End of Theory + Comp Details
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@ -98,19 +98,23 @@
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\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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%pbeuegXiCAS
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\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
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\newcommand{\pbeuegXi}{\text{SPBE-UEG}}
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\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}(\br{})}
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%pbeontxiCAS
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\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta}
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\newcommand{\pbeontxi}{\text{SPBE-OT}}
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\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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%pbeontXiCAS
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\newcommand{\pbeontXi}{\text{PBE-ot-}\tilde{\zeta}}
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\newcommand{\pbeontXi}{\text{SPBE-OT}}
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\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})}
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%pbeont0xiCAS
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\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta}
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\newcommand{\pbeuegns}{\text{PBE-UEG}}
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\newcommand{\argpbeuegns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegns}[0]{\den(\br{}),0,s(\br{}),\ntwo^{\text{UEG}}(\br{}),\mu_{\text{}}^{}(\br{})}
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%pbeont0xiCAS
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\newcommand{\pbeontns}{\text{PBE-OT}}
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\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{}}^{}(\br{})}
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@ -289,7 +293,7 @@
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\begin{abstract}
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We extend to strongly correlated systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, \href{https://doi.org/10.1063/1.5052714}{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. This allows to use RSDFT-type complementary functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. 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 explore various approximations of complementary functionals suited to describe strong correlation. These short-range correlation functionals fulfill two very desirable properties: \titou{spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$)} and size consistency. Specifically, we systematically investigate the dependence of the functionals on different flavors of on-top pair densities and spin polarizations. The key result of this study 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.
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We extend to strongly correlated systems the recently introduced basis-set incompleteness correction based on density-functional theory (DFT) [E. Giner \textit{et al.}, \href{https://doi.org/10.1063/1.5052714}{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. This allows to use RSDFT-type \titou{complementary functionals} to recover the dominant part of the short-range correlation effects missing in this finite basis. 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 explore various approximations of \titou{complementary functionals} suited to describe strong correlation. These short-range correlation functionals fulfill two very desirable properties: \titou{spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$)} and size consistency. Specifically, we systematically investigate the dependence of the functionals on different flavors of on-top pair densities and spin polarizations. The key result of this study 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.
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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 here. Also, the present basis-set incompleteness correction provides smooth curves along the whole potential energy surfaces.
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\end{abstract}
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@ -300,10 +304,10 @@ In the general context of multiconfigurational DFT, this finding shows that one
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%%%%%%%%%%%%%%%%%%%%%%%%
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The general goal of quantum chemistry is to provide reliable theoretical tools to explore the rich area of chemistry. More specifically, developments in quantum chemistry primarily aim at accurately computing the electronic structure of molecular systems, but despite intense developments, no definitive solution to this problem has been found. The theoretical challenge to tackle belongs to the quantum many-body problem, due the intrinsic quantum nature of the electrons and the Coulomb repulsion between them. This so-called electronic correlation problem corresponds to finding a solution to the Schr\"odinger equation for a $N$-electron system, and two main roads have emerged to approximate this solution: wave-function theory (WFT) \cite{Pop-RMP-99} and density-functional theory (DFT). \cite{Koh-RMP-99} Although both WFT and DFT spring from the same Schr\"odinger equation, they rely on very different formalisms, as the former deals with the complicated $N$-electron wave function whereas the latter focuses on the much simpler one-electron density. In its Kohn-Sham (KS) formulation, \cite{KohSha-PR-65} the computational cost of DFT is very appealing since it is a simple mean-field procedure. Therefore, although continued efforts have been done to reduce the computational cost of WFT, DFT still remains the workhorse of quantum chemistry.
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The difficulty of obtaining 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. 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 (near the electron-electron coalescence point) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit 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 exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems.
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The difficulty of obtaining 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. 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 (near the electron-electron coalescence point) or long range (London dispersion interactions). The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining issue for these systems is to push the limit 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 exhibit a much more complex electronic structure. For example, transition metal complexes, low-spin open-shell systems, covalent bond breaking situations have all in common that they cannot be even qualitatively described by a single electronic configuration. It is now clear that the usual semilocal density-functional approximations of KS DFT fail to accurately describe these situations and WFT is king for the treatment of strongly correlated systems.
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\PFL{I think we should add some references in the paragraph above.}
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In practice, WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within this basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra eigenvalue problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: 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 system requires also to account for weak correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two objectives is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as \titou{spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$)} and size consistency.
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In practice, WFT uses a finite one-particle basis set (here denoted as $\basis$) to project the Schr\"odinger equation. The exact solution within this basis set is then provided by full configuration interaction (FCI) which consists in a linear-algebra eigenvalue problem with a dimension scaling exponentially with the system size. Due to this exponential growth of the FCI computational cost, introducing approximations is necessary, with at least two difficulties for strongly correlated systems: 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 system requires also to account for weak-correlation effects which involve many other electronic configurations with typically much smaller weights in the wave function. Addressing these two objectives is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as \titou{spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$)} and size consistency.
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%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.
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@ -382,7 +386,7 @@ As a simple non-self-consistent version of this approach, we can approximate the
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\label{eq:e0approx}
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E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
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\end{equation}
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where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating strong correlation regimes. Two key requirements for this purpose are i) size consistency, and ii) spin-multiplet degeneracy.
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where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for treating the strong-correlation regime. Two key requirements for this purpose are i) size consistency, and ii) spin-multiplet degeneracy.
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\subsection{Effective interaction in a finite basis}
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\label{sec:wee}
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@ -571,55 +575,60 @@ An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$,
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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, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$. 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 CAS wave function is sufficient to recover this property. \titou{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.}
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\subsection{Different types of approximations for the functional}
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\label{sec:final_def_func}
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\subsubsection{Definition of the protocol to design functionals}
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\subsection{\titou{Complementary density functional approximations}}
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\label{sec:def_func}
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%\subsubsection{Definition of the protocol to design functionals}
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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.
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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 two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density.
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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.
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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} and calculated from the CAS wave function, and ii) a \textit{zero} spin polarization.
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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
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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
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\begin{equation}
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\label{eq:def_n2ueg}
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\ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n),
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\end{equation}
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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 CAS wave function through $\tilde{\zeta}$.
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Another approach to approximate the exact on top pair density consists in using directly the on-top pair density of the CAS wave function. Following the work of some of the previous authors~\cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density $\ntwoextrap$ as
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Another approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CAS wave function. Following the work of some of the previous authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density
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\begin{equation}
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\label{eq:def_n2extrap}
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\ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo
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\ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo,
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\end{equation}
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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''.
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\subsubsection{Definition of functionals with good formal properties}
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\label{sec:def_func}
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%\subsubsection{Definition of functionals with good formal properties}
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%\label{sec:def_func}
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We define the following functionals:\\
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We then define \titou{four} functionals:
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i) PBE-UEG-$\tilde{\zeta}$ which uses the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg} and the effective spin polarization of Eq.~\eqref{eq:def_effspin}
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\begin{equation}
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\begin{itemize}
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\item[i)] $\pbeuegXi$ which combines the UEG on-top pair density defined in Eq.~\eqref{eq:def_n2ueg} and the effective spin polarization of Eq.~\eqref{eq:def_effspin}:
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\begin{multline}
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\label{eq:def_pbeueg_i}
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\begin{aligned}
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\bar{E}^\Bas_{\pbeuegXi} = \int \d\br{} \,\denr \ecmd(\argrpbeuegXi),
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\end{aligned}
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\end{equation}
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ii) PBE-ot-$\tilde{\zeta}$ which uses the on-top pair density of Eq.~\eqref{eq:def_n2extrap}
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\bar{E}^\Bas_{\pbeuegXi}
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\\
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= \int \d\br{} \,\denr \ecmd(\argrpbeuegXi),
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\end{multline}
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\item[ii)] $\pbeontXi$ which combines the on-top pair density of Eq.~\eqref{eq:def_n2extrap} and the effective spin polarization of Eq.~\eqref{eq:def_effspin}:
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\begin{equation}
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\label{eq:def_pbeueg_ii}
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\begin{aligned}
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\bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi),
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\end{aligned}
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\end{equation}
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iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair density of Eq.~\eqref{eq:def_n2extrap}
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\item[iii)] \titou{$\pbeuegns$ which combines the UEG on-top pair density of Eq.~\eqref{eq:def_n2ueg} and zero spin polarization:}
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\begin{equation}
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\label{eq:def_pbeueg_iii}
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\begin{aligned}
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\bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns).
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\end{aligned}
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\bar{E}^\Bas_{\pbeuegns} = \int \d\br{} \,\denr \ecmd(\argrpbeuegns),
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\end{equation}
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\item[iv)] $\pbeontns$ which combines the on-top pair density of Eq.~\eqref{eq:def_n2extrap} and zero spin polarization:
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\begin{equation}
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\label{eq:def_pbeueg_iii}
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\bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns).
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\end{equation}
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\end{itemize}
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The performance of each of these \titou{four} functionals is tested below.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}
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@ -634,7 +643,7 @@ iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair den
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\includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat.eps}
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\includegraphics[width=0.45\linewidth]{data/H10/DFT_vqzE_relat_zoom.eps}
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\caption{
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Potential energy curves of the H$_{10}$ chain with equally distant atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top), cc-pVTZ (middle), and cc-pVQZ (bottom) basis sets.
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Potential energy curves of the H$_{10}$ chain with equally-spaced atoms calculated with MRCI+Q and basis-set corrected MRCI+Q using the cc-pVDZ (top), cc-pVTZ (middle), and cc-pVQZ (bottom) basis sets.
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The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}.
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\label{fig:H10}}
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\end{figure*}
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@ -642,18 +651,18 @@ iii) PBE-ot-$0{\zeta}$ where uses zero spin polarization and the on-top pair den
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\subsection{Computational details}
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The purpose of the present paper being the study of the basis-set correction in the regime of strong correlation, we study the potential energy curves up to the dissociation limit of a H$_{10}$ chain with equally distant atoms and the C$_2$, N$_2$, O$_2$, and F$_2$ molecules. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$.
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The purpose of the present paper being the study of the basis-set correction in the regime of strong correlation, we study the potential energy curves up to the dissociation limit of a \ce{H10} chain with equally-spaced atoms and the \ce{C2}, \ce{N2}, \ce{O2}, and \ce{F2} diatomics. In a given basis set, in order to compute the approximation of the exact ground-state energy using Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$.
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In the case of the C$_2$, N$_2$, O$_2$, and F$_2$ molecules, approximations to the FCI energies are obtained using converged frozen-core (1s orbitals are kept frozen) CIPSI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the Quantum Package software~\cite{QP2}. The estimated exact potential energy curves are obtained from Ref.~\onlinecite{LieCle-JCP-74a}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be below 0.5 mH. In the case of the H$_{10}$ chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space).
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In the case of the \ce{C2}, \ce{N2}, \ce{O2}, and \ce{F2} molecules, approximations to the FCI energies are obtained using converged frozen-core ($1s$ orbitals are kept frozen) selected CI calculations and the extrapolation scheme for the perturbative correction of Umrigar \textit{et. al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more details) using the latest version of Quantum Package. \cite{QP2} The estimated exact potential energy curves are obtained from Ref.~\onlinecite{LieCle-JCP-74a}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be below $0.5$ mHa. In the case of the \ce{H10} chain, the approximation to the FCI energies together with the estimated exact potential energy curves are obtained from the data of Ref.~\onlinecite{h10_prx} where the authors performed MRCI+Q calculations with a minimal valence active space as reference (see below for the description of the active space).
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Regarding the \titou{complementary basis functional}, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional (density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$) together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. These CASSCF wave functions correspond to the following active spaces: 10 electrons in 10 orbitals for H$_{10}$, 8 electrons in 8 electrons for C$_2$, 10 electrons in 8 orbitals for N$_2$, 12 electrons in 8 orbitals for O$_2$, and 14 electrons in 8 orbitals for F$_2$.
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Regarding the \titou{complementary basis functional}, we first perform full-valence complete-active-space self-consistent-field (CASSCF) calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-like quantities involved in the functional [density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation function $\mu(\br{})$ of Eq.~\eqref{eq:def_mur} are calculated with this full-valence CASSCF wave function. The CASSCF calculations have been performed with the following active spaces: (10e,10o) for \ce{H10}, (8e,8o) for \ce{C2}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}.
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Also, as the frozen core approximation is used in all our CIPSI calculations, we use the corresponding valence-only complementary functionals. Therefore, all density-like quantities exclude any contribution from the core 1s orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}.
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Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only \titou{complementary functionals}. Therefore, all density-like quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function is taken as the one defined in Eq.~\eqref{eq:def_mur_val}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{table*}
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\label{tab:d0}
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\caption{Atomization energies $D_0$ (in mH) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with various basis sets.}
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\caption{Atomization energies $D_0$ (in mHa) and associated errors (in square brackets) with respect to the estimated exact values computed at different approximation levels with various basis sets.}
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\begin{ruledtabular}
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\begin{tabular}{llcccc}
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@ -703,11 +712,11 @@ Also, as the frozen core approximation is used in all our CIPSI calculations, we
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The study of the \ce{H10} 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).
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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.
|
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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.
|
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|
||||
More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, 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.
|
||||
More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an error below 1.4 mHa) 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.
|
||||
|
||||
Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mH), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density.
|
||||
Regarding in more details the performance of the different types of approximate functionals, the results show that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and they give slightly more accurate results than PBE-UEG-$\tilde{\zeta}$. These findings bring two important clues on the role of the different physical ingredients used in the functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function (see Eq.~\eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see Eq.~\eqref{eq:def_n2ueg}); ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. Point ii) is important as it shows that spin polarization in density-functional approximations essentially plays the same role as that of the on-top pair density.
|
||||
|
||||
\subsection{Dissociation of diatomics}
|
||||
|
||||
@ -759,13 +768,13 @@ Regarding in more details the performance of the different types of approximate
|
||||
\end{figure*}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
The \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of \ce{H10}, but in addition these molecules 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 interactions which are medium to long-range weak correlation effects. Also, \ce{O2} has a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of the different types of functionals proposed here.
|
||||
The \ce{C2}, \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} system for the present study as the level of strong correlation increases while stretching the bond similarly to the case of \ce{H10}, but in addition these molecules 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 interactions which are medium to long-range weak-correlation effects. Also, \ce{O2} has a triplet ground state and is therefore a good check for the performance of the dependence on the spin polarization of the different types of functionals proposed here.
|
||||
|
||||
We report in Figure \ref{fig:C2}, \ref{fig:N2}, \ref{fig:O2}, and \ref{fig:F2} the potential energy curves computed using the aug-cc-pVDZ and aug-cc-pVTZ basis sets of C$_2$, N$_2$, O$_2$, and N$_2$, respectively, for different approximation levels. The computation of the atomization energies $D_0$ at each level of theory used here is reported in Table \ref{tab:d0}.
|
||||
|
||||
Just as in the case of \ce{H10}, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for a high-spin system like \ce{O2}. More quantitatively, an error below 1.0 mH on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of \ce{C2} with 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 in the \ce{C2} molecule.
|
||||
Just as in the case of \ce{H10}, the quality of $D_0$ are globally improved by adding the basis-set correction and it is remarkable that PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ give very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for a high-spin system like \ce{O2}. More quantitatively, an error below 1.0 mHa on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the PBE-ot-$0{\zeta}$ functional, whereas such a result is far from reach within the same basis set at the near-FCI level. In the case of \ce{C2} with the aug-cc-pVTZ basis set, an error of about 5.5 mHa 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 in the \ce{C2} molecule.
|
||||
|
||||
Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the \titou{complementary basis functionals} can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is expected.
|
||||
Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the \titou{complementary basis functionals} can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron cusp: the local range-separation function $\mu(\br{})$ is designed by looking at the electron-electron coalescence point and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is expected.
|
||||
|
||||
\section{Conclusion}
|
||||
\label{sec:conclusion}
|
||||
@ -776,7 +785,7 @@ The DFT-based basis-set correction relies on three aspects: i) the definition of
|
||||
|
||||
The development of new $S_z$-independent and size-consistent 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 loss of accuracy. This avoids the commonly used effective spin polarization calculated from a multideterminant wave function originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly become complex-valued for some multideterminant wave functions. From a more fundamental aspect, this shows that the spin polarization in DFT-related frameworks only mimics the role of the on-top pair density.
|
||||
|
||||
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 \ce{C2}, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of \ce{C2}, 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.
|
||||
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 \ce{C2}, whereas the uncorrected near-FCI results are far from that accuracy within the same basis set. In the case of \ce{C2}, an error of 5.5 mHa 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 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.
|
||||
|
||||
|
||||
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