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valence OK

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Pierre-Francois Loos 2019-04-12 12:31:47 +02:00
parent 717463bad3
commit de11198b1d
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45
Manuscript/G2-srDFT.tex
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@ -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 $\CBS$.