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Manuscript/G2-srDFT.tex
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@ -275,21 +275,17 @@ is the complementary density functional defined in Eq.~(8) of Ref.~\onlinecite{G
In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ is the wave function obtained from the $\Nel$-electron Hilbert space spanned by $\Bas$, and $\wf{}{}$ is a general $\Nel$-electron wave function being obtained in a complete basis.
Both wave functions yield the same target density $\n{}{}$.
\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
An important aspect of such a 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 $\n{}{}$,
\begin{equation}
\label{eq:limitfunc}
\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0,
\end{equation}
which implies that
An important aspect of such a 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}
\label{eq:limitfunc}
\lim_{\Bas \rightarrow \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\wf{\modX}{\Bas}}{}] ) = \E{\modX}{\infty} \approx E,
\end{equation}
where $\E{\modX}{\infty}$ is the energy associated with the method $\modX$ in complete basis set.
In the case of $\modX = \FCI$, we $\E{\FCI}{\infty} = E$.
\alert{T2 stopped here.}
Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, at this stage, the only source of error lies in the potential approximate nature of the method $\modX$.
%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 $\emodel$.
%As any wave function model is necessary an approximation to the FCI model, one can write
@ -386,26 +382,28 @@ In the case of $\modX = \FCI$, we $\E{\FCI}{\infty} = E$.
%=================================================================
%\subsection{General scheme for the approximation of the unknown complementary functional $\efuncbasis$}
%=================================================================
The functional $\bE{}{\Bas}[\n{}{}]$ is not universal as it depends on the basis set $\Bas$ used and a simple analytical form for such a functional is of course not known.
Following the work of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ in two steps which guarantee the correct behaviour in the limit of a complete basis set [see Eq.~\eqref{eq:limitfunc}].
First, we define a real-space representation of the coulomb interaction projected in $\Bas$, which is then fitted with a long-range interaction thanks to a range-separation parameter $\mu(r)$ varying in space (see \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 the density $\denmodel$ provided by the model (see \ref{sec:ecmd}) and with the range-separation parameter $\mu(r)$ varying in space.
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 parameter $\mu(\br)$ varying in 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 $\n{\modX}{\Bas}$ (see Sec.~\ref{sec:ecmd}) with $\mu(\br)$.
%=================================================================
\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 use of an incomplete basis-set $\Bas$ is that the wave function does not present a cusp near the electron coalescence point, which means that all derivatives of the wave function are continuous. As the exact electronic cusp originates from the divergence of the coulomb interaction at the electron coalescence point, a cusp-free wave function could also originate from an Hamiltonian with a non-divergent electron-electron interaction. Therefore, the impact of the incompleteness of a finite basis-set $\Bas$ can be thought as a cutting of the divergence of the coulomb interaction at the electron coalescence point.
%=================================================================
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 coalescence points.
As the electron-electron 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 electron-electron interaction.
Therefore, the impact of the incompleteness of a finite basis set $\Bas$ 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 interaction $\wbasis$ which i) is finite at the electron-electron 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 present paragraph briefly describes how to obtain an effective interaction $\wbasis$ which:
\begin{itemize}
\item is non-divergent at the electron coalescence point as long as an incomplete basis set $\Bas$ is used,
\item tends to the regular $1/r_{12}$ interaction in the limit of a complete basis set $\Bas$.
\end{itemize}
\subsubsection{General definition of an effective interaction for the basis set $\Bas$}
Consider the coulomb operator projected in the basis-set $\Bas$
%----------------------------------------------------------------
%\subsubsection{General definition of an effective interaction for the basis set $\Bas$}
%----------------------------------------------------------------
Consider the Coulomb operator projected in $\Bas$
\begin{equation}
\begin{aligned}
\weeopbasis = \frac{1}{2}\,\, \sum_{ijkl\,\,\in\,\,\Bas} \,\, \vijkl \,\, \aic{k}\aic{l}\ai{j}\ai{i},
\weeopbasis = \frac{1}{2} \sum_{ijkl \in \Bas} \vijkl \aic{k}\aic{l}\ai{j}\ai{i},
\end{aligned}
\end{equation}
where the indices run over all orthonormal spin-orbitals in $\Bas$ and $\vijkl$ are the usual coulomb two-electron integrals.
@ -413,45 +411,37 @@ Consider now the expectation value of $\weeopbasis$ over a general wave function
After a few mathematical work (see appendix A of \onlinecite{GinPraFerAssSavTou-JCP-18} for a detailed derivation), such an expectation value can be rewritten as an integral over the two-electron spin and space coordinates:
\begin{equation}
\label{eq:expectweeb}
\elemm{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis\,\,,
\mel*{\psibasis}{\weeopbasis}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasis\,\,,
\end{equation}
where the function $\fbasis$ is
where
\begin{equation}
\label{eq:fbasis}
\begin{aligned}
\fbasis = \sum_{ijklmn\,\,\in\,\,\Bas} & \vijkl \,\, \gammaklmn{\psibasis} \\& \phix{n}{2} \phix{m}{1} \phix{i}{1} \phix{j}{2}\,\,,
\end{aligned}
\end{equation}
$\gammamnpq{\psibasis}$ is the two-body density tensor of $\psibasis$
and
\begin{equation}
\gammamnpq{\psibasis} = \elemm{\psibasis}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\psibasis}\,\,,
\gammamnpq{\psibasis} = \mel*{\psibasis}{ \aic{p}\aic{q}\ai{n}\ai{m} }{\psibasis}\,\,,
\end{equation}
and $\bfr{}$ collects the space and spin variables,
\begin{equation}
\label{eq:define_x}
\begin{aligned}
&\bfr{} = \left({\bf r},\sigma \right)\qquad {\bf r} \in {\rm I\!R}^3,\,\, \sigma = \pm \frac{1}{2}\\
&\int \, \dr{} = \sum_{\sigma = \pm \frac{1}{2}}\,\int_{{\rm I\!R}^3} \, \text{d}{\bf r}\,\,.
\end{aligned}
\end{equation}
is the two-body density tensor of $\psibasis$ and $\bfr{} = \qty(\br,\sigma)$ collects the space and spin variables, $\int \, \dr{} = \sum_{\sigma}\,\int_{{\rm I\!R}^3} \, \text{d}{\bf r}$.
Then, consider the expectation value of the exact coulomb operator over $\psibasis$
\begin{equation}
\label{eq:expectwee}
\elemm{\psibasis}{\weeop}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \frac{1}{r_{12}} \twodmrdiagpsi\,\,
\mel*{\psibasis}{\weeop}{\psibasis} = \frac{1}{2} \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}
\end{equation}
where $\twodmrdiagpsi$ is the two-body density associated to $\psibasis$.
Because $\psibasis$ belongs to $\Bas$, such an expectation value coincides with the expectation value of $\weeopbasis$
\begin{equation}
\elemm{\psibasis}{\weeopbasis}{\psibasis} = \elemm{\psibasis}{\weeop}{\psibasis},
\mel*{\psibasis}{\weeopbasis}{\psibasis} = \mel*{\psibasis}{\weeop}{\psibasis},
\end{equation}
which can be rewritten as:
\begin{equation}
\begin{aligned}
\label{eq:int_eq_wee}
& \iint \dr{1}\,\dr{2} \,\, \wbasis \,\, \twodmrdiagpsi \\ = &\iint \dr{1}\,\dr{2} \,\,\frac{1}{\norm{\bfrb{1} - \bfrb{2} }} \,\, \twodmrdiagpsi.
\end{aligned}
\end{equation}
\begin{multline}
\label{eq:int_eq_wee}
\iint \wbasis \twodmrdiagpsi \dr{1} \dr{2}
\\
= \iint r_{12}^{-1} \twodmrdiagpsi \dr{1} \dr{2}.
\end{multline}
where we introduced $\wbasis$
\begin{equation}
\label{eq:def_weebasis}
@ -463,13 +453,15 @@ As already discussed in \onlinecite{GinPraFerAssSavTou-JCP-18}, such an effectiv
Also, as demonstrated in the appendix B of \onlinecite{GinPraFerAssSavTou-JCP-18}, $\wbasis$ tends to the regular coulomb interaction $1/r_{12}$ for all points $(\bfr{1},\bfr{2})$ and any choice of $\psibasis$ in the limit of a complete basis set $\Bas$.
%----------------------------------------------------------------
\subsubsection{Definition of a valence effective interaction}
%----------------------------------------------------------------
As most of the WFT calculations are done using a frozen core approximation, it is important to define an effective interaction within a general subset of molecular orbitals that we refer as $\Basval$.
According to \eqref{eq:def_weebasis} and \eqref{eq:expectweeb}, the effective interaction is defined by the expectation value of the coulomb operator over a wave function $\psibasis$. Therefore, to define an effective interaction accounting only for the valence electrons, one needs to define a function $\fbasisval$ satisfying
\begin{equation}
\label{eq:expectweebval}
\elemm{\psibasis}{\weeopbasisval}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
\mel*{\psibasis}{\weeopbasisval}{\psibasis} = \frac{1}{2}\,\,\iint \dr{1}\,\dr{2} \,\, \fbasisval,
\end{equation}
where $\weeopbasisval$ is the valence coulomb operator defined as
\begin{equation}