DD

Data/DD_n2.dat

@@ -1,3 +1,8 @@
+168.1946        44.0662         12.0035         3.4747  1.0896  0.3719  0.1367  
+330.2471        85.0890         22.5112         6.2247  1.8355  0.5845  0.2006  
+809.9972        204.9840        52.4777         13.7252 3.7248  1.0696  0.3300  
+162.0525        41.0228         10.5078         2.7500  0.7458  0.2125  0.0639  
+641.8026        160.9177        40.4743         10.2505 2.6352  0.6977  0.1933  
 -0.0397	-0.0391	-0.0380	-0.0361	-0.0323	-0.0236	
 0.0215	0.0213	0.0210	0.0200	0.0159	0.0102	
 -0.0426	-0.0425	-0.0419	-0.0387	-0.0250	-0.0045	

Data/DD_n3.dat

@@ -1,3 +1,8 @@
+475.6891        125.7776        34.8248 10.3536 3.3766  1.2126  0.4721
+702.8330        183.3370        49.5922 14.2255 4.4269  1.5105  0.5606
+1379.3128       353.5967        92.7398 25.3135 7.3546  2.3203  0.7990
+227.1438        57.5594         14.7674 3.8720  1.0504  0.2979  0.0885
+903.6236        227.8191        57.9150 14.9599 3.9780  1.1077  0.3269
 -0.0481	-0.0478	-0.0473	-0.0463	-0.0446	-0.0387	-0.0257
 0.0343	0.0336	0.0321	0.0292	0.0220	0.0084	0.0008 
 0.0277	0.0267	0.0247	0.0216	0.0187	0.0208	0.0209 

Data/DD_n4.dat

@@ -1,3 +1,8 @@
+1020.3778       270.0849        74.9426         22.3790 7.3595  2.6798  1.0633  
+1312.2776       344.0184        93.8936         27.3398 8.7021  3.0600  1.1764  
+2183.4399       563.5949        149.6753        41.7213 12.5052 4.1033  1.4749  
+291.8998        73.9335         18.9510         4.9608  1.3426  0.3802  0.1131  
+1163.0621       293.5099        74.7326         19.3423 5.1457  1.4235  0.4116  
 -0.0541	-0.0539	-0.0537	-0.0534	-0.0529	-0.0504	-0.0386
 0.0413	0.0406	0.0390	0.0362	0.0304	0.0159	0.0008 
 0.0642	0.0622	0.0586	0.0517	0.0399	0.0254	0.0149 

Data/DD_n5.dat

@@ -1,11 +1,13 @@
+1867.6344	493.6760	136.7020	40.7244		13.3763	4.8811	1.9492	
+2224.11488	583.8981	159.7957	46.7553		15.0029	5.3399	2.0855	
+3289.2022	852.4249	228.0415	64.3597		19.6613	6.6206	2.4547	
+356.4804	90.2221		23.0937		6.0308		1.6266	0.4588	0.1363	
+1421.56773	358.7489	91.3395		23.6352		6.2850	1.7395	0.5055	
 -0.0587	-0.0586	-0.0587	-0.0588	-0.0591	-0.0590	-0.0506	
 0.0457	0.0450	0.0435	0.0409	0.0362	0.0241	0.0033	
 0.0861	0.0838	0.0793	0.0712	0.0571	0.0377	0.0196	
 0.1044	0.1036	0.1022	0.0997	0.0953	0.0830	0.0540	
 0.1447	0.1424 	0.1380	0.1300	0.1162	0.0966	0.0703	
--0.1356 -0.1342 -0.1314 -0.1264 -0.1176 -0.1037 -0.0849
--0.1356 -0.1342 -0.1314 -0.1263 -0.1174 -0.1031 -0.0834
--0.1356 -0.1341 -0.1314 -0.1262 -0.1172 -0.1026 -0.0824
 0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	
 0.0172	0.0163	0.0147	0.0117	0.0067	-0.0004	-0.0080	
 0.0416	0.0402	0.0376	0.0329	0.0253	0.0140	0.0005	
@@ -14,9 +16,6 @@
 0.1080	0.1056	0.1011	0.0925	0.0772	0.0538	0.0325	
 0.0092	0.0081	0.0060	0.0022	-0.0035	-0.0081	-0.0047	
 0.1010	0.0986	0.0943	0.0862	0.0719	0.0523	0.0373	
--0.1160 -0.1154 -0.1141 -0.1115 -0.1069 -0.0991 -0.0870
--0.1160 -0.1154 -0.1140 -0.1115 -0.1069 -0.0991 -0.0872
--0.1160 -0.1153 -0.1140 -0.1115 -0.1069 -0.0991 -0.0871
 -0.0196	-0.0188	-0.0174	-0.0148	-0.0106	-0.0044	0.0029	
 -0.0024	-0.0025	-0.0027	-0.0032	-0.0040	-0.0050	-0.0056	
 0.0220	0.0213	0.0201	0.0180	0.0146	0.0093	0.0027	

Data/DD_n6.dat

@@ -1,3 +1,8 @@
+3082.5386       813.0910        224.3734        66.5257 21.7454 7.9136  3.1633  
+3503.4911       919.5487        251.5842        73.6145 23.6504 8.4487  3.3217  
+4762.0921       1236.8257       332.1993        94.3988 29.1455 9.9582  3.7572  
+420.9525        106.4577        27.2108         7.0888  1.9050  0.5351  0.1583  
+1679.5536       423.7347        107.8259        27.8731 7.4001  2.0446  0.5938  
 -0.0626	-0.0627	-0.0628	-0.0632	-0.0641	-0.0654	-0.0612	
 0.0486	0.0477	0.0465	0.0440	0.0400	0.0308	0.0078	
 0.1017	0.0992	0.0946	0.0862	0.0718	0.0507	0.0271	

Data/DD_n7.dat

@@ -1,3 +1,8 @@
+4729.98018      1244.7753       342.1796        100.8943        32.7728 11.8683 4.7359  
+5215.3307       1367.4316       373.4897        109.0326        34.9524 12.4779 4.9156  
+6667.18516      1733.3319       466.4133        132.9686        41.2715 14.2096 5.4146  
+485.3505        122.6563        31.3101         8.1382          2.1796  0.6096  0.1797  
+1937.2050       488.5566        124.2336        32.0743         8.4987  2.3413  0.6787  
 -0.0664	-0.0666	-0.0667	-0.0672	-0.0684	-0.0707	-0.0702
 0.0502	0.0495	0.0482	0.0459	0.0423	0.0355	0.0131 
 0.1122	0.1104	0.1061	0.0979	0.0836	0.0635	0.0360 

Manuscript/eDFT.tex

@@ -897,19 +897,20 @@ while the uncorrected KS-eLDA ensemble energy obtained via Eq.~\eqref{eq:min_wit
 %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
 The corresponding excitation energies are
 \beq\label{eq:Om-eLDA}
-\begin{split}
 	\Ex{eLDA}{(I)} 
-	& = 
+	= 
-        \Ex{HF}{(I)}
+	\Ex{HF}{(I)}
-	\\
+	+ \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} 
-	& + \int \fdv{\E{c}{\bw}[\n{\bGam{\bw}}{}]}{\n{}{}(\br{})} 
 	\qty[ \n{\bGam{(I)}}{}(\br{}) - \n{\bGam{(0)}}{}(\br{}) ] d\br{} 
-	\\
+	+ \DD{c}{(I)},
-	& + \int \n{\bGam{\bw}}{}(\br{})
-	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{},
-\end{split}
 \eeq
-with $\Ex{HF}{(I)} = \E{HF}{(I)} - \E{HF}{(0)}$, where the last term is the ensemble correlation derivative contribution to the excitation energy.
+with $\Ex{HF}{(I)} = \E{HF}{(I)} - \E{HF}{(0)}$, and where 
+\beq\label{eq:DD-eLDA}
+	\DD{c}{(I)}
+	= \int \n{\bGam{\bw}}{}(\br{})
+	\left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{I}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
+\eeq
+is the ensemble correlation derivative contribution to the excitation energy.
 }
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \section{Density-functional approximations for ensembles}
@@ -1307,30 +1308,25 @@ electrons.
 	\includegraphics[width=\linewidth]{EvsL_5_HF}
 	\caption{
 	\label{fig:EvsLHF}
-	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. 
+	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. 
-	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. 
+	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. 
 	}
 \end{figure}
 %%% %%% %%%
 
-\titou{T2: there is a micmac with the derivative discontinuity as it is
+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.
-only defined at zero weight. We should clean up this.}\manu{I will!}
+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 $L$
-It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
+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}].
-To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage
+%\manu{Manu: there is something I do not understand. If you want to
-(with respect to FCI) on the excitation energies obtained at the KS-eLDA
+%evaluate the importance of the ensemble correlation derivatives you
-and HF\manu{-like} levels [see Eqs.~\eqref{eq:EI-eLDA} and
+%should only remove the following contribution from the $K$th KS-eLDA
-\eqref{eq:ind_HF-like_ener}, respectively] as a function of the box
+%excitation energy:  
-length $L$ in the case of 5-boxium.
+%\beq\label{eq:DD_term_to_compute}
-\manu{Manu: there is something I do not understand. If you want to
+%\int  \n{\bGam{\bw}}{}(\br{})
-evaluate the importance of the ensemble correlation derivatives you
+%      \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
-should only remove the following contribution from the $K$th KS-eLDA
+%\eeq
-excitation energy:  
+%%rather than $E^{(I)}_{\rm HF}$
-\beq\label{eq:DD_term_to_compute}
+%}
-\int  \n{\bGam{\bw}}{}(\br{})
-      \left. \pdv{\e{c}{\bw}(\n{}{})}{\ew{K}} \right|_{\n{}{}=\n{\bGam{\bw}}{}(\br{})} d\br{}
-\eeq
-%rather than $E^{(I)}_{\rm HF}$
-}
 The influence of the ensemble correlation derivative is clearly more important in the strong correlation regime.
 Its contribution is also significantly larger in the case of the single
 excitation; the ensemble correlation derivative hardly influences the double excitation.
@@ -1339,11 +1335,12 @@ derivative is much smaller in the case of equal-weight calculations (as compared
 This could explain why equiensemble calculations are clearly more
 accurate as it reduces the influence of the ensemble correlation derivative:
 for a given method, equiensemble orbitals partially remove the burden
-of modeling properly the ensemble correlation derivative.\manu{Manu: well, we
+of modeling properly the ensemble correlation derivative.
-would need the exact derivative value to draw such a conclusion. We can
+%\manu{Manu: well, we
-only speculate. Let us first see how important the contribution in
+%would need the exact derivative value to draw such a conclusion. We can
-Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
+%only speculate. Let us first see how important the contribution in
-updated in the light of the new results.}
+%Eq.~\eqref{eq:DD_term_to_compute} is. What follows should also be
+%updated in the light of the new results.}
 
 %%% FIG 6 %%%
 \begin{figure}
@@ -1352,7 +1349,7 @@ updated in the light of the new results.}
 	\label{fig:EvsN_HF}
 	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. 
 	Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, black and red lines) and
-equal-weight (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported. 
+equi-weight (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported. 
 	}
 \end{figure}
 %%% %%% %%%