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41
Manuscript/srDFT_SC.bbl
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@ -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
{NoStop}%
\bibinfo {pages} {032506} (\bibinfo {year} {2006})}\BibitemShut {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

37
Manuscript/srDFT_SC.tex
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@ -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.
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}).
which guarantees an unaltered CBS limit.
\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}
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.
\subsubsection{Requirements: size-consistency and spin-multiplet degeneracy}
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}