diff --git a/Manuscript/SI/srDFT_SC-SI.tex b/Manuscript/SI/srDFT_SC-SI.tex index a3d9b95..a4d0d87 100644 --- a/Manuscript/SI/srDFT_SC-SI.tex +++ b/Manuscript/SI/srDFT_SC-SI.tex @@ -70,7 +70,8 @@ \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}}^{A+B}[#1]} +\newcommand{\efuncdenpbe}[1]{\bar{E}_{\text{}}^\Bas[#1]} +\newcommand{\efuncdenpbeAB}[1]{\bar{E}_{\text{A}+\text{B}}^\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]} @@ -85,7 +86,7 @@ \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{\ecmd}[0]{\bar{\varepsilon}_{\text{c,md}}^{\text{sr},\text{PBE}}} \newcommand{\psibasis}[0]{\Psi^{\basis}} \newcommand{\BasFC}{\mathcal{A}} @@ -117,12 +118,12 @@ %%%%%% arguments \newcommand{\argepbe}[0]{\den,\zeta,s} -\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu_{\Psi^{A+B}}} +\newcommand{\argebasis}[0]{\den,\zeta,\ntwo,\mu} \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(\br{}),\ntwo(\br{}),\mu_{\Psi^{A+B}}(\br{})} +\newcommand{\argrebasis}[0]{\denr,\zeta(\br{}),s(\br{}),\ntwo(\br{}),\mu(\br{})} \newcommand{\argrebasisab}[0]{\denr,\zeta(\br{}),s,\ntwo(\br{}),\mu_{\Psi^{\basis}}(\br{})} @@ -282,25 +283,63 @@ \maketitle + \section{Size consistency of the basis-set correction} + \subsection{General considerations} -The following paragraph proposes a proof 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, + +The following paragraph gives a proof of the size consistency of the basis-set correction. The basis-set correction is expressed as an integral in real space \begin{equation} \label{eq:def_ecmdpbebasis} \begin{aligned} & \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 $B$ +where all the local quantities $\argrebasis$ are obtained from the same wave function $\Psi$. In the limit of two dissociated fragments $\text{A}+\text{B}$, this integral can be rewritten as the sum of the integral over the region $\Omega_\text{A}$ and the integral over the region $\Omega_\text{B}$ \begin{equation} \label{eq:def_ecmdpbebasis} \begin{aligned} - & \efuncdenpbe{\argebasis} = \\ & \int_{ \br{} \in A} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{ \br{} \in B} \text{d}\br{} \,\denr \ecmd(\argrebasis), + & \efuncdenpbeAB{\argebasis} = \\ & \int_{\Omega_\text{A}} \text{d}\br{} \,\denr \ecmd(\argrebasis) \\ & + \int_{\Omega_\text{B}} \text{d}\br{} \,\denr \ecmd(\argrebasis). \end{aligned} \end{equation} -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 $A+B$. +Therefore, a sufficient condition to obtain size consistency is that all the local quantities $\argrebasis$ are \textit{intensive}, i.e. that they \textit{locally} coincide in the supersystem $\text{A}+\text{B}$ and in each isolated fragment $\text{X}=\text{A}$ or $\text{B}$. Hence, for $\br{} \in \Omega_\text{X}$, we should have +\begin{subequations} +\begin{equation} +n_\text{A+B}(\br{}) = n_\text{X}(\br{}), +\label{nAB} +\end{equation} +\begin{equation} +\zeta_\text{A+B}(\br{}) = \zeta_\text{X}(\br{}), +\label{zAB} +\end{equation} +\begin{equation} +s_\text{A+B}(\br{}) = s_\text{X}(\br{}), +\label{sAB} +\end{equation} +\begin{equation} +n_{2,\text{A+B}}(\br{}) = n_{2,\text{X}}(\br{}), +\label{n2AB} +\end{equation} +\begin{equation} +\mu_{\text{A+B}}(\br{}) = \mu_{\text{X}}(\br{}), +\label{muAB} +\end{equation} +\end{subequations} +where the left-hand-side quantities are for the supersystem and the right-hand-side quantities for an isolated fragment. Such conditions can be difficult to fulfil in the presence of degeneracies since the system X can be in a different mixed state (i.e. ensemble) in the supersystem $\text{A}+\text{B}$ and in the isolated fragment~\cite{Sav-CP-09}. Here, we will consider the simple case where the wave function of the supersystem is multiplicatively separable, i.e. +\begin{equation} + \ket{\wf{\text{A}+\text{B}}{}} = \ket{\wf{\text{A}}{}} \otimes \ket{\wf{\text{B}}{}}, +\end{equation} +where $\otimes$ is the antisymmetric tensor product. In this case, it is well known that Eqs.~(\ref{nAB})-(\ref{sAB}) are valid and it remains to show that Eqs.~(\ref{n2AB}) and~(\ref{muAB}) are also valid. + +For spatial degeneracies such as different p states, by selected the same member of the ensemble in the supersystem and the isolated fragements. + +applies to the systems treated in this work. + +Even though this does deal with the case of degeneracies, + +To avoid difficulties arising from spin-multiplet degeneracy, we can use the effective spin polarization $\tilde{\zeta}(n(\br{}),n_{2}(\br{}))$ depending only on $n(\br{})$ and $n_{2}(\br{})$ or a zero spin polarization $\zeta(\br{}) = 0$. + + 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. diff --git a/Manuscript/srDFT_SC.tex b/Manuscript/srDFT_SC.tex index d32f6fe..fc15b1f 100644 --- a/Manuscript/srDFT_SC.tex +++ b/Manuscript/srDFT_SC.tex @@ -565,8 +565,11 @@ Therefore, following other authors, \cite{MieStoSav-MP-97,LimCarLuoMaOlsTruGag-J An alternative way to eliminate the $S_z$ dependency is to simply set $\zeta=0$, \ie, to resort to the spin-unpolarized functional. This lowers the accuracy for open-shell systems at $\mu=0$, \ie, for the usual PBE correlation functional $\varepsilon_{\text{c}}^{\text{PBE}}(\argepbe)$. Nevertheless, we argue that, for sufficiently large $\mu$, it is a viable option. Indeed, the purpose of introducing the spin polarization in semilocal density-functional approximations is to mimic the exact on-top pair density, \cite{PerSavBur-PRA-95} but our functional $\ecmd(\argecmd)$ already explicitly depends on the on-top pair density [see Eqs.~\eqref{eq:def_ecmdpbe} and \eqref{eq:def_beta}]. The dependencies on $\zeta$ and $n_2$ can thus be expected to be largely redundant. Consequently, we propose here to test the $\ecmd$ functional with \textit{a zero spin polarization}. This ensures its $S_z$ independence and, as will be numerically demonstrated, very weakly affects the complementary functional accuracy. + \subsubsection{Size consistency} +cite~\cite{Sav-CP-09} here + Since $\efuncdenpbe{\argebasis}$ is computed via a single integral over $\mathbb{R}^3$ [see Eq.~\eqref{eq:def_ecmdpbebasis}] which involves only local quantities [$n(\br{})$, $\zeta(\br{})$, $s(\br{})$, $n_2(\br{})$, and $\mu(\br{})$], in the case of non-overlapping fragments \ce{A\bond{...}B}, it can be written as the sum of two local contributions: one coming from the integration over the region of subsystem \ce{A} and the other one from the region of subsystem \ce{B}. Therefore, a sufficient condition for size consistency is that these local quantities coincide in the isolated systems and in the subsystems of the supersystem \ce{A\bond{...}B}. Since these local quantities are calculated from the wave function $\psibasis$, a sufficient condition is that the wave function is multiplicatively separable in the limit of non-interacting fragments, \ie, $\Psi_{\ce{A\bond{...}B}}^{\basis} = \Psi_{\ce{A}}^{\basis} \Psi_{\ce{B}}^{\basis}$. We refer the interested reader to the {\SI} for a detailed proof of the latter statement. In the case where the two subsystems \ce{A} and \ce{B} dissociate in closed-shell systems, a simple RHF wave function ensures this property, but when one or several covalent bonds are broken, a properly chosen CASSCF wave function is sufficient to recover this property. The underlying active space must however be chosen in such a way that it leads to size-consistent energies in the limit of dissociated fragments. diff --git a/new/F2_avdz/plots/DFT_avdzE_error.eps b/new/F2_avdz/plots/DFT_avdzE_error.eps deleted file mode 100644 index 183317e..0000000 --- a/new/F2_avdz/plots/DFT_avdzE_error.eps +++ /dev/null @@ -1,832 +0,0 @@ -%!PS-Adobe-3.0 EPSF-3.0 -%%Creator: cairo 1.14.6 (http://cairographics.org) -%%CreationDate: Sat Nov 23 16:42:37 2019 -%%Pages: 1 -%%DocumentData: Clean7Bit -%%LanguageLevel: 2 -%%BoundingBox: 0 0 360 216 -%%EndComments -%%BeginProlog -save -50 dict begin -/q { gsave } bind def -/Q { grestore } bind def -/cm { 6 array astore concat } bind def -/w { setlinewidth } bind def -/J { setlinecap } bind def -/j { setlinejoin } bind def -/M { setmiterlimit } bind def -/d { setdash } bind def -/m { moveto } bind def -/l { lineto } bind def -/c { curveto } bind def -/h { closepath } bind def -/re { exch dup neg 3 1 roll 5 3 roll moveto 0 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0 -1 0 216 cm -324.398 37.699 m 324.398 39.699 321.398 39.699 321.398 37.699 c 321.398 - 35.699 324.398 35.699 324.398 37.699 c S Q -0 g -q 1 0 0 -1 0 216 cm -38.75 10.75 306.398 187.148 re S Q -Q Q -showpage -%%Trailer -end restore -%%EOF diff --git a/new/F2_avdz/plots/plot.sh b/new/F2_avdz/plots/plot.sh index fe74090..b0812d5 100755 --- a/new/F2_avdz/plots/plot.sh +++ b/new/F2_avdz/plots/plot.sh @@ -42,7 +42,7 @@ plot '${FILE}' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 p replot '${FILE}' using 1:3 smooth cspline notitle lt 9 , "" using 1:3 w p lt 9 ps 1 title "${WF}+PBE-UEG~{/Symbol z}{.8-}/$BASIS" replot '${FILE}' using 1:4 smooth cspline notitle lt 4 , "" using 1:4 w p lt 4 ps 1 title "${WF}+PBEot~{/Symbol z}{.8-}/$BASIS" replot '${FILE}' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "${WF}+PBEot0{/Symbol z}/$BASIS" -replot '${FILE}' using 1:6 smooth cspline notitle lt 7 , "" using 1:6 w p lt 7 ps 1 title "Exact" +replot '${FILE}' using 1:6 smooth acsplines notitle lt 7 , "" using 1:6 w p lt 7 ps 1 title "Exact" set terminal eps enhanced linewidth 6 set output "${OUT}.eps" replot @@ -51,6 +51,7 @@ EOF fi +exit gnuplot pouet.gp TYPE=error diff --git a/new/F2_avdz/plots/pouet.gp b/new/F2_avdz/plots/pouet.gp index c441878..8c8787c 100644 --- a/new/F2_avdz/plots/pouet.gp +++ b/new/F2_avdz/plots/pouet.gp @@ -1,9 +1,17 @@ -set xrange [:7] -plot 'data_DFT_avdzE_error' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "FCI/avdz" -replot 'data_DFT_avdzE_error' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "FCI+PBEot{/Symbol z}/avdz" -replot 'data_DFT_avdzE_error' using 1:4 smooth cspline notitle lt 4 , "" using 1:4 w p lt 4 ps 1 title "FCI+PBEot/avdz" -replot 'data_DFT_avdzE_error' using 1:3 smooth cspline notitle lt 9 , "" using 1:3 w p lt 9 ps 1 title "FCI+PBE/avdz" +set xrange [2.5:3.5] +set key left + +set xlabel "Inter atomic distance (a.u.) " +set ylabel "Energy (Hartree)" +set key font ",15" +set key spacing "1,8" + +plot 'data_DFT_avdzE_relat' using 1:2 smooth cspline notitle lt 2 , "" using 1:2 w p lt 2 ps 1 title "FCI/avdz" +replot 'data_DFT_avdzE_relat' using 1:3 smooth cspline notitle lt 9 , "" using 1:3 w p lt 9 ps 1 title "FCI+PBE-UEG~{/Symbol z}{.8-}/avdz" +replot 'data_DFT_avdzE_relat' using 1:4 smooth cspline notitle lt 4 , "" using 1:4 w p lt 4 ps 1 title "FCI+PBEot~{/Symbol z}{.8-}/avdz" +replot 'data_DFT_avdzE_relat' using 1:5 smooth cspline notitle lt 8 , "" using 1:5 w p lt 8 ps 1 title "FCI+PBEot0{/Symbol z}/avdz" +replot 'data_DFT_avdzE_relat' using 1:6 smooth cspline notitle lt 7 , "" using 1:6 w p lt 7 ps 1 title "Exact" set terminal eps enhanced linewidth 6 -set output "DFT_avdzE_error.eps" +set output "DFT_avdzE_relat_zoom.eps" replot