2024-06-17 10:07:19 +02:00
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\newcommand { \T } [1]{ #1^ { \intercal } }
% methods
\newcommand { \evGW } { ev$ GW $ }
\newcommand { \qsGW } { qs$ GW $ }
\newcommand { \scGW } { sc$ GW $ }
\newcommand { \GOWO } { $ G _ 0 W _ 0 $ }
\newcommand { \GOW } { $ G _ 0 W $ }
\newcommand { \GWO } { $ GW _ 0 $ }
\newcommand { \GW } { $ GW $ }
\newcommand { \GT } { $ GT $ }
\newcommand { \GOWOSOSEX } { { \GOWO } +SOSEX}
\newcommand { \GWSOSEX } { { \GW } +SOSEX}
\newcommand { \GnWn } [1]{ $ G _ { # 1 } W _ { # 1 } $ }
\newcommand { \GOF } { $ G _ 0 F 2 $ }
\newcommand { \GF } { $ GF 2 $ }
\newcommand { \KS } { \text { KS} }
\renewcommand { \HF } { \text { HF} }
\newcommand { \RPA } { \text { RPA} }
\newcommand { \RPAx } { \text { RPAx} }
\newcommand { \BSE } { \text { BSE} }
\newcommand { \TDA } { \text { TDA} }
\newcommand { \xc } { \text { xc} }
\newcommand { \Ha } { \text { H} }
\newcommand { \co } { \text { c} }
\newcommand { \x } { \text { x} }
% operators
\newcommand { \hH } { \Hat { H} }
% energies
\newcommand { \Ec } { E_ \text { c} }
\newcommand { \EHF } { E_ \text { HF} }
\newcommand { \EcK } { E_ \text { c} ^ \text { Klein} }
\newcommand { \EcRPA } { E_ \text { c} ^ \text { RPA} }
\newcommand { \EcGM } { E_ \text { c} ^ \text { GM} }
\newcommand { \EcGMGW } { E_ \text { c} ^ \text { GM@GW} }
\newcommand { \EcGMGF } { E_ \text { c} ^ \text { GM@GF2} }
\newcommand { \EcGMGWSOSEX } { E_ \text { c} ^ \text { GM@GW+SOSEX} }
\newcommand { \EcMP } { E_ c^ \text { MP2} }
\newcommand { \EcGF } { E_ c^ \text { \GF } }
\newcommand { \EcGOF } { E_ c^ \text { \GOF } }
\newcommand { \Eg } [1]{ E_ \text { g} ^ { #1} }
\newcommand { \IP } { \text { IP} }
\newcommand { \EA } { \text { EA} }
% orbital energies
\newcommand { \nSat } [1]{ N_ { #1} ^ \text { sat} }
\newcommand { \eSat } [2]{ \epsilon _ { #1,#2} }
\newcommand { \e } [2]{ \epsilon _ { #1} ^ { #2} }
\newcommand { \eHF } [1]{ \epsilon ^ \text { HF} _ { #1} }
\newcommand { \eKS } [1]{ \epsilon ^ \text { KS} _ { #1} }
\newcommand { \eQP } [1]{ \epsilon ^ \text { QP} _ { #1} }
\newcommand { \eGOWO } [1]{ \epsilon ^ \text { \GOWO } _ { #1} }
\newcommand { \eGW } [1]{ \epsilon ^ \text { \GW } _ { #1} }
\newcommand { \eGnWn } [2]{ \epsilon ^ \text { \GnWn { #2} } _ { #1} }
\newcommand { \eGF } [1]{ \epsilon ^ \text { \GF } _ { #1} }
\newcommand { \eGOF } [1]{ \epsilon ^ \text { \GOF } _ { #1} }
\newcommand { \de } [1]{ \Delta \epsilon _ { #1} }
\newcommand { \deHF } [1]{ \Delta \epsilon ^ \text { HF} _ { #1} }
\newcommand { \deKS } [1]{ \Delta \epsilon ^ \text { KS} _ { #1} }
\newcommand { \Om } [2]{ \Omega _ { #1} ^ { #2} }
\newcommand { \eHOMO } [1]{ \epsilon _ \text { HOMO} ^ { #1} }
\newcommand { \eLUMO } [1]{ \epsilon _ \text { LUMO} ^ { #1} }
\newcommand { \cHF } [1]{ c^ \text { HF} _ { #1} }
\newcommand { \cKS } [1]{ c^ \text { KS} _ { #1} }
% Matrix elements
\newcommand { \A } [2]{ A_ { #1} ^ { #2} }
\newcommand { \tA } [2]{ \Tilde { A} _ { #1} ^ { #2} }
\newcommand { \B } [2]{ B_ { #1} ^ { #2} }
\newcommand { \tB } [2]{ \Tilde { B} _ { #1} ^ { #2} }
\renewcommand { \S } [1]{ S_ { #1} }
\newcommand { \ABSE } [1]{ A^ \text { BSE} _ { #1} }
\newcommand { \BBSE } [1]{ B^ \text { BSE} _ { #1} }
\newcommand { \ARPA } [1]{ A^ \text { RPA} _ { #1} }
\newcommand { \BRPA } [1]{ B^ \text { RPA} _ { #1} }
\newcommand { \dABSE } [1]{ \delta A^ \text { BSE} _ { #1} }
\newcommand { \dBBSE } [1]{ \delta B^ \text { BSE} _ { #1} }
\newcommand { \G } [1]{ G_ { #1} }
\newcommand { \Po } [1]{ P_ { #1} }
\newcommand { \W } [1]{ W_ { #1} }
\newcommand { \Wc } [1]{ W^ \text { c} _ { #1} }
\newcommand { \vc } [1]{ v_ { #1} }
\newcommand { \SigX } [1]{ \Sigma ^ \text { x} _ { #1} }
\newcommand { \SigC } [1]{ \Sigma ^ \text { c} _ { #1} }
\newcommand { \Sig } [2]{ \Sigma _ { #1} ^ { #2} }
\newcommand { \SigGW } [1]{ \Sigma ^ \text { \GW } _ { #1} }
\newcommand { \SigGWSOSEX } [1]{ \Sigma ^ \text { \GWSOSEX } _ { #1} }
\newcommand { \SigGF } [1]{ \Sigma ^ \text { \GF } _ { #1} }
\newcommand { \Z } [1]{ Z_ { #1} }
% excitation energies
\newcommand { \OmRPA } [1]{ \Omega ^ \text { RPA} _ { #1} }
\newcommand { \OmCIS } [1]{ \Omega ^ \text { CIS} _ { #1} }
\newcommand { \OmTDHF } [1]{ \Omega ^ \text { TDHF} _ { #1} }
\newcommand { \OmBSE } [1]{ \Omega ^ \text { BSE} _ { #1} }
\newcommand { \spinup } { \downarrow }
\newcommand { \spindw } { \uparrow }
\newcommand { \singlet } { \uparrow \downarrow }
\newcommand { \triplet } { \uparrow \uparrow }
\newcommand { \Oms } [1]{ { } ^ { 1} \Omega _ { #1} }
\newcommand { \OmsRPA } [1]{ { } ^ { 1} \Omega ^ \text { RPA} _ { #1} }
\newcommand { \OmsCIS } [1]{ { } ^ { 1} \Omega ^ \text { CIS} _ { #1} }
\newcommand { \OmsTDHF } [1]{ { } ^ { 1} \Omega ^ \text { TDHF} _ { #1} }
\newcommand { \OmsBSE } [1]{ { } ^ { 1} \Omega ^ \text { BSE} _ { #1} }
\newcommand { \Omt } [1]{ { } ^ { 3} \Omega _ { #1} }
\newcommand { \OmtRPA } [1]{ { } ^ { 3} \Omega ^ \text { RPA} _ { #1} }
\newcommand { \OmtCIS } [1]{ { } ^ { 3} \Omega ^ \text { CIS} _ { #1} }
\newcommand { \OmtTDHF } [1]{ { } ^ { 3} \Omega ^ \text { TDHF} _ { #1} }
\newcommand { \OmtBSE } [1]{ { } ^ { 3} \Omega ^ \text { BSE} _ { #1} }
\newcommand { \MO } [1]{ \phi _ { #1} }
\newcommand { \ERI } [2]{ (#1|#2)}
\newcommand { \rbra } [1]{ (#1|}
\newcommand { \rket } [1]{ |#1)}
\newcommand { \sERI } [2]{ [#1|#2]}
\newcommand { \sig } { \sigma }
\newcommand { \sigp } { \sigma '}
% Matrices
\newcommand { \bE } { \bm { E} }
\newcommand { \bG } { \bm { G} }
\newcommand { \bF } { \bm { F} }
\newcommand { \bFHF } { \bm { F} ^ \text { HF} }
\newcommand { \bH } { \bm { H} }
\newcommand { \bh } { \bm { h} }
\newcommand { \bvc } { \bm { v} }
\newcommand { \bSig } { \bm { \Sigma } }
\newcommand { \bSigX } { \bm { \Sigma } ^ \text { x} }
\newcommand { \bSigC } { \bm { \Sigma } ^ \text { c} }
\newcommand { \bSigGW } { \bm { \Sigma } ^ \text { \GW } }
\newcommand { \bSigGWSOSEX } { \bm { \Sigma } ^ \text { \GWSOSEX } }
\newcommand { \bSigGF } { \bm { \Sigma } ^ \text { \GF } }
\newcommand { \be } { \bm { \epsilon } }
\newcommand { \bDelta } { \bm { \Delta } }
\newcommand { \beHF } { \bm { \epsilon } ^ \text { HF} }
\newcommand { \beKS } { \bm { \epsilon } ^ \text { KS} }
\newcommand { \bcHF } { \bm { c} ^ \text { HF} }
\newcommand { \bcKS } { \bm { c} ^ \text { KS} }
\newcommand { \beGW } { \bm { \epsilon } ^ \text { \GW } }
\newcommand { \beGnWn } [1]{ \bm { \epsilon } ^ \text { \GnWn { #1} } }
\newcommand { \bcGnWn } [1]{ \bm { c} ^ \text { \GnWn { #1} } }
\newcommand { \beGF } { \bm { \epsilon } ^ \text { \GF } }
\newcommand { \bde } { \bm { \Delta \epsilon } }
\newcommand { \bdeHF } { \bm { \Delta \epsilon } ^ \text { HF} }
\newcommand { \bdeGW } { \bm { \Delta \epsilon } ^ \text { GW} }
\newcommand { \bdeGF } { \bm { \Delta \epsilon } ^ \text { GF2} }
\newcommand { \bO } { \bm { 0} }
\newcommand { \bI } { \bm { 1} }
\newcommand { \bOm } [2]{ \bm { \Omega } _ { #1} ^ { #2} }
\newcommand { \bA } [2]{ \bm { A} _ { #1} ^ { #2} }
\newcommand { \btA } [2]{ \bm { \Tilde { A} } _ { #1} ^ { #2} }
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\newcommand { \bY } [2]{ \bm { Y} _ { #1} ^ { #2} }
\newcommand { \bZ } [2]{ \bm { Z} _ { #1} ^ { #2} }
\newcommand { \bK } [2]{ \blue { \bm { K} } _ { #1} ^ { #2} }
\newcommand { \bP } [2]{ \red { \bm { P} } _ { #1} ^ { #2} }
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% *************
% * HEAD DATA *
% *************
\title [Green's function-based methods in chemistry] {
Green's function-based methods in chemistry
}
\author [PF Loos (\url{https://pfloos.github.io/WEB_LOOS})] { Pierre-Fran\c { c} ois LOOS}
\date { ISTPC 2024}
\institute [CNRS@LCPQ] {
Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\
Universit\' e de Toulouse, CNRS, UPS, Toulouse, France.
}
\titlegraphic {
\includegraphics [width=0.3\textwidth] { fig/jarvis}
\\
\vspace { 0.05\textheight }
\includegraphics [height=0.05\textwidth] { fig/UPS}
\hspace { 0.2\textwidth }
\includegraphics [height=0.05\textwidth] { fig/ERC}
\hspace { 0.2\textwidth }
\includegraphics [height=0.05\textwidth] { fig/LCPQ}
\hspace { 0.2\textwidth }
\includegraphics [height=0.05\textwidth] { fig/CNRS}
}
\begin { document}
%-----------------------------------------------------
\begin { frame}
\titlepage
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Today's program}
\begin { itemize}
\item \textbf { Charged excitations}
\begin { itemize}
\item One-shot $ GW $ (\GOWO )
\item Partially self-consistent eigenvalue $ GW $ (\evGW )
\item Quasiparticle self-consistent $ GW $ (\qsGW )
\item Other self-energies (GF2, SOSEX, T-matrix, etc)
\end { itemize}
\bigskip
\item \textbf { Neutral excitations}
\begin { itemize}
\item Random-phase approximation (RPA)
\item Configuration interaction with singles (CIS)
\item Time-dependent Hartree-Fock (TDHF) or RPA with exchange (RPAx)
\item Time-dependent density-functional theory (TDDFT)
\item Bethe-Salpeter equation (BSE) formalism
\end { itemize}
\bigskip
\item \textbf { Correlation energy}
\begin { itemize}
\item Plasmon (or trace) formula
\item Galitski-Migdal formulation
\item Adiabatic connection fluctuation-dissipation theorem (ACFDT)
\end { itemize}
\end { itemize}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section { Motivations}
\begin { frame}
\tableofcontents [currentsection]
\end { frame}
%-----------------------------------------------------
\begin { frame} { L\" owdin partitioning technique}
\begin { block} { Folding or dressing process}
\begin { equation}
\underbrace { \bH { } { } \cdot \bc = \yo \, \bc } _ { \text { A large linear system with $ N $ solutions\ldots } }
\qq { $ \Rightarrow $ }
\begin { pmatrix}
\overbrace { \bH _ 0} ^ { N_ 0 \times N_ 0} & \T { \bh } \\
\bh & \underbrace { \bH _ 1} _ { N_ 1 \times N_ 1} \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bc _ 0 \\
\bc _ 1 \\
\end { pmatrix}
= \yo
\begin { pmatrix}
\bc _ 0 \\
\bc _ 1 \\
\end { pmatrix}
\qquad N = N_ 0 + N_ 1
\end { equation}
\begin { align}
\qq * { \bf Row \# 2:}
& \bh \cdot \bc _ 0 + \bH _ 1 \cdot \bc _ 1 = \yo \, \bc _ 1
& \qq { $ \Rightarrow $ }
& \bc _ 1 = (\yo \, \bI - \bH _ 1)^ { -1} \cdot \bh \cdot \bc _ 0
\\
\qq * { \bf Row \# 1:}
& \bH _ 0 \cdot \bc _ 0 + \T { \bh } \cdot \bc _ 1 = \yo \, \bc _ 0
& \qq { $ \Rightarrow $ }
& \underbrace { \Tilde { \bH } _ 0(\yo ) \cdot \bc _ 0 = \yo \, \bc _ 0} _ { \text { A smaller non-linear system with $ N $ solutions\ldots } }
\end { align}
\begin { equation}
\boxed {
\underbrace { \Tilde { \bH } _ 0(\yo )} _ { \text { Effective Hamitonian} }
= \bH _ 0 + \underbrace { \T { \bh } \cdot (\yo \, \bI - \bH _ 1)^ { -1} \cdot \bh } _ { \text { Self-Energy $ \bSig ( \yo ) $ } }
}
\end { equation}
\begin { equation}
\qq * { Static approx. (e.g.~$ \yo = 0 $ ):}
\underbrace { \Tilde { \bH } _ 0(\yo = 0)} _ { \text { A smaller linear system with $ N _ 0 $ solutions\ldots } }
= \bH _ 0 - \underbrace { \T { \bh } \cdot \bH _ 1^ { -1} \cdot \bh } _ { \text { approximations possible...} }
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
\begin { frame} { Green's Function}
\begin { block} { Many-Body Green's Function}
\begin { equation}
\boxed { \qty ( \yo \bI - \bH ) \cdot \bG = \bI }
\end { equation}
\end { block}
\begin { block} { Dyson equation}
\begin { equation}
\Tilde { \bH } _ 0(\yo ) \cdot \bc _ 0 = \yo \bc _ 0
\qq { $ \Rightarrow $ }
\qty [ \bH_0 + \bSig(\yo) ] \cdot \bc _ 0 = \yo \bc _ 0
\qq { $ \Rightarrow $ }
\underbrace { \qty [ \yo \bI - \bH_0 - \bSig(\yo) ] } _ { \bG ^ { -1} (\yo )} \cdot \bc _ 0 = \bO
\end { equation}
\begin { align}
\bG ^ { -1} (\yo ) = \underbrace { \yo \bI - \bH _ 0} _ { \bG _ 0^ { -1} (\yo )} - \bSig (\yo )
& \qq { $ \Rightarrow $ }
\bG ^ { -1} (\yo ) = \bG _ 0^ { -1} (\yo ) - \bSig (\yo )
\\
& \qq { $ \Rightarrow $ }
\boxed { \bG (\yo ) = \bG _ 0(\yo ) + \bG _ 0(\yo ) \cdot \bSig (\yo ) \cdot \bG (\yo )}
\\
& \qq { $ \Rightarrow $ }
\bG (\yo ) = \qty [ \bI - \bG_0(\yo) \cdot \bSig(\yo) ] ^ { -1} \bG _ 0(\yo )
\end { align}
\end { block}
\end { frame}
%-----------------------------------------------------
\begin { frame} { Non-Interacting Green's Function}
\begin { block} { Matrix representation}
\begin { equation}
\bH _ 0 \cdot \bc = \bc \cdot \bE
\qq { $ \Rightarrow $ }
\bH _ 0 \cdot \underbrace { \bc \cdot \bc ^ \dag } _ { \bI } = \bc _ 0 \cdot \bE \cdot \bc ^ \dag
\qq { $ \Rightarrow $ }
\bH _ 0 = \bc \cdot \bE \cdot \bc ^ \dag
\end { equation}
\begin { equation}
\yo \bI - \bH _ 0 = \bc \cdot \qty ( \yo \bI - \bE ) \cdot \bc ^ \dag
\qq { $ \Rightarrow $ }
\underbrace { \qty ( \yo \bI - \bH _ 0 )^ { -1} } _ { \bG _ 0} = \bc \cdot \qty ( \yo \bI - \bE )^ { -1} \cdot \bc ^ \dag
\end { equation}
\begin { equation}
\bG _ 0 = \bc \cdot \qty ( \yo \bI - \bE )^ { -1} \cdot \bc ^ \dag
\qq { $ \Rightarrow $ }
(\bG _ 0)_ { pq} = \sum _ { r} \frac { c_ { pr} c_ { qr} ^ *} { \yo - E_ r}
\end { equation}
\end { block}
\begin { block} { Hartree-Fock Green's function}
\begin { equation}
(\bG _ \text { \HF } )_ { pq}
= \sum _ { r} \frac { c_ { pr} c_ { qr} ^ *} { \yo - \e { r} { \HF } }
= \underbrace { \sum _ { i} \frac { c_ { pi} c_ { qi} ^ *} { \yo - \e { i} { \HF } } } _ { \text { removal} }
+ \underbrace { \sum _ { a} \frac { c_ { pa} c_ { qa} ^ *} { \yo - \e { a} { \HF } } } _ { \text { addition} }
\end { equation}
\end { block}
\end { frame}
\begin { frame} { Solving Dyson's Equation}
We're looking for the poles of $ \bG ( \yo ) $ :
\begin { equation}
\boxed { \bG ^ { -1} (\yo ) = \bG _ 0^ { -1} (\yo ) - \bSig (\yo )}
\qq { $ \Rightarrow $ }
\bG _ 0^ { -1} (\yo ) - \bSig (\yo ) = \bO
\qq { $ \Rightarrow $ }
\yo \bI - \be - \bSig (\yo ) = \bO
\end { equation}
\begin { block} { Diagonal approximation}
\begin { equation}
\yo \bI - \be - \bSig (\yo ) = \bO
\qq { $ \Rightarrow $ }
\yo - \e { p} { \HF } - \Sig { pp} { } (\yo ) = 0
\end { equation}
\end { block}
\begin { block} { Linearization}
\begin { equation}
\Sig { pp} { } (\yo ) \approx \Sig { pp} { } (\yo = \e { p} { \HF } ) + \qty (\yo - \e { p} { \HF } ) \eval { \pdv { \Sig { pp} { } (\yo )} { \yo } } _ { \yo = \e { p} { \HF } }
\qq { $ \Rightarrow $ }
\e { p} { } = \e { p} { \HF } + Z_ p \Sig { pp} { } (\yo )
\end { equation}
\begin { equation}
\qq * { Renormalization Factor:} Z_ p = \frac { 1} { 1 - \eval { \pdv { \Sig { pp} { } (\yo )} { \yo } } _ { \yo = \e { p} { \HF } } }
\end { equation}
\end { block}
\end { frame}
\begin { frame} { Spectral Function}
\begin { equation}
\bSig (\yo ) = \Re \bSig (\yo ) + i \Im \bSig (\yo )
\end { equation}
\begin { equation}
\bA { } { } (\yo )
= - \frac { 1} { \pi } \Im \abs { \bG (\yo )}
= - \frac { 1} { \pi } \frac { \abs { \Im \bSig (\yo )} } { \qty [\yo \bI - \be - \Re \bSig(\yo)] ^ 2 + \qty [ \Im \bSig(\yo)] ^ 2}
\end { equation}
\end { frame}
2024-06-17 17:47:30 +02:00
2024-06-17 10:07:19 +02:00
%-----------------------------------------------------
\section { Context}
\begin { frame}
\tableofcontents [currentsection]
\end { frame}
%-----------------------------------------------------
\begin { frame} { Assumptions \& Notations}
\begin { block} { Let's talk about notations}
\begin { itemize}
\item We consider \blue { closed-shell systems} (2 opposite-spin electrons per orbital)
\item We only deal with \blue { singlet excited states} but \purple { triplets} can also be obtained
\bigskip
\item Number of \green { occupied orbitals} $ O $
\item Number of \alert { vacant orbitals} $ V $
\item \violet { Total number of orbitals} $ N = O + V $
\bigskip
\item $ \MO { p } ( \br ) $ is a (real) \blue { spatial orbital}
\item $ i,j,k,l $ are \green { occupied orbitals}
\item $ a,b,c,d $ are \alert { vacant orbitals}
\item $ p,q,r,s $ are \violet { arbitrary (occupied or vacant) orbitals}
\item $ \mu , \nu , \lambda , \sigma $ are \purple { basis function indexes}
\bigskip
\item $ m $ indexes \purple { the $ OV $ single excitations} ($ i \to a $ )
\end { itemize}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Useful papers/programs}
\begin { itemize}
\item \red { mol$ GW $ :} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
\bigskip
\item \green { Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528
\bigskip
\item \violet { Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
\bigskip
\item \purple { FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
\bigskip
\item \orange { Reviews \& Books:}
\begin { itemize}
\item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344
\item Onida et al. Rev. Mod. Phys. 74 (2002) 601
\item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022
\item Golze et al. Front. Chem. 7 (2019) 377
\item Blase et al. JPCL 11 (2020) 7371
\item Martin, Reining \& Ceperley \textit { Interacting Electrons} (Cambridge University Press)
\end { itemize}
\bigskip
\item \red { $ GW $ 100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url { http://gw100.wordpress.com} )
\end { itemize}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Fundamental and optical gaps (\copyright ~Bruno Senjean)}
\begin { center}
\includegraphics [width=\textwidth] { fig/gaps}
\end { center}
\begin { equation}
\underbrace { \Eg { \KS } } _ { \text { KS gap} } = \eLUMO { \KS } - \eHOMO { \KS } \ll \underbrace { \green { \Eg { GW} } } _ { \text { \green { { \GW } gap} } } = \eLUMO { GW} - \eHOMO { GW}
\end { equation}
\begin { equation}
\underbrace { \blue { \Eg { \text { opt} } } } _ { \text { \blue { optical gap} } } = E_ 1^ N - E_ 0^ N = \underbrace { \red { \Eg { \text { fund} } } } _ { \text { \red { fundamental gap} } } + \underbrace { \purple { E_ \text { B} } } _ { \text { \purple { excitonic effect} } }
\end { equation}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Hedin's pentagon}
\begin { columns}
\begin { column} { 0.4\textwidth }
\centering
\includegraphics [width=0.8\linewidth] { fig/pentagon}
\\
\pub { Hedin, Phys Rev 139 (1965) A796}
\end { column}
\begin { column} { 0.6\textwidth }
\begin { block} { What can you calculate with $ GW $ ?}
\begin { itemize}
\item Ionization potentials (IPs) given by occupied MO energies
\item Electron affinities (EAs) given by virtual MO energies
\item Fundamental (HOMO-LUMO) gap (or band gap in solids)
\item Correlation and total energies
\end { itemize}
\end { block}
\begin { block} { What can you calculate with BSE?}
\begin { itemize}
\item Singlet and triplet optical excitations (vertical absorption energies)
\item Oscillator strengths (absorption intensities)
\item Correlation and total energies
\end { itemize}
\end { block}
\end { column}
\end { columns}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { The MBPT chain of actions}
\begin { center}
\includegraphics [width=0.7\textwidth] { fig/BSE-GW}
\\
\bigskip
\pub { Blase et al. JPCL 11 (2020) 7371}
\end { center}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Photochemistry: Jablonski diagram}
% colors
\definecolor { turquoise} { rgb} { 0 0.41 0.41}
\definecolor { rouge} { rgb} { 0.79 0.0 0.1}
\definecolor { vert} { rgb} { 0.15 0.4 0.1}
\definecolor { mauve} { rgb} { 0.6 0.4 0.8}
\definecolor { violet} { rgb} { 0.58 0. 0.41}
\definecolor { orange} { rgb} { 0.8 0.4 0.2}
\definecolor { bleu} { rgb} { 0.39, 0.58, 0.93}
\begin { center}
\begin { tikzpicture} [scale=0.7]
% styles
\tikzstyle { elec} = [line width=2pt,draw=black!80]
\tikzstyle { vib} = [thick,draw=black!30]
\tikzstyle { trans} = [line width=2pt,->]
\tikzstyle { transCI} = [trans,dashed,draw=vert]
\tikzstyle { transCS} = [trans,dashed,draw=violet]
\tikzstyle { relax} = [draw=orange,ultra thick,decorate,decoration=snake]
\tikzstyle { rv} = [rotate=90,text=orange,pos=0.5,yshift=3mm]
% fondamental
\path [elec] (0,0) -- ++ (14,0)
node[below,pos=0.5,yshift=-1mm] { Ground state $ S _ 0 $ } ;
\path [vib] (0,0.2) -- ++ (14,0);
\path [vib] (0,0.4) -- ++ (13,0);
\foreach \i in { 1,2,...,30} {
\path [vib] (0,0.4 + \i * 0.2) -- ++ ({ 2 + 10*exp(-0.2*\i )} ,0);
}
% T1
\path [elec] (11,4) -- ++ (3,0) node[anchor=south west] { $ T _ 1 $ } ;
\foreach \i in { 1,2,...,6} {
\path [vib] (11,4 + \i * 0.2) -- ++ (3,0);
}
% S1
\path [elec] (4,5) node[anchor=south east] { $ S _ 1 $ } -- ++ (5,0);
\foreach \i in { 1,2,...,6} {
\path [vib] (4,5 + \i * 0.2) -- ++ (5,0);
}
\foreach \i in { 1,2,...,12} {
\path [vib] ({ 7.5 - 1*exp(-0.3*\i )} ,6.2+\i * 0.2) -- (9,6.2+\i * 0.2);
}
% S2
\path [elec] (4,8) node[anchor=south east] { $ S _ 2 $ } -- ++ (2,0);
\foreach \i in { 1,2,...,6} {
\path [vib] (4,8 + \i * 0.2) -- ++ (2,0);
}
% absorption
\path [trans,draw=turquoise] (4.5,0) -- ++(0,9)
node[rotate=90,pos=0.35,text=turquoise,yshift=-3mm] { \small Absorption} ;
% fluo
\path [trans,draw=rouge] (7,5) -- ++(0,-4.4)
node[rotate=90,pos=0.5,text=rouge,yshift=-3mm] { \small Fluorescence} ;
% phosphorescence
\path [trans,draw=mauve] (13,4) -- ++(0,-3.4)
node[rotate=90,pos=0.5,text=mauve,yshift=-3mm] { \small Phosphorescence} ;
% Conversion interne
\path [transCI] (4,5) -- ++(-1.9,0) node[below,pos=0.5,text=vert] { \small IC} ;
\path [transCI] (6,8) -- ++(1.3,0) node[above,pos=0.5,text=vert] { \small IC} ;
% Croisement intersysteme
\path [transCS] (9,5) -- ++(2,0) node[below,pos=0.5,text=violet] { \small ISC} ;
\path [transCS] (11,4) -- ++(-2.5,0) node[below,pos=0.5,text=violet] { \small ISC} ;
% relaxation vib
\path [relax] (5.5,8.8) -- ++(0,-0.8) node[rv] { \small \textbf { VR} } ;
\path [relax] (8,8) -- ++(0,-3) node[rv] { \small \textbf { VR} } ;
\path [relax] (1,5) -- ++(0,-5) node[rv] { \small \textbf { VR} } ;
\path [relax] (11.5,5) -- ++(0,-1) node[rv] { \small \textbf { VR} } ;
\end { tikzpicture}
\end { center}
%\tiny
%\begin{itemize}
% \item[] \tikz {\path[line width=2pt,->,dashed,draw=vert]
% (0,0) -- (1,0) node[above,pos=0.5,text=vert] {IC};} Internal Conversion,
% $S_i\,\longrightarrow\,S_j$ non radiative transition.
%
% \item[] \tikz {\path[line width=2pt,->,dashed,draw=violet]
% (0,0) -- (1,0) node[above,pos=0.5,text=violet] {ISC};} InterSystem Crossing,
% $S_i\,\longrightarrow\,T_j$ non radiative transition.
%
% \item[] \tikz {\path[line width=2pt,draw=orange,ultra thick,
% decorate,decoration=snake] (0,0) -- (1,0) node[above,pos=0.5,text=orange] {RV};}
% Vibrationnal Relaxation.
%\end{itemize}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Photochemistry: absorption, emission, and 0-0}
\begin { center}
\includegraphics [width=0.5\textwidth] { fig/0-0}
\\
\textbf { \alert { Vertical excitation energies cannot be computed experimentally!!!} }
\end { center}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section { Charged excitations}
\begin { frame}
\tableofcontents [currentsection]
\end { frame}
%-----------------------------------------------------
\begin { frame} { Green's function and dynamical screening}
\begin { block} { One-body Green's function}
\begin { equation}
\blue { G} (\br _ 1,\br _ 2;\yo )
= \underbrace { \sum _ i \frac { \MO { i} (\br _ 1) \MO { i} (\br _ 2)} { \yo - \e { i} { } - i\eta } } _ { \text { \green { removal part = IPs} } }
+ \underbrace { \sum _ a \frac { \MO { a} (\br _ 1) \MO { a} (\br _ 2)} { \yo - \e { a} { } + i\eta } } _ { \text { \red { addition part = EAs} } }
\end { equation}
\end { block}
\begin { block} { Polarizability}
\begin { equation}
P(\br _ 1,\br _ 2;\yo ) = - \frac { i} { \pi } \int \blue { G} (\br _ 1,\br _ 2;\yo +\omega ') \blue { G} (\br _ 1,\br _ 2;\omega ') d\omega '
\end { equation}
\end { block}
\begin { block} { Dielectric function and dynamically-screened Coulomb potential}
\begin { equation}
\epsilon (\br _ 1,\br _ 2;\yo ) = \delta (\br _ 1 - \br _ 2) - \int \frac { P(\br _ 1,\br _ 3;\yo ) } { \abs { \br _ 2 - \br _ 3} } d\br _ 3
\end { equation}
\begin { equation}
\highlight { W} (\br _ 1,\br _ 2;\yo ) = \int \frac { \epsilon ^ { -1} (\br _ 1,\br _ 3;\yo ) } { \abs { \br _ 2 - \br _ 3} } d\br _ 3
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Dynamical screening in the orbital basis}
\begin { block} { Spectral representation of $ W $ }
\begin { equation}
\begin { split}
\highlight { W} _ { pq,rs} (\yo )
& = \iint \MO { p} (\br _ 1) \MO { q} (\br _ 1) \highlight { W} (\br _ 1,\br _ 2;\yo ) \MO { r} (\br _ 2) \MO { s} (\br _ 2) d\br _ 1 d\br _ 2
\\
& = \underbrace { \ERI { pq} { rs} } _ { \text { (static) exchange part} }
+ \underbrace { 2 \sum _ m \violet { \ERI { pq} { m} } \violet { \ERI { rs} { m} }
\qty [ \frac{1}{\yo - \orange{\Om{m}{\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\RPA}} - i \eta} ] } _ { \text { (dynamical) correlation part } \highlight { W} ^ { \co } _ { pq,rs} (\yo )}
\end { split}
\end { equation}
\end { block}
\begin { block} { Electron repulsion integrals (ERIs)}
\begin { equation}
\ERI { pq} { rs} = \iint \frac { \MO { p} (\br _ 1) \MO { q} (\br _ 1) \MO { r} (\br _ 2) \MO { s} (\br _ 2)} { \abs { \br _ 1 - \br _ 2} } d\br _ 1 d\br _ 2
\end { equation}
\end { block}
\begin { block} { Screened ERIs (or spectral weights)}
\begin { equation}
\violet { \ERI { pq} { m} } = \sum _ { ia} \ERI { pq} { ia} (\orange { \bX { m} { \RPA } +\bY { m} { \RPA } } )_ { ia}
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Computation of the dynamical screening}
\begin { block} { Direct (ph-)RPA calculation (pseudo-hermitian linear problem)}
\begin { equation}
\begin { pmatrix}
\bA { } { \RPA } & \bB { } { \RPA } \\
-\bB { } { \RPA } & -\bA { } { \RPA } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\orange { \bX { m} { \RPA } } \\
\orange { \bY { m} { \RPA } } \\
\end { pmatrix}
=
\orange { \Om { m} { \RPA } }
\begin { pmatrix}
\orange { \bX { m} { \RPA } } \\
\orange { \bY { m} { \RPA } } \\
\end { pmatrix}
\end { equation}
\begin { equation}
\qq * { For singlet states:} \A { ia,jb} { \RPA } = \delta _ { ij} \delta _ { ab} (\e { a} { } - \e { i} { } ) + 2\ERI { ia} { bj}
\qquad
\B { ia,jb} { \RPA } = 2\ERI { ia} { jb}
\end { equation}
\end { block}
\begin { block} { Non-hermitian to hermitian}
\begin { equation}
(\bA { } { } - \bB { } { } )^ { 1/2} \cdot (\bA { } { } + \bB { } { } ) \cdot (\bA { } { } - \bB { } { } )^ { 1/2} \cdot \bZ { m} { } = \Om { m} { 2} \, \bZ { m} { }
\end { equation}
\begin { gather}
(\bX { m} { } + \bY { m} { } ) = \Om { m} { -1/2} (\bA { } { } - \bB { } { } )^ { +1/2} \cdot \bZ { m} { }
\\
(\bX { m} { } - \bY { m} { } ) = \Om { m} { +1/2} (\bA { } { } - \bB { } { } )^ { -1/2} \cdot \bZ { m} { }
\end { gather}
\end { block}
\begin { block} { Tamm-Dancoff approximation (TDA)}
\begin { equation}
\bB { } { } = \bO \quad \Rightarrow \quad \bA { } { } \cdot \orange { \bX { m} { } } = \orange { \Om { m} { \TDA } \bX { m} { } }
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { The self-energy}
\begin { block} { $ GW $ Self-energy}
\begin { equation}
\underbrace { \red { \Sig { } { \xc } } (\br _ 1,\br _ 2;\yo )} _ { \text { $ GW $ self-energy} }
= \underbrace { \purple { \Sig { } { \x } } (\br _ 1,\br _ 2)} _ { \text { \purple { exchange} } }
+ \underbrace { \red { \Sig { } { \co } } (\br _ 1,\br _ 2;\yo )} _ { \text { \red { correlation} } }
= \frac { i} { 2\pi } \int \blue { G} (\br _ 1,\br _ 2;\yo +\omega ') \highlight { W} (\br _ 1,\br _ 2;\omega ') e^ { i \eta \omega '} d\omega '
\end { equation}
\end { block}
\begin { block} { Exchange part of the (static) self-energy}
\begin { equation}
\purple { \Sig { pq} { \x } } = - \sum _ { i} \ERI { pi} { iq}
\end { equation}
\end { block}
\begin { block} { Correlation part of the (dynamical) self-energy}
\begin { equation}
\red { \Sig { pq} { \co } } (\yo )
= 2 \sum _ { im} \frac { \violet { \ERI { pi} { m} } \violet { \ERI { qi} { m} } } { \yo - \e { i} { } + \orange { \Om { m} { \RPA } } - i \eta }
+ 2 \sum _ { am} \frac { \violet { \ERI { pa} { m} } \violet { \ERI { qa} { m} } } { \yo - \e { a} { } - \orange { \Om { m} { \RPA } } + i \eta }
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Quasiparticle equation}
\begin { block} { Dyson equation}
\begin { equation}
\qty [ \blue{G}(\br_1,\br_2;\yo) ] ^ { -1}
= \underbrace { \qty [ G_{\KS}(\br_1,\br_2;\yo) ] ^ { -1} } _ { \text { KS Green's function} }
+ \red { \Sig { } { \xc } } (\br _ 1,\br _ 2;\yo ) - \underbrace { \upsilon ^ { \xc } (\br _ 1)} _ { \text { KS potential} } \delta (\br _ 1 - \br _ 2)
\end { equation}
\end { block}
\begin { block} { Non-linear quasiparticle (QP) equation}
\begin { equation}
\yo = \eKS { p} + \red { \Sig { pp} { \xc } } (\yo ) - V_ { p} ^ { \xc }
\qq { with}
V_ { p} ^ { \xc } = \int \MO { p} (\br ) \upsilon ^ { \xc } (\br ) \MO { p} (\br ) d\br
\end { equation}
\end { block}
\begin { block} { Linearized QP equation}
\begin { equation}
\red { \Sig { pp} { \xc } } (\yo ) \approx \red { \Sig { pp} { \xc } } (\eKS { p} ) + (\yo - \eKS { p} ) \left . \pdv { \red { \Sig { pp} { \xc } } (\yo )} { \yo } \right |_ { \yo = \eKS { p} }
\qq { $ \Rightarrow $ }
\blue { \eGW { p} } = \eKS { p} + \green { Z_ { p} } [\red { \Sig { pp} { \xc } } (\eKS { p} ) - V_ { p} ^ { \xc } ]
\end { equation}
\begin { equation}
\underbrace { \green { Z_ { p} } } _ { \text { renormalization factor} } = \qty [ 1 - \left. \pdv{\red{\Sig{pp}{\xc}}(\yo)}{\yo} \right|_{\yo = \eKS{p}} ] ^ { -1}
\qq { with} 0 \le \green { Z_ { p} } \le 1
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Solutions of the non-linear QP equation: { \evGW } @HF/6-31G for \ce { H2} at $ R = 1 $ bohr}
\begin { columns}
\begin { column} { 0.5\textwidth }
\begin { center}
\includegraphics [width=0.7\textwidth] { fig/QP}
\\
\bigskip
\pub { V\' eril et al, JCTC 14 (2018) 5220}
\end { center}
\end { column}
\begin { column} { 0.5\textwidth }
\begin { center}
\includegraphics [width=\textwidth] { fig/GWSph}
\\
\bigskip
\pub { Loos et al, JCTC 14 (2018) 3071}
\end { center}
\end { column}
\end { columns}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { $ GW $ flavours}
\begin { block} { Acronyms}
\begin { itemize}
\bigskip
\item perturbative $ GW $ , one-shot $ GW $ , or \green { \GOWO }
\bigskip
\item \orange { \evGW } or eigenvalue-only (partially) self-consistent $ GW $
\bigskip
\item \red { \qsGW } or quasiparticle (partially) self-consistent $ GW $
\bigskip
\item \violet { \scGW } or (fully) self-consistent $ GW $
\bigskip
\end { itemize}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Perturbative { \GW } with linearized solution}
\begin { block} { }
\begin { algorithmic}
\Procedure { { \GOWO } lin@KS} { }
\State Perform KS calculation to get $ \beKS $ , $ \bcKS $ , and $ \bm { V } ^ { \xc } $
\State AO to MO transformation for ERIs: $ \ERI { \mu \nu } { \lambda \sigma } \stackrel { \bcKS } { \rightarrow } \ERI { pq } { rs } $
\State Construct RPA matrices $ \orange { \bA { } { \RPA } } $ and $ \orange { \bB { } { \RPA } } $ from $ \beKS $ and $ \ERI { pq } { rs } $
\State Compute RPA eigenvalues $ \orange { \bOm { } { \RPA } } $ and eigenvectors $ \orange { \bX { } { \RPA } + \bY { } { \RPA } } $
\Comment { \alert { This is a $ \order * { N ^ 6 } $ step!} }
\State Form screened ERIs $ \violet { \ERI { pq } { m } } $
\For { $ p = 1 , \ldots ,N $ }
\State Compute diagonal of the self-energy $ \red { \SigC { pp } } ( \yo ) $ at $ \yo = \eKS { p } $
\State Compute renornalization factors \green { $ \Z { p } $ }
\State Evaluate $ \blue { \eGOWO { p } } = \eKS { p } + \green { \Z { p } } \qty { \Re [ \red { \SigC { pp } } ( \eKS { p } ) ] - V _ { p } ^ { \xc } } $
\EndFor
\EndProcedure
\end { algorithmic}
\end { block}
\bigskip
For contour deformation technique, see, for example, \pub { Duchemin \& Blase, JCTC 16 (2020) 1742}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Example from \texttt { QuAcK} (\ce { Ne} /cc-pVDZ)}
\begin { center}
\includegraphics [width=0.55\textwidth] { fig/G0W0}
\\
\bigskip
\pub { https://github.com/pfloos/QuAcK}
\end { center}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Perturbative { \GW } with graphical solution}
\begin { block} { }
\begin { algorithmic}
\Procedure { { \GOWO } graph@KS} { }
\State Perform KS calculation to get $ \beKS $ , $ \bcKS $ , and $ \bm { V } ^ { \xc } $
\State AO to MO transformation for ERIs: $ \ERI { \mu \nu } { \lambda \sigma } \stackrel { \bcKS } { \rightarrow } \ERI { pq } { rs } $
\State Construct RPA matrices $ \orange { \bA { } { \RPA } } $ and $ \orange { \bB { } { \RPA } } $ from $ \beKS $ and $ \ERI { pq } { rs } $
\State Compute RPA eigenvalues $ \orange { \Om { } { \RPA } } $ and eigenvectors $ \orange { \bX { } { \RPA } + \bY { } { \RPA } } $
\Comment { \alert { This is a $ \order * { N ^ 6 } $ step!} }
\State Form screened ERIs $ \violet { \ERI { pq } { m } } $
\For { $ p = 1 , \ldots ,N $ }
\State Compute diagonal of the self-energy $ \red { \SigC { pp } } ( \yo ) $
\State Solve $ \yo = \eKS { p } + \Re [ \red { \SigC { pp } } ( \yo ) ] - V _ { p } ^ { \xc } $ to get $ \blue { \eGOWO { p } } $ via Newton's method
\EndFor
\EndProcedure
\end { algorithmic}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Newton's method}
\centering
\url { https://en.wikipedia.org/wiki/Newton\% 27s_ method}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Partially self-consistent eigenvalue \GW }
\begin { block} { }
\begin { algorithmic}
\Procedure { { \evGW } @KS} { }
\State Perform KS calculation to get $ \beKS $ , $ \bcKS $ , and $ \bm { V } ^ { \xc } $
\State AO to MO transformation for ERIs: $ \ERI { \mu \nu } { \lambda \sigma } \stackrel { \bcKS } { \rightarrow } \ERI { pq } { rs } $
\State Set $ \blue { \beGnWn { - 1 } } = \beKS $ and $ n = 0 $
\While { $ \max { \abs { \bDelta } } > \tau $ }
\State Construct RPA matrices $ \orange { \bA { } { \RPA } } $ and $ \orange { \bB { } { \RPA } } $ from $ \blue { \beGnWn { n - 1 } } $ and $ \ERI { pq } { rs } $
\State Compute RPA eigenvalues $ \orange { \Om { } { \RPA } } $ and eigenvectors $ \orange { \bX { } { \RPA } + \bY { } { \RPA } } $
\Comment { \alert { This is a $ \order * { N ^ 6 } $ step!} }
\State Form screened ERIs $ \violet { \ERI { pq } { m } } $
\For { $ p = 1 , \ldots ,N $ }
\State Compute diagonal of the self-energy $ \red { \SigC { pp } } ( \yo ) $
\State Solve $ \yo = \eKS { p } + \Re [ \red { \SigC { pp } } ( \yo ) ] - V _ { p } ^ { \xc } $ to get $ \blue { \eGnWn { p } { n } } $
\EndFor
\State $ \bDelta = \blue { \beGnWn { n } } - \blue { \beGnWn { n - 1 } } $
\State $ n \leftarrow n + 1 $
\EndWhile
\EndProcedure
\end { algorithmic}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Example from \texttt { QuAcK} (\ce { Ne} /cc-pVDZ)}
\begin { center}
\includegraphics [width=0.5\textwidth] { fig/evGW}
\\
\bigskip
\pub { https://github.com/pfloos/QuAcK}
\end { center}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Quasiparticle self-consistent { \GW } (\qsGW )}
\begin { block} { }
\begin { algorithmic}
\Procedure { { \qsGW } } { }
\State Perform HF calculation to get $ \beHF $ and $ \bcHF $ \green { (optional)}
\State Set $ \blue { \beGnWn { - 1 } } = \beHF $ , $ \blue { \bcGnWn { - 1 } } = \bcHF $ and $ n = 0 $
\While { $ \max { \abs { \bDelta } } > \tau $ }
\State AO to MO transformation for ERIs: $ \ERI { \mu \nu } { \lambda \sigma } \stackrel { \blue { \bcGnWn { n - 1 } } } { \rightarrow } \ERI { pq } { rs } $
\Comment { \alert { This is a $ \order * { N ^ 5 } $ step!} }
\State Construct RPA matrices $ \orange { \bA { } { \RPA } } $ and $ \orange { \bB { } { \RPA } } $ from $ \blue { \beGnWn { n - 1 } } $ and $ \ERI { pq } { rs } $
\State Compute RPA eigenvalues $ \orange { \Om { } { \RPA } } $ and eigenvectors $ \orange { \bX { } { \RPA } + \bY { } { \RPA } } $
\Comment { \alert { This is a $ \order * { N ^ 6 } $ step!} }
\State Form screened ERIs $ \violet { \ERI { pq } { m } } $
\State Evaluate $ \red { \bSigC } ( \blue { \beGnWn { n - 1 } } ) $ and form
$ \red { \Tilde { \Sigma } ^ { \co } } \leftarrow \qty [ \red { \bSigC } ( \blue { \beGnWn { n - 1 } } ) ^ \dag + \red { \bSigC } ( \blue { \beGnWn { n - 1 } } ) ] / 2 $
\State Form $ \bFHF $ from $ \blue { \bcGnWn { n - 1 } } $ and then $ \purple { \Tilde { \bF } } = \bFHF + \red { \Tilde { \Sigma } ^ { \co } } $
\State Diagonalize $ \purple { \Tilde { \bF } } $ to get $ \blue { \beGnWn { n } } $ and $ \blue { \bcGnWn { n } } $
\State $ \bDelta = \blue { \beGnWn { n } } - \blue { \beGnWn { n - 1 } } $
\State $ n \leftarrow n + 1 $
\EndWhile
\EndProcedure
\end { algorithmic}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Example from \texttt { QuAcK} (\ce { Ne} /cc-pVDZ)}
\begin { center}
\includegraphics [width=0.45\textwidth] { fig/qsGW1}
\hspace { 0.1\textwidth }
\includegraphics [width=0.4\textwidth] { fig/qsGW2}
\\
\bigskip
\pub { https://github.com/pfloos/QuAcK}
\end { center}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Other self-energies}
\begin { columns}
\begin { column} { 0.7\textwidth }
\begin { block} { Second-order Green's function (GF2) \pub { [Hirata et al. JCP 147 (2017) 044108]} }
\begin { equation}
\Sig { pq} { \text { GF2} } (\yo )
= \frac { 1} { 2} \sum _ { iab} \frac { \mel { iq} { } { ab} \mel { ab} { } { ip} } { \yo + \e { i} { } - \e { a} { } - \e { b} { } }
+ \frac { 1} { 2} \sum _ { ija} \frac { \mel { aq} { } { ij} \mel { ij} { } { ap} } { \yo + \e { a} { } - \e { i} { } - \e { j} { } }
\end { equation}
\end { block}
\begin { block} { T-matrix \pub { [Romaniello et al. PRB 85 (2012) 155131; Zhang et al. JPCL 8 (2017) 3223]} }
\begin { equation}
\Sig { pq} { GT} (\omega )
= \sum _ { im} \frac { \braket * { pi} { \green { \chi _ m^ { N+2} } } \braket * { qi} { \green { \chi _ m^ { N+2} } } } { \yo + \e { i} { } - \green { \Om { m} { N+2} } }
+ \sum _ { am} \frac { \braket * { pa} { \blue { \chi _ m^ { N-2} } } \braket * { qa} { \blue { \chi _ m^ { N-2} } } } { \yo + \e { i} { } - \blue { \Om { m} { N-2} } }
\end { equation}
\begin { gather}
\braket * { pi} { \green { \chi _ m^ { N+2} } } = \sum _ { c<d} \mel { pi} { } { cd} \green { X_ { cd} ^ { N+2,m} } + \sum _ { k<l} \mel { pi} { } { kl} \green { Y_ { kl} ^ { N+2,m} }
\\
\braket * { pa} { \blue { \chi _ m^ { N-2} } } = \sum _ { c<d} \mel { pa} { } { cd} \blue { X_ { cd} ^ { N-2,m} } + \sum _ { k<l} \mel { pa} { } { kl} \blue { Y_ { kl} ^ { N-2,m} }
\end { gather}
\begin { equation}
\qq * { \purple { pp-RPA problem:} }
\begin { pmatrix}
\bA { } { } & \bB { } { }
\\
-\bB { } { \intercal } & -\bC { } { }
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { m} { N\pm 2}
\\
\bY { m} { N\pm 2}
\end { pmatrix}
=
\Om { m} { N\pm 2}
\begin { pmatrix}
\bX { m} { N\pm 2}
\\
\bY { m} { N\pm 2}
\end { pmatrix}
\end { equation}
\end { block}
\end { column}
\begin { column} { 0.35\textwidth }
\includegraphics [width=\textwidth] { fig/Sigma}
\\
\bigskip
\includegraphics [width=\textwidth] { fig/Tmatrix}
\\
\pub { Martin, Reining \& Ceperley, Interacting Electrons (Cambridge University Press)}
\end { column}
\end { columns}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section { Neutral excitations}
\begin { frame}
\tableofcontents [currentsection]
\end { frame}
%-----------------------------------------------------
\begin { frame} { Dynamical vs static kernels}
\begin { block} { A non-linear BSE problem \pub { [Strinati, Riv.~Nuovo Cimento 11 (1988) 1]} }
\begin { equation}
\begin { pmatrix}
\bA { } { } (\yo ) & \bB { } { } (\yo )
\\
-\bB { } { } (-\yo ) & -\bA { } { } (-\yo )
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { } { }
\\
\bY { } { }
\end { pmatrix}
=
\yo
\begin { pmatrix}
\bX { } { }
\\
\bY { } { }
\end { pmatrix}
\qq { \alert { \bf Hard to solve!} }
\end { equation}
\end { block}
\begin { block} { Static BSE vs dynamic BSE for \ce { HeH+} /STO-3G}
\begin { columns}
\begin { column} { 0.5\textwidth }
\begin { center}
\includegraphics [width=0.7\textwidth] { fig/dyn}
\end { center}
\end { column}
\begin { column} { 0.5\textwidth }
Dynamical kernels can give you more than static kernels... Sometimes, too much...
\end { column}
\end { columns}
\bigskip
\center
\pub { Authier \& Loos, JCP 153 (2020) 184105} [see also \pub { Romaniello et al, JCP 130 (2009) 044108} ]
\end { block}
\end { frame}
%-----------------------------------------------------
\begin { frame} { TD-DFT and BSE in practice: Casida-like equations}
\begin { block} { Linear response problem}
\begin { equation*}
\boxed { \begin { pmatrix}
\red { \bA { } { } } & \orange { \bB { } { } }
\\
\orange { -\bB { } { } } & \red { -\bA { } { } }
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { m} { }
\\
\bY { m} { }
\end { pmatrix}
=
\highlight { \Om { m} { } }
\begin { pmatrix}
\bX { m} { }
\\
\bY { m} { }
\end { pmatrix} }
\end { equation*}
\end { block}
%
\begin { block} { Blue pill: TD-DFT within the \alert { adiabatic} approximation}
\begin { gather}
\red { A} _ { ia,jb} = \qty ( \e { a} { \green { \KS } } - \e { i} { \green { \KS } } ) \delta _ { ij} \delta _ { ab} + 2 \blue { (ia|bj)} + \yellow { f} ^ { \yellow { xc} } _ { ia,bj}
\qquad
\orange { B} _ { ia,jb} = 2 \blue { (ia|jb)} + \yellow { f} ^ { \yellow { xc} } _ { ia,jb}
\\
\yellow { f} ^ { \yellow { xc} } _ { ia,bj} = \iint \phi _ { i} (\br { } )\phi _ { a} (\br { } ) \frac { \delta ^ 2 E^ { xc} } { \delta \rho (\br { } ) \delta \rho (\br { } ')} \phi _ { b} (\br { } )\phi _ { j} (\br { } ) d\br { } d\br { } '
\end { gather}
\end { block}
%
\begin { block} { Red pill: BSE within the \alert { static} approximation}
\begin { gather}
\red { A} _ { ia,jb} = \qty ( \e { a} { \green { GW} } - \e { i} { \green { GW} } ) \delta _ { ij} \delta _ { ab} + 2 \blue { (ia|bj)} - \purple { W} ^ \text { stat} _ { ij,ba}
\qquad
\orange { B} _ { ia,jb} = 2 \blue { (ia|jb)} - \purple { W} ^ \text { stat} _ { ib,ja}
\\
\purple { W} ^ \text { stat} _ { ij,ab} \equiv \purple { W} _ { ij,ab} (\omega = 0) = (ij|ab) - W^ { c} _ { ij,ab} (\omega = 0)
\end { gather}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { The bridge between TD-DFT and BSE}
\begin { block} { }
\begin { center}
\begin { tabular} { lcr}
\hline
\bf \red { TD-DFT} & \bf \purple { Connection} & \bf \violet { BSE}
\\
\hline
\\
\red { One-point density} & & \violet { Two-point Green's function}
\\
$ \rho ( 1 ) $ & $ \rho ( 1 ) = - iG ( 11 ^ { + } ) $ & $ G ( 12 ) $
\\
\\
\red { Two-point susceptibility} & & \violet { Four-point susceptibility}
\\
$ \chi ( 12 ) = \pdv { \rho ( 1 ) } { U ( 2 ) } $ & $ \chi ( 12 ) = - i L ( 12 ; 1 ^ + 2 ^ + ) $ & $ L ( 12 ; 34 ) = \pdv { G ( 13 ) } { U ( 42 ) } $
\\
\\
\red { Two-point kernel} & & \violet { Four-point kernel}
\\
$ K ( 12 ) = v ( 12 ) + \pdv { V ^ { xc } ( 1 ) } { \rho ( 2 ) } $ & & $ i \Xi ( 1234 ) = v ( 13 ) \delta ( 12 ) \delta ( 34 ) - \pdv { \Sigma ^ { xc } ( 12 ) } { G ( 34 ) } $ \\
\hline
\end { tabular}
\end { center}
\end { block}
\bigskip
For dynamical correction within BSE, see, for example, \pub { Loos \& Blase, JCP 153 (2020) 114120}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { BSE in a computer}
\begin { block} { Vertical excitation energies from BSE}
\begin { algorithmic}
\Procedure { BSE@GW} { }
\State Compute $ GW $ quasiparticle energies \blue { $ \eGW { p } $ } at the { \GOWO } , { \evGW } , or { \qsGW } level
\State Compute static screening $ \highlight { W ^ \text { stat } _ { pq,rs } } $
\State Construct BSE matrices $ \orange { \bA { } { \BSE } } $ and $ \orange { \bB { } { \BSE } } $ from \blue { $ \eGW { p } $ } , $ \ERI { pq } { rs } $ , and $ \highlight { W ^ \text { stat } _ { pq,rs } } $
\State Compute lowest eigenvalues $ \orange { \Om { m } { \BSE } } $ and eigenvectors $ \orange { \bX { m } { \BSE } + \bY { m } { \BSE } } $ \green { (optional)}
\Comment { \alert { This is a $ \order * { N ^ 4 } $ step!} }
\EndProcedure
\end { algorithmic}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Removing the correlation part: TDHF and CIS}
\begin { block} { Linear response problem}
\begin { equation*}
\boxed { \begin { pmatrix}
\red { \bA { } { } } & \orange { \bB { } { } }
\\
\orange { -\bB { } { } } & \red { -\bA { } { } }
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { m} { }
\\
\bY { m} { }
\end { pmatrix}
=
\highlight { \Om { m} { } }
\begin { pmatrix}
\bX { m} { }
\\
\bY { m} { }
\end { pmatrix} }
\end { equation*}
\end { block}
%
\begin { block} { TDHF = RPA with exchange (RPAx)}
\begin { align}
\red { A} _ { ia,jb} & = \qty ( \e { a} { \green { \HF } } - \e { i} { \green { \HF } } ) \delta _ { ij} \delta _ { ab} + 2 \blue { (ia|bj)} - \yellow { (ij|ba)}
&
\orange { B} _ { ia,jb} & = 2 \blue { (ia|jb)} - \yellow { (ib|ja)}
\end { align}
\end { block}
%
\begin { block} { Linear response problem within the Tamm-Dancoff approximation}
\begin { equation}
\boxed { \red { \bA { } { } } \cdot \bX { m} { } = \highlight { \Om { m} { } } \, \bX { m} { } }
\end { equation}
\end { block}
%
\begin { block} { TDHF within TDA = CIS}
\begin { equation}
\red { A} _ { ia,jb}
= \qty ( \e { a} { \green { \HF } } - \e { i} { \green { \HF } } ) \delta _ { ij} \delta _ { ab}
+ 2 \blue { (ia|bj)} - \yellow { (ij|ba)}
\end { equation}
\end { block}
%
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Relationship between CIS, TDHF, DFT and TDDFT}
\center
\begin { tikzpicture}
\usetikzlibrary { shapes.misc}
\begin { scope} [very thick,
node distance=3cm,on grid,>=stealth',
box/.style={ rectangle,draw,fill=green!40} ],
\node [box, align=center] (CIS) { \textbf { CIS} } ;
\node [box, align=center] (HF) [left=of CIS, yshift=1cm] { \textbf { HF} } ;
\node [box, align=center] (TDHF) [right=of CIS, yshift=1cm] { \textbf { TDHF} } ;
\node [box, align=center] (DFT) [below=of HF] { \textbf { DFT} } ;
\node [box, align=center] (TDDFT) [below=of TDHF] { \textbf { TDDFT} } ;
\node [box, align=center] (TDA) [below=of CIS] { \textbf { TDA} } ;
\path
(CIS) edge [<-] node[below,sloped]{ CI} (HF)
(CIS) edge [<-] node[below,sloped]{ $ \bB { } { } = \bO $ } (TDHF)
(HF) edge [->] node[above]{ linear response} (TDHF)
(HF) edge [<->] node[left]{ $ \upsilon _ \text { x } ^ \text { HF } $ vs $ \upsilon _ \text { xc } $ } (DFT)
(TDHF) edge [<->] node[right]{ $ \upsilon _ \text { x } ^ \text { HF } $ vs $ \upsilon _ \text { xc } $ } (TDDFT)
(DFT) edge [->] node[above]{ linear response} (TDDFT)
(DFT) edge [->] node[below,sloped]{ CI} node[strike out,sloped]{ \alert { $ \cross $ } } (TDA)
(TDDFT) edge [->] node[below,sloped]{ $ \bB { } { } = \bO { } { } $ } (TDA)
;
\end { scope}
\end { tikzpicture}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Linear response}
\begin { block} { General linear response problem}
\begin { algorithmic}
\Procedure { Linear response} { }
\State Compute $ \red { \bA { } { } } $ matrix at a given level of theory (RPA, RPAx, TD-DFT, BSE, etc)
\If { $ \TDA $ }
\State Diagonalize $ \red { \bA { } { } } $ to get $ \highlight { \Om { m } { \TDA } } $ and $ \bX { m } { \TDA } $
\Else
\State Compute \orange { $ \bB { } { } $ } matrix at a given level of theory
\State Diagonalize $ \red { \bA { } { } } - \orange { \bB { } { } } $ to form $ ( \red { \bA { } { } } - \orange { \bB { } { } } ) ^ { 1 / 2 } $
\State Form and diagonalize $ ( \red { \bA { } { } } - \orange { \bB { } { } } ) ^ { 1 / 2 } \cdot ( \red { \bA { } { } } + \orange { \bB { } { } } ) \cdot ( \red { \bA { } { } } - \orange { \bB { } { } } ) ^ { 1 / 2 } $
to get $ \highlight { \Om { m } { 2 } } $ and $ \bZ { m } { } $
\State Compute $ \sqrt { \highlight { \Om { m } { 2 } } } $ and $ ( \bX { m } { } + \bY { m } { } ) = \highlight { \Om { m } { - 1 / 2 } } ( \red { \bA { } { } } - \orange { \bB { } { } } ) ^ { 1 / 2 } \cdot \bZ { m } { } $
\EndIf
\EndProcedure
\end { algorithmic}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Form linear response matrices}
\begin { block} { Linear-response matrices for BSE}
\begin { algorithmic}
\Procedure { Form $ \red { \bA { } { } } $ for singlet states} { }
\State Set $ \red { \bA { } { } } = \bO $
\State $ ia \gets 0 $
\For { $ i = 1 , \ldots , O $ }
\For { $ a = 1 , \ldots , V $ }
\State $ ia \gets ia + 1 $
\State $ jb \gets 0 $
\For { $ j = 1 , \ldots , O $ }
\For { $ b = 1 , \ldots , V $ }
\State $ jb \gets jb + 1 $
\State $ \red { A _ { ia,jb } } = \delta _ { ij } \delta _ { ab } ( \e { a } { \green { GW } } - \e { i } { \green { GW } } )
+ 2\blue { (ia|bj)} - \yellow { (ij|ba)} + \purple { W^ { \co } _ { ij,ba} } (\omega = 0)$
\EndFor
\EndFor
\EndFor
\EndFor
\EndProcedure
\end { algorithmic}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Properties}
\begin { block} { Oscillator strength (length gauge)}
\begin { equation}
\boxed { \green { f_ m} = \frac { 2} { 3} \orange { \Om { m} { } } \qty [ (\blue{\mu_m^x})^2 + (\blue{\mu_m^y})^2 + (\blue{\mu_m^z})^2 ] }
\end { equation}
\end { block}
\begin { block} { Transition dipole}
\begin { equation}
\boxed { \blue { \mu _ m^ x} = \sum _ { ia} \red { (i|x|a)} \orange { (\bX { m} { } + \bY { m} { } )_ { ia} } }
\qquad
\red { (p|x|q)} = \int \MO { p} (\br ) \, x\, \MO { q} (\br ) d\br
\end { equation}
\end { block}
\begin { block} { Monitoring possible spin contamination \pub { [Monino \& Loos, JCTC 17 (2021) 2852]} }
\begin { equation}
\boxed { \purple { \expval { \hat { S} ^ 2} _ m} = \violet { \expval { \hat { S} ^ 2} _ 0} + \underbrace { \Delta \expval { \hat { S} ^ 2} _ m} _ { \text { \pub { JCP 134101 (2011) 134} } } }
\qquad
\violet { \expval { \hat { S} ^ 2} _ 0} = \frac { n_ \alpha - n_ \beta } { 2} \qty ( \frac { n_ \alpha - n_ \beta } { 2} + 1 ) + n_ \beta + \sum _ p (p_ \alpha |p_ \beta )
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Example from \texttt { QuAcK} (\ce { H2O} /cc-pVDZ)}
\begin { center}
\includegraphics [height=0.45\textwidth] { fig/BSE1}
\hspace { 0.05\textwidth }
\includegraphics [height=0.45\textwidth] { fig/BSE3}
\\
\bigskip
\pub { https://github.com/pfloos/QuAcK}
\end { center}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Open-shell systems and double excitations}
\begin { block} { Spin-flip formalism (H2/cc-pVQZ)}
\begin { center}
\includegraphics [width=0.28\textwidth] { fig/SFBSE}
\includegraphics [width=0.4\textwidth] { fig/H2}
\includegraphics [width=0.3\textwidth] { fig/H2_ QuAcK}
\\
\bigskip
\pub { Monino \& Loos, JCTC 17 (2021) 2852}
\end { center}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\section { Correlation energy}
\begin { frame}
\tableofcontents [currentsection]
\end { frame}
%-----------------------------------------------------
\begin { frame} { Correlation energy at the $ GW $ or BSE level}
\begin { block} { RPA@$ GW $ correlation energy: plasmon (or trace) formula}
\begin { equation*}
\label { eq:Ec-RPA}
\green { \EcRPA }
= \frac { 1} { 2} \qty [ \sum_{p} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
= \frac { 1} { 2} \sum _ { m} \qty ( \orange { \Om { m} { \RPA } } - \orange { \Om { m} { \TDA } } )
\end { equation*}
\end { block}
\begin { block} { Galitskii-Migdal functional}
\begin { equation*}
\label { eq:GM}
\green { \EcGM }
= \frac { -i} { 2} \sum _ { pq} ^ { \infty } \int \frac { d\omega } { 2\pi } \red { \SigC { pq} } (\omega ) \blue { \G { pq} } (\omega ) e^ { i\omega \eta }
= 4 \sum _ { ia} \sum _ { m} \frac { \violet { \ERI { ai} { m} } ^ 2} { \blue { \eGW { a} } - \blue { \eGW { i} } + \orange { \Om { m} { \RPA } } }
\end { equation*}
\end { block}
\begin { block} { ACFDT@BSE@$ GW $ correlation energy from the adiabatic connection}
\begin { equation}
\green { \Ec ^ \text { ACFDT} } = \frac { 1} { 2} \int _ { \red { 0} } ^ { \red { 1} } \Tr ( \bK { } { } \bP { } { \la } ) d\la
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Adiabatic connection fluctuation dissipation theorem (ACFDT)}
\begin { block} { Adiabatic connection}
\begin { equation}
\boxed {
\green { \Ec ^ \text { ACFDT} }
= \frac { 1} { 2} \int _ { \red { 0} } ^ { \red { 1} } \Tr ( \bK { } { } \bP { } { \la } ) d\la
\stackrel { \blue { \text { quad} } } { \approx } \frac { 1} { 2} \sum _ { k=1} ^ { K} \purple { w_ k} \Tr ( \bK { } { } \bP { } { \violet { \lambda _ k} } )
}
\end { equation}
$ \la $ is the \textbf { strength} of the electron-electron interaction:
\begin { itemize}
\item $ \la = 0 $ for the \green { non-interacting system}
\item $ \la = 1 $ for the \alert { physical system}
\end { itemize}
\end { block}
\begin { block} { Interaction kernel}
\begin { equation}
\bK { } { } =
\begin { pmatrix}
\btA { } { } & \btB { } { }
\\
\btB { } { } & \btA { } { }
\end { pmatrix}
\qquad
\tA { ia,jb} { } = 2\ERI { ia} { bj}
\qquad
\tB { ia,jb} { } = 2\ERI { ia} { jb}
\end { equation}
\end { block}
\begin { block} { Correlation part of the two-particle density matrix}
\begin { equation}
\bP { } { \la } =
\begin { pmatrix}
\bY { } { \la } \cdot \T { (\bY { } { \la } )} & \bY { } { \la } \cdot \T { (\bX { } { \la } )}
\\
\bX { } { \la } \cdot \T { (\bY { } { \la } )} & \bX { } { \la } \cdot \T { (\bX { } { \la } )}
\end { pmatrix}
-
\begin { pmatrix}
\bO & \bO
\\
\bO & \bI
\end { pmatrix}
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Gaussian quadrature}
\begin { block} { Numerical integration by quadrature}
\textit { ``A $ K $ -point \orange { Gaussian quadrature} rule is a quadrature rule constructed to yield an exact result for polynomials up to degree $ 2 K - 1 $ by a suitable choice of the \violet { roots $ x _ k $ } and \purple { weights $ w _ k $ } for $ k = 1 , \ldots , K $ .''}
\begin { equation}
\boxed { \int _ { \red { a} } ^ { \red { b} } f(x) \purple { w(x)} dx \approx \sum _ k^ { K} \underbrace { \purple { w_ k} } _ { \text { \purple { weights} } } f(\underbrace { \violet { x_ k} } _ { \text { \violet { roots} } } )}
\end { equation}
\end { block}
\begin { block} { Quadrature rules}
\begin { center}
\small
\begin { tabular} { llll}
\hline
\red { Interval $ [ a,b ] $ } & \purple { Weight function $ w ( x ) $ } & \violet { Orthogonal polynomials} & \orange { Name} \\
\hline
$ [ - 1 , 1 ] $ & $ 1 $ & Legendre $ P _ n ( x ) $ & Gauss-Legendre \\
$ ( - 1 , 1 ) $ & $ ( 1 - x ) ^ \alpha ( 1 + x ) ^ \beta , \quad \alpha , \beta > - 1 $ & Jacobi $ P _ n ^ { \alpha , \beta } ( x ) $ & Gauss-Jacobi \\
$ ( - 1 , 1 ) $ & $ 1 / \sqrt { 1 - x ^ 2 } $ & Chebyshev (1st kind) $ T _ n ( x ) $ & Gauss-Chebyshev \\
$ [ - 1 , 1 ] $ & $ \sqrt { 1 - x ^ 2 } $ & Chebyshev (2nd kind) $ U _ n ( x ) $ & Gauss-Chebyshev \\
$ [ 0 , \infty ) $ & $ \exp ( - x ) $ & Laguerre $ L _ n ( x ) $ & Gauss-Laguerre \\
$ [ 0 , \infty ) $ & $ x ^ \alpha \exp ( - x ) , \quad \alpha > - 1 $ & Generalized Laguerre $ L _ n ^ \alpha ( x ) $ & Gauss-Laguerre \\
$ ( - \infty , \infty ) $ & $ \exp ( - x ^ 2 ) $ & Hermite $ H _ n ( x ) $ & Gauss-Hermite \\
\hline
\end { tabular}
\\
\url { https://en.wikipedia.org/wiki/Gaussian_ quadrature}
\end { center}
\end { block}
\end { frame}
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%-----------------------------------------------------
\begin { frame} { ACFDT at the RPA/RPAx level}
\begin { block} { RPA matrix elements}
\begin { equation}
\orange { \A { ia,jb} { \la ,\RPA } } = \delta _ { ij} \delta _ { ab} (\violet { \eHF { a} } - \violet { \eHF { i} } ) + 2\la \ERI { ia} { bj}
\qquad
\orange { \B { ia,jb} { \la ,\RPA } } = 2\la \ERI { ia} { jb}
\end { equation}
\begin { equation}
\boxed {
\green { \Ec ^ \RPA }
= \frac { 1} { 2} \int _ { \red { 0} } ^ { \red { 1} } \Tr ( \bK { } { } \bP { } { \la } ) d\la
= \frac { 1} { 2} \qty [ \sum_{m} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
}
\end { equation}
\end { block}
\begin { block} { RPAx matrix elements}
\begin { equation}
\orange { \A { ia,jb} { \la ,\RPAx } } = \delta _ { ij} \delta _ { ab} (\violet { \eHF { a} } - \violet { \eHF { i} } ) + \la \qty [2 \ERI{ia}{bj} - \ERI{ij}{ab} ]
\qquad
\orange { \B { ia,jb} { \la ,\RPAx } } = \la \qty [2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
\end { equation}
\begin { equation}
\boxed {
\green { \Ec ^ \RPAx }
= \frac { 1} { 2} \int _ { \red { 0} } ^ { \red { 1} } \Tr ( \bK { } { } \bP { } { \la } ) d\la
\alert { \neq } \frac { 1} { 2} \qty [ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
}
\end { equation}
If exchange added to kernel, i.e., $ \bK { } { } = \bK { } { \x } $ , then \pub { [Angyan et al. JCTC 7 (2011) 3116]}
\begin { equation}
\green { \Ec ^ \RPAx }
= \frac { 1} { 4} \int _ { \red { 0} } ^ { \red { 1} } \Tr ( \bK { } { \x } \bP { } { \la } ) d\la
\alert { =} \frac { 1} { 4} \qty [ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { ACFDT at the BSE level}
\begin { block} { BSE matrix elements}
\begin { equation}
\orange { \A { ia,jb} { \la ,\BSE } } = \delta _ { ij} \delta _ { ab} (\violet { \eGW { a} } - \violet { \eGW { i} } ) + \la \qty [2 \ERI{ia}{bj} - \highlight{W}_{ij,ab}^{\la}(\omega = 0) ]
\qquad
\orange { \B { ia,jb} { \la ,\BSE } } = \la \qty [2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)]
\end { equation}
\begin { equation}
\boxed {
\green { \Ec ^ \BSE }
= \frac { 1} { 2} \int _ { \red { 0} } ^ { \red { 1} } \Tr ( \bK { } { } \bP { } { \la } ) d\la
\alert { \neq } \frac { 1} { 2} \qty [ \sum_{m} \orange{\Om{m}{\BSE}} - \Tr(\orange{\bA{}{\BSE}}) ]
}
\end { equation}
\end { block}
\begin { block} { $ \la $ -dependent screening}
\begin { equation}
\highlight { W} _ { pq,rs} ^ { \la } (\omega )
= \ERI { pq} { rs}
+ 2 \sum _ m \violet { \ERI { pq} { m} ^ { \la } } \violet { \ERI { rs} { m} ^ { \la } }
\qty [ \frac{1}{\omega - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\omega + \orange{\Om{m}{\la,\RPA}} - i \eta} ]
\end { equation}
\begin { equation}
\violet { \ERI { pq} { m} ^ { \la } } = \sum _ { ia} \ERI { pq} { ia} (\orange { \bX { m} { \la ,\RPA } +\bY { m} { \la ,\RPA } } )_ { ia}
\end { equation}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { ACFDT in a computer}
\begin { block} { ACFDT correlation energy from BSE}
\begin { algorithmic}
\Procedure { ACFDT for BSE} { }
\State Compute $ GW $ quasiparticle energies $ \blue { \beGW } $ and interaction kernel $ \bK { } { } $
\State Get Gauss-Legendre weights and roots $ \{ \purple { w _ k } , \violet { \lambda _ k } \} _ { 1 \le k \le K } $
\State $ \green { \Ec } \gets 0 $
\For { $ k = 1 , \ldots ,K $ }
\State Compute static screening elements $ \highlight { W } _ { pq,rs } ^ { \violet { \lambda _ k } } ( \omega = 0 ) $
\State Perform BSE calculation at $ \la = \violet { \lambda _ k } $ to get $ \bX { } { \violet { \lambda _ k } } $ and $ \bY { } { \violet { \lambda _ k } } $
\Comment { \alert { This is a $ \order * { N ^ 6 } $ step done many times!} }
\State Form two-particle density matrix $ \bP { } { \violet { \lambda _ k } } $
\State $ \green { \Ec } \gets \green { \Ec } + \purple { w _ k } \Tr ( \bK { } { } \bP { } { \violet { \lambda _ k } } ) $
\EndFor
\EndProcedure
\end { algorithmic}
\end { block}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame}
\begin { center}
\includegraphics [width=0.7\textwidth] { fig/TOC_ BSE}
\\
\pub { Loos et al. JPCL 11 (2020) 3536}
\end { center}
\end { frame}
%-----------------------------------------------------
%-----------------------------------------------------
\begin { frame} { Useful papers/programs}
\begin { itemize}
\item \red { mol$ GW $ :} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
\bigskip
\item \green { Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528
\bigskip
\item \violet { Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
\bigskip
\item \purple { FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
\bigskip
\item \orange { Reviews \& Books:}
\begin { itemize}
\item Reining, WIREs Comput Mol Sci 2017, e1344. doi: 10.1002/wcms.1344
\item Onida et al. Rev. Mod. Phys. 74 (2002) 601
\item Blase et al. Chem. Soc. Rev. , 47 (2018) 1022
\item Golze et al. Front. Chem. 7 (2019) 377
\item Blase et al. JPCL 11 (2020) 7371
\item Martin, Reining \& Ceperley \textit { Interacting Electrons} (Cambridge University Press)
\end { itemize}
\bigskip
\item \red { $ GW $ 100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url { http://gw100.wordpress.com} )
\end { itemize}
\end { frame}
%-----------------------------------------------------
\end { document}
