2019-12-16 20:04:02 +01:00
\documentclass [aip,jcp,reprint,noshowkeys] { revtex4-1}
%\documentclass[aip,jcp,noshowkeys]{revtex4-1}
2019-12-16 20:36:45 +01:00
\usepackage { graphicx,dcolumn,bm,xcolor,microtype,multirow,amsmath,amssymb,amsfonts,physics,mhchem,xspace}
2019-12-16 20:04:02 +01:00
\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] }
2019-12-16 20:36:45 +01:00
\newcommand { \efuncdenpbe } [1]{ \bar { E} _ { \text { X} } ^ { A+B} [#1]}
2019-12-16 20:04:02 +01:00
\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}
2019-12-16 20:36:45 +01:00
\newcommand { \argebasis } [0]{ \den ,\zeta ,s,\ntwo ,\mu _ { \Psi ^ { A+B} } }
2019-12-16 20:04:02 +01:00
\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 } } }
2019-12-16 20:36:45 +01:00
\newcommand { \argrebasis } [0]{ \denr ,\zeta (\br { } ),s,\ntwo (\br { } ),\mu _ { \Psi ^ { A+B} } (\br { } )}
2019-12-16 20:04:02 +01:00
\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 } }
2019-12-18 14:16:14 +01:00
\newcommand { \murpsibas } [0]{ \mu _ { \wf { } { \Bas } } ({ \bf r} )}
2019-12-16 20:04:02 +01:00
\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} )}
2019-12-16 21:51:47 +01:00
\newcommand { \twodmrdiagpsiaad } [0]{ \ntwo _ { \wf { } { AA} } (\rr { } { } )}
2019-12-16 20:04:02 +01:00
\newcommand { \twodmrdiagpsibb } [0]{ \ntwo _ { \wf { } { BB} } (\rr { 1} { 2} )}
2019-12-16 21:51:47 +01:00
\newcommand { \twodmrdiagpsibbd } [0]{ \ntwo _ { \wf { } { BB} } (\rr { } { } )}
2019-12-16 20:04:02 +01:00
\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}
2020-01-05 12:47:54 +01:00
\title { A density-based basis-set correction for weak and strong correlation}
2019-12-16 20:04:02 +01:00
\begin { abstract}
2020-01-05 12:47:54 +01:00
2019-12-16 20:04:02 +01:00
\end { abstract}
\maketitle
\section { Extensivity of the basis set correction}
2019-12-16 20:36:45 +01:00
\subsection { General considerations}
2019-12-16 20:04:02 +01:00
The following paragraph proposes a demonstration of the size consistency of the basis set correction in the limit of dissociated fragments.
2019-12-16 20:36:45 +01:00
The present basis set correction being an integral in real space,
2019-12-16 20:04:02 +01:00
\begin { equation}
2019-12-16 20:36:45 +01:00
\label { eq:def_ ecmdpbebasis}
2019-12-16 20:04:02 +01:00
\begin { aligned}
2019-12-16 20:36:45 +01:00
& \efuncdenpbe { \argebasis } = \\ & \int \text { d} \br { } \, \denr \ecmd (\argrebasis ),
\end { aligned}
\end { equation}
where all the quantities $ \argrebasis $ are obtained from the same wave function $ \Psi ^ { A + B } $ .
Such an integral can be rewritten as the sum of the contribution on $ A $ and $ B $
\begin { equation}
2019-12-16 20:04:02 +01:00
\label { eq:def_ ecmdpbebasis}
2019-12-16 20:36:45 +01:00
\begin { aligned}
& \efuncdenpbe { \argebasis } = \\ & \int _ { \br { } \in A} \text { d} \br { } \, \denr \ecmd (\argrebasis ) \\ & + \int _ { \br { } \in B} \text { d} \br { } \, \denr \ecmd (\argrebasis ),
2019-12-16 20:04:02 +01:00
\end { aligned}
\end { equation}
2019-12-16 21:51:47 +01:00
Therefore, a sufficient condition to obtain size extensivity in the limit of dissociated fragments is that all arguments entering in the function $ \ecmd ( \argrebasis ) $ are \textit { intensive} , which means that they \textit { locally} coincide in the system $ A $ and in the sub system $ A $ of the super system $ A + B $ .
2019-12-16 20:36:45 +01:00
Regarding the density and its gradients, these are necessary intensive quantities. The remaining questions are therefore the local range-separation parameter $ \murpsi $ and the on-top pair density.
2019-12-16 20:04:02 +01:00
\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}
2020-01-05 12:47:54 +01:00
\label { eq:def_ n2}
2019-12-16 20:04:02 +01:00
\ntwo _ { \wf { } { } } (\br { } ) = \sum _ { pqrs} \SO { p} { } \SO { q} { } \Gam { pq} { rs} \SO { r} { } \SO { s} { } ,
\end { equation}
2019-12-16 21:51:47 +01:00
with $ \Gam { pq } { rs } = 2 \mel * { \wf { } { } } { \aic { r _ \downarrow } \aic { s _ \uparrow } \ai { q _ \uparrow } \ai { p _ \downarrow } } { \wf { } { } } $ .
Assume now that the wave function $ \wf { A + B } { } $ of the super system $ A + B $ can be written as a product of two wave functions defined on two non-overlapping and non-interacting fragments $ A $ and $ B $
2019-12-16 20:04:02 +01:00
\begin { equation}
\ket { \wf { A+B} { } } = \ket { \wf { A} { } } \times \ket { \wf { B} { } } .
\end { equation}
2019-12-16 21:51:47 +01:00
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 density operator as
2019-12-16 20:04:02 +01:00
\begin { equation}
\begin { aligned}
2019-12-16 21:51:47 +01:00
\hat { \Gamma } (\br { 1} ,\br { 2} ) = \hat { \Gamma } _ { AA} { } (\br { 1} ,\br { 2} ) + \hat { \Gamma } _ { BB} { } (\br { 1} ,\br { 2} ) + \hat { \Gamma } _ { AB} { } (\br { 1} ,\br { 2} )
2019-12-16 20:04:02 +01:00
\end { aligned}
\end { equation}
with
\begin { equation}
\begin { aligned}
2019-12-16 21:51:47 +01:00
\hat { \Gamma } _ { AA} (\br { 1} ,\br { 2} ) = \sum _ { p_ A,q_ A,r_ A,s_ A} & \SO { r_ A} { 1} \SO { s_ A} { 2} \SO { p_ A} { 1} \SO { q_ A} { 2} \\ & \aic { r_ { A,\downarrow } } \aic { s_ { A,\uparrow } } \ai { q_ { A,\uparrow } } \ai { p_ { A,\downarrow } } ,
2019-12-16 20:04:02 +01:00
\end { aligned}
\end { equation}
2019-12-16 20:36:45 +01:00
(and equivalently for $ B $ ),
%\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}
2019-12-16 20:04:02 +01:00
and
\begin { equation}
\begin { aligned}
2019-12-16 21:51:47 +01:00
\hat { \Gamma } _ { AB} = \sum _ { p_ A,q_ B,r_ A,s_ B} & \SO { r_ A} { 1} \SO { s_ B} { 2} \SO { p_ A} { 1} \SO { q_ B} { 2} \\ & \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 ) .
2019-12-16 20:04:02 +01:00
\end { aligned}
\end { equation}
Therefore, one can express the two-body density as
\begin { equation}
\twodmrdiagpsitot = \twodmrdiagpsiaa + \twodmrdiagpsibb + \twodmrdiagpsiab
\end { equation}
2019-12-16 20:36:45 +01:00
where $ \twodmrdiagpsiaa $ and $ \twodmrdiagpsibb $ are the two-body densities of the isolated fragments
2019-12-16 20:04:02 +01:00
\begin { equation}
\begin { aligned}
2019-12-16 21:51:47 +01:00
& \twodmrdiagpsiaa = \bra { \wf { A} { } } \hat { \Gamma } _ { AA} (\br { 1} ,\br { 2} ) \ket { \wf { A} { } }
2019-12-16 20:04:02 +01:00
\end { aligned}
\end { equation}
2019-12-16 20:36:45 +01:00
(and equivalently for $ B $ ),
and $ \twodmrdiagpsiab $ is simply the product of the one body densities of the sub systems
2019-12-16 20:04:02 +01:00
\begin { equation}
\begin { aligned}
2019-12-16 20:36:45 +01:00
& \twodmrdiagpsiab = n_ { A} (\br { 1} ) n_ B(\br { 2} ) + n_ { B} (\br { 1} ) n_ A(\br { 2} ),
2019-12-16 20:04:02 +01:00
\end { aligned}
\end { equation}
\begin { equation}
\begin { aligned}
2019-12-16 21:51:47 +01:00
& 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} { } ,
2019-12-16 20:04:02 +01:00
\end { aligned}
\end { equation}
2019-12-16 21:51:47 +01:00
(and equivalently for $ B $ ).
As the densities of $ A $ and $ B $ are by definition non overlapping, one can express the on-top pair density as the sum of the on-top pair densities of the isolated systems
2019-12-16 20:04:02 +01:00
\begin { equation}
2019-12-16 20:36:45 +01:00
\begin { aligned}
2019-12-16 21:51:47 +01:00
\ntwo _ { \wf { A+B} { } } (\br { } ) = \twodmrdiagpsiaad + \twodmrdiagpsibbd
2019-12-16 20:36:45 +01:00
\end { aligned}
2019-12-16 20:04:02 +01:00
\end { equation}
2019-12-16 21:51:47 +01:00
As $ \ntwo _ { \wf { } { A / A } } ( \br { } ) = 0 \text { if } \br { } \in B $ (and equivalently for $ \ntwo _ { \wf { } { B / B } } ( \br { } ) $ on $ A $ ), one can conclude that provided that the wave function is multiplicative, the on-top pair density is a local intensive quantity.
2019-12-16 20:04:02 +01:00
\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}
2020-01-05 12:47:54 +01:00
f_ { \wf { } { } } (\bfr { } ,\bfr { } ) = \sum _ { pqrstu\in \Bas } \SO { p} { } \SO { q} { } \V { pq} { rs} \Gam { rs} { tu} \SO { t} { } \SO { u} { } .
2019-12-16 20:04:02 +01:00
\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}
2019-12-16 20:36:45 +01:00
(and equivalently for $ B $ ).
%\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}
2019-12-16 21:51:47 +01:00
As a consequence, the local range-separation parameter in the super system $ A + B $
2019-12-16 20:04:02 +01:00
\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}
2019-12-16 21:51:47 +01:00
which, in the case of a multiplicative wave function is nothing but
2019-12-16 20:04:02 +01:00
\begin { equation}
\label { eq:def_ mur}
2019-12-16 21:51:47 +01:00
\murpsi = \murpsia + \murpsib .
2019-12-16 20:04:02 +01:00
\end { equation}
2019-12-16 21:51:47 +01:00
As $ \murpsia = 0 \text { if } \br { } \in B $ (and equivalently for $ \murpsib $ on $ B $ ), $ \murpsi $ 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.
\section { Computational considerations}
2020-01-04 15:44:41 +01:00
The computational cost of the present approach is driven by two quantities: the computation of the on-top pair density and the $ \murpsibas $ on the real-space grid. Within a blind approach, for each grid point the computational cost is of order $ n _ { \Bas } ^ 4 $ and $ n _ { \Bas } ^ 6 $ for the on-top pair density $ \ntwo _ { \wf { \Bas } { } } ( \br { } ) $ and the local range separation parameter $ \murpsibas $ , respectively.
2020-01-05 12:47:54 +01:00
Nevertheless, using CASSCF wave functions to compute these quantities leads to significant simplifications which can substantially reduce the CPU time.
2020-01-04 15:44:41 +01:00
\subsection { Computation of the on-top pair density for a CASSCF wave function}
Given a generic wave function developed on a basis set with $ n _ { \Bas } $ basis functions, the evaluation of the on-top pair density is of order $ \left ( n _ { \Bas } \right ) ^ 4 $ .
Nevertheless, assuming that the wave function $ \Psi ^ { \Bas } $ is of CASSCF type, a lot of simplifications happen.
2020-01-05 12:47:54 +01:00
If the active space is referred as the set of spatial orbitals $ \mathcal { A } $ which are labelled by the indices $ t,u,v,w $ , and the doubly occupied orbitals are the set of spatial orbitals $ \mathcal { C } $ labeled by the indices $ i,j $ , one can write the on-top pair density of a CASSCF wave function as
2020-01-04 15:44:41 +01:00
\begin { equation}
\label { def_ n2_ good}
\ntwo _ { \wf { \Bas } { } } (\br { } ) = \ntwo _ { \mathcal { A} } (\br { } ) + n_ { \mathcal { C} } (\br { } ) n_ { \mathcal { A} } (\br { } ) + \left ( n_ { \mathcal { C} } (\br { } )\right )^ 2
\end { equation}
where
\begin { equation}
\label { def_ n2_ act}
2020-01-05 12:47:54 +01:00
\ntwo _ { \mathcal { A} } (\br { } ) = \sum _ { t,u,v,w \, \in \mathcal { A} } 2 \mel * { \wf { } { \Bas } } { \aic { t_ \downarrow } \aic { u_ \uparrow } \ai { v_ \uparrow } \ai { w_ \downarrow } } { \wf { } { \Bas } } \phi _ t (\br { } ) \phi _ u (\br { } ) \phi _ v (\br { } ) \phi _ w (\br { } )
2020-01-04 15:44:41 +01:00
\end { equation}
is the purely active part of the on-top pair density,
\begin { equation}
n_ { \mathcal { C} } (\br { } ) = \sum _ { i\, \in \mathcal { C} } \left (\phi _ i (\br { } ) \right )^ 2,
\end { equation}
and
\begin { equation}
2020-01-05 12:47:54 +01:00
n_ { \mathcal { A} } (\br { } ) = \sum _ { t,u\, \in \mathcal { A} } \phi _ t (\br { } ) \phi _ u (\br { } )
\mel * { \wf { } { \Bas } } { \aic { t_ \downarrow } \ai { u_ \downarrow } + \aic { t_ \uparrow } \ai { u_ \uparrow } } { \wf { } { \Bas } }
2020-01-04 15:44:41 +01:00
\end { equation}
is the purely active one-body density.
2020-01-05 12:47:54 +01:00
Written as in eq. \eqref { def_ n2_ good} , the leading computational cost is the evaluation of $ \ntwo _ { \mathcal { A } } ( \br { } ) $ which, according to eq. \eqref { def_ n2_ act} , scales as $ \left ( n _ { \mathcal { A } } \right ) ^ 4 $ where $ n _ { \mathcal { A } } $ is the number of active orbitals which is much smaller than the number of basis functions $ n _ { \Bas } $ . Therefore, the final computational scaling of the on-top pair density for a CASSCF wave function over the whole real-space grid is of $ \left ( n _ { \mathcal { A } } \right ) ^ 4 n _ G $ , where $ n _ G $ is the number of grid points.
2020-01-04 15:44:41 +01:00
\subsection { Computation of $ \murpsibas $ }
2020-01-05 12:47:54 +01:00
At a given grid point, the computation of $ \murpsibas $ needs the computation of $ f _ { \wf { } { } } ( \bfr { } , \bfr { } ) $ defined in eq. \eqref { eq:def_ f} and the on-top pair density defined in eq. \eqref { eq:def_ n2} . In the previous paragraph we gave an explicit form of the on-top pair density in the case of a CASSCF wave function with a computational scaling of $ \left ( n _ { \mathcal { A } } \right ) ^ 4 $ . In the present paragraph we focus on simplifications that one can obtain for the computation of $ f _ { \wf { } { } } ( \bfr { } , \bfr { } ) $ in the case of a CASSCF wave function.
One can rewrite $ f _ { \wf { } { } } ( \bfr { } , \bfr { } ) $ as
\begin { equation}
\label { eq:f_ good}
f_ { \wf { } { } } (\bfr { } ,\bfr { } ) = \sum _ { r,s \in \Bas } \mathcal { V} _ r^ s(\bfr { } ) \, \mathcal { N} _ { r} ^ s(\bfr { } )
\end { equation}
where
\begin { equation}
\mathcal { V} _ r^ s(\bfr { } ) = \sum _ { p,q \in \Bas } V_ { pq} ^ { rs} \phi _ p(\br { } ) \phi _ q(\br { } )
\end { equation}
and
\begin { equation}
\mathcal { N} _ { r} ^ s(\bfr { } ) = \sum _ { p,q \in \Bas } \Gam { pq} { rs} \phi _ p(\br { } ) \phi _ q(\br { } ) .
\end { equation}
\textit { A priori} , for a given grid point, the computational scaling of eq. \eqref { eq:f_ good} is of $ \left ( n _ { \Bas } \right ) ^ 4 $ and the total computational cost over the whole grid scales therefore as $ \left ( n _ { \Bas } \right ) ^ 4 n _ G $ .
In the case of a CASSCF wave function, it is interesting to notice that $ \Gam { pq } { rs } $ vanishes if one index $ p,q,r,s $ does not belong
to the set of of doubly occupied or active orbitals $ \mathcal { C } \cup \mathcal { A } $ . Therefore, at a given grid point, the matrix $ \mathcal { N } _ { r } ^ s ( \bfr { } ) $ has only at most $ \left ( n _ { \mathcal { A } } + n _ { \mathcal { C } } \right ) ^ 2 $ non-zero elements, which is usually much smaller than $ \left ( n _ { \Bas } \right ) ^ 2 $ .
As a consequence, in the case of a CASSCF wave function one can rewrite $ f _ { \wf { } { } } ( \bfr { } , \bfr { } ) $ as
\begin { equation}
f_ { \wf { } { } } (\bfr { } ,\bfr { } ) = \sum _ { r,s \in \mathcal { C} \cup \mathcal { A} } \mathcal { V} _ r^ s(\bfr { } ) \, \mathcal { N} _ { r} ^ s(\bfr { } ).
\end { equation}
Therefore the final computational cost of $ f _ { \wf { } { } } ( \bfr { } , \bfr { } ) $ is dominated by that of $ \mathcal { V } _ r ^ s ( \bfr { } ) $ , which scales as $ \left ( n _ { \mathcal { A } } + n _ { \mathcal { C } } \right ) ^ 2 \left ( n _ { \Bas } \right ) ^ 2 n _ G $ , which is much weaker than $ \left ( n _ { \Bas } \right ) ^ 4 n _ G $ .
2019-12-16 20:04:02 +01:00
\bibliography { ../srDFT_ SC}
\end { document}
