From de11198b1d65fa5035e124c4982dcad11e58f2b6 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 12 Apr 2019 12:31:47 +0200 Subject: [PATCH] valence OK --- Manuscript/G2-srDFT.tex | 45 ++++++++++++++++++++++++----------------- 1 file changed, 27 insertions(+), 18 deletions(-) diff --git a/Manuscript/G2-srDFT.tex b/Manuscript/G2-srDFT.tex index d023cdc..d23e272 100644 --- a/Manuscript/G2-srDFT.tex +++ b/Manuscript/G2-srDFT.tex @@ -61,6 +61,8 @@ \newcommand{\sr}{\text{sr}} \newcommand{\Ne}{N} +\newcommand{\NeUp}{\Ne^{\uparrow}} +\newcommand{\NeDw}{\Ne^{\downarrow}} \newcommand{\Nb}{N_{\Bas}} \newcommand{\Ng}{N_\text{grid}} \newcommand{\nocca}{n_{\text{occ}^{\alpha}}} @@ -87,7 +89,11 @@ % basis sets \newcommand{\Bas}{\mathcal{B}} -\newcommand{\Val}{\text{val}} +\newcommand{\BasFC}{\Bar{\mathcal{B}}} +\newcommand{\FC}{\text{FC}} +\newcommand{\occ}{\text{occ}} +\newcommand{\virt}{\text{virt}} +\newcommand{\val}{\text{val}} \newcommand{\Cor}{\mathcal{C}} % operators @@ -370,12 +376,12 @@ 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 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 $\Cor \bigcap \Val = \O$. -Therefore, we define the valence-only effective interaction +We then naturally split the basis set as $\Bas = \Cor \bigcup \BasFC$, where $\Cor$ is the set of core spinorbitals. +Therefore, we define the FC version of the effective interaction as \begin{equation} - \W{\Bas}{\Val}(\br{1},\br{2}) = + \W{\Bas}{\FC}(\br{1},\br{2}) = \begin{cases} - \f{\Bas}{\Val}(\br{1},\br{2})/\n{2}{\Val}(\br{1},\br{2}), & \text{if $\n{2}{\Val}(\br{1},\br{2})\ne 0$}, + \f{\Bas}{\FC}(\br{1},\br{2})/\n{2}{\FC}(\br{1},\br{2}), & \text{if $\n{2}{\FC}(\br{1},\br{2})\ne 0$}, \\ \infty, & \text{otherwise,} \end{cases} @@ -384,21 +390,21 @@ with \begin{subequations} \begin{gather} \label{eq:fbasisval} - \f{\Bas}{\Val}(\br{1},\br{2}) - = \sum_{pq \in \Bas} \sum_{rstu \in \Val} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, + \f{\Bas}{\FC}(\br{1},\br{2}) + = \sum_{pq \in \Bas} \sum_{rstu \in \BasFC} \SO{p}{1} \SO{q}{2} \V{pq}{rs} \Gam{rs}{tu} \SO{t}{1} \SO{u}{2}, \\ - \n{2}{\Val}(\br{1},\br{2}) - = \sum_{pqrs \in \Val} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, + \n{2}{\FC}(\br{1},\br{2}) + = \sum_{pqrs \in \BasFC} \SO{p}{1} \SO{q}{2} \Gam{pq}{rs} \SO{r}{1} \SO{s}{2}, \end{gather} \end{subequations} -and the corresponding valence range-separation function +and the corresponding FC range-separation function \begin{equation} \label{eq:muval} - \rsmu{\Bas}{\Val}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\Val}(\br{},\br{}). + \rsmu{\Bas}{\FC}(\br{}) = \frac{\sqrt{\pi}}{2} \W{\Bas}{\FC}(\br{},\br{}). \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}. +It is worth noting that, within the present definition, $\W{\Bas}{\FC}(\br{1},\br{2})$ still satisfies Eq.~\eqref{eq:lim_W}. -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 $\bE{\PBE}{\sr}[\n{\modZ}{\Val}(\br{}),\rsmu{\Bas}{\Val}(\br{})]$. +Defining $\n{\modZ}{\FC}$ as the FC (i.e.~valence-only) one-electron density obtained with the model $\modZ$, the FC contribution of the complementary functional $\bE{}{\FC}[\n{\modZ}{\FC}]$ is then evaluated as $\bE{\LDA}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$ or $\bE{\PBE}{\sr}[\n{\modZ}{\FC}(\br{}),\rsmu{\Bas}{\FC}(\br{})]$. %================================================================= %\subsection{Computational considerations} @@ -410,13 +416,16 @@ for $\Gam{rs}{tu}$ and therefore the computational bottleneck reduces to the eva at each quadrature grid point of \begin{equation} \label{eq:fcoal} - \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{i \in \nocca} \sum_{j\in \noccb} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{} + \f{\Bas}{\HF}(\br{}) = \sum_{pq \in \Bas} \sum_{ij}^{\occ} \SO{p}{} \SO{q}{} \V{pq}{ij} \SO{i}{} \SO{j}{}, \end{equation} -which scales as $\Nb^2\times N_{elec}^2 \times \Ng$ and is embarassingly parallel. Within the present formulation, the bottleneck is the four-index transformation to obtain the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}. Nevertheless, this step has in general to be performed before a correlated WFT calculations and therefore it represent a minor limitation. +which scales as $\Nb^2\times \Ne^2 \times \Ng$ and is embarrassingly parallel. +Within the present formulation, the bottleneck is the four-index transformation to obtain the two-electron integrals on the MO basis which appear in \eqref{eq:fcoal}. Nevertheless, this step has in general to be performed before a correlated WFT calculations and therefore it represent a minor limitation. When the four-index transformation become prohibitive, by performing successive matrix multiplications, one could rewrite the equations directly in the AO basis where it scales formally as $\order{\Ng \Nb^4}$ but where one can take advantage of the sparsity atomic-orbital-based algorithms to significantly speed up the calculations. -To conclude this theory session, it is important to notice that the basis set correction proposed here has the folowing properties whatever the approximations made in the DFT part: i) it can be applied to any WFT model that provides an energy and a density, ii) it vanishes for one-electron systems, -iii) it vanishes in the limit of a complete basis set and thus garentees the correct CBS limit of the WFT model. +To conclude this session, we point out that the present basis set correction has, independently of the DFT functional, the following properties: +i) it can be applied to any WFT model that provides an energy and a density, +ii) it does not correct one-electron systems, and +iii) it vanishes in the limit of a complete basis set, hence guaranteeing an unaltered CBS limit for the given WFT model. %%%%%%%%%%%%%%%%%%%%%%%% \section{Results} @@ -529,7 +538,7 @@ RS-DFT and exFCI calculations are performed with {\QP}. \cite{QP2} For the numerical quadrature, we employ the SG-2 grid. \cite{DasHer-JCC-17} Except for the carbon dimer where we have taken the experimental equilibrium bond length (\InAA{1.2425}), all geometries have been extracted from Ref.~\onlinecite{HauJanScu-JCP-09} and have been obtained at the B3LYP/6-31G(2df,p) level of theory. Frozen core calculations are defined as such: an \ce{He} core is frozen from \ce{Li} to \ce{Ne}, while a \ce{Ne} core is frozen from \ce{Na} to \ce{Ar}. -In the context of the basis set correction, the set of valence spinorbitals $\Val$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals. +In the context of the basis set correction, the set of valence spinorbitals $\val$ involved in the definition of the effective interaction refers to the non-frozen spinorbitals. The ``valence'' correction was used consistently when the FC approximation was applied. In order to estimate the complete basis set (CBS) limit for each model, we employed the two-point extrapolation proposed in Ref.~\onlinecite{HalHelJorKloKocOlsWil-CPL-98} for the correlation energies. We refer to these atomization energies as $\CBS$.