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Julien Toulouse 2019-10-24 21:47:53 +02:00
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Manuscript/GW-srDFT.tex
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@ -399,7 +399,7 @@ with
B_{ia,jb} & = 2 (ia|jb),
\end{align}
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which represent the poles of the screened Coulomb potential $\W{}{}$.
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{m}$ which represent the poles of the screened Coulomb interaction $\W{}{}(\omega)$.
The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of the (non-linear) quasiparticle equation \cite{Hybertsen_1985a, vanSetten_2013, Veril_2018}
\begin{equation}
@ -431,7 +431,7 @@ where $\eGOWO{\HOMO}$ and $\eGOWO{\LUMO}$ are the HOMO and LUMO orbital energies
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, these RS-DFT functionals requires a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial non-homogeneity of the basis-set incompleteness error and is computed using the one-electron density and opposite-spin on-top pair density originating from the HF or KS orbitals obtained in the basis set $\Bas$.
Additionally to the one-electron density calculated from the HF or KS orbitals, these RS-DFT functionals requires a range-separation function $\rsmu{}{\Bas}(\br{})$ which automatically adapts to the spatial non-homogeneity 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 and the explicit expressions of these two short-range correlation functionals are provided.
The basis set corrected {\GOWO} quasiparticle energies are thus given by