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Julien Toulouse 2019-10-21 12:03:40 +02:00
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19
Manuscript/GW-srDFT.tex
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@ -327,7 +327,7 @@ The Dyson equation \eqref{eq:Dyson} can be written with an arbitrary reference
(\G{}{\Bas})^{-1} = (\G{\text{ref}}{\Bas})^{-1} - \qty( \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \Sig{\text{ref}}{\Bas} ) - \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
\end{equation}
where $(\G{\text{ref}}{\Bas})^{-1} = (\G{0}{\Bas})^{-1} - \Sig{\text{ref}}{\Bas}$.
For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \Sig{\Hx,\HF}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham ($\KS$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \pot{\Hxc}{\Bas}(\br{}) \delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \Sig{\Hx}{\Bas}(\br{},\br{}')$ is the $\HF$ nonlocal self-energy, and if the reference is Kohn-Sham ($\KS$), $\Sig{\text{ref}}{\Bas}(\br{},\br{}') = \pot{\Hxc}{\Bas}(\br{}) \delta(\br{}-\br{}')$ is the local $\Hxc$ potential.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@ -338,6 +338,7 @@ In this subsection, we provide the minimal set of equations required to describe
More details can be found, for example, in Refs.~\citenum{vanSetten_2013, Kaplan_2016, Bruneval_2016}.
For sake of generality, we consider a $\KS$ reference.
The one-electron energies $\e{p}$ and their corresponding (real-valued) orbitals $\MO{p}(\br{})$ (which defines the basis set $\Bas$) are then $\KS$ energies and orbitals.
\jt{equations are for closed-shell systems}
Within the {\GW} approximation, the correlation part of the self-energy reads
\begin{equation}
@ -351,7 +352,7 @@ Within the {\GW} approximation, the correlation part of the self-energy reads
& + 2 \sum_{a}^\text{virt} \sum_{x} \frac{[pa|x]^2}{\omega - \e{a} - \Om{x} + i \eta},
\end{split}
\end{equation}
where $\eta$ is a positive infinitesimal.
\jt{x should be defined} where $\eta$ is a positive infinitesimal.
The screened two-electron integrals
\begin{equation}
[pq|x] = \sum_{ia} (pq|ia) (\bX+\bY)_{ia}^{x}
@ -385,6 +386,7 @@ with
&
B_{ia,jb} & = 2 (ia|bj),
\end{align}
\jt{the integral in the B term could be written as in the A term since we use real-valued orbitals}
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which represent the poles of the screened Coulomb potential $\W{}{}$.
@ -393,6 +395,7 @@ The {\GOWO} quasiparticle energies $\eGOWO{p}$ are provided by the solution of t
\label{eq:QP-G0W0}
\omega = \e{p} - \Pot{\xc,p}{\Bas} + \Sig{\text{x},p}{\Bas} + \Re[\Sig{\text{c},p}{\Bas}(\omega)].
\end{equation}
\jt{an equation involving $Z_p$ seems to be missing!}
with the largest renormalization weight (or factor)
\begin{equation}
\label{eq:Z}
@ -401,7 +404,7 @@ with the largest renormalization weight (or factor)
Because of sum rules, \cite{Martin_1959, Baym_1961, Baym_1962, vonBarth_1996} the other solutions, known as satellites, share the remaining weight.
In Eq.~\eqref{eq:QP-G0W0}, $\Sig{\text{x},p}{\Bas} = \mel*{\MO{p}}{\Sig{\text{x}}{\Bas}}{\MO{p}}$ is the (static) exchange part of the self-energy and
\begin{equation}
\Pot{\xc}{\Bas} = \int \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{})^2 \dbr{}.
\Pot{\xc,p}{\Bas} = \int \MO{p}(\br{}) \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{}) \dbr{}.
\end{equation}
In particular, the ionization potential (IP) and electron affinity (EA) are extracted thanks to the following relationships: \cite{SzaboBook}
\begin{align}
@ -417,21 +420,21 @@ 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{PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) inspired correlation functional \cite{FerGinTou-JCP-19} which interpolates between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
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{TouGorSav-TCA-05,PazMorGorBac-PRB-06} and a short-range Perdew-Burke-Ernzerhof ($\srPBE$) correlation functional \cite{FerGinTou-JCP-19} which interpolates between the usual PBE functional \cite{PerBurErn-PRL-96} at $\mu = 0$ and the exact large-$\mu$ behavior. \cite{TouColSav-PRA-04, GorSav-PRA-06, PazMorGorBac-PRB-06}
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 opposite-spin on-top pair density.
We refer the interested reader to Refs.~\onlinecite{GinPraFerAssSavTou-JCP-18, LooPraSceTouGin-JPCL-19, GinSceTouLoo-JCP-19} 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
\begin{equation}
\beGOWO{p} = \eGOWO{p} + \bPot{}{\Bas}
\beGOWO{p} = \eGOWO{p} + \bPot{p}{\Bas}
\label{eq:QP-corrected}
\end{equation}
with
\begin{equation}
\begin{split}
\bPot{}{\Bas}
& = \int \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}) \MO{p}(\br{}') \dbr{} \dbr{}'
\bPot{p}{\Bas}
& = \int \MO{p}(\br{}) \bSig{}{\Bas}[\n{}{}](\br{},\br{}') \MO{p}(\br{}') \dbr{} \dbr{}'
\\
& = \int \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{})^2 \dbr{}.
& = \int \MO{p}(\br{}) \bpot{}{\Bas}[\n{}{}](\br{}) \MO{p}(\br{}) \dbr{}.
\end{split}
\end{equation}
As evidenced by Eq.~\eqref{eq:QP-corrected}, the present basis set correction is a non-self-consistent, \textit{post}-GW correction.