changes
This commit is contained in:
Julien Toulouse
parent
6910e77c60
commit
3edbfe838c
@ -91,16 +91,16 @@
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%pbeuegxiHF
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\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas}
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\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}}
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\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})}
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\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,n_{2}^{\text{UEG}},\mu_{\text{HF}}^{\basis}}
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\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})}
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%pbeuegxiCAS
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\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas}
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\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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%pbeuegXiCAS
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\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}}
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\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,n_{2}^{\text{UEG}},\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),n_{2}^{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})}
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%pbeontxiCAS
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\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta}
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\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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@ -117,12 +117,12 @@
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%%%%%% arguments
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\newcommand{\argepbe}[0]{\den,\zeta,s}
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\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{A+B}}}
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\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu_{\Psi^{A+B}}}
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\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu}
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\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
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\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
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\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}}
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\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}}
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\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{A+B}}(\br{})}
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\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo^{\text{UEG}},\mu_{\Psi^{\basis}}}
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\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu_{\Psi^{A+B}}(\br{})}
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\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})}
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@ -141,7 +141,7 @@
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\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})}
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\newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})}
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\newcommand{\murpsib}[0]{\mu({\bf r};\wf{}{B})}
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\newcommand{\ntwo}[0]{n^{(2)}}
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\newcommand{\ntwo}[0]{n_{2}}
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\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}}
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\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}}
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\newcommand{\ntwoextrap}[0]{\mathring{n}^{(2)}_{\psibasis}}
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@ -160,20 +160,20 @@
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\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})}
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\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})}
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\newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)}
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\newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsitot}[0]{ \ntwo_{\wf{}{A+B}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsiaa}[0]{ \ntwo_{\wf{}{AA}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsiaad}[0]{ \ntwo_{\wf{}{AA}}(\rr{}{})}
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\newcommand{\twodmrdiagpsibb}[0]{ \ntwo_{\wf{}{BB}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsibbd}[0]{ \ntwo_{\wf{}{BB}}(\rr{}{})}
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\newcommand{\twodmrdiagpsiab}[0]{ \ntwo_{\wf{}{AB}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
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\newcommand{\twodmrpsi}[0]{ n^{2,\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})}
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\newcommand{\twodmrdiagpsi}[0]{ n_{2,{\wf{}{\Bas}}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsitot}[0]{ n_{2,\wf{}{A+B}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsiaa}[0]{ n_{2,\wf{}{AA}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsiaad}[0]{ n_{2,\wf{}{AA}}(\rr{}{})}
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\newcommand{\twodmrdiagpsibb}[0]{ n_{2,\wf{}{BB}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsibbd}[0]{ n_{2,\wf{}{BB}}(\rr{}{})}
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\newcommand{\twodmrdiagpsiab}[0]{ n_{2\wf{}{AB}}(\rr{1}{2})}
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\newcommand{\twodmrdiagpsival}[0]{ n_{2\wf{}{\Bas},\,\text{val}}(\rr{1}{2})}
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\newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]}
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\newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}}
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\newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]}
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%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})}
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\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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\newcommand{\ontoppsi}[1]{ n_{2,\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})}
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\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})}
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@ -282,10 +282,10 @@
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\maketitle
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\section{Extensivity of the basis set correction}
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\section{Size consistency of the basis-set correction}
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\subsection{General considerations}
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The following paragraph proposes a demonstration of the size consistency of the basis set correction in the limit of dissociated fragments.
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The present basis set correction being an integral in real space,
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The following paragraph proposes a proof of the size consistency of the basis-set correction in the limit of dissociated fragments.
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The present basis-set correction being an integral in real space,
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\begin{equation}
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\label{eq:def_ecmdpbebasis}
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\begin{aligned}
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@ -308,7 +308,7 @@ Regarding the density and its gradients, these are necessary intensive quantitie
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A crucial ingredient in the type of functionals used in the present paper together with the definition of the local-range separation parameter is the on-top pair density defined as
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\begin{equation}
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\label{eq:def_n2}
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\ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
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n_{2,\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{},
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\end{equation}
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with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{}}$.
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Assume now that the wave function $\wf{A+B}{}$ of the super system $A+B$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$
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@ -365,10 +365,10 @@ and $\twodmrdiagpsiab$ is simply the product of the one body densities of the su
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As the densities of $A$ and $B$ are by definition non overlapping, one can express the on-top pair density as the sum of the on-top pair densities of the isolated systems
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\begin{equation}
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\begin{aligned}
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\ntwo_{\wf{A+B}{}}(\br{}) = \twodmrdiagpsiaad + \twodmrdiagpsibbd
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n_{2,\wf{A+B}{}}(\br{}) = \twodmrdiagpsiaad + \twodmrdiagpsibbd
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\end{aligned}
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\end{equation}
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As $\ntwo_{\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for $\ntwo_{\wf{}{B/B}}(\br{}) $ on $A$), one can conclude that provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity.
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As $n_{2,\wf{}{A/A}}(\br{}) = 0 \text{ if }\br{} \in B$ (and equivalently for $n_{2,\wf{}{B/B}}(\br{}) $ on $A$), one can conclude that provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity.
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\subsection{Property of the local-range separation parameter}
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The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator
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\begin{equation}
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@ -399,7 +399,7 @@ with
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As a consequence, the local range-separation parameter in the super system $A+B$
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\begin{equation}
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\label{eq:def_mur}
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\murpsi = \frac{\sqrt{\pi}}{2} \frac{f_{\wf{A+B}{}}(\bfr{},\bfr{})}{\ntwo_{\wf{A+B}{}}(\br{})}
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\murpsi = \frac{\sqrt{\pi}}{2} \frac{f_{\wf{A+B}{}}(\bfr{},\bfr{})}{n_{2,\wf{A+B}{}}(\br{})}
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\end{equation}
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which, in the case of a multiplicative wave function is nothing but
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\begin{equation}
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@ -409,7 +409,7 @@ which, in the case of a multiplicative wave function is nothing but
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As $\murpsia = 0 \text{ if }\br{} \in B$ (and equivalently for $\murpsib $ on $B$), $\murpsi$ is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent.
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\section{Computational considerations}
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The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $\ntwo_{\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively.
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The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $\murpsibas$ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $n_{\Bas}^4$ and $n_{\Bas}^6$ for the on-top pair density $n_{2,\wf{\Bas}{}}(\br{})$ and the local range separation parameter $\murpsibas$, respectively.
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Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time.
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\subsection{Computation of the on-top pair density for a CASSCF wave function}
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Given a generic wave function developed on a basis set with $n_{\Bas}$ basis functions, the evaluation of the on-top pair density is of order $\left(n_{\Bas}\right)^4$.
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@ -417,12 +417,12 @@ Nevertheless, assuming that the wave function $\Psi^{\Bas}$ is of CASSCF type, a
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If the active space is referred as the set of spatial orbitals $\mathcal{A}$ which are labelled by the indices $t,u,v,w$, and the doubly occupied orbitals are the set of spatial orbitals $\mathcal{C}$ labeled by the indices $i,j$, one can write the on-top pair density of a CASSCF wave function as
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\begin{equation}
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\label{def_n2_good}
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\ntwo_{\wf{\Bas}{}}(\br{}) = \ntwo_{\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2
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n_{2,\wf{\Bas}{}}(\br{}) = n_{2,\mathcal{A}}(\br{}) + n_{\mathcal{C}}(\br{}) n_{\mathcal{A}}(\br{}) + \left( n_{\mathcal{C}}(\br{})\right)^2
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\end{equation}
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where
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\begin{equation}
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\label{def_n2_act}
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\ntwo_{\mathcal{A}}(\br{}) = \sum_{t,u,v,w \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\aic{u_\uparrow}\ai{v_\uparrow}\ai{w_\downarrow}}{\wf{}{\Bas}} \phi_t (\br{}) \phi_u (\br{}) \phi_v (\br{}) \phi_w (\br{})
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n_{2,\mathcal{A}}(\br{}) = \sum_{t,u,v,w \, \in \mathcal{A}} 2 \mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\aic{u_\uparrow}\ai{v_\uparrow}\ai{w_\downarrow}}{\wf{}{\Bas}} \phi_t (\br{}) \phi_u (\br{}) \phi_v (\br{}) \phi_w (\br{})
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\end{equation}
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is the purely active part of the on-top pair density,
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\begin{equation}
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@ -434,7 +434,7 @@ and
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\mel*{\wf{}{\Bas}}{ \aic{t_\downarrow}\ai{u_\downarrow} + \aic{t_\uparrow}\ai{u_\uparrow}}{\wf{}{\Bas}}
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\end{equation}
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is the purely active one-body density.
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Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $\ntwo_{\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $\left( n_{\mathcal{A}}\right) ^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $\left( n_{\mathcal{A}}\right) ^4 n_G$, where $n_G$ is the number of grid points.
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Written as in eq. \eqref{def_n2_good}, the leading computational cost is the evaluation of $n_{2,\mathcal{A}}(\br{})$ which, according to eq. \eqref{def_n2_act}, scales as $\left( n_{\mathcal{A}}\right) ^4$ where $n_{\mathcal{A}}$ is the number of active orbitals which is much smaller than the number of basis functions $n_{\Bas}$. Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $\left( n_{\mathcal{A}}\right) ^4 n_G$, where $n_G$ is the number of grid points.
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\subsection{Computation of $\murpsibas$}
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At a given grid point, the computation of $\murpsibas$ needs the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ defined in eq. \eqref{eq:def_f} and the on-top pair density defined in eq. \eqref{eq:def_n2}. In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $\left( n_{\mathcal{A}}\right)^4$. In the present paragraph we focus on simplifications that one can obtain for the computation of $f_{\wf{}{}}(\bfr{},\bfr{}) $ in the case of a CASSCF wave function.
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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 electron-electron Coulomb interaction projected in the finite basis set. This allows one to use RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To study both weak and strong correlation regimes we consider the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit, and we explore various approximations of complementary density functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we systematically investigate the dependence of the functional on different types of on-top pair densities and spin polarizations. The key result of this study is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy.
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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 one to use RSDFT-type complementary density functionals to recover the dominant part of the short-range correlation effects missing in this finite basis. To study both weak and strong correlation regimes we consider the potential energy curves of the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit, and we explore various approximations of complementary functionals fulfilling two key properties: spin-multiplet degeneracy (\ie, independence of the energy with respect to the spin projection $S_z$) and size consistency. Specifically, we systematically investigate the dependence of the functional on different types of on-top pair densities and spin polarizations. The key result of this study is that the explicit dependence on the on-top pair density allows one to completely remove the dependence on any form of spin polarization without any significant loss of accuracy.
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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. [JT: I don't like this sentence in the abstract in particular because it is not clear what the "effective" spin polarization is.]
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Quantitatively, we show that the basis-set correction reaches chemical accuracy on atomization energies with triple-$\zeta$ quality basis sets for most of the systems studied here. Also, the present basis-set incompleteness correction provides smooth potential energy curves along the whole distance range.
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\end{abstract}
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@ -323,7 +323,7 @@ Then, in Sec.~\ref{sec:results}, we apply the method to the calculation of the p
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\section{Theory}
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\label{sec:theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary density functional to a basis set. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective \textit{local} range-separation parameter in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual functionals used in this work are introduced in Sec.~\ref{sec:def_func}.
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As the theory behind the present basis-set correction has been exposed in details in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we only briefly recall the main equations and concepts needed for this study in Secs.~\ref{sec:basic}, \ref{sec:wee}, and \ref{sec:mur}. More specifically, in Sec.~\ref{sec:basic}, we recall the basic mathematical framework of the present theory by introducing the complementary functional to a basis set. Section \ref{sec:wee} introduces the effective non-divergent interaction in the basis set, which leads us to the definition of the effective \textit{local} range-separation parameter in Sec.~\ref{sec:mur}. Then, Sec.~\ref{sec:functional} exposes the new approximate RSDFT-based complementary correlation functionals. The generic form of such functionals is exposed in Sec.~\ref{sec:functional_form}, their properties in the context of the basis-set correction are discussed in Sec.~\ref{sec:functional_prop}, and the specific requirements for strong correlation are discussed in Sec.~\ref{sec:requirements}. Finally, the actual functionals used in this work are introduced in Sec.~\ref{sec:def_func}.
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\subsection{Basic theory}
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\label{sec:basic}
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@ -361,7 +361,7 @@ 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 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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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 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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@ -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 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 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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\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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@ -469,7 +469,7 @@ It is also noteworthy that, with the present definition, $\wbasisval$ still tend
|
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\subsubsection{Generic approximate 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 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 $\mur \equiv\murpsi$.
|
||||
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 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}
|
||||
@ -513,10 +513,10 @@ Finally, the function $\ecmd(\argecmd)$ vanishes when $\mu \to \infty$ like all
|
||||
\lim_{\mu \to \infty} \ecmd(\argecmd) = 0.
|
||||
\end{equation}
|
||||
|
||||
\subsubsection{Two limits where the complementary density functional vanishes}
|
||||
\subsubsection{Two limits where the complementary functional vanishes}
|
||||
\label{sec:functional_prop}
|
||||
|
||||
Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary density functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.
|
||||
Within the definitions of Eqs.~\eqref{eq:def_mur} and \eqref{eq:def_ecmdpbebasis}, any approximate complementary functional $\efuncdenpbe{\argebasis}$ satisfies two important properties.
|
||||
|
||||
First, thanks to the properties in Eqs.~\eqref{eq:cbs_mu} and~\eqref{eq:lim_muinf}, $\efuncdenpbe{\argebasis}$ vanishes in the CBS limit, independently of the type of wave function $\psibasis$ used to define the local range-separation parameter $\mu(\br{})$ in a given basis set $\Bas$,
|
||||
\begin{equation}
|
||||
@ -563,7 +563,7 @@ Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-J
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
|
||||
An alternative way to eliminate the $S_z$ dependency 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 density functional accuracy.
|
||||
An alternative way to eliminate the $S_z$ dependency 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.
|
||||
|
||||
\subsubsection{Size consistency}
|
||||
|
||||
@ -709,7 +709,7 @@ In the case of the \ce{H10} chain, the approximation to the FCI energies togethe
|
||||
|
||||
Regarding the complementary functional, we first perform full-valence CASSCF calculations with the GAMESS-US software\cite{gamess} to obtain the wave function $\psibasis$. Then, all density-related quantities involved in the functional [density $n(\br{})$, spin polarization $\zeta(\br{})$, reduced density gradient $s(\br{})$, and on-top pair density $n_2(\br{})$] together with the local range-separation parameter $\mu(\br{})$ 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}.
|
||||
|
||||
Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary density functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the $1s$ core orbitals, and the range-separation parameter follows the definition given in Eq.~\eqref{eq:def_mur_val}.
|
||||
Also, as the frozen-core approximation is used in all our selected CI calculations, we use the corresponding valence-only complementary functionals (see Subsec.~\ref{sec:FC}). Therefore, all density-related quantities exclude any contribution from the $1s$ core orbitals, and the range-separation parameter follows the definition given in Eq.~\eqref{eq:def_mur_val}.
|
||||
|
||||
Regarding the computational cost of the present approach, it should be stressed (see {\SI} for additional details) that the basis-set correction represents, for all systems and basis sets studied here, a much smaller computational cost than any of the selected CI calculations.
|
||||
%We thus believe that this approach is a significant step towards the routine calculation of near-CBS energetic quantities in strongly correlated systems.
|
||||
@ -723,7 +723,7 @@ We report in Fig.~\ref{fig:H10} the potential energy curves computed using the c
|
||||
In other words, smooth potential energy curves are obtained with the present basis-set correction.
|
||||
More quantitatively, the values of $D_0$ are within chemical accuracy (\ie, an error below $1.4$ mHa) from the cc-pVTZ basis set when using the $\pbeontXi$ and $\pbeontns$ functionals, whereas such an accuracy is not even reached at the standard MRCI+Q/cc-pVQZ level of theory.
|
||||
|
||||
Analyzing more carefully the performance of the different types of approximate density functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients used in these functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function [see Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [see Eq.~\eqref{eq:def_n2ueg}] which is somehow understandable, and ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density.
|
||||
Analyzing more carefully the performance of the different types of approximate functionals, the results show that $\pbeontXi$ and $\pbeontns$ are very similar (the maximal difference on $D_0$ being 0.3 mHa), and that they give slightly more accurate results than $\pbeuegXi$. These findings provide two important clues on the role of the different physical ingredients used in these functionals: i) the explicit use of the on-top pair density coming from the CASSCF wave function [Eq.~\eqref{eq:def_n2extrap}] is preferable over the use of the UEG on-top pair density [Eq.~\eqref{eq:def_n2ueg}] which is somewhat understandable, and ii) removing the dependency on any kind of spin polarization does not lead to significant loss of accuracy providing that one employs a qualitatively correct on-top pair density. The latter point is crucial as it shows that the spin polarization in density-functional approximations essentially plays the same role as the on-top pair density.
|
||||
This could have significant implications for the construction of more robust families of density-functional approximations within DFT.
|
||||
|
||||
Finally, the reader would have noticed that we did not report the performance of the $\pbeuegns$ functional as its performance are significantly inferior than the three other functionals. The main reason behind this comes from the fact that $\pbeuegns$ has no direct or indirect knowledge of the on-top pair density of the system. Therefore, it yields a non-zero correlation energy for the totally dissociated \ce{H10} chain even if the on-top pair density is vanishingly small. This necessary lowers the value of $D_0$. Therefore, from hereon, we simply discard the $\pbeuegns$ functional.
|
||||
@ -789,7 +789,7 @@ We report in Figs.~\ref{fig:N2} and \ref{fig:O2} the potential energy curves of
|
||||
Just as in \ce{H10}, the quality of $D_0$ is globally improved by adding the basis-set correction and it is remarkable that $\pbeontXi$ and $\pbeontns$ provide again very similar results. The latter observation confirms that the dependency on the on-top pair density allows one to remove the dependency of any kind of spin polarization for a quite wide range of electron density and also for open-shell systems like \ce{O2}. More quantitatively, an error below 1.0 mHa on the estimated exact valence-only $D_0$ is found for \ce{N2}, \ce{O2}, and \ce{F2} with the aug-cc-pVTZ basis set using the $\pbeontns$ functional, whereas such a feat is far from being reached within the same basis set at the near-FCI level. In the case of \ce{F2} it is clear that the addition of diffuse functions in the double- and triple-$\zeta$ basis sets strongly improves the results, a result that can be anticipated due to the strong breathing-orbital effect induced by the ionic valence bond forms in this molecule. \cite{HibHumByrLen-JCP-94}
|
||||
It should be also noticed that when reaching the aug-cc-pVQZ basis set for \ce{N2}, the quality of $D_0$ slightly deteriorates for the $\pbeontXi$ and $\pbeontns$ functionals, but it remains nevertheless more accurate than the estimated FCI $D_0$ and very close to chemical accuracy.
|
||||
|
||||
Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary density functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is designed thanks to the universal condition provided by the electron-electron cusp and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected and predictable.
|
||||
Regarding now the performance of the basis-set correction along the whole potential energy curve, it is interesting to notice that it fails to provide a noticeable improvement far from the equilibrium geometry. Acknowledging that the weak-correlation effects in these regions are dominated by dispersion interactions which are long-range effects, the failure of the present approximations for the complementary functionals can be understood easily. Indeed, the whole scheme designed here is based on the physics of correlation near the electron-electron coalescence point: the local range-separation function $\mu(\br{})$ is designed thanks to the universal condition provided by the electron-electron cusp and the ECMD functionals are suited for short-range correlation effects. Therefore, the failure of the present basis-set correction to describe dispersion interactions is theoretically expected and predictable.
|
||||
We hope to report further on this in the near future.
|
||||
|
||||
\section{Conclusion}
|
||||
@ -797,7 +797,7 @@ We hope to report further on this in the near future.
|
||||
|
||||
In the present paper we have extended the recently proposed DFT-based basis-set correction to strongly correlated systems. We studied the \ce{H10}, \ce{N2}, \ce{O2}, and \ce{F2} molecules up to the dissociation limit at near-FCI level in increasing-large basis sets, and investigated how the basis-set correction affects the convergence toward the CBS limit of the potential energy curves of these molecular systems.
|
||||
|
||||
The density-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb interaction projected into an incomplete basis set $\basis$, ii) the fit of this effective interaction with the long-range interaction used in RS-DFT, and iii) the use of a short-range, complementary density functional borrowed from RS-DFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling spin-multiplet degeneracy and size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities.
|
||||
The density-based basis-set correction relies on three aspects: i) the definition of an effective non-divergent electron-electron interaction obtained from the expectation value over a wave function $\psibasis$ of the Coulomb interaction projected into an incomplete basis set $\basis$, ii) the fit of this effective interaction with the long-range interaction used in RS-DFT, and iii) the use of a short-range, complementary functional borrowed from RS-DFT. In the present paper, we investigated i) and iii) in the context of strong correlation and focused on potential energy curves and atomization energies. More precisely, we proposed a new scheme to design functionals fulfilling spin-multiplet degeneracy and size consistency. To fulfil such requirements we proposed to use CASSCF wave functions leading to size-consistent energies, and to develop functionals using only $S_z$-independent density-like quantities.
|
||||
|
||||
The development of new $S_z$-independent and size-consistent functionals has lead us to investigate the role of two related quantities: the spin polarization and the on-top pair density. One important result of the present study is that by using functionals \textit{explicitly} depending on the on-top pair density, one can eschew its spin polarization dependency without loss of accuracy. This avoids the commonly used effective spin polarization originally proposed in Ref.~\onlinecite{BecSavSto-TCA-95} which has the disadvantage of possibly becoming complex-valued in the multideterminant case. From a more fundamental aspect, this shows that, in a DFT framework, the spin polarization mimics the role of the on-top pair density.
|
||||
Consequently, we believe that one could potentially develop new families of density functional approximations where the spin polarization is abondonned and replaced by the on-top pair density.
|
||||
|
||||
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