Theory
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@ -293,7 +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 density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To model 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 density functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$) and size consistency. Specifically, we systematically investigate the functional dependence 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.
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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 electron-electron Coulomb interaction projected in the finite basis set. This allows to use RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To model 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 density functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, invariance with respect to the spin operator $S_z$) and size consistency. Specifically, we systematically investigate the functional dependence 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.
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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 in certain cases. 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 surface.
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\end{abstract}
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@ -309,7 +309,7 @@ The difficulty of obtaining a reliable theoretical description of a given chemic
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In practice, WFT uses a finite one-particle basis set (here denoted as $\basis$). 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 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, invariance with respect to the spin operator $S_z$) and size consistency.
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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 electronic 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 Coulomb electron-electron 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-arxiv-19} 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 distances. 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- and four-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, Ref.~\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}
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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 electronic 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-arxiv-19} 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 distances. 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- and four-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, Ref.~\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}
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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 $\mu$.
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% which can be seen as an empirical parameter.
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@ -328,7 +328,7 @@ As the theory behind the present basis-set correction has been exposed in detail
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\subsection{Basic equations}
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\label{sec:basic}
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The exact ground-state energy $E_0$ of a $N$-electron system can in principle be obtained in DFT by a minimization over $N$-electron density $\denr$
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The exact ground-state energy $E_0$ of a $N$-electron system can, in principle, be obtained in DFT by a minimization over \titou{$N$-electron density} $\denr$
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\begin{equation}
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\label{eq:levy}
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E_0 = \min_{\den} \bigg\{ F[\den] + \int \d \br{} v_{\text{ne}} (\br{}) \denr \bigg\},
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@ -339,7 +339,7 @@ where $v_{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is the uni
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F[\den] = \min_{\Psi \to \den} \mel{\Psi}{\kinop +\weeop}{\Psi},
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\end{equation}
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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 $\den$.
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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 Hilbert space generated by the basis set $\Bas$. In the following, we always implicitly consider only such densities representable in $\Bas$. With this restriction, Eq.~\eqref{eq:levy} gives then 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, $E_0^\Bas \approx E_0$.
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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 Hilbert space generated by the 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, $E_0^\Bas \approx E_0$.
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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
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\begin{equation}
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@ -361,21 +361,21 @@ Introducing the decomposition in Eq.~\eqref{eq:def_levy_bas} back into Eq.~\eqre
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\\
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+ \int \d \br{} v_{\text{ne}} (\br{}) \den_{\Psi^{\Bas}}(\br{}) \bigg\},
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\end{multline}
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where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density extracted from $\wf{}{\Bas}$. Therefore, thanks to Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional accounting for the correlation effects that are not included in the basis set.
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where the minimization is only over wave functions $\wf{}{\Bas}$ restricted to the basis set $\basis$ and $\den_{{\Psi^{\Bas}}}(\br{})$ refers to the density generated from $\wf{}{\Bas}$. Therefore, thanks to Eq.~\eqref{eq:E0basminPsiB}, one can properly combine a WFT calculation in a finite basis set with a density functional (hereafter referred to as complementary density functional) accounting for the correlation effects that are not included in the basis set.
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As a simple non-self-consistent version of this approach, we can approximate the minimizing wave function $\Psi^{\Bas}$ in Eq.~\eqref{eq:E0basminPsiB} by the ground-state FCI wave function $\psifci$ within $\Bas$, and we then obtain the following approximation for the exact ground-state energy [see Eqs.~(12)--(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}]
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\begin{equation}
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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 the strong-correlation regime. 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 strong correlation. 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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As originally shown by Kato, \cite{Kat-CPAM-57} the electron-electron cusp of the exact wave function originates from the divergence of the Coulomb interaction at the coalescence point. Therefore, a cuspless wave function $\wf{}{\Bas}$ could also be obtained from a Hamiltonian with a non-divergent electron-electron interaction.
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In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual Coulomb electron-electron interaction.
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In other words, the impact of the basis set incompleteness can be understood as the removal of the divergence of the usual electron-electron Coulomb interaction.
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As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see Sec.~II D~and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the Coulomb electron-electron interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
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As originally derived in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} (see Sec.~II D~and Appendices), one can obtain an effective non-divergent electron-electron interaction, here referred to as $\wbasis$, which reproduces the expectation value of the electron-electron Coulomb interaction operator over a given wave function $\wf{}{\Bas}$. As we are interested in the behaviour at the coalescence point, we focus on the opposite-spin part of the electron-electron interaction. More specifically, the effective electron-electron interaction associated to a given wave function $\wf{}{\Bas}$ is defined as
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\begin{equation}
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\label{eq:wbasis}
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\wbasis =
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@ -389,13 +389,13 @@ where $\twodmrdiagpsi$ is the opposite-spin pair density associated with $\wf{}{
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\begin{equation}
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\twodmrdiagpsi = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
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\end{equation}
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and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$, and
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and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ its associated tensor in a basis of spatial orthonormal orbitals $\{\SO{p}{}\}$,
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\begin{equation}
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\label{eq:fbasis}
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\fbasis
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= \sum_{pqrstu \in \Bas} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
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\end{equation}
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with the usual two-electron Coulomb integrals $\V{pq}{rs}= \braket{pq}{rs}$.
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and $\V{pq}{rs}= \braket{pq}{rs}$ are the usual two-electron Coulomb integrals.
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With such a definition, one can show that $\wbasis$ satisfies
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\begin{multline}
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\frac{1}{2}\iint \dr{1} \dr{2} \wbasis \twodmrdiagpsi =
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@ -434,7 +434,7 @@ Because of the very definition of $\wbasis$, one has the following property in t
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which is again fundamental to guarantee the correct behavior of the theory in the CBS limit.
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\subsubsection{Frozen-core approximation}
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As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the complementary density functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals, respectively, and define the valence-only local range-separation parameter as
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As all WFT calculations in this work are performed within the frozen-core approximation, we use a valence-only version of the various quantities needed for the complementary density functional introduced in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}. We partition the basis set as \titou{$\Bas = \Cor \bigcup \BasFC$, where $\Cor$ and $\BasFC$ are the sets of core and active orbitals}, respectively, and define the valence-only local range-separation parameter as
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\begin{equation}
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\label{eq:def_mur_val}
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\murpsival = \frac{\sqrt{\pi}}{2} \wbasiscoalval{},
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@ -462,13 +462,13 @@ is the valence-only effective interaction and
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One would note the restrictions of the sums to the set of active orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
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It is also noteworthy that, with the present definition, $\wbasisval$ still tends to the usual Coulomb interaction as $\Bas \to \CBS$.
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\subsection{Density functional approximations for short-range correlation}
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\subsection{Complementary density functionals}
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\label{sec:functional}
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||||
|
||||
\subsubsection{Generic form}
|
||||
\label{sec:functional_form}
|
||||
|
||||
As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary density functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}\cite{TouGorSav-TCA-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the total density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$, the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$.
|
||||
As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary density functional $\efuncden{\den}$ by using the so-called correlation energy functional with multideterminant reference (ECMD) introduced by Toulouse \textit{et al.}\cite{TouGorSav-TCA-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to use a Perdew-Burke-Ernzerhof (PBE)-like functional which uses the one-electron density $\denr$, the spin polarization $\zeta(\br{})=[n_\uparrow(\br{})-n_\downarrow(\br{})]/\denr$ (where $n_\uparrow(\br{})$ and $n_\downarrow(\br{})$ are the spin-up and spin-down densities), the reduced density gradient $s(\br{}) = \nabla \denr/\denr^{4/3}$, and the on-top pair density $\ntwo(\br{})\equiv \ntwo(\br{},\br{})$. In the present work, all these quantities are computed with the same wave function $\psibasis$ used to define $\mur \equiv\murpsi$.
|
||||
Therefore, $\efuncden{\den}$ has the following generic form
|
||||
\begin{multline}
|
||||
\label{eq:def_ecmdpbebasis}
|
||||
@ -479,12 +479,12 @@ Therefore, $\efuncden{\den}$ has the following generic form
|
||||
where
|
||||
\begin{equation}
|
||||
\label{eq:def_ecmdpbe}
|
||||
\ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe,\titou{\ntwo}) \; \mu^3},
|
||||
\ecmd(\argecmd) = \frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{1+ \beta(\argepbe,\ntwo) \; \mu^3},
|
||||
\end{equation}
|
||||
is the correlation energy per particle, with
|
||||
\begin{equation}
|
||||
\label{eq:def_beta}
|
||||
\beta(\argepbe,\titou{\ntwo}) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den},
|
||||
\beta(\argepbe,\ntwo) = \frac{3}{2\sqrt{\pi}(1 - \sqrt{2})}\frac{\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)}{\ntwo/\den},
|
||||
\end{equation}
|
||||
where $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$ is the usual PBE correlation energy per particle. \cite{PerBurErn-PRL-96} Before introducing the different flavors of approximate functionals that we will use here (see Sec.~\ref{sec:def_func}), we would like to give some motivations for this choice of functional form.
|
||||
|
||||
@ -499,12 +499,13 @@ which is relevant in the weak-correlation (or high-density) limit. In the large-
|
||||
\end{equation}
|
||||
which is the exact large-$\mu$ behavior of the exact ECMD correlation energy. \cite{PazMorGorBac-PRB-06,FerGinTou-JCP-18} Of course, for a specific system, the large-$\mu$ behavior will be exact only if one uses for $n_2$ the \textit{exact} on-top pair density of this system. This large-$\mu$ limit in Eq.~\eqref{eq:lim_mularge} is relevant in the strong-correlation (or low-density) limit. In the context of RSDFT, some of the present authors have illustrated in Ref.~\onlinecite{FerGinTou-JCP-18} that the on-top pair density involved in Eq.~\eqref{eq:def_ecmdpbe} plays indeed a crucial role when reaching the strong-correlation regime. The importance of the on-top pair density in the strong-correlation regime have been also recently acknowledged by Gagliardi and coworkers \cite{CarTruGag-JPCA-17} and Pernal and coworkers.\cite{GritMeePer-PRA-18}
|
||||
|
||||
Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes
|
||||
Note also that $\ecmd(\argecmd)$ vanishes when $\ntwo$ vanishes, \ie,
|
||||
\begin{equation}
|
||||
\label{eq:lim_n2}
|
||||
\lim_{\ntwo \to 0} \ecmd(\argecmd) = 0,
|
||||
\end{equation}
|
||||
which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems. Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \to \infty$ like all RSDFT short-range functionals \begin{equation}
|
||||
which is expected for systems with a vanishing on-top pair density, such as the totally dissociated H$_2$ molecule which is the archetype of strongly correlated systems. Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \to \infty$ like all RSDFT short-range functionals, \ie,
|
||||
\begin{equation}
|
||||
\label{eq:lim_muinf}
|
||||
\lim_{\mu \to \infty} \ecmd(\argecmd) = 0.
|
||||
\end{equation}
|
||||
@ -520,12 +521,12 @@ First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muin
|
||||
\lim_{\basis \to \text{CBS}} \efuncdenpbe{\argebasis} = 0, \quad \forall\, \psibasis,
|
||||
\end{equation}
|
||||
|
||||
Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ [see Eq.~\eqref{eq:wbasis}] together with the condition in Eq.~\eqref{eq:lim_muinf}, ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
|
||||
Second, the fact that $\efuncdenpbe{\argebasis}$ vanishes for systems with vanishing on-top pair density guarantees the correct limit for one-electron systems and for the stretched H$_2$ molecule. This property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ [see Eq.~\eqref{eq:wbasis}] together with the condition in Eq.~\eqref{eq:lim_muinf}, and ii) the fact that $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
|
||||
|
||||
\subsection{Requirements for strong correlation}
|
||||
\label{sec:requirements}
|
||||
|
||||
An important requirement for any electronic-structure method is size-consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space-self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies.
|
||||
An important requirement for any electronic-structure method is size-consistency, \ie, the additivity of the energies of non-interacting fragments, which is mandatory to avoid any ambiguity in computing interaction energies. When two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, as in the case of weak intermolecular interactions for instance, spin-restricted Hartree-Fock (RHF) is size-consistent. When the two subsystems dissociate in open-shell systems, such as in covalent bond breaking, it is well known that the RHF approach fails and an alternative is to use a complete-active-space self-consistent-field (CASSCF) wave function which, provided that the active space has been properly chosen, leads to additive energies.
|
||||
|
||||
Another important requirement is spin-multiplet degeneracy, \ie, the independence of the energy with respect to the $S_z$ component of a given spin state, which is also a property of any exact wave function. Such a property is also important in the context of covalent bond breaking where the ground state of the supersystem $\ce{A + B}$ is generally of lower spin than the corresponding ground states of the fragments (\ce{A} and \ce{B}) which can have multiple $S_z$ components.
|
||||
|
||||
@ -564,23 +565,23 @@ In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shel
|
||||
|
||||
As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between two flavors of functionals are only due to the type of i) spin polarization, and ii) on-top pair density.
|
||||
|
||||
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 CASSCF wave function, and ii) a \textit{zero} spin polarization. \manu{When using the \textit{effective} spin polarization $\tilde{\zeta}$, we refer the functional with "SP" which stands for "spin polarized"}.
|
||||
Regarding the spin polarization that enters into $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, two different types of $S_z$-independent formulations are considered: i) \titou{the \textit{effective} spin polarization $\tilde{\zeta}$} defined in Eq.~\eqref{eq:def_effspin} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization. \titou{In the latter case, the functional is referred as to ``SU'' which stands for ``spin unpolarized''}.
|
||||
|
||||
Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use two different approximations. The first one is based on the uniform electron gas (UEG) and reads
|
||||
\begin{equation}
|
||||
\label{eq:def_n2ueg}
|
||||
\ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n),
|
||||
\end{equation}
|
||||
where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$. \manu{When using $\ntwo^{\text{UEG}}(n,\zeta)$ in a functional, we will refer it as ``UEG''. }
|
||||
where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin} in order to ensure $S_z$ independence. Thus, $\ntwo^{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function through $\tilde{\zeta}$. When using $\ntwo^{\text{UEG}}(n,\tilde{\zeta})$ in a functional, we will refer to it as ``UEG''.
|
||||
|
||||
Another approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the previous authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density
|
||||
Another approach to approximate the exact on-top pair density consists in using directly the on-top pair density of the CASSCF wave function. Following the work of some of the present authors, \cite{FerGinTou-JCP-18,GinSceTouLoo-JCP-19} we introduce the extrapolated on-top pair density
|
||||
\begin{equation}
|
||||
\label{eq:def_n2extrap}
|
||||
\ntwoextrap(\ntwo,\mu) = \bigg( 1 + \frac{2}{\sqrt{\pi}\mu} \bigg)^{-1} \; \ntwo,
|
||||
\end{equation}
|
||||
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. When using $\ntwoextrap(\ntwo,\mu)$ in a functional, we will simply refer it as ``ot''\manu{, which stands for "on-top"}.
|
||||
which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. When using $\ntwoextrap(\ntwo,\mu)$ in a functional, we will simply refer it as ``OT'', which stands for "on-top".
|
||||
|
||||
We then define \titou{four} functionals:
|
||||
We then define four functionals:
|
||||
|
||||
\begin{itemize}
|
||||
|
||||
@ -596,7 +597,7 @@ We then define \titou{four} functionals:
|
||||
\label{eq:def_pbeueg_ii}
|
||||
\bar{E}^\Bas_{\pbeontXi} = \int \d\br{} \,\denr \ecmd(\argrpbeontXi),
|
||||
\end{equation}
|
||||
\item[iii)] \titou{$\pbeuegns$ which combines a zero spin polarization and the UEG on-top pair density of Eq.~\eqref{eq:def_n2ueg}:}
|
||||
\item[iii)] $\pbeuegns$ which combines a zero spin polarization and the UEG on-top pair density of Eq.~\eqref{eq:def_n2ueg}:
|
||||
\begin{equation}
|
||||
\label{eq:def_pbeueg_iii}
|
||||
\bar{E}^\Bas_{\pbeuegns} = \int \d\br{} \,\denr \ecmd(\argrpbeuegns),
|
||||
@ -607,10 +608,10 @@ We then define \titou{four} functionals:
|
||||
\bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns).
|
||||
\end{equation}
|
||||
\end{itemize}
|
||||
The performance of each of these \titou{four} functionals is tested below.
|
||||
The performance of each of these four functionals is tested below.
|
||||
|
||||
|
||||
DFT: BLACK BOX and not CASSCF
|
||||
%DFT: BLACK BOX and not CASSCF
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Results}
|
||||
\label{sec:results}
|
||||
@ -632,7 +633,7 @@ DFT: BLACK BOX and not CASSCF
|
||||
|
||||
The purpose of the present paper being the study of the basis-set correction in regimes of both weak and/or 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. 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 $\efuncbasisFCI$.
|
||||
|
||||
In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVXZ (X=D,T), 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} (exFCI), and the correlation energy extrapolation by intrinsic scaling\cite{BytNagGorRue-JCP-07} (CEEIS) in the case of \ce{F2} for the cc-pVXZ (X=D,T,Q) basis set. The estimated exact potential energy curves are obtained from experimental data in Ref.~\onlinecite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from extrapolated 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 on 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 estimations of the FCI dissociation energies in these basis sets.
|
||||
In the case of the \ce{N2}, \ce{O2}, and \ce{F2} molecules for the aug-cc-pVXZ (X=D,T), 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 \textsc{QUANTUM PACKAGE} \cite{QP2} (exFCI), and the correlation energy extrapolation by intrinsic scaling\cite{BytNagGorRue-JCP-07} (CEEIS) in the case of \ce{F2} for the cc-pVXZ (X=D,T,Q) basis set. The estimated exact potential energy curves are obtained from experimental data in Ref.~\onlinecite{LieCle-JCP-74a} for the \ce{N2} and \ce{O2} molecules, and from extrapolated 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 on 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 estimations of the FCI dissociation energies in these basis sets.
|
||||
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 density 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}, (10e,8o) for \ce{N2}, (12e,8o) for \ce{O2}, and (14e,8o) for \ce{F2}.
|
||||
|
||||
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