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Pierre-Francois Loos 2020-01-06 14:11:02 +01:00
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Manuscript/srDFT_SC.tex
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@ -328,7 +328,7 @@ As the theory behind the present basis-set correction has been exposed in detail
\subsection{Basic equations} \subsection{Basic equations}
\label{sec:basic} \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 \titou{one-electron densities} $\denr$ The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over one-electron densities $\denr$
\begin{equation} \begin{equation}
\label{eq:levy} \label{eq:levy}
E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\}, E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\},
@ -434,7 +434,7 @@ Because of the very definition of $\wbasis$, one has the following property in t
which is again fundamental to guarantee the correct behavior of the theory in the CBS limit. which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
\subsubsection{Frozen-core approximation} \subsubsection{Frozen-core approximation}
As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the complementary density functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as \titou{$\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals}, respectively, and define the valence-only local range-separation parameter as As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the complementary density functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
\begin{equation} \begin{equation}
\label{eq:def_mur_val} \label{eq:def_mur_val}
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{}, \murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
@ -534,7 +534,7 @@ Another important requirement is spin-multiplet degeneracy, \ie, the independenc
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 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}].
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}) \manu{is zero if square roots is complex} 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})
\begin{equation} \begin{equation}
\label{eq:def_effspin} \label{eq:def_effspin}
\tilde{\zeta}(n,n_{2}) = \tilde{\zeta}(n,n_{2}) =
@ -544,10 +544,16 @@ To do so, it has been proposed to substitute the dependency on the spin polariza
% \end{cases} % \end{cases}
\end{equation} \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 spin polarization [see Eq.~\eqref{eq:def_effspin}]. \cite{BecSavSto-TCA-95}
Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-JCTC-14,GarBulHenScu-JCP-15} we use the following definition
%The advantage of this approach are at least two folds: i) the effective spin polarization $\tilde{\zeta}$ is independent from $S_z$ since the on-top pair density is independent from $S_z$, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation. \begin{equation}
%Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $1 - 2 \; n_{2}/n^2 < 0 $ and also \label{eq:def_effspin-0}
%the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo^{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions. \tilde{\zeta}(n,n_{2}) =
\begin{cases}
\sqrt{ 1 - 2 \; n_{2}/n^2 }, & \text{if } n^2 \ge 4 n_{2},
\\
0, & \text{otherwise.}
\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 $S_z$ invariance and, as will be numerically demonstrated, very weakly affects the complementary density functional accuracy.
@ -565,19 +571,19 @@ In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shel
As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density. As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density.
Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) \titou{the \textit{effective} spin polarization $\tilde{\zeta}$} defined in Eq.~\eqref{eq:def_effspin} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization. \titou{In the latter case, the functional is referred as to ``SU'' which stands for ``spin unpolarized''}. Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin-0} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization. In the latter case, the functional is referred as to ``SU'' which stands for ``spin unpolarized''.
Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
\begin{equation} \begin{equation}
\label{eq:def_n2ueg} \label{eq:def_n2ueg}
\ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n), \ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n),
\end{equation} \end{equation}
where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$. When using $\ntwo^{\text{UEG}}(n,\tilde{\zeta})$ in a functional, we will refer to it as ``UEG''. where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin-0} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$. When using $\ntwo^{\text{UEG}}(n,\tilde{\zeta})$ in a functional, we will refer to it as ``UEG''.
Another approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the present authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density Another approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the present authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density
\begin{equation} \begin{equation}
\label{eq:def_n2extrap} \label{eq:def_n2extrap}
\ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo, \ntwoextrap(\ntwo,\mu) = \qty( 1 + \frac{2}{\sqrt{\pi}\mu} )^{-1} \; \ntwo,
\end{equation} \end{equation}
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. When using $\ntwoextrap(\ntwo,\mu)$ in a functional, we will simply refer it as ``OT'', which stands for "on-top". which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. When using $\ntwoextrap(\ntwo,\mu)$ in a functional, we will simply refer it as ``OT'', which stands for "on-top".
@ -585,14 +591,14 @@ We then define four functionals:
\begin{itemize} \begin{itemize}
\item[i)] $\pbeuegXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin} and the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg}: \item[i)] $\pbeuegXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin-0} and the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg}:
\begin{multline} \begin{multline}
\label{eq:def_pbeueg_i} \label{eq:def_pbeueg_i}
\bar{E}^\Bas_{\pbeuegXi} \bar{E}^\Bas_{\pbeuegXi}
\\ \\
= \int \d\br{} \,\denr \ecmd(\argrpbeuegXi), = \int \d\br{} \,\denr \ecmd(\argrpbeuegXi),
\end{multline} \end{multline}
\item[ii)] $\pbeontXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin} and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}: \item[ii)] $\pbeontXi$ which combines the effective spin polarization of Eq.~\eqref{eq:def_effspin-0} and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}:
\begin{equation} \begin{equation}
\label{eq:def_pbeueg_ii} \label{eq:def_pbeueg_ii}
\bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi), \bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi),