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@ -188,6 +188,7 @@
\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})}
\newcommand{\denfci}[0]{\den_{\psifci}}
\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}}
\newcommand{\denbas}[0]{{P}^{\Bas}(\den)}
\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas}
\newcommand{\denrfci}[0]{\denr_{\psifci}}
\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})}
@ -295,7 +296,7 @@
\begin{abstract}
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 electron-electron Coulomb interaction projected in the finite basis set. This allows one to use RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To study both weak and strong correlation regimes we consider the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit, and we explore various approximations of complementary functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we systematically investigate the dependence of the functional on different types 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.
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 electron-electron Coulomb interaction projected in the finite basis set. This \alert{enables the use of} RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To study both weak and strong correlation regimes we consider the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit, and we explore various approximations of complementary functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we systematically investigate the dependence of the functional on different types 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.
%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 in certain cases. [JT: I don't like this sentence in the abstract in particular because it is not clear what the "effective" spin polarization is.]
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 potential energy curves along the whole range of distances.
\end{abstract}
@ -305,13 +306,13 @@ Quantitatively, we show that the basis-set correction reaches chemical accuracy
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%
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. 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 made to reduce the computational cost of WFT, DFT still remains the workhorse of quantum computational chemistry.
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. 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 \alert{to} 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 made to reduce the computational cost of WFT, DFT still remains the workhorse of quantum computational chemistry.
The difficulty of obtaining a reliable theoretical description of a given chemical system can be roughly categorized by the strength of the electronic correlation. 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 points) \cite{HatKloKohTew-CR-12} or long range (London dispersion interactions). \cite{AngDobJanGou-BOOK-20} The theoretical description of weakly correlated systems is one of the most concrete achievement of quantum chemistry, and the main remaining challenge for these systems is to push the limit of the chemical system size that can be treated. The case of the so-called strongly correlated systems, which are ubiquitous in chemistry, is 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 \cite{GorSeiSav-PCCP-08,GagTruLiCarHoyBa-ACR-17} and WFT is king for the treatment of strongly correlated systems.
In practice, WFT uses a finite one-electron basis set. The exact solution of the Schr\"odinger equation 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 simultaneously these two issues is a rather complicated task for a given approximate WFT method, especially if one adds the requirement of satisfying formal properties, such as spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency.
Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energy (and related properties) with respect to the size of the basis set. As initially shown by the seminal work of Hylleraas \cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers, \cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94} the main convergence problem originates from the divergence of the electron-electron Coulomb interaction at the 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 one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties are strongly affected. To alleviate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis, \cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98} or more recently based on perturbative arguments. \cite{IrmHulGru-PRL-19,IrmGru-JCP-2019} A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called explicitly correlated F12 (or R12) methods \cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function depending explicitly on the interelectronic distance. This ensures a correct representation of the Coulomb correlation hole around the electron-electron coalescence point, and leads to a much faster convergence of the energy than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] in a triple-$\zeta$ basis set is equivalent to using a quintuple-$\zeta$ basis set with the usual CCSD(T) method, \cite{TewKloNeiHat-PCCP-07} although a computational overhead is introduced by the auxiliary basis set needed to compute the three-electron integrals involved in F12 theory. \cite{BarLoo-JCP-17} In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicit correlation (see, for example, Refs.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods. \cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}
Beside the difficulties of accurately describing the molecular electronic structure within a given basis set, a crucial limitation of WFT methods is the slow convergence of the energy (and related properties) with respect to the size of the \alert{one-electron} basis set. As initially shown by the seminal work of Hylleraas \cite{Hyl-ZP-29} and further developed by Kutzelnigg and coworkers, \cite{Kut-TCA-85,KutKlo-JCP-91, NogKut-JCP-94} the main convergence problem originates from the divergence of the electron-electron Coulomb interaction at the 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 one-electron basis set is impossible and, as a consequence, the convergence of the computed energies and properties are strongly affected. To alleviate this problem, extrapolation techniques have been developed, either based on a partial-wave expansion analysis, \cite{HelKloKocNog-JCP-97,HalHelJorKloKocOlsWil-CPL-98} or more recently based on perturbative arguments. \cite{IrmHulGru-PRL-19,IrmGru-JCP-2019} A more rigorous approach to tackle the basis-set convergence problem is provided by the so-called explicitly correlated F12 (or R12) methods \cite{Ten-TCA-12,TenNog-WIREs-12,HatKloKohTew-CR-12, KonBisVal-CR-12, GruHirOhnTen-JCP-17, MaWer-WIREs-18} which introduce a geminal function depending explicitly on the interelectronic distance. This ensures a correct representation of the Coulomb correlation hole around the electron-electron coalescence point, and leads to a much faster convergence of the energy than usual WFT methods. For instance, using the explicitly correlated version of coupled cluster with singles, doubles, and perturbative triples [CCSD(T)] in a triple-$\zeta$ basis set is equivalent to using a quintuple-$\zeta$ basis set with the usual CCSD(T) method, \cite{TewKloNeiHat-PCCP-07} although a computational overhead is introduced by the auxiliary basis set needed to compute the three-electron integrals involved in F12 theory. \cite{BarLoo-JCP-17} In addition to the computational cost, a possible drawback of F12 theory is its rather complex formalism which requires non-trivial developments for adapting it to a new method. For strongly correlated systems, several multi-reference methods have been extended to explicit correlation (see, for example, Refs.~\onlinecite{Ten-CPL-07,ShiWer-JCP-10,TorKniWer-JCP-11,DemStanMatTenPitNog-PCCP-12,GuoSivValNee-JCP-17}), including approaches based on the so-called universal F12 theory which are potentially applicable to any electronic-structure computational methods. \cite{TorVal-JCP-09,KonVal-JCP-11,HauMaoMukKlo-CPL-12,BooCleAlaTew-JCP-12}
An alternative way to improve the convergence towards the complete basis set (CBS) limit is to treat the short-range correlation effects within DFT and to use WFT methods to deal only with the long-range and/or strong correlation effects. A rigorous approach achieving this mixing of DFT and WFT is range-separated DFT (RSDFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) which relies on a decomposition of the electron-electron Coulomb interaction in terms of the interelectronic distance thanks to a range-separation parameter $\mu$. The advantage of this approach is at least two-fold: i) the DFT part deals primarily with the short-range part of the Coulomb interaction, and consequently the usual semilocal density-functional approximations are more accurate than for standard KS DFT; ii) the WFT part deals only with a smooth non-divergent interaction, and consequently the wave function has no electron-electron cusp \cite{GorSav-PRA-06} and the basis-set convergence is much faster. \cite{FraMusLupTou-JCP-15} A number of approximate RSDFT schemes have been developed involving single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09, TouZhuSavJanAng-JCP-11, MusReiAngTou-JCP-15,KalTou-JCP-18,KalMusTou-JCP-19} and multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, HedTouJen-JCP-18, FerGinTou-JCP-18} WFT methods. Nevertheless, there are still some open issues in RSDFT, such as remaining fractional-charge and fractional-spin errors in the short-range density functionals \cite{MusTou-MP-17} or the dependence of the quality of the results on the value of the range-separation parameter $\mu$.
@ -340,18 +341,18 @@ where $v_\text{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is th
F[\den] = \min_{\Psi \to \den} \mel{\Psi}{\kinop +\weeop}{\Psi},
\end{equation}
where $\kinop$ and $\weeop$ are the kinetic and electron-electron Coulomb operators, and the notation $\Psi \to \den$ means that the wave function $\Psi$ yields the density $\den$.
The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, we always implicitly consider only such representable-in-$\Bas$ densities. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$.
The minimizing density $n_0$ in Eq.~\eqref{eq:levy} is the exact ground-state density. Nevertheless, in practical calculations, the accessible densities are necessarily restricted to the set of densities ``representable in a basis set $\Bas$'', \ie, densities coming from wave functions expandable in the $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, \alert{we always consider only such representable-in-$\Bas$ densities}. With this restriction, Eq.~\eqref{eq:levy} then gives an upper bound $E_0^\Bas$ of the exact ground-state energy. Since the density has a faster convergence with the size of the basis set than the wave function, this restriction is a rather weak one and we can consider that $E_0^\Bas$ is an acceptable approximation to the exact ground-state energy, \ie, $E_0^\Bas \approx E_0$.
In the present context, it is important to notice that the wave functions $\Psi$ defined in Eq.~\eqref{eq:levy_func} are not restricted to a finite basis set, \ie, they should be expanded in a complete basis set. In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, it was then proposed to decompose $F[\den]$ as
\begin{equation}
\label{eq:def_levy_bas}
F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den},
F[\den] = \min_{\wf{}{\Bas} \to \denbas} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den},
\end{equation}
where $\wf{}{\Bas}$ are wave functions expandable in the $N$-electron Hilbert space generated by $\basis$, and
where $\wf{}{\Bas}$ are wave functions expandable in the $N$-electron Hilbert space generated by $\basis$, \alert{$\denbas$ is the projection of the density $n$ in the set of representable-in-$\Bas$ densities}, and
\begin{equation}
\begin{aligned}
\efuncden{\den} = \min_{\Psi \to \den} \mel*{\Psi}{\kinop +\weeop }{\Psi}
- \min_{\Psi^{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}
- \min_{\Psi^{\Bas} \to \denbas} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}}
\end{aligned}
\end{equation}
is the complementary density functional to the basis set $\Bas$.
@ -629,6 +630,7 @@ The performance of each of these functionals is tested in the following. Note th
\caption{
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, labelled vdz), cc-pVTZ (middle, labelled as ``vtz''), and cc-pVQZ (bottom, labelled as ``vqz'') basis sets.
The MRCI+Q energies and the estimated exact energies have been extracted from Ref.~\onlinecite{h10_prx}.
% \alert{The reported energy is divided by the number of atoms in the chain.}
\label{fig:H10}}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -693,16 +695,20 @@ The performance of each of these functionals is tested in the following. Note th
\subsection{Computational details}
The purpose of the present paper being to investigate the performance of the density-based basis-set correction in regimes of both weak and 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{N2}, \ce{O2}, and \ce{F2} diatomics. For a given basis set, in order to compute the ground-state energy in Eq.~\eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the basis-set correction $\efuncbasisFCI$.
\alert{We present potential energy curves of small molecules up to the dissociation limit
to investigate the performance of the basis-set correction in regimes of both weak and strong correlation.
The considered systems are the \ce{H10} linear chain with equally-spaced atoms, and the \ce{N2}, \ce{O2}, and \ce{F2} diatomics.}
In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVDZ and aug-cc-pVTZ basis sets,~\cite{KenDunHar-JCP-92} approximations to the FCI energies are obtained using frozen-core selected CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \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). All these calculations are performed with the latest version of \textsc{QUANTUM PACKAGE}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} method as an estimate of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$ mHa. For the \ce{N2}, \ce{O2}, and \ce{F2} molecules, we also performed single-point exFCI calculations in the aug-cc-pVQZ basis set at the equilibrium geometry to obtain reliable estimates of the FCI/CBS dissociation energy.
\alert{The computation of the ground-state energy in Eq.~\eqref{eq:e0approx} in a given basis set requires approximations to the FCI energy $\efci$ and to the basis-set correction $\efuncbasisFCI$.
For diatomics with the aug-cc-pVDZ and aug-cc-pVTZ basis sets},~\cite{KenDunHar-JCP-92} energies are obtained using frozen-core selected-CI calculations (using the CIPSI algorithm) followed by the extrapolation scheme proposed by Holmes \textit{et al.} (see Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJac-JCTC-19, QP2} for more detail). All these calculations are performed with the latest version of \textsc{Quantum Package}, \cite{QP2} and will be labeled as exFCI in the following. In the case of \ce{F2}, we also use the correlation energy extrapolated by intrinsic scaling (CEEIS) \cite{BytNagGorRue-JCP-07} method as an estimate of the FCI correlation energy with the cc-pVXZ (X $=$ D, T, and Q) basis sets.~\cite{Dun-JCP-89} The estimated exact potential energy curves are obtained from experimental data \cite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from CEEIS calculations in the case of \ce{F2}. For all geometries and basis sets, the error with respect to the exact FCI energies are estimated to be of the order of $0.5$~mHa.
\alert{For the three diatomics, we performed an additional exFCI calculation with the aug-cc-pVQZ basis set at the equilibrium geometry} to obtain reliable estimates of the FCI/CBS dissociation energy.
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).
Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, 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}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}.
Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software~\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, effective spin polarization $\tilde{\zeta}(\br{})$, 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 \alert{are} performed with the following active spaces: (10e,10o) for \ce{H10}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}.
Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the $1s$ core orbitals, and the range-separation function follows the definition given in Eq.~\eqref{eq:def_mur_val}.
Regarding the computational cost of the present approach, it should be stressed (see Appendix~\ref{app:computational} for additional details) that the basis-set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations.
\alert{It should be stressed that the computational cost of the basis-set correction (see Appendix~\ref{app:computational}) is negligible compared to the cost the selected-CI calculations}.
%We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -719,13 +725,16 @@ Regarding the computational cost of the present approach, it should be stressed
\subsection{H$_{10}$ chain}
The \ce{H10} chain with equally-spaced atoms is a 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 system).
The \ce{H10} chain with equally-spaced atoms is a prototype of strongly-correlated systems as it consists in the simultaneous breaking of \alert{10 interacting covalent $\sigma$ bonds}.
\alert{As it is a relatively small system, benchmark calculations at near-CBS values are available (see Ref.~\onlinecite{h10_prx} for a detailed study of this system)}.
We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation. The computation of the atomization energies $D_0$ for each level of theory 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, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime.
\alert{We report in Fig.~\ref{fig:H10} the potential energy curves computed using the cc-pVXZ (X $=$ D, T, and Q) basis sets for different levels of approximation, and the corresponding atomization energies $D_0$ are reported in Table \ref{tab:d0}.
As a general trend, the addition of the basis-set correction globally improves
the quality of the potential energy curves}, independently of the approximation level of $\efuncbasis$. Also, no erratic behavior is found when stretching the bonds, which shows that the present procedure (\ie, the determination of the range-separation function and the definition of the functionals) is robust when reaching the strong-correlation regime.
In other words, smooth potential energy curves are obtained with the present basis-set correction.
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 $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not even reached at the standard MRCI+Q/cc-pVQZ level of theory.
More quantitatively, the values of $D_0$ are within \alert{the} chemical accuracy (\ie, an error below $1.4$ mHa) \alert{with} the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not \alert{yet} reached at the standard MRCI+Q/cc-pVQZ level of theory.
Analyzing more carefully the performance of the different types of approximate functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients included in these functionals: i) the explicit use of the on-top pair density originating from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependence on any kind of spin polarization does not lead to significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density.
Analyzing more carefully the performance of the different types of approximate functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the \alert{maximum} difference on $D_0$ being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients included in these functionals: i) the explicit use of the on-top pair density originating from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependence on any kind of spin polarization does not lead to a significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density.
This could have significant implications for the construction of more robust families of density-functional approximations within DFT.
%Finally, the reader would have noticed that we did not report the performance of the $\pbeuegns$ functional as its performance are significantly inferior than the three other functionals. The main reason behind this comes from the fact that $\pbeuegns$ has no direct or indirect knowledge of the on-top pair density of the system. Therefore, it yields a non-zero correlation energy for the totally dissociated \ce{H10} chain even if the on-top pair density is vanishingly small. This necessary lowers the value of $D_0$. Therefore, from hereon, we simply discard the $\pbeuegns$ functional.
@ -789,7 +798,7 @@ The \ce{N2}, \ce{O2} and \ce{F2} molecules are complementary to the \ce{H10} sys
We report in Figs.~\ref{fig:N2} and \ref{fig:O2} the potential energy curves of \ce{N2} and \ce{O2} computed at various approximation levels using the aug-cc-pVDZ and aug-cc-pVTZ basis sets. Figure \ref{fig:F2} reports the potential energy curve of \ce{F2} using the cc-pVXZ (X $=$ D, T, and Q) basis sets. The value of $D_0$ for each level of theory is reported in Table \ref{tab:d0}.
Just as in \ce{H10}, the quality of $D_0$ is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of electron density and also for an open-shell 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 $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the results, a result that can be anticipated due to the strong breathing-orbital effect induced by the ionic valence bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
Just as in \ce{H10}, the quality of $D_0$ is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependence on the on-top pair density allows one to remove the dependence of any kind of spin polarization for a quite wide range of \alert{electron densities} and also for an open-shell 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 $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the \alert{quality of the curves}, a result that can be anticipated due to the strong breathing-orbital effect induced by the ionic valence bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the quality of $D_0$ slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI $D_0$ and very close to chemical accuracy.
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 complementary functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is based on the universal condition provided by the electron-electron cusp 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 theoretically expected.
@ -798,7 +807,7 @@ We hope to report further on this in the near future.
\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 \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasingly large basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems.
In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We \alert{have applied the method to} the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasingly large basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems.
The density-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 electron-electron interaction projected into an incomplete basis set $\basis$; ii) the fit of this effective interaction with the long-range interaction used in RSDFT; and iii) the use of a short-range, complementary functional borrowed from RSDFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling spin-multiplet degeneracy and size consistency. To fulfill such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and we developed functionals using only $S_z$-independent density-like quantities.

13
new/F2_avdz/plots/plot.sh
View File

@ -1,6 +1,8 @@
#!/bin/bash
WF=FCI
BASIS=avdz
BASIS2=aug-cc-pVDZ
METHOD=DFT
TYPE=relat
@ -12,17 +14,16 @@ cat << EOF > pouet.gp
set xrange [:5]
set key bottom
set xlabel "Inter nuclear distance (a.u.) "
set xlabel "Internuclear distance (a.u.) "
set ylabel "Energy (Hartree)"
set key font ",15"
set key spacing "1,8"
plot '${FILE}' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "${WF}/$BASIS"
replot '${FILE}' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "${WF}+SU-PBE-OT/$BASIS"
set terminal pdf
set output "${OUT}.pdf"
plot '${FILE}' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "${WF}/$BASIS2"
replot '${FILE}' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "${WF}+SU-PBE-OT/$BASIS2"
replot '${FILE2}' using 1:2 smooth cspline notitle lt 7 , "" using 1:2 w p lt 7 ps 1 title "Exact"
set terminal eps enhanced linewidth 6
set output "${OUT}.eps"
replot
EOF

15
new/F2_avdz/plots/plot_diff.sh
View File

@ -1,6 +1,7 @@
WF=FCI
BASIS=avdz
BASIS2=aug-cc-pVDZ
METHOD=DFT
TYPE=relat
@ -16,12 +17,11 @@ set ylabel "Energy (Hartree)"
set key font ",15"
set key spacing "1,8"
plot '${FILE}' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "${WF}/$BASIS"
replot '${FILE}' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "${WF}+PBEot0{/Symbol z}/$BASIS"
set term pdf
set output "${OUT}.pdf"
plot '${FILE}' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "${WF}/$BASIS2"
replot '${FILE}' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "${WF}+PBEot0{/Symbol z}/$BASIS2"
replot '${FILE}' using 1:6 smooth cspline notitle lt 7 , "" using 1:6 w p lt 7 ps 1 title "Exact"
set terminal eps enhanced linewidth 6
set output "${OUT}.eps"
replot
EOF
@ -38,14 +38,13 @@ set ylabel "Energy (Hartree)"
set key font ",15"
set key spacing "1,8"
set term pdf
set output "${OUT}.pdf"
plot '${FILE}' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "${WF}/$BASIS"
replot '${FILE}' using 1:3 smooth cspline notitle lt 9 , "" using 1:3 w p lt 9 ps 1 title "${WF}+PBE-UEG~{/Symbol z}{.8-}/$BASIS"
replot '${FILE}' using 1:4 smooth cspline notitle lt 4 , "" using 1:4 w p lt 4 ps 1 title "${WF}+PBEot~{/Symbol z}{.8-}/$BASIS"
replot '${FILE}' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "${WF}+PBEot0{/Symbol z}/$BASIS"
replot '${FILE}' using 1:6 smooth cspline notitle lt 7 , "" using 1:6 w p lt 7 ps 1 title "Exact"
set terminal eps enhanced linewidth 6
set output "${OUT}.eps"
replot
EOF

18
new/F2_avdz/plots/pouet.gp
View File

@ -1,15 +1,9 @@
set xrange [:5]
set key bottom
set xlabel "Inter nuclear distance (a.u.) "
set ylabel "Energy (Hartree)"
set key font ",15"
set key spacing "1,8"
plot 'E_relat_dft' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "FCI/avdz"
replot 'E_relat_dft' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "FCI+SU-PBE-OT/avdz"
replot 'E_relat_exact' using 1:2 smooth cspline notitle lt 7 , "" using 1:2 w p lt 7 ps 1 title "Exact"
set xrange [:7]
plot 'data_DFT_avdzE_error' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "FCI/avdz"
replot 'data_DFT_avdzE_error' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "FCI+PBEot{/Symbol z}/avdz"
replot 'data_DFT_avdzE_error' using 1:4 smooth cspline notitle lt 4 , "" using 1:4 w p lt 4 ps 1 title "FCI+PBEot/avdz"
replot 'data_DFT_avdzE_error' using 1:3 smooth cspline notitle lt 9 , "" using 1:3 w p lt 9 ps 1 title "FCI+PBE/avdz"
set terminal eps enhanced linewidth 6
set output "DFT_avdzE_relat.eps"
set output "DFT_avdzE_error.eps"
replot