From b9a1b73ebf9586bb9bdeee1d9b2649523e4d38bd Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sun, 16 Feb 2020 14:26:23 +0100 Subject: [PATCH] add eHF results to SI --- Manuscript/SI/eDFT-SI.tex | 78 +- Notebooks/eDFT_FUEG.nb | 3497 +++++++++++++++++++------------------ 2 files changed, 1824 insertions(+), 1751 deletions(-) diff --git a/Manuscript/SI/eDFT-SI.tex b/Manuscript/SI/eDFT-SI.tex index f232bf4..c624ded 100644 --- a/Manuscript/SI/eDFT-SI.tex +++ b/Manuscript/SI/eDFT-SI.tex @@ -579,10 +579,10 @@ The numerical values of the correlation energy for various $R$ are reported in T Based on these highly-accurate calculations, one can write down, for each state, an accurate analytical expression of the reduced correlation energy \cite{Loos_2013a, Loos_2014a} via the following Pad\'e approximant \begin{equation} \label{eq:ec} - \e{c}{(I)}(n) = \frac{c_1^{(I)}\,n}{n + c_2^{(I)} \sqrt{n} + c_3^{(I)}}, + \e{c}{(I)}(n) = \frac{a_1^{(I)}\,n}{n + a_2^{(I)} \sqrt{n} + a_3^{(I)}}, \end{equation} -where $c_2^{(I)}$ and $c_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript. -The value of $c_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a} +where $a_2^{(I)}$ and $a_3^{(I)}$ are state-specific fitting parameters, which are provided in Table I of the manuscript. +The value of $a_1^{(I)}$ is obtained via the exact high-density expansion of the correlation energy. \cite{Loos_2013a, Loos_2014a} Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside the data gathered in Table \ref{tab:Ref}. @@ -647,6 +647,18 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside \\ TDLDA & & $\Ex{(1)}$ & 0.0015 & 0.0006 & -0.0018 & -0.0106 & -0.0370 & -0.0518 & \tabc{DNC} \\ \\ + eHF & $(0,0)$ & $\E{(0)}$ & 0.0137 & 0.0131 & 0.0119 & 0.0098 & 0.0067 & 0.0033 & 0.0010 \\ + & & $\E{(1)}$ & 0.0685 & 0.0678 & 0.0664 & 0.0637 & 0.0588 & 0.0500 & 0.0372 \\ + & & $\E{(2)}$ & -0.0052 & -0.0055 & -0.0058 & -0.0055 & -0.0022 & 0.0086 & 0.0271 \\ + & & $\Ex{(1)}$ & 0.1082 & 0.1069 & 0.1044 & 0.0998 & 0.0911 & 0.0736 & \tabc{DNC} \\ + & & $\Ex{(2)}$ & 0.0345 & 0.0336 & 0.0322 & 0.0306 & 0.0302 & 0.0321 & \tabc{DNC} \\ + \\ + eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.0565 & 0.0557 & 0.0540 & 0.0509 & 0.0455 & 0.0372 & 0.0269 \\ + & & $\E{(1)}$ & 0.0560 & 0.0553 & 0.0538 & 0.0510 & 0.0461 & 0.0384 & 0.0287 \\ + & & $\E{(2)}$ & 0.0371 & 0.0366 & 0.0357 & 0.0342 & 0.0316 & 0.0277 & 0.0223 \\ + & & $\Ex{(1)}$ & 0.0529 & 0.0517 & 0.0494 & 0.0456 & 0.0418 & 0.0409 & \tabc{DNC} \\ + & & $\Ex{(2)}$ & 0.0340 & 0.0330 & 0.0314 & 0.0288 & 0.0274 & 0.0303 & \tabc{DNC} \\ + \\ eLDA & $(0,0)$ & $\E{(0)}$ & -0.0397 & -0.0391 & -0.0380 & -0.0361 & -0.0323 & -0.0236 & \tabc{DNC}\\ & & $\E{(1)}$ & 0.0215 & 0.0213 & 0.0210 & 0.0200 & 0.0159 & 0.0102 & \tabc{DNC}\\ & & $\E{(2)}$ & -0.0426 & -0.0425 & -0.0419 & -0.0387 & -0.0250 & -0.0045 & \tabc{DNC}\\ @@ -697,6 +709,18 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside \\ TDLDA & & $\Ex{(1)}$ & 0.0262 & 0.0264 & 0.0264 & 0.0269 & 0.0273 & 0.0206 & -0.0116 \\ \\ + eHF & $(0,0)$ & $\E{(0)}$ & 0.0327 & 0.0316 & 0.0297 & 0.0261 & 0.0203 & 0.0126 & 0.0055 \\ + & & $\E{(1)}$ & 0.1051 & 0.1038 & 0.1013 & 0.0965 & 0.0875 & 0.0708 & 0.0440 \\ + & & $\E{(2)}$ & 0.0841 & 0.0829 & 0.0806 & 0.0766 & 0.0704 & 0.0625 & 0.0517 \\ + & & $\Ex{(1)}$ & 0.1532 & 0.1516 & 0.1486 & 0.1428 & 0.1320 & 0.1095 & 0.0698 \\ + & & $\Ex{(2)}$ & 0.1322 & 0.1307 & 0.1279 & 0.1229 & 0.1149 & 0.1012 & 0.0774 \\ + \\ + eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.0886 & 0.0874 & 0.0850 & 0.0805 & 0.0725 & 0.0593 & 0.0404 \\ + & & $\E{(1)}$ & 0.0880 & 0.0865 & 0.0836 & 0.0783 & 0.0692 & 0.0552 & 0.0374 \\ + & & $\E{(2)}$ & 0.1209 & 0.1200 & 0.1182 & 0.1147 & 0.1078 & 0.0953 & 0.0747 \\ + & & $\Ex{(1)}$ & 0.0802 & 0.0785 & 0.0754 & 0.0699 & 0.0611 & 0.0529 & 0.0397 \\ + & & $\Ex{(2)}$ & 0.1131 & 0.1120 & 0.1100 & 0.1062 & 0.0997 & 0.0930 & 0.0770 \\ + \\ eLDA & $(0,0)$ & $\E{(0)}$ & -0.0481 & -0.0478 & -0.0473 & -0.0463 & -0.0446 & -0.0387 & -0.0257 \\ & & $\E{(1)}$ & 0.0343 & 0.0336 & 0.0321 & 0.0292 & 0.0220 & 0.0084 & 0.0008 \\ & & $\E{(2)}$ & 0.0277 & 0.0267 & 0.0247 & 0.0216 & 0.0187 & 0.0208 & 0.0209 \\ @@ -748,6 +772,18 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside \\ TDLDA & & $\Ex{(1)}$ & 0.0008 & 0.0007 & 0.0004 & -0.0006 & -0.0074 & -0.0360 & -0.0653 \\ \\ + eHF & $(0,0)$ & $\E{(0)}$ & 0.0541 & 0.0529 & 0.0504 & 0.0459 & 0.0381 & 0.0266 & 0.0139 \\ + & & $\E{(1)}$ & 0.1359 & 0.1346 & 0.1320 & 0.1271 & 0.1186 & 0.1041 & 0.0804 \\ + & & $\E{(2)}$ & 0.1393 & 0.1374 & 0.1335 & 0.1260 & 0.1126 & 0.0910 & 0.0604 \\ + & & $\Ex{(1)}$ & 0.1900 & 0.1885 & 0.1857 & 0.1806 & 0.1715 & 0.1545 & 0.1189 \\ + & & $\Ex{(2)}$ & 0.1934 & 0.1913 & 0.1872 & 0.1795 & 0.1656 & 0.1414 & 0.0990 \\ + \\ + eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.1167 & 0.1153 & 0.1125 & 0.1073 & 0.0977 & 0.0818 & 0.0592 \\ + & & $\E{(1)}$ & 0.1110 & 0.1093 & 0.1059 & 0.0996 & 0.0888 & 0.0723 & 0.0517 \\ + & & $\E{(2)}$ & 0.1688 & 0.1669 & 0.1630 & 0.1553 & 0.1400 & 0.1118 & 0.0705 \\ + & & $\Ex{(1)}$ & 0.1025 & 0.1008 & 0.0975 & 0.0915 & 0.0816 & 0.0702 & 0.0532 \\ + & & $\Ex{(2)}$ & 0.1603 & 0.1584 & 0.1546 & 0.1471 & 0.1328 & 0.1097 & 0.0720 \\ + \\ eLDA & $(0,0)$ & $\E{(0)}$ & -0.0541 & -0.0539 & -0.0537 & -0.0534 & -0.0529 & -0.0504 & -0.0386 \\ & & $\E{(1)}$ & 0.0413 & 0.0406 & 0.0390 & 0.0362 & 0.0304 & 0.0159 & 0.0008 \\ & & $\E{(2)}$ & 0.0642 & 0.0622 & 0.0586 & 0.0517 & 0.0399 & 0.0254 & 0.0149 \\ @@ -798,6 +834,18 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside \\ TDLDA & & $\Ex{(1)}$ & 0.0005 & 0.0005 & 0.0004 & 0.0000 & -0.0033 & -0.0248 & -0.0650 \\ \\ + eHF & $(0,0)$ & $\E{(0)}$ & 0.0769 & 0.0755 & 0.0728 & 0.0676 & 0.0584 & 0.0439 & 0.0257 \\ + & & $\E{(1)}$ & 0.1642 & 0.1628 & 0.1603 & 0.1556 & 0.1475 & 0.1344 & 0.1120 \\ + & & $\E{(2)}$ & 0.1800 & 0.1777 & 0.1732 & 0.1645 & 0.1493 & 0.1260 & 0.0988 \\ + & & $\Ex{(1)}$ & 0.2228 & 0.2215 & 0.2189 & 0.2143 & 0.2066 & 0.1933 & 0.1626 \\ + & & $\Ex{(2)}$ & 0.2387 & 0.2364 & 0.2318 & 0.2233 & 0.2084 & 0.1850 & 0.1495 \\ + \\ + eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.1426 & 0.1411 & 0.1382 & 0.1326 & 0.1225 & 0.1059 & 0.0823 \\ + & & $\E{(1)}$ & 0.1347 & 0.1329 & 0.1295 & 0.1233 & 0.1124 & 0.0957 & 0.0737 \\ + & & $\E{(2)}$ & 0.2019 & 0.1996 & 0.1950 & 0.1860 & 0.1696 & 0.1429 & 0.1070 \\ + & & $\Ex{(1)}$ & 0.1277 & 0.1260 & 0.1228 & 0.1169 & 0.1071 & 0.0943 & 0.0785 \\ + & & $\Ex{(2)}$ & 0.1949 & 0.1926 & 0.1882 & 0.1797 & 0.1643 & 0.1414 & 0.1119 \\ + \\ eLDA & $(0,0)$ & $\E{(0)}$ & -0.0587 & -0.0586 & -0.0587 & -0.0588 & -0.0591 & -0.0590 & -0.0506 \\ & & $\E{(1)}$ & 0.0457 & 0.0450 & 0.0435 & 0.0409 & 0.0362 & 0.0241 & 0.0033 \\ & & $\E{(2)}$ & 0.0861 & 0.0838 & 0.0793 & 0.0712 & 0.0571 & 0.0377 & 0.0196 \\ @@ -848,6 +896,18 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside \\ TDLDA & & $\Ex{(1)}$ & 0.0002 & 0.0000 & 0.0002 & 0.0000 & -0.0016 & -0.0162 & -0.0612 \\ \\ + eHF & $(0,0)$ & $\E{(0)}$ & 0.1004 & 0.0989 & 0.0960 & 0.0903 & 0.0802 & 0.0634 & 0.0403 \\ + & & $\E{(1)}$ & 0.1908 & 0.1893 & 0.1870 & 0.1824 & 0.1747 & 0.1627 & 0.1422 \\ + & & $\E{(2)}$ & 0.2144 & 0.2120 & 0.2073 & 0.1985 & 0.1830 & 0.1595 & 0.1333 \\ + & & $\Ex{(1)}$ & 0.2534 & 0.2520 & 0.2498 & 0.2456 & 0.2387 & 0.2281 & 0.2035 \\ + & & $\Ex{(2)}$ & 0.2770 & 0.2747 & 0.2701 & 0.2617 & 0.2470 & 0.2249 & 0.1946 \\ + \\ + eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.1676 & 0.1660 & 0.1631 & 0.1574 & 0.1471 & 0.1300 & 0.1049 \\ + & & $\E{(1)}$ & 0.1580 & 0.1561 & 0.1528 & 0.1464 & 0.1352 & 0.1173 & 0.0924 \\ + & & $\E{(2)}$ & 0.2294 & 0.2270 & 0.2222 & 0.2130 & 0.1966 & 0.1702 & 0.1336 \\ + & & $\Ex{(1)}$ & 0.1534 & 0.1516 & 0.1485 & 0.1425 & 0.1321 & 0.1167 & 0.0991 \\ + & & $\Ex{(2)}$ & 0.2249 & 0.2225 & 0.2179 & 0.2091 & 0.1935 & 0.1696 & 0.1404 \\ + \\ eLDA & $(0,0)$ & $\E{(0)}$ & -0.0626 & -0.0627 & -0.0628 & -0.0632 & -0.0641 & -0.0654 & -0.0612 \\ & & $\E{(1)}$ & 0.0486 & 0.0477 & 0.0465 & 0.0440 & 0.0400 & 0.0308 & 0.0078 \\ & & $\E{(2)}$ & 0.1017 & 0.0992 & 0.0946 & 0.0862 & 0.0718 & 0.0507 & 0.0271 \\ @@ -899,6 +959,18 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside \\ TDLDA & & $\Ex{(1)}$ & 0.0000 & 0.0001 & 0.0000 & -0.0001 & -0.0009 & -0.0107 & -0.0539 \\ \\ + eHF & $(0,0)$ & $\E{(0)}$ & 0.1918 & 0.1902 & 0.1873 & 0.1817 & 0.1713 & 0.1538 & 0.1273 \\ + & & $\E{(1)}$ & 0.1810 & 0.1792 & 0.1758 & 0.1694 & 0.1579 & 0.1392 & 0.1122 \\ + & & $\E{(2)}$ & 0.2532 & 0.2514 & 0.2470 & 0.2381 & 0.2220 & 0.1969 & 0.1584 \\ + & & $\Ex{(1)}$ & 0.2473 & 0.2458 & 0.2425 & 0.2366 & 0.2263 & 0.2099 & 0.1825 \\ + & & $\Ex{(2)}$ & 0.3196 & 0.3180 & 0.3137 & 0.3054 & 0.2904 & 0.2676 & 0.2286 \\ + \\ + eHF & $(1/3,1/3)$ & $\E{(0)}$ & 0.1918 & 0.1902 & 0.1873 & 0.1817 & 0.1713 & 0.1538 & 0.1273 \\ + & & $\E{(1)}$ & 0.1810 & 0.1792 & 0.1758 & 0.1694 & 0.1579 & 0.1392 & 0.1122 \\ + & & $\E{(2)}$ & 0.2532 & 0.2514 & 0.2470 & 0.2381 & 0.2220 & 0.1969 & 0.1584 \\ + & & $\Ex{(1)}$ & 0.1796 & 0.1780 & 0.1746 & 0.1685 & 0.1576 & 0.1406 & 0.1202 \\ + & & $\Ex{(2)}$ & 0.2518 & 0.2501 & 0.2458 & 0.2372 & 0.2217 & 0.1982 & 0.1663 \\ + \\ eLDA & $(0,0)$ & $\E{(0)}$ & -0.0664 & -0.0666 & -0.0667 & -0.0672 & -0.0684 & -0.0707 & -0.0702 \\ & & $\E{(1)}$ & 0.0502 & 0.0495 & 0.0482 & 0.0459 & 0.0423 & 0.0355 & 0.0131 \\ & & $\E{(2)}$ & 0.1122 & 0.1104 & 0.1061 & 0.0979 & 0.0836 & 0.0635 & 0.0360 \\ diff --git a/Notebooks/eDFT_FUEG.nb b/Notebooks/eDFT_FUEG.nb index 7e1b8df..f18e4fb 100644 --- a/Notebooks/eDFT_FUEG.nb +++ b/Notebooks/eDFT_FUEG.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 9073989, 197297] -NotebookOptionsPosition[ 8936808, 195033] -NotebookOutlinePosition[ 8937199, 195050] 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