up to RSDFT

Manuscript/G2-srDFT.tex

@@ -82,8 +82,8 @@
 \newcommand{\V}[2]{V_{#1}^{#2}}
 \newcommand{\SO}[2]{\phi_{#1}(\br{#2})}
 
-\newcommand{\modX}{\mathcal{X}}
+\newcommand{\modY}{Y}
-\newcommand{\modY}{\mathcal{Y}}
+\newcommand{\modZ}{Z}
 
 % basis sets 
 \newcommand{\Bas}{\mathcal{B}}
@@ -184,16 +184,13 @@ The present basis set correction relies on the RS-DFT formalism to capture the m
 Here, we only provide the main working equations. 
 We refer the interested reader to Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18} for a more formal derivation.  
 
-%=================================================================
+Let us assume we have both the energy $\E{\modY}{\Bas}$ and density $\n{\modZ}{\Bas}$ of a $\Ne$-electron system described by two methods $\modY$ and $\modZ$ (potentially identical) in an incomplete basis set $\Bas$. 
-%\subsection{Correcting the basis set error of a general WFT model}
+According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modY}{\Bas}$ and $\n{\modZ}{\Bas}$ are reasonable approximations of the \alert{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
-%=================================================================
-Let us assume we have both the energy $\E{\modX}{\Bas}$ and density $\n{\modY}{\Bas}$ of a $\Ne$-electron system described by two methods $\modX$ and $\modY$ (potentially identical) in an incomplete basis set $\Bas$. 
-According to Eq.~(15) of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, assuming that $\E{\modX}{\Bas}$ and $\n{\modY}{\Bas}$ are reasonable approximations of the \alert{FCI} energy and density within $\Bas$, the exact ground state energy $\E{}{}$ may be written as
 \begin{equation}
 	\label{eq:e0basis}
 	\E{}{} 
-	\approx \E{\modX}{\Bas} 
+	\approx \E{\modY}{\Bas} 
-	+ \bE{}{\Bas}[\n{\modY}{\Bas}],
+	+ \bE{}{\Bas}[\n{\modZ}{\Bas}],
 \end{equation}
 where
 \begin{equation}
@@ -206,21 +203,19 @@ is the basis-dependent complementary density functional, $\hT$ is the kinetic op
 In Eq.~\eqref{eq:E_funcbasis}, $\wf{}{\Bas}$ and $\wf{}{}$ are two general $\Ne$-electron wave functions belonging to the Hilbert space spanned by $\Bas$ and the complete basis, respectively.
 Both wave functions yield the same target density $\n{}{}$. 
 
-%\alert{Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only approximation performed in \eqref{eq:e0basis} is that the FCI density $\n{\FCI}{\Bas}$ coincides with the exact ground state density, which in general is a reasonable approximation as the density converges rapidly with the basis set.}
-
 Importantly, in the limit of a complete basis set (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$, which implies that
 \begin{equation}
   \label{eq:limitfunc}
-        \lim_{\Bas \to \infty} \qty( \E{\modX}{\Bas} + \bE{}{\Bas}[\n{\modY}{\Bas}] ) = \E{\modX}{} \approx E,
+        \lim_{\Bas \to \infty} \qty( \E{\modY}{\Bas} + \bE{}{\Bas}[\n{\modZ}{\Bas}] ) = \E{\modY}{} \approx E,
 \end{equation}
-where  $\E{\modX}{}$ is the energy associated with the method $\modX$ in the complete basis set.
+where  $\E{\modY}{}$ is the energy associated with the method $\modY$ in the complete basis set.
-In the case $\modX = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
+In the case $\modY = \FCI$, we have a strict equality as $\E{\FCI}{} = \E{}{}$.
-Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modX$ and $\modY$ for the \titou{FCI} energy and density within $\Bas$, respectively.
+Provided that the functional $\bE{}{\Bas}[\n{}{}]$ is known exactly, the only source of error at this stage lies in the potential approximate nature of the methods $\modY$ and $\modZ$ for the \titou{FCI} energy and density within $\Bas$, respectively.
 
 Rigorously speaking,  the functional $\bE{}{\Bas}[\n{}{}]$ is obviously \textit{not} universal as it depends on $\Bas$. 
 Moreover, as $\bE{}{\Bas}[\n{}{}]$ aims at fixing the incompleteness of $\Bas$, its main role is to correct 
 for the lack of cusp in $\wf{}{\Bas}$ (i.e.~a discontinuous derivative) at the e-e coalescence points, a universal condition of exact wave functions. 
-As the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$. 
+Because the e-e cusp originates from the divergence of the Coulomb operator at $r_{12} = 0$, a cuspless wave function could equivalently originate from a Hamiltonian with a non-divergent Coulomb interaction at $r_{12} = 0$. 
 Therefore, as we shall do later on, it feels natural to  approximate $\bE{}{\Bas}[\n{}{}]$ with short-range density functionals which deal with a smooth long-range electron interaction.
 Contrary to the conventional RS-DFT scheme which requires a range-separated \textit{parameter} $\rsmu{}{}$, here we use a range-separated \textit{function} $\rsmu{\Bas}{}(\br{})$ which automatically adapts to quantify the incompleteness of $\Bas$ in $\mathbb{R}^3$.  
 
@@ -229,6 +224,9 @@ The present definition ensures that $\W{\Bas}{}(\br{1},\br{2})$ is finite at the
 In a second step, we shall link $\W{\Bas}{}(\br{1},\br{2})$ to $\rsmu{\Bas}{}(\br{})$.
 In the final step, we employ short-range density functionals \cite{TouGorSav-TCA-05} with $\rsmu{\Bas}{}(\br{})$ as range separation. 
 
+%=================================================================
+%\subsection{Effective Coulomb operator}
+%=================================================================
 We define the effective operator as 
 \begin{equation}
 	\label{eq:def_weebasis}
@@ -243,15 +241,15 @@ where
 \begin{equation}
 	\label{eq:n2basis}
 	\n{2}{}(\br{1},\br{2})
-	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}
+	= \sum_{pqrs \in \Bas}  \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2},
 \end{equation}
-and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ are the opposite-spin two-body density and density tensor (respectively) associated with $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital,
+and $\Gam{pq}{rs} = \mel*{\wf{}{\Bas}}{ \aic{r}\aic{s}\ai{p}\ai{q} }{\wf{}{\Bas}}$ are the opposite-spin pair density and density tensor (respectively) associated with $\wf{}{\Bas}$, $\SO{p}{}$ is a spinorbital,
 \begin{equation}
 	\label{eq:fbasis}
 	\f{\Bas}{}(\br{1},\br{2}) 
 	= \sum_{pqrstu \in \Bas}  \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2},
 \end{equation}
-and $\V{pq}{rs}$ are the usual Coulomb two-electron integrals. 
+and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
 Note that the divergence condition of $\W{\Bas}{}(\br{1},\br{2})$ in Eq.~\eqref{eq:def_weebasis} ensures that one-electron systems do not have any basis set correction. 
 \PFL{I don't agree with this. There must be a correction for one-electron system.
 However, it does not come from the e-e cusp but from the e-n cusp.}
@@ -266,37 +264,39 @@ Because Eq.~\eqref{eq:int_eq_wee} can be rewritten as
   \iint r_{12}^{-1} \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2} =  \iint \W{\Bas}{}(\br{1},\br{2}) \n{2}{}(\br{1},\br{2}) \dbr{1} \dbr{2},
 \end{equation}
 it intuitively motivates $\W{\Bas}{}(\br{1},\br{2})$ as a potential candidate for an effective interaction. 
-As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational nor rotational invariant if $\Bas$ does not have such symmetries. 
+As already discussed in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, $\W{\Bas}{}(\br{1},\br{2})$ is symmetric, \textit{a priori} non translational, nor rotational invariant if $\Bas$ does not have such symmetries. 
 An important quantity to define in the present context is the value of the effective interaction at coalescence of opposite-spin electrons
 \begin{equation}
 	\label{eq:wcoal}
 	\W{\Bas}{}(\br{}) =  \W{\Bas}{}(\br{},{\br{}}),
 \end{equation}
-and which is necessarily \textit{finite} for an incomplete basis set as long as the on-top two-body density is non vanishing. 
+which is necessarily \textit{finite} for an incomplete basis set as long as the on-top pair density $\n{2}{}(\br{},\br{})$ is non vanishing. 
 
 Of course, there exists \textit{a priori} an infinite set of functions in $\mathbb{R}^6$ satisfying \eqref{eq:int_eq_wee}, but thanks to its definition one can show that (see Appendix B of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}) 
 \begin{equation}
 	\label{eq:lim_W}
 	\lim_{\Bas \to \infty}\W{\Bas}{}(\br{1},\br{2}) = r_{12}^{-1}\   
 \end{equation}
-for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$, which guarantees a physically satisfying limit. 
+for any $(\br{1},\br{2})$ such that $\n{2}{}(\br{1},\br{2}) \ne 0$ and for any $\wf{}{\Bas}$.%, which guarantees a physically satisfying limit. 
 %An important point here is that, with the present definition of $\W{\Bas}{}(\br{1},\br{2})$, one can quantify the effect of the incompleteness of $\Bas$ on the Coulomb operator itself as a removal of the divergence of the two-electron interaction near the electron coalescence.  
 %As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, choosing a HF wave function as $\wf{}{\Bas}$ to define the effective interaction $\W{\Bas}{}(\br{1},\br{2})$ already provides a quantitative representation of the incompleteness of $\Bas$ for weakly correlated systems. 
 
-Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators. 
+\alert{Because the Coulomb operator within a basis set $\Bas$ is a non divergent two-electron interaction, we can straightforwardly link the present theory with the RS-DFT which uses the so-called long-range interaction which are smooth bounded two-electron operators.}
-To do so, we choose a range-separation \textit{function}
+Although this choice is not unique, we choose here the range-separation function
 \begin{equation}
 	\label{eq:mu_of_r}
- \rsmu{\Bas}{}(\br{})  = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{}) 
+	\rsmu{\Bas}{}(\br{})  = \frac{\sqrt{\pi}}{2} \W{\Bas}{}(\br{}) ,
 \end{equation}
 such that the long-range interaction
 \begin{equation}
         \w{}{\lr,\rsmu{\Bas}{}}(\br{1},\br{2}) = \frac{1}{2} \qty{ \frac{\erf[ \rsmu{\Bas}{}(\br{1}) r_{12}]}{r_{12}} + \frac{\erf[ \rsmu{\Bas}{}(\br{2}) r_{12}]}{ r_{12}} }
 \end{equation}
-\PFL{This expression looks like a cheap spherical average.}
+\PFL{This expression looks like a cheap spherical average.
+What about $\rsmu{\Bas}{}(\br{1},\br{2}) = \sqrt{\rsmu{\Bas}{}(\br{1}) \rsmu{\Bas}{}(\br{2})}$ and a proper spherical average to get $\rsmu{\Bas}{}(r_{12})$?}
 coincides with the effective interaction $\W{\Bas}{}(\br{})$ at coalescence, i.e.~$\w{}{\lr,\rsmu{\Bas}{}}(\br{},\br{}) = \W{\Bas}{}(\br{})$.
 
-Once defined the range-separation function $\rsmu{\Bas}{}(\br{})$, we can use the functionals defined in the field of RS-DFT to approximate $\bE{}{\Bas}[\n{}{}]$. As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
+Once defined, $\rsmu{\Bas}{}(\br{})$ can be used in RS-DFT functionals defined to approximate $\bE{}{\Bas}[\n{}{}]$. 
+As in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, we approximate $\bE{}{\Bas}[\n{}{}]$ using a specific class of short-range correlation functionals known as ECMD whose general definition reads \cite{TouGorSav-TCA-05}
 \begin{multline}
   \label{eq:ec_md_mu}
   \bE{}{\sr}[\n{}{}(\br{}),\rsmu{}{}] = \min_{\wf{}{} \to \n{}{}(\br{})} \mel*{\Psi}{\hT + \hWee{}}{\wf{}{}}
@@ -356,7 +356,7 @@ Therefore, the PBE complementary functional reads
 	\label{eq:def_pbe_tot}
         \bE{\PBE}{\sr}[\n{}{}(\br{}),\rsmu{}{}(\br{})] = \int \be{\PBE}{\sr}\big(\n{}{}(\br{}),\nabla \n{}{}(\br{}),\rsmu{}{}(\br{})\big) \n{}{}(\br{}) \dbr{}.
 \end{equation}
-Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modY}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
+Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n{\modZ}{}]$ is then equal to $\bE{\LDA}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{}(\br{}),\rsmu{\Bas}{}(\br{})]$ where $\rsmu{\Bas}{}(\br{})$ is given by Eq.~\eqref{eq:mu_of_r}.
 
 As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a general subset of molecular orbitals. 
 We then naturally split the basis set as $\Bas = \Cor \bigcup \Val$, where $\Cor$ and $\Val$ are its core and valence parts, respectively.% and $\Cor \bigcap \Val = \O$.
@@ -388,11 +388,11 @@ and the corresponding valence range separation function
 \end{equation}
 It is worth noting that, within the present definition, $\W{\Bas}{\Val}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}.
 
-Defining $\n{\modY}{\Val}$ as the valence one-electron density obtained with the model $\modY$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modY}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modY}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
+Defining $\n{\modZ}{\Val}$ as the valence one-electron density obtained with the model $\modZ$, the valence part of the complementary functional $\bE{}{\Val}[\n{\modZ}{\Val}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$.
 
 Regarding now the main computational source of the present approach, it consists in the evaluation 
 of $\W{\Bas}{}(\br{})$ [See Eqs.~\eqref{eq:wcoal}] at each quadrature grid point. 
-All through this paper, we use two-body density matrix of a single Slater determinant (typically HF) 
+All through this paper, we use pair density matrix of a single Slater determinant (typically HF) 
 for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the evaluation 
 at each quadrature grid point of 
 \begin{equation}
@@ -505,10 +505,10 @@ In the case of \ce{C2} and \ce{N2}, we also perform calculations with the cc-pCV
 In a second time, we compute the entire atomization energies of the G2 set \cite{CurRagTruPop-JCP-91} composed by 55 molecules with the cc-pVXZ family of basis sets.
 This molecular set has been exhausively studied in the last 20 years (see, for example, Refs.~\onlinecite{FelPetDix-JCP-08,Gro-JCP-09,FelPet-JCP-09,NemTowNee-JCP-10,FelPetHil-JCP-11,PetTouUmr-JCP-12,FelPet-JCP-13,KesSylKohTewMar-JCP-18}).
 %The reference values for the atomization energies are extracted from Ref.~\onlinecite{HauKlo-JCP-12} and corresponds to frozen-core non-relativistic atomization energies obtained at the CCSD(T)(F12)/cc-pVQZ-F12 level of theory corrected for higher-excitation contributions ($E_\text{CCSDT(Q)/cc-pV(D+d)Z} - E_\text{CCSD(T)/cc-pV(D+d)Z})$.
-As a method $\modX$ we employ either CCSD(T) or exFCI.
+As a method $\modY$ we employ either CCSD(T) or exFCI.
 Here, exFCI stands for extrapolated FCI energies computed with the CIPSI algorithm. \cite{HurMalRan-JCP-73, GinSceCaf-CJC-13, GinSceCaf-JCP-15}
 We refer the interested reader to Refs.~\onlinecite{HolUmrSha-JCP-17, SceGarCafLoo-JCTC-18, LooSceBloGarCafJac-JCTC-18, SceBenJacCafLoo-JCP-18, LooBogSceCafJAc-JCTC-19} for more details.
-In the case of the CCSD(T) calculations, we have $\modY = \HF$ as we use the restricted open-shell Hartree-Fock (ROHF) one-electron density to compute the complementary energy. 
+In the case of the CCSD(T) calculations, we have $\modZ = \HF$ as we use the restricted open-shell Hartree-Fock (ROHF) one-electron density to compute the complementary energy. 
 \titou{For exFCI, we use the density of a converged variational wave function.} 
 For the definition of the interaction, we use a single Slater determinant built in the ROHF basis for the CCSD(T) calculations, and built with the natural orbitals of the converged variational wave function for the exFCI calculations. 
 The CCSD(T) calculations have been performed with Gaussian09 with standard threshold values. \cite{g09}