Manu: saving work.

Manuscript/eDFT.tex

@@ -219,7 +219,8 @@ They are normalized, \ie,
 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}. 
-In GOK-DFT, the ensemble energy is determined variationally as follows~\cite{Gross_1988b}:
+In the KS formulation of GOK-DFT, \manu{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}:
 \beq\label{eq:var_ener_gokdft}
 	\E{}{\bw}
  	= \min_{\opGam{\bw}}
@@ -1025,12 +1026,22 @@ We use as basis functions the (orthonormal) orbitals of the one-electron system,
 	\end{cases}
 \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 $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw} \bS - \bS \bGam{\bw} \bF{\bw}}}$ is set to $10^{-5}$.
+\manu{The convergence threshold $\tau = \max{ \abs{ \bF{\bw} \bGam{\bw}
-For KS-DFT, KS-eDFT and TDDFT calculations, a 51-point Gauss-Legendre quadrature is employed to compute the various integrals that cannot be performed in closed form.
+\bS - \bS \bGam{\bw} \bF{\bw}}}$ [see Eq.~(\ref{eq:commut_F_AO})] is set
+to $10^{-5}$. For comparison, regular HF and KS-DFT calculations 
+are performed with the same threshold.
+In order to compute the various density-functional
+integrals that cannot be performed in closed form, 
+a 51-point Gauss-Legendre quadrature is employed.}
 
 In order to test the present eLDA functional we perform various sets of calculations.
 To get reference excitation energies for both the single and double excitations, we compute full configuration interaction (FCI) energies with the Knowles-Handy FCI program described in Ref.~\onlinecite{Knowles_1989}.
-For the single excitations, we also perform 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}
+For the single excitations, we also perform 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}\manu{Manu: has been
+studied previously (if so why do you mention this?) or will be discussed
+in the present work?}
 Concerning the KS-eDFT and eHF calculations, two sets of weight are tested: the zero-weight limit where $\bw = (0,0)$ and the equi-ensemble (or state-averaged) limit where $\bw = (1/3,1/3)$.
 
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