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@ -325,7 +325,7 @@ Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the p
%%%%%%%%%%%%%%%%%%%%%%%%
As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary density functional to a basis set. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective \textit{local} range-separation parameter in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual functionals used in this work are introduced in Sec.~\ref{sec:def_func}.
\subsection{Basic equations}
\subsection{Basic theory}
\label{sec:basic}
The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over $N$-representable one-electron densities $\denr$
@ -463,10 +463,10 @@ is the valence-only effective interaction and
One would note the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
\subsection{Complementary density functionals}
\subsection{General form of the complementary density functional}
\label{sec:functional}
\subsubsection{Generic form}
\subsubsection{Generic approximate form}
\label{sec:functional_form}
As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary density functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}\cite{TouGorSav-TCA-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the one-electron density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$ (where $n_\uparrow(\br{})$ and $n_\downarrow(\br{})$ are the spin-up and spin-down densities), the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$.
@ -505,13 +505,15 @@ Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes, \ie,
\label{eq:lim_n2}
\lim_{\ntwo \to 0} \ecmd(\argecmd) = 0,
\end{equation}
which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems. Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \to \infty$ like all RSDFT short-range functionals, \ie,
which is expected for systems with a vanishing on-top pair density.
%, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems.
Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \to \infty$ like all RSDFT short-range functionals, \ie,
\begin{equation}
\label{eq:lim_muinf}
\lim_{\mu \to \infty} \ecmd(\argecmd) = 0.
\end{equation}
\subsubsection{Properties}
\subsubsection{Two limits where the complementary density functional vanishes}
\label{sec:functional_prop}
Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary density functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.
@ -519,23 +521,28 @@ Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis
First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, independently of the type of wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$ in a given basis set $\Bas$,
\begin{equation}
\label{eq:lim_ebasis}
\lim_{\basis \to \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis,
\lim_{\basis \to \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis.
\end{equation}
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}, and ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
Second, $\efuncdenpbe{\argebasis}$ correctly vanishes for systems with uniformily vanishing on-top pair density, such as one-electron systems and for the stretched H$_2$ molecule,
\begin{equation}
\label{eq:lim_ebasis}
\lim_{n_2 \to 0} \efuncdenpbe{\argebasis} = 0.
\end{equation}
This property is doubly guaranteed by i) the choice of setting the effective interaction $\wbasis$ at $\infty$ for a vanishing pair density [see Eq.~\eqref{eq:wbasis}] leading to $\mu(\br{}) \to \infty$ and thus a vanishing $\ecmd(\argecmd)$ according to Eq.~\eqref{eq:lim_muinf}, and ii) the fact that $\ecmd(\argecmd)$ vanishes anyway when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
\subsection{Requirements for strong correlation}
\subsection{Requirements on the complementary density functional for strong correlation}
\label{sec:requirements}
An important requirement for any electronic-structure method is size-consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{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 self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies.
An important requirement for any electronic-structure method is size consistency, \ie$\,$ the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{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 self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies.
Another important requirement is spin-multiplet degeneracy, \ie, 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 $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components.
Another important requirement is spin-multiplet degeneracy, \ie$\,$ 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 $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components.
\subsubsection{Spin-multiplet degeneracy}
A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$. In the case of the functional $\ecmd(\argecmd)$, this means removing the dependency on the spin polarization $\zeta(\br{})$ originating from the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}].
A sufficient condition to achieve spin-multiplet degeneracy is to eliminate all dependencies on $S_z$. In the case of the function $\ecmd(\argecmd)$, this means removing the dependency on the spin polarization $\zeta(\br{})$ originating from the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ [see Eq.~\eqref{eq:def_ecmdpbe}].
To do so, it has been proposed to substitute the dependency on the spin polarization by the dependency on the on-top pair density. Most often, it is done by introducing an effective spin polarization~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01})
To do so, it has been proposed to replace the dependency on the spin polarization by the dependency on the on-top pair density. Most often, it is done by introducing an effective spin polarization~\cite{MosSan-PRA-91,BecSavSto-TCA-95,Sav-INC-96a,Sav-INC-96,MieStoSav-MP-97,TakYamYam-CPL-02,TakYamYam-IJQC-04,GraCre-MP-05,TsuScuSav-JCP-10,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15,GarBulHenScu-PCCP-15,CarTruGag-JCTC-15,GagTruLiCarHoyBa-ACR-17} (see, also, Refs.~\onlinecite{PerSavBur-PRA-95,StaDav-CPL-01})
\begin{equation}
\label{eq:def_effspin}
\tilde{\zeta}(n,n_{2}) =
@ -544,7 +551,7 @@ To do so, it has been proposed to substitute the dependency on the spin polariza
% 0 & \text{otherwise.}
% \end{cases}
\end{equation}
expressed as a function of the density $n$ and the on-top pair density $n_2$, calculated from a given wave function. The advantage of this approach is that this effective spin polarization $\tilde{\zeta}$ is independent from $S_z$, since the on-top pair density is $S_z$-independent. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents some disadvantages since this expression was derived for a single-determinant wave function. Hence, it does not appear justified to use it for a multideterminant wave function. More particularly, it may happen, in the multideterminant case, that $1 - 2 \; n_{2}/n^2 < 0 $ which results in a complex-valued spin polarization [see Eq.~\eqref{eq:def_effspin}]. \cite{BecSavSto-TCA-95}
expressed as a function of the density $n$ and the on-top pair density $n_2$, calculated from a given wave function. The advantage of this approach is that this effective spin polarization $\tilde{\zeta}$ is independent from $S_z$, since the on-top pair density is $S_z$-independent. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents some disadvantages since this expression was derived for a single-determinant wave function. Hence, it does not appear justified to use it for a multideterminant wave function. More particularly, it may happen, in the multideterminant case, that $1 - 2 \; n_{2}/n^2 < 0 $ which results in a complex-valued effective spin polarization. \cite{BecSavSto-TCA-95}
Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15} we use the following definition
\begin{equation}
\label{eq:def_effspin-0}
@ -556,16 +563,16 @@ Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-J
\end{cases}
\end{equation}
An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures its $S_z$ invariance and, as will be numerically demonstrated, very weakly affects the complementary density functional accuracy.
An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures its $S_z$ independence and, as will be numerically demonstrated, very weakly affects the complementary density functional accuracy.
\subsubsection{Size consistency}
Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$.
We refer the interested reader to the {\SI} for a detailed demonstration of the latter statement.
We refer the interested reader to the {\SI} for a detailed proof of the latter statement.
In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function is sufficient to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.
\subsection{Complementary density functional approximations}
\subsection{Actual approximations used for the complementary density functional}
\label{sec:def_func}
%\subsubsection{Definition of the protocol to design functionals}