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changes in sr functionals

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Julien Toulouse 2019-12-08 22:04:34 +01:00
parent 5a6fdfd263
commit f02786aed4
1 changed files with 22 additions and 8 deletions
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JCTC_revision/GW-srDFT.tex
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@ -71,6 +71,8 @@
\newcommand{\PBEO}{\text{PBE0}}
\newcommand{\srLDA}{\text{srLDA}}
\newcommand{\srPBE}{\text{srPBE}}
\newcommand{\be}[2]{\Bar{\varepsilon}_{#1}^{#2}}
% orbital energies
\newcommand{\e}[1]{\epsilon_{#1}}
@ -500,13 +502,23 @@ Since the present basis-set correction employs complementary short-range correla
%%%%%%%%%%%%%%%%%%%%%%%%
The frequency-independent local self-energy $\bSig{}{\Bas}[\n{}{}](\br{},\br{}') = \bpot{}{\Bas}[\n{}{}](\br{}) \delta(\br{}-\br{}')$ originates from the functional derivative of complementary basis-correction density functionals $\bpot{}{\Bas}[\n{}{}](\br{}) = \delta \bE{}{\Bas}[\n{}{}] / \delta \n{}{}(\br{})$.
Here, we employ two types of complementary, short-range correlation functionals $\bE{}{\Bas}[\n{}{}]$: a short-range local-density approximation ($\srLDA$) functional with multideterminant reference \cite{Toulouse_2005, Paziani_2006} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{Ferte_2019, Loos_2019} which interpolates between the usual PBE functional \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior \cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density from the uniform-electron gas. \cite{Loos_2019}
Additionally to the one-electron density calculated from the HF or KS orbitals, these RS-DFT functionals require a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial inhomogeneity of the basis-set incompleteness error and is computed using the HF or KS opposite-spin pair-density matrix in the basis set $\Bas$.
We refer the interested reader to Refs.~\onlinecite{Giner_2018, Loos_2019, Giner_2019} where our procedure is thoroughly detailed.
\titou{The explicit expressions of these two short-range correlation functionals, as well as their corresponding potentials, are provided in the {\SI}.}
\jt{In this work, we have tested two complementary density functionals coming from two approximations to the short-range correlation functional with multideterminant reference of RS-DFT~\cite{Toulouse_2005}. The first one is a short-range local-density approximation ($\srLDA$)~\cite{Toulouse_2005,Paziani_2006}
\begin{equation}
\label{eq:def_lda_tot}
\bE{\srLDA}{\Bas}[\n{}{}] =
\int \n{}{}(\br{}) \be{\text{c,md}}{\srLDA}\qty(\n{}{}(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation}
where the correlation energy per particle $\be{\text{c,md}}{\srLDA}\qty(\n{}{},\rsmu{}{})$ has been parametrized from uniform-electron gas calculations in Ref.~\onlinecite{Paziani_2006}. The second one is a short-range Perdew-Burke-Ernzerhof ($\srPBE$) approximation \cite{Ferte_2019, Loos_2019}
\begin{equation}
\label{eq:def_pbe_tot}
\bE{\srPBE}{\Bas}[\n{}{}] =
\int \n{}{}(\br{}) \be{\text{c,md}}{\srPBE}\qty(\n{}{}(\br{}),s(\br{}),\rsmu{}{\Bas}(\br{})) \dbr{},
\end{equation}
where $s(\br{})=\nabla n(\br{})/n(\br{})^{4/3}$ is the reduced density gradient and the correlation energy per particle $\be{\text{c,md}}{\srPBE}\qty(\n{}{},s,\rsmu{}{})$ interpolates between the usual PBE correlation energy per particle \cite{Perdew_1996} at $\mu = 0$ and the exact large-$\mu$ behavior \cite{Toulouse_2004, Gori-Giorgi_2006, Paziani_2006} using the on-top pair density of the Coulomb uniform-electron gas (see Ref.~\onlinecite{Loos_2019}). Note that the information on the local basis-set incompleteness error is provided to these RS-DFT functionals through the range-separation function $\rsmu{}{\Bas}(\br{})$.
}
The basis-set corrected {\GOWO} quasiparticle energies are thus given by
\jt{From these energy functionals, we generate the potentials $\bpot{\srLDA}{\Bas}[\n{}{}](\br{}) = \delta \bE{\srLDA}{\Bas}[\n{}{}]/\delta \n{}{}(\br{})$ and $\bpot{\srPBE}{\Bas}[\n{}{}](\br{}) = \delta \bE{\srPBE}{\Bas}[\n{}{}]/\delta \n{}{}(\br{})$ (considering $\rsmu{}{\Bas}(\br{})$ as being fixed) which are then used to obtain the basis-set corrected {\GOWO} quasiparticle energies
\begin{equation}
\beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas}
\label{eq:QP-corrected}
@ -515,11 +527,13 @@ with
\begin{equation}
\begin{split}
\bPot{p}{\Bas}
& = \int \MO{p}(\br{}) \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}') \dbr{} \dbr{}'
\\
& = \int \MO{p}(\br{}) \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{}) \dbr{}.
& = \int \MO{p}(\br{}) \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{}) \dbr{},
\end{split}
\end{equation}
where $\bpot{}{\Bas}[\n{}{}](\br{})=\bpot{\srLDA}{\Bas}[\n{}{}](\br{})$ or $\bpot{\srPBE}{\Bas}[\n{}{}](\br{})$ and the density is calculated from the HF or KS orbitals.
}
\titou{The explicit expressions of these srLDA and srPBE correlation potentials are provided in the {\SI}.}
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis-set correction is a non-self-consistent, \textit{post}-{\GW} correction.
Although outside the scope of this study, various other strategies can be potentially designed, for example, within linearized {\GOWO} or self-consistent {\GW} calculations.