Bread on the board
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@ -159,6 +159,9 @@
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% methods
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\newcommand{\UEG}{\text{UEG}}
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\newcommand{\LDA}{\text{LDA}}
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\newcommand{\PBE}{\text{PBE}}
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\newcommand{\FCI}{\text{FCI}}
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\newcommand{\CCSDT}{\text{CCSD(T)}}
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\newcommand{\lr}{\text{lr}}
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@ -177,7 +180,7 @@
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\newcommand{\SO}[2]{\phi_{#1}(\bx{#2})}
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\newcommand{\modX}{\text{X}}
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\newcommand{\modY}{Y}
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\newcommand{\modY}{\text{Y}}
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% basis sets
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\newcommand{\Bas}{\mathcal{B}}
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@ -267,13 +270,13 @@ We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} fo
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%=================================================================
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%\subsection{Correcting the basis set error of a general WFT model}
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%=================================================================
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Let us assume we have both the density $\n{\modX}{\Bas}$ and energy $\E{\modX}{\Bas}$ of a $\Nel$-electron system described by a method $\modX$ in an incomplete basis set $\Bas$.
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According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\n{\modX}{\Bas}$ is a resonable approximation of the \textit{exact} ground state density $\n{}{}$, one may approximate the \textit{exact} ground state energy as
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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 $\Bas$.
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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 $\n{}{}$, one may write
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\begin{equation}
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\label{eq:e0basis}
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\E{}{}
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\approx \E{\modX}{\Bas}
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+ \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}],
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+ \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}],
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\end{equation}
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where
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\begin{equation}
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@ -291,11 +294,11 @@ Both wave functions yield the same target density $\n{}{}$.
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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
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\begin{equation}
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\label{eq:limitfunc}
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\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{} \approx E,
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\lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modY}{\Bas}}{}] ) = \E{\modX}{} \approx E,
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\end{equation}
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where $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
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In the case of $\modX = \FCI$, we have as strict equality as $\E{\FCI}{} = E$.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the method $\modX$.
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Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$.
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%Here we propose to generalize such approach to a general WFT model, referred here as $\model$, projected in a basis set $\Bas$ which must provides a density $\denmodel$ and an energy $\emodel$.
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%As any wave function model is necessary an approximation to the FCI model, one can write
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@ -415,7 +418,7 @@ In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ de
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\label{eq:int_eq_wee}
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\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},
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\end{equation}
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(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 $\int \dbx{} = \sum_{\sigma}\,\int_{\mathbb{R}^3} \dbr{}$), it is key to realise that
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(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
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\begin{equation}
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\mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}},
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\end{equation}
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@ -424,7 +427,7 @@ which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal
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\label{eq:WeeB}
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\hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i}
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\end{equation}
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over the same wave function $\wf{}{\Bas}$, where the indices run over all \alert{occupied} spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
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over the same wave function $\wf{}{\Bas}$, where the indices run over all spinorbitals $\SO{i}{}$ in $\Bas$ and $\vijkl$ are the usual two-electron Coulomb integrals.
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Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that
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\begin{subequations}
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\begin{align}
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@ -440,7 +443,7 @@ where
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\label{eq:fbasis}
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\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})
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\\
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= \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{pq}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1},
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= \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{kl}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1},
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\end{multline}
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and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\wf{}{\Bas}}$ is the two-body density tensor of $\wf{}{\Bas}$, it comes naturally that
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\begin{equation}
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@ -448,7 +451,7 @@ and $\Gam{mn}{pq}[\wf{}{\Bas}] = \mel*{\wf{}{\Bas}}{ \aic{p}\aic{q}\ai{n}\ai{m}
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\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}.
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\end{equation}
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set $\Bas$.
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As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries and is necessarily \textit{finite} at $r_{12} = 0$ for an incomplete basis set.
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Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$.
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@ -495,81 +498,47 @@ and therefore
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%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, .
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\end{equation}
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%=================================================================
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\subsection{Valence effective interaction}
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%=================================================================
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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.
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We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
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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 $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying
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\begin{equation}
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\label{eq:expectweebval}
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\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
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%\begin{equation}
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% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
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%\end{equation}
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Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as
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\begin{multline}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
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\\
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= \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
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\end{multline}
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and, the valence part of the effective interaction is
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})},
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\end{equation}
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where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
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%\begin{equation}
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% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
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%\end{equation}
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%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$.
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It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ and $\murpsival$ fulfils Eqs.~\eqref{eq:lim_W} and \eqref{eq:lim_mur}.
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%=================================================================
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\subsection{Complementary functional}
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%=================================================================
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\label{sec:ecmd}
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In Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} the authors have 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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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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\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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\\
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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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where the wave function $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ is defined by the constrained minimization
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where $\wf{}{\rsmu{}{}}[\n{}{}(\br{})]$ 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{}{}}[\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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where
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and
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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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\end{equation}
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is the long-range electron-electron interaction operator with
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is the long-range Coulomb operator with
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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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\end{equation}
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and the pair-density operator $\hn{}{(2)}(\br{1},\br{2}) =\hn{}{}(\br{1}) \hn{}{}(\br{2}) - \delta (\br{1}-\br{2}) \hn{}{}(\br{1})$.
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The ECMD functionals admit two limits as function of $\mu$
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\begin{equation}
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\label{eq:large_mu_ecmd}
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\lim_{\mu \rightarrow \infty} \ecmubis = 0 \quad \forall\,\,\denr
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\end{equation}
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\begin{equation}
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\label{eq:small_mu_ecmd}
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\lim_{\mu \rightarrow 0} \ecmubis = E_{\text{c}}[\denr]\quad \forall\,\,\denr
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\end{equation}
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where $E_{\text{c}}[\denr]$ is the usual universal correlation functional defined in the Kohn-Sham DFT.
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These functionals differ from the standard RSDFT correlation functional by the fact that the reference is not the Kohn-Sham Slater determinant but a multi determinant wave function, which makes them much more adapted in the present context where one aims at correcting the general multi-determinant WFT model.
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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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\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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\\
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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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\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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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.
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The general scheme for estimating $\ecompmodel$ is the following. Consider a given approximated ECMD functional $\ecmuapprox$ labelled by ECMD-$\mathcal{X}$.
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Such a functional of the density $\denr$ (and potentially its derivatives $\nabla \denr$) is defined for any value of the range-separation parameter $\mu$.
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@ -588,17 +557,18 @@ for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to
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%--------------------------------------------
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\subsubsection{Local density approximation}
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%--------------------------------------------
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As done in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, one can define an LDA-like approximation for $\ecompmodel$ as
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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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\ecompmodellda = \int \, \text{d}{\bf r} \,\, \denmodelr \,\, \emuldamodel\,,
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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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\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}. In practice, for open-shell systems, we use the spin-polarized version of this functional (i.e., depending on the spin densities) but for simplicity we will continue to use only the notation of the spin-unpolarized case.
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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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%--------------------------------------------
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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 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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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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@ -641,9 +611,40 @@ Therefore, we propose this approximation for the complementary functional $\ecom
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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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\end{equation}
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%--------------------------------------------
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\subsection{Valence-only approximation for the complementary functional}
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%--------------------------------------------
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%=================================================================
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\subsection{Valence effective interaction}
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%=================================================================
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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.
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We then split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively, and $\Cor \bigcap \Val = \O$.
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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 $\wf{}{\Bas}$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ satisfying
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\begin{equation}
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\label{eq:expectweebval}
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\mel*{\wf{}{\Bas}}{\hWee{\Val}}{\wf{}{\Bas}} = \frac{1}{2} \iint \f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) \dbx{1} \dbx{2},
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\end{equation}
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where $\hWee{\Val}$, the valence part of the Coulomb operator, has a similar expression as $\hWee{\Bas}$ in Eq.~\eqref{eq:WeeB}.
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%\begin{equation}
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% \hWee{\Val} = \frac{1}{2} \sum_{ijkl \in \Val} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
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%\end{equation}
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Following the spirit of Eq.~\eqref{eq:fbasis}, the function $\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ can be defined as
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\begin{multline}
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\label{eq:fbasisval}
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\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})
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\\
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= \sum_{ij \in \Bas} \sum_{klmn \in \Val} \SO{i}{1} \SO{j}{2} \vijkl \gammaklmn{\wf{}{\Bas}} \SO{n}{2} \SO{m}{1}.
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\end{multline}
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and, the valence part of the effective interaction is
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\begin{equation}
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\label{eq:def_weebasis}
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\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2}) }{\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})},
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\end{equation}
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where $\n{\wf{}{\Bas},\Val}{(2)}(\bx{1},\bx{2})$ is the two body density associated to the valence electrons.
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%\begin{equation}
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% \twodmrdiagpsival = \sum_{klmn \in \Val} \SO{m}{1} \SO{n}{2} \gammamnkl[\wf{}{\Bas}] \SO{k}{1} \SO{l}{2} .
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%\end{equation}
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%It is worth noting that, in Eq.~\eqref{eq:fbasisval} the difference between the set of orbitals for the indices $(i,j)$, which span the full set of MOs within $\Bas$, and the $(k,l,m,n)$, which span only the valence space $\Basval$. Only with such a definition, one can show (see annex) that $\fbasisval$ fulfills \eqref{eq:expectweebval} and tends to the exact interaction $1/r_{12}$ in the limit of a complete basis set $\Bas$, whatever the choice of subset $\Basval$.
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It is worth noting that, within the present definition, $\W{\wf{}{\Bas}}{\Val}(\bx{1},\bx{2})$ and $\murpsival$ fulfils Eqs.~\eqref{eq:lim_W} and \eqref{eq:lim_mur}.
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We now introduce a valence-only approximation for the complementary functional which is needed to correct for frozen core WFT models.
|
||||
Defining the valence one-body spin density matrix as
|
||||
\begin{equation}
|
||||
|
||||
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Block a user