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@ -163,7 +163,7 @@
\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
\newcommand{\wbasiscoalval}{W_{\wf{}{\Bas}}^{\text{val}}({\bf r},{\bf r})}
\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
@ -265,12 +265,12 @@
\title{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds}
\begin{abstract}
The present work proposes an extension to prototypes of strongly correlated systems of the recently proposed basis set correction based on density functional theory (DFT).
The present work proposes an application and extension to strongly correlated systems of the recently proposed basis set correction based on density functional theory (DFT).
We study the potential energy surfaces (PES) of the H$_{10}$, C$_2$, N$_2$, O$_2$ and F$_2$ molecules up to full dissociation limit in increasing basis sets at near full configuration interaction (FCI) level with and without the present basis set correction.
Such basis set correction relies on a mapping between range-separated DFT (RSDFT) and wave function calculations in a finite basis set through the definition of an effective non-divergent interaction mimicking the coulomb operator projected in a finite basis set. From that mapping, RSDFT-types functionals are used to recover the dominant the correlation effects missing in a finite basis set.
Such basis set correction relies on a mapping between range-separated DFT (RSDFT) and wave function calculations in a finite basis set through the definition of an effective non-divergent interaction mimicking the coulomb operator projected in a finite basis set. From that mapping, RSDFT-types functionals are used to recover the dominant the short-range correlation effects missing in a finite basis set.
The scope of the present work is to develop new 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 extensivity.
In that context, 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.
In the general context of multi-configurational DFT, such findings show that one can avoid the use of effective spin polarization whose mathematical definition is rather \textit{ad hoc} and which can become complex valued.
In the general context of multi-configurational DFT, such findings show 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 allows chemical accuracy on atomization energies in a triple-zeta quality for most of the systems studied. Also, the present basis set correction provides smooth curves all along the PES.
\end{abstract}
@ -284,30 +284,30 @@ The accurate computation of the electronic structure of molecular systems plays
but despite intense developments, no definitive solution to that problem have been found.
The theoretical challenge to be overcome falls back in the category of the quantum many-body problem due the intrinsic quantum nature
of the electrons and the coulomb repulsion between them, inducing the so-called electronic correlation problem.
Tackling this problem translate to solving the Schroedinger equation for a $N$~-~electron system, and two roads have emerged to approximate the solution to this formidably complex mathematical problem: the wave function theory (WFT) and density functional theory (DFT).
Although both WFT and DFT spring from the same problem, their formalisms are very different as the former deals with the complex
Tackling this problem translates into solving the Schroedinger equation for a $N$~-~electron system, and two roads have emerged to approximate the solution to this formidably complex mathematical problem: the wave function theory (WFT) and density functional theory (DFT).
Although both WFT and DFT spring from the same equation, their formalisms are very different as the former deals with the complex
$N$~-~body wave function whereas the latter handles the much simpler one~-~body density.
The computational cost of DFT is very appealing as in its Kohn-Sham (KS) formulation it can be recast in a mean-field procedure.
In its Kohn-Sham (KS) formulation, the computational cost of DFT is very appealing as it can be recast in a mean-field procedure.
Therefore, although constant efforts are performed to reduce the computational cost of WFT, DFT remains still the workhorse of quantum chemistry.
From the theoretician point of view, the complexity of description of a given chemical system can be roughly
The complexity of a reliable theoretical description of a given chemical system can be roughly
categorized by the strength of the electronic correlation appearing in its electronic structure.
Weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by the avoidance effects when electron are near the electron coalescence point, which are often called short-range correlation effects,
or far from each other, typically dispersion forces. The theoretical description of weakly correlated systems is one of the more concrete achievement
The so-called weakly correlated systems, such as closed-shell organic molecules near their equilibrium geometry, are typically dominated by correlation effects which do not affect the qualitative mean-field picture of the system. These weak correlation effects can be either short-range when electron are near the electron coalescence point, or long-range
with dispersion forces. The theoretical description of weakly correlated systems is one of the more concrete achievement
of quantum chemistry, and the main remaining issue for these systems is to push the limit in terms of the size of the chemical systems that can be treated.
The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is much more problematic as they exhibits
a much more exotic electronic structure.
Transition metals containing systems, low-spin open shell systems, covalent bond breaking or excited states
have all in common that they cannot be even qualitatively described by a single electronic configuration.
It is now clear that the usual approximations in KS-DFT fails in giving an accurate description of these situations and WFT has become
It is now clear that the usual semi-local approximations in KS-DFT fail in giving an accurate description of these situations and WFT has become
the standard for the treatment of strongly correlated systems.
In practice WFT uses a finite one-particle basis set (here after as $\basis$) to project the Schroedinger equation whose exact solution becomes clear: the full configuration interaction (FCI) which consists in a linear algebra problem whose dimension scales exponentially with the system size.
In practice WFT uses a finite one-particle basis set (here referred as $\basis$) to project the Schroedinger equation whose exact solution becomes clear: the full configuration interaction (FCI) which consists in a linear algebra problem whose dimension scales exponentially with the system size.
Because of the exponential growth of the FCI, many approximations have appeared and in that regard the complexity of the strong correlation problem is, at least, two-fold:
i) the qualitative description of the wave function is determined by a primary set of electronic configurations (whose size can scale exponentially in many cases) among which near degeneracies and/or strong interactions appear in the Hamiltonian matrix,
ii) the quantitative description of the systems must take into account weak correlation effects which requires to take into account many
other electronic configurations with typically much smaller weights in the wave function.
Fulfilling these two objectives is a rather complicated task for a given approximated approach, specially if one adds the requirement of verifying other formal properties, such as size-extensivity and additivity of the computed energy in the case of non interacting fragments, or $S_z$ invariance.
Fulfilling these two objectives is a rather complicated task for a given approximated approach, specially if one adds the requirement of satisfying formal properties, such $S_z$ invariance or additivity of the computed energy in the case of non interacting fragments.
%To tackle this complicated problem, many methods have been proposed and an exhaustive review of the zoology of methods for strong correlation goes beyond the scope and purpose of this article.
@ -327,14 +327,18 @@ Fulfilling these two objectives is a rather complicated task for a given approxi
%Among the SCI algorithms, the CI perturbatively selected iteratively (CIPSI) can be considered as a pioneer. The main idea of the CIPSI and other related SCI algorithms is to iteratively select the most important Slater determinants thanks to perturbation theory in order to build a MRCI zeroth-order wave function which automatically concentrate the strongly interacting part of the wave function. On top of this MRCI zeroth-order wave function, a rather simple MRPT approach is used to recover the missing weak correlation and the process is iterated until reaching a given convergence criterion. It is important to notice that in the SCI algorithms, neither the SCI or the MRPT are size extensive \textit{per se}, but the extensivity property is almost recovered by approaching the FCI limit.
%When the SCI are affordable, their clear advantage are that they provide near FCI wave functions and energies, whatever the level of knowledge of the user on the specific physical/chemical problem considered. The drawback of SCI is certainly their \textit{intrinsic} exponential scaling due to their linear parametrisation. Nevertheless, such an exponential scaling is lowered by the smart selection of the zeroth-order wave function together with the MRPT calculation.
Besides the difficulties of accurately describing the electronic structure within a given basis set, a crucial component of the limitations of applicability of WFT concerns the slow convergence of the energies and properties with respect to the quality of the basis set. As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg \textit{et. al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the coulomb interaction at the electron coalescence point, which induces a discontinuity in the first-derivative of the wave function (the so-called electron-electron cusp). Describing such a discontinuity with an incomplete basis set is impossible and as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques has been developed, either based on the Hylleraas's expansion of the coulomb operator\cite{HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis set convergence problem has been proposed by the so-called R12 and F12 methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a function explicitly depending on the interelectronic coordinates ensuring the correct cusp condition in the wave function, and the resulting correlation energies converge much faster than the usual WFT. For instance, using the explicitly correlated version of coupled cluster with single, double and perturbative triple substitution (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to a quintuple-$\zeta$ quality of the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although inherent computational overhead are introduced by the auxiliary basis sets needed to resolve the rather complex three- and four-electron integrals involved in the F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires a quite involved mathematical development to adapt to a new theory. In the context of strong correlation, several multi-reference methods have been extended to explicitly correlation (see for instance Ref. \cite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on so-called universal F12 which are potentially applicable to any electronic structure approaches\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}.
Besides the difficulties of accurately describing the electronic structure within a given basis set, a crucial component of the limitations of applicability of WFT concerns the slow convergence of the energies and properties with respect to the quality of the basis set.
As initially shown by the seminal work of Hylleraas\cite{Hyl-ZP-29} and further developed by Kutzelnigg \textit{et. al.}\cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94}, the main convergence problem originates from the divergence of the coulomb interaction at the electron coalescence point, which induces a discontinuity in the first-derivative of the exact wave function (the so-called electron-electron cusp).
Describing such a discontinuity with an incomplete basis set is impossible and as a consequence, the convergence of the computed energies and properties can be strongly affected. To attenuate this problem, extrapolation techniques has been developed, either based on the Hylleraas's expansion of the coulomb operator\cite{HalHelJorKloKocOlsWil-CPL-98}, or more recently based on perturbative arguments\cite{IrmHulGru-arxiv-19}. A more rigorous approach to tackle the basis set convergence problem has been proposed by the so-called R12 and F12 methods\cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a function explicitly depending on the interelectronic coordinates ensuring the correct cusp condition in the wave function, and the resulting correlation energies converge much faster than the usual WFT. For instance, using the explicitly correlated version of coupled cluster with single, double and perturbative triple substitution (CCSD(T)) in a triple-$\zeta$ quality basis set is equivalent to a quintuple-$\zeta$ quality of the usual CCSD(T) method\cite{TewKloNeiHat-PCCP-07}, although inherent computational overhead are introduced by the auxiliary basis sets needed to resolve the rather complex three- and four-electron integrals involved in the F12 theory. In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires a quite involved mathematical development to adapt to a new theory. In the context of strong correlation, several multi-reference methods have been extended to explicitly correlation (see for instance Ref. \cite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on so-called universal F12 which are potentially applicable to any electronic structure approaches\cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}.
An alternative point of view to improve the convergence towards the CBS limit is to leave the short-range correlation effects to DFT and to use WFT to deal only with the long-range and/or strong-correlation effects. A rigorous approach to mix DFT and WFT is the range-separated DFT (RSDFT) formalism (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which rely on a splitting of the coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of such approach is at least two-folds: i) the DFT part deals only with the short-range part of the coulomb interaction, and therefore the usual semi-local approximations to the unknown exchange-correlation functional are more suited to that correlation regime, ii) as the WFT part deals with a smooth non divergent interaction, the exact wave function has no cusp\cite{GorSav-PRA-06} and therefore the basis set convergence is much faster\cite{FraMusLupTou-JCP-15}.
Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches. Nevertheless, there are still some open issues in RSDFT, such as the dependence of the quality of the results on the value of the range separation $\mu$ which can be seen as an empirical parameter, and the remaining self-interaction errors.
Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT approaches.
Nevertheless, there are still some open issues in RSDFT, such remaining self-interaction errors or
the dependence of the quality of the results on the value of the range separation $\mu$ which can be seen as an empirical parameter.
Following this path, a very recent solution to the basis set convergence problem has been proposed by some of the preset authors\cite{GinPraFerAssSavTou-JCP-18} where they proposed to use RSDFT to take into account only the correlation effects outside a given basis set. The key idea in such a work is to realize that as a wave function developed in an incomplete basis set is cusp-less, it could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, the authors proposed a mapping with RSDFT through the introduction of an effective non-divergent interaction representing the usual coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried recently\cite{LooPraSceTouGin-JCPL-19} together with the first attempt to generalize this approach to excited states\cite{GinSceTouLoo-JCP-19}.
Following this path, a very recent solution to the basis set convergence problem has been proposed by some of the preset authors\cite{GinPraFerAssSavTou-JCP-18} where they proposed to use RSDFT to take into account only the correlation effects outside a given basis set. The key idea in such a work is to realize that a wave function developed in an incomplete basis set is cusp-less could also come from a Hamiltonian with a non divergent electron-electron interaction. Therefore, the authors proposed a mapping with RSDFT through the introduction of an effective non-divergent interaction representing the usual coulomb interaction projected in an incomplete basis set. First applications to weakly correlated molecular systems have been successfully carried recently\cite{LooPraSceTouGin-JCPL-19} together with the first attempt to generalize this approach to excited states\cite{GinSceTouLoo-JCP-19}.
The goal of the present work is to push the development of this new theory toward the description of strongly correlated systems.
The paper is organized as follows: in section \ref{sec:theory} we recall the mathematical framework of the basis set correction and we propose a practical extension for strongly correlated systems. Within the present development, two important formal properties are imposed: the extensivity of the correlation energies together with the $S_z$ independence of the results.
The paper is organized as follows: in section \ref{sec:theory} we recall the mathematical framework of the basis set correction and we expose the extension for strongly correlated systems. Within the present development, two important formal properties are imposed: the extensivity of the correlation energies together with the $S_z$ independence of the results.
Then in section \ref{sec:results} we discuss the potential energy surfaces (PES) of the C$_2$, N$_2$, O$_2$, F$_2$ and H$_{10}$ molecules up to full dissociation as a prototype of strongly correlated problems. Finally, we conclude in section \ref{sec:conclusion}.
%%%%%%%%%%%%%%%%%%%%%%%%
@ -342,7 +346,9 @@ Then in section \ref{sec:results} we discuss the potential energy surfaces (PES)
\label{sec:theory}
%%%%%%%%%%%%%%%%%%%%%%%%
As the theoretical framework of the basis set correction has been exposed in details in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we briefly recall the main equations and concepts needed for this study in sections \ref{sec:basic}, \ref{sec:wee} and \ref{sec:mur}.
More specifically, in section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$. Then in section \ref{sec:wee} we introduce an effective non divergent interaction in a basis set $\Bas$, which leads us to the definition of an effective range separation parameter varying in space in section \ref{sec:mur}. Thanks to the range separation parameter, we make a mapping with a specific class of RSDFT functionals and propose practical approximations for the unknown density functional complementary to a basis set $\Bas$, for which new approximations for the strong correlation regime are given in section \ref{sec:functional}.
More specifically, in section \ref{sec:basic} we recall the basic mathematical framework of the present theory by introducing the density functional complementary to a basis set $\Bas$.
Then in section \ref{sec:wee} we introduce an effective non divergent interaction in a basis set $\Bas$, which leads us to the definition of an effective range separation parameter varying in space in section \ref{sec:mur}.
Then, in section \ref{sec:functional} we expose the new approximated functionals complementary to a basis set $\Bas$ based on RSDFT functionals. The generic form of such functionals is exposed in section \ref{sec:functional_form}, their properties in the context of the basis set correction is discussed in \ref{sec:functional_prop}, and the requirements for strong correlation is discussed in section \ref{sec:requirements}. Then, the actual form of the functionals used in this work are introduced in section \ref{sec:final_def_func}.
\subsection{Basic formal equations}
\label{sec:basic}
The exact ground state energy $E_0$ of a $N-$electron system can be obtained by an elegant mathematical framework connecting WFT and DFT, that is the Levy-Lieb constrained search formalism which reads
@ -356,7 +362,7 @@ where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given den
F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
\end{equation}
The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density.
Nevertheless, in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$, we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
Nevertheless, in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$ and we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
In the present context it is important to notice that in order to recover the \textit{exact} ground state energy, the wave functions $\Psi$ involved in the definition of eq. \eqref{eq:levy_func} must be developed in a complete basis set.
An important step proposed originally by some of the present authors in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}
@ -365,7 +371,7 @@ was to propose to split the minimization in the definition of $F[\denr]$ using
\label{eq:def_levy_bas}
F[\denr] = \min_{\wf{}{\Bas} \rightarrow \denr} \elemm{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\denr},
\end{equation}
which leads to the following definition of $\efuncden{\denr}$ which is the the density functional complementary to the basis set $\Bas$
which leads to the following definition of $\efuncden{\denr}$ which is the density functional complementary to the basis set $\Bas$
\begin{equation}
\begin{aligned}
\efuncden{\denr} =& \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi} \\ 
@ -374,7 +380,7 @@ which leads to the following definition of $\efuncden{\denr}$ which is the the d
\end{equation}
Therefore thanks to eq. \eqref{eq:def_levy_bas} one can properly connect the DFT formalism with the basis set error in WFT calculations. In other terms, the existence of $\efuncden{\denr}$ means that the correlation effects not taken into account in $\basis$ can be formulated as a density functional.
Assuming that the density $\denFCI$ associated to the ground state FCI wave function $\psifci$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state density (see equations 12-15 of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
Assuming that the density $\denFCI$ associated to the ground state FCI wave function $\psifci$ is a good approximation of the exact density, one obtains the following approximation for the exact ground state energy (see equations 12-15 of Ref. \onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
\label{eq:e0approx}
E_0 = \efci + \efuncbasisFCI
@ -384,7 +390,7 @@ The whole purpose of this paper is to determine approximations for $\efuncbasisF
\subsection{Definition of an effective interaction within $\Bas$}
\label{sec:wee}
As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, a cusp less wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non divergent electron-electron interaction. In other words, the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual coulomb interaction at the electron coalescence point.
As it was originally shown by Kato\cite{kato}, the cusp in the exact wave function originates from the divergence of the coulomb interaction at the coalescence point. Therefore, a cusp less wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non divergent electron-electron interaction. In other words, the impact of the incompleteness of a finite basis set can be understood as the removal of the divergence of the usual coulomb interaction at the electron coalescence point.
As it was originally derived in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18} (see section D and annexes), one can obtain an effective non divergent interaction, here referred as $\wbasis$, which reproduces the expectation value of the coulomb operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite spin part of the electron-electron interaction.
@ -428,7 +434,7 @@ As the effective interaction within a basis set $\wbasis$ is non divergent, one
\label{eq:weelr}
w_{ee}^{\lr}(\mu;r_{12}) = \frac{\text{erf}\big(\mu \,r_{12} \big)}{r_{12}}.
\end{equation}
As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we introduce a range-separation parameter $\murpsi$ varying in real space
As originally proposed in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we use a range-separation parameter $\murpsi$ varying in real space
\begin{equation}
\label{eq:def_mur}
\murpsi = \frac{\sqrt{\pi}}{2} \wbasiscoal
@ -445,12 +451,12 @@ Because of the very definition of $\wbasis$, one has the following properties at
which is fundamental to guarantee the good behaviour of the theory at the CBS limit.
\subsubsection{Frozen core density approximation}
As all WFT calculations for the purpose of that work are performed within the frozen core approximation, we define the valence-only versions of the various quantities needed for the complementary basis set functional.
As all WFT calculations for the purpose of that work are performed within the frozen core approximation, we use the valence-only versions of the various quantities needed for the complementary basis set functional introduced in Ref. \cite{LooPraSceTouGin-JCPL-19}.
We split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs, respectively)
and define the valence only range separation parameter
\begin{equation}
\label{eq:def_mur_val}
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval,
\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
\end{equation}
where $\wbasisval$ is the valence-only effective interaction defined as
\begin{equation}
@ -473,11 +479,13 @@ and $\twodmrdiagpsival$
\label{eq:twordm_val}
\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{equation}
Notice the summations on the active set of orbitals in equations \eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
It is noteworthy that, within the present definition, $\wbasisval$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
\label{sec:functional}
\subsubsection{Generic form of the approximated functionals}
\label{sec:functional_form}
As originally proposed and motivated in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary basis set functional $\efuncden{\denr}$ by using the so-called multi-determinant correlation functional (ECMD) introduced by Toulouse and co-workers\cite{TouGorSav-TCA-05}.
Following the recent work of some of the present authors\cite{LooPraSceTouGin-JCPL-19}, we propose to use a PBE-like functional which uses the total density $\denr$, spin polarisation $\zeta(\br{})$, reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$ and the on-top pair density $\ntwo(\br{})$. In the present work, all the density-related quantities are computed with the same wave function $\psibasis$ used to define $\murpsi$.
Therefore, a given approximation X of $\efuncden{\denr}$ have the following generic form
@ -523,6 +531,7 @@ Also, the function $\ecmd(\argecmd)$ vanishes when $\mu \rightarrow \infty$ as a
\end{equation}
\subsubsection{Properties of approximated 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 $\mu_{\Psi^{\basis}}$:
\begin{equation}
@ -534,6 +543,7 @@ Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pa
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}
\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.
@ -566,15 +576,15 @@ An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\
In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ is an integral over $\mathbb{R}^3$ of local quantities, in the case of non overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the sub-system $A$ and the other one from the region of the sub-system $B$.
Therefore, a sufficient condition for the extensivity is that these quantities coincide in the isolated systems and in the subsystem of the super system $A\ldots B$.
As $\efuncdenpbe{\argebasis}$ depends only on quantities which are properties of the wave function $\psibasis$, a sufficient condition for the extensivity of these quantities is that the function factorise in the limit of non-interacting fragments, that is $\Psi_{A\ldots B}^{\basis} = \Psi_A^{\basis} \Psi_B^{\basis}$.
In the case where the two subsystems $A$ and $B$ dissociate in closed shell systems, a simple HF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CASSCF wave function is sufficient to recover this property, as will be numerically illustrated in section \ref{sec:separability}.
In the case where the two subsystems $A$ and $B$ dissociate in closed shell systems, a simple HF wave function ensures this property, but when one or several covalent bonds are broken, the use of a properly chosen CASSCF wave function is sufficient to recover this property.
The condition for the active space involved in the CASSCF wave function is that it has to lead to extensive energies in the limit of dissociated fragments.
\subsection{Different types of approximations for the functional}
\label{sec:final_def_func}
\subsubsection{Definition of the protocol to design functionals}
\label{sec:def_func}
As the present work focusses on the strong correlation regime, we propose here to investigate only approximated functionals which are $S_z$ invariant and size extensive in the case of covalent bond breaking. Therefore, the wave function $\psibasis$ used throughout this paper are of CASSCF type in order to ensure extensivity of all density related quantities.
The difference between the different flavours of functionals are only on the i) the type of on-top pair density used, and ii) the type of spin polarisation used.
The difference between the different flavours of functionals are only on i) the type of on-top pair density used, and ii) the type of spin polarisation used.
Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
@ -590,11 +600,12 @@ Another approach to approximate of the exact on top pair density consists in tak
\label{eq:def_n2extrap}
\ntwoextrap(\ntwo_{\psibasis},\mu,\br{}) = \ntwo_{\wf{}{\Bas}}(\br{}) \bigg( 1 + \frac{2}{\sqrt{\pi}\murpsi} \bigg)^{-1}
\end{equation}
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06}.
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT.
When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply refer it as "ot".
\subsubsection{Definition of functionals with good formal properties}
\label{sec:def_func}
We define the following functionals:
i) The PBE-UEG-$\tilde{\zeta}$ which uses the UEG-like on-top pair density defined in equation \eqref{eq:def_n2ueg}, the effective spin polarization of equation \eqref{eq:def_effspin} and which reads
\begin{equation}
@ -639,7 +650,7 @@ Also, as the frozen core approximation is used in all near FCI calculations, we
The study of equally distant H$_{10}$ chains is a good prototype for the study of strong correlation regime 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 can be performed at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for detailed study of that problem).
We report in figures \ref{fig:H10_vdz}, \ref{fig:H10_vtz}, \ref{fig:H10_vqz} the PES computed using the cc-pVXZ (X=D,T,Q) basis sets of H$_{10}$, 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 surfaces are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no divergence of bizarre behaviour are found when stretching the bonds, which show that the functionals are robust when reaching the strong correlation regime.
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 surfaces are globally improved by adding the basis-set correction, whatever the level of approximation used for the functional $\efuncbasisFCI$. Also, no divergence of bizarre behaviour 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 the chemical accuracy (\textit{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 accuracy is not reached at the cc-pVQZ basis set using MRCI+Q.
@ -649,14 +660,14 @@ i) the explicit use of the on-top pair density coming from the CASSCF wave funct
ii) removing the dependence on any kind of spin polarizations does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. The point ii) is important as it shows that the use of the spin-polarization in density functional approximations (DFA) essentially plays the role of the effect of the on-top pair density.
\subsection{Dissociation of C$_2$, N$_2$, O$_2$ and F$_2$}
The study of C$_2$, N$_2$, O$_2$ and F$_2$ molecules are complementary to the H$_{10}$ system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of H$_{10}$, but also these systems exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion forces which are medium to long-range weak correlation effects.
Also, O$_2$ exhibit a triplet ground state and therefore is good check for the performance of the dependence on the spin polarization of various types of functionals proposed here.
The C$_2$, N$_2$, O$_2$ and F$_2$ molecules are complementary to the H$_{10}$ system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of H$_{10}$, but also these systems exhibit more important and versatile types of weak correlations due to the larger number of electrons. Indeed, the short-range correlation effects are known to play a strong differential effect on the computation of $D_0$, while the shape of the curve far from the equilibrium geometry is governed by dispersion forces which are medium to long-range weak correlation effects.
Also, O$_2$ exhibit a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of various types of functionals proposed here.
We report in figures \ref{fig:C2_avdz}, \ref{fig:N2_avdz}, \ref{fig:O2_avdz} and \ref{fig:F2_avdz} (\ref{fig:C2_avtz}, \ref{fig:N2_avtz}, \ref{fig:O2_avtz} and \ref{fig:F2_avtz}) the potential energy curves computed using the aug-cc-pVDZ (aug-cc-pVTZ) basis sets of C$_2$, N$_2$, O$_2$ and N$_2$, respectively, for different levels of computations. The computation of the atomization energies $D_0$ at each level of theory used here is reported in table \ref{tab:d0}.
Just as the case of H$_{10}$, the quality of $D_0$ are globally improved by adding the basis set correction and it is remarkable that the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ functionals give very similar results.
The latter observation confirms that the dependence on the on-top pair density allows to remove the dependence of any kind of spin polarizations for a quite wide spread of electron density and also for purely high spin systems as O$_2$.
The latter observation confirms that the dependence on the on-top pair density allows to remove the dependence of any kind of spin polarizations for a quite wide spread of electron density and also for high spin systems as O$_2$.
More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for N$_2$, O$_2$ and F$_2$ in aug-cc-pVTZ with the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at near FCI level.
In the case of C$_2$ in the aug-cc-pVTZ basis set, an error of about 5.5 mH is found with respect to the estimated exact $D_0$. Such an error is remarkably large with respect to the other diatomic molecules studied here and might be associated to the level of strong correlation of the C$_2$ molecule.
@ -808,17 +819,17 @@ F$_2$, aug-cc-pvtz & 59.3$/$2.9 & 61.2$/$1.0 &
\section{Conclusion}
\label{sec:conclusion}
In the present paper we have extended the recently proposed DFT-based basis set correction to strongly correlated systems.
We studied the H$_{10}$, C$_2$, N$_2$, O$_2$ and F$_2$ linear molecules up to full dissociation limits at near FCI level in increasing basis sets, and investigated how the basis set correction affect the convergence toward the CBS limits of the PES of these molecular systems.
We studied the H$_{10}$, C$_2$, N$_2$, O$_2$ and F$_2$ linear molecules up to full dissociation limits at near FCI level in increasing basis sets, and investigated how the basis set correction affects the convergence toward the CBS limits of the PES of these molecular systems.
The DFT-based basis set correction rely on three aspects:
The DFT-based basis set correction relies on three aspects:
i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the coulomb interaction projected into an incomplete basis set $\basis$,
ii) the fitting of such effective interaction with a long-range interaction used in RS-DFT,
iii) the use of complementary correlation functional of RS-DFT.
In the present paper, we investigated points i) and iii) in the context of strong correlation and focussed on atomization energies.
In the present paper, we investigated points i) and iii) in the context of strong correlation and focussed on PES and atomization energies.
More precisely, we proposed a new scheme to design functionals fulfilling a) $S_z$ invariance, b) size extensivity. To achieve such requirements we proposed to use CASSCF wave functions leading to extensive energies, and to develop functionals using only $S_z$ invariant density-related quantities.
The development of new $S_z$ invariant and size extensive functionals has lead us to investigate the role of two related quantities: the spin-polarization and the on-top pair density.
One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin-polarization without loos of accuracy.
One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can avoid dependence to any form of spin-polarization without loss of accuracy.
This avoids the commonly used effective spin-polarization with multi-configurational wave function whose mathematical definition originally proposed by Perdrew and co-workers in Ref. \cite{PerSavBur-PRA-95} has only a clear mathematical ground for a single Slater determinant and can be become complex-valued in the case of multi-configurational wave functions. From a more fundamental aspect, this shows that the spin-polarization in DFT-related frameworks only mimic's the role of the on-top density.
Regarding the results of the present approach, the basis set correction systematically improves the near FCI calculation in a given basis set. More quantitatively, it is shown that the atomization energy $D_0$ is within the chemical accuracy for all systems but C$_2$ within a triple zeta quality basis set, whereas the near FCI values are far from that accuracy within the same basis set.