Done for T2

Manuscript/eDFT.tex

@@ -254,7 +254,7 @@ Atomic units are used throughout.
 In this section we give a brief review of GOK-DFT and discuss the
 extraction of individual energy levels \cite{Deur_2019,Fromager_2020} with a particular focus on exact
 individual exchange energies. 
-Let us start by introducing the GOK ensemble energy~\cite{Gross_1988a}:
+Let us start by introducing the GOK ensemble energy: \cite{Gross_1988a}
 \beq\label{eq:exact_GOK_ens_ener}
 	\E{}{\bw}=\sum_{K \geq 0} \ew{K} \E{}{(K)},
 \eeq
@@ -270,9 +270,9 @@ They are normalized, \ie,
 \eeq
 so that only the weights $\bw \equiv \qty( \ew{1}, \ew{2}, \ldots, \ew{K}, \ldots )$ assigned to the excited states can vary independently.  
 For simplicity we will assume in the following that the energies are not degenerate. 
-Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states~\cite{Gross_1988b}. 
+Note that the theory can be extended to multiplets simply by assigning the same ensemble weight to all degenerate states.\cite{Gross_1988b}
 In the KS formulation of GOK-DFT, {which is simply referred to as
-KS ensemble DFT (KS-eDFT) in the following}, the ensemble energy is determined variationally as follows~\cite{Gross_1988b}:
+KS ensemble DFT (KS-eDFT) in the following}, the ensemble energy is determined variationally as follows:\cite{Gross_1988b}
 \beq\label{eq:var_ener_gokdft}
 	\E{}{\bw}
  	= \min_{\opGam{\bw}}
@@ -305,7 +305,7 @@ The (approximate) description of the correlation part is discussed in
 Sec.~\ref{sec:eDFA}.
 
 In practice, the ensemble energy is not the most interesting quantity, and one is more concerned with excitation energies or individual energy levels (for geometry optimizations, for example). 
-As pointed out recently in Ref.~\cite{Deur_2019}, the latter can be extracted
+As pointed out recently in Ref.~\onlinecite{Deur_2019}, the latter can be extracted
 exactly from a single ensemble calculation as follows:
 \beq\label{eq:indiv_ener_from_ens}
 	\E{}{(I)} = \E{}{\bw} + \sum_{K>0} \qty(\delta_{IK} - \ew{K} )
@@ -349,7 +349,7 @@ auxiliary double-weight ensemble density reads
 Since, for given ensemble weights $\bw$ and $\bxi$, the ensemble densities $\n{}{\bxi,\bxi}$ and $\n{}{\bw,\bxi}$ are generated from the \textit{same} KS potential (which is unique up to a constant), it comes
 from the exact expression in Eq.~(\ref{eq:exact_ens_Hx}) that 
 \beq
-	\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}}
+	\E{Hx}{\bxi}[\n{}{\bxi,\bxi}] = \sum_{K \geq 0} \xi_K \mel*{\Det{(K),\bxi}}{\hWee}{\Det{(K),\bxi}},
 \eeq  
 and
 \beq
@@ -376,7 +376,7 @@ Since, according to Eqs.~(\ref{eq:var_ener_gokdft}) and (\ref{eq:exact_ens_Hx}),
 \eeq 
 with $\Det{(K)} = \Det{(K),\bw}$ [note that, when the minimum is reached in Eq.~\eqref{eq:var_ener_gokdft}, $\n{\opGam{\bw}}{} = \n{}{\bw,\bw}$], 
 we finally recover from Eqs.~\eqref{eq:KS_ens_density} and
-\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\cite{Fromager_2020} for the $I$th energy level:
+\eqref{eq:indiv_ener_from_ens} the {\it exact} expression of Ref.~\onlinecite{Fromager_2020} for the $I$th energy level:
 \beq\label{eq:exact_ener_level_dets}
 \begin{split}
 	\E{}{(I)} 
@@ -419,7 +419,7 @@ C_{\rm c}[n]=\dfrac{\E{c}{}[n]
 \fdv{\E{c}{}[n]}{\n{}{}(\br{})}n(\br{})d\br{}}{\int n(\br{})d\br{}}
 \eeq
 is the correlation component of
-Levy--Zahariev's constant shift in potential~\cite{Levy_2014}. 
+Levy--Zahariev's constant shift in potential.\cite{Levy_2014}
 Similarly, the excited-state ($I>0$) energy level expressions
 can be recast as follows: 
 \beq\label{eq:excited_ener_level_gs_lim}
@@ -451,7 +451,7 @@ as basic variables, rather than Slater determinants.
 As the theory is applied later on to {\it spin-polarized}
 systems, we drop spin indices in the density matrices, for convenience.
 If we expand the
-ensemble KS orbitals [from which the determinants are constructed] in an atomic orbital (AO) basis,
+ensemble KS orbitals (from which the determinants are constructed) in an atomic orbital (AO) basis,
 \beq
 	\MO{p}{}(\br{}) =  \sum_{\mu} \cMO{\mu p}{} \AO{\mu}(\br{}),
 \eeq
@@ -535,7 +535,7 @@ where
 \beq
 	\bh \equiv h_{\mu\nu} = \mel*{\AO{\mu}}{\hh}{\AO{\nu}}	
 \eeq
-denotes the one-electron integrals matrix.
+denotes the matrix of the one-electron integrals.
 The exact individual Hx energies are obtained from the following trace formula
 \beq
 	\Tr[\bGam{(K)} \bG \bGam{(L)}]
@@ -650,7 +650,7 @@ w}_K
 
 
 In the following, GOK-DFT will be applied
-to one-dimension
+to 1D
 spin-polarized systems where 
 Hartree and exchange energies cannot be separated. 
 For that reason, we will substitute the Hartree--Fock (HF) density-matrix-functional interaction energy,
@@ -809,11 +809,11 @@ Note that, within the approximation of Eq.~(\ref{eq:min_with_HF_ener_fun}), the
 optimized with a non-local exchange potential rather than a
 density-functional local one, as expected from
 Eq.~\eqref{eq:var_ener_gokdft}. This procedure is actually general, \ie,
-applicable to not-necessarily spin polarized and real (higher-dimension) systems. 
+applicable to not-necessarily spin polarized and real (higher-dimensional) systems. 
 As readily seen from Eq.~\eqref{eq:eHF-dens_mat_func}, inserting the
 ensemble density matrix into the HF interaction energy functional
-introduces unphysical \textit{ghost interaction} errors~\cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
-as well as {\it curvature}~\cite{Alam_2016,Alam_2017}:
+introduces unphysical \textit{ghost interaction} errors \cite{Gidopoulos_2002, Pastorczak_2014, Alam_2016, Alam_2017, Gould_2017}
+as well as \textit{curvature}:\cite{Alam_2016,Alam_2017}
 \beq\label{eq:WHF}
 \begin{split}
 	\WHF[\bGam{\bw}] 
@@ -828,7 +828,7 @@ These errors are essentially removed when evaluating the individual energy
 levels on the basis of Eq.~\eqref{eq:exact_ind_ener_rdm}.
 
 Turning to the density-functional ensemble correlation energy, the
-following ensemble local density approximation (eLDA) will be employed:  
+following ensemble local density \textit{approximation} (eLDA) will be employed:  
 \beq\label{eq:eLDA_corr_fun}
 	\E{c}{\bw}[\n{}{}]\approx \int \n{}{}(\br{}) \e{c}{\bw}(\n{}{}(\br{})) d\br{},
 \eeq
@@ -870,9 +870,9 @@ of Eq.~\eqref{eq:EI-eLDA} can be simplified as follows through first order in
 $\n{\bGam{\bw}}{}(\br{})-\n{\bGam{(I)}}{}(\br{})$: 
 \beq
 \int \e{c}{\bw}(\n{\bGam{(I)}}{}(\br{})) \n{\bGam{(I)}}{}(\br{}) d\br{}
-+\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right),
++\mathcal{O}\left([\n{\bGam{\bw}}{}-\n{\bGam{(I)}}{}]^2\right).
 \eeq
-and it can therefore be identified as 
+Therefore, it can be identified as 
 an individual-density-functional correlation energy where the density-functional
 correlation energy per particle is approximated by the ensemble one for
 all the states within the ensemble. 
@@ -1222,11 +1222,11 @@ However, when the box gets larger (\ie, $L$ increases), there is a strong mixing
 In particular, the single and double excitations strongly mix, which makes their assignment as single or double excitations more discutable. \cite{Loos_2019}
 This can be clearly evidenced by the weights of the different configurations in the FCI wave function.
 % TITOU: shall we keep the paragraph below?
-Therefore, it is paramount to construct a two-weight correlation functional
-(\ie, a triensemble functional, as we have done here) which
-allows the mixing of singly- and doubly-excited configurations.
-Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
-\titou{Titou will add results for the biensemble to illustrate this.}
+%Therefore, it is paramount to construct a two-weight correlation functional
+%(\ie, a triensemble functional, as we have done here) which
+%allows the mixing of singly- and doubly-excited configurations.
+%Using a single-weight (\ie, a biensemble) functional where only the ground state and the lowest singly-excited states are taken into account, one would observe a neat deterioration of the excitation energies (as compared to FCI) when the box gets larger.
+%\titou{Titou might add results for the biensemble to illustrate this.}
 %\manu{Well, neglecting the second excited state is not the same as
 %considering the $w_2=0$ limit. I thought you were referring to an
 %approximation where the triensemble calculation is performed with
@@ -1314,9 +1314,9 @@ electrons.
 \end{figure}
 %%% %%% %%%
 
-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.
+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.
 To do so, we have reported in Fig.~\ref{fig:EvsL_DD}, in the case of 3-boxium, the error percentage (with respect to FCI) as a function of the box length $L$
-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}].
+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}].
 %\manu{Manu: there is something I do not understand. If you want to
 %evaluate the importance of the ensemble correlation derivatives you
 %should only remove the following contribution from the $K$th KS-eLDA
@@ -1336,6 +1336,7 @@ 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.
+\titou{Note also that, in our case, the second term in Eq.~\eqref{eq:Om-eLDA}, which involves the weight-dependent correlation potential and the density difference between ground and excited states, has a negligible effect on the excitation energies.}
 %\manu{Manu: well, we
 %would need the exact derivative value to draw such a conclusion. We can
 %only speculate. Let us first see how important the contribution in
@@ -1354,7 +1355,7 @@ of modeling properly the ensemble correlation derivative.
 %%% %%% %%%
 
 Finally, in Fig.~\ref{fig:EvsN_DD}, we report the same quantities as a function of the electron number for a box of length $8\pi$ (\ie, in the strong correlation regime). 
-The difference between the solid and dashed lines and KS-eLDA excitation energies
+The difference between the solid and dashed curves
 undoubtedly show that, even in the strong correlation regime, the
 ensemble correlation derivative has a small impact on the double
 excitations with a slight tendency of worsening the excitation energies
@@ -1384,11 +1385,12 @@ from the finite uniform electron gas eLDA is
 combined with eLDA, delivers accurate excitation energies for both
 single and double excitations, especially when an equiensemble is used.
 In the latter case, the same weights are assigned to each state belonging to the ensemble.
-{\it We have observed that, although the derivative discontinuity has a
+\titou{The improvement on the excitation energies brought by the KS-eLDA scheme is particularly impressive in the strong correlation regime where usual methods, such as TDLDA, fail.
+We have observed that, although the ensemble correlation discontinuity has a
 non-negligible effect on the excitation energies (especially for the
 single excitations), its magnitude can be significantly reduced by
-performing state-averaged calculations instead of zero-weight
-calculations.}\manu{to be updated ...}
+performing equiweight calculations instead of zero-weight
+calculations.}
 
 Let us finally stress that the present methodology can be extended
 straightforwardly to other types of ensembles like, for example, the