changes in theory

Manuscript/srDFT_SC.bbl

@@ -307,14 +307,13 @@
   }\textbf {\bibinfo {volume} {70}},\ \bibinfo {pages} {062505} (\bibinfo
   {year} {2004})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
-  {Savin}(2006{\natexlab{a}})}]{GorSav-PRA-06}%
+  {Savin}(2006)}]{GorSav-PRA-06}%
   \BibitemOpen
   \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
   {Gori-Giorgi}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
   {Savin}},\ }\href {\doibase 10.1103/PhysRevA.73.032506} {\bibfield  {journal}
   {\bibinfo  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {73}},\
-  \bibinfo {pages} {032506} (\bibinfo {year} {2006}{\natexlab{a}})}\BibitemShut
+  \bibinfo {pages} {032506} (\bibinfo {year} {2006})}\BibitemShut {NoStop}%
-  {NoStop}%
 \bibitem [{\citenamefont {Franck}\ \emph {et~al.}(2015)\citenamefont {Franck},
   \citenamefont {Mussard}, \citenamefont {Luppi},\ and\ \citenamefont
   {Toulouse}}]{FraMusLupTou-JCP-15}%
@@ -579,24 +578,6 @@
   }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {Phys. Rev. B}\
   }\textbf {\bibinfo {volume} {73}},\ \bibinfo {pages} {155111} (\bibinfo
   {year} {2006})}\BibitemShut {NoStop}%
-\bibitem [{\citenamefont {Gori-Giorgi}\ and\ \citenamefont
-  {Savin}(2006{\natexlab{b}})}]{GoriSav-PRA-06}%
-  \BibitemOpen
-  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
-  {Gori-Giorgi}}\ and\ \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
-  {Savin}},\ }\href {\doibase 10.1103/PhysRevA.73.032506} {\bibfield  {journal}
-  {\bibinfo  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {73}},\
-  \bibinfo {pages} {032506} (\bibinfo {year} {2006}{\natexlab{b}})}\BibitemShut
-  {NoStop}%
-\bibitem [{\citenamefont {Gritsenko}, \citenamefont {van Meer},\ and\
-  \citenamefont {Pernal}(2018)}]{GritMeePer-PRA-18}%
-  \BibitemOpen
-  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {O.~V.}\ \bibnamefont
-  {Gritsenko}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {van Meer}},
-  \ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Pernal}},\ }\href
-  {\doibase 10.1103/PhysRevA.98.062510} {\bibfield  {journal} {\bibinfo
-  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo
-  {pages} {062510} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Carlson}, \citenamefont {Truhlar},\ and\
   \citenamefont {Gagliardi}(2017)}]{CarTruGag-JPCA-17}%
   \BibitemOpen
@@ -606,6 +587,24 @@
   {Gagliardi}},\ }\href@noop {} {\bibfield  {journal} {\bibinfo  {journal} {J.
   Phys. Chem. A}\ }\textbf {\bibinfo {volume} {121}},\ \bibinfo {pages} {5540}
   (\bibinfo {year} {2017})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Gritsenko}, \citenamefont {van Meer},\ and\
+  \citenamefont {Pernal}(2018)}]{GritMeePer-PRA-18}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {O.~V.}\ \bibnamefont
+  {Gritsenko}}, \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {van Meer}},
+  \ and\ \bibinfo {author} {\bibfnamefont {K.}~\bibnamefont {Pernal}},\ }\href
+  {\doibase 10.1103/PhysRevA.98.062510} {\bibfield  {journal} {\bibinfo
+  {journal} {Phys. Rev. A}\ }\textbf {\bibinfo {volume} {98}},\ \bibinfo
+  {pages} {062510} (\bibinfo {year} {2018})}\BibitemShut {NoStop}%
+\bibitem [{\citenamefont {Becke}, \citenamefont {Savin},\ and\ \citenamefont
+  {Stoll}(1995)}]{BecSavSto-TCA-95}%
+  \BibitemOpen
+  \bibfield  {author} {\bibinfo {author} {\bibfnamefont {A.~D.}\ \bibnamefont
+  {Becke}}, \bibinfo {author} {\bibfnamefont {A.}~\bibnamefont {Savin}}, \ and\
+  \bibinfo {author} {\bibfnamefont {H.}~\bibnamefont {Stoll}},\ }\href@noop {}
+  {\bibfield  {journal} {\bibinfo  {journal} {Theoret. Chim. Acta}\ }\textbf
+  {\bibinfo {volume} {{91}}},\ \bibinfo {pages} {147} (\bibinfo {year}
+  {1995})}\BibitemShut {NoStop}%
 \bibitem [{\citenamefont {Perdew}, \citenamefont {Savin},\ and\ \citenamefont
   {Burke}(1995)}]{PerSavBur-PRA-95}%
   \BibitemOpen

Manuscript/srDFT_SC.tex

@@ -510,30 +510,33 @@ which is expected for systems with a vanishing on-top pair density, such as the
   \lim_{\mu \rightarrow \infty} \ecmd(\argecmd) = 0. 
 \end{equation}
 
-\subsubsection{Properties of approximated functionals}
+\subsubsection{Properties of approximate functionals}
 \label{sec:functional_prop}
-Within the definition of \eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximated complementary basis set functionals $\efuncdenpbe{\argecmd}$ satisfies two important properties. 
+
-Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncdenpbe{\argecmd}$ vanishes when reaching the complete basis set limit, whatever the wave function $\psibasis$ used to define the range separation parameter $\mu_{\Psi^{\basis}}$: 
+Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary basis functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.
+
+First, thanks to the properties in Eq.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, whatever the wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$, 
 \begin{equation}
  \label{eq:lim_ebasis}
- \lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argecmd} = 0\quad \forall\, \psibasis, 
+ \lim_{\basis \rightarrow \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis, 
 \end{equation}
-which guarantees an unaltered limit when reaching the CBS limit. 
+which guarantees an unaltered CBS limit. 
-Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system. 
-Such a property is guaranteed independently by i) the definition of the effective interaction  $\wbasis$ (see equation \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see equation \eqref{eq:lim_n2}). 
 
-\subsection{Requirements for the approximated functionals in the strong correlation regime}
+Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see Eq.~\eqref{eq:wbasis}) together with the condition in Eq.~\eqref{eq:lim_muinf}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see Eq.~\eqref{eq:lim_n2}). 
+
+\subsection{Requirements for the approximate functionals in the strong-correlation regime}
 \label{sec:requirements}
-\subsubsection{Requirements: separability of the energies and $S_z$ invariance}
-An important requirement for any electronic structure method is the extensivity of the energy, \textit{i. e.} the additivity of the energies in the case of non interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. 
-When two subsystems $A$ and $B$ dissociate in closed shell systems, as in the case of weak interactions for instance, a simple RHF wave function leads to extensive energies. 
-When the two subsystems dissociate in open shell systems, such as in covalent bond breaking, it is well known that the RHF approach fail and an alternative is to use a CASSCF wave function which, provided that the active space has been properly chosen, leads to additives energies.  
-Another important requirement is the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function.   
-Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.
 
-\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
+\subsubsection{Requirements: size-consistency and spin-multiplet degeneracy}
-A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}). 
+
-As originally shown by Perdew and co-workers\cite{PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density. 
+An important requirement for any electronic-structure method is size-consistency, i.e. the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems $A$ and $B$ dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space (CAS) wave function which, provided that the active space has been properly chosen, leads to additive energies.  
+
+Another important requirement is spin-multiplet degeneracy, i.e. the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $A+B$ is generally low spin while the ground states of the fragments $A$ and $B$ are high spin and can have multiple $S_z$ components.
+
+\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain spin-multiplet degeneracy}
+
+A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$, which in the case of the functional $\ecmd(\argecmd)$ means removing the dependence on the spin polarization $\zeta(\br{})$ used in the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}). 
+As originally shown by Perdew and co-workers\cite{BecSavSto-TCA-95,PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density. 
 Based on this reasoning, a similar approach has been used in the context of multi configurational DFT in order to remove the $S_z$ dependency. 
 In practice, these approaches introduce the effective spin polarisation 
 \begin{equation}