revision in progress

JCTC_revision/GW-srDFT.tex

@@ -332,6 +332,7 @@ and
 The solution of the Dyson equation \eqref{eq:Dyson} gives the Green function $\G{}{\Bas}(\br{},\br{}',\omega)$ which is not exact (even using the exact complementary basis-correction density functional $\bSig{}{\Bas}[\n{}{}]$) but should converge more rapidly with the basis set thanks to the presence of the basis-set correction $\bSig{}{\Bas}$. 
 Of course, in the CBS limit, the basis-set correction vanishes and the Green function becomes exact, \ie,
 \begin{align}
+	\label{eq:limSig}
 	\lim_{\Bas \to \CBS} \bSig{}{\Bas} & = 0,
 	 &
 	\lim_{\Bas \to \CBS} \G{}{\Bas} & = \G{}{}.
@@ -433,19 +434,20 @@ where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies
 \subsection{Short-range correlation functionals}
 \label{sec:srDFT}
 %%%%%%%%%%%%%%%%%%%%%%%%
-The fundamental idea behind the present basis set correction is to recongnise that the two-electron interaction $\abs*{\br{} - \br{}'}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $r_{12} = 0$.
-We can therefore define an effective two-electron interaction which ``mimicks'' the Coulomb operator in an incomplete basis, \ie,
+The fundamental idea behind the present basis set correction is to recognize that the two-electron interaction $\abs*{\br{} - \br{}'}^{-1}$ projected in a finite basis $\Bas$ is a finite, non-divergent quantity at $\abs*{\br{} - \br{}'} = 0$.
+We can therefore define an effective two-electron interaction which equals the Coulomb operator in an incomplete basis, \ie,
 \begin{equation}
 	\iint \frac{\n{2}{\Bas}(\br{},\br{}')}{\abs*{\br{} - \br{}'}} d\br{} d\br{}'
 	= 
-	\iint \n{2}{\Bas}(\br{},\br{}') W(\br{},\br{}') d\br{} d\br{}'
+	\iint \n{2}{\Bas}(\br{},\br{}') W(\br{},\br{}') d\br{} d\br{}'.
 \end{equation}
 A convenient choice is, for example,
 \begin{equation}
-	\rsmu{}{\Bas}(\br{}) = \frac{\sqrt{\pi}}{2} 
+\label{eq:W}
+	W(\br{},\br{}') =  
 	\begin{cases}
 		f^{\Bas}(\br{})/\n{2}{\Bas}(\br{}),		&	\n{2}{\Bas}(\br{})	\neq 0,		\\
-		\infty,									&	\text{otherwise}				\\
+		\infty,									&	\text{otherwise},				\\
 	\end{cases}
 \end{equation}
 where, in the case of a single-determinant method (such as HF and KS-DFT),
@@ -457,6 +459,16 @@ and
 	\n{2}{\Bas}(\br{}) = \frac{[\n{}{\Bas}(\br{})]^2}{4} 
 \end{equation}
 is the opposite-spin on-top pair density.
+The effective operator $W(\br{},\br{}')$ has some interesting properties.
+For example, we have
+\begin{equation}
+	\lim_{\Bas \to \CBS}  W(\br{},\br{}') = \abs*{\br{} - \br{}'}^{-1}
+\end{equation}
+which means that, in the limit of a complete basis, one recovers the genuine (divergent) Coulomb operator.
+Consequently, the magnitude of the correction tends to zero (see Eq.~\eqref{eq:limSig}).
+Note also that the divergence condition of $W(\br{},\br{}')$ in Eq.~\eqref{eq:W} evidences that one-electron systems are free of correction.
+
+Because the value of $W(\br{},\br{}')$ at coalescence, $W(\br{},\br{})$, is necessarily finite in a finite basis $\Bas$, one can map its value to a non-divergent, long-range interaction of the form $\abs*{\br{} - \br{}'}^{-1} \erf(\rsmu{}{} \abs*{\br{} - \br{}'})$.
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Short-range correlation functionals}
@@ -700,7 +712,8 @@ The extrapolation error is reported in parenthesis.
 %%%%%%%%%%%%%%%%%%%%%%%%
 In order to check the transferability of the present observations to larger systems, we have computed the values of the IPs of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) at the {\GOWO}@PBE level of theory with a different basis set family. \cite{Weigend_2003a, Weigend_2005a}
 The numerical values are reported in Table \ref{tab:DNA_IP}, and their error with respect to the {\GOWO}@PBE/def2-TQZVP extrapolated values \cite{vanSetten_2015} (obtained via extrapolation of the def2-TZVP and def2-QZVP results) are shown in Fig.~\ref{fig:DNA_IP}.
-The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.} \cite{Krause_2015} as well as the experimental results extracted from Ref.~\onlinecite{vanSetten_2015} are reported for comparison purposes.
+\titou{Table \ref{tab:DNA_IP} also contains extrapolated IPs obtained with plane wave basis sets with two different software packages. \cite{Maggio_2017,Govoni_2018}
+The CCSD(T)/def2-TZVPP computed by Krause \textit{et al.} \cite{Krause_2015} on the same geometries, the CCSD(T)//CCSD/aug-cc-pVDZ results from Ref.~\onlinecite{Roca-Sanjuan_2006}, as well as the experimental results extracted from Ref.~\onlinecite{vanSetten_2015} are reported for comparison purposes.}
 
 For these five systems, the IPs are all of the order of $8$ or $9$ eV with an amplitude of roughly $1$ eV between the smallest basis set (def2-SVP) and the CBS value.
 The conclusions that we have drawn in the previous subsection do apply here as well.
@@ -715,9 +728,11 @@ This is quite remarkable as the number of basis functions jumps from $371$ to $7
 In the present manuscript, we have shown that the density-based basis set correction developed by some of the authors in Ref.~\onlinecite{Giner_2018} and applied recently to ground- and excited-state properties \cite{Loos_2019, Giner_2019} can also be successfully applied to Green function methods such as {\GW}.
 In particular, we have evidenced that the present basis set correction (which relies on LDA- or PBE-based short-range correlation functionals) significantly speeds up the convergence of IPs for small and larger molecules towards the CBS limit.
 These findings have been observed for different {\GW} starting points (HF, PBE, and PBE0).
+\titou{We have observed that the performance of the two short-range correlation functionals (srLDA and srPBE) are quite similar with a slight edge for srPBE over srLDA.
+Therefore, because srPBE is only slightly more computationally expensive than srLDA, we do recommend the use of the former.}
 
 As mentioned earlier, the present basis set correction can be straightforwardly applied to other properties of interest such as electron affinities or fundamental gap.
-It is also applicable to other flavors of {\GW} such as the partially self-consistent {\evGW} or {\qsGW} methods.
+It is also applicable to other flavors of {\GW} such as the partially self-consistent {\evGW} or {\qsGW} methods, \titou{and more generally to any approximation of the self-energy.}
 We are currently investigating the performance of the present approach within linear response theory in order to speed up the convergence of excitation energies obtained within the RPA and Bethe-Salpeter equation (BSE) \cite{Strinati_1988, Leng_2016, Blase_2018} formalisms.  
 We hope to report on this in the near future.
 
@@ -725,6 +740,7 @@ We hope to report on this in the near future.
 \section*{Supporting Information}
 %%%%%%%%%%%%%%%%%%%%%%%%
 See {\SI} for additional graphs reporting the convergence of the ionization potentials of the GW20 subset with respect to the size of the basis set.
+\titou{The numerical data of Tables \ref{tab:GW20_HF} and \ref{tab:GW20_PBE0} are provided in txt and json format.}
 
 %%%%%%%%%%%%%%%%%%%%%%%%
 \begin{acknowledgements}