changes in theory

Manuscript/srDFT_SC.tex

@@ -188,7 +188,7 @@
 \newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
 \newcommand{\denfci}[0]{\den_{\psifci}}
 \newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}}
-\newcommand{\denbas}[0]{\alert{{P}^{\Bas}(\den)}}
+\newcommand{\denbas}[0]{\den^\Bas}
 \newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas}
 \newcommand{\denrfci}[0]{\denr_{\psifci}}
 \newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})}
@@ -339,18 +339,18 @@ where $v_\text{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is th
  F[\den] = \min_{\Psi \to \den} \mel{\Psi}{\kinop +\weeop}{\Psi},
 \end{equation}
 where $\kinop$ and $\weeop$ are the kinetic and electron-electron Coulomb operators, and the notation $\Psi \to \den$ means that the wave function $\Psi$ yields the density $\den$. 
-The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, \alert{we always consider only such representable-in-$\Bas$ densities}. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$.
+The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, we always consider only such representable-in-$\Bas$ densities. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$.
 
-In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then proposed to decompose $F[\den]$ as
+In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then proposed to decompose $F[\den]$ as (for a representable-in-$\Bas$ density $\den$)
 \begin{equation}
  \label{eq:def_levy_bas}
- F[\den] = \min_{\wf{}{\Bas} \to \denbas} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, 
+ F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den}, 
 \end{equation}
-where $\wf{}{\Bas}$ are wave functions expandable in the $N$-electron Hilbert space generated by $\basis$, \alert{$\denbas$ is the projection of the density $n$ in the set of representable-in-$\Bas$ densities}, and 
+where $\wf{}{\Bas}$ are wave functions expandable in the $N$-electron Hilbert space generated by $\basis$, and 
 \begin{equation}
  \begin{aligned}
  \efuncden{\den} = \min_{\Psi \to \den} \mel*{\Psi}{\kinop +\weeop }{\Psi} 
-                  - \min_{\Psi^{\Bas} \to \denbas} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} 
+                  - \min_{\Psi^{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} 
  \end{aligned}
 \end{equation}
 is the complementary density functional to the basis set $\Bas$.
@@ -368,7 +368,7 @@ As a simple non-self-consistent version of this approach, we can approximate the
  \label{eq:e0approx}
  E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
 \end{equation}
-where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation. Two key requirements for this purpose are i) spin-multiplet degeneracy, and ii) size consistency.
+where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for strongly correlated situations. Two key requirements for this purpose are i) spin-multiplet degeneracy, and ii) size consistency.
 
 \subsection{Effective interaction in a finite basis}
 \label{sec:wee}