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Pierre-Francois Loos 2020-02-19 14:52:47 +01:00
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Manuscript/SI/eDFT-SI.tex
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@ -659,7 +659,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\Ex{(1)}$ & 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}\\
KS-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}\\
& & $\Ex{(1)}$ & 0.0612 & 0.0604 & 0.0590 & 0.0561 & 0.0483 & 0.0337 & \tabc{DNC}\\
@ -668,7 +668,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\DD{c}{(1)}$ & 0.0064 & 0.0056 & 0.0043 & 0.0022 & -0.0007 & -0.0037 & \tabc{DNC}\\
& & $\DD{c}{(2)}$ & 0.0159 & 0.0147 & 0.0126 & 0.0093 & 0.0046 & -0.0009 & \tabc{DNC}\\
\\
eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0031 & 0.0036 & 0.0044 & 0.0054 & 0.0042 & -0.0025 & \tabc{DNC}\\
KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0031 & 0.0036 & 0.0044 & 0.0054 & 0.0042 & -0.0025 & \tabc{DNC}\\
& & $\E{(1)}$ & 0.0090 & 0.0087 & 0.0083 & 0.0076 & 0.0070 & 0.0071 & \tabc{DNC}\\
& & $\E{(2)}$ & -0.0005 & -0.0009 & -0.0015 & -0.0023 & -0.0030 & -0.0026 & \tabc{DNC}\\
& & $\Ex{(1)}$ & 0.0058 & 0.0052 & 0.0039 & 0.0022 & 0.0028 & 0.0096 & \tabc{DNC}\\
@ -721,7 +721,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\Ex{(1)}$ & 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 \\
KS-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 \\
& & $\Ex{(1)}$ & 0.0824 & 0.0814 & 0.0794 & 0.0755 & 0.0666 & 0.0471 & 0.0266 \\
@ -730,7 +730,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\DD{c}{(1)}$ & 0.0100 & 0.0092 & 0.0077 & 0.0051 & 0.0012 & -0.0034 & -0.0072 \\
& & $\DD{c}{(2)}$ & 0.0244 & 0.0231 & 0.0208 & 0.0168 & 0.0108 & 0.0029 & -0.0050 \\
\\
eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0078 & 0.0080 & 0.0082 & 0.0085 & 0.0081 & 0.0024 & -0.0022 \\
KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0078 & 0.0080 & 0.0082 & 0.0085 & 0.0081 & 0.0024 & -0.0022 \\
& & $\E{(1)}$ & 0.0172 & 0.0162 & 0.0144 & 0.0112 & 0.0064 & 0.0019 & 0.0004 \\
& & $\E{(2)}$ & 0.0645 & 0.0636 & 0.0621 & 0.0590 & 0.0530 & 0.0420 & 0.0300 \\
& & $\Ex{(1)}$ & 0.0094 & 0.0083 & 0.0062 & 0.0027 & -0.0018 & -0.0004 & 0.0026 \\
@ -784,7 +784,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\Ex{(1)}$ & 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 \\
KS-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 \\
& & $\Ex{(1)}$ & 0.0954 & 0.0945 & 0.0927 & 0.0896 & 0.0833 & 0.0663 & 0.0394 \\
@ -793,7 +793,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\DD{c}{(1)}$ & 0.0136 & 0.0127 & 0.0111 & 0.0083 & 0.0038 & -0.0022 & -0.0080 \\
& & $\DD{c}{(2)}$ & 0.0330 & 0.0316 & 0.0291 & 0.0248 & 0.0178 & 0.0080 & -0.0028 \\
\\
eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0085 & 0.0085 & 0.0084 & 0.0082 & 0.0072 & 0.0021 & -0.0015 \\
KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0085 & 0.0085 & 0.0084 & 0.0082 & 0.0072 & 0.0021 & -0.0015 \\
& & $\E{(1)}$ & 0.0164 & 0.0152 & 0.0129 & 0.0087 & 0.0020 & -0.0050 & -0.0044 \\
& & $\E{(2)}$ & 0.0936 & 0.0917 & 0.0880 & 0.0807 & 0.0664 & 0.0434 & 0.0300 \\
& & $\Ex{(1)}$ & 0.0079 & 0.0067 & 0.0045 & 0.0006 & -0.0051 & -0.0071 & -0.0029 \\
@ -846,7 +846,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\Ex{(1)}$ & 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 \\
KS-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 \\
& & $\Ex{(1)}$ & 0.1044 & 0.1036 & 0.1022 & 0.0997 & 0.0953 & 0.0830 & 0.0540 \\
@ -855,7 +855,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\DD{c}{(1)}$ & 0.0172 & 0.0163 & 0.0147 & 0.0117 & 0.0067 & -0.0004 & -0.0080 \\
& & $\DD{c}{(2)}$ & 0.0416 & 0.0402 & 0.0376 & 0.0329 & 0.0253 & 0.0140 & 0.0005 \\
\\
eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0070 & 0.0070 & 0.0068 & 0.0063 & 0.0053 & 0.0015 & -0.0049 \\
KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0070 & 0.0070 & 0.0068 & 0.0063 & 0.0053 & 0.0015 & -0.0049 \\
& & $\E{(1)}$ & 0.0162 & 0.0151 & 0.0128 & 0.0086 & 0.0018 & -0.0066 & -0.0095 \\
& & $\E{(2)}$ & 0.1080 & 0.1056 & 0.1011 & 0.0925 & 0.0772 & 0.0538 & 0.0325 \\
& & $\Ex{(1)}$ & 0.0092 & 0.0081 & 0.0060 & 0.0022 & -0.0035 & -0.0081 & -0.0047 \\
@ -908,7 +908,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\Ex{(1)}$ & 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 \\
KS-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 \\
& & $\Ex{(1)}$ & 0.1112 & 0.1104 & 0.1093 & 0.1072 & 0.1041 & 0.0962 & 0.0690 \\
@ -917,7 +917,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\DD{c}{(1)}$ & 0.0208 & 0.0199 & 0.0182 & 0.0151 & 0.0098 & 0.0018 & -0.0075 \\
& & $\DD{c}{(2)}$ & 0.0503 & 0.0488 & 0.0460 & 0.0412 & 0.0330 & 0.0205 & 0.0046 \\
\\
eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0046 & 0.0045 & 0.0043 & 0.0039 & 0.0031 & 0.0006 & -0.0067 \\
KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0046 & 0.0045 & 0.0043 & 0.0039 & 0.0031 & 0.0006 & -0.0067 \\
& & $\E{(1)}$ & 0.0157 & 0.0144 & 0.0123 & 0.0080 & 0.0009 & -0.0091 & -0.0160 \\
& & $\E{(2)}$ & 0.1167 & 0.1142 & 0.1095 & 0.1007 & 0.0853 & 0.0616 & 0.0355 \\
& & $\Ex{(1)}$ & 0.0112 & 0.0099 & 0.0080 & 0.0041 & -0.0022 & -0.0097 & -0.0093 \\
@ -971,7 +971,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\Ex{(1)}$ & 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 \\
KS-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 \\
& & $\Ex{(1)}$ & 0.1165 & 0.1161 & 0.1149 & 0.1131 & 0.1108 & 0.1062 & 0.0834 \\
@ -980,7 +980,7 @@ Equation \eqref{eq:ec} is depicted in Fig.~\ref{fig:Ec} for each state alongside
& & $\DD{c}{(1)}$ & 0.0244 & 0.0235 & 0.0218 & 0.0186 & 0.0130 & 0.0043 & -0.0065 \\
& & $\DD{c}{(2)}$ & 0.0589 & 0.0574 & 0.0546 & 0.0496 & 0.0410 & 0.0275 & 0.0095 \\
\\
eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0014 & 0.0013 & 0.0012 & 0.0009 & 0.0003 & -0.0013 & -0.0079 \\
KS-eLDA & $(1/3,1/3)$ & $\E{(0)}$ & 0.0014 & 0.0013 & 0.0012 & 0.0009 & 0.0003 & -0.0013 & -0.0079 \\
& & $\E{(1)}$ & 0.0149 & 0.0138 & 0.0115 & 0.0072 & -0.0001 & -0.0110 & -0.0209 \\
& & $\E{(2)}$ & 0.1217 & 0.1198 & 0.1154 & 0.1069 & 0.0917 & 0.0691 & 0.0389 \\
& & $\Ex{(1)}$ & 0.0135 & 0.0125 & 0.0103 & 0.0063 & -0.0005 & -0.0096 & -0.0130 \\

101
Manuscript/eDFT.tex
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@ -672,7 +672,7 @@ and curvature~\cite{} will be introduced in the Hx energy:
\eeq
These errors will be removed when computing individual energies according to Eq.~\eqref{eq:exact_ind_ener_rdm}.
Turning to the density-functional ensemble correlation energy, the following eLDA will be employed:
Turning to the density-functional ensemble correlation energy, the following eLDA functional will be employed:
\beq\label{eq:eLDA_corr_fun}
\E{c}{\bw}[\n{}{}] = \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
\eeq
@ -681,7 +681,7 @@ Its construction from a finite uniform electron gas model is discussed in detail
\titou{Manu, I think we should clearly define here what the expression of the ensemble energy with and without GOC.
What do you think?}
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within eLDA:
Combining Eq.~\eqref{eq:exact_ind_ener_rdm} with Eq.~\eqref{eq:eLDA_corr_fun} leads to our final energy level expression within KS-eLDA:
\beq\label{eq:EI-eLDA}
\begin{split}
\E{\titou{eLDA}}{(I)}
@ -737,7 +737,7 @@ All these states have the same (uniform) density $\n{}{} = 2/(2\pi R)$ where $R$
We refer the interested reader to Refs.~\onlinecite{Loos_2012, Loos_2013a, Loos_2014b} for more details about this paradigm.
Generalization to a larger number of states is straightforward and is left for future work.
To ensure the GOK variational principle, \cite{Gross_1988a} the weights must fulfil the following conditions:
$0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$.
$0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$ [or $\ew{2} \le \ew{1} \le (1-\ew{2})/2$].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Weight-dependent correlation functional}
@ -795,6 +795,7 @@ where
\end{equation}
The local-density approximation (LDA) correlation functional,
\begin{equation}
\label{eq:LDA}
\e{c}{\text{LDA}}(\n{}{})
= a_1^\text{LDA} F\qty[1,\frac{3}{2},a_3^\text{LDA}, \frac{a_1^\text{LDA}(1-a_3^\text{LDA})}{a_2^\text{LDA}} {\n{}{}}^{-1}],
\end{equation}
@ -849,11 +850,11 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
\end{equation}
with $ \mu = 1,\ldots,\nBas$ and $\nBas = 30$ for all calculations.
For the self-consistent calculations (such as HF, KS-DFT or KS-eDFT), the convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ been set to $10^{-5}$.
For KS-DFT and KS-eDFT calculations, a Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form.
For KS-DFT, KS-eDFT and TDDFT calculations, a Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form.
In order to test the present eLDA functional we have performed various sets of calculations.
To get reference excitation energies for both the single and double excitations, we have performed full configuration interaction (FCI) calculations with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
For the single excitations, we have also performed time-dependent LDA (TDLDA) calculations, and the quality of the Tamm-Dancoff approximation (TDA) has been also investigated.\cite{Dreuw_2005}
For the single excitations, we have also performed time-dependent LDA (TDLDA) calculations [\ie, TDDFT with the LDA functional defined in Eq.~\eqref{eq:LDA}], and the effect of the Tamm-Dancoff approximation (TDA) has been also investigated. \cite{Dreuw_2005}
Concerning the KS-eDFT and eHF calculations, two sets of weight have been tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -863,32 +864,40 @@ Concerning the KS-eDFT and eHF calculations, two sets of weight have been tested
First, we discuss the linearity of the ensemble energy.
To do so, we consider 5-boxium with box lengths of $L = \pi/8$, $L = \pi$, and $L = 8\pi$, which correspond (qualitatively at least) to the weak, intermediate, and strong correlation regimes, respectively.
The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while fulfilling the restrictions on the ensemble weights to ensure the GOK variational principle (\ie, $0 \le \ew{1} \le 1/3$ and $\ew{2} \le \ew{1} \le (1-\ew{2})/2$).
The three-state ensemble energy $\E{}{(\ew{1},\ew{2})}$ is represented in Fig.~\ref{fig:EvsW} as a function of both $\ew{1}$ and $\ew{2}$ while fulfilling the restrictions on the ensemble weights to ensure the GOK variational principle [\ie, $0 \le \ew{1} \le 1/3$ and $0 \le \ew{2} \le \ew{1}$].
To illustrate the magnitude of the ghost interaction error (GIE), we have reported the ensemble energy with and without ghost interaction correction (GIC) as explained above [see Eqs.~\eqref{eq:WHF} and \eqref{eq:EI-eLDA}].
As one can see, the linearity of the GOC-free ensemble energy deteriorates when $L$ gets larger, while the GOC makes the ensemble energy almost perfectly linear.
As one can see in Fig.~\ref{fig:EvsW}, the linearity of the GOC-free ensemble energy deteriorates when $L$ gets larger, while the GOC makes the ensemble energy almost perfectly linear.
In other words, the GIE increases as the correlation gets stronger.
Because the GIE can be easily computed via Eq.~\eqref{eq:WHF} even for real, three-dimensional systems, this provides a cheap way of quantifying strong correlation in a given electronic system.
It is important to note that, even though the GIC removes the explicit quadratic terms from the ensemble energy, a weak non-linearity remains in the ensemble energy due to the optimization of the ensemble KS orbitals in presence of GIE.
However, this ``density-driven''-type of error is extremely small (in our case at least).
It is important to note that, even though the GIC removes the explicit quadratic terms from the ensemble energy, a weak non-linearity remains in the GIC ensemble energy due to the optimization of the ensemble KS orbitals in presence of GIE.
However, this ``density-driven''-type error is extremely small (in our case at least) as the correlation part of the ensemble KS potential $\delta \E{c}{\bw}[\n{}{}] /\delta \n{}{}(\br{})$ is relatively small compared to the Hx contribution.
%%% FIG 1 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{EvsW_n5}
\caption{
\label{fig:EvsW}
Weight dependence of the ensemble energy $\E{}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
Weight dependence of the KS-eLDA ensemble energy $\E{}{(\ew{1},\ew{2})}$ with (dashed lines) and without (solid lines) ghost interaction correction (GIC) for 5-boxium (\ie, $\nEl = 5$) with a box of length $L = \pi/8$ (left), $L = \pi$ (center), and $L = 8\pi$ (right).
}
\end{figure*}
%%% %%% %%%
In Fig.~\ref{fig:EvsL}, we report the excitation energies (multiplied by $L^2$) for various methods and box sizes in the case of 5-boxium (\ie, $\nEl = 5$).
Similar graphs are obtained for the other $\nEl$ values and they can be found in the {\SI} alongside the numerical data associated with each method.
For small $L$, the single and double excitations can be labeled as ``pure''.
In other words, each excitation is dominated by a sole, well-defined reference Slater determinant.
However, when the box gets larger (\ie, $L$ increases), there is a strong mixing between the different excitation degrees.
In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
This is clearly evidenced if one looks at the weights of the different configurations in the FCI wave function.
Therefore, it is paramount to construct a two-weight functional (as we have done here) which allows the mixing of single and double configurations.
Using a single-weight (\ie, a biensemble) functional, where only the ground state and the lowest singly-excited states are taken into account, evidences a quick deterioration of the excitation energies (as compared to FCI) when the box gets larger.
\titou{Shall we add results for $\ew{2} = 0$ to illustrate this?}
In the weakly correlated regime (\ie, small $L$), all methods provide accurate estimates of the excitation energies.
When the box gets larger, they start to deviate.
For the single excitation, TDLDA is extremely accurate up to $L = 2\pi$, but yields more significant errors at larger $L$ by underestimating the excitation energies.
TDA-TDLDA slightly corrects this trend thanks to error compensation.
Concerning the eLDA functional, our results clearly evidence that the equi-weights [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
Concerning the eLDA functional, our results clearly evidence that the equiweight [\ie, $\bw = (1/3,1/3)$] excitation energies are much more accurate than the ones obtained in the zero-weight limit [\ie, $\bw = (0,0)$].
This is especially true for the single excitation which is significantly improved by using state-averaged weights.
The effect on the double excitation is less pronounced.
Overall, one clearly sees that, with state-averaged weights, the eLDA functional yields accurate excitation energies for both single and double excitations.
@ -905,32 +914,14 @@ This conclusion is verified for smaller and larger number of electrons (see {\SI
\end{figure}
%%% %%% %%%
It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage (with respect to FCI) on the excitation energies obtained at the eLDA and eHF levels [see Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:EI-eHF}, respectively] as a function of the box length $L$ in the case of 5-boxium.
\titou{The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
Its contribution is also significantly larger in the case of the single excitation; the derivative discontinuity hardly influence the double excitation.
Importantly, one realises that the magnitude of the derivative discontinuity is much smaller in the case of state-averaged calculations (as compared to the zero-weight calculations).
This could explain why equiensemble calculations are clearly more accurate as it reduces the influence of the derivative discontinuity: for a given method, state-averaged orbitals partially remove the burden of modelling properly the derivative discontinuity.
}
Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\nEl$ for three values of $L$ ($L=\pi/8$, $\pi$, and $8\pi$).
We draw similar conclusions as above: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations, with a very significant improvement brought by the state-averaged KS-eLDA orbitals as compared to their zero-weight analogs.
As a rule of thumb, in the weak and intermediate correlation regimes, we see that KS-eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$.
Finally, we note that, in the strong correlation regime (left graph of Fig.~\ref{fig:EvsN}), the single excitation energies obtained at the state-averaged KS-eLDA level remain in good agreement with FCI and are much more accurate than the TDLDA and TDA-TDLA excitation energies which can deviate by up to $60 \%$.
This also applies to double excitations, the discrepancy between FCI and KS-eLDA remaining of the order of a few percents in the strong correlation regime.
These observations nicely illustrate the robustness of the present state-averaged GOK-DFT scheme in any correlation regime for both single and double excitations.
%%% FIG 3 %%%
\begin{figure}
\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 eLDA (solid lines) and eHF (dashed lines) levels of theory.
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.
}
\end{figure}
%%% %%% %%%
Figure \ref{fig:EvsN} reports the error (in \%) in excitation energies (as compared to FCI), for the same methods, as a function of $\nEl$ and fixed $L$ (in this case $L=\pi$).
The graphs associated with other $L$ values are reported as {\SI}.
Again, the graph for $L=\pi$ is quite typical and we draw similar conclusions as in the previous paragraph: irrespectively of the number of electrons, the eLDA functional with state-averaged weights is able to accurately model single and double excitations.
As a rule of thumb, we see that eLDA single excitations are of the same quality as the ones obtained in the linear response formalism (such as TDLDA), while double excitations only deviates from the FCI values by a few tenth of percent for $L=\pi$.
Even for larger boxes, the discrepancy between FCI and eLDA for double excitations is only few percents.
%%% FIG 4 %%%
\begin{figure*}
\includegraphics[width=\linewidth]{EvsN}
\caption{
@ -940,17 +931,37 @@ Even for larger boxes, the discrepancy between FCI and eLDA for double excitatio
\end{figure*}
%%% %%% %%%
\titou{For small $L$, the single and double excitations are ``pure''. In other words, the excitation is dominated by a single reference Slater determinant.
However, when the box gets larger, there is a strong mixing between different degree of excitations.
In particular, the single and double excitations strongly mix.
This is clearly evidenced if one looks at the weights of the different configurations in the FCI wave function.
In one hand, if one does construct a eDFA with a single state (either single or double), one clearly sees that the results quickly deteriorates when the box gets larger.
On the other hand, building a functional which does mix singles and doubles corrects this by allowing configuration mixing.}
It is also interesting to investigate the influence of the derivative discontinuity on both the single and double excitations.
To do so, we have reported in Fig.~\ref{fig:EvsLHF} the error percentage (with respect to FCI) on the excitation energies obtained at the KS-eLDA and eHF levels [see Eqs.~\eqref{eq:EI-eLDA} and \eqref{eq:EI-eHF}, respectively] as a function of the box length $L$ in the case of 5-boxium.
The influence of the derivative discontinuity is clearly more important in the strong correlation regime.
Its contribution is also significantly larger in the case of the single excitation; the derivative discontinuity hardly influence the double excitation.
Importantly, one realizes that the magnitude of the derivative discontinuity is much smaller in the case of state-averaged calculations (as compared to the zero-weight calculations).
This could explain why equiensemble calculations are clearly more accurate as it reduces the influence of the derivative discontinuity: for a given method, state-averaged orbitals partially remove the burden of modeling properly the derivative discontinuity.
\manu{It might be useful to add eHF results where one switch off the correlation part.
For both zero weight and state-averaged weights?
It would highlight the contribution of the derivative discontinuity.}
%%% FIG 4 %%%
\begin{figure}
\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 of theory.
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.
}
\end{figure}
%%% %%% %%%
Finally, in Fig.~\ref{fig:EvsN_HF}, we report the same quantities as a function of the number of electrons for a box of length $8\pi$ (\ie, in the strongly-correlated regime).
The difference between the eHF and KS-eLDA excitation energies undoubtedly show that, even in the strongly-correlated regime, the derivative discontinuity has i) a small impact on the double excitations and has a slight tendency of worsening the excitation energies in the case of state-averaged weights, and ii) a rather large influence on the single excitation energies obtained in the zero-weight limit, showing once again that the usage of state-averaged weights has the benefit of significantly reducing the magnitude of the derivative discontinuity.
%%% FIG 5 %%%
\begin{figure}
\includegraphics[width=\linewidth]{EvsN_HF}
\caption{
\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 of theory.
Zero-weight (\ie, $\ew{1} = \ew{2} = 0$, black and red lines) and state-averaged (\ie, $\ew{1} = \ew{2} = 1/3$, blue and green lines) calculations are reported.
}
\end{figure}
%%% %%% %%%
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\section{Concluding remarks}

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Notebooks/eDFT_FUEG.nb
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