From 9386e1bd2da058bf1dd7f087015c3cdeb9a9ecbe Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Mon, 14 Oct 2019 10:30:17 +0200 Subject: [PATCH] clean up theory --- Manuscript/GW-srDFT.tex | 19 ++++++++++--------- 1 file changed, 10 insertions(+), 9 deletions(-) diff --git a/Manuscript/GW-srDFT.tex b/Manuscript/GW-srDFT.tex index 7141ff6..87bf4c6 100644 --- a/Manuscript/GW-srDFT.tex +++ b/Manuscript/GW-srDFT.tex @@ -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}