From e142c34d2c515932ae3d1adfabbce75569cf4e12 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 5 Apr 2019 15:23:07 +0200 Subject: [PATCH] getting there --- Manuscript/G2-srDFT.tex | 192 ++++++++++++++++++++-------------------- 1 file changed, 97 insertions(+), 95 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index 4d206ba..0e9fae3 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -394,25 +394,114 @@ Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only so % \EexFCIinfty \approx \EexFCIbasis + \efuncden{\dencipsi} %\end{equation} +However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. +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 $r_{12} = 0$. +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} $\rsmu{}{}(\br{})$. + +%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}]. +%First, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) . +%Second, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). %================================================================= %\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$} %================================================================= -However, in addition of being unknown, the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. -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}]. -First, we define a real-space representation of the Coulomb operator projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation \textit{function} $\mu(\br{})$ defined in real space. %(see Sec.~\ref{sec:weff}). -Then, we choose a specific class of short-range density functionals, namely the short-range correlation functionals with multi-determinantal reference (ECMD) introduced by Toulouse \textit{et al.} \cite{TouGorSav-TCA-05} that we evaluate at $\n{\modX}{\Bas}$ alongside $\mu(\br{})$.% (see Sec.~\ref{sec:ecmd}) . %================================================================= %\subsection{Definition of a real-space representation of the coulomb operator truncated in a basis-set $\Bas$} %\label{sec:weff} %================================================================= -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 also originate from a Hamiltonian with a non-divergent Coulomb interaction. -Therefore, the impact of the incompleteness of $\Bas$ can be viewed as a removal of the divergence of the Coulomb interaction at $r_{12} = 0$. -The present paragraph briefly describes how to obtain 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. +%================================================================= +\subsection{Effective Coulomb operator} +%================================================================= +%The present section briefly describes how to obtain 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 order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that +\begin{equation} + \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}, +\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 +\begin{equation} + \mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}}, +\end{equation} +which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$ +\begin{equation} +\label{eq:WeeB} + \hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i} +\end{equation} +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. +Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that +\begin{subequations} +\begin{align} + \label{eq:expectweeb} + \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} & = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, + \\ + \label{eq:expectwee} + \mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} & = \frac{1}{2} \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, +\end{align} +\end{subequations} +where +\begin{multline} + \label{eq:fbasis} + \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) + \\ + = \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{kl}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1}, +\end{multline} +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 +\begin{equation} + \label{eq:def_weebasis} + \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}. +\end{equation} + +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. +Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. + + +%================================================================= +\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 fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}. +More precisely, if we define the value of the interaction at coalescence as +\begin{equation} + \label{eq:def_wcoal} + \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}). +\end{equation} +where $(\bfr{},\bar{{\bf x}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$, +we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction: +\begin{equation} + \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{}) +\end{equation} +where the long-range-like interaction is defined as +\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}} }. +\end{equation} +Equation \eqref{eq:def_wcoal} is equivalent to the following condition +\begin{equation} + \label{eq:mu_of_r} + \rsmu{\wf{}{\Bas}}{}(\br{}) = \W{\wf{}{\Bas}}{}(\br{}) +\end{equation} +%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as +%\begin{equation} +% \label{eq:mu_of_r_val} +% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, . +%\end{equation} +An important point to notice is that, in the limit of a complete basis set $\Bas$, as +\begin{equation} +\label{eq:lim_W} + \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\bx{1},\bx{2}) +% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , +\end{equation} +one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$ +% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, +and therefore +\begin{equation} +\label{eq:lim_mur} + \lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty +%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . +\end{equation} @@ -509,93 +598,6 @@ It is important to notice that in the limit of a complete basis set, according t \end{equation} for whatever choice of density $\denmodel$, wave function $\wf{}{\Bas}$ used to define the interaction, and ECMD functional used to approximate the exact ECMD. -%================================================================= -\subsection{Effective Coulomb operator} -%================================================================= -In order to compute the effective operator $\W{\wf{}{\Bas}}{}(\bx{1},\bx{2})$ defined such that -\begin{equation} - \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}, -\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 -\begin{equation} - \mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} = \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}}, -\end{equation} -which states that the expectation value of $\hWee{}$ over $\wf{}{\Bas}$ is equal to the expectation value of its projected version in $\Bas$ -\begin{equation} -\label{eq:WeeB} - \hWee{\Bas} = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i} -\end{equation} -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. -Because one can show (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) that -\begin{subequations} -\begin{align} - \label{eq:expectweeb} - \mel*{\wf{}{\Bas}}{\hWee{\Bas}}{\wf{}{\Bas}} & = \frac{1}{2} \iint \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, - \\ - \label{eq:expectwee} - \mel*{\wf{}{\Bas}}{\hWee{}}{\wf{}{\Bas}} & = \frac{1}{2} \iint r_{12}^{-1} \n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2}) \dbx{1} \dbx{2}, -\end{align} -\end{subequations} -where -\begin{multline} - \label{eq:fbasis} - \f{\wf{}{\Bas}}{}(\bx{1},\bx{2}) - \\ - = \sum_{ijklmn \in \Bas} \SO{i}{1} \SO{j}{2} \vijkl \Gam{mn}{kl}[\wf{}{\Bas}] \SO{n}{2} \SO{m}{1}, -\end{multline} -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 -\begin{equation} - \label{eq:def_weebasis} - \W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = \frac{\f{\wf{}{\Bas}}{}(\bx{1},\bx{2})}{\n{\wf{}{\Bas}}{(2)}(\bx{1},\bx{2})}. -\end{equation} - -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. -Also, as demonstrated in Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\lim_{\Bas \to \infty}\W{\wf{}{\Bas}}{}(\bx{1},\bx{2}) = r_{12}^{-1}$. - - -%================================================================= -\subsubsection{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 fit the effective interaction with a long-range interaction having a range-separation parameter \textit{varying in space}. -More precisely, if we define the value of the interaction at coalescence as -\begin{equation} - \label{eq:def_wcoal} - \wbasiscoal{} = W_{\wf{}{\Bas}}(\bfr{},\bar{{\bf x}}_{}). -\end{equation} -where $(\bfr{},\bar{{\bf x}}_{})$ means a couple of anti-parallel spins at the same point in $\bfrb{}$, -we propose a fit for each point in $\rnum^3$ of $\wbasiscoal{ }$ with a long-range-like interaction: -\begin{equation} - \wbasiscoal{} = \w{}{\lr,\rsmu{\wf{}{\Bas}}{}}(\bfrb{},\bfrb{}) -\end{equation} -where the long-range-like interaction is defined as -\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}} }. -\end{equation} -Equation \eqref{eq:def_wcoal} is equivalent to the following condition -\begin{equation} - \label{eq:mu_of_r} - \rsmu{\wf{}{\Bas}}{}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\wf{}{\Bas}}{}(\br{}) -\end{equation} -%As we defined an effective interaction for the valence electrons, we also introduce a valence range-separation parameter as -%\begin{equation} -% \label{eq:mu_of_r_val} -% \murpsival = \frac{\sqrt{\pi}}{2} \, \wbasiscoalval{} \, . -%\end{equation} -An important point to notice is that, in the limit of a complete basis set $\Bas$, as -\begin{equation} -\label{eq:lim_W} - \lim_{\Bas \rightarrow \infty}\wbasis = r_{12}^{-1} \quad \forall (\br{1},\br{2}) -% &\lim_{\Bas \rightarrow \infty}\wbasisval = 1/r_{12} \,\,\,\,\forall \,\, (\bfr{1},\bfr{2})\,\, , -\end{equation} -one has $\lim_{\Bas \rightarrow \infty} \wbasiscoal{} = \infty$ -% &\lim_{\Bas \rightarrow \infty} \wbasiscoalval{} = +\infty\,\,, -and therefore -\begin{equation} -\label{eq:lim_mur} - \lim_{\Bas \rightarrow \infty} \rsmu{\wf{}{\Bas}}{}(\br{}) = \infty -%\lim_{\Bas \rightarrow \infty} \murpsival = +\infty \,\, . -\end{equation} %================================================================= \subsection{Valence effective interaction}