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Pierre-Francois Loos 2019-04-17 09:19:59 +02:00
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Manuscript/G2-srDFT.tex
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@ -168,7 +168,7 @@ Present-day DFT calculations are almost exclusively done within the so-called Ko
DFT's attractiveness originates from its very favorable cost/efficiency ratio as it can provide accurate energies and properties at a relatively low computational cost.
Thanks to this, KS-DFT \cite{HohKoh-PR-64, KohSha-PR-65} has become the workhorse of electronic structure calculations for atoms, molecules and solids. \cite{ParYan-BOOK-89}
Although there is no clear way on how to systematically improve density-functional approximations, \cite{Bec-JCP-14} climbing the Jacob's ladder of DFT is potentially the most satisfactory way forward. \cite{PerSch-AIPCP-01, PerRuzTaoStaScuCso-JCP-05}
In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the \trashMG{size} \manu{quality} of the basis set. \cite{FraMusLupTou-JCP-15}
In the context of the present work, one of the interesting feature of density-based methods is their much faster convergence with respect to the size of the basis set. \cite{FraMusLupTou-JCP-15}
Progress toward unifying WFT and DFT are on-going.
In particular, range-separated DFT (RS-DFT) (see Ref.~\onlinecite{TouColSav-PRA-04} and references therein) rigorously combines these two approaches via a decomposition of the electron-electron (e-e) interaction into a smooth long-range part and a (complementary) short-range part treated with WFT and DFT, respectively.
@ -176,7 +176,7 @@ As the WFT method is relieved from describing the short-range part of the correl
Therefore, a number of approximate RS-DFT schemes have been developed within single-reference \cite{AngGerSavTou-PRA-05, GolWerSto-PCCP-05, TouGerJanSavAng-PRL-09,JanHenScu-JCP-09} or multi-reference \cite{LeiStoWerSav-CPL-97, FroTouJen-JCP-07, FroCimJen-PRA-10, HedKneKieJenRei-JCP-15, FerGinTou-JCP-18} WFT approaches.
Very recently, a major step forward has been taken by some of the present authors thanks to the development of a density-based basis set correction for WFT methods. \cite{GinPraFerAssSavTou-JCP-18}
The present work proposes an extension of this new methodological development \trashMG{together} alongside the first numerical tests on molecular systems.
The present work proposes an extension of this new methodological development alongside the first numerical tests on molecular systems.
%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Theory}
@ -201,10 +201,10 @@ where
- \min_{\wf{}{\Bas} \to \n{}{}} \mel*{\wf{}{\Bas}}{\hT + \hWee{}}{\wf{}{\Bas}}
\end{equation}
is the basis-dependent complementary density functional, $\hT$ is the kinetic operator and $\hWee{} = \sum_{i<j} r_{ij}^{-1}$ is the interelectronic repulsion operator.
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 set \manu{(CBS)}, respectively.
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 set (CBS), respectively.
Both wave functions yield the same target density $\n{}{}$.
Importantly, \trashMG{in the limit of a complete basis set} \manu{when reaching the CBS limit} (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
Importantly, in the CBS limit (which we refer to as $\Bas \to \infty$), we have, for any density $\n{}{}$, $\lim_{\Bas \to \infty} \bE{}{\Bas}[\n{}{}] = 0$.
This implies that
\begin{equation}
\label{eq:limitfunc}
@ -250,7 +250,7 @@ and $\Gam{pq}{rs} =\mel*{\wf{}{\Bas}}{ \aic{r_\downarrow}\aic{s_\uparrow}\ai{p_\
\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}
\trashMG{and} $\V{pq}{rs}$ are the usual two-electron Coulomb integrals \manu{and $\wf{}{\Bas}$ is a wave function belonging to the $N-$electron Hilbert space of $\Bas$ and which will be defined later on. }
and $\V{pq}{rs}$ are the usual two-electron Coulomb integrals.
With such a definition, $\W{\Bas}{}(\br{1},\br{2})$ satisfies (see Appendix A of Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18})
\begin{equation}
\label{eq:int_eq_wee}
@ -379,7 +379,7 @@ Depending on the functional choice, the complementary functional $\bE{}{\Bas}[\n
%=================================================================
As most WFT calculations are performed within the frozen-core (FC) approximation, it is important to define an effective interaction within a subset of MOs.
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ is the set of core MOs) and define the FC version of the effective interaction as
We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$ (where $\Cor$ and $\BasFC$ are the sets of core and active MOs respectively) and define the FC version of the effective interaction as
\begin{equation}
\W{\Bas}{\FC}(\br{1},\br{2}) =
\begin{cases}
@ -412,8 +412,8 @@ Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obt
%\subsection{Computational considerations}
%=================================================================
The most computationally intensive task of the present approach is the evaluation of $\W{\Bas}{}(\br{})$ [see Eq.~\eqref{eq:wcoal}] at each quadrature grid point.
Yet embarrassingly parallel, this step scales, in the general (\manu{\textit{i.e.} $\wf{}{\Bas}$ is multi-determinantal in \eqref{eq:n2basis} and \eqref{eq:fbasis} }) case, as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ in the case of a single Slater determinant \manu{for $\wf{}{\Bas}$}.
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this \trashMG{simple choice} \manu{simplification for $\wf{}{\Bas}$} already provides, for weakly correlated systems, a quantitative representation of the \trashMG{basis set} incompleteness \manu{of the basis set $\Bas$}.
Yet embarrassingly parallel, this step scales, in the general case (i.e.~$\wf{}{\Bas}$ is a determinant expansion), as $\Ng \Nb^4$ (where $\Nb$ is the number of basis functions in $\Bas$) but is reduced to $\order*{ \Ng \Ne^2 \Nb^2}$ when $\wf{}{\Bas}$ is a single Slater determinant.
As shown in Ref.~\onlinecite{GinPraFerAssSavTou-JCP-18}, this choice for $\wf{}{\Bas}$ already provides, for weakly correlated systems, a quantitative representation of the incompleteness of $\Bas$.
Hence, unless otherwise stated, we will stick to this choice throughout the current study.
In our current implementation, the computational bottleneck is the four-index transformation to get the two-electron integrals in the MO basis which appear in Eqs.~\eqref{eq:n2basis} and \eqref{eq:fbasis}.
Nevertheless, this step usually has to be performed for most correlated WFT calculations.
@ -422,7 +422,7 @@ Modern integral decomposition techniques (such as density fitting \cite{Whi-JCP-
To conclude this section, we point out that, thanks to the definitions \eqref{eq:def_weebasis} and \eqref{eq:mu_of_r} as well as the properties \eqref{eq:lim_W} and \eqref{eq:large_mu_ecmd}, independently of the DFT functional, the present basis set correction
i) can be applied to any WFT model that provides an energy and a density,
ii) does not correct one-electron systems, and
iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered \trashMG{complete basis set} CBS limit for a given WFT model.
iii) vanishes in the limit of a complete basis set, hence guaranteeing an unaltered CBS limit for a given WFT model.
%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Results}
@ -514,7 +514,7 @@ Nevertheless, the deviation observed for the largest basis set is typically with
In most cases, the basis set corrected triple-$\zeta$ atomization energies are on par with the uncorrected quintuple-$\zeta$ ones.
Importantly, the sensitivity with respect to the SR-DFT functional is quite large for the double- and triple-$\zeta$ basis sets, where clearly the PBE functional performs better.
However, from the quadruple-$\zeta$ basis, the LDA and PBE functionals agree within a few tenths of a {\kcal}.
Such weak sensitivity \manu{to the approximated DFT functional} when reaching large basis sets shows the robustness of the approach.
Such weak sensitivity to the density-functional approximation when reaching large basis sets shows the robustness of the approach.
As a second set of numerical examples, we compute the error (with respect to the CBS values) of the atomization energies from the G2 test set with $\modY=\CCSDT$, $\modZ=\ROHF$ and the cc-pVXZ basis sets.
Here, all atomization energies have been computed with the same near-CBS HF/cc-pV5Z energies; only the correlation energy contribution varies from one method to the other.
@ -530,7 +530,7 @@ Already at the CCSD(T)+LDA/cc-pVDZ and CCSD(T)+PBE/cc-pVDZ levels, the MAD is re
With the triple-$\zeta$ basis, the MAD of CCSD(T)+PBE/cc-pVTZ is already below 1 {\kcal} with 36 cases (out of 55) where we achieve chemical accuracy.
CCSD(T)+LDA/cc-pVQZ and CCSD(T)+PBE/cc-pVQZ return MAD of 0.33 and 0.31 kcal/mol (respectively) while CCSD(T)/cc-pVQZ still yields a fairly large MAD of 2.50 {\kcal}.
Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction provides significant basis set \manu{error} reduction and recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
Therefore, similar to F12 methods, \cite{TewKloNeiHat-PCCP-07} we can safely claim that the present basis set correction provides significant basis set reduction and recovers quintuple-$\zeta$ quality atomization and correlation energies with triple-$\zeta$ basis sets for a much cheaper computational cost.
Encouraged by these promising results, we are currently pursuing various avenues toward basis set reduction for strongly correlated systems and electronically excited states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%