small changes in theory
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Julien Toulouse
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@ -341,7 +341,7 @@ where $v_\text{ne}(\br{})$ is the nuclei-electron potential, and $F[\den]$ is th
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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 $N$-electron Hilbert space generated by the one-electron basis set $\Bas$. In the following, 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, $E_0^\Bas \approx E_0$.
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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 (for a representable-in-$\Bas$ density $\den$)
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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, for a representable-in-$\Bas$ density $\den$,
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\begin{equation}
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\label{eq:def_levy_bas}
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F[\den] = \min_{\wf{}{\Bas} \to \den} \mel*{\wf{}{\Bas}}{\kinop +\weeop}{\wf{}{\Bas}} + \efuncden{\den},
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@ -368,7 +368,7 @@ As a simple non-self-consistent version of this approach, we can approximate the
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\label{eq:e0approx}
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E_0 \approx E_0^\Bas \approx \efci + \efuncbasisFCI,
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\end{equation}
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where $\efci$ and $n_\text{FCI}^\Bas$ are the ground-state FCI energy and density, respectively. As it was originally shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further emphasized in Refs.~\onlinecite{LooPraSceTouGin-JCPL-19,GinSceTouLoo-JCP-19}, the main role of $\efuncbasisFCI$ is to correct for the basis-set incompleteness error, a large part of which originating from the lack of electron-electron cusp in the wave function expanded in an incomplete basis set. The whole purpose of this work is to determine approximations for $\efuncbasisFCI$ which are suitable for strongly correlated situations. Two key requirements for this purpose are i) spin-multiplet degeneracy, and ii) size consistency.
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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 strongly correlated molecular systems. Two key requirements for this purpose are i) spin-multiplet degeneracy, and ii) size consistency.
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\subsection{Effective interaction in a finite basis}
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\label{sec:wee}
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@ -435,7 +435,7 @@ which is again fundamental to guarantee the correct behavior of the theory in th
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\subsubsection{Frozen-core approximation}
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\label{sec:FC}
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As all WFT calculations in this work are performed within the frozen-core approximation, we use a \alert{``valence-only'' (or no-core) version} of the various quantities needed for the complementary 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 valence (\ie, no-core) orbitals, respectively, and define the valence-only local range-separation function as
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As all WFT calculations in this work are performed within the frozen-core approximation, we use a ``valence-only'' (or no-core) version of the various quantities needed for the complementary 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 ``valence'' (\ie, non-core) orbitals, respectively, and define the valence-only local range-separation function 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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@ -460,7 +460,7 @@ is the valence-only effective interaction and
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\twodmrdiagpsival
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= \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}.
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\end{gather}
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One would note the restrictions of the sums to the set of valence orbitals in Eqs.~\eqref{eq:fbasis_val} and \eqref{eq:twordm_val}.
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One would note the restrictions of the sums to the set $\BasFC$ 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{General form of the complementary functional}
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@ -469,7 +469,7 @@ It is also noteworthy that, with the present definition, $\wbasisval$ still tend
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\subsubsection{Generic approximate form}
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\label{sec:functional_form}
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As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary 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 consider 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 $\mur \equiv\murpsi$.
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As originally proposed and motivated in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate the complementary 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,Tou-THESIS-05} Following the recent work in Ref.~\onlinecite{LooPraSceTouGin-JCPL-19}, we propose to consider 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 $\mur \equiv\murpsi$.
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Therefore, $\efuncden{\den}$ has the following generic form
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\begin{multline}
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\label{eq:def_ecmdpbebasis}
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@ -489,7 +489,7 @@ is the correlation energy per particle, with
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\end{equation}
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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.
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The functional form of $\ecmd(\argecmd)$ in Eq.~\ref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional, \ie,
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The form of $\ecmd(\argecmd)$ in Eq.~\eqref{eq:def_ecmdpbe} has been originally proposed in Ref.~\onlinecite{FerGinTou-JCP-18} in the context of RSDFT. In the $\mu\to 0$ limit, it reduces to the usual PBE correlation functional, \ie,
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\begin{equation}
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\lim_{\mu \to 0} \ecmd(\argecmd) = \varepsilon_{\text{c}}^{\text{PBE}}(\argepbe),
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\end{equation}
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@ -527,7 +527,7 @@ Second, $\efuncdenpbe{\argebasis}$ correctly vanishes for systems with uniformly
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\begin{equation}
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\lim_{n_2 \to 0} \efuncdenpbe{\argebasis} = 0.
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\end{equation}
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This property is doubly guaranteed by i) the choice of setting $\wbasis = +\infty$ for a vanishing pair density [see Eq.~\eqref{eq:wbasis}], which leads to $\mu(\br{}) \to \infty$ and thus a vanishing $\ecmd(\argecmd)$ [see Eq.~\eqref{eq:lim_muinf}], and ii) the fact that $\ecmd(\argecmd)$ vanishes anyway when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
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This property is doubly guaranteed by i) the choice of setting $\wbasis = \infty$ for a vanishing pair density [see Eq.~\eqref{eq:wbasis}], which leads to $\mu(\br{}) \to \infty$ and thus a vanishing $\ecmd(\argecmd)$ [see Eq.~\eqref{eq:lim_muinf}], and ii) the fact that $\ecmd(\argecmd)$ vanishes anyway when the on-top pair density vanishes [see Eq.~\eqref{eq:lim_n2}].
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\subsection{Requirements on the complementary functional for strong correlation}
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\label{sec:requirements}
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@ -557,19 +557,19 @@ Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-J
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\end{cases}
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\end{equation}
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An alternative way to eliminate the $S_z$ dependence is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures its $S_z$ independence and, as will be numerically demonstrated, very weakly affects the complementary functional accuracy.
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An alternative way to eliminate the $S_z$ dependence is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the use of $\ecmd$ with \textit{a zero spin polarization}. This ensures its $S_z$ independence and, as will be numerically demonstrated, very weakly affects the complementary functional accuracy.
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\subsubsection{Size consistency}
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Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments $\text{A}+\text{B}$, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem $\text{A}+\text{B}$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\ket*{\Psi_{\text{A}+\text{B}}^{\basis}} = \ket*{\Psi_{\ce{A}}^{\basis}} \otimes \ket*{\Psi_{\ce{B}}^{\basis}}$. We refer the interested reader to Appendix~\ref{app:sizeconsistency} for a detailed proof and discussion of the latter statement.
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In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function is sufficient to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.
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Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments $\text{A}+\text{B}$, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these quantities locally coincide in the isolated fragments and in the supersystem $\text{A}+\text{B}$. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\ket*{\Psi_{\text{A}+\text{B}}^{\basis}} = \ket*{\Psi_{\ce{A}}^{\basis}} \otimes \ket*{\Psi_{\ce{B}}^{\basis}}$. We refer the interested reader to Appendix~\ref{app:sizeconsistency} for a detailed proof and discussion of the latter statement.
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In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function can be used to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments.
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\subsection{Actual approximations used for the complementary functional}
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\label{sec:def_func}
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As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are 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.
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As the present work focuses on the strong-correlation regime, we propose here to investigate only approximate functionals which are $S_z$ independent and size-consistent in the case of covalent bond breaking. Therefore, the wave functions $\psibasis$ used throughout this paper are CASSCF wave functions in order to ensure size consistency of all local quantities. The difference between the different flavors of functionals are only due to the types of spin polarization and on-top pair density used.
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Regarding the spin polarization that enters into the function $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$, 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-0} and calculated from the CASSCF wave function, and ii) a \textit{zero} spin polarization. In the latter case, the functional is referred as to ``SU'' which stands for ``spin unpolarized''.
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@ -578,14 +578,14 @@ Regarding the on-top pair density entering in Eq.~\eqref{eq:def_beta}, we use tw
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\label{eq:def_n2ueg}
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\ntwo^{\text{UEG}}(n,\zeta) \approx n^2\big(1-\zeta^2\big)g_0(n),
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\end{equation}
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where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin-0} 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''.
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where the pair-distribution function $g_0(n)$ is taken from Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. As the spin polarization appears in Eq.~\eqref{eq:def_n2ueg}, we use the effective spin polarization $\tilde{\zeta}$ of Eq.~\eqref{eq:def_effspin-0} 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 $\ntwo^{\text{UEG}}(\br{}) \equiv \ntwo^{\text{UEG}}(n(\br{}),\tilde{\zeta}(\br{}))$ in a functional, we will refer to it as ``UEG''.
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The second 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
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\begin{equation}
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\label{eq:def_n2extrap}
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\ntwoextrap(\ntwo,\mu) = \qty( 1 + \frac{2}{\sqrt{\pi}\mu} )^{-1} \; \ntwo,
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\end{equation}
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which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. When using $\ntwoextrap(\ntwo,\mu)$ in a functional, we will simply refer it as ``OT'', which stands for "on-top".
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which directly follows from the large-$\mu$ extrapolation of the exact on-top pair density derived by Gori-Giorgi and Savin\cite{GorSav-PRA-06} in the context of RSDFT. Thus, the extrapolated on-top pair density $\ntwoextrap$ depends on the local range-separation function $\mu$. When using $\ntwoextrap(\br{}) \equiv \ntwoextrap(\ntwo(\br{}),\mu(\br{}))$ in a functional, we will simply refer it as ``OT'', which stands for "on-top".
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We then define three complementary functionals:
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\begin{itemize}
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@ -613,7 +613,7 @@ We then define three complementary functionals:
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\bar{E}^\Bas_{\pbeontns} = \int \d\br{} \,\denr \ecmd(\argrpbeontns).
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\end{equation}
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\end{itemize}
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The performance of each of these functionals is tested in the following. Note that we did not define a spin-unpolarized version of the PBE-UEG functional because it would have been significantly inferior (in terms of performance) compared to the three other functionals. Indeed, because to the lack of knowledge on the spin polarization or on the accurate on-top pair density, such a functional would be inaccurate. This assumption has been numerically confirmed by preliminary calculations.
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The performance of each of these functionals is tested in the following. Note that we did not define a spin-unpolarized version of the PBE-UEG functional because it would have been significantly inferior (in terms of performance) compared to the three other functionals. Indeed, because to the lack of knowledge on the spin polarization or on the accurate on-top pair density, such a functional would be inaccurate in particular for open-shell systems. This assumption has been numerically confirmed by calculations.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Results}
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