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Manuscript/G2-srDFT.tex
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@ -259,11 +259,11 @@ Although there is no clear way on how to systematically improve density-function
In the present context, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15} In the present context, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
Progress toward unifying these two approaches are on-going. Progress toward unifying these two approaches are on-going.
Using accurate and rigorous WFT methods, some of us have developed radical generalisations of DFT that are free of the well-known limitations of conventional DFT. Using accurate and rigorous WFT methods, some of us have developed radical generalizations of DFT that are free of the well-known limitations of conventional DFT.
In that respect range-separated DFT (RS-DFT) is particularly promising as it allows to perform multi-configurational DFT calculations within a rigorous mathematical framework. In that respect range-separated DFT (RS-DFT) is particularly promising as it allows to perform multi-configurational DFT calculations within a rigorous mathematical framework.
Range-separated hybrids, i.e.~single-determinant approximations of RS-DFT, correct for the wrong long-range behavior of the usual hybrid approximations thanks to the inclusion of the long-range part of the Hartree-Fock (HF) exchange. Range-separated hybrids, i.e.~single-determinant approximations of RS-DFT, correct for the wrong long-range behavior of the usual hybrid approximations thanks to the inclusion of the long-range part of the Hartree-Fock (HF) exchange.
%The present manuscript is organised as follows. %The present manuscript is organized as follows.
Unless otherwise stated, atomic used are used. Unless otherwise stated, atomic used are used.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
@ -278,12 +278,12 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
%\subsection{Correcting the basis set error of a general WFT model} %\subsection{Correcting the basis set error of a general WFT model}
%================================================================= %=================================================================
Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Nel$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$. Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Nel$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$.
According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and \alert{$\n{\modY}{\Bas}$} are reasonable approximations of the \textit{exact} ground state energy $\E{}{}$ and density $\n{}{}$, respectively, one may write According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \textit{exact} ground state energy $\E{}{}$ and density $\n{}{}$, respectively, one may write
\begin{equation} \begin{equation}
\label{eq:e0basis} \label{eq:e0basis}
\E{}{} \E{}{}
\approx \E{\modX}{\Bas} \approx \E{\modX}{\Bas}
+ \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}], + \bE{}{\Bas}[\n{\modY}{\Bas}],
\end{equation} \end{equation}
where where
\begin{equation} \begin{equation}
@ -301,7 +301,7 @@ Both wave functions yield the same target density $\n{}{}$.
An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that An important aspect of such theory is that, in the limit of a complete basis set $\Bas$ (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
\begin{equation} \begin{equation}
\label{eq:limitfunc} \label{eq:limitfunc}
\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}] ) = \E{\modX}{} \approx E, \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
\end{equation} \end{equation}
where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set. where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
In the case $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$. In the case $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$.
@ -401,9 +401,9 @@ However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is o
One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points. One of the consequences of the incompleteness of $\Bas$ is that $\wf{}{\Bas}$ does not have a cusp (i.e.~a discontinuous derivative) at the electron-electron (e-e) coalescence points.
As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$. As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$.
Therefore, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals. Therefore, it feels natural to evaluate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals.
Contrary to conventional RS-DFT schemes which require a range-separated parameter $\rsmu{}{}$, we must know the value of $\rsmu{}{}$ at any point in space due to the spatial inhomogeneity of $\Bas$, hence defining a range-separated \textit{function} $\rsmu{}{}(\br{})$. Contrary to conventional RS-DFT schemes which require a range-separated parameter $\rsmu{}{}$, we must know the value of $\rsmu{}{}$ at any point in space due to the spatial inhomogeneity of $\Bas$, hence defining a \textit{range-separated function} $\rsmu{}{}(\br{})$.
The first step of our basis set correction consists in obtaining the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set. The first step of our basis set correction consists in obtaining an effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ which i) is finite at the e-e coalescence points as long as an incomplete basis set is used, and ii) tends to the genuine, unbounded $r_{12}^{-1}$ Coulomb operator in the limit of a complete basis set.
In a second step, we shall link $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ to $\rsmu{}{}(\br{})$. In a second step, we shall link $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ to $\rsmu{}{}(\br{})$.
The final step employs $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05} The final step employs $\rsmu{}{}(\br{})$ within short-range density functionals. \cite{TouGorSav-TCA-05}
%Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}]. %Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ following a two-step procedure which guarantees the correct behaviour in the limit $\Bas \to \infty$ [see Eq.~\eqref{eq:limitfunc}].
@ -428,7 +428,7 @@ In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ de
\label{eq:int_eq_wee} \label{eq:int_eq_wee}
\mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} = \iint \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
\end{equation} \end{equation}
(where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realise that (where $\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})$ is the two-body density associated with $\wf{}{\Bas}$, $\bx{} = \qty(\br{},\sigma)$ collects space and spin variables, and $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), one must realize that
\begin{equation} \begin{equation}
\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}}, \mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
\end{equation} \end{equation}
@ -472,7 +472,7 @@ Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-1
%================================================================= %=================================================================
%\subsection{Range-separation function} %\subsection{Range-separation function}
%================================================================= %=================================================================
To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we associate the effective interaction to a long-range interaction characterised by a range-separation function $\rsmu{}{}(\br{})$. To be able to approximate the complementary functional $\bE{}{\Bas}[\n{}{}]$ thanks to functionals developed in the field of RS-DFT, we associate the effective interaction to a long-range interaction characterized by a range-separation function $\rsmu{}{}(\br{})$.
Although this choice is not unique, the long-range interaction we have chosen is Although this choice is not unique, the long-range interaction we have chosen is
\begin{equation} \begin{equation}
\w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }. \w{}{\lr,\rsmu{}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \murr{1} \, r_{12}]}{r_{12}} + \frac{\erf[ \murr{2} r_{12}]}{ r_{12}} }.
@ -559,7 +559,7 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT. where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general multi-determinant WFT model. These functionals differ from the standard RS-DFT correlation functional by the fact that the reference is not the KS Slater determinant but a multi-determinantal wave function, which makes them much more adapted in the present context where one aims at correcting a general WFT method.
%-------------------------------------------- %--------------------------------------------
%\subsubsection{Local density approximation} %\subsubsection{Local density approximation}
@ -576,21 +576,21 @@ where $\be{\UEG}{\sr}(\n{}{},\rsmu{}{})$ is the multi-determinant short-range co
%\subsubsection{New PBE functional} %\subsubsection{New PBE functional}
%-------------------------------------------- %--------------------------------------------
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 $\mu$. 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 $\mu$.
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 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-$\rsmu{}{}$ behavior, \cite{TouColSav-PRA-04, GoriSav-PRA-06, PazMorGori-PRB-06} yielding
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
\label{eq:epsilon_cmdpbe} \label{eq:epsilon_cmdpbe}
\be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \rsmu{}{})\rsmu{}{3} } \be{\PBE}{\sr}(\n{}{},\nabla \n{}{},\rsmu{}{}) = \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{1 + \beta(n,\nabla n, \rsmu{}{})\rsmu{}{3} },
\\ \\
\label{eq:epsilon_cmdpbe} \label{eq:epsilon_cmdpbe}
\beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}. \beta(n,\nabla n,\rsmu{}{}) = \frac{3}{2\sqrt{\pi} (1 - \sqrt{2} )} \frac{\e{\PBE}{}(\n{}{},\nabla \n{}{})}{\n{\UEG}{(2)}(\n{}{})}.
\end{gather} \end{gather}
\end{subequations} \end{subequations}
The difference between the ECMD PBE functional defined in Ref.~\onlinecite{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.~$\n{}{(2)} \approx \n{\UEG}{(2)} = \n{}{2} g_0(\n{}{})$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}. The difference between the ECMD PBE functional defined in Ref.~\onlinecite{FerGinTou-JCP-18} and the present expression is that we approximate the \textit{exact} ground-state on-top pair density of the system $\n{}{(2)}$ by its UEG version, i.e.~$\n{}{(2)} \approx \n{\UEG}{(2)} = \n{}{2} g_0(\n{}{})$, where $g_0(\n{}{})$ is the UEG correlation factor whose parametrization can be found in Eq.~(46) of Ref.~\onlinecite{GorSav-PRA-06}.
Therefore, the PBE complementary function reads Therefore, the PBE complementary functional reads
\begin{equation} \begin{equation}
\label{eq:def_lda_tot} \label{eq:def_lda_tot}
\bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{} \bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}[\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})] \n{}{}(\br{}) \dbr{}.
\end{equation} \end{equation}
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ where $\rsmu{\wf{}{\Bas}}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}. Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\wf{}{\Bas}}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{}(\br{}),\rsmu{\wf{}{\Bas}}{}(\br{})]$ where $\rsmu{\wf{}{\Bas}}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
@ -613,7 +613,7 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
%\subsection{Valence effective interaction} %\subsection{Valence effective interaction}
%================================================================= %=================================================================
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals. As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals.
We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$. We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
%According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$. %According to Eqs.~\eqref{eq:expectweeb} and \eqref{eq:def_weebasis} , the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\wf{}{\Bas}$.
Accounting solely for the valence electrons, Eq.~\eqref{eq:expectweeb} becomes Accounting solely for the valence electrons, Eq.~\eqref{eq:expectweeb} becomes
@ -668,7 +668,7 @@ It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\
% \label{eq:def_lda_tot} % \label{eq:def_lda_tot}
% \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval) % \ecompmodelpbeval = \int \, \text{d}{\bf r} \,\, \bar{e}_{\text{c,md}}^\text{PBE}(\denval({\bf r}),\nabla \denval({\bf r});\,\murval)
%\end{equation} %\end{equation}
Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluate as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$. Defining $\n{\wf{}{\Bas}}{\Val}$ as the valence one-electron density, the valence part of the complementary functional $\bE{}{\Val}[\n{\wf{}{\Bas}}{\Val}]$ is then evaluate as $\bE{\LDA}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\wf{}{\Bas}}{\Val}(\br{}),\rsmu{\wf{}{\Bas}}{\Val}(\br{})]$.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
%\section{Results} %\section{Results}