From 8297f46b6896917bc56eb37d7db3d9288bb2ee0b Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 12 Apr 2019 10:28:00 +0200 Subject: [PATCH] valence OK --- Manuscript/G2-srDFT.tex | 35 ++++++++++++++++++++++++----------- 1 file changed, 24 insertions(+), 11 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index e6d21df..d023cdc 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -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)