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clean up theory

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Pierre-Francois Loos 2019-10-14 10:30:17 +02:00
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19
Manuscript/GW-srDFT.tex
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@ -259,7 +259,8 @@ In this expression,
\F{}{}[n] = \min_{\wf{}{} \rightsquigarrow \n{}{}} \mel*{\wf{}{}}{\hT + \hWee{}}{\wf{}{}}
\end{equation}
is the exact Levy-Lieb universal density functional, where the notation $\wf{}{} \rightsquigarrow \n{}{}$ in Eq.~\eqref{eq:E0B} states that $\wf{}{}$ yields the one-electron density $\n{}{}$.
$\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators, which is then decomposed as
$\hT$ and $\hWee{}$ are the kinetic and electron-electron interaction operators.
The exact Levy-Lieb universal density functional is then decomposed as
\begin{equation}
\F{}{}[\n{}{}] = \F{}{\Bas}[\n{}{}] + \bE{}{\Bas}[\n{}{}],
\label{eq:Fn}
@ -303,7 +304,7 @@ The stationary condition from Eq.~\eqref{eq:E0BGB} gives the following Dyson equ
(\G{}{\Bas})^{-1} = (\G{0}{\Bas})^{-1}- \Sig{\Hxc}{\Bas}[\G{}{\Bas}]- \bSig{}{\Bas}[\n{\G{}{\Bas}}{}],
\label{eq:Dyson}
\end{equation}
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $\vne(\br{})$
where $(\G{0}{\Bas})^{-1}$ is the basis projection of the inverse non-interacting Green's function with potential $\vne(\br{})$, \textit{i.e.},
\begin{equation}
(\G{0}{})^{-1}(\br{},\br{}',\omega)= \qty(\omega + \frac{\nabla_{\br{}}^2}{2} + \vne(\br{}) + \lambda) \delta(\br{}-\br{}')
\end{equation}
@ -332,12 +333,12 @@ For example, if the reference is Hartree-Fock ($\HF$), $\Sig{\text{ref}}{\Bas}(\
\subsection{The {\GW} Approximation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section, we provide the minimal set of equations required to describe {\GOWO}.
In this subsection, we provide the minimal set of equations required to describe {\GOWO}.
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 orbitals $\MO{p}(\br{})$ (which defines our basis set $\Bas$) are then $\KS$ energies and orbitals.
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.
For a given (occupied or virtual) orbital $p$, the correlation part of the self-energy read, within the {\GW} approximation,
Within the {\GW} approximation, the correlation part of the self-energy reads
\begin{equation}
\label{eq:SigC}
\begin{split}
@ -358,7 +359,7 @@ are obtained via the contraction of the bare two-electron integrals \cite{Gill_1
\begin{equation}
(pq|rs) = \iint \MO{p}(\br{}) \MO{q}(\br{}) \frac{1}{r_{12}} \MO{r}(\br{}') \MO{s}(\br{}') \dbr{} \dbr{}',
\end{equation}
and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
and the transition densities $(\bX+\bY)_{ia}^{x}$ originating from a (direct) random phase approximation (RPA) calculation \cite{Casida_1995, Dreuw_2005}
\begin{equation}
\label{eq:LR}
\begin{pmatrix}
@ -384,7 +385,7 @@ with
B_{ia,jb} & = 2 (ia|bj),
\end{align}
and $\delta_{pq}$ is the Kronecker delta. \cite{NISTbook}
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which are used to build the screened Coulomb potential $\W{}{}$.
Equation \eqref{eq:LR} also provides the RPA neutral excitation energies $\Om{x}$ which represent the poles of the screened Coulomb potential $\W{}{}$.
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}
@ -401,7 +402,7 @@ In Eq.~\eqref{eq:QP-G0W0}, $\Sig{\text{x},p}{\Bas} = \mel*{\MO{p}}{\Sig{\text{x}
\begin{equation}
\Pot{\xc}{\Bas} = \int \pot{\xc}{\Bas}(\br{}) \MO{p}(\br{})^2 \dbr{}.
\end{equation}
In particular, the ionization potential (IP) and electron affinity (EA) are defined as \cite{SzaboBook}
In particular, the ionization potential (IP) and electron affinity (EA) are extracted thanks to the following relationships: \cite{SzaboBook}
\begin{align}
\IP & = -\eGOWO{\HOMO},
&
@ -416,7 +417,7 @@ 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{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}
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.
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}