Theory
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\newcommand{\ISCD}{Institut des Sciences du Calcul et des Donn\'ees, Sorbonne Universit\'e, Paris, France}
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@ -193,10 +167,98 @@
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\label{sec:intro}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Theory}
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\label{sec:theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Here, we provide self-contained summary of the main equations and quantities behind {\GOWO} and {\evGW}.
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More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
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For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy is conveniently split in its hole (h) and particle (p) contributions
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\begin{equation}
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\label{eq:SigC}
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\SigC{p}(\omega) = \SigCp{p}(\omega) + \SigCh{p}(\omega),
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\end{equation}
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which, within the GW approximation, read
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\begin{subequations}
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\begin{align}
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\label{eq:SigCh}
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\SigCh{p}(\omega)
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& = 2 \sum_{i}^\text{occ} \sum_{x} \frac{[pi|x]^2}{\omega - \e{i} + \Om{x} - i \eta},
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\\
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\label{eq:SigCp}
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\SigCp{p}(\omega)
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& = 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
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\end{align}
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\end{subequations}
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where $\eta$ is a positive infinitesimal.
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The screened two-electron integrals
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\begin{equation}
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[pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x}
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\end{equation}
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are obtained via the contraction of the bare two-electron integrals \cite{Gill_1994} $(pq|rs)$ and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
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\begin{equation}
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\label{eq:LR}
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\begin{pmatrix}
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\bA & \bB \\
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\bB & \bA \\
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\end{pmatrix}
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\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix}
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=
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\bOm
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\begin{pmatrix}
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\boldsymbol{1} & \boldsymbol{0} \\
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\boldsymbol{0} & \boldsymbol{-1} \\
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\end{pmatrix}
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\begin{pmatrix}
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\bX \\
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\bY \\
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\end{pmatrix},
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\end{equation}
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with
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\begin{align}
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\label{eq:RPA}
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A_{ia,jb} & = \delta_{ij} \delta_{ab} (\epsilon_a - \epsilon_i) + 2 (ia|jb),
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&
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B_{ia,jb} & = 2 (ia|bj),
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\end{align}
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and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
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The one-electron energies $\epsilon_p$ in \eqref{eq:SigCh}, \eqref{eq:SigCp} and \eqref{eq:RPA} are either the HF or the GW quasiparticle energies.
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Equation \eqref{eq:LR} also provides the neutral excitation energies $\Om{x}$.
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In practice, there exist two ways of determining the {\GOWO} QP energies. \cite{Hybertsen_1985a, vanSetten_2013}
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In its ``graphical'' version, they are provided by one of the many solutions of the (non-linear) QP equation
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\begin{equation}
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\label{eq:QP-G0W0}
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\omega = \eHF{p} + \Re[\SigC{p}(\omega)].
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\end{equation}
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In this case, special care has to be taken in order to select the ``right'' solution, known as the QP solution.
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In particular, it is usually worth calculating its renormalization weight (or factor), $\Z{p}(\eHF{p})$, where
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\begin{equation}
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\label{eq:Z}
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\Z{p}(\omega) = \qty[ 1 - \pdv{\Re[\SigC{p}(\omega)]}{\omega} ]^{-1}.
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\end{equation}
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Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
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In a well-behaved case (belonging to the weakly correlated regime), the QP weight is much larger than the sum of the satellite weights, and of the order of $0.7$-$0.9$.
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Within the linearized version of {\GOWO}, one assumes that
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\begin{equation}
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\label{eq:SigC-lin}
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\SigC{p}(\omega) \approx \SigC{p}(\eHF{p}) + (\omega - \eHF{p}) \left. \pdv{\SigC{p}(\omega)}{\omega} \right|_{\omega = \eHF{p}},
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\end{equation}
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that is, the self-energy behaves linearly in the vicinity of $\omega = \eHF{p}$.
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Substituting \eqref{eq:SigC-lin} into \eqref{eq:QP-G0W0} yields
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\begin{equation}
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\label{eq:QP-G0W0-lin}
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\eGOWO{p} = \eHF{p} + \Z{p}(\eHF{p}) \Re[\SigC{p}(\eHF{p})].
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\end{equation}
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Unless otherwise stated, in the remaining of this paper, the {\GOWO} QP energies are determined via the linearized method.
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In the case of {\evGW}, the QP energy, $\eGW{p}$, are obtained via Eq.~\eqref{eq:QP-G0W0}, which has to be solved self-consistently due to the QP energy dependence of the self-energy [see Eq.~\eqref{eq:SigC}]. \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011}
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At least in the weakly correlated regime where a clear QP solution exists, we believe that, within {\evGW}, the self-consistent algorithm should select the solution of the QP equation \eqref{eq:QP-G0W0} with the largest renormalization weight $\Z{p}(\eGW{p})$.
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%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Computational details}
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@ -225,6 +287,6 @@ This work was performed using HPC resources from GENCI-TGCC (Grant No.~2018-A004
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\end{acknowledgements}
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%%%%%%%%%%%%%%%%%%%%%%%%
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\bibliography{GW-srDFT,GW-srDFT-control}
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\bibliography{GW,GW-srDFT,GW-srDFT-control}
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\end{document}
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]
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*)
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||||
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||||
Loading…
Reference in New Issue
Block a user