From 1e6e14c43d3667a16abdddde4febbdf06d448078 Mon Sep 17 00:00:00 2001
From: Pierre-Francois Loos <pierrefrancois.loos@gmail.com>
Date: Tue, 25 Jun 2019 16:33:33 +0200
Subject: [PATCH] Done for T2

---
 Data/.DS_Store          | Bin 6148 -> 6148 bytes
 Manuscript/Ex-srDFT.tex |  71 ++++++++++++++++++++--------------------
 2 files changed, 35 insertions(+), 36 deletions(-)

diff --git a/Data/.DS_Store b/Data/.DS_Store
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diff --git a/Manuscript/Ex-srDFT.tex b/Manuscript/Ex-srDFT.tex
index 099dd97..8b16a76 100644
--- a/Manuscript/Ex-srDFT.tex
+++ b/Manuscript/Ex-srDFT.tex
@@ -66,6 +66,7 @@
 \newcommand{\ROHF}{\text{ROHF}}
 \newcommand{\LDA}{\text{LDA}}
 \newcommand{\PBE}{\text{PBE}}
+\newcommand{\PBEUEG}{\text{PBE-UEG}}
 \newcommand{\PBEot}{\text{PBEot}}
 \newcommand{\FCI}{\text{FCI}}
 \newcommand{\CBS}{\text{CBS}}
@@ -264,8 +265,8 @@ In other words, the correction vanishes in the CBS limit, hence guaranteeing an
 \label{sec:rs}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-As initially proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further developed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference \manu{taken from RS-DFT}. \cite{TouGorSav-TCA-05}
-The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, \manu{depends on the range-separation parameter $\mu$ and} admits, for any $\n{}{}$, the following two limits \manu{as a function of $\mu$}
+As initially proposed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} and further developed in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, we have shown that one can efficiently approximate $\bE{}{\Bas}[\n{}{}]$ by short-range correlation functionals with multi-determinantal (ECMD) reference borrowed from RS-DFT. \cite{TouGorSav-TCA-05}
+The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, is a function of the range-separation parameter $\mu$ and admits, for any $\n{}{}$, the following two limits
 \begin{align}
 	\label{eq:large_mu_ecmd}
 	\lim_{\mu \to \infty}  \bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}] & = 0,
@@ -275,12 +276,12 @@ The ECMD functional, $\bE{\text{c,md}}{\sr}[\n{}{},\rsmu{}{}]$, \manu{depends on
 which correspond to the WFT limit ($\mu = \infty$) and the DFT limit ($\mu = 0$).
 In Eq.~\eqref{eq:large_mu_ecmd}, $\Ec[\n{}{}]$ is the usual universal correlation density functional defined in Kohn-Sham DFT. \cite{HohKoh-PR-64, KohSha-PR-65}
 
-The key ingredient \manu{that allows us to use the ECMD to correct for the basis set incompleteness error is} the range-separated function
+The key ingredient that allows to exploit ECMD functionals for correcting the basis set incompleteness error is the range-separated function
 \begin{equation}
 	\label{eq:def_mu}
-	\rsmu{}{\Bas}(\br{}) =  \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{})
+	\rsmu{}{\Bas}(\br{}) =  \frac{\sqrt{\pi}}{2} \W{}{\Bas}(\br{},\br{}),
 \end{equation}
-\manu{which} automatically adapts to the spatial non-homogeneity of the basis set incompleteness error.
+which 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 $\br{}$. \cite{GinPraFerAssSavTou-JCP-18}
 The explicit expression of $\W{}{\Bas}(\br{1},\br{2})$ is given by 
 \begin{equation}
@@ -317,23 +318,23 @@ We refer the interested readers to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18,L
 \label{sec:func}
 %%%%%%%%%%%%%%%%%%%%%%%%
 
-The local-density approximation (LDA) of the ECMD complementary functional is defined as 
+The local-density approximation ($\LDA$) of the ECMD complementary functional is defined as 
 \begin{equation}
 	\label{eq:def_lda_tot}
 	\bE{\LDA}{\Bas}[\n{}{},\rsmu{}{\Bas}] = \int \n{}{}(\br{}) \be{\text{c,md}}{\sr,\LDA}\qty(\n{}{}(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
 \end{equation}
 where $\zeta = (\n{\uparrow}{} - \n{\downarrow}{})/\n{}{}$ is the spin polarization and $\be{\text{c,md}}{\sr,\LDA}(\n{}{},\zeta,\rsmu{}{})$ is the ECMD short-range correlation energy per electron of the uniform electron gas (UEG) \cite{LooGil-WIRES-16} parameterized in Ref.~\citenum{PazMorGorBac-PRB-06}.
 
-The $\be{\text{c,md}}{\sr,\LDA}$ used in \eqref{eq:def_lda_tot} presents two main defects: i) at small $\mu$, it overestimates the correlation energy, and ii) UEG-based quantities are hardly transferable when the system becomes strongly correlated or multi-configurational. 
+The functional $\be{\text{c,md}}{\sr,\LDA}$ from Eq.~\eqref{eq:def_lda_tot} presents two main defects: i) at small $\mu$, it overestimates the correlation energy, and ii) UEG-based quantities are hardly transferable when the system becomes strongly correlated and/or multi-configurational. 
 An attempt to solve these problems was suggested by some of the authors in the context of RS-DFT. \cite{FerGinTou-JCP-18}  
-They proposed to interpolate between the usual Perdew-Burke-Ernzerhof (PBE) correlation functional \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
+They proposed to interpolate between the usual Perdew-Burke-Ernzerhof ($\PBE$) correlation functional \cite{PerBurErn-PRL-96} $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
 In the context of RS-DFT, the large-$\mu$ behavior corresponds to an extremely short-range interaction. 
 In this regime, the ECMD energy
 \begin{align}
   \label{eq:exact_large_mu}
   \bE{\text{c,md}}{\sr}[\n{2}{},\rsmu{}{}] \propto \frac{1}{\mu^3} \int \dbr{} \n{2}{}(\br{}) + \order*{\mu^{-4}}
 \end{align}
-only depends on the \textit{exact} on-top pair density  $\n{2}{}(\br{}) \equiv \n{2}{}(\br{},\br{})$ which is obtained from the \textit{exact} ground state wave function $\Psi$ belonging to the Hilbert space spanned by \manu{a} complete basis set.
+only depends on the \textit{exact} on-top pair density  $\n{2}{}(\br{}) \equiv \n{2}{}(\br{},\br{})$ which is obtained from the \textit{exact} ground state wave function $\Psi$ belonging to the Hilbert space spanned by a complete basis set.
 
 Obviously, an exact quantity such as $\n{2}{}(\br{})$ is out of reach in practical calculations and must be approximated by a function referred here as $\tn{2}{}(\br{})$. 
 For a given $\tn{2}{}(\br{})$, some of the authors proposed the following functional form in order to interpolate between $\e{\text{c}}{\PBE}(\n{}{},s,\zeta)$ at $\mu = 0$ and Eq.~\eqref{eq:exact_large_mu} as $\mu \to \infty$: \cite{FerGinTou-JCP-18} 
@@ -347,8 +348,7 @@ For a given $\tn{2}{}(\br{})$, some of the authors proposed the following functi
 \end{gather}
 \end{subequations}
 As illustrated in the context of RS-DFT, \cite{FerGinTou-JCP-18} such a functional form is able to treat both weakly and strongly correlated systems thanks to the explicit inclusion of $\e{\text{c}}{\PBE}$ and $\tn{2}{}$, respectively.
-
-Therefore, in the present context, we consider the explicit form of Eqs.~\eqref{eq:epsilon_cmdpbe} and \eqref{eq:beta_cmdpbe} with $\rsmu{}{\Bas}$ and introduce the general form of the PBE-based complementary functional \manu{for the basis set $\Bas$}: 
+Therefore, in the present context, we introduce the general form of the $\PBE$-based complementary functional within a given basis set $\Bas$ 
 \begin{multline}
 	\label{eq:def_pbe_tot}
 	\bE{\PBE}{\Bas}[\n{}{},\tn{2}{},\rsmu{}{\Bas}] =
@@ -356,39 +356,38 @@ Therefore, in the present context, we consider the explicit form of Eqs.~\eqref{
 	\\
 	\times \be{\text{c,md}}{\sr,\PBE}\qty(\n{}{}(\br{}),\tn{2}{}(\br{}),s(\br{}),\zeta(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
 \end{multline}
-which depends on the approximation level of $\tn{2}{}$.
+which has an explicit dependency on both the range-separation function $\rsmu{}{\Bas}(\br{})$ (instead of the range-separation parameter in RS-DFT) and the approximation level of $\tn{2}{}$.
 
-In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a version of the PBE-based functional, here-referred as \titou{PBE-UEG}:
+In Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, some of the authors introduced a version of the $\PBE$-based functional, here-referred as $\PBEUEG$
 \begin{equation}
 	\label{eq:def_pbe_tot}
-	\bE{\titou{\PBE\text{-}\UEG}}{\Bas} 
+	\bE{\PBEUEG}{\Bas} 
 	\equiv
 	\bE{\PBE}{\Bas}[\n{}{},\n{2}{\UEG},\rsmu{}{\Bas}],
 \end{equation}
 in which the on-top pair density was approximated by its UEG version, i.e., $\tn{2}{}(\br{}) = \n{2}{\UEG}(\br{})$, with
 \begin{equation}
- \n{2}{\UEG}(\br{}) = n(\br{})^2 (1-\zeta(\br{})^2) g_0(n(\br{})), 
+ \n{2}{\UEG}(\br{}) = n(\br{})^2 [1-\zeta(\br{})^2] g_0(n(\br{})), 
 \end{equation}
 and where $g_0(n)$ is the UEG on-top pair distribution function [see Eq.~(46) of Ref.~\citenum{GorSav-PRA-06}].
-As illustrated in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19} for weakly correlated systems, this PBE-based functional has clearly shown to improve energetics over the pure UEG-based functional $\bE{\LDA}{\Bas}$ (see \eqref{eq:def_lda_tot}) thanks to the inclusion of the PBE functional.
+As illustrated in Ref.~\onlinecite{LooPraSceTouGin-JPCL-19}, the $\PBEUEG$ functional has clearly shown, for weakly correlated systems, to improve energetics over the pure UEG-based functional $\bE{\LDA}{\Bas}$ [see Eq.~\eqref{eq:def_lda_tot}] thanks to the leverage brought by the $\PBE$ functional in the small-$\mu$ regime.
 
 However, the underlying UEG on-top pair density might not be suited for the treatment of excited states and/or strongly correlated systems. 
-Besides, in the context of the present basis set correction, we have to compute $\n{2}{\Bas}(\br{})$ to obtain $\rsmu{}{\Bas}(\br{})$ [see Eqs.~\eqref{eq:def_mu} and \eqref{eq:def_weebasis}]. 
-Therefore we can use $\n{2}{\Bas}(\br{})$ to have an approximation of the exact on-top pair density.}
-More precisely, we propose an approximation of the on-top pair density $\ttn{2}{\Bas}(\br{})$ defined as
+Besides, in the context of the present basis set correction, $\n{2}{\Bas}(\br{})$, the on-top pair density in $\Bas$, must be computed anyway to obtain $\rsmu{}{\Bas}(\br{})$ [see Eqs.~\eqref{eq:def_mu} and \eqref{eq:def_weebasis}].
+Therefore, we define a better approximation of the exact on-top pair density as
 \begin{equation}
+	\label{eq:ot-extrap}
   \ttn{2}{\Bas}(\br{}) = \n{2}{\Bas}(\br{}) \qty( 1 + \frac{2}{\sqrt{\pi}\rsmu{}{\Bas}(\br{})})^{-1}
 \end{equation}
 which is inspired by the large-$\mu$ extrapolation of the exact on-top pair density proposed by Gori-Giorgi and Savin \cite{GorSav-PRA-06} in the context of RS-DFT. 
-
-Therefore, we propose here the "PBE-ontop" (PBEot) functional which uses the on-top pair density computed in the basis set $\Bas$ as ingredient
+Using this new ingredient, we propose here the ``$\PBE$-ontop'' (\PBEot) functional
 \begin{equation}
 	\label{eq:def_pbe_tot}
 	\bE{\PBEot}{\Bas}
 	\equiv
 	\bE{\PBE}{\Bas}[\n{}{},\ttn{2}{\Bas},\rsmu{}{\Bas}].
 \end{equation}
-The sole distinction between \titou{PBE-UEG} and PBEot is the level of approximation of the exact on-top pair density.
+The sole distinction between $\PBEUEG$ and $\PBEot$ is the level of approximation of the exact on-top pair density.
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%
@@ -435,10 +434,10 @@ These results are illustrated in Fig.~\ref{fig:CH2} and reported in Table \ref{t
 Total energies for each state can be found in the {\SI}.
 Figure \ref{fig:CH2} clearly shows that, for the double-$\zeta$ basis, the exFCI adiabatic energies are far from being chemically accurate with errors as high as 0.015 eV.
 From the triple-$\zeta$ basis onward, the exFCI excitation energies are chemically-accurate though, and converge steadily to the CBS limit when one increases the size of the basis set.
-Concerning the basis set correction, already at the double-$\zeta$ level, the PBEot correction returns chemically accurate excitation energies.
-The performance of the PBE-UEG and LDA functionals (which does not require the computation of the on-top density of each state) is less impressive. 
+Concerning the basis set correction, already at the double-$\zeta$ level, the $\PBEot$ correction returns chemically accurate excitation energies.
+The performance of the $\PBEUEG$ and $\LDA$ functionals (which does not require the computation of the on-top density of each state) is less impressive. 
 Yet, they still yield significant reductions of the basis set incompleteness error, hence representing a good compromise between computational cost and accuracy.
-Note that the results for the PBE-UEG functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the LDA ones (similar considerations apply to the other systems studied below).
+Note that the results for the $\PBEUEG$ functional are not represented in Fig.~\ref{fig:CH2} as they are very similar to the $\LDA$ ones (similar considerations apply to the other systems studied below).
 It is also quite evident that, the basis set correction has the tendency of over-correcting the excitation energies via an over-stabilization of the excited states compared to the ground state.
 This trend is quite systematic as we shall see below.
 
@@ -479,7 +478,7 @@ This trend is quite systematic as we shall see below.
 								&	$1.390$	
 								&	$2.504$	\\	
 								\\
-	exFCI+PBEot	&	AVDZ					
+	exFCI+$\PBEot$	&	AVDZ					
 								&	$0.347$ [$-0.042$]	
 								&	$1.401$ [$+0.011$]	
 								&	$2.511$ [$+0.007$]	\\	
@@ -492,7 +491,7 @@ This trend is quite systematic as we shall see below.
 								&	$1.378$ [$-0.011$]	
 								&	$2.489$ [$-0.016$]	\\	
 								\\
-	exFCI+PBE-UEG	&	AVDZ					
+	exFCI+$\PBEUEG$	&	AVDZ					
 								&	$0.308$ [$-0.080$]	
 								&	$1.388$ [$-0.002$]	
 								&	$2.560$ [$+0.056$]	\\	
@@ -505,7 +504,7 @@ This trend is quite systematic as we shall see below.
 								&	$1.375$ [$-0.015$]	
 								&	$2.498$ [$-0.006$]	\\	
 								\\
-	exFCI+LDA	&	AVDZ					
+	exFCI+$\LDA$	&	AVDZ					
 								&	$0.337$ [$-0.051$]	
 								&	$1.420$ [$+0.030$]	
 								&	$2.586$ [$+0.082$]	\\	
@@ -588,9 +587,9 @@ However, these results also clearly evidence that special care has to be taken f
 	\\
 	\cline{5-16}
 				&				&					&				&	\mc{3}{c}{exFCI}	
-																	&	\mc{3}{c}{exFCI+PBEot}	
-																	&	\mc{3}{c}{exFCI+PBE-UEG}	
-																	&	\mc{3}{c}{exFCI+LDA}
+																	&	\mc{3}{c}{exFCI+$\PBEot$}	
+																	&	\mc{3}{c}{exFCI+$\PBEUEG$}	
+																	&	\mc{3}{c}{exFCI+$\LDA$}
 	\\
 	\cline{5-7}	\cline{8-10}	\cline{11-13}	\cline{14-16}
 	Molecule	&	Transition		&	Nature		&	\tabc{TBE}	&	\tabc{AVDZ}	&	\tabc{AVTZ}	&	\tabc{AVQZ}	
@@ -747,9 +746,9 @@ In order to have a miscellaneous test set of excitations, in a third time, we pr
 These two valence excitations --- $1\,^{1}\Sigma_g^+ \ra 1\,^{1}\Delta_g$ and $1\,^{1}\Sigma_g^+ \ra 2\,^{1}\Sigma_g^+$ --- are both of $(\pi,\pi) \ra (\si,\si)$ character.
 They have been recently studied with state-of-the-art methods, and have been shown to be ``pure'' doubly-excited states as they do not involve single excitations. \cite{LooBogSceCafJac-JCTC-19} 
 The vertical excitation energies associated with these transitions are reported in Table \ref{tab:Mol} and represented in Fig.~\ref{fig:C2}.
-An interesting point here is that one really needs to consider the PBEot functional to get chemically-accurate absorption energies with the AVDZ atomic basis set.
+An interesting point here is that one really needs to consider the $\PBEot$ functional to get chemically-accurate absorption energies with the AVDZ atomic basis set.
 We believe that the present result is a direct consequence of the multireference character of the \ce{C2} molecule.
-In other words, the UEG on-top density used in the LDA and PBE-UEG functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density for the present system.
+In other words, the UEG on-top density used in the $\LDA$ and $\PBEUEG$ functionals (see Sec.~\ref{sec:func}) is a particularly bad approximation of the true on-top density for the present system.
 
 %%% FIG 5 %%%
 \begin{figure}
@@ -801,9 +800,9 @@ As a final example, we consider the ethylene molecule, yet another system which
 We refer the interested reader to the work of Feller et al.\cite{FelPetDav-JCP-14} for an exhaustive investigation dedicated to the excited states of ethylene using state-of-the-art CI calculations.
 In the present context, ethylene is a particularly interesting system as it contains a mixture of valence and Rydberg excited states. 
 Our basis set corrected vertical excitation energies are gathered in Table \ref{tab:Mol} and depicted in Fig.~\ref{fig:C2H4}.
-Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+PBEot/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis.
+Except for one particular excitation (the lowest singlet-triplet excitation $1\,^{1}A_{1g} \ra 1\,^{3}B_{1u}$), the exFCI+$\PBEot$/AVDZ excitation energies are chemically accurate and the errors drop further when one goes to the triple-$\zeta$ basis.
 %(Note that one cannot afford exFCI/AVQZ calculations for ethylene.)
-Consistently with the previous examples, the LDA and PBE-UEG functionals are slightly less accurate, although they still correct the excitation energies in the right direction.
+Consistently with the previous examples, the $\LDA$ and $\PBEUEG$ functionals are slightly less accurate, although they still correct the excitation energies in the right direction.
 
 %%% FIG 6 %%%
 \begin{figure}
@@ -822,7 +821,7 @@ Consistently with the previous examples, the LDA and PBE-UEG functionals are sli
 We have shown that, by employing the recently proposed density-based basis set correction developed by some of the authors, \cite{GinPraFerAssSavTou-JCP-18} one can obtain, using sCI methods, chemically-accurate excitation energies with typically augmented double-$\zeta$ basis sets.
 This nicely complements our recent investigation on ground-state properties, \cite{LooPraSceTouGin-JPCL-19} which has evidenced that one recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets.
 The present study clearly shows that, for very diffuse excited states, the present correction relying on short-range correlation functionals from RS-DFT might not be enough to catch the radial incompleteness of the one-electron basis set.
-Also, in the case of multireference systems, we have evidenced that the PBEot functional is more appropriate than the LDA and PBE-UEG functionals relying on the UEG on-top density.
+Also, in the case of multireference systems, we have evidenced that the $\PBEot$ functional is more appropriate than the $\LDA$ and $\PBEUEG$ functionals relying on the UEG on-top density.
 We are currently investigating the performance of the present basis set correction for strongly correlated systems and we hope to report on this in the near future.
 
 %%%%%%%%%%%%%%%%%%%%%%%%