DD
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@ -1,3 +1,8 @@
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168.1946 44.0662 12.0035 3.4747 1.0896 0.3719 0.1367
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330.2471 85.0890 22.5112 6.2247 1.8355 0.5845 0.2006
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809.9972 204.9840 52.4777 13.7252 3.7248 1.0696 0.3300
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162.0525 41.0228 10.5078 2.7500 0.7458 0.2125 0.0639
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641.8026 160.9177 40.4743 10.2505 2.6352 0.6977 0.1933
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-0.0397 -0.0391 -0.0380 -0.0361 -0.0323 -0.0236
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0.0215 0.0213 0.0210 0.0200 0.0159 0.0102
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-0.0426 -0.0425 -0.0419 -0.0387 -0.0250 -0.0045
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@ -1,3 +1,8 @@
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475.6891 125.7776 34.8248 10.3536 3.3766 1.2126 0.4721
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702.8330 183.3370 49.5922 14.2255 4.4269 1.5105 0.5606
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1379.3128 353.5967 92.7398 25.3135 7.3546 2.3203 0.7990
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227.1438 57.5594 14.7674 3.8720 1.0504 0.2979 0.0885
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903.6236 227.8191 57.9150 14.9599 3.9780 1.1077 0.3269
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-0.0481 -0.0478 -0.0473 -0.0463 -0.0446 -0.0387 -0.0257
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0.0343 0.0336 0.0321 0.0292 0.0220 0.0084 0.0008
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0.0277 0.0267 0.0247 0.0216 0.0187 0.0208 0.0209
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@ -1,3 +1,8 @@
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1020.3778 270.0849 74.9426 22.3790 7.3595 2.6798 1.0633
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1312.2776 344.0184 93.8936 27.3398 8.7021 3.0600 1.1764
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2183.4399 563.5949 149.6753 41.7213 12.5052 4.1033 1.4749
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291.8998 73.9335 18.9510 4.9608 1.3426 0.3802 0.1131
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1163.0621 293.5099 74.7326 19.3423 5.1457 1.4235 0.4116
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-0.0541 -0.0539 -0.0537 -0.0534 -0.0529 -0.0504 -0.0386
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0.0413 0.0406 0.0390 0.0362 0.0304 0.0159 0.0008
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0.0642 0.0622 0.0586 0.0517 0.0399 0.0254 0.0149
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@ -1,11 +1,13 @@
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1867.6344 493.6760 136.7020 40.7244 13.3763 4.8811 1.9492
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2224.11488 583.8981 159.7957 46.7553 15.0029 5.3399 2.0855
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3289.2022 852.4249 228.0415 64.3597 19.6613 6.6206 2.4547
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356.4804 90.2221 23.0937 6.0308 1.6266 0.4588 0.1363
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1421.56773 358.7489 91.3395 23.6352 6.2850 1.7395 0.5055
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-0.0587 -0.0586 -0.0587 -0.0588 -0.0591 -0.0590 -0.0506
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0.0457 0.0450 0.0435 0.0409 0.0362 0.0241 0.0033
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0.0861 0.0838 0.0793 0.0712 0.0571 0.0377 0.0196
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0.1044 0.1036 0.1022 0.0997 0.0953 0.0830 0.0540
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0.1447 0.1424 0.1380 0.1300 0.1162 0.0966 0.0703
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-0.1356 -0.1342 -0.1314 -0.1264 -0.1176 -0.1037 -0.0849
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-0.1356 -0.1342 -0.1314 -0.1263 -0.1174 -0.1031 -0.0834
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-0.1356 -0.1341 -0.1314 -0.1262 -0.1172 -0.1026 -0.0824
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0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
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0.0172 0.0163 0.0147 0.0117 0.0067 -0.0004 -0.0080
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0.0416 0.0402 0.0376 0.0329 0.0253 0.0140 0.0005
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@ -14,9 +16,6 @@
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0.1080 0.1056 0.1011 0.0925 0.0772 0.0538 0.0325
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0.0092 0.0081 0.0060 0.0022 -0.0035 -0.0081 -0.0047
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0.1010 0.0986 0.0943 0.0862 0.0719 0.0523 0.0373
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-0.1160 -0.1154 -0.1141 -0.1115 -0.1069 -0.0991 -0.0870
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-0.1160 -0.1154 -0.1140 -0.1115 -0.1069 -0.0991 -0.0872
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-0.1160 -0.1153 -0.1140 -0.1115 -0.1069 -0.0991 -0.0871
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-0.0196 -0.0188 -0.0174 -0.0148 -0.0106 -0.0044 0.0029
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-0.0024 -0.0025 -0.0027 -0.0032 -0.0040 -0.0050 -0.0056
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0.0220 0.0213 0.0201 0.0180 0.0146 0.0093 0.0027
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@ -1,3 +1,8 @@
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3082.5386 813.0910 224.3734 66.5257 21.7454 7.9136 3.1633
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3503.4911 919.5487 251.5842 73.6145 23.6504 8.4487 3.3217
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4762.0921 1236.8257 332.1993 94.3988 29.1455 9.9582 3.7572
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420.9525 106.4577 27.2108 7.0888 1.9050 0.5351 0.1583
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1679.5536 423.7347 107.8259 27.8731 7.4001 2.0446 0.5938
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-0.0626 -0.0627 -0.0628 -0.0632 -0.0641 -0.0654 -0.0612
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0.0486 0.0477 0.0465 0.0440 0.0400 0.0308 0.0078
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0.1017 0.0992 0.0946 0.0862 0.0718 0.0507 0.0271
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@ -1,3 +1,8 @@
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4729.98018 1244.7753 342.1796 100.8943 32.7728 11.8683 4.7359
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5215.3307 1367.4316 373.4897 109.0326 34.9524 12.4779 4.9156
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6667.18516 1733.3319 466.4133 132.9686 41.2715 14.2096 5.4146
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485.3505 122.6563 31.3101 8.1382 2.1796 0.6096 0.1797
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1937.2050 488.5566 124.2336 32.0743 8.4987 2.3413 0.6787
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-0.0664 -0.0666 -0.0667 -0.0672 -0.0684 -0.0707 -0.0702
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0.0502 0.0495 0.0482 0.0459 0.0423 0.0355 0.0131
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0.1122 0.1104 0.1061 0.0979 0.0836 0.0635 0.0360
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@ -897,19 +897,20 @@ while the uncorrected KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_wit
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%This shows clearly that there is a correction due to the correlation functional itself as well as a correction due to the ensemble correlation derivative
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The corresponding excitation energies are
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\beq\label{eq:Om-eLDA}
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\begin{split}
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\Ex{eLDA}{(I)}
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& =
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\Ex{HF}{(I)}
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\\
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& + \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})}
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=
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\Ex{HF}{(I)}
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+ \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})}
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\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{}
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\\
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& + \int \n{\bGam{\bw}}{}(\br{})
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\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{},
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\end{split}
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+ \DD{c}{(I)},
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\eeq
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with $\Ex{HF}{(I)} = \E{HF}{(I)} - \E{HF}{(0)}$, where the last term is the ensemble correlation derivative contribution to the excitation energy.
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with $\Ex{HF}{(I)} = \E{HF}{(I)} - \E{HF}{(0)}$, and where
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\beq\label{eq:DD-eLDA}
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\DD{c}{(I)}
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= \int \n{\bGam{\bw}}{}(\br{})
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\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
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\eeq
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is the ensemble correlation derivative contribution to the excitation energy.
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}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Density-functional approximations for ensembles}
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@ -1307,30 +1308,25 @@ electrons.
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\includegraphics[width=\linewidth]{EvsL_5_HF}
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\caption{
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\label{fig:EvsLHF}
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Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 5-boxium at the KS-eLDA (solid lines) and eHF (dashed lines) levels.
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Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and state-averaged (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported.
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Error with respect to FCI (in \%) associated with the single excitation $\Ex{}{(1)}$ (bottom) and double excitation $\Ex{}{(2)}$ (top) as a function of the box length $L$ for 3-boxium at the KS-eLDA (solid lines) and eHF (dashed lines) levels.
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Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, red lines) and equi-weight (\ie, $\ew{1} = \ew{2} = 1/3$, blue lines) calculations are reported.
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}
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\end{figure}
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%%% %%% %%%
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\titou{T2: there is a micmac with the derivative discontinuity as it is
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only defined at zero weight. We should clean up this.}\manu{I will!}
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It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
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To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage
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(with respect to FCI) on the excitation energies obtained at the KS-eLDA
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and HF\manu{-like} levels [see Eqs.~\eqref{eq:EI-eLDA} and
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\eqref{eq:ind_HF-like_ener}, respectively] as a function of the box
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length $L$ in the case of 5-boxium.
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\manu{Manu: there is something I do not understand. If you want to
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evaluate the importance of the ensemble correlation derivatives you
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should only remove the following contribution from the $K$th KS-eLDA
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excitation energy:
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\beq\label{eq:DD_term_to_compute}
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\int \n{\bGam{\bw}}{}(\br{})
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\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
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\eeq
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%rather than $E^{(I)}_{\rm HF}$
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}
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It is also interesting to investigate the influence of the ensemble correlation derivative $\DD{c}{I}$ [defined in Eq.~\eqref{eq:DD-eLDA}] on both the single and double excitations.
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To do so, we have reported in Fig.~\ref{fig:EvsLHF}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$
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on the excitation energies obtained at the KS-eLDA with and without $\DD{c}{I}$ [\ie, the last term in Eq.~\eqref{eq:Om-eLDA}].
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%\manu{Manu: there is something I do not understand. If you want to
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%evaluate the importance of the ensemble correlation derivatives you
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%should only remove the following contribution from the $K$th KS-eLDA
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%excitation energy:
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%\beq\label{eq:DD_term_to_compute}
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%\int \n{\bGam{\bw}}{}(\br{})
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% \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
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%\eeq
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%%rather than $E^{(I)}_{\rm HF}$
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%}
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The influence of the ensemble correlation derivative is clearly more important in the strong correlation regime.
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Its contribution is also significantly larger in the case of the single
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excitation; the ensemble correlation derivative hardly influences the double excitation.
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@ -1339,11 +1335,12 @@ derivative is much smaller in the case of equal-weight calculations (as compared
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This could explain why equiensemble calculations are clearly more
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accurate as it reduces the influence of the ensemble correlation derivative:
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for a given method, equiensemble orbitals partially remove the burden
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of modeling properly the ensemble correlation derivative.\manu{Manu: well, we
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would need the exact derivative value to draw such a conclusion. We can
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only speculate. Let us first see how important the contribution in
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Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
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updated in the light of the new results.}
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of modeling properly the ensemble correlation derivative.
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%\manu{Manu: well, we
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%would need the exact derivative value to draw such a conclusion. We can
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%only speculate. Let us first see how important the contribution in
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%Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
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%updated in the light of the new results.}
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%%% FIG 6 %%%
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\begin{figure}
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@ -1352,7 +1349,7 @@ updated in the light of the new results.}
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\label{fig:EvsN_HF}
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Error with respect to FCI in single and double excitation energies for $\nEl$-boxium (with a box length of $L=8\pi$) as a function of the number of electrons $\nEl$ at the KS-eLDA (solid lines) and eHF (dashed lines) levels.
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Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, black and red lines) and
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equal-weight (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported.
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equi-weight (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported.
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}
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\end{figure}
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%%% %%% %%%
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11148
Notebooks/eDFT_FUEG.nb
11148
Notebooks/eDFT_FUEG.nb
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