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Pierre-Francois Loos 2019-05-29 10:19:32 +02:00
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Manuscript/Ex-srDFT.tex
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@ -224,8 +224,8 @@ is the basis-dependent complementary density functional,
& &
\hWee{} & = \sum_{i<j}^{\Ne} r_{ij}^{-1}, \hWee{} & = \sum_{i<j}^{\Ne} r_{ij}^{-1},
\end{align} \end{align}
are the kinetic and interelectronic repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set, respectively. are the kinetic and electron-electron repulsion operators, respectively, and $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron normalized wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis set, respectively.
The notation $\wf{}{} \rightsquigarrow \n{}{}$ states that $\wf{}{}$ yield the density $\n{}{}$. The notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E_funcbasis} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
Hence, the CBS excitation energy associated with the $k$th excited state reads Hence, the CBS excitation energy associated with the $k$th excited state reads
\begin{equation} \begin{equation}
@ -248,6 +248,7 @@ An important property of the present correction is
\lim_{\Bas \to \CBS} \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = 0. \lim_{\Bas \to \CBS} \DbE{}{\Bas}[\n{0}{\Bas},\n{k}{\Bas}] = 0.
\end{equation} \end{equation}
In other words, the correction vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit. \cite{LooPraSceTouGin-JPCL-19} In other words, the correction vanishes in the CBS limit, hence guaranteeing an unaltered CBS limit. \cite{LooPraSceTouGin-JPCL-19}
%In the following, we will drop the state index $k$ and focus on the quantity $\bE{}{\Bas}[\n{}{}]$.
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Range-separation function} \subsection{Range-separation function}
@ -265,8 +266,9 @@ The ECMD functionals admit, for any $\n{}{}$, the following two limits
where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65} where $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65}
The key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- automatically adapts to the spatial non-homogeneity of the basis set incompleteness error. The key ingredient --- the range-separated function $\rsmu{}{\Bas}(\br{})$ --- automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$, with \cite{GinPraFerAssSavTou-JCP-18} It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{12}) = \erf( \mu r_{12})/r_{12}$, coincides, at coalescence, with an effective two-electron interaction $\W{}{\Bas}(\br{1},\br{2})$ ``mimicking'' the Coulomb operator in an incomplete basis $\Bas$, i.e.~$\w{}{\lr,\rsmu{}{\Bas}(\br{})}(0) = \W{}{\Bas}(\br{},\br{})$ at any $\br{}$. \cite{GinPraFerAssSavTou-JCP-18}
\begin{gather} The explicit expression of $\W{}{\Bas}(\br{1},\br{2})$ is given by
\begin{equation}
\label{eq:def_weebasis} \label{eq:def_weebasis}
\W{}{\Bas}(\br{1},\br{2}) = \W{}{\Bas}(\br{1},\br{2}) =
\begin{cases} \begin{cases}
@ -274,11 +276,13 @@ It is defined such that the long-range interaction of RS-DFT, $\w{}{\lr,\mu}(r_{
\\ \\
\infty, & \text{otherwise,} \infty, & \text{otherwise,}
\end{cases} \end{cases}
\\ \end{equation}
where
\begin{equation}
\label{eq:n2basis} \label{eq:n2basis}
\n{2}{\Bas}(\br{1},\br{2}) \n{2}{\Bas}(\br{1},\br{2})
= \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, = \sum_{pqrs \in \Bas} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
\end{gather} \end{equation}
and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO), and $\Gam{pq}{rs} = 2 \mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$ are the opposite-spin pair density associated with $\wf{}{\Bas}$ and its corresponding tensor, respectively, $\SO{p}{}$ is a (real-valued) molecular orbital (MO),
\begin{equation} \begin{equation}
\label{eq:fbasis} \label{eq:fbasis}
@ -307,7 +311,7 @@ To go beyond the LDA and cure its over correlation at small $\mu$, some of the a
\int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{}, \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation} \end{equation}
where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient. where $s = \abs{\nabla \n{}{}}/\n{}{4/3}$ is the reduced density gradient.
$\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{}=0$ and the exact large-$\rsmu{}{}$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06} $\be{\text{c,md}}{\sr,\PBE}\qty(\n{}{},s,\zeta,\rsmu{}{})$ interpolates between the usual PBE correlation functional, \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$, at $\rsmu{}{} = 0$ (DFT limit) and the exact large-$\rsmu{}{}$ behavior (WFT limit). \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
It reads It reads
\begin{subequations} \begin{subequations}
\begin{gather} \begin{gather}
@ -328,7 +332,7 @@ This computationally-lighter functional will be refered to as PBE.
\label{sec:compdetails} \label{sec:compdetails}
%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%
In the present study, we compute the ground- and excited-state energies, one-electron and on-top densities with a selected CI method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15} In the present study, we compute the ground- and excited-state energies, one-electron and on-top densities with a selected CI method known as CIPSI (Configuration Interaction using a Perturbative Selection made Iteratively). \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI energies. \cite{QP2} The total energy of each state is obtained via an efficient extrapolation procedure of the sCI energies designed to reach near-FCI accuracy. \cite{QP2}
These energies will be labeled exFCI in the following. These energies will be labeled exFCI in the following.
We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19, QP2} for more details. We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19, QP2} for more details.
The one-electron and on-top densities are computed from a very large CIPSI expansion containing several million determinants. The one-electron and on-top densities are computed from a very large CIPSI expansion containing several million determinants.