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Emmanuel Giner
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@ -373,7 +373,7 @@ where $(v_{ne}(\br{})|\denr)$ is the nuclei-electron interaction for a given den
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\label{eq:levy_func}
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F[\denr] = \min_{\Psi \rightarrow \denr} \elemm{\Psi}{\kinop +\weeop }{\Psi}.
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\end{equation}
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The minimizing density $n_0$ of equation \eqref{eq:levy} is the exact ground state density.
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The minimizing density $n_0$ of eq. \eqref{eq:levy} is the exact ground state density.
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Nevertheless, in practical calculations the minimization is performed over the set $\setdenbasis$ which are the densities representable in a basis set $\Bas$ and we assume from thereon that the densities used in the equations belong to $\setdenbasis$.
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In the present context it is important to notice that in order to recover the \textit{exact} ground state energy, the wave functions $\Psi$ involved in the definition of eq. \eqref{eq:levy_func} must be developed in a complete basis set.
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@ -436,7 +436,7 @@ As it was shown in Ref. \onlinecite{GinPraFerAssSavTou-JCP-18}, the effective in
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\label{eq:cbs_wbasis}
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\lim_{\Bas \rightarrow \text{CBS}} \wbasis = \frac{1}{|\br{1}-\br{2}|}\quad \forall\,\psibasis.
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\end{equation}
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The condition of equation \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set.
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The condition of eq. \eqref{eq:cbs_wbasis} is fundamental as it guarantees the good behaviour of all the theory in the limit of a complete basis set.
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\subsection{Definition of a range-separation parameter varying in real space}
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\label{sec:mur}
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@ -491,7 +491,7 @@ and $\twodmrdiagpsival$
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\label{eq:twordm_val}
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\twodmrdiagpsival = \sum_{pqrs \in \BasFC} \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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Notice the summations on the active set of orbitals in equations \eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
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Notice the summations on the active set of orbitals in eqs. \eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
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It is noteworthy that, within the present definition, $\wbasisval$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
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\subsection{Generic form and properties of the approximations for $\efuncden{\denr}$ }
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@ -552,7 +552,7 @@ Because of the properties \eqref{eq:cbs_mu} and \eqref{eq:lim_muinf}, $\efuncden
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\end{equation}
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which guarantees an unaltered limit when reaching the CBS limit.
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Also, the $\efuncdenpbe{\argecmd}$ vanishes for systems with vanishing on-top pair density, which guarantees the good limit in the case of stretched H$_2$ and for one-electron system.
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Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see equation \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see equation \eqref{eq:lim_n2}).
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Such a property is guaranteed independently by i) the definition of the effective interaction $\wbasis$ (see eq. \eqref{eq:wbasis}) together with the condition \eqref{eq:lim_muinf}, ii) the fact that the $\ecmd(\argecmd)$ vanishes when the on-top pair density vanishes (see eq. \eqref{eq:lim_n2}).
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\subsection{Requirements for the approximated functionals in the strong correlation regime}
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\label{sec:requirements}
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@ -564,7 +564,7 @@ Another important requirement is the independence of the energy with respect to
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Such a property is also important in the context of covalent bond breaking where the ground state of the super system $A+B$ is in general of low spin while the ground states of the fragments $A$ and $B$ are in high spin which can have multiple $S_z$ components.
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\subsubsection{Condition for the functional $\efuncdenpbe{\argebasis}$ to obtain $S_z$ invariance}
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A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see equation \eqref{eq:def_ecmdpbe}).
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A sufficient condition to achieve $S_z$ invariance is to eliminate all dependency to $S_z$, which in the case of $\ecmd(\argecmd)$ is the spin polarisation $\zeta(\br{})$ involved in the correlation energy density $\varepsilon_{\text{c,PBE}}(\argepbe)$ (see eq. \eqref{eq:def_ecmdpbe}).
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As originally shown by Perdew and co-workers\cite{PerSavBur-PRA-95}, the dependence on the spin polarisation in the KS-DFT framework can be removed by the rewriting the spin polarisation of a single Slater determinant with only the on-top pair density and the total density. In other terms, the spin density dependence usually introduced in the correlation functionals of KS-DFT tries to mimic the effect of the on-top pair density.
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Based on this reasoning, a similar approach has been used in the context of multi configurational DFT in order to remove the $S_z$ dependency.
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In practice, these approaches introduce the effective spin polarisation
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@ -580,9 +580,9 @@ which uses the on-top pair density $\ntwo_{\psibasis}$ of a given wave function
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The advantages of this approach are at least two folds: i) the effective spin polarisation $\tilde{\zeta}$ is $S_z$ invariant, ii) it introduces an indirect dependency on the on-top pair density of the wave function $\psibasis$ which usually improves the treatment of strong correlation.
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Nevertheless, the use of $\tilde{\zeta}$ presents several disadvantages as it can become complex when $n^2 - 4 \ntwo_{\psibasis}<0$ and also
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the formula of equation \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
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the formula of eq. \eqref{eq:def_effspin} is exact only when the density $n$ and on-top pair density $\ntwo_{\psibasis}$ are obtained from a single determinant\cite{PerSavBur-PRA-95}, but it is applied to multi configurational wave functions.
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An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see equations \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
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An alternative to eliminate the $S_z$ dependency would be to simply set $\zeta(\br{})=0$, but this would lower the accuracy of the usual correlation functional, such as the PBE correlation functional used here $\varepsilon_{\text{c,PBE}}(\argepbe)$. Nevertheless, as the spin polarisation usually tries to mimic the on-top pair density and the function $\ecmd(\argecmd)$ explicitly depends on the on-top pair density (see eqs. \eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}), we propose here to use the $\ecmd$ functional with \textit{a zero spin polarisation}. This ensures a $S_z$ invariance and, as will be numerically shown, very weakly affect the accuracy of the functional.
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\subsubsection{Conditions on $\psibasis$ for the extensivity}
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In the case of the present basis set correction, as $\efuncdenpbe{\argebasis}$ is an integral over $\mathbb{R}^3$ of local quantities, in the case of non overlapping fragments $A\ldots B$ it can be written as the sum of two local contributions: one coming from the integration over the region of the sub-system $A$ and the other one from the region of the sub-system $B$.
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@ -598,14 +598,14 @@ The condition for the active space involved in the CASSCF wave function is that
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As the present work focusses on the strong correlation regime, we propose here to investigate only approximated functionals which are $S_z$ invariant and size extensive in the case of covalent bond breaking. Therefore, the wave function $\psibasis$ used throughout this paper are of CASSCF type in order to ensure extensivity of all density related quantities.
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The difference between the different flavours of functionals are only on i) the type of on-top pair density used, and ii) the type of spin polarisation used.
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Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in equation \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
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Regarding the spin polarisation that enters into $\varepsilon_{\text{c,PBE}}(\argepbe)$, two different types of $S_z$ invariant formulations are used: i) the \textit{effective} spin polarization $\tilde{\zeta}$ defined in eq. \eqref{eq:def_effspin}, and iii) a \textit{zero} spin polarization.
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Regarding the approximation to the \textit{exact} on-top pair density entering in equation \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 approximation to the \textit{exact} 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,\br{}) = n(\br{})^2\big(1-\zeta(\br{})\big)g_0\big(n(\br{})\big)
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\end{equation}
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where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in equation \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of equation \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo_{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
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where the pair-distribution function $g_0(n)$ is taken from equation (46) of Ref. \onlinecite{GorSav-PRA-06}. As some spin polarization appear in eq. \eqref{eq:def_n2ueg}, we use the effective spin density $\tilde{\zeta}$ of eq. \eqref{eq:def_effspin} in order to ensure $S_z$ invariance. Notice that, as we use a CASSCF wave function and $\tilde{\zeta}$ as spin polarization, the $\ntwo_{\text{UEG}}$ will depend indirectly on the on-top pair density of the CASSCF wave function as $\tilde{\zeta}$ depends on the on-top pair density.
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Another approach to approximate of the exact on top pair density consists in taking advantage of the on-top pair density of the wave function $\psibasis$. 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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\begin{equation}
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@ -619,21 +619,21 @@ When using $\ntwoextrap(\ntwo,\mu,\br{})$ in a functional, we will refer simply
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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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i) The PBE-UEG-$\tilde{\zeta}$ which uses the UEG-like on-top pair density defined in equation \eqref{eq:def_n2ueg}, the effective spin polarization of equation \eqref{eq:def_effspin} and which reads
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i) The PBE-UEG-$\tilde{\zeta}$ which uses the UEG-like on-top pair density defined in eq. \eqref{eq:def_n2ueg}, the effective spin polarization of eq. \eqref{eq:def_effspin} and which reads
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\begin{equation}
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\label{eq:def_pbeueg}
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\begin{aligned}
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\pbeuegXi = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeuegXi),
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\end{aligned}
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\end{equation}
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ii) the PBE-ot-$\tilde{\zeta}$ where the on-top pair density of equation \eqref{eq:def_n2extrap} is used and which reads
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ii) the PBE-ot-$\tilde{\zeta}$ where the on-top pair density of eq. \eqref{eq:def_n2extrap} is used and which reads
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\begin{equation}
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\label{eq:def_pbeueg}
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\begin{aligned}
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\pbeontXi = &\int d\br{} \,\denr \\ & \ecmd(\argrpbeontXi),
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\end{aligned}
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\end{equation}
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iii) and the PBE-ot-$0{\zeta}$ where no spin polarization is used and which therefore uses only the total density and the on-top pair density of equation \eqref{eq:def_n2extrap} and which reads
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iii) and the PBE-ot-$0{\zeta}$ where no spin polarization is used and which therefore uses only the total density and the on-top pair density of eq. \eqref{eq:def_n2extrap} and which reads
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\begin{equation}
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\label{eq:def_pbeueg}
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\begin{aligned}
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@ -646,17 +646,17 @@ iii) and the PBE-ot-$0{\zeta}$ where no spin polarization is used and which ther
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\label{sec:results}
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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 propose to study the potential energy surfaces (PES) until dissociation of an equally distant H$_{10}$ chain, together with the C$_2$, N$_2$, O$_2$ and F$_2$ molecules.
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In a given basis set, to compute the approximation of the exact ground state energy using equation \eqref{eq:e0approx}, one needs an approximation to both the FCI energy $\efci$ and the complementary basis set energy functional $\efuncbasisFCI$.
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In a given basis set, 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 complementary basis set energy functional $\efuncbasisFCI$.
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In the case of C$_2$, N$_2$, O$_2$ and F$_2$, the approximation 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.}
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(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 PES are obtained from Ref. \onlinecite{LieCle-JCP-74a}.
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For all geometry and basis sets, the error with respect to actual FCI energies are estimated to be below 0.5 mH.
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In the case of H$_{10}$, the approximation to $\efci$ together with the estimated exact curves are obtained from the data from 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 complementary basis set energy functional, we use a full valence CASSCF wave functions computed with the GAMESS-US software\cite{gamess} to obtain the wave functions $\psibasis$. Therefore, all density related quantities (such as the total densities, different flavors of spin polarizations and on-top pair densities) together with the $\murpsi$ of equation \eqref{eq:def_mur} are obtained at full valence CASSCF level.
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Regarding the complementary basis set energy functional, we use a full valence CASSCF wave functions computed with the GAMESS-US software\cite{gamess} to obtain the wave functions $\psibasis$. Therefore, all density related quantities (such as the total densities, different flavors of spin polarizations and on-top pair densities) together with the $\murpsi$ of eq. \eqref{eq:def_mur} are obtained at full valence CASSCF level.
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These CASSCF wave functions correspond to the following active spaces: ten electrons in ten orbitals for H$_{10}$, 8 electrons in 8 electrons for C$_2$, 10 electrons in 8 orbitals for N$_2$, twelve electrons in eight orbitals for O$_2$ and forteen electrons in eight orbitals for F$_2$.
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Also, as the frozen core approximation is used in all near FCI calculations, we use the corresponding valence-only complementary functionals. Therefore, all density related quantities exclude any contribution from the core $1s$ orbitals, and the range-separation parameter is taken as the one defined in equation \eqref{eq:def_mur_val}.
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Also, as the frozen core approximation is used in all near FCI calculations, we use the corresponding valence-only complementary functionals. Therefore, all density related quantities exclude any contribution from the core $1s$ orbitals, and the range-separation parameter is taken as the one defined in eq. \eqref{eq:def_mur_val}.
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\subsection{Dissociation of equally distant H$_{10}$ chains}
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The study of equally distant H$_{10}$ chains is a good prototype for the study of strong correlation regime as it consists in the simultaneous breaking of 10 covalent $\sigma$ bonds which all interact with each other. Also, being a relatively small system, benchmark calculations can be performed at near CBS values can be obtained (see Ref. \onlinecite{h10_prx} for detailed study of that problem).
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@ -668,7 +668,7 @@ More quantitatively, the values of $D_0$ are within the chemical accuracy (\text
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Regarding in more details the performance of the different types of approximated functionals, the results show that the PBE-ot-$\tilde{\zeta}$ and PBE-ot-$0{\zeta}$ are very similar (the maximal difference being 0.3 mH on $D_0$), and they give slightly more accurate than the PBE-UEG-$\tilde{\zeta}$.
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These observations bring two important clues on the role of the different physical ingredients used in the functionals:
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i) the explicit use of the on-top pair density coming from the CASSCF wave function (see equation \eqref{eq:def_n2extrap}) is preferable to the use of the on-top pair density based on the UEG (see equation \eqref{eq:def_n2ueg}),
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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}),
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ii) removing the dependence on any kind of spin polarizations does not lead to significant loss of accuracy provided that one uses a qualitatively correct on-top pair density. The point ii) is important as it shows that the use of the spin-polarization in density functional approximations (DFA) essentially plays the role of the effect of the on-top pair density.
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\subsection{Dissociation of C$_2$, N$_2$, O$_2$ and F$_2$}
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