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Julien Toulouse
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@ -10,7 +10,7 @@
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\juju}[1]{\textcolor{purple}{#1}}
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\newcommand{\jt}[1]{\textcolor{purple}{#1}}
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\newcommand{\manu}[1]{\textcolor{darkgreen}{#1}}
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\newcommand{\toto}[1]{\textcolor{brown}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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@ -197,8 +197,11 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
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%Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$.
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%According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as
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Let us assume we have both the energy \titou{$\E{\CCSDT}{\Bas}$ and density $\n{\HF}{\Bas}$ of a $\Ne$-electron system in an incomplete basis set $\Bas$.}
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that \titou{$\E{\CCSDT}{\Bas}$ and $\n{\HF}{\Bas}$} are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as
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Let us assume
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\titou{that we have reasonable approximations of the FCI energy and density of a $\Ne$-electron system in an incomplete basis set $\Bas$, say the CCSD(T) energy $\E{\CCSDT}{\Bas}$ and the Hartree-Fock (HF) density $\n{\HF}{\Bas}$. According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, the exact ground-state energy $\E{}{}$ may be approximated as
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}
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%we have both the energy \titou{$\E{\CCSDT}{\Bas}$ and density $\n{\HF}{\Bas}$ of a $\Ne$-electron system in an incomplete basis set $\Bas$.}
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%According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that \titou{$\E{\CCSDT}{\Bas}$ and $\n{\HF}{\Bas}$} are reasonable approximations of the FCI energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be approximated as
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\begin{equation}
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\label{eq:e0basis}
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\titou{\E{}{}
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@ -215,17 +218,17 @@ where
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is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
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In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set (CBS), respectively.
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Both wave functions yield the same target density $\n{}{}$ (assumed to be representable in $\Bas$).
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Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
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Importantly, in the CBS limit (which we refer to as $\Bas \to \CBS$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \CBS} \bE{}{\Bas}[\n{}{}] = 0$.
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This implies that
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\begin{equation}
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\label{eq:limitfunc}
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\titou{\lim_{\Bas \to \infty} \qty( \E{\CCSDT}{\Bas} + \bE{}{\Bas}[\n{\HF}{\Bas}] ) = \E{\CCSDT}{} \approx \E{}{},}
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\titou{\lim_{\Bas \to \CBS} \qty( \E{\CCSDT}{\Bas} + \bE{}{\Bas}[\n{\HF}{\Bas}] ) = \E{\CCSDT}{\CBS} \approx \E{}{},}
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\end{equation}
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%where $\E{\modY}{}$ is the energy associated with the method $\modY$ in the CBS limit.
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where \titou{$\E{\CCSDT}{}$ is the $\CCSDT$ energy} in the CBS limit.
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where \titou{$\E{\CCSDT}{\CBS}$ is the $\CCSDT$ energy} in the CBS limit.
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\titou{Of course, the above holds true for any method that provides a good approximation to the energy and density, not just CCSD(T) and HF.}
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%In the case where $\modY = \FCI$ in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
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In the case where \titou{$\CCSDT$ is replaced by $\FCI$} in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
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In the case where \titou{$\CCSDT$ is replaced by $\FCI$} in Eq.~\eqref{eq:limitfunc}, we have a strict equality as $\E{\FCI}{\CBS} = \E{}{}$.
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%Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the potential approximate nature of the methods $\modY$ and $\modZ$, and the lack of self-consistency in the present scheme.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only sources of error at this stage lie in the approximate nature of the \titou{$\CCSDT$ and $\HF$ methods}, and the lack of self-consistency of the present scheme.
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@ -278,7 +281,7 @@ As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{}{\Bas}
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Thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
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\begin{equation}
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\label{eq:lim_W}
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\lim_{\Bas \to \infty}\W{}{\Bas}(\br{1},\br{2}) = r_{12}^{-1},
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\lim_{\Bas \to \CBS}\W{}{\Bas}(\br{1},\br{2}) = \frac{1}{r_{12}},
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\end{equation}
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for any $(\br{1},\br{2})$ such that $\n{2}{\Bas}(\br{1},\br{2}) \ne 0$.
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@ -305,15 +308,15 @@ As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific c
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\label{eq:ec_md_mu}
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\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]
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= \min_{\wf{}{} \to \n{}{}} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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- \mel*{\wf{}{\rsmu{}{}}}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}},
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- \mel*{\wf{}{\rsmu{}{}}[n]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[n]},
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\end{equation}
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where $\wf{}{\rsmu{}{}}$ is defined by the constrained minimization
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where $\wf{}{\rsmu{}{}}[n]$ is defined by the constrained minimization
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\begin{equation}
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\label{eq:argmin}
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\wf{}{\rsmu{}{}} = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\wf{}{\rsmu{}{}}[n] = \arg \min_{\wf{}{} \to \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\end{equation}
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with $\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})$.
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The ECMD functionals admit, for any $\n{}{}$, the following two limiting forms
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The ECMD functionals admit, for any $\n{}{}$, the following two limits
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\begin{align}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \to \infty} \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
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@ -354,7 +357,7 @@ The difference between the ECMD functional defined in Ref.~\onlinecite{FerGinTou
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This represents a major computational saving without loss of accuracy for weakly correlated systems as we eschew the computation of $\n{2}{\Bas}(\br{},\br{})$.
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%Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{\Bas}]$ is approximated by $\bE{\LDA}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\n{\modZ}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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\titou{The complementary functional $\bE{}{\Bas}[\n{\HF}{\Bas}]$ is approximated by $\bE{\PBE}{\Bas}[\n{\HF}{\Bas},\rsmu{}{\Bas}]$ where $\rsmu{}{\Bas}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.}
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\titou{The local-density approximation (LDA) version of the ECMD functional is discussed in the {\SI}.}
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\titou{The slightly simpler local-density approximation (LDA) version of the ECMD functional is discussed in the {\SI}.}
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%=================================================================
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%\subsection{Frozen-core approximation}
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%=================================================================
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@ -382,7 +385,7 @@ with
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\end{gather}
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\end{subequations}
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and the corresponding FC range-separation function $\rsmuFC{}{\Bas}(\br{}) = (\sqrt{\pi}/2) \WFC{}{\Bas}(\br{},\br{})$.
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It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \infty$.
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It is noteworthy that, within the present definition, $\WFC{}{\Bas}(\br{1},\br{2})$ still tends to the regular Coulomb interaction as $\Bas \to \CBS$.
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%Defining $\nFC{\modZ}{\Bas}$ as the FC (i.e.~valence-only) one-electron density obtained with a method $\modZ$ in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\LDA}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$ or $\bE{\PBE}{\Bas}[\nFC{\modZ}{\Bas},\rsmuFC{}{\Bas}]$.
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\titou{Defining $\nFC{\HF}{\Bas}$ as the FC (i.e.~valence-only) $\HF$ one-electron density in $\Bas$, the FC contribution of the complementary functional is then approximated by $\bE{\PBE}{\Bas}[\nFC{\HF}{\Bas},\rsmuFC{}{\Bas}]$}.
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@ -463,22 +466,22 @@ In most cases, the basis-set corrected triple-$\zeta$ atomization energies are o
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\begin{figure}
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\includegraphics[width=0.5\linewidth]{fig2}
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\caption{
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\titou{$\rsmu{}{\Bas}$ (top) and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ (bottom) along the molecular axis ($z$) for \ce{N2} in various basis sets.
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The two nitrogen atoms are located at $z=0$ and $z=2.076$ bohr.}
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\titou{$\rsmu{}{\Bas}$ (top) and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ (bottom) along the molecular axis ($z$) for \ce{N2} for various basis sets.
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The two nitrogen nuclei are located at $z=0$ and $z=2.076$ bohr.}
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\label{fig:N2}}
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\end{figure}
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\titou{The fundamental quantity of the present basis set correction is $\rsmu{}{\Bas}(\br{})$.
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\titou{The fundamental quantity of the present basis-set correction is $\rsmu{}{\Bas}(\br{})$.
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As it grows when one gets closer to the CBS limit, the value of $\rsmu{}{\Bas}(\br{})$ quantifies the quality of a given basis set at a given $\br{}$.
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Another important quantity closely related to $\rsmu{}{\Bas}(\br{})$ is the local energetic correction, $\n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{}))$, which integrates to the total basis set correction $\bE{\PBE}{\Bas}[\n{}{},\rsmu{}{\Bas}]$ [see Eq.~\eqref{eq:def_pbe_tot}].
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Such quantity essentially depends on the local values of both $\rsmu{}{\Bas}(\br{})$ and $\n{}{}(\br{})$.
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In order to qualitatively illustrate how the basis set correction operates, we report, in Figure \ref{fig:N2}, $\rsmu{}{\Bas}$ and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ along the molecular axis ($z$) of \ce{N2} for $\Bas=\{\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}\}$.
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Such a quantity essentially depends on the local values of both $\rsmu{}{\Bas}(\br{})$ and $\n{}{}(\br{})$.
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In order to qualitatively illustrate how the basis-set correction operates, we report, in Figure \ref{fig:N2}, $\rsmu{}{\Bas}$ and $\n{}{} \times \be{\text{c,md}}{\sr,\PBE}$ along the molecular axis ($z$) of \ce{N2} for $\Bas=\text{cc-pVDZ, cc-pVTZ, cc-pVQZ}$.
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This figure illustrates several general trends:
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i) the global value of $\rsmu{}{\Bas}(z)$ is much larger than 0.5 which is the standard value used in RS-DFT,
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ii) $\rsmu{}{\Bas}(z)$ is highly non uniform in space, illustrating the non-homogeneity of basis set quality in quantum chemistry,
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iii) $\rsmu{}{\Bas}(z)$ is significantly larger close to the nuclei, a signature that atom-centered basis sets better describe these high-density regions than the bonding regions,
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v) the global value of the energy correction get smaller as one improves the basis set quality, and the reduction is spectacular close to the nuclei, and
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iv) a large energetic contribution comes from the bonding regions highlighting the poor description of correlation effects in these region with Gaussian basis sets.}
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i) the value of $\rsmu{}{\Bas}(z)$ tends to be much larger than 0.5 bohr$^{-1}$ which is the common value used in RS-DFT,
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ii) $\rsmu{}{\Bas}(z)$ is highly non-uniform in space, illustrating the non-homogeneity of basis-set quality in quantum chemistry,
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iii) $\rsmu{}{\Bas}(z)$ is significantly larger close to the nuclei, a signature that nucleus-centered basis sets better describe these high-density regions than the bonding regions,
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v) the value of the energy correction gets smaller as one improves the basis-set quality, the reduction being spectacular close to the nuclei, and
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iv) a large energetic contribution comes from the bonding regions, highlighting the imperfect description of correlation effects in these regions with Gaussian basis sets.}
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%%% TABLE II %%%
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\begin{table}
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@ -517,7 +520,7 @@ iv) a large energetic contribution comes from the bonding regions highlighting t
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Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with \titou{various basis sets for CCSD(T) (top) and CCSD(T)+PBE (bottom).}
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% Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
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The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
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\titou{Note the difference in scaling of the vertical axes.}
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\titou{Note the different scales of the vertical axes.}
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See {\SI} for raw data \titou{and the corresponding LDA results}.
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\label{fig:G2_Ec}}
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\end{figure*}
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@ -545,7 +548,7 @@ Encouraged by these promising results, we are currently pursuing various avenues
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section*{Supporting Information Available}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set \titou{as well as the definition of the LDA ECMD functionals (and the corresponding numerical results).}
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See {\SI} for raw data associated with the atomization energies of the four diatomic molecules and the G2 set \titou{as well as the definition of the LDA ECMD functional (and the corresponding numerical results).}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{acknowledgements}
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@ -139,7 +139,7 @@
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Local-density approximation}
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%\section{Local-density approximation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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The local-density approximation (LDA) of the ECMD complementary functional is defined as
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\begin{equation}
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@ -147,7 +147,7 @@ The local-density approximation (LDA) of the ECMD complementary functional is de
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\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
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\end{equation}
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where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to over correlate for small $\mu$.
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The short-range LDA correlation functional relies on the transferability of the physics of the UEG which is certainly valid for large $\mu$ but is known to overcorrelate for small $\mu$.
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The sensitivity with respect to the RS-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
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However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
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@ -163,7 +163,7 @@ Such weak sensitivity to the density-functional approximation when reaching larg
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\hspace{1cm}
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\includegraphics[width=0.30\linewidth]{fig1d}
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\caption{
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Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left) and \ce{F2} (bottom right) obtained with various methods and basis sets.
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Deviation (in \kcal) from CBS atomization energies of \ce{C2} (top left), \ce{O2} (top right), \ce{N2} (bottom left), and \ce{F2} (bottom right) obtained with various methods and basis sets.
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The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
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\label{fig:diatomics}}
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\end{figure*}
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@ -201,7 +201,7 @@ Such weak sensitivity to the density-functional approximation when reaching larg
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\includegraphics[width=\linewidth]{fig2b}
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\includegraphics[width=\linewidth]{fig2c}
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\caption{
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Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center) and cc-pVQZ (bottom) basis sets.
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Deviation (in \kcal) from the CCSD(T)/CBS atomization energy obtained with various methods with the cc-pVDZ (top), cc-pVTZ (center), and cc-pVQZ (bottom) basis sets.
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The green region corresponds to chemical accuracy (i.e.~error below 1 {\kcal}).
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\label{fig:G2_Ec}}
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\end{figure*}
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@ -210,7 +210,7 @@ Such weak sensitivity to the density-functional approximation when reaching larg
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\begin{table*}
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\caption{
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\label{tab:diatomics}
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Frozen-core atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2} and \ce{F2} computed with various methods and basis sets.
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Frozen-core atomization energies (in {\kcal}) of \ce{C2}, \ce{O2}, \ce{N2}, and \ce{F2} computed with various methods and basis sets.
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The deviations with respect to the corresponding CBS values are reported in parenthesis.
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See main text for more details.
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}
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