From 9997f27fc5778033dc0422c56fb4bd6cf7a24806 Mon Sep 17 00:00:00 2001 From: eginer Date: Mon, 16 Dec 2019 20:04:02 +0100 Subject: [PATCH] added SI --- Manuscript/SI/srDFT_SC-SI.tex | 411 +++++++++++++++++++++++++++++++++ new/C2_avdz/data/pouet | 24 +- new/O2_avtz/data/data_DFT_avtz | 2 +- new/O2_avtz/data/exact-O2 | 1 + 4 files changed, 436 insertions(+), 2 deletions(-) create mode 100644 Manuscript/SI/srDFT_SC-SI.tex diff --git a/Manuscript/SI/srDFT_SC-SI.tex b/Manuscript/SI/srDFT_SC-SI.tex new file mode 100644 index 0000000..8875914 --- /dev/null +++ b/Manuscript/SI/srDFT_SC-SI.tex @@ -0,0 +1,411 @@ + +\documentclass[aip,jcp,reprint,noshowkeys]{revtex4-1} + +%\documentclass[aip,jcp,noshowkeys]{revtex4-1} +\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,mhchem,longtable} + +\usepackage{mathpazo,libertine} +\usepackage[normalem]{ulem} +\newcommand{\alert}[1]{\textcolor{red}{#1}} +\definecolor{darkgreen}{RGB}{0, 180, 0} +\newcommand{\beurk}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\trash}[1]{\textcolor{red}{\sout{#1}}} +\usepackage{xspace} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=blue, + filecolor=blue, + urlcolor=blue, + citecolor=blue +} +\newcommand{\cdash}{\multicolumn{1}{c}{---}} +\newcommand{\mc}{\multicolumn} +\newcommand{\fnm}{\footnotemark} +\newcommand{\fnt}{\footnotetext} +\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}} +\newcommand{\mr}{\multirow} +\newcommand{\SI}{\textcolor{blue}{supporting information}} + +% second quantized operators +\newcommand{\psix}[1]{\hat{\Psi}\left({\bf X}_{#1}\right)} +\newcommand{\psixc}[1]{\hat{\Psi}^{\dagger}\left({\bf X}_{#1}\right)} +\newcommand{\ai}[1]{\hat{a}_{#1}} +\newcommand{\aic}[1]{\hat{a}^{\dagger}_{#1}} +\newcommand{\vijkl}[0]{V_{ij}^{kl}} +\newcommand{\phix}[2]{\phi_{#1}(\bfr{#2})} +\newcommand{\phixprim}[2]{\phi_{#1}(\bfr{#2}')} + +\newcommand{\CBS}{\text{CBS}} + + +%operators +\newcommand{\elemm}[3]{{\ensuremath{\bra{#1}{#2}\ket{#3}}\xspace}} +\newcommand{\ovrlp}[2]{{\ensuremath{\langle #1|#2\rangle}\xspace}} + +%\newcommand{\ket}[1]{{\ensuremath{|#1\rangle}\xspace}} +%\newcommand{\bra}[1]{{\ensuremath{\langle #1|}\xspace}} + +% + + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EPT}{E_\text{PT2}} +\newcommand{\EsCI}{E_\text{sCI}} +\newcommand{\EDMC}{E_\text{DMC}} +\newcommand{\EexFCI}{E_\text{exFCI}} +\newcommand{\EexFCIbasis}{E_\text{exFCI}^{\Bas}} +\newcommand{\EexFCIinfty}{E_\text{exFCI}^{\infty}} +\newcommand{\EexDMC}{E_\text{exDMC}} +\newcommand{\Ead}{\Delta E_\text{ad}} +\newcommand{\efci}[0]{E_{\text{FCI}}^{\Bas}} +\newcommand{\emodel}[0]{E_{\model}^{\Bas}} +\newcommand{\emodelcomplete}[0]{E_{\model}^{\infty}} +\newcommand{\efcicomplete}[0]{E_{\text{FCI}}^{\infty}} +\newcommand{\ecccomplete}[0]{E_{\text{CCSD(T)}}^{\infty}} +\newcommand{\ecc}[0]{E_{\text{CCSD(T)}}^{\Bas}} +\newcommand{\efuncbasisFCI}[0]{\bar{E}^\Bas[\denFCI]} +\newcommand{\efuncbasisfci}[0]{\bar{E}^\Bas[\denfci]} +\newcommand{\efuncbasis}[0]{\bar{E}^\Bas[\den]} +\newcommand{\efuncden}[1]{\bar{E}^\Bas[#1]} +\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{X}}^\Bas[#1]} +\newcommand{\ecompmodel}[0]{\bar{E}^\Bas[\denmodel]} +\newcommand{\ecmubis}[0]{\bar{E}_{\text{c,md}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmubisldapbe}[0]{\bar{E}_{\text{c,md}\,\text{PBE}}^{\text{sr}}[\denr;\,\mu]} +\newcommand{\ecmuapprox}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mu]} +\newcommand{\ecmuapproxmur}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\den;\,\mur]} +\newcommand{\ecmuapproxmurfci}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denfci;\,\mur]} +\newcommand{\ecmuapproxmurmodel}[0]{\bar{E}_{\text{c,md-}\mathcal{X}}^{\text{sr}}[\denmodel;\,\mur]} +\newcommand{\ecompmodellda}[0]{\bar{E}_{\text{LDA}}^{\Bas,\wf{}{\Bas}}[\denmodel]} +\newcommand{\ecompmodelldaval}[0]{\bar{E}_{\text{LDA, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbe}[0]{\bar{E}_{\text{PBE}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\ecompmodelpbeval}[0]{\bar{E}_{\text{PBE, val}}^{\Bas,\wf{}{\Bas}}[\den]} +\newcommand{\emulda}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldamodel}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denmodelr;\mu({\bf r};\wf{}{\Bas})\right)} +\newcommand{\emuldaval}[0]{\bar{\varepsilon}^{\text{sr},\text{unif}}_{\text{c,md}}\left(\denval ({\bf r});\murval;\wf{}{\Bas})\right)} +\newcommand{\ecmd}[0]{\varepsilon^{\text{c,md}}_{\text{PBE}}} +\newcommand{\psibasis}[0]{\Psi^{\basis}} +\newcommand{\BasFC}{\mathcal{A}} + +%pbeuegxiHF +\newcommand{\pbeuegxihf}{\text{PBE-UEG-}\zeta\text{-HF}^\Bas} +\newcommand{\argpbeuegxihf}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{HF}}^{\basis}} +\newcommand{\argrpbeuegxihf}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{HF}}^{\basis}(\br{})} +%pbeuegxiCAS +\newcommand{\pbeuegxi}{\text{PBE-UEG-}\zeta\text{-CAS}^\Bas} +\newcommand{\argpbeuegxicas}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegxicas}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeuegXiCAS +\newcommand{\pbeuegXi}{\text{PBE-UEG-}\tilde{\zeta}} +\newcommand{\argpbeuegXi}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeuegXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwo_{\text{UEG}}(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontxiCAS +\newcommand{\pbeontxi}{\text{PBE-ot-}\zeta} +\newcommand{\argpbeontxi}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontxi}[0]{\den(\br{}),\zeta(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeontXiCAS +\newcommand{\pbeontXi}{\text{PBE-ot-}\tilde{\zeta}} +\newcommand{\argpbeontXi}[0]{\den,\tilde{\zeta},s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontXi}[0]{\den(\br{}),\tilde{\zeta}(\br{}),s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} +%pbeont0xiCAS +\newcommand{\pbeontns}{\text{PBE-ot-}0\zeta} +\newcommand{\argpbeontns}[0]{\den,0,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argrpbeontns}[0]{\den(\br{}),0,s(\br{}),\ntwoextrapcas(\br{}),\mu_{\text{CAS}}^{\basis}(\br{})} + +%%%%%% arguments + +\newcommand{\argepbe}[0]{\den,\zeta,s} +\newcommand{\argebasis}[0]{\den,\zeta,s,\ntwo,\mu_{\Psi^{\basis}}} +\newcommand{\argecmd}[0]{\den,\zeta,s,\ntwo,\mu} +\newcommand{\argepbeueg}[0]{\den,\zeta,s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argepbeontxicas}[0]{\den,\zeta,s,\ntwoextrapcas,\mu_{\text{CAS}}^{\basis}} +\newcommand{\argepbeuegXihf}[0]{\den,\tilde{\zeta},s,\ntwo_{\text{UEG}},\mu_{\Psi^{\basis}}} +\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} +\newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} + + +% numbers +\newcommand{\rnum}[0]{{\rm I\!R}} +\newcommand{\bfr}[1]{{\bf r}_{#1}} +\newcommand{\dr}[1]{\text{d}\bfr{#1}} +\newcommand{\rr}[2]{\bfr{#1}, \bfr{#2}} +\newcommand{\rrrr}[4]{\bfr{#1}, \bfr{#2},\bfr{#3},\bfr{#4} } + + + +% effective interaction +\newcommand{\twodm}[4]{\elemm{\Psi}{\psixc{#4}\psixc{#3} \psix{#2}\psix{#1}}{\Psi}} +\newcommand{\murpsi}[0]{\mu({\bf r};\wf{}{A+B})} +\newcommand{\murpsia}[0]{\mu({\bf r};\wf{}{A})} +\newcommand{\murpsib}[0]{\mu({\bf r};\wf{}{B})} +\newcommand{\ntwo}[0]{n^{(2)}} +\newcommand{\ntwohf}[0]{n^{(2),\text{HF}}} +\newcommand{\ntwophi}[0]{n^{(2)}_{\phi}} +\newcommand{\ntwoextrap}[0]{\mathring{n}^{(2)}_{\psibasis}} +\newcommand{\ntwoextrapcas}[0]{\mathring{n}^{(2)\,\basis}_{\text{CAS}}} +\newcommand{\mur}[0]{\mu({\bf r})} +\newcommand{\murr}[1]{\mu({\bf r}_{#1})} +\newcommand{\murval}[0]{\mu_{\text{val}}({\bf r})} +\newcommand{\murpsival}[0]{\mu_{\text{val}}({\bf r};\wf{}{\Bas})} +\newcommand{\murrval}[1]{\mu_{\text{val}}({\bf r}_{#1})} +\newcommand{\weeopmu}[0]{\hat{W}_{\text{ee}}^{\text{lr},\mu}} + + +\newcommand{\wbasis}[0]{W_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\wbasiscoal}[0]{W_{\wf{}{\Bas}}(\bfr{},\bfr{})} +\newcommand{\wbasisval}[0]{W_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} +\newcommand{\fbasis}[0]{f_{\wf{}{\Bas}}(\bfr{1},\bfr{2})} +\newcommand{\fbasisval}[0]{f_{\wf{}{\Bas}}^{\text{val}}(\bfr{1},\bfr{2})} + \newcommand{\ontop}[2]{ n^{(2)}_{#1}({\bf #2}_1)} + \newcommand{\twodmrpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rrrr{1}{2}{2}{1})} + \newcommand{\twodmrdiagpsi}[0]{ \ntwo_{\wf{}{\Bas}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsitot}[0]{ \ntwo_{\wf{}{A+B}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsiaa}[0]{ \ntwo_{\wf{}{AA}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsibb}[0]{ \ntwo_{\wf{}{BB}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsiab}[0]{ \ntwo_{\wf{}{AB}}(\rr{1}{2})} + \newcommand{\twodmrdiagpsival}[0]{ \ntwo_{\wf{}{\Bas},\,\text{val}}(\rr{1}{2})} + \newcommand{\gammamnpq}[1]{\Gamma_{mn}^{pq}[#1]} + \newcommand{\gammamnkl}[0]{\Gamma_{mn}^{kl}} + \newcommand{\gammaklmn}[1]{\Gamma_{kl}^{mn}[#1]} +%\newcommand{\wbasiscoal}[1]{W_{\wf{}{\Bas}}({\bf r}_{#1})} +\newcommand{\ontoppsi}[1]{ n^{(2)}_{\wf{}{\Bas}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} +\newcommand{\wbasiscoalval}[1]{W_{\wf{}{\Bas}}^{\text{val}}({\bf r}_{#1})} +\newcommand{\ontoppsival}[1]{ n^{(2)}_{\wf{}{\Bas}}^{\text{val}}(\bfr{#1},\barr{#1},\barr{#1},\bfr{#1})} + + + +\newcommand{\ex}[4]{$^{#1}#2_{#3}^{#4}$} +\newcommand{\ra}{\rightarrow} +\newcommand{\De}{D_\text{e}} + +% MODEL +\newcommand{\model}[0]{\mathcal{Y}} + +% densities +\newcommand{\denmodel}[0]{\den_{\model}^\Bas} +\newcommand{\denmodelr}[0]{\den_{\model}^\Bas ({\bf r})} +\newcommand{\denfci}[0]{\den_{\psifci}} +\newcommand{\denFCI}[0]{\den^{\Bas}_{\text{FCI}}} +\newcommand{\denhf}[0]{\den_{\text{HF}}^\Bas} +\newcommand{\denrfci}[0]{\denr_{\psifci}} +\newcommand{\dencipsir}[0]{{n}_{\text{CIPSI}}^\Bas({\bf r})} +\newcommand{\dencipsi}[0]{{n}_{\text{CIPSI}}^\Bas} +\newcommand{\den}[0]{{n}} +\newcommand{\denval}[0]{{n}^{\text{val}}} +\newcommand{\denr}[0]{{n}({\bf r})} +\newcommand{\onedmval}[0]{\rho_{ij,\sigma}^{\text{val}}} + +% wave functions +\newcommand{\psifci}[0]{\Psi^{\Bas}_{\text{FCI}}} +\newcommand{\psimu}[0]{\Psi^{\mu}} +% operators +\newcommand{\weeopbasis}[0]{\hat{W}_{\text{ee}}^\Bas} +\newcommand{\kinop}[0]{\hat{T}} + +\newcommand{\weeopbasisval}[0]{\hat{W}_{\text{ee}}^{\Basval}} +\newcommand{\weeop}[0]{\hat{W}_{\text{ee}}} + + +% units +\newcommand{\IneV}[1]{#1 eV} +\newcommand{\InAU}[1]{#1 a.u.} +\newcommand{\InAA}[1]{#1 \AA} + + +% methods +\newcommand{\UEG}{\text{UEG}} +\newcommand{\LDA}{\text{LDA}} +\newcommand{\PBE}{\text{PBE}} +\newcommand{\FCI}{\text{FCI}} +\newcommand{\CCSDT}{\text{CCSD(T)}} +\newcommand{\lr}{\text{lr}} +\newcommand{\sr}{\text{sr}} + +\newcommand{\Nel}{N} +\newcommand{\V}[2]{V_{#1}^{#2}} + + +\newcommand{\n}[2]{n_{#1}^{#2}} +\newcommand{\E}[2]{E_{#1}^{#2}} +\newcommand{\bE}[2]{\Bar{E}_{#1}^{#2}} +\newcommand{\bEc}[1]{\Bar{E}_\text{c}^{#1}} +\newcommand{\e}[2]{\varepsilon_{#1}^{#2}} +\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}} +\newcommand{\bec}[1]{\Bar{e}^{#1}} +\newcommand{\wf}[2]{\Psi_{#1}^{#2}} +\newcommand{\W}[2]{W_{#1}^{#2}} +\newcommand{\w}[2]{w_{#1}^{#2}} +\newcommand{\hn}[2]{\Hat{n}_{#1}^{#2}} +\newcommand{\rsmu}[2]{\mu_{#1}^{#2}} +\newcommand{\SO}[2]{\phi_{#1}(\br{#2})} + +\newcommand{\modX}{\text{X}} +\newcommand{\modY}{\text{Y}} + +% basis sets +\newcommand{\setdenbasis}{\mathcal{N}_{\Bas}} +\newcommand{\Bas}{\mathcal{B}} +\newcommand{\basis}{\mathcal{B}} +\newcommand{\Basval}{\mathcal{B}_\text{val}} +\newcommand{\Val}{\mathcal{V}} +\newcommand{\Cor}{\mathcal{C}} + +% operators +\newcommand{\hT}{\Hat{T}} +\newcommand{\hWee}[1]{\Hat{W}_\text{ee}^{#1}} +\newcommand{\f}[2]{f_{#1}^{#2}} +\newcommand{\Gam}[2]{\Gamma_{#1}^{#2}} + +% coordinates +\newcommand{\br}[1]{{\mathbf{r}_{#1}}} +\newcommand{\bx}[1]{\mathbf{x}_{#1}} +\newcommand{\dbr}[1]{d\br{#1}} +\newcommand{\PBEspin}{PBEspin} +\newcommand{\PBEueg}{PBE-UEG-{$\tilde{\zeta}$}} + +\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France} +\newcommand{\LCT}{Laboratoire de Chimie Th\'eorique, Universit\'e Pierre et Marie Curie, Sorbonne Universit\'e, CNRS, Paris, France} + +\begin{document} + +\title{Mixing density functional theory and wave function theory for strong correlation: the best of both worlds} + +\begin{abstract} +bla bla bla youpi tralala +\end{abstract} + +\maketitle + +\section{Extensivity of the basis set correction} +The following paragraph proposes a demonstration of the size consistency of the basis set correction in the limit of dissociated fragments. +The present basis set correction being an integral in real space, in the limit of two dissociated fragments $A\ldots B$, one can split the contribution in +\begin{equation} + \begin{aligned} + \label{eq:def_ecmdpbebasis} + \efuncdenpbe{\argebasis} = &\int d\br{} \,\denr \\ & \ecmd(\argrebasis) + \end{aligned} +\end{equation} +\subsection{Property of the on-top pair density} +A crucial ingredient in the type of functionals used in the present paper together with the definition of the local-range separation parameter is the on-top pair density defined as +\begin{equation} + \ntwo_{\wf{}{}}(\br{}) = \sum_{pqrs} \SO{p}{} \SO{q}{} \Gam{pq}{rs} \SO{r}{} \SO{s}{}, +\end{equation} +with $\Gam{pq}{rs} = 2 \mel*{\wf{}{}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{q_\uparrow}\ai{p_\downarrow}}{\wf{}{\Bas}}$. +Assume now that the wave function $\wf{}{\Bas}$ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $A$ and $B$ +\begin{equation} + \ket{\wf{A+B}{}} = \ket{\wf{A}{}} \times \ket{\wf{B}{}}. +\end{equation} +Labelling the orbitals of fragment $A$ as $p_A,q_A,r_A,s_A$ and of fragment $B$ as $p_B,q_B,r_B,s_B$ and assuming that they don't overlap, one can split the two-body operator as +\begin{equation} + \begin{aligned} + \hat{\Gamma} = \hat{\Gamma}_{AA}{} + \hat{\Gamma}_{BB}{} + \hat{\Gamma}_{AB}{} + \end{aligned} +\end{equation} +with +\begin{equation} + \begin{aligned} + \hat{\Gamma}_{AA} = \sum_{p_A,q_A,r_A,s_A} \aic{r_{A,\downarrow}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ai{p_{A,\downarrow}}, + \end{aligned} +\end{equation} +\begin{equation} + \begin{aligned} + \hat{\Gamma}_{BB} = \sum_{p_B,q_B,r_B,s_B} \aic{r_{B,\downarrow}}\aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}}\ai{p_{B,\downarrow}}, + \end{aligned} +\end{equation} +and +\begin{equation} + \begin{aligned} + \hat{\Gamma}_{AB} = \sum_{p_A,q_B,r_A,s_B} \left( \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} + \aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}} \aic{r_{A,\downarrow}}\ai{p_{A,\downarrow}} \right) . + \end{aligned} +\end{equation} +Therefore, one can express the two-body density as +\begin{equation} + \twodmrdiagpsitot = \twodmrdiagpsiaa + \twodmrdiagpsibb + \twodmrdiagpsiab +\end{equation} +where +\begin{equation} + \begin{aligned} + & \twodmrdiagpsiaa = \\ & \sum_{p_A q_A r_A s_A} \SO{p_A}{1} \SO{q_A}{2} \bra{\wf{A}{}}\hat{\Gamma}_{AA}\ket{\wf{A}{}} \SO{r_A}{1} \SO{s_A}{2}, + \end{aligned} +\end{equation} +\begin{equation} + \begin{aligned} + & \twodmrdiagpsibb = \\ & \sum_{p_B q_B r_B s_B} \SO{p_B}{1} \SO{q_B}{2} \bra{\wf{B}{}}\hat{\Gamma}_{BB}\ket{\wf{B}{}} \SO{r_B}{1} \SO{s_B}{2}, + \end{aligned} +\end{equation} +and +\begin{equation} + \begin{aligned} + & \twodmrdiagpsiab = n_{A}(\br{1}) n_B(\br{2}) + n_{B}(\br{1}) n_A(\br{2}) + \end{aligned} +\end{equation} +where $n_{A}(\br{})$ and $n_{A}(\br{})$ are the one body density of the sub systems +\begin{equation} + \begin{aligned} + & n_{B}(\br{}) = \sum_{p_B r_B} \SO{p_B}{} \bra{\wf{B}{}}\aic{s_{B,\uparrow}}\ai{q_{B,\uparrow}}\ket{\wf{B}{}} \SO{r_B}{} + \end{aligned} +\end{equation} +\begin{equation} + \begin{aligned} + & n_{A}(\br{}) = \sum_{p_A r_A} \SO{p_A}{} \bra{\wf{A}{}}\aic{s_{A,\uparrow}}\ai{q_{A,\uparrow}}\ket{\wf{A}{}} \SO{r_A}{}. + \end{aligned} +\end{equation} +Based on these considerations, one can express the on-top pair density is simply +\begin{equation} + \begin{aligned} + \ntwo_{\wf{A+B}{}}(\br{}) = \ntwo_{\wf{A/A}{}}(\br{}) + \ntwo_{\wf{B/B}{}}(\br{}) + \end{aligned} +\end{equation} +with +\begin{equation} + \ntwo_{\wf{}{A/A}}(\br{}) = \sum_{p_A q_A r_A s_A} \SO{p_A}{} \SO{q_A}{} \Gam{p_A q_A}{r_A s_A} \SO{r_A}{} \SO{s_A}{} +\end{equation} +and +\begin{equation} + \ntwo_{\wf{}{B/B}}(\br{}) = \sum_{p_B q_B r_B s_B} \SO{p_B}{} \SO{q_B}{} \Gam{p_B q_B}{r_B s_B} \SO{r_B}{} \SO{s_B}{} +\end{equation} +being the on-top pair densities of the local fragments associated to $\ket{\wf{A}{}}$ and $\ket{\wf{B}{}}$. Therefore, provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity. +\subsection{Property of the local-range separation parameter} +The local range separation parameter depends on the on-top pair density at a given point $\br{}$ and on the numerator +\begin{equation} + \label{eq:def_f} + f_{\wf{A+B}{}}(\bfr{},\bfr{}) = \sum_{pqrstu\in \Bas} \SO{p}{ } \SO{q}{ } \V{pq}{rs} \Gam{rs}{tu} \SO{t}{ } \SO{u}{ }. +\end{equation} +As the summations run over all orbitals in the basis set $\Bas$, the quantity $f_{\wf{}{\Bas}}(\bfr{},\bfr{})$ is orbital invariant and therefore can be expressed in terms of localized orbitals. +In the limit of dissociated fragments, the coulomb interaction is vanishing between $A$ and $B$ and therefore any two-electron integral involving orbitals on both the system $A$ and $B$ vanishes. +Therefore, one can rewrite eq. \eqref{eq:def_f} as +\begin{equation} + \label{eq:def_fa+b} + f_{\wf{A+B}{}}(\bfr{},\bfr{}) = f_{\wf{AA}{}}(\bfr{},\bfr{}) + f_{\wf{BB}{}}(\bfr{},\bfr{}), +\end{equation} +with +\begin{equation} +\begin{aligned} + \label{eq:def_faa} + & f_{\wf{AA}{}}(\bfr{},\bfr{}) = \\ & \sum_{p_A q_A r_A s_A t_A u_A} \SO{p_A }{ } \SO{q_A}{ } \V{p_A q_A}{r_A s_A} \Gam{r_A s_A}{t_A u_A} \SO{t_A}{ } \SO{u_A}{ }, +\end{aligned} +\end{equation} + +\begin{equation} + \begin{aligned} + \label{eq:def_faa} + & f_{\wf{BB}{}}(\bfr{},\bfr{}) = \\ &\sum_{p_B q_B r_B s_B t_B u_B} \SO{p_B }{ } \SO{q_B}{ } \V{p_B q_B}{r_B s_B} \Gam{r_B s_B}{t_B u_B} \SO{t_B}{ } \SO{u_B}{ }. + \end{aligned} +\end{equation} +As a consequence, the local range-separation parameter in the super system $A+B$ with a multiplicative function is simply +\begin{equation} + \label{eq:def_mur} + \murpsi = \frac{\sqrt{\pi}}{2} \frac{f_{\wf{A+B}{}}(\bfr{},\bfr{})}{\ntwo_{\wf{A+B}{}}(\br{})} +\end{equation} +which is nothing but +\begin{equation} + \label{eq:def_mur} + \murpsi = \murpsia + \murpsib +\end{equation} +and therefore is an intensive quantity. The conclusion of this paragraph is that, provided that the wave function for the system $A+B$ is multiplicative in the limit of the dissociated fragments, all quantities used for the basis set correction are intensive and therefore the basis set correction is size consistent. + +\bibliography{../srDFT_SC} + +\end{document} diff --git a/new/C2_avdz/data/pouet b/new/C2_avdz/data/pouet index 55ebfe8..a928eaf 100644 --- a/new/C2_avdz/data/pouet +++ b/new/C2_avdz/data/pouet @@ -1 +1,23 @@ -1.00 1.25 1.50 1.75 2.25 2.50 4.50 5.50 6.00 6.50 +1.00 -71.3879730560 +1.25 -73.5380510155 +1.50 -74.7142790078 +1.75 -75.2833714343 +2.00 -75.5308796868 +2.10 -75.5784329315 +2.25 -75.6156406185 +2.3474 -75.6241944773 +2.5 -75.6212632036 +2.50 -75.6212632036 +2.63 -75.6084922721 +2.80 -75.5838967745 +3.00 -75.5502277197 +3.50 -75.4872252787 +4.00 -75.4459181176 +4.50 -75.4242926650 +5.00 -75.4039255031 +5.50 -75.4029433725 +6.00 -75.4028645487 +6.50 -75.4029826858 +7.00 -75.4031244020 +10.00 -75.4035442012 + diff --git a/new/O2_avtz/data/data_DFT_avtz b/new/O2_avtz/data/data_DFT_avtz index b10f7b0..58fc73b 100644 --- a/new/O2_avtz/data/data_DFT_avtz +++ b/new/O2_avtz/data/data_DFT_avtz @@ -7,4 +7,4 @@ 3.00 -150.06324132 -0.0451765580 -0.0521176567 -0.0521957399 4.00 -149.88767371 -0.0435573953 -0.0499577563 -0.0501613581 5.00 -149.88756789 -0.0432114786 -0.0495348356 -0.0497456437 - +10.0 -149.95830994 diff --git a/new/O2_avtz/data/exact-O2 b/new/O2_avtz/data/exact-O2 index f3d8293..935380c 100644 --- a/new/O2_avtz/data/exact-O2 +++ b/new/O2_avtz/data/exact-O2 @@ -6,3 +6,4 @@ 3.00 -150.2314 4.00 -150.1476 5.00 -150.1274 +10.0 -150.1214