From f02786aed4e54c99528b411a5dd2da86f1710ecb Mon Sep 17 00:00:00 2001 From: Julien Toulouse Date: Sun, 8 Dec 2019 22:04:34 +0100 Subject: [PATCH] changes in sr functionals --- JCTC_revision/GW-srDFT.tex | 30 ++++++++++++++++++++++-------- 1 file changed, 22 insertions(+), 8 deletions(-) diff --git a/JCTC_revision/GW-srDFT.tex b/JCTC_revision/GW-srDFT.tex index 77cc62d..50e83e1 100644 --- a/JCTC_revision/GW-srDFT.tex +++ b/JCTC_revision/GW-srDFT.tex @@ -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.