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Manuscript/srDFT_SC.tex
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@ -281,7 +281,7 @@
\begin{abstract}
We extend to strongly correlated systems the recently introduced basis-set correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, J. Chem. Phys. \textbf{149}, 194301 (2018)]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the Coulomb electron-electron interaction projected in the finite basis set, allowing one to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in a finite basis set. Using as test cases the potential energy curves of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit, we systematically explore different approximations for the complementary functionals which are suited to describe strong-correlation regimes and which fulfill two very desirable properties: $S_z$ invariance and size consistency. Specifically, we investigate the dependence of the functionals on different flavours of on-top pair densities and spin polarizations. An important result is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy.
We extend to strongly correlated systems the recently introduced basis-set correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, J. Chem. Phys. \textbf{149}, 194301 (2018)]. This basis-set correction relies on a mapping between wave-function calculations in a finite basis set and range-separated DFT (RSDFT) through the definition of an effective non-divergent interaction corresponding to the Coulomb electron-electron interaction projected in the finite basis set, allowing one to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in a finite basis set. Using as test cases the potential energy curves of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit, we systematically explore different approximations for the complementary functionals which are suited to describe strong-correlation regimes and which fulfill two very desirable properties: $S_z$ invariance and size consistency. Specifically, we investigate the dependence of the functionals on different flavors of on-top pair densities and spin polarizations. An important result is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy.
In the general context of multiconfigurational DFT, this finding shows that one can avoid the effective spin polarization whose mathematical definition is rather \textit{ad hoc} and which can become complex valued. Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-zeta quality basis sets for most of the systems studied. Also, the present basis-set correction provides smooth curves along the whole potential energy curves.
%We study the potential energy surfaces (PES) of the H$_{10}$, C$_2$, N$_2$, O$_2$, and F$_2$ molecules up to the dissociation limit using increasing basis sets at near full configuration interaction (FCI) level with and without the present basis-set correction.
\end{abstract}
@ -413,17 +413,17 @@ The condition in Eq.~\eqref{eq:cbs_wbasis} is fundamental as it guarantees the c
\subsection{Definition of a local range-separation parameter}
\label{sec:mur}
\subsubsection{General definition}
As the effective interaction within a basis set, $\wbasis$, is non divergent, it ressembles the long-range interaction used in RSDFT
As the effective interaction within a basis set, $\wbasis$, is non divergent, it resembles the long-range interaction used in RSDFT
\begin{equation}
\label{eq:weelr}
w_\text{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}},
\end{equation}
where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondance between these two interactions by using the local range-separation parameter $\murpsi$
where $\mu$ is the range-separation parameter. As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we make the correspondence between these two interactions by using the local range-separation parameter $\murpsi$
\begin{equation}
\label{eq:def_mur}
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal,
\end{equation}
such that the interactions coincide at the electron-electron colescence point for each $\br{}$
such that the interactions coincide at the electron-electron coalescence point for each $\br{}$
\begin{equation}
w_\text{ee}^{\lr}(\murpsi;0) = \wbasiscoal, \quad \forall \, \br{}.
\end{equation}
@ -470,7 +470,7 @@ It is noteworthy that, with the present definition, $\wbasisval$ still tends to
\subsubsection{Generic form of the approximate functionals}
\label{sec:functional_form}
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis 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 total density $\denr$, the spin polarisation $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, 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 $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis 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 total density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, 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 $\mur \equiv\murpsi$. Therefore, a given approximation X of $\efuncden{\den}$ will have the following generic local form
\begin{equation}
\begin{aligned}
\label{eq:def_ecmdpbebasis}
@ -537,7 +537,7 @@ Another important requirement is spin-multiplet degeneracy, i.e. the independenc
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 dependency on the spin polarization $\zeta(\br{})$ use the PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ (see Eq. \eqref{eq:def_ecmdpbe}).
It has been proposed to replace in functionals 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 polarisation~\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})
It has been proposed to replace in functionals 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}) =
@ -546,13 +546,13 @@ It has been proposed to replace in functionals the dependency on the spin polari
% 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 polarisation $\tilde{\zeta}$ is independent from $S_z$, since the on-top pair density is independent from $S_z$. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents the disadvantage that since this expression was derived for a single-determinant wave function, and it does not appear well justified to use it for a multideterminant wave function as well. In particular, for a multideterminant wave function, it may happen that $1 - 2 \; n_{2}/n^2 < 0 $ and thus in this cas Eq.~\eqref{eq:def_effspin} gives a complex value of $\tilde{\zeta}$~\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 independent from $S_z$. Nevertheless, the use of $\tilde{\zeta}$ in Eq.~\eqref{eq:def_effspin} presents the disadvantage that since this expression was derived for a single-determinant wave function, and it does not appear well justified to use it for a multideterminant wave function as well. In particular, for a multideterminant wave function, it may happen that $1 - 2 \; n_{2}/n^2 < 0 $ and thus in this cas Eq.~\eqref{eq:def_effspin} gives a complex value of $\tilde{\zeta}$~\cite{BecSavSto-TCA-95}.
%The advantage of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ 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.
%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.
%Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $1 - 2 \; n_{2}/n^2 < 0 $ and also
%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.
An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, i.e. to always use the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, i.e. for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for $\mu$ sufficiently large, it is a viable option. Indeed, the purpose of introducing the spin polarisation 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 Eq.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}). The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Therefore, we propose here to also test the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ independence and, as will be numerically shown, very weakly affects the accuracy of the functional.
An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, i.e. to always use the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, i.e. for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for $\mu$ sufficiently large, 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 Eq.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}). The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Therefore, we propose here to also test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures a $S_z$ independence and, as will be numerically shown, very weakly affects the accuracy of the functional.
\subsubsection{Conditions for size consistency}
@ -564,9 +564,9 @@ Since $\efuncdenpbe{\argebasis}$ is a single integral over $\mathbb{R}^3$ of loc
\label{sec:final_def_func}
\subsubsection{Definition of the protocol to design functionals}
As the present work focusses 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 CAS wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only on i) the type of spin polarisation used, and ii) the type of on-top pair density used.
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 CAS wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only on i) the type of spin polarization used, and ii) the type of on-top pair density used.
Regarding the spin polarisation that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CAS wave function, and ii) a \textit{zero} spin polarization.
Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in Eq.~\eqref{eq:def_effspin} and calculated from the CAS wave function, and ii) a \textit{zero} spin polarization.
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}
@ -695,7 +695,7 @@ $^b$ From the extrapolated valence-only non-relativistic calculations of Ref. \o
The study of the H$_{10}$ chain with equally distant atoms is a good prototype of strongly-correlated systems as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for a detailed study of this problem).
We report in Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy cruves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime.
We report in Figure \ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X=D,T,Q) basis sets for different levels of approximations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}. A general trend that can be observed from these data is that, in a given basis set, the quality of the potential energy curves are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no bizarre behaviors are found when stretching the bonds, which shows that the functionals are robust when reaching the strong correlation regime.
More quantitatively, the values of $D_0$ are within chemical accuracy (i.e., an error below 1.4 mH) from the cc-pVTZ basis set when using the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals, whereas such an accuracy is not reached at the cc-pVQZ basis set using standard MRCI+Q.
@ -770,7 +770,7 @@ The development of new $S_z$-independent and size-consistent functionals has lea
Regarding the results of the present approach, the basis-set correction systematically improves the near-FCI calculations in a given basis set. More quantitatively, it is shown that with only triple-zeta quality basis sets chemically accurate atomization energies $D_0$ are obtained for all systems but C$_2$, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of C$_2$, an error of 5.5 mH is obtained with respect to the estimated exact $D_0$, and we leave for further study the detailed investigation of the reasons of this relatively unusual poor performance of the basis-set correction.
Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long interatomic distances it cannot recover the dispersion interactions missing because of the incompleteness of the basis set. This behaviour is actually expected as the dispersion interactions are long-range correlation effects and the present approach was designed to only recover electron correlation effects near the electron-electron coalescence.
Also, it is shown that the basis-set correction gives substantial differential contribution to potential energy curves close to the equilibrium geometries, but at long internuclear distances it cannot recover the dispersion interactions missing because of the incompleteness of the basis set. This behaviour is actually expected as the dispersion interactions are long-range correlation effects and the present approach was designed to only recover electron correlation effects near the electron-electron coalescence.
Finally, regarding the computational cost of the present approach, it should be stressed (see supplementary information) that it is minor with respect to WFT methods for all systems and basis sets studied here. We thus believe that this approach is a significant step towards calculations near the CBS limit for strongly correlated systems.