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Generalized Eigenvalue Problems
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GeneralizedEigenvalueProblems.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 一般化固有値問題\n",
"\n",
"JuliaのLinearAlgebra.jlを利用して一般化固有値問題を解く. 一般化固有値問題の例として[J.M.ティッセン著『計算物理学』(丸善出版, 2012)](https://www.maruzen-publishing.co.jp/item/b294596.html)から水素原子の変分計算を取り上げる.\n",
"\n",
"© 2021 Shuhei Ohno\n",
"<br>Source: https://gist.github.com/ohno\n",
"<br>License: https://opensource.org/licenses/MIT"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Julia Version 1.6.2\n",
"Commit 1b93d53fc4 (2021-07-14 15:36 UTC)\n",
"Platform Info:\n",
" OS: Windows (x86_64-w64-mingw32)\n",
" CPU: Intel(R) Core(TM) i7-4650U CPU @ 1.70GHz\n",
" WORD_SIZE: 64\n",
" LIBM: libopenlibm\n",
" LLVM: libLLVM-11.0.1 (ORCJIT, haswell)\n"
]
}
],
"source": [
"# using Pkg\n",
"# Pkg.add(\"Plots\")\n",
"# Pkg.add(\"Optim\")\n",
"using Plots\n",
"using Optim\n",
"using LinearAlgebra\n",
"versioninfo()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 一般化固有値問題\n",
"\n",
"時間に依存しないSchrödinger方程式\n",
"\n",
"$$\n",
" \\hat{H}\\psi(\\boldsymbol{r}) = E\\psi(\\boldsymbol{r})\n",
"$$\n",
"\n",
"を非直交な基底関数で展開すると, 一般化固有値問題\n",
"\n",
"$$\n",
" \\boldsymbol{H}\\boldsymbol{C} = E\\boldsymbol{S}\\boldsymbol{C}\n",
"$$\n",
"\n",
"に帰着できる. 直交基底を用いれば$\\boldsymbol{S}=\\boldsymbol{I}$となり, 標準固有値問題$\\boldsymbol{H}\\boldsymbol{C} = E\\boldsymbol{C}$となる. 導出の詳細は[J.M.ティッセン著『計算物理学』(丸善出版, 2012) 3.1](https://www.maruzen-publishing.co.jp/item/b294596.html)または[A.ザボ, Ｎ.Ｓ.オストランド著『新しい量子化学 上 電子構造の理論入門』(東京大学出版会, 1987) 3.2.1](http://www.utp.or.jp/book/b302128.html)を参照されたい."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 水素原子\n",
"\n",
"水素原子の時間に依存しないSchrödinger方程式は\n",
"\n",
"$$\n",
" \\left[ -\\frac{\\hbar^2}{2m_\\mathrm{e}}\\nabla_\\mathrm{e}^2 -\\frac{\\hbar^2}{2m_\\mathrm{p}}\\nabla_\\mathrm{p}^2 - \\frac{e^2}{4\\pi\\varepsilon_0 r} \\right] \\psi_\\mathrm{tot}(\\boldsymbol{r}_\\mathrm{e},\\boldsymbol{r}_\\mathrm{p}) = E_\\mathrm{tot} \\psi_\\mathrm{tot}(\\boldsymbol{r}_\\mathrm{e},\\boldsymbol{r}_\\mathrm{p})\n",
"$$\n",
"\n",
"である. これを換算質量$\\frac{1}{\\mu}:=\\frac{1}{m_\\mathrm{e}}+\\frac{1}{m_\\mathrm{p}}$と相対運動$\\boldsymbol{r}:=\\boldsymbol{r}_\\mathrm{e}-\\boldsymbol{r}_\\mathrm{p}$を用いて並進運動の自由度を分離すると\n",
"\n",
"$$\n",
" \\left[ -\\frac{\\hbar^2}{2\\mu}\\nabla^2 - \\frac{e^2}{4\\pi\\varepsilon_0 r} \\right] \\psi(\\boldsymbol{r}) = E \\psi(\\boldsymbol{r})\n",
"$$\n",
"\n",
"が得られる. 分離の詳細は[原田義也『量子化学 下巻』(裳華房, 2007) §A2.3](https://www.shokabo.co.jp/mybooks/ISBN978-4-7853-3074-3.htm), [高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22](http://www.asakura.co.jp/books/isbn/978-4-254-13681-4/), [河合光路, 吉田思郎『朝倉物理学大系 19 原子核反応論』(朝倉書店, 2002) pp.8-12,25-38](http://www.asakura.co.jp/books/isbn/978-4-254-13689-0/), [Frank Jensen『Introduction to Computational Chemistry, 2nd Edition』(John Wiley & Sons, 2007) pp.14-17](https://www.wiley.com/en-us/Introduction+to+Computational+Chemistry%2C+2nd+Edition-p-9780470058046)などを参照されたい. さらに$m_\\mathrm{p}\\rightarrow\\infty$を仮定し, 原子単位を用いると\n",
"\n",
"$$\n",
" \\left[ -\\frac{1}{2}\\nabla^{*2} - \\frac{1}{r^*} \\right] \\psi^*(\\boldsymbol{r^*}) = E^* \\psi^*(\\boldsymbol{r}^*)\n",
"$$\n",
"\n",
"のように無次元化できる. 以降はこのアスタリスクを省略する. 無次元化については[IUPAC GreenBook 3.9.2](https://unit.aist.go.jp/nmij/public/report/translation/IUPAC/)を参照されたい."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 変分計算\n",
"\n",
"ここでは, [J.M.ティッセン著『計算物理学』(丸善出版, 2012) 3.2.2](https://www.maruzen-publishing.co.jp/item/b294596.html)より水素原子の例を取り上げる. 任意の方位量子数$l$や2体以上の系を考える場合は, [Hiyama,Kino,Kamimura(2003)](https://doi.org/10.1016/S0146-6410(03)90015-9)または[Hiyama,Kamimura(2018)](https://doi.org/10.1007/s11467-018-0828-5)を参照されたい. 以下では, $l=0$の場合に制限し, ガウス関数の線形結合として表現する.\n",
"\n",
"$$\n",
" \\phi(\\boldsymbol{r}) = \\sum_{i=1}^{N} C_i \\exp(-\\alpha_i r^2)\n",
"$$\n",
"\n",
"このとき, [J.M.ティッセン著『計算物理学』(丸善出版, 2012) 3.2.2](https://www.maruzen-publishing.co.jp/item/b294596.html)によれば, ハミルトニアン行列$\\boldsymbol{H}$, 運動エネルギー行列$\\boldsymbol{T}$, クーロン行列$\\boldsymbol{V}$, 重なり行列$\\boldsymbol{S}$の各要素はそれぞれ\n",
"\n",
"$$\n",
"\\begin{align}\n",
" H_{ij} &= \\langle\\phi|\\hat{H}|\\phi\\rangle = T_{ij} + V_{ij}\\\\\n",
" T_{ij} &= \\langle\\phi|\\hat{T}|\\phi\\rangle = \\frac{3\\pi^{3/2} \\alpha_i \\alpha_j}{(\\alpha_i + \\alpha_j)^{5/2}} \\\\\n",
" V_{ij} &= \\langle\\phi|\\hat{V}|\\phi\\rangle = -\\frac{2\\pi}{\\alpha_i + \\alpha_j}\\\\\n",
" S_{ij} &= \\langle\\phi|\\phi\\rangle = \\left(\\frac{\\pi}{\\alpha_i + \\alpha_j}\\right)^{3/2}\\\\\n",
"\\end{align}\n",
"$$\n",
"\n",
"である. この行列$\\boldsymbol{H}$と$\\boldsymbol{S}$を`eigen()`に渡すと$E$と$\\boldsymbol{C}$が得られる. 文献に従って$\\alpha_1=13.00773$, $\\alpha_2=1.962079$, $\\alpha_3=0.444529$, $\\alpha_4=0.1219492$を用いると$E_\\mathrm{1s}= -0.499 278 E_\\mathrm{h}$が得られる. 厳密解$E_\\mathrm{1s}= -0.5 E_\\mathrm{h}$に近い値である."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"GeneralizedEigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
"values:\n",
"4-element Vector{Float64}:\n",
" -0.499278405667485\n",
" 0.11321392045798685\n",
" 2.5922995719598094\n",
" 21.144365190122507\n",
"vectors:\n",
"4×4 Matrix{Float64}:\n",
" 0.0961015 0.119454 -0.0103621 -6.1551\n",
" 0.163017 0.0813295 1.74489 1.2402\n",
" 0.185587 0.496216 -0.629196 -0.226412\n",
" 0.0737008 -0.205916 0.0977745 0.0307798"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# 軌道指数\n",
"α = [13.00773, 1.962079, 0.444529, 0.1219492]\n",
"\n",
"# 配列宣言\n",
"n_max = length(α)\n",
"H = zeros(n_max, n_max)\n",
"T = zeros(n_max, n_max)\n",
"V = zeros(n_max, n_max)\n",
"S = zeros(n_max, n_max)\n",
"\n",
"# 行列要素\n",
"for j in 1:n_max\n",
" for i in 1:n_max\n",
" S[i,j] = (pi/(α[i]+α[j]))^(3/2)\n",
" T[i,j] = 3*pi^(3/2)*α[i]*α[j]/(α[i]+α[j])^(5/2)\n",
" V[i,j] = -2*pi/(α[i]+α[j])\n",
" H[i,j] = T[i,j] + V[i,j]\n",
" end\n",
"end\n",
"\n",
"# 固有値計算\n",
"resalt = eigen(Symmetric(H),Symmetric(S))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"さらに, エネルギーの固有値$E_n$と, 固有ベクトルから計算できる期待値$\\langle H\\rangle_n = \\sum_{i,j} C_{in}^*H_{ij}C_{jn}$が倍精度に由来する誤差の範疇では一致し, ビリアル定理$2\\langle T\\rangle+\\langle V\\rangle=0$も成り立っていることも確認できた."
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"E \t= -0.499278405667485\n",
"<H> \t= -0.49927840566748566\n",
"2<T>+<V>\t= -3.6997480412637174e-8\n"
]
}
],
"source": [
"E = resalt.values\n",
"C = resalt.vectors\n",
"\n",
"println(\"E \\t= \", E[1])\n",
"println(\"<H> \\t= \", C[:,1]'*H*C[:,1])\n",
"println(\"2<T>+<V>\\t= \", 2*C[:,1]'*T*C[:,1] + C[:,1]'*V*C[:,1])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"固有ベクトルが直交していることも確認できた."
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"4×4 Matrix{Float64}:\n",
" 1.0 1.73724e-16 3.88247e-16 -1.51252e-16\n",
" 7.51925e-17 1.0 -5.90027e-16 -3.41603e-16\n",
" 4.07307e-16 -5.63134e-16 1.0 5.35582e-16\n",
" -1.57765e-16 -2.60359e-16 4.76444e-16 1.0"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"C'*S*C"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"波動関数および動径分布も1s軌道の厳密解(Slater型関数)と非常に近い形をしていることも確認できた. なお, 線形代数のソルバ(LAPACK, NUMPACなど)で得られる固有ベクトルは正負が逆になることがある（符号が逆でも解であることには違いないが, 適当な基準で符号を決めないと描写する上で困る）ため, 動径波動関数を描写する際は正負の値を確認するか, 絶対値を取るなどの工夫が必要である. ここでは絶対値を取って描写した."
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
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},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y(θ,φ) = 1/sqrt(pi)\n",
"R(r) = exp(-r)\n",
"ψ(r,θ,φ) = R(r) * Y(θ,φ)\n",
"ρ(r,θ,φ) = r^2*ψ(r,θ,φ)^2\n",
"data2ψ(α,C,r,θ,φ) = sum(C[:,1] .* exp.(-α*r^2))\n",
"data2ρ(α,C,r,θ,φ) = r^2*data2ψ(α,C,r,θ,φ)^2\n",
"\n",
"plot(xlims=(0,4), xlabel=\"\\$r\\$\", ylabel=\"\\$|\\\\psi_\\\\mathrm{1s}(r,\\\\theta,\\\\phi)|\\$\")\n",
"plot!(0:0.01:4,r->abs(data2ψ(α,C,r,0,0)), label=\"STO-4G\")\n",
"plot!(r->ψ(r,0,0), label=\"exact\", lc=\"#000000\")"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
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" 1439.72,1282.44 1443.23,1284.14 1446.74,1285.82 1450.25,1287.49 1453.76,1289.13 1457.27,1290.75 1460.79,1292.36 1464.3,1293.94 1467.81,1295.51 1471.32,1297.06 \n",
" 1474.83,1298.6 1478.35,1300.11 1481.86,1301.61 1485.37,1303.09 1488.88,1304.56 1492.39,1306 1495.9,1307.43 1499.42,1308.85 1502.93,1310.25 1506.44,1311.63 \n",
" 1509.95,1313 1513.46,1314.35 1516.97,1315.68 1520.49,1317 1524,1318.31 1527.51,1319.6 1531.02,1320.87 1534.53,1322.13 1538.04,1323.38 1541.56,1324.61 \n",
" 1545.07,1325.83 1548.58,1327.03 1552.09,1328.22 1555.6,1329.4 1559.11,1330.56 1562.63,1331.71 1566.14,1332.85 1569.65,1333.97 1573.16,1335.08 1576.67,1336.17 \n",
" 1580.18,1337.26 1583.7,1338.33 1587.21,1339.39 1590.72,1340.43 1594.23,1341.47 1597.74,1342.49 1601.25,1343.5 1604.77,1344.49 1608.28,1345.48 1611.79,1346.45 \n",
" 1615.3,1347.42 1618.81,1348.37 1622.32,1349.31 1625.84,1350.23 1629.35,1351.15 1632.86,1352.06 1636.37,1352.95 1639.88,1353.84 1643.39,1354.71 1646.91,1355.57 \n",
" 1650.42,1356.43 1653.93,1357.27 1657.44,1358.1 1660.95,1358.92 1664.46,1359.73 1667.98,1360.53 1671.49,1361.32 1675,1362.1 1678.51,1362.88 1682.02,1363.64 \n",
" 1685.53,1364.39 1689.05,1365.13 1692.56,1365.86 1696.07,1366.59 1699.58,1367.3 1703.09,1368.01 1706.6,1368.7 1710.12,1369.39 1713.63,1370.07 1717.14,1370.74 \n",
" 1720.65,1371.4 1724.16,1372.05 1727.68,1372.7 1731.19,1373.33 1734.7,1373.96 1738.21,1374.58 1741.72,1375.19 1745.23,1375.79 1748.75,1376.38 1752.26,1376.97 \n",
" 1755.77,1377.55 1759.28,1378.12 1762.79,1378.68 1766.3,1379.24 1769.82,1379.78 1773.33,1380.32 1776.84,1380.86 1780.35,1381.38 1783.86,1381.9 1787.37,1382.41 \n",
" 1790.89,1382.91 1794.4,1383.41 1797.91,1383.9 1801.42,1384.38 1804.93,1384.86 1808.44,1385.32 1811.96,1385.79 1815.47,1386.24 1818.98,1386.69 1822.49,1387.13 \n",
" 1826,1387.57 1829.51,1388 1833.03,1388.42 1836.54,1388.84 1840.05,1389.25 1843.56,1389.65 1847.07,1390.05 1850.58,1390.44 1854.1,1390.83 1857.61,1391.21 \n",
" 1861.12,1391.59 1864.63,1391.96 1868.14,1392.32 1871.65,1392.68 1875.17,1393.03 1878.68,1393.38 1882.19,1393.72 1885.7,1394.06 1889.21,1394.39 1892.72,1394.71 \n",
" 1896.24,1395.03 1899.75,1395.35 1903.26,1395.66 1906.77,1395.97 1910.28,1396.27 1913.79,1396.56 1917.31,1396.86 1920.82,1397.14 1924.33,1397.42 1927.84,1397.7 \n",
" 1931.35,1397.97 1934.86,1398.24 1938.38,1398.51 1941.89,1398.77 1945.4,1399.02 1948.91,1399.27 1952.42,1399.52 1955.93,1399.76 1959.45,1400 1962.96,1400.24 \n",
" 1966.47,1400.47 1969.98,1400.7 1973.49,1400.92 1977.01,1401.14 1980.52,1401.35 1984.03,1401.57 1987.54,1401.77 1991.05,1401.98 1994.56,1402.18 1998.08,1402.38 \n",
" 2001.59,1402.57 2005.1,1402.76 2008.61,1402.95 2012.12,1403.13 2015.63,1403.31 2019.15,1403.49 2022.66,1403.66 2026.17,1403.84 2029.68,1404 2033.19,1404.17 \n",
" 2036.7,1404.33 2040.22,1404.49 2043.73,1404.65 2047.24,1404.8 2050.75,1404.95 2054.26,1405.1 2057.77,1405.24 2061.29,1405.38 2064.8,1405.52 2068.31,1405.66 \n",
" 2071.82,1405.8 2075.33,1405.93 2078.84,1406.06 2082.36,1406.18 2085.87,1406.31 2089.38,1406.43 2092.89,1406.55 2096.4,1406.67 2099.91,1406.78 2103.43,1406.9 \n",
" 2106.94,1407.01 2110.45,1407.12 2113.96,1407.22 2117.47,1407.33 2120.98,1407.43 2124.5,1407.53 2128.01,1407.63 2131.52,1407.73 2135.03,1407.82 2138.54,1407.91 \n",
" 2142.05,1408 2145.57,1408.09 2149.08,1408.18 2152.59,1408.27 2156.1,1408.35 2159.61,1408.43 2163.12,1408.51 2166.64,1408.59 2170.15,1408.67 2173.66,1408.74 \n",
" 2177.17,1408.82 2180.68,1408.89 2184.19,1408.96 2187.71,1409.03 2191.22,1409.1 2194.73,1409.17 2198.24,1409.23 2201.75,1409.3 2205.26,1409.36 2208.78,1409.42 \n",
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" 2317.64,1410.78 2321.15,1410.81 2324.66,1410.84 2328.17,1410.87 2331.69,1410.9 2335.2,1410.93 2338.71,1410.95 2342.22,1410.98 2345.73,1411 2349.24,1411.03 \n",
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},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(xlims=(0,6), xlabel=\"\\$r\\$\", ylabel=\"\\$r^2 |\\\\psi_\\\\mathrm{1s}(r,\\\\theta,\\\\phi)|^2\\$\")\n",
"plot!(0:0.01:6,r->data2ρ(α,C,r,0,0), label=\"STO-4G\")\n",
"plot!(r->ρ(r,0,0), label=\"exact\", lc=\"#000000\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 最適化\n",
"\n",
"教科書に書いてあるパラメータを使ったところ, 厳密値の$E_\\mathrm{1s}= -0.5 E_\\mathrm{h}$に近い値が得られた. このパラメータはエネルギーを目的関数として, 各軌道指数を説明変数とした最適化計算を行うことで得られる. ここでは`Optim.jl`の[Nelder-Mead法(ANMS)](https://julianlsolvers.github.io/Optim.jl/stable/#algo/nelder_mead/)を用いて最適化を行う. パラメータの最適化にあたって最低でも3までの安全対策を施す必要がある.\n",
"\n",
"1. $0<\\alpha_i+\\alpha_j$を判定すること\n",
"2. `try` `catch` `end`でソルバが異常終了した場合に停止しないよう対策すること\n",
"3. `eigen()`に渡す前に`Symmetric()`を通して対称行列の型を指定すること\n",
"4. 固有ベクトルから計算した期待値が固有値と`1e-8`以内の誤差で一致するか確認すること"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"GeneralizedEigen{Float64, Float64, Matrix{Float64}, Vector{Float64}}\n",
"values:\n",
"4-element Vector{Float64}:\n",
" -0.499278405667485\n",
" 0.11321392045798685\n",
" 2.5922995719598094\n",
" 21.144365190122507\n",
"vectors:\n",
"4×4 Matrix{Float64}:\n",
" 0.0961015 0.119454 -0.0103621 -6.1551\n",
" 0.163017 0.0813295 1.74489 1.2402\n",
" 0.185587 0.496216 -0.629196 -0.226412\n",
" 0.0737008 -0.205916 0.0977745 0.0307798"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# using Plots\n",
"# using LinearAlgebra\n",
"\n",
"function STOnG(α)\n",
" #準備\n",
" n_max = length(α)\n",
" H = zeros(n_max, n_max)\n",
" T = zeros(n_max, n_max)\n",
" V = zeros(n_max, n_max)\n",
" S = zeros(n_max, n_max)\n",
"\n",
" # 行列要素\n",
" for j in 1:n_max\n",
" for i in 1:n_max\n",
" if 0<(α[i]+α[j])\n",
" S[i,j] = (pi/(α[i]+α[j]))^(3/2)\n",
" T[i,j] = 3*pi^(3/2)*α[i]*α[j]/(α[i]+α[j])^(5/2)\n",
" V[i,j] = -2*pi/(α[i]+α[j])\n",
" H[i,j] = T[i,j] + V[i,j]\n",
" end\n",
" end\n",
" end\n",
"\n",
" # 計算 # LAPACK.sygvd!(1, 'V', 'U', H, S)\n",
" resalt = try; eigen(Symmetric(H),Symmetric(S)); catch; (values=fill(Inf,n_max), vectors=zeros(n_max, n_max)); end\n",
" E = resalt.values\n",
" C = resalt.vectors\n",
"\n",
" # 検証\n",
" if 1e-9 < abs(E[1] - transpose(C[:,1])*H*C[:,1])\n",
" resalt.values .= Inf\n",
" end\n",
"\n",
" return resalt\n",
"end\n",
"\n",
"STOnG([13.00773, 1.962079, 0.444529, 0.1219492])"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"(values = [-0.37837916709551256], vectors = [0.4237772081237576])"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"function STO1G(α)\n",
" α = α[1]\n",
" S = (pi/(α+α))^(3/2)\n",
" T = 3*pi^(3/2)*α*α/(α+α)^(5/2)\n",
" V = -2*pi/(α+α)\n",
" H = T + V\n",
" return (values=[H/S]::Vector{Float64}, vectors=1/sqrt(S).*ones(Float64, 1, 1)::Matrix{Float64})\n",
"end\n",
"\n",
"STO1G(0.5)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"scrolled": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"STO-nG\n",
"n\tenergy\t\t\tcalls\n",
"1\t-0.4244131815783876\t11\n",
"2\t-0.48581271661627506\t116\n",
"3\t-0.4969792527050511\t255\n",
"4\t-0.4992784057143485\t4457\n",
"5\t-0.4998098322318914\t3653\n",
"6\t-0.49994557039665577\t2292\n",
"7\t-0.4999832977891709\t2012\n",
"8\t-0.4999945613797857\t1563\n",
"9\t-0.4999980709736585\t1500\n",
"10\t-0.49999926122327554\t1545\n",
"11\t-0.49999966691605624\t1523\n",
"12\t-0.49999974883313014\t1567\n",
"13\t-0.49999986092017573\t1476\n",
"14\t-0.49999996389813534\t1453\n",
"15\t-0.4999999815370859\t1540\n",
"16\t-0.4999999860830632\t1454\n",
"17\t-0.4999999903186757\t1521\n",
"18\t-0.4999999911930606\t2725\n",
"19\t-0.49999999347274715\t11328\n",
"20\t-0.4999999936544641\t14065\n",
"∞ \t-0.5\n"
]
}
],
"source": [
"αs = []\n",
"Cs = []\n",
"\n",
"println(\"STO-nG\")\n",
"println(\"n\\tenergy\\t\\t\\tcalls\")\n",
"\n",
"res = optimize(X->STO1G(X).values[1], [0.5], method=GradientDescent())\n",
"α = Optim.minimizer(res)\n",
"C = STO1G(α).vectors\n",
"println(1, \"\\t\", Optim.minimum(res), \"\\t\", Optim.f_calls(res))\n",
"push!(αs, α)\n",
"push!(Cs, C)\n",
"\n",
"α = [0.00001, 0.0001]\n",
"for i in 2:20\n",
" res = optimize(X->STOnG(X).values[1], α, g_tol=1e-16)\n",
" α = Optim.minimizer(res)\n",
" C = STOnG(α).vectors\n",
" println(i, \"\\t\", Optim.minimum(res), \"\\t\", Optim.f_calls(res))\n",
" push!(αs, α[:])\n",
" push!(Cs, C)\n",
" push!(α, α[end]*0.99)\n",
"end\n",
"\n",
"println(\"∞ \\t\", -0.50000000000000000)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"かなり低いエネルギーが得られているが, いずれも評価回数が非常に大きいことがわかる. 2体より大きな系では, 小さな基底でない限り全ての軌道指数を最適化するのはほぼ不可能であるから, 何らかの制約を課すなどしてパラメータを減らす必要がある. つまり全ての軌道指数を完璧に最適化した場合に比べればエネルギーを妥協することになる.\n",
"\n",
"先ほどと同様に動径波動関数および動径分布関数を描写する. 下図の比較対象としては[J.M.ティッセン著『計算物理学』(丸善出版, 2012) 図4.3](https://www.maruzen-publishing.co.jp/item/b294596.html)や[A.ザボ, Ｎ.Ｓ.オストランド著『新しい量子化学 上 電子構造の理論入門』(東京大学出版会, 1987) 図3.3](http://www.utp.or.jp/book/b302128.html)を挙げる. 基底関数を増やすほどにカスプ($r=0$の点)の記述が良くなっていくことが見て取れる."
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"scrolled": false
},
"outputs": [
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},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Y(θ,φ) = 1/sqrt(pi)\n",
"R(r) = exp(-r)\n",
"ψ(r,θ,φ) = R(r) * Y(θ,φ)\n",
"ρ(r,θ,φ) = r^2*ψ(r,θ,φ)^2\n",
"data2ψ(α,C,r,θ,φ) = sum(C[:,1] .* exp.(-α*r^2))\n",
"data2ρ(α,C,r,θ,φ) = r^2*data2ψ(α,C,r,θ,φ)^2\n",
"\n",
"plot(xlims=(0,4), xlabel=\"\\$r\\$\", ylabel=\"\\$|\\\\psi_\\\\mathrm{1s}(r,\\\\theta,\\\\phi)|\\$\")\n",
"\n",
"for i in 1:10\n",
" plot!(0:0.01:4, r->abs(data2ψ(αs[i],Cs[i],r,0,0)), label=\"STO-$(i)G\")\n",
"end\n",
"\n",
"plot!(r->ψ(r,0,0), label=\"exact\", lc=\"#000000\")"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
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" 1193.9,1121.24 1197.41,1124.48 1200.92,1127.7 1204.43,1130.89 1207.94,1134.05 1211.46,1137.18 1214.97,1140.3 1218.48,1143.38 1221.99,1146.45 1225.5,1149.49 \n",
" 1229.01,1152.5 1232.53,1155.49 1236.04,1158.46 1239.55,1161.4 1243.06,1164.32 1246.57,1167.22 1250.09,1170.09 1253.6,1172.94 1257.11,1175.77 1260.62,1178.57 \n",
" 1264.13,1181.36 1267.64,1184.12 1271.16,1186.86 1274.67,1189.57 1278.18,1192.27 1281.69,1194.94 1285.2,1197.59 1288.71,1200.22 1292.23,1202.83 1295.74,1205.42 \n",
" 1299.25,1207.99 1302.76,1210.53 1306.27,1213.06 1309.78,1215.56 1313.3,1218.05 1316.81,1220.51 1320.32,1222.95 1323.83,1225.37 1327.34,1227.77 1330.85,1230.15 \n",
" 1334.37,1232.51 1337.88,1234.85 1341.39,1237.17 1344.9,1239.47 1348.41,1241.75 1351.92,1244.01 1355.44,1246.25 1358.95,1248.47 1362.46,1250.67 1365.97,1252.85 \n",
" 1369.48,1255.01 1372.99,1257.15 1376.51,1259.28 1380.02,1261.38 1383.53,1263.46 1387.04,1265.52 1390.55,1267.57 1394.06,1269.59 1397.58,1271.59 1401.09,1273.58 \n",
" 1404.6,1275.55 1408.11,1277.49 1411.62,1279.42 1415.13,1281.33 1418.65,1283.22 1422.16,1285.09 1425.67,1286.95 1429.18,1288.78 1432.69,1290.59 1436.2,1292.39 \n",
" 1439.72,1294.17 1443.23,1295.93 1446.74,1297.67 1450.25,1299.39 1453.76,1301.09 1457.27,1302.78 1460.79,1304.45 1464.3,1306.09 1467.81,1307.73 1471.32,1309.34 \n",
" 1474.83,1310.93 1478.35,1312.51 1481.86,1314.07 1485.37,1315.61 1488.88,1317.14 1492.39,1318.64 1495.9,1320.13 1499.42,1321.6 1502.93,1323.06 1506.44,1324.49 \n",
" 1509.95,1325.91 1513.46,1327.32 1516.97,1328.7 1520.49,1330.07 1524,1331.42 1527.51,1332.76 1531.02,1334.08 1534.53,1335.38 1538.04,1336.66 1541.56,1337.93 \n",
" 1545.07,1339.19 1548.58,1340.42 1552.09,1341.65 1555.6,1342.85 1559.11,1344.04 1562.63,1345.21 1566.14,1346.37 1569.65,1347.51 1573.16,1348.64 1576.67,1349.75 \n",
" 1580.18,1350.85 1583.7,1351.93 1587.21,1353 1590.72,1354.05 1594.23,1355.09 1597.74,1356.11 1601.25,1357.12 1604.77,1358.11 1608.28,1359.09 1611.79,1360.06 \n",
" 1615.3,1361.01 1618.81,1361.95 1622.32,1362.87 1625.84,1363.78 1629.35,1364.68 1632.86,1365.56 1636.37,1366.43 1639.88,1367.29 1643.39,1368.14 1646.91,1368.97 \n",
" 1650.42,1369.79 1653.93,1370.59 1657.44,1371.39 1660.95,1372.17 1664.46,1372.94 1667.98,1373.69 1671.49,1374.44 1675,1375.17 1678.51,1375.89 1682.02,1376.6 \n",
" 1685.53,1377.3 1689.05,1377.99 1692.56,1378.66 1696.07,1379.33 1699.58,1379.98 1703.09,1380.62 1706.6,1381.26 1710.12,1381.88 1713.63,1382.49 1717.14,1383.09 \n",
" 1720.65,1383.68 1724.16,1384.26 1727.68,1384.83 1731.19,1385.39 1734.7,1385.94 1738.21,1386.48 1741.72,1387.01 1745.23,1387.54 1748.75,1388.05 1752.26,1388.55 \n",
" 1755.77,1389.05 1759.28,1389.53 1762.79,1390.01 1766.3,1390.48 1769.82,1390.94 1773.33,1391.39 1776.84,1391.84 1780.35,1392.27 1783.86,1392.7 1787.37,1393.12 \n",
" 1790.89,1393.53 1794.4,1393.93 1797.91,1394.33 1801.42,1394.72 1804.93,1395.1 1808.44,1395.47 1811.96,1395.84 1815.47,1396.2 1818.98,1396.55 1822.49,1396.9 \n",
" 1826,1397.24 1829.51,1397.57 1833.03,1397.9 1836.54,1398.22 1840.05,1398.53 1843.56,1398.84 1847.07,1399.14 1850.58,1399.44 1854.1,1399.73 1857.61,1400.01 \n",
" 1861.12,1400.29 1864.63,1400.56 1868.14,1400.83 1871.65,1401.09 1875.17,1401.34 1878.68,1401.59 1882.19,1401.84 1885.7,1402.08 1889.21,1402.31 1892.72,1402.54 \n",
" 1896.24,1402.77 1899.75,1402.99 1903.26,1403.2 1906.77,1403.42 1910.28,1403.62 1913.79,1403.82 1917.31,1404.02 1920.82,1404.22 1924.33,1404.41 1927.84,1404.59 \n",
" 1931.35,1404.77 1934.86,1404.95 1938.38,1405.12 1941.89,1405.29 1945.4,1405.46 1948.91,1405.62 1952.42,1405.78 1955.93,1405.94 1959.45,1406.09 1962.96,1406.24 \n",
" 1966.47,1406.38 1969.98,1406.52 1973.49,1406.66 1977.01,1406.8 1980.52,1406.93 1984.03,1407.06 1987.54,1407.18 1991.05,1407.31 1994.56,1407.43 1998.08,1407.54 \n",
" 2001.59,1407.66 2005.1,1407.77 2008.61,1407.88 2012.12,1407.99 2015.63,1408.09 2019.15,1408.19 2022.66,1408.29 2026.17,1408.39 2029.68,1408.49 2033.19,1408.58 \n",
" 2036.7,1408.67 2040.22,1408.76 2043.73,1408.84 2047.24,1408.93 2050.75,1409.01 2054.26,1409.09 2057.77,1409.17 2061.29,1409.24 2064.8,1409.32 2068.31,1409.39 \n",
" 2071.82,1409.46 2075.33,1409.53 2078.84,1409.59 2082.36,1409.66 2085.87,1409.72 2089.38,1409.79 2092.89,1409.85 2096.4,1409.91 2099.91,1409.96 2103.43,1410.02 \n",
" 2106.94,1410.07 2110.45,1410.13 2113.96,1410.18 2117.47,1410.23 2120.98,1410.28 2124.5,1410.33 2128.01,1410.37 2131.52,1410.42 2135.03,1410.46 2138.54,1410.51 \n",
" 2142.05,1410.55 2145.57,1410.59 2149.08,1410.63 2152.59,1410.67 2156.1,1410.7 2159.61,1410.74 2163.12,1410.78 2166.64,1410.81 2170.15,1410.85 2173.66,1410.88 \n",
" 2177.17,1410.91 2180.68,1410.94 2184.19,1410.97 2187.71,1411 2191.22,1411.03 2194.73,1411.06 2198.24,1411.09 2201.75,1411.11 2205.26,1411.14 2208.78,1411.16 \n",
" 2212.29,1411.19 2215.8,1411.21 2219.31,1411.23 2222.82,1411.26 2226.34,1411.28 2229.85,1411.3 2233.36,1411.32 2236.87,1411.34 2240.38,1411.36 2243.89,1411.38 \n",
" 2247.41,1411.4 2250.92,1411.41 2254.43,1411.43 2257.94,1411.45 2261.45,1411.46 2264.96,1411.48 2268.48,1411.5 2271.99,1411.51 2275.5,1411.52 2279.01,1411.54 \n",
" 2282.52,1411.55 2286.03,1411.57 2289.55,1411.58 2293.06,1411.59 2296.57,1411.6 2300.08,1411.61 2303.59,1411.63 2307.1,1411.64 2310.62,1411.65 2314.13,1411.66 \n",
" 2317.64,1411.67 2321.15,1411.68 2324.66,1411.69 2328.17,1411.7 2331.69,1411.71 2335.2,1411.71 2338.71,1411.72 2342.22,1411.73 2345.73,1411.74 2349.24,1411.75 \n",
" 2352.76,1411.75 \n",
" \"/>\n",
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" 245.742,1411.98 249.253,1411.26 252.765,1409.12 256.277,1405.57 259.788,1400.63 263.3,1394.35 266.812,1386.76 270.324,1377.94 273.835,1367.92 277.347,1356.8 \n",
" 280.859,1344.63 284.37,1331.5 287.882,1317.48 291.394,1302.65 294.905,1287.11 298.417,1270.91 301.929,1254.16 305.44,1236.91 308.952,1219.23 312.464,1201.21 \n",
" 315.975,1182.89 319.487,1164.34 322.999,1145.6 326.511,1126.72 330.022,1107.74 333.534,1088.7 337.046,1069.63 340.557,1050.54 344.069,1031.48 347.581,1012.45 \n",
" 351.092,993.465 354.604,974.542 358.116,955.69 361.627,936.917 365.139,918.23 368.651,899.636 372.163,881.139 375.674,862.747 379.186,844.463 382.698,826.295 \n",
" 386.209,808.248 389.721,790.331 393.233,772.552 396.744,754.919 400.256,737.443 403.768,720.134 407.279,703.005 410.791,686.067 414.303,669.335 417.815,652.821 \n",
" 421.326,636.539 424.838,620.504 428.35,604.731 431.861,589.233 435.373,574.025 438.885,559.12 442.396,544.532 445.908,530.274 449.42,516.358 452.931,502.794 \n",
" 456.443,489.594 459.955,476.766 463.466,464.32 466.978,452.262 470.49,440.598 474.002,429.335 477.513,418.476 481.025,408.024 484.537,397.982 488.048,388.35 \n",
" 491.56,379.128 495.072,370.315 498.583,361.91 502.095,353.911 505.607,346.313 509.118,339.112 512.63,332.304 516.142,325.883 519.654,319.844 523.165,314.18 \n",
" 526.677,308.885 530.189,303.95 533.7,299.369 537.212,295.133 540.724,291.236 544.235,287.668 547.747,284.421 551.259,281.488 554.77,278.859 558.282,276.527 \n",
" 561.794,274.482 565.306,272.717 568.817,271.222 572.329,269.991 575.841,269.015 579.352,268.286 582.864,267.796 586.376,267.537 589.887,267.503 593.399,267.686 \n",
" 596.911,268.078 600.422,268.674 603.934,269.466 607.446,270.448 610.957,271.614 614.469,272.957 617.981,274.473 621.493,276.155 625.004,277.999 628.516,279.998 \n",
" 632.028,282.149 635.539,284.447 639.051,286.887 642.563,289.464 646.074,292.176 649.586,295.017 653.098,297.985 656.609,301.075 660.121,304.285 663.633,307.61 \n",
" 667.145,311.049 670.656,314.599 674.168,318.255 677.68,322.016 681.191,325.88 684.703,329.844 688.215,333.905 691.726,338.062 695.238,342.311 698.75,346.653 \n",
" 702.261,351.083 705.773,355.601 709.285,360.205 712.797,364.893 716.308,369.662 719.82,374.513 723.332,379.442 726.843,384.449 730.355,389.531 733.867,394.688 \n",
" 737.378,399.917 740.89,405.218 744.402,410.588 747.913,416.027 751.425,421.532 754.937,427.102 758.448,432.737 761.96,438.433 765.472,444.19 768.984,450.006 \n",
" 772.495,455.88 776.007,461.81 779.519,467.794 783.03,473.832 786.542,479.92 790.054,486.058 793.565,492.245 797.077,498.477 800.589,504.755 804.1,511.075 \n",
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" 842.729,582.932 846.241,589.63 849.752,596.347 853.264,603.082 856.776,609.833 860.287,616.598 863.799,623.375 867.311,630.162 870.823,636.958 874.334,643.76 \n",
" 877.846,650.568 881.358,657.378 884.869,664.19 888.381,671.001 891.893,677.81 895.404,684.615 898.916,691.414 902.428,698.205 905.939,704.988 909.451,711.76 \n",
" 912.963,718.52 916.475,725.265 919.986,731.995 923.498,738.709 927.01,745.403 930.521,752.078 934.033,758.731 937.545,765.361 941.056,771.967 944.568,778.548 \n",
" 948.08,785.101 951.591,791.627 955.103,798.123 958.615,804.589 962.127,811.023 965.638,817.424 969.15,823.791 972.662,830.124 976.173,836.42 979.685,842.68 \n",
" 983.197,848.901 986.708,855.085 990.22,861.228 993.732,867.331 997.243,873.393 1000.76,879.413 1004.27,885.391 1007.78,891.325 1011.29,897.216 1014.8,903.062 \n",
" 1018.31,908.863 1021.83,914.619 1025.34,920.328 1028.85,925.992 1032.36,931.608 1035.87,937.177 1039.38,942.699 1042.9,948.172 1046.41,953.598 1049.92,958.975 \n",
" 1053.43,964.303 1056.94,969.582 1060.45,974.812 1063.97,979.994 1067.48,985.125 1070.99,990.208 1074.5,995.241 1078.01,1000.22 1081.52,1005.16 1085.04,1010.04 \n",
" 1088.55,1014.88 1092.06,1019.66 1095.57,1024.4 1099.08,1029.09 1102.59,1033.72 1106.11,1038.31 1109.62,1042.85 1113.13,1047.34 1116.64,1051.79 1120.15,1056.18 \n",
" 1123.66,1060.53 1127.18,1064.83 1130.69,1069.08 1134.2,1073.28 1137.71,1077.44 1141.22,1081.55 1144.73,1085.61 1148.25,1089.63 1151.76,1093.6 1155.27,1097.52 \n",
" 1158.78,1101.4 1162.29,1105.24 1165.8,1109.03 1169.32,1112.78 1172.83,1116.48 1176.34,1120.14 1179.85,1123.76 1183.36,1127.33 1186.87,1130.86 1190.39,1134.35 \n",
" 1193.9,1137.8 1197.41,1141.21 1200.92,1144.57 1204.43,1147.9 1207.94,1151.18 1211.46,1154.43 1214.97,1157.64 1218.48,1160.81 1221.99,1163.94 1225.5,1167.03 \n",
" 1229.01,1170.09 1232.53,1173.11 1236.04,1176.09 1239.55,1179.04 1243.06,1181.95 1246.57,1184.82 1250.09,1187.66 1253.6,1190.47 1257.11,1193.24 1260.62,1195.98 \n",
" 1264.13,1198.69 1267.64,1201.36 1271.16,1204 1274.67,1206.61 1278.18,1209.18 1281.69,1211.73 1285.2,1214.24 1288.71,1216.73 1292.23,1219.18 1295.74,1221.6 \n",
" 1299.25,1224 1302.76,1226.37 1306.27,1228.7 1309.78,1231.01 1313.3,1233.3 1316.81,1235.55 1320.32,1237.78 1323.83,1239.98 1327.34,1242.15 1330.85,1244.3 \n",
" 1334.37,1246.42 1337.88,1248.52 1341.39,1250.59 1344.9,1252.64 1348.41,1254.66 1351.92,1256.66 1355.44,1258.64 1358.95,1260.59 1362.46,1262.52 1365.97,1264.43 \n",
" 1369.48,1266.31 1372.99,1268.17 1376.51,1270.01 1380.02,1271.83 1383.53,1273.63 1387.04,1275.41 1390.55,1277.16 1394.06,1278.9 1397.58,1280.61 1401.09,1282.31 \n",
" 1404.6,1283.98 1408.11,1285.64 1411.62,1287.27 1415.13,1288.89 1418.65,1290.49 1422.16,1292.07 1425.67,1293.63 1429.18,1295.17 1432.69,1296.7 1436.2,1298.21 \n",
" 1439.72,1299.7 1443.23,1301.17 1446.74,1302.63 1450.25,1304.07 1453.76,1305.5 1457.27,1306.9 1460.79,1308.29 1464.3,1309.67 1467.81,1311.03 1471.32,1312.37 \n",
" 1474.83,1313.7 1478.35,1315.02 1481.86,1316.31 1485.37,1317.6 1488.88,1318.87 1492.39,1320.12 1495.9,1321.36 1499.42,1322.59 1502.93,1323.8 1506.44,1325 \n",
" 1509.95,1326.18 1513.46,1327.35 1516.97,1328.51 1520.49,1329.66 1524,1330.79 1527.51,1331.91 1531.02,1333.01 1534.53,1334.1 1538.04,1335.18 1541.56,1336.25 \n",
" 1545.07,1337.31 1548.58,1338.35 1552.09,1339.38 1555.6,1340.4 1559.11,1341.41 1562.63,1342.4 1566.14,1343.39 1569.65,1344.36 1573.16,1345.32 1576.67,1346.27 \n",
" 1580.18,1347.21 1583.7,1348.14 1587.21,1349.06 1590.72,1349.96 1594.23,1350.86 1597.74,1351.75 1601.25,1352.62 1604.77,1353.49 1608.28,1354.34 1611.79,1355.18 \n",
" 1615.3,1356.02 1618.81,1356.84 1622.32,1357.66 1625.84,1358.46 1629.35,1359.26 1632.86,1360.04 1636.37,1360.82 1639.88,1361.58 1643.39,1362.34 1646.91,1363.09 \n",
" 1650.42,1363.83 1653.93,1364.56 1657.44,1365.28 1660.95,1365.99 1664.46,1366.69 1667.98,1367.39 1671.49,1368.07 1675,1368.75 1678.51,1369.42 1682.02,1370.08 \n",
" 1685.53,1370.73 1689.05,1371.37 1692.56,1372.01 1696.07,1372.64 1699.58,1373.25 1703.09,1373.87 1706.6,1374.47 1710.12,1375.07 1713.63,1375.65 1717.14,1376.23 \n",
" 1720.65,1376.81 1724.16,1377.37 1727.68,1377.93 1731.19,1378.48 1734.7,1379.02 1738.21,1379.56 1741.72,1380.09 1745.23,1380.61 1748.75,1381.13 1752.26,1381.63 \n",
" 1755.77,1382.14 1759.28,1382.63 1762.79,1383.12 1766.3,1383.6 1769.82,1384.07 1773.33,1384.54 1776.84,1385 1780.35,1385.46 1783.86,1385.91 1787.37,1386.35 \n",
" 1790.89,1386.79 1794.4,1387.22 1797.91,1387.64 1801.42,1388.06 1804.93,1388.47 1808.44,1388.88 1811.96,1389.28 1815.47,1389.67 1818.98,1390.06 1822.49,1390.44 \n",
" 1826,1390.82 1829.51,1391.19 1833.03,1391.56 1836.54,1391.92 1840.05,1392.28 1843.56,1392.63 1847.07,1392.97 1850.58,1393.31 1854.1,1393.65 1857.61,1393.98 \n",
" 1861.12,1394.3 1864.63,1394.62 1868.14,1394.94 1871.65,1395.25 1875.17,1395.55 1878.68,1395.86 1882.19,1396.15 1885.7,1396.44 1889.21,1396.73 1892.72,1397.01 \n",
" 1896.24,1397.29 1899.75,1397.56 1903.26,1397.83 1906.77,1398.1 1910.28,1398.36 1913.79,1398.62 1917.31,1398.87 1920.82,1399.12 1924.33,1399.36 1927.84,1399.6 \n",
" 1931.35,1399.84 1934.86,1400.07 1938.38,1400.3 1941.89,1400.53 1945.4,1400.75 1948.91,1400.97 1952.42,1401.18 1955.93,1401.39 1959.45,1401.6 1962.96,1401.8 \n",
" 1966.47,1402 1969.98,1402.2 1973.49,1402.39 1977.01,1402.58 1980.52,1402.77 1984.03,1402.95 1987.54,1403.13 1991.05,1403.31 1994.56,1403.48 1998.08,1403.66 \n",
" 2001.59,1403.82 2005.1,1403.99 2008.61,1404.15 2012.12,1404.31 2015.63,1404.47 2019.15,1404.62 2022.66,1404.77 2026.17,1404.92 2029.68,1405.07 2033.19,1405.21 \n",
" 2036.7,1405.35 2040.22,1405.49 2043.73,1405.62 2047.24,1405.76 2050.75,1405.89 2054.26,1406.01 2057.77,1406.14 2061.29,1406.26 2064.8,1406.38 2068.31,1406.5 \n",
" 2071.82,1406.62 2075.33,1406.73 2078.84,1406.85 2082.36,1406.96 2085.87,1407.06 2089.38,1407.17 2092.89,1407.27 2096.4,1407.38 2099.91,1407.48 2103.43,1407.57 \n",
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" 983.197,851.897 986.708,857.699 990.22,863.463 993.732,869.188 997.243,874.875 1000.76,880.522 1004.27,886.131 1007.78,891.7 1011.29,897.229 1014.8,902.72 \n",
" 1018.31,908.171 1021.83,913.582 1025.34,918.954 1028.85,924.286 1032.36,929.578 1035.87,934.831 1039.38,940.044 1042.9,945.218 1046.41,950.352 1049.92,955.446 \n",
" 1053.43,960.5 1056.94,965.515 1060.45,970.491 1063.97,975.427 1067.48,980.323 1070.99,985.181 1074.5,989.999 1078.01,994.778 1081.52,999.517 1085.04,1004.22 \n",
" 1088.55,1008.88 1092.06,1013.5 1095.57,1018.09 1099.08,1022.63 1102.59,1027.14 1106.11,1031.61 1109.62,1036.04 1113.13,1040.43 1116.64,1044.78 1120.15,1049.1 \n",
" 1123.66,1053.38 1127.18,1057.62 1130.69,1061.82 1134.2,1065.99 1137.71,1070.11 1141.22,1074.2 1144.73,1078.26 1148.25,1082.27 1151.76,1086.25 1155.27,1090.2 \n",
" 1158.78,1094.1 1162.29,1097.97 1165.8,1101.81 1169.32,1105.6 1172.83,1109.37 1176.34,1113.09 1179.85,1116.78 1183.36,1120.44 1186.87,1124.06 1190.39,1127.64 \n",
" 1193.9,1131.19 1197.41,1134.7 1200.92,1138.18 1204.43,1141.63 1207.94,1145.04 1211.46,1148.42 1214.97,1151.76 1218.48,1155.07 1221.99,1158.34 1225.5,1161.58 \n",
" 1229.01,1164.79 1232.53,1167.96 1236.04,1171.11 1239.55,1174.21 1243.06,1177.29 1246.57,1180.33 1250.09,1183.34 1253.6,1186.32 1257.11,1189.27 1260.62,1192.18 \n",
" 1264.13,1195.07 1267.64,1197.92 1271.16,1200.74 1274.67,1203.53 1278.18,1206.29 1281.69,1209.02 1285.2,1211.72 1288.71,1214.38 1292.23,1217.02 1295.74,1219.63 \n",
" 1299.25,1222.21 1302.76,1224.75 1306.27,1227.27 1309.78,1229.76 1313.3,1232.22 1316.81,1234.66 1320.32,1237.06 1323.83,1239.44 1327.34,1241.79 1330.85,1244.11 \n",
" 1334.37,1246.4 1337.88,1248.66 1341.39,1250.9 1344.9,1253.11 1348.41,1255.3 1351.92,1257.46 1355.44,1259.59 1358.95,1261.69 1362.46,1263.77 1365.97,1265.83 \n",
" 1369.48,1267.86 1372.99,1269.86 1376.51,1271.84 1380.02,1273.79 1383.53,1275.72 1387.04,1277.62 1390.55,1279.51 1394.06,1281.36 1397.58,1283.2 1401.09,1285.01 \n",
" 1404.6,1286.79 1408.11,1288.56 1411.62,1290.3 1415.13,1292.01 1418.65,1293.71 1422.16,1295.38 1425.67,1297.04 1429.18,1298.67 1432.69,1300.28 1436.2,1301.86 \n",
" 1439.72,1303.43 1443.23,1304.98 1446.74,1306.5 1450.25,1308.01 1453.76,1309.49 1457.27,1310.96 1460.79,1312.4 1464.3,1313.83 1467.81,1315.24 1471.32,1316.62 \n",
" 1474.83,1317.99 1478.35,1319.34 1481.86,1320.67 1485.37,1321.99 1488.88,1323.28 1492.39,1324.56 1495.9,1325.82 1499.42,1327.07 1502.93,1328.29 1506.44,1329.5 \n",
" 1509.95,1330.69 1513.46,1331.87 1516.97,1333.03 1520.49,1334.17 1524,1335.3 1527.51,1336.41 1531.02,1337.51 1534.53,1338.59 1538.04,1339.65 1541.56,1340.7 \n",
" 1545.07,1341.74 1548.58,1342.76 1552.09,1343.77 1555.6,1344.76 1559.11,1345.74 1562.63,1346.7 1566.14,1347.65 1569.65,1348.59 1573.16,1349.52 1576.67,1350.43 \n",
" 1580.18,1351.32 1583.7,1352.21 1587.21,1353.08 1590.72,1353.94 1594.23,1354.79 1597.74,1355.63 1601.25,1356.45 1604.77,1357.26 1608.28,1358.06 1611.79,1358.85 \n",
" 1615.3,1359.63 1618.81,1360.39 1622.32,1361.15 1625.84,1361.89 1629.35,1362.63 1632.86,1363.35 1636.37,1364.06 1639.88,1364.77 1643.39,1365.46 1646.91,1366.14 \n",
" 1650.42,1366.81 1653.93,1367.48 1657.44,1368.13 1660.95,1368.77 1664.46,1369.41 1667.98,1370.03 1671.49,1370.65 1675,1371.26 1678.51,1371.86 1682.02,1372.45 \n",
" 1685.53,1373.03 1689.05,1373.6 1692.56,1374.16 1696.07,1374.72 1699.58,1375.27 1703.09,1375.81 1706.6,1376.34 1710.12,1376.87 1713.63,1377.39 1717.14,1377.9 \n",
" 1720.65,1378.4 1724.16,1378.89 1727.68,1379.38 1731.19,1379.86 1734.7,1380.34 1738.21,1380.8 1741.72,1381.26 1745.23,1381.72 1748.75,1382.17 1752.26,1382.61 \n",
" 1755.77,1383.04 1759.28,1383.47 1762.79,1383.89 1766.3,1384.31 1769.82,1384.72 1773.33,1385.12 1776.84,1385.52 1780.35,1385.91 1783.86,1386.3 1787.37,1386.68 \n",
" 1790.89,1387.06 1794.4,1387.43 1797.91,1387.79 1801.42,1388.15 1804.93,1388.51 1808.44,1388.85 1811.96,1389.2 1815.47,1389.54 1818.98,1389.87 1822.49,1390.2 \n",
" 1826,1390.53 1829.51,1390.85 1833.03,1391.16 1836.54,1391.48 1840.05,1391.78 1843.56,1392.08 1847.07,1392.38 1850.58,1392.68 1854.1,1392.97 1857.61,1393.25 \n",
" 1861.12,1393.53 1864.63,1393.81 1868.14,1394.08 1871.65,1394.35 1875.17,1394.62 1878.68,1394.88 1882.19,1395.14 1885.7,1395.39 1889.21,1395.64 1892.72,1395.89 \n",
" 1896.24,1396.13 1899.75,1396.37 1903.26,1396.61 1906.77,1396.84 1910.28,1397.07 1913.79,1397.3 1917.31,1397.52 1920.82,1397.74 1924.33,1397.96 1927.84,1398.17 \n",
" 1931.35,1398.38 1934.86,1398.59 1938.38,1398.8 1941.89,1399 1945.4,1399.2 1948.91,1399.39 1952.42,1399.59 1955.93,1399.78 1959.45,1399.97 1962.96,1400.15 \n",
" 1966.47,1400.33 1969.98,1400.51 1973.49,1400.69 1977.01,1400.86 1980.52,1401.04 1984.03,1401.21 1987.54,1401.37 1991.05,1401.54 1994.56,1401.7 1998.08,1401.86 \n",
" 2001.59,1402.02 2005.1,1402.17 2008.61,1402.33 2012.12,1402.48 2015.63,1402.63 2019.15,1402.78 2022.66,1402.92 2026.17,1403.06 2029.68,1403.2 2033.19,1403.34 \n",
" 2036.7,1403.48 2040.22,1403.61 2043.73,1403.75 2047.24,1403.88 2050.75,1404 2054.26,1404.13 2057.77,1404.26 2061.29,1404.38 2064.8,1404.5 2068.31,1404.62 \n",
" 2071.82,1404.74 2075.33,1404.86 2078.84,1404.97 2082.36,1405.08 2085.87,1405.19 2089.38,1405.3 2092.89,1405.41 2096.4,1405.52 2099.91,1405.62 2103.43,1405.73 \n",
" 2106.94,1405.83 2110.45,1405.93 2113.96,1406.03 2117.47,1406.12 2120.98,1406.22 2124.5,1406.31 2128.01,1406.41 2131.52,1406.5 2135.03,1406.59 2138.54,1406.68 \n",
" 2142.05,1406.77 2145.57,1406.85 2149.08,1406.94 2152.59,1407.02 2156.1,1407.1 2159.61,1407.18 2163.12,1407.26 2166.64,1407.34 2170.15,1407.42 2173.66,1407.5 \n",
" 2177.17,1407.57 2180.68,1407.65 2184.19,1407.72 2187.71,1407.79 2191.22,1407.86 2194.73,1407.93 2198.24,1408 2201.75,1408.07 2205.26,1408.14 2208.78,1408.2 \n",
" 2212.29,1408.27 2215.8,1408.33 2219.31,1408.39 2222.82,1408.45 2226.34,1408.51 2229.85,1408.57 2233.36,1408.63 2236.87,1408.69 2240.38,1408.75 2243.89,1408.8 \n",
" 2247.41,1408.86 2250.92,1408.91 2254.43,1408.97 2257.94,1409.02 2261.45,1409.07 2264.96,1409.12 2268.48,1409.17 2271.99,1409.22 2275.5,1409.27 2279.01,1409.32 \n",
" 2282.52,1409.37 2286.03,1409.41 2289.55,1409.46 2293.06,1409.5 2296.57,1409.55 2300.08,1409.59 2303.59,1409.63 2307.1,1409.68 2310.62,1409.72 2314.13,1409.76 \n",
" 2317.64,1409.8 2321.15,1409.84 2324.66,1409.88 2328.17,1409.92 2331.69,1409.95 2335.2,1409.99 2338.71,1410.03 2342.22,1410.06 2345.73,1410.1 2349.24,1410.13 \n",
" 2352.76,1410.17 \n",
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" 280.859,1342.62 284.37,1329.73 287.882,1316.01 291.394,1301.52 294.905,1286.34 298.417,1270.52 301.929,1254.13 305.44,1237.26 308.952,1219.94 312.464,1202.25 \n",
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" 1053.43,961.475 1056.94,966.556 1060.45,971.595 1063.97,976.591 1067.48,981.545 1070.99,986.457 1074.5,991.326 1078.01,996.152 1081.52,1000.94 1085.04,1005.68 \n",
" 1088.55,1010.38 1092.06,1015.04 1095.57,1019.65 1099.08,1024.23 1102.59,1028.76 1106.11,1033.25 1109.62,1037.7 1113.13,1042.11 1116.64,1046.47 1120.15,1050.8 \n",
" 1123.66,1055.08 1127.18,1059.33 1130.69,1063.53 1134.2,1067.7 1137.71,1071.82 1141.22,1075.9 1144.73,1079.94 1148.25,1083.95 1151.76,1087.91 1155.27,1091.84 \n",
" 1158.78,1095.72 1162.29,1099.57 1165.8,1103.38 1169.32,1107.15 1172.83,1110.89 1176.34,1114.58 1179.85,1118.24 1183.36,1121.86 1186.87,1125.44 1190.39,1128.99 \n",
" 1193.9,1132.5 1197.41,1135.98 1200.92,1139.42 1204.43,1142.82 1207.94,1146.19 1211.46,1149.52 1214.97,1152.81 1218.48,1156.08 1221.99,1159.3 1225.5,1162.5 \n",
" 1229.01,1165.66 1232.53,1168.78 1236.04,1171.88 1239.55,1174.94 1243.06,1177.96 1246.57,1180.96 1250.09,1183.92 1253.6,1186.85 1257.11,1189.74 1260.62,1192.61 \n",
" 1264.13,1195.44 1267.64,1198.25 1271.16,1201.02 1274.67,1203.76 1278.18,1206.47 1281.69,1209.15 1285.2,1211.8 1288.71,1214.43 1292.23,1217.02 1295.74,1219.58 \n",
" 1299.25,1222.12 1302.76,1224.62 1306.27,1227.1 1309.78,1229.55 1313.3,1231.97 1316.81,1234.36 1320.32,1236.73 1323.83,1239.07 1327.34,1241.38 1330.85,1243.66 \n",
" 1334.37,1245.92 1337.88,1248.16 1341.39,1250.36 1344.9,1252.54 1348.41,1254.7 1351.92,1256.83 1355.44,1258.93 1358.95,1261.01 1362.46,1263.07 1365.97,1265.1 \n",
" 1369.48,1267.1 1372.99,1269.09 1376.51,1271.04 1380.02,1272.98 1383.53,1274.89 1387.04,1276.78 1390.55,1278.64 1394.06,1280.49 1397.58,1282.31 1401.09,1284.11 \n",
" 1404.6,1285.88 1408.11,1287.64 1411.62,1289.37 1415.13,1291.08 1418.65,1292.77 1422.16,1294.44 1425.67,1296.09 1429.18,1297.71 1432.69,1299.32 1436.2,1300.91 \n",
" 1439.72,1302.47 1443.23,1304.02 1446.74,1305.55 1450.25,1307.06 1453.76,1308.54 1457.27,1310.01 1460.79,1311.46 1464.3,1312.9 1467.81,1314.31 1471.32,1315.71 \n",
" 1474.83,1317.08 1478.35,1318.44 1481.86,1319.78 1485.37,1321.11 1488.88,1322.41 1492.39,1323.7 1495.9,1324.98 1499.42,1326.23 1502.93,1327.47 1506.44,1328.69 \n",
" 1509.95,1329.9 1513.46,1331.09 1516.97,1332.27 1520.49,1333.43 1524,1334.57 1527.51,1335.7 1531.02,1336.81 1534.53,1337.91 1538.04,1338.99 1541.56,1340.06 \n",
" 1545.07,1341.11 1548.58,1342.15 1552.09,1343.18 1555.6,1344.19 1559.11,1345.18 1562.63,1346.17 1566.14,1347.14 1569.65,1348.09 1573.16,1349.04 1576.67,1349.97 \n",
" 1580.18,1350.88 1583.7,1351.79 1587.21,1352.68 1590.72,1353.56 1594.23,1354.43 1597.74,1355.28 1601.25,1356.12 1604.77,1356.95 1608.28,1357.77 1611.79,1358.58 \n",
" 1615.3,1359.38 1618.81,1360.16 1622.32,1360.93 1625.84,1361.7 1629.35,1362.45 1632.86,1363.19 1636.37,1363.92 1639.88,1364.64 1643.39,1365.35 1646.91,1366.05 \n",
" 1650.42,1366.74 1653.93,1367.42 1657.44,1368.09 1660.95,1368.75 1664.46,1369.4 1667.98,1370.04 1671.49,1370.67 1675,1371.29 1678.51,1371.91 1682.02,1372.51 \n",
" 1685.53,1373.11 1689.05,1373.69 1692.56,1374.27 1696.07,1374.84 1699.58,1375.4 1703.09,1375.95 1706.6,1376.5 1710.12,1377.04 1713.63,1377.57 1717.14,1378.09 \n",
" 1720.65,1378.6 1724.16,1379.11 1727.68,1379.61 1731.19,1380.1 1734.7,1380.58 1738.21,1381.06 1741.72,1381.53 1745.23,1381.99 1748.75,1382.44 1752.26,1382.89 \n",
" 1755.77,1383.34 1759.28,1383.77 1762.79,1384.2 1766.3,1384.62 1769.82,1385.04 1773.33,1385.45 1776.84,1385.85 1780.35,1386.25 1783.86,1386.64 1787.37,1387.03 \n",
" 1790.89,1387.41 1794.4,1387.78 1797.91,1388.15 1801.42,1388.52 1804.93,1388.87 1808.44,1389.23 1811.96,1389.57 1815.47,1389.92 1818.98,1390.25 1822.49,1390.59 \n",
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" 912.963,726.897 916.475,733.369 919.986,739.817 923.498,746.238 927.01,752.632 930.521,758.999 934.033,765.337 937.545,771.646 941.056,777.926 944.568,784.175 \n",
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" 1053.43,960.51 1056.94,965.586 1060.45,970.623 1063.97,975.62 1067.48,980.577 1070.99,985.493 1074.5,990.369 1078.01,995.205 1081.52,1000 1085.04,1004.76 \n",
" 1088.55,1009.47 1092.06,1014.15 1095.57,1018.78 1099.08,1023.38 1102.59,1027.93 1106.11,1032.45 1109.62,1036.92 1113.13,1041.36 1116.64,1045.75 1120.15,1050.11 \n",
" 1123.66,1054.42 1127.18,1058.7 1130.69,1062.93 1134.2,1067.13 1137.71,1071.28 1141.22,1075.4 1144.73,1079.48 1148.25,1083.52 1151.76,1087.52 1155.27,1091.48 \n",
" 1158.78,1095.41 1162.29,1099.29 1165.8,1103.14 1169.32,1106.95 1172.83,1110.72 1176.34,1114.45 1179.85,1118.14 1183.36,1121.8 1186.87,1125.42 1190.39,1129 \n",
" 1193.9,1132.55 1197.41,1136.06 1200.92,1139.53 1204.43,1142.97 1207.94,1146.37 1211.46,1149.73 1214.97,1153.06 1218.48,1156.35 1221.99,1159.61 1225.5,1162.83 \n",
" 1229.01,1166.02 1232.53,1169.18 1236.04,1172.29 1239.55,1175.38 1243.06,1178.43 1246.57,1181.45 1250.09,1184.43 1253.6,1187.38 1257.11,1190.3 1260.62,1193.18 \n",
" 1264.13,1196.03 1267.64,1198.85 1271.16,1201.64 1274.67,1204.39 1278.18,1207.12 1281.69,1209.81 1285.2,1212.47 1288.71,1215.1 1292.23,1217.7 1295.74,1220.27 \n",
" 1299.25,1222.81 1302.76,1225.33 1306.27,1227.81 1309.78,1230.26 1313.3,1232.68 1316.81,1235.07 1320.32,1237.44 1323.83,1239.78 1327.34,1242.09 1330.85,1244.37 \n",
" 1334.37,1246.62 1337.88,1248.85 1341.39,1251.05 1344.9,1253.22 1348.41,1255.37 1351.92,1257.49 1355.44,1259.59 1358.95,1261.66 1362.46,1263.7 1365.97,1265.72 \n",
" 1369.48,1267.71 1372.99,1269.68 1376.51,1271.63 1380.02,1273.55 1383.53,1275.45 1387.04,1277.32 1390.55,1279.17 1394.06,1281 1397.58,1282.8 1401.09,1284.58 \n",
" 1404.6,1286.34 1408.11,1288.08 1411.62,1289.79 1415.13,1291.49 1418.65,1293.16 1422.16,1294.81 1425.67,1296.44 1429.18,1298.05 1432.69,1299.63 1436.2,1301.2 \n",
" 1439.72,1302.75 1443.23,1304.28 1446.74,1305.79 1450.25,1307.28 1453.76,1308.74 1457.27,1310.2 1460.79,1311.63 1464.3,1313.04 1467.81,1314.43 1471.32,1315.81 \n",
" 1474.83,1317.17 1478.35,1318.51 1481.86,1319.84 1485.37,1321.14 1488.88,1322.43 1492.39,1323.7 1495.9,1324.96 1499.42,1326.2 1502.93,1327.42 1506.44,1328.63 \n",
" 1509.95,1329.82 1513.46,1330.99 1516.97,1332.15 1520.49,1333.3 1524,1334.42 1527.51,1335.54 1531.02,1336.64 1534.53,1337.72 1538.04,1338.79 1541.56,1339.85 \n",
" 1545.07,1340.89 1548.58,1341.91 1552.09,1342.93 1555.6,1343.93 1559.11,1344.91 1562.63,1345.89 1566.14,1346.85 1569.65,1347.79 1573.16,1348.73 1576.67,1349.65 \n",
" 1580.18,1350.56 1583.7,1351.46 1587.21,1352.34 1590.72,1353.21 1594.23,1354.07 1597.74,1354.92 1601.25,1355.76 1604.77,1356.59 1608.28,1357.4 1611.79,1358.2 \n",
" 1615.3,1359 1618.81,1359.78 1622.32,1360.55 1625.84,1361.31 1629.35,1362.06 1632.86,1362.8 1636.37,1363.53 1639.88,1364.24 1643.39,1364.95 1646.91,1365.65 \n",
" 1650.42,1366.34 1653.93,1367.02 1657.44,1367.69 1660.95,1368.35 1664.46,1369 1667.98,1369.65 1671.49,1370.28 1675,1370.9 1678.51,1371.52 1682.02,1372.13 \n",
" 1685.53,1372.72 1689.05,1373.31 1692.56,1373.9 1696.07,1374.47 1699.58,1375.03 1703.09,1375.59 1706.6,1376.14 1710.12,1376.68 1713.63,1377.22 1717.14,1377.74 \n",
" 1720.65,1378.26 1724.16,1378.77 1727.68,1379.28 1731.19,1379.77 1734.7,1380.26 1738.21,1380.75 1741.72,1381.22 1745.23,1381.69 1748.75,1382.15 1752.26,1382.61 \n",
" 1755.77,1383.06 1759.28,1383.5 1762.79,1383.94 1766.3,1384.37 1769.82,1384.79 1773.33,1385.21 1776.84,1385.62 1780.35,1386.02 1783.86,1386.42 1787.37,1386.81 \n",
" 1790.89,1387.2 1794.4,1387.58 1797.91,1387.96 1801.42,1388.33 1804.93,1388.7 1808.44,1389.06 1811.96,1389.41 1815.47,1389.76 1818.98,1390.1 1822.49,1390.44 \n",
" 1826,1390.78 1829.51,1391.11 1833.03,1391.43 1836.54,1391.75 1840.05,1392.07 1843.56,1392.38 1847.07,1392.68 1850.58,1392.98 1854.1,1393.28 1857.61,1393.57 \n",
" 1861.12,1393.86 1864.63,1394.14 1868.14,1394.42 1871.65,1394.69 1875.17,1394.97 1878.68,1395.23 1882.19,1395.49 1885.7,1395.75 1889.21,1396.01 1892.72,1396.26 \n",
" 1896.24,1396.51 1899.75,1396.75 1903.26,1396.99 1906.77,1397.22 1910.28,1397.46 1913.79,1397.69 1917.31,1397.91 1920.82,1398.13 1924.33,1398.35 1927.84,1398.57 \n",
" 1931.35,1398.78 1934.86,1398.99 1938.38,1399.19 1941.89,1399.39 1945.4,1399.59 1948.91,1399.79 1952.42,1399.98 1955.93,1400.17 1959.45,1400.36 1962.96,1400.54 \n",
" 1966.47,1400.73 1969.98,1400.9 1973.49,1401.08 1977.01,1401.25 1980.52,1401.42 1984.03,1401.59 1987.54,1401.76 1991.05,1401.92 1994.56,1402.08 1998.08,1402.24 \n",
" 2001.59,1402.39 2005.1,1402.55 2008.61,1402.7 2012.12,1402.84 2015.63,1402.99 2019.15,1403.13 2022.66,1403.27 2026.17,1403.41 2029.68,1403.55 2033.19,1403.68 \n",
" 2036.7,1403.82 2040.22,1403.95 2043.73,1404.08 2047.24,1404.2 2050.75,1404.33 2054.26,1404.45 2057.77,1404.57 2061.29,1404.69 2064.8,1404.8 2068.31,1404.92 \n",
" 2071.82,1405.03 2075.33,1405.14 2078.84,1405.25 2082.36,1405.36 2085.87,1405.47 2089.38,1405.57 2092.89,1405.67 2096.4,1405.78 2099.91,1405.88 2103.43,1405.97 \n",
" 2106.94,1406.07 2110.45,1406.17 2113.96,1406.26 2117.47,1406.35 2120.98,1406.44 2124.5,1406.53 2128.01,1406.62 2131.52,1406.7 2135.03,1406.79 2138.54,1406.87 \n",
" 2142.05,1406.95 2145.57,1407.04 2149.08,1407.11 2152.59,1407.19 2156.1,1407.27 2159.61,1407.35 2163.12,1407.42 2166.64,1407.49 2170.15,1407.57 2173.66,1407.64 \n",
" 2177.17,1407.71 2180.68,1407.78 2184.19,1407.84 2187.71,1407.91 2191.22,1407.98 2194.73,1408.04 2198.24,1408.1 2201.75,1408.17 2205.26,1408.23 2208.78,1408.29 \n",
" 2212.29,1408.35 2215.8,1408.41 2219.31,1408.46 2222.82,1408.52 2226.34,1408.58 2229.85,1408.63 2233.36,1408.69 2236.87,1408.74 2240.38,1408.79 2243.89,1408.84 \n",
" 2247.41,1408.89 2250.92,1408.94 2254.43,1408.99 2257.94,1409.04 2261.45,1409.09 2264.96,1409.14 2268.48,1409.18 2271.99,1409.23 2275.5,1409.27 2279.01,1409.31 \n",
" 2282.52,1409.36 2286.03,1409.4 2289.55,1409.44 2293.06,1409.48 2296.57,1409.52 2300.08,1409.56 2303.59,1409.6 2307.1,1409.64 2310.62,1409.68 2314.13,1409.72 \n",
" 2317.64,1409.75 2321.15,1409.79 2324.66,1409.82 2328.17,1409.86 2331.69,1409.89 2335.2,1409.93 2338.71,1409.96 2342.22,1409.99 2345.73,1410.03 2349.24,1410.06 \n",
" 2352.76,1410.09 \n",
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" 1158.78,1095.45 1162.29,1099.32 1165.8,1103.16 1169.32,1106.96 1172.83,1110.72 1176.34,1114.44 1179.85,1118.12 1183.36,1121.77 1186.87,1125.38 1190.39,1128.95 \n",
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" 1334.37,1246.49 1337.88,1248.72 1341.39,1250.92 1344.9,1253.1 1348.41,1255.25 1351.92,1257.38 1355.44,1259.48 1358.95,1261.55 1362.46,1263.6 1365.97,1265.63 \n",
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" 1509.95,1329.9 1513.46,1331.07 1516.97,1332.23 1520.49,1333.38 1524,1334.51 1527.51,1335.62 1531.02,1336.72 1534.53,1337.81 1538.04,1338.88 1541.56,1339.94 \n",
" 1545.07,1340.98 1548.58,1342.01 1552.09,1343.02 1555.6,1344.02 1559.11,1345.01 1562.63,1345.98 1566.14,1346.94 1569.65,1347.89 1573.16,1348.82 1576.67,1349.74 \n",
" 1580.18,1350.65 1583.7,1351.55 1587.21,1352.43 1590.72,1353.3 1594.23,1354.16 1597.74,1355.01 1601.25,1355.84 1604.77,1356.67 1608.28,1357.48 1611.79,1358.28 \n",
" 1615.3,1359.07 1618.81,1359.85 1622.32,1360.62 1625.84,1361.38 1629.35,1362.13 1632.86,1362.87 1636.37,1363.59 1639.88,1364.31 1643.39,1365.02 1646.91,1365.71 \n",
" 1650.42,1366.4 1653.93,1367.08 1657.44,1367.75 1660.95,1368.41 1664.46,1369.06 1667.98,1369.7 1671.49,1370.33 1675,1370.95 1678.51,1371.56 1682.02,1372.17 \n",
" 1685.53,1372.76 1689.05,1373.35 1692.56,1373.93 1696.07,1374.5 1699.58,1375.07 1703.09,1375.62 1706.6,1376.17 1710.12,1376.71 1713.63,1377.24 1717.14,1377.76 \n",
" 1720.65,1378.28 1724.16,1378.79 1727.68,1379.29 1731.19,1379.79 1734.7,1380.27 1738.21,1380.76 1741.72,1381.23 1745.23,1381.7 1748.75,1382.16 1752.26,1382.61 \n",
" 1755.77,1383.06 1759.28,1383.5 1762.79,1383.93 1766.3,1384.36 1769.82,1384.78 1773.33,1385.2 1776.84,1385.61 1780.35,1386.01 1783.86,1386.41 1787.37,1386.8 \n",
" 1790.89,1387.19 1794.4,1387.57 1797.91,1387.94 1801.42,1388.31 1804.93,1388.67 1808.44,1389.03 1811.96,1389.39 1815.47,1389.74 1818.98,1390.08 1822.49,1390.42 \n",
" 1826,1390.75 1829.51,1391.08 1833.03,1391.4 1836.54,1391.72 1840.05,1392.03 1843.56,1392.34 1847.07,1392.65 1850.58,1392.95 1854.1,1393.24 1857.61,1393.54 \n",
" 1861.12,1393.82 1864.63,1394.1 1868.14,1394.38 1871.65,1394.66 1875.17,1394.93 1878.68,1395.19 1882.19,1395.46 1885.7,1395.71 1889.21,1395.97 1892.72,1396.22 \n",
" 1896.24,1396.47 1899.75,1396.71 1903.26,1396.95 1906.77,1397.19 1910.28,1397.42 1913.79,1397.65 1917.31,1397.87 1920.82,1398.09 1924.33,1398.31 1927.84,1398.53 \n",
" 1931.35,1398.74 1934.86,1398.95 1938.38,1399.15 1941.89,1399.36 1945.4,1399.56 1948.91,1399.75 1952.42,1399.95 1955.93,1400.14 1959.45,1400.32 1962.96,1400.51 \n",
" 1966.47,1400.69 1969.98,1400.87 1973.49,1401.05 1977.01,1401.22 1980.52,1401.39 1984.03,1401.56 1987.54,1401.72 1991.05,1401.89 1994.56,1402.05 1998.08,1402.21 \n",
" 2001.59,1402.36 2005.1,1402.52 2008.61,1402.67 2012.12,1402.81 2015.63,1402.96 2019.15,1403.11 2022.66,1403.25 2026.17,1403.39 2029.68,1403.52 2033.19,1403.66 \n",
" 2036.7,1403.79 2040.22,1403.92 2043.73,1404.05 2047.24,1404.18 2050.75,1404.31 2054.26,1404.43 2057.77,1404.55 2061.29,1404.67 2064.8,1404.79 2068.31,1404.9 \n",
" 2071.82,1405.02 2075.33,1405.13 2078.84,1405.24 2082.36,1405.35 2085.87,1405.45 2089.38,1405.56 2092.89,1405.66 2096.4,1405.76 2099.91,1405.86 2103.43,1405.96 \n",
" 2106.94,1406.06 2110.45,1406.16 2113.96,1406.25 2117.47,1406.34 2120.98,1406.43 2124.5,1406.52 2128.01,1406.61 2131.52,1406.7 2135.03,1406.78 2138.54,1406.87 \n",
" 2142.05,1406.95 2145.57,1407.03 2149.08,1407.11 2152.59,1407.19 2156.1,1407.27 2159.61,1407.34 2163.12,1407.42 2166.64,1407.49 2170.15,1407.57 2173.66,1407.64 \n",
" 2177.17,1407.71 2180.68,1407.78 2184.19,1407.85 2187.71,1407.91 2191.22,1407.98 2194.73,1408.04 2198.24,1408.11 2201.75,1408.17 2205.26,1408.23 2208.78,1408.29 \n",
" 2212.29,1408.35 2215.8,1408.41 2219.31,1408.47 2222.82,1408.53 2226.34,1408.58 2229.85,1408.64 2233.36,1408.69 2236.87,1408.75 2240.38,1408.8 2243.89,1408.85 \n",
" 2247.41,1408.9 2250.92,1408.95 2254.43,1409 2257.94,1409.05 2261.45,1409.1 2264.96,1409.15 2268.48,1409.19 2271.99,1409.24 2275.5,1409.28 2279.01,1409.33 \n",
" 2282.52,1409.37 2286.03,1409.41 2289.55,1409.45 2293.06,1409.5 2296.57,1409.54 2300.08,1409.58 2303.59,1409.62 2307.1,1409.65 2310.62,1409.69 2314.13,1409.73 \n",
" 2317.64,1409.77 2321.15,1409.8 2324.66,1409.84 2328.17,1409.87 2331.69,1409.91 2335.2,1409.94 2338.71,1409.97 2342.22,1410.01 2345.73,1410.04 2349.24,1410.07 \n",
" 2352.76,1410.1 \n",
" \"/>\n",
"<polyline clip-path=\"url(#clip1502)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
" 245.742,1411.98 248.314,1411.53 250.886,1410.21 253.458,1408.06 256.03,1405.11 258.602,1401.4 261.174,1396.97 263.746,1391.84 266.318,1386.06 278.21,1351.68 \n",
" 290.101,1306.8 301.992,1253.92 313.884,1195.22 337.666,1067.48 361.449,935.312 385.232,806.98 409.015,688.093 432.797,582.207 456.58,491.335 470.872,444.345 \n",
" 485.165,403.139 499.457,367.66 513.749,337.784 528.042,313.328 542.334,294.069 556.627,279.749 570.919,270.088 595.087,263.518 619.255,267.94 643.423,281.867 \n",
" 667.591,303.856 693.845,335.284 720.098,372.99 746.351,415.508 772.605,461.536 821.011,551.889 869.418,644.636 921.315,742.012 973.211,833.915 1033.34,930.88 \n",
" 1093.47,1016.23 1146.9,1082.11 1200.34,1138.95 1247.19,1181.87 1294.05,1218.89 1345.14,1253.22 1396.22,1282.01 1447.61,1306.13 1499,1326.09 1612.64,1358.49 \n",
" 1720.76,1378.33 1833.19,1391.43 1927.82,1398.52 2043.36,1404.02 2151.97,1407.16 2325.98,1409.85 2352.76,1410.1 \n",
" \"/>\n",
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"M1822.71 856.524 L2280.76 856.524 L2280.76 130.764 L1822.71 130.764 Z\n",
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" 1822.71,856.524 2280.76,856.524 2280.76,130.764 1822.71,130.764 1822.71,856.524 \n",
" \"/>\n",
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"</g>\n",
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" 1846.71,433.164 1990.71,433.164 \n",
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"</g>\n",
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"</g>\n",
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" 1846.71,614.604 1990.71,614.604 \n",
" \"/>\n",
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"<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2014.71, 632.104)\" x=\"2014.71\" y=\"632.104\">STO-8G</text>\n",
"</g>\n",
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" 1846.71,675.084 1990.71,675.084 \n",
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"</g>\n",
"</svg>\n"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"plot(xlims=(0,6), xlabel=\"\\$r\\$\", ylabel=\"\\$r^2 |\\\\psi_\\\\mathrm{1s}(r,\\\\theta,\\\\phi)|^2\\$\")\n",
"\n",
"for i in 1:10\n",
" plot!(0:0.01:6, r->data2ρ(αs[i],Cs[i],r,0,0), label=\"STO-$(i)G\")\n",
"end\n",
"\n",
"plot!(r->ρ(r,0,0), label=\"exact\", lc=\"#000000\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 参考文献\n",
"\n",
"- [J.M.ティッセン著『計算物理学』(丸善出版, 2012) 3.1](https://www.maruzen-publishing.co.jp/item/b294596.html)\n",
"- [A.ザボ, Ｎ.Ｓ.オストランド著『新しい量子化学 上 電子構造の理論入門』(東京大学出版会, 1987) 3.2.1, 3.2.2](http://www.utp.or.jp/book/b302128.html)\n",
"- [原田義也『量子化学 下巻』(裳華房, 2007) §A2.3](https://www.shokabo.co.jp/mybooks/ISBN978-4-7853-3074-3.htm)\n",
"- [高柳和夫『朝倉物理学大系 11 原子分子物理学』(2000, 朝倉書店) pp.11-22](http://www.asakura.co.jp/books/isbn/978-4-254-13681-4/)\n",
"- [河合光路, 吉田思郎『朝倉物理学大系 19 原子核反応論』(朝倉書店, 2002) pp.8-12,25-38](http://www.asakura.co.jp/books/isbn/978-4-254-13689-0/)\n",
"- [Frank Jensen『Introduction to Computational Chemistry, 2nd Edition』(John Wiley & Sons, 2007) pp.14-17](https://www.wiley.com/en-us/Introduction+to+Computational+Chemistry%2C+2nd+Edition-p-9780470058046)\n",
"- [IUPAC GreenBook 3.9.2](https://unit.aist.go.jp/nmij/public/report/translation/IUPAC/)\n",
"- HIYAMA, E., KINO, Y., KAMIMURA, M. (2003) Prog. Part. Nucl. Phys. 51.1: 223-307. https://doi.org/10.1016/S0146-6410(03)90015-9\n",
"- HIYAMA, E., KAMIMURA, M. (2018) Front. Phys. 13, 132106 https://doi.org/10.1007/s11467-018-0828-5\n",
"- https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/\n",
"- https://julianlsolvers.github.io/Optim.jl/stable/#algo/nelder_mead/"
]
}
],
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