\u308c\u306e\u96fb\u5b50\u306f\u3001\u6709\u52b9\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb\\(v_{s}\\)\u3092\u611f\u3058\u306a\u304c\u3089\u904b\u52d5\u3057\u3066\u3044\u308b\u3068\u8003\u3048\u307e\u3059\u3002\u300c\u76f8\u4e92\u4f5c\u7528\u306e\u306a\u3044\u300d\u3068\u306f\u3001\u3053\u306e\u53c2\u7167\u7cfb\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306b\u3001\u5b9f\u969b\u306e\u96fb\u5b50\u7cfb\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u306b\u542b\u307e\u308c\u308b\u4e8c\u4f53\u6f14\u7b97\u5b50\uff08\u4f8b\u3048\u3070\u30d8\u30ea\u30a6\u30e0\u539f\u5b50\u3067\u306f\u96fb\u5b50\u9593\u53cd\u767a\u6f14\u7b97\u5b50\u306f\\(\\frac{1}{r_{12}}\\)\uff09\u306a\u3069\u304c\u542b\u307e\u308c\u306a\u3044\u3001\u3068\u3044\u3046\u610f\u5473\u3067\u3059\u3002\u300c\u53b3\u5bc6\u306a\u4ea4\u63db\u30fb\u76f8\u95a2\u6c4e\u95a2\u6570\uff08\u4e0a\u306e\\(W_{nc}[\\rho]\\)\u3067\u306f\u306a\u304f\u5f8c\u3067\u5b9a\u7fa9\u3055\u308c\u308b\\(E_{xc}[\\rho]\\)\uff09\u3092\u4f7f\u3063\u3066\u3001\u9069\u5207\u306a\u53c2\u7167\u7cfb\u304c\u5b58\u5728\u3059\u308b\u300d\u3068\u3044\u3046\u6761\u4ef6\u306e\u4e0b\u3067\u306f\u3001\u57fa\u5e95\u72b6\u614b\u306e\u89e3\u306f\u5b9f\u969b\u306e\u7cfb\u306e\u96fb\u5b50\u5bc6\u5ea6\u3092\u518d\u73fe\u3057\u307e\u3059\u3002<\/p>\n\n\n\n \n \u6587\u732e\u306b\u3088\u3063\u3066\u300c\u88dc\u52a9\u7cfb\uff08auxiliary system\uff09\u300d\u3068\u3044\u3046\u8a00\u3044\u65b9\u3068\u300c\u53c2\u7167\u7cfb\uff08reference system\uff09\u300d\u3068\u3044\u3046\u8a00\u3044\u65b9\uff08\u3042\u308b\u3044\u306f\u4e21\u65b9 “auxiliary, noninteracting, \u201cKohn-Sham\u201d reference system<\/em>“\uff09\u304c\u4f7f\u308f\u308c\u3066\u3044\u307e\u3059\u3002\u30cb\u30e5\u30a2\u30f3\u30b9\u3068\u3057\u3066\u306f\u3001\u300c\u88dc\u52a9\u7cfb\u300d\u304c\u8a08\u7b97\u4e0a\u306e\u4eee\u60f3\u7684\u306a\u7cfb\u3067\u3042\u308b\u3053\u3068\u3092\u5f37\u8abf\u3057\u305f\u8a00\u3044\u65b9\u3067\u3001\u300c\u53c2\u7167\u7cfb\u300d\u306f\u5b9f\u5728\u306e\u7cfb\u3068\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u5dee\u3092\u5b9a\u7fa9\u3059\u308b\u57fa\u6e96\u306e\u7cfb\u3067\u3042\u308b\u3053\u3068\u3092\u5f37\u8abf\u3057\u305f\u8a00\u3044\u65b9\u3067\u3057\u3087\u3046\u304b\u3002<\/p>\n<\/div>\n\n\n\n\u53c2\u7167\u7cfb\u306e\u4e00\u96fb\u5b50\u8ecc\u9053\u3092\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u8ecc\u9053\u3068\u547c\u3073\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u3053\u308c\u3089\u306e\u8ecc\u9053\u304b\u3089\u5358\u4e00\u306e\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f \\(\\Phi_s\\) \u3092\u4f5c\u308c\u308b\u5834\u5408\u3092\u8003\u3048\u307e\u3059\u3002\u3053\u306e\u884c\u5217\u5f0f\u3092\u4f7f\u3063\u3066\u3001\u53c2\u7167\u7cfb\u306e\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc \\(T_s\\) \u3092\u8a08\u7b97\u3057\u307e\u3059\u3002\u76f8\u4e92\u4f5c\u7528\u306e\u306a\u3044\u53c2\u7167\u7cfb\uff08\u30b3\u30fc\u30f3\u30fb\u30b7\u30e3\u30e0\u53c2\u7167\u7cfb\uff09\u3068\u5b9f\u969b\u306e\u7cfb\u3067\u306f\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u304c\u4e00\u822c\u306b\u7570\u306a\u308a\u307e\u3059\u3002\u305d\u306e\u5dee\u3092\\[\\Delta T = T – T_s\\]\u3068\u5b9a\u7fa9\u3057\u307e\u3059\u3002\u3053\u306e\u5dee\u3092\u4ed6\u306e\u90e8\u5206\u3067\u88dc\u6b63\u3057\u306a\u304f\u3066\u306f\u3044\u3051\u307e\u305b\u3093\u3002<\/p>\n\n\n\n \u3053\u3053\u3067\u3001\u96fb\u5b50\u9593\u76f8\u4e92\u4f5c\u7528\u306e\u975e\u53e4\u5178\u7684\u30a8\u30cd\u30eb\u30ae\u30fc\\(W_{nc}[\\rho_0]\\)\u3092\u4ea4\u63db\u9805\u3068\u76f8\u95a2\u306e\u96fb\u5b50\u9593\u53cd\u767a\u6210\u5206\u306b\u5206\u3051\u307e\u3059\u3002\\[W_{nc}[\\rho_0] = E_x[\\rho_0] + U_c[\\rho_0].\\]\n\n\n\n \\(E_x[\\rho_0] \\) \u306f\u3001KS\u8ecc\u9053\u304b\u3089\u4f5c\u3063\u305f\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f \\(\\Phi_s\\) \u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u305f\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u3068\u30cf\u30fc\u30c8\u30ea\u30fc\u30a8\u30cd\u30eb\u30ae\u30fc\\(J[\\rho_0]\\)\u306e\u5dee\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\\[E_x[\\rho_0] = \\langle\\Phi_s|\\hat V_{ee}|\\Phi_s\\rangle – J[\\rho_0]. \\] \u3053\u306e\u4ea4\u63db\u9805\u306b\u3088\u3063\u3066\u3001\u30cf\u30fc\u30c8\u30ea\u30fc\u9805\u306b\u542b\u307e\u308c\u308b\u5404\u5360\u6709\u30b9\u30d4\u30f3\u8ecc\u9053\u306e\u81ea\u5df1\u76f8\u4e92\u4f5c\u7528\u304c\u76f8\u6bba\u3055\u308c\u307e\u3059\uff08\u3053\u306e\u76f8\u6bba\u306f\u4e0a\u8a18\u306e\u53b3\u5bc6\u4ea4\u63db\u306b\u3064\u3044\u3066\u6210\u308a\u7acb\u3061\u307e\u3059\u304c\u3001LDA\u306a\u3069\u306e\u8fd1\u4f3c\u4ea4\u63db\u6c4e\u95a2\u6570\u3067\u306f\u3001\u4e00\u822c\u306b\u81ea\u5df1\u76f8\u4e92\u4f5c\u7528\u306e\u8aa4\u5dee\u304c\u6b8b\u308a\u307e\u3059\u3002\uff09\u3002<\/p>\n\n\n\n \\(U_c[\\rho_0]\\)\u306f\u6b8b\u308a\u306e\u96fb\u5b50\u9593\u53cd\u767a\u306e\u88dc\u6b63\u30a8\u30cd\u30eb\u30ae\u30fc\u3067\u3001\u5b9f\u969b\u306e\u7cfb\u306e\u6ce2\u52d5\u95a2\u6570 \\(\\Psi_0\\) \u3092\u4f7f\u3063\u305f\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u3068KS\u8ecc\u9053\u304b\u3089\u4f5c\u3063\u305f\u30b9\u30ec\u30fc\u30bf\u30fc\u884c\u5217\u5f0f \\(\\Phi_s\\) \u3092\u4f7f\u3063\u3066\u8a08\u7b97\u3057\u305f\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u671f\u5f85\u5024\u306e\u5dee\\[U_c[\\rho_0]=\\langle\\Psi_0|\\hat V_{ee}|\\Psi_0\\rangle-\\langle\\Phi_s|\\hat V_{ee}|\\Phi_s\\rangle\\]\u3068\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u3055\u3089\u306b\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5dee\u3082\u542b\u3081\u3066\u3001\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002\\[E_c[\\rho_0] = \\underbrace{\\left(T[\\rho_0]-T_s[\\rho_0]\\right)}_{\\text{\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5dee}} + \\underbrace{U_c[\\rho_0]}_{\\text{\u96fb\u5b50\u9593\u53cd\u767a\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5dee}}\\]\u5b9f\u969b\u306b\u3067\u304d\u308b\u304b\u306f\u3068\u3082\u304b\u304f\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5dee\u3092\u96fb\u5b50\u5bc6\u5ea6\u304b\u3089\u8a08\u7b97\u3059\u308b\u6c4e\u95a2\u6570\\(\\Delta T[\\rho_0]\\)\u304c\u7406\u8ad6\u4e0a\u306f\u5b58\u5728\u3057\u307e\u3059\u3002\u3053\u3053\u3067\u5b9a\u7fa9\u3057\u305f \\(E_c\\) \u304c\u3001KS\u53c2\u7167\u7cfb\u3092\u57fa\u6e96\u3068\u3059\u308b\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u3067\u3059\u3002KS\u884c\u5217\u5f0f\u3068HF\u884c\u5217\u5f0f\u306f\u4e00\u822c\u306b\u7570\u306a\u308b\u305f\u3081\u3001WFT\u3067\u7528\u3044\u3089\u308c\u308b\u96fb\u5b50\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5b9a\u7fa9 \\(E_{\\mathrm{exact}}-E_{\\mathrm{HF}}\\) \u3068\u306f\u3001\u4e00\u822c\u306b\u306f\u4e00\u81f4\u3057\u307e\u305b\u3093\u3002<\/p>\n\n\n\n \u4ee5\u4e0a\u304b\u3089\u3001\u96fb\u5b50\u7cfb\u306e\u5168\u30a8\u30cd\u30eb\u30ae\u30fc\u306f\u3001\\[\\begin{align}E[\\rho_0] &= T_s + E_{ext}+ J + E_{x} + E_c \\\\
&= T_s + E_{ext}+ J + E_{x} + \\left(T-T_s\\right) + (V_{ee} – J – E_{x})\\\\
&= T + E_{\\mathrm{ext}} + V_{ee}.\\end{align}\\]\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306b\u3001\u904b\u52d5\u30a8\u30cd\u30eb\u30ae\u30fc\u306e\u5dee\\(T – T_s\\)\u3068\u96fb\u5b50\u9593\u76f8\u4e92\u4f5c\u7528\u306e\u975e\u53e4\u5178\u7684\u306a\u52b9\u679c\u3092\u4ea4\u63db\u76f8\u95a2\u6c4e\u95a2\u6570\u306b\u307e\u3068\u3081\u308b\u3053\u3068\u3067\u3001\u5143\u306e\u96fb\u5b50\u7cfb\u306e\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u53b3\u5bc6\u306b\u66f8\u304d\u76f4\u305b\u307e\u3059\u3002<\/p>\n\n\n\n \u6c4e\u95a2\u6570\u306f\u57fa\u5e95\u72b6\u614b\u4ee5\u5916\u306e\u8a66\u884c\u5bc6\u5ea6\u306b\u5bfe\u3057\u3066\u3082\u5b9a\u7fa9\u3055\u308c\u3066\u3044\u307e\u3059\u3002\\[E_{xc}[\\rho] = F[\\rho] – T_s[\\rho] – J[\\rho]\\]\\[E[\\rho] = T_s[\\rho]+ E_{\\mathrm{ext}}[\\rho] + J[\\rho] + E_{xc}[\\rho]\\]\n\n\n\n \u96fb\u5b50\u6570\u3092\u56fa\u5b9a\u3059\u308b\u6761\u4ef6 \\(\\int\\rho(\\mathbf r)\\,d\\mathbf r=N\\) \u306e\u3082\u3068\u3067\\(E[\\rho]\\)\u3092\u5909\u5206\u3059\u308b\u3068\u3001\u5fae\u5206\u53ef\u80fd\u306a\u5834\u5408\u3001\u57fa\u5e95\u72b6\u614b\u5bc6\u5ea6\u3067\u306f\\[\\left.\\frac{\\delta E[\\rho]}{\\delta\\rho(\\mathbf r)}\\right|_{\\rho=\\rho_0}=\\mu\\]\u3068\u306a\u308a\u307e\u3059\u3002\u5c55\u958b\u3059\u308b\u3068\u3001\\[\\left.\\frac{\\delta T_s[\\rho]}{\\delta\\rho(\\mathbf r)}\\right|_{\\rho_0}+v_{\\mathrm{ext}}(\\mathbf r)+v_{\\mathrm H}[\\rho_0](\\mathbf r)+v_{\\mathrm{xc}}[\\rho_0](\\mathbf r)=\\mu,\\qquad v_{\\mathrm{xc}}[\\rho](\\mathbf r)=\\frac{\\delta E_{\\mathrm{xc}}[\\rho]}{\\delta\\rho(\\mathbf r)}.\\] \u3053\u3053\u3067\u3001\\(E_{\\mathrm{xc}}=E_x+E_c\\)\u3001 \\(\\mu\\) \u306f\u96fb\u5b50\u6570\u3092\u56fa\u5b9a\u3059\u308b\u305f\u3081\u306e\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u672a\u5b9a\u4e57\u6570\u3067\u3059\u3002<\/p>\n\n\n\n \u554f\u984c\u306f\u3001\u4f55\u5ea6\u3082\u300c\u53b3\u5bc6\u306a\u4ea4\u63db\u30fb\u76f8\u95a2\u6c4e\u95a2\u6570\u3092\u4f7f\u3048\u3070\u300d\u3068\u66f8\u3044\u3066\u3044\u308b\u901a\u308a\u3067\u3001\u53b3\u5bc6\u306a\u6c4e\u95a2\u6570\u306e\u5177\u4f53\u7684\u306a\u5f62\u306f\u4e00\u822c\u306b\u5206\u304b\u3063\u3066\u3044\u306a\u3044\u70b9\u306b\u3042\u308a\u307e\u3059\u3002\u3053\u306e\u8a18\u4e8b\u3067\u306f\u3001\u57fa\u672c\u7684\u306a\u8fd1\u4f3c\u3067\u3042\u308b\u5c40\u6240\u5bc6\u5ea6\u8fd1\u4f3c\uff08LDA\u3001\u5404\u70b9\u306e\u5bc6\u5ea6\u306b\u3001\u305d\u306e\u5bc6\u5ea6\u306e\u5747\u4e00\u96fb\u5b50\u30ac\u30b9\u306e\u4ea4\u63db\u76f8\u95a2\u30a8\u30cd\u30eb\u30ae\u30fc\u3092\u5f53\u3066\u306f\u3081\u308b\u8fd1\u4f3c\uff09\u306b\u57fa\u3065\u304f\u4ea4\u63db\u30fb\u76f8\u95a2\u6c4e\u95a2\u6570\u3092\u4f7f\u3044\u307e\u3057\u305f\u3002\u3053\u308c\u3089\u306f\u3001\u80cc\u666f\u306b\u4e00\u69d8\u306a\u6b63\u96fb\u8377\u304c\u3042\u3063\u3066\u3001\u305d\u306e\u4e2d\u3092\u591a\u6570\u306e\u96fb\u5b50\u304c\u904b\u52d5\u3057\u3066\u3044\u308b\u3068\u8003\u3048\u308b\u5747\u4e00\u96fb\u5b50\u30ac\u30b9\uff08homogeneous electron gas\uff09\u3068\u3044\u3046\u7406\u60f3\u7684\u306a\u7cfb\u304b\u3089\u5c0e\u304b\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":" \u672c\u8a18\u4e8b\u306e\u4f5c\u6210\u306b\u3042\u305f\u308a\u3001\u8aac\u660e\u30fb\u6570\u5f0f\u306e\u78ba\u8a8d\u3068\u4fee\u6b63\u306bChatGPT …<\/p>\n","protected":false},"author":1,"featured_media":9505,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_locale":"ja","_original_post":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/?p=9508","footnotes":""},"categories":[5],"tags":[],"class_list":["post-9508","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-5","ja"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t | | | | |
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![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2026\/10\/3-21G_contracted_basis.png\")