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Pierre-Francois Loos 2019-04-12 10:28:00 +02:00
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Manuscript/G2-srDFT.tex
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@ -87,8 +87,7 @@
% basis sets
\newcommand{\Bas}{\mathcal{B}}
\newcommand{\Basval}{\mathcal{B}_\text{val}}
\newcommand{\Val}{\mathcal{V}}
\newcommand{\Val}{\text{val}}
\newcommand{\Cor}{\mathcal{C}}
% operators
@ -281,6 +280,10 @@ for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $
%An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.
%As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems.
%=================================================================
%\subsection{Range-separation function}
%=================================================================
\alert{Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators.}
Although this choice is not unique, we choose here the range-separation function
\begin{equation}
@ -295,8 +298,11 @@ such that the long-range interaction
What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?}
coincides with the effective interaction at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
%=================================================================
%\subsection{Short-range correlation functionals}
%=================================================================
Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals to approximate $\bE{}{\Bas}[\n{}{}]$.
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we consider here a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
\begin{multline}
\label{eq:ec_md_mu}
\bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
@ -319,19 +325,20 @@ The ECMD functionals admit, for any density $\n{}{}(\br{})$, the two following l
\begin{subequations}
\begin{align}
\label{eq:large_mu_ecmd}
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = 0,
\lim_{\mu \to \infty} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = 0,
\\
\label{eq:small_mu_ecmd}
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] & = \Ec[\n{}{}(\br{})],
\lim_{\mu \to 0} \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] & = \Ec[\n{}{}(\br{})],
\end{align}
\end{subequations}
where $\Ec[\n{}{}(\br{})]$ is the usual universal correlation functional defined in KS-DFT.
The choice of the ECMD as the functionals to be used in this scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [see equation \eqref{eq:E_funcbasis}] and that of the ECMD functionals [see equation \eqref{eq:ec_md_mu}].
The choice of ECMD in the present scheme is motivated by the analogy between the definition of $\bE{}{\Bas}[\n{}{}]$ [Eq.~\eqref{eq:E_funcbasis}] and that of the ECMD functionals [Eq.~\eqref{eq:ec_md_mu}].
Indeed, provided that $\w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) \approx \W{\Bas}{}(\br{1},\br{2})$, then the wave function $\wf{}{\rsmu{\Bas}{}}$ coincides with $\wf{}{\Bas}$.
%The ECMD 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.
%This makes them particularly well adapted to the present context where one aims at correcting a general WFT method.
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$. Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
Following Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ by the ECMD functionals evaluated with the range separation function $\rsmu{\Bas}{}(\br{})$.
Therefore, we define the LDA version of $\bE{}{\Bas}[\n{}{}]$ as
\begin{equation}
\label{eq:def_lda_tot}
\bE{\LDA}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\LDA}{\sr}\big(\n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{},
@ -358,10 +365,13 @@ Therefore, the PBE complementary functional reads
\end{equation}
Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
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 naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
%=================================================================
%\subsection{Valence approximation}
%=================================================================
We therefore define the valence-only 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.
%We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
Therefore, we define the valence-only effective interaction
\begin{equation}
\W{\Bas}{\Val}(\br{1},\br{2}) =
\begin{cases}
@ -381,7 +391,7 @@ with
= \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{gather}
\end{subequations}
and the corresponding valence range separation function
and the corresponding valence range-separation function
\begin{equation}
\label{eq:muval}
\rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}).
@ -390,6 +400,9 @@ It is worth noting that, within the present definition, $\W{\Bas}{\Val}(\br{1},\
Defining $\n{\modZ}{\Val}$ as the valence one-electron density obtained with the model $\modZ$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modZ}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
%=================================================================
%\subsection{Computational considerations}
%=================================================================
Regarding now the main computational source of the present approach, it consists in the evaluation
of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point.
All through this paper, we use pair density matrix of a single Slater determinant (typically HF)