LDA and PBE
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@ -172,6 +172,10 @@
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\newcommand{\n}[2]{n_{#1}^{#2}}
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\newcommand{\E}[2]{E_{#1}^{#2}}
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\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}}
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\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}}
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\newcommand{\e}[2]{\varepsilon_{#1}^{#2}}
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\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
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\newcommand{\bec}[1]{\Bar{e}^{#1}}
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\newcommand{\wf}[2]{\Psi_{#1}^{#2}}
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\newcommand{\W}[2]{W_{#1}^{#2}}
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\newcommand{\w}[2]{w_{#1}^{#2}}
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@ -504,10 +508,10 @@ and therefore
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%=================================================================
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\label{sec:ecmd}
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In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} the authors proposed to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals, namely the ECMD whose general definition is \cite{TouGorSav-TCA-05}
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we propose here to approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of SR-DFT energy functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
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\begin{multline}
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\label{eq:ec_md_mu}
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\ecmubis = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
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\\
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- \mel*{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]}{\hT + \hWee{}}{\wf{}{\rsmu{}{}}[\n{}{}(\br{})]},
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\end{multline}
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@ -516,25 +520,25 @@ where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimizati
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\label{eq:argmin}
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\wf{}{\rsmu{}{}}[\n{}{}(\br{})] = \arg \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\wf{}{}}{\hT + \hWee{\lr,\rsmu{}{}}}{\wf{}{}},
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\end{equation}
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and
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with
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\begin{equation}
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\label{eq:weemu}
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\hWee{\lr,\rsmu{}{}} = \frac{1}{2} \iint \w{}{\lr,\rsmu{}{}}(r_{12}) \hn{}{(2)}(\br{1},\br{2}) \dbr{1} \dbr{2},
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\hWee{\lr,\rsmu{}{}} = \sum_{i<j} \w{}{\lr,\rsmu{}{}}(r_{ij})
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\end{equation}
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is the long-range Coulomb operator with
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and
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\begin{equation}
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\label{eq:erf}
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\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}},
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\w{}{\lr,\rsmu{}{}}(r_{12}) = \frac{\erf(\rsmu{}{} r_{12})}{r_{12}}.
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\end{equation}
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and $\hn{}{(2)}(\br{1},\br{2}) =\hn{}{}(\br{1}) \hn{}{}(\br{2}) - \delta (\br{1}-\br{2}) \hn{}{}(\br{1})$ is the pair-density operator.
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The ECMD functionals admit two limits as function of $\rsmu{}{}$
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is the long-range part of the Coulomb operator.
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The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following limiting forms:
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\begin{subequations}
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\begin{align}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \rightarrow \infty} \ecmubis & = 0 \quad & \forall \n{}{}(\br{})
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\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
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\\
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\label{eq:small_mu_ecmd}
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\lim_{\mu \to 0} \ecmubis & = \Ec[\denr] \quad & \forall \n{}{}(\br{})
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\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
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\end{align}
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\end{subequations}
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where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
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@ -555,60 +559,40 @@ It is important to notice that in the limit of a complete basis set, according t
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for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD.
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%--------------------------------------------
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\subsubsection{Local density approximation}
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%\subsubsection{Local density approximation}
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%--------------------------------------------
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Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we define the LDA version of ECMD as
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\begin{equation}
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\label{eq:def_lda_tot}
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\bE{\LDA}{\Bas,\wf{}{\Bas}}[\n{}{}(\br{})] = \int \n{}{}(\br{}) \emuldamodel \dbr{}
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\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
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\end{equation}
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where $\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}(n,\mu)$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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where $\be{\UEG}{\sr}[\n{}{}(\br{}),\rsmu{}{}]$ is the multi-determinant short-range correlation energy per particle of the uniform electron gas (UEG) for which a parametrization can be found in Ref.~\onlinecite{PazMorGorBac-PRB-06}.
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%In practice, for open-shell systems, we use the spin-resolved version of this functional (i.e., depending on both spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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%--------------------------------------------
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\subsubsection{New PBE functional}
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%\subsubsection{New PBE functional}
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%--------------------------------------------
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The LDA-like functional defined in \eqref{eq:def_lda_tot} relies only on the transferability of the physics of the uniform electron gas (UEG) which is certainly valid for large values of $\mu$ but which is known to over correlate for small values of $\mu$.
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In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors\cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional when $\mu \rightarrow 0$ and the exact behaviour which is known when $\mu \rightarrow \infty$.
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Thanks to the study of the behaviour in the large $\mu$ limit of the various quantities appearing in the ECMD\cite{TouColSav-PRA-04,GoriSav-PRA-06,PazMorGori-PRB-06}, one can have an analytical expression of $\ecmubis$ in that regime
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\begin{equation}
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\label{eq:ecmd_large_mu}
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\ecmubis = \frac{2\sqrt{\pi}\left(1 - \sqrt{2}\right)}{3\,\mu^3} \int \text{d}{\bf r} \,\, n^{(2)}({\bf} r)
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\end{equation}
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where $ n^{(2)}({\bf r}) $ is the \textit{exact} on-top pair density for the ground state of the system.
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As the exact ground state on-top pair density $n^{(2)}({\bf} r)$ is not known, we propose here to approximate it by that of the UEG at the density of the system:
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The short-range LDA correlation functional defined in Eq.~\eqref{eq:def_lda_tot} 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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In order to correct such a defect, we propose here a new ECMD functional inspired by the recently proposed functional of some of the present authors \cite{FerGinTou-JCP-18} which interpolates between the usual PBE correlation functional $\e{}{\PBE}(\n{}{},\nabla \n{}{})$ for small $\rsmu{}{}$ and the exact large-$\rsmu{}{}$ behaviour, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
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\begin{subequations}
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\begin{gather}
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\label{eq:epsilon_cmdpbe}
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\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \mu)\mu^3 }
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\\
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\label{eq:epsilon_cmdpbe}
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\beta(n,\nabla n;\,\mu) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
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\end{gather}
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\end{subequations}
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The difference between the ECMD PBE functional defined in Ref.~\cite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}(\br{})$ by its UEG version, i.e.
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\begin{equation}
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\label{eq:ueg_ontop}
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n^{(2)}({\bf} r) \approx n^{(2)}_{\text{UEG}}(n_{\uparrow}({\bf} r) , \, n_{\downarrow}({\bf} r))
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\n{}{(2)}(\br{}) \approx \n{\UEG}{(2)}(\br{}) = \n{}{}(\br{})^2 g_0[\n{}{}(\br{})]
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\end{equation}
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where $n_{\uparrow}({\bf} r)$ and $ n_{\downarrow}({\bf} r)$ are, respectively, the up and down spin densities of the physical system at ${\bf} r$, $n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow})$ is the UEG on-top pair density
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where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
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Therefore, the PBE complementary function reads
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\begin{equation}
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\label{eq:ueg_ontop}
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n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow}) = 4\, n_{\uparrow} \, n_{\downarrow} \, g_0(n_{\uparrow},\, n_{\downarrow})
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\end{equation}
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and $g_0(n_{\uparrow} ,\, n_{\downarrow})$ is the correlation factor of the UEG whose parametrization can be found in equation (46) of \onlinecite{GorSav-PRA-06}.
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As the form in \eqref{eq:ecmd_large_mu} diverges for small values of $\mu$ as $1/\mu^3$, we follow the work proposed in \cite{FerGinTou-JCP-18} and interpolate between the large-$\mu$ limit and the $\mu = 0$ limit where the $\ecmubis$ reduces to the Kohn-Sham correlation functional (see equation \eqref{eq:small_mu_ecmd}), for which we take the PBE approximation as in \cite{FerGinTou-JCP-18}.
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More precisely, we propose the following expression for the
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\begin{equation}
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\label{eq:ecmd_large_mu}
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\ecmubis = \int \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf} r),\nabla n({\bf} r);\,\mu)
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\end{equation}
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with
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\begin{equation}
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\label{eq:epsilon_cmdpbe}
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\bar{e}_{\text{c,md}}^\text{PBE}(n,\nabla n;\,\mu) = \frac{e_c^{PBE}(n,\nabla n)}{1 + \beta_{\text{c,md}\,\text{PBE}}(n,\nabla n;\,\mu)\mu^3 }
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\end{equation}
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\begin{equation}
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\label{eq:epsilon_cmdpbe}
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\beta(n,\nabla n;\,\mu) = \frac{3 e_c^{PBE}(n,\nabla n)}{2\sqrt{\pi}\left(1 - \sqrt{2}\right)n^{(2)}_{\text{UEG}}(n_{\uparrow} , n_{\downarrow})}.
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\end{equation}
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Therefore, we propose this approximation for the complementary functional $\ecompmodel$:
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\begin{equation}
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\label{eq:def_lda_tot}
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\ecompmodelpbe = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(n({\bf r}),\nabla n({\bf r});\,\mur)
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\label{eq:def_lda_tot}
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\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}] \n{}{}(\br{}) \dbr{}
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\end{equation}
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%=================================================================
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