almost OK for lecture 2
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@ -211,10 +211,11 @@
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\newcommand{\bX}[2]{\bm{X}_{#1}^{#2}}
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\newcommand{\bY}[2]{\bm{Y}_{#1}^{#2}}
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\newcommand{\bZ}[2]{\bm{Z}_{#1}^{#2}}
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\newcommand{\bK}[2]{\bm{K}_{#1}^{#2}}
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\newcommand{\bP}[2]{\bm{P}_{#1}^{#2}}
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\newcommand{\bK}[2]{\blue{\bm{K}}_{#1}^{#2}}
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\newcommand{\bP}[2]{\red{\bm{P}}_{#1}^{#2}}
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\newcommand{\yo}{\yellow{\omega}}
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\newcommand{\la}{\yellow{\lambda}}
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\newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}}
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@ -917,12 +918,19 @@ decoration={snake,
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\end{equation}
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\end{block}
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\begin{block}{Static BSE vs dynamic BSE for \ce{HeH+}/STO-3G}
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\begin{center}
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\includegraphics[width=0.4\textwidth]{fig/dyn}
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\\
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\bigskip
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\pub{Authier \& Loos, JCP 153 (2020) 184105} [see also \pub{Romaniello et al, JCP 130 (2009) 044108}]
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\end{center}
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\begin{columns}
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\begin{column}{0.5\textwidth}
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\begin{center}
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\includegraphics[width=0.7\textwidth]{fig/dyn}
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\end{center}
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\end{column}
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\begin{column}{0.5\textwidth}
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Dynamical kernels can give you more than static kernels... Sometimes, too much...
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\end{column}
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\end{columns}
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\bigskip
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\center
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\pub{Authier \& Loos, JCP 153 (2020) 184105} [see also \pub{Romaniello et al, JCP 130 (2009) 044108}]
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\end{block}
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\end{frame}
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@ -947,18 +955,17 @@ decoration={snake,
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\end{pmatrix}
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\qquad N = N_1 + N_2
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\end{equation}
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\begin{equation}
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\qq*{\bf Row \#2:}
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\bh \cdot \bc_1 + \bH_2 \cdot \bc_2 = \yo \, \bc_2
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\qq{$\Rightarrow$}
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\bc_2 = (\yo \, \bI - \bH_2)^{-1} \cdot \bh \cdot \bc_1
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\end{equation}
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\begin{equation}
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\begin{align}
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\qq*{\bf Row \#2:}
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& \bh \cdot \bc_1 + \bH_2 \cdot \bc_2 = \yo \, \bc_2
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& \qq{$\Rightarrow$}
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& \bc_2 = (\yo \, \bI - \bH_2)^{-1} \cdot \bh \cdot \bc_1
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\\
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\qq*{\bf Row \#1:}
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\bH_1 \cdot \bc_1 + \T{\bh} \cdot \bc_2 = \yo \, \bc_1
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\qq{$\Rightarrow$}
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\underbrace{\Tilde{\bH}_1(\yo) \cdot \bc_1 = \yo \, \bc_1}_{\text{A smaller non-linear system with $N$ solutions\ldots}}
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\end{equation}
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& \bH_1 \cdot \bc_1 + \T{\bh} \cdot \bc_2 = \yo \, \bc_1
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& \qq{$\Rightarrow$}
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& \underbrace{\Tilde{\bH}_1(\yo) \cdot \bc_1 = \yo \, \bc_1}_{\text{A smaller non-linear system with $N$ solutions\ldots}}
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\end{align}
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\begin{equation}
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\boxed{
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\underbrace{\Tilde{\bH}_1(\yo)}_{\text{Effective Hamitonian}}
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@ -1063,7 +1070,7 @@ decoration={snake,
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\State Compute $GW$ quasiparticle energies \blue{$\eGW{p}$} at the {\GOWO}, {\evGW}, or {\qsGW} level
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\State Compute static screening $\highlight{W^\text{stat}_{pq,rs}}$
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\State Construct BSE matrices $\orange{\bA{}{\BSE}}$ and $\orange{\bB{}{\BSE}}$ from \blue{$\eGW{p}$}, $\ERI{pq}{rs}$, and $\highlight{W^\text{stat}_{pq,rs}}$
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\State Compute lowest BSE eigenvalues $\orange{\Om{m}{\BSE}}$ and eigenvectors $\orange{\bX{m}{\BSE}+\bY{m}{\BSE}}$ \green{(optional)}
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\State Compute lowest eigenvalues $\orange{\Om{m}{\BSE}}$ and eigenvectors $\orange{\bX{m}{\BSE}+\bY{m}{\BSE}}$ \green{(optional)}
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\Comment{\alert{This is a $\order*{N^4}$ step!}}
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\EndProcedure
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\end{algorithmic}
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@ -1155,7 +1162,7 @@ decoration={snake,
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\begin{block}{General linear response problem}
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\begin{algorithmic}
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\Procedure{Linear response}{}
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\State Compute $\red{\bA{}{}}$ matrix at a given level of theory
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\State Compute $\red{\bA{}{}}$ matrix at a given level of theory (RPA, RPAx, TD-DFT, BSE, etc)
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\If{$\TDA$}
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\State Diagonalize $\red{\bA{}{}}$ to get $\highlight{\Om{m}{\TDA}}$ and $\bX{m}{\TDA}$
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\Else
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@ -1251,7 +1258,7 @@ decoration={snake,
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%-----------------------------------------------------
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\begin{frame}{Correlation energy at the $GW$ level}
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\begin{block}{RPA correlation energy: plasmon formula}
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\begin{block}{RPA correlation energy: plasmon (or trace) formula}
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\begin{equation*}
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\label{eq:Ec-RPA}
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\EcRPA
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@ -1267,6 +1274,12 @@ decoration={snake,
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= 4 \sum_{ia} \sum_{m} \frac{\violet{\ERI{ai}{m}}^2}{\e{a}{} - \e{i}{} + \orange{\Om{m}{\RPA}}}
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\end{equation*}
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\end{block}
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\begin{block}{Adiabatic connection}
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\begin{equation}
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\Ec^\text{ACFDT} = \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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\end{equation}
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\end{block}
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\end{frame}
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%-----------------------------------------------------
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@ -1275,9 +1288,9 @@ decoration={snake,
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\begin{block}{Adiabatic connection}
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\begin{equation}
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\boxed{
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\Ec^\text{ACFDT}
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
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\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} w_k \Tr( \bK{}{} \bP{}{\lambda_k})
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\green{\Ec^\text{ACFDT}}
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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\stackrel{\blue{\text{quad}}}{\approx} \frac{1}{2} \sum_k^{N_\text{grid}} \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})
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}
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\end{equation}
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\end{block}
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@ -1290,18 +1303,18 @@ decoration={snake,
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\btB{}{} & \btA{}{}
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\end{pmatrix}
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\qquad
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\tA{ia,jb}{} = 2\lambda\ERI{ia}{bj}
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\tA{ia,jb}{} = 2\la\ERI{ia}{bj}
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\qquad
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\tB{ia,jb}{} = 2\lambda\ERI{ia}{jb}
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\tB{ia,jb}{} = 2\la\ERI{ia}{jb}
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\end{equation}
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\end{block}
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\begin{block}{Correlation part of the two-particle density matrix}
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\begin{equation}
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\bP{}{\lambda} =
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\bP{}{\la} =
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\begin{pmatrix}
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\bY{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bY{}{\lambda} \cdot \T{(\bX{}{\lambda})}
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\bY{}{\la} \cdot \T{(\bY{}{\la})} & \bY{}{\la} \cdot \T{(\bX{}{\la})}
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\\
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\bX{}{\lambda} \cdot \T{(\bY{}{\lambda})} & \bX{}{\lambda} \cdot \T{(\bX{}{\lambda})}
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\bX{}{\la} \cdot \T{(\bY{}{\la})} & \bX{}{\la} \cdot \T{(\bX{}{\la})}
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\end{pmatrix}
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-
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\begin{pmatrix}
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@ -1318,14 +1331,14 @@ decoration={snake,
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\begin{frame}{Gaussian quadrature}
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\begin{block}{Numerical integration by quadrature}
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\begin{equation}
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\boxed{\int_a^b f(x) w(x) dx \approx \sum_k \underbrace{w_k}_{\text{weights}} f(\underbrace{x_k}_{\text{roots}})}
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\boxed{\int_{\red{a}}^{\red{b}} f(x) \purple{w(x)} dx \approx \sum_k \underbrace{\purple{w_k}}_{\text{\purple{weights}}} f(\underbrace{\violet{x_k}}_{\text{\violet{roots}}})}
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\end{equation}
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\end{block}
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\begin{block}{Quadrature rules}
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\begin{center}
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\begin{tabular}{llll}
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\hline
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Interval $[a,b]$ & Weight function $w(x)$ & Orthogonal polynomials & Name \\
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\red{Interval $[a,b]$} & \purple{Weight function $w(x)$} & \violet{Orthogonal polynomials} & \orange{Name} \\
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\hline
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$[-1,1]$ & $1$ & Legendre $P_n(x)$ & Gauss-Legendre \\
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$(-1,1)$ & $(1-x)^\alpha(1+x)^\beta, \quad \alpha,\beta > -1$ & Jacobi $P_n^{\alpha,\beta}(x)$ & Gauss-Jacobi \\
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@ -1348,30 +1361,30 @@ decoration={snake,
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\begin{frame}{ACFDT at the RPA/RPAx level}
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\begin{block}{RPA matrix elements}
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\begin{equation}
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\A{ia,jb}{\lambda,\RPA} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\lambda\ERI{ia}{bj}
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\orange{\A{ia,jb}{\la,\RPA}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + 2\la\ERI{ia}{bj}
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\qquad
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\B{ia,jb}{\lambda,\RPA} = 2\lambda\ERI{ia}{jb}
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\orange{\B{ia,jb}{\la,\RPA}} = 2\la\ERI{ia}{jb}
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\end{equation}
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\begin{equation}
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\boxed{
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\Ec^\RPA
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
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= \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPA} - \Tr(\bA{}{\RPA}) ]
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\green{\Ec^\RPA}
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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= \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPA}} - \Tr(\orange{\bA{}{\RPA}}) ]
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}
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\end{equation}
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\end{block}
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\begin{block}{RPAx matrix elements}
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\begin{equation}
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\A{ia,jb}{\lambda,\RPAx} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \lambda \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]
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\orange{\A{ia,jb}{\la,\RPAx}} = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + \la \qty[2 \ERI{ia}{bj} - \ERI{ij}{ab} ]
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\qquad
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\B{ia,jb}{\lambda,\RPAx} = \lambda \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
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\orange{\B{ia,jb}{\la,\RPAx}} = \la \qty[2 \ERI{ia}{jb} - \ERI{ib}{aj} ]
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\end{equation}
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\begin{equation}
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\boxed{
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\Ec^\RPAx
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
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\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \Om{m}{\RPAx} - \Tr(\bA{}{\RPAx}) ]
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\green{\Ec^\RPAx}
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\RPAx}} - \Tr(\orange{\bA{}{\RPAx}}) ]
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}
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\end{equation}
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\end{block}
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@ -1384,28 +1397,28 @@ decoration={snake,
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\begin{frame}{ACFDT at the BSE level}
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\begin{block}{BSE matrix elements}
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\begin{equation}
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\A{ia,jb}{\lambda,\BSE} = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \lambda \qty[2 \ERI{ia}{bj} - W_{ij,ab}^{\lambda}(\omega = 0) ]
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\orange{\A{ia,jb}{\la,\BSE}} = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \la \qty[2 \ERI{ia}{bj} - \highlight{W}_{ij,ab}^{\la}(\omega = 0) ]
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\qquad
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\B{ia,jb}{\lambda,\BSE} = \lambda \qty[2 \ERI{ia}{jb} - W_{ib,ja}^{\lambda}(\omega = 0)]
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\orange{\B{ia,jb}{\la,\BSE}} = \la \qty[2 \ERI{ia}{jb} - \highlight{W}_{ib,ja}^{\la}(\omega = 0)]
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\end{equation}
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\begin{equation}
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\boxed{
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\Ec^\BSE
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\lambda}) d\lambda
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\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \Om{m}{\BSE} - \Tr(\bA{}{\BSE}) ]
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\green{\Ec^\BSE}
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= \frac{1}{2} \int_0^1 \Tr( \bK{}{} \bP{}{\la}) d\la
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\alert{\neq} \frac{1}{2} \qty[ \sum_{m} \orange{\Om{m}{\BSE}} - \Tr(\orange{\bA{}{\BSE}}) ]
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}
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\end{equation}
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\end{block}
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\begin{block}{$\lambda$-dependent screening}
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\begin{block}{$\la$-dependent screening}
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\begin{equation}
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\highlight{W}_{pq,rs}^{\lambda}(\yo)
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\highlight{W}_{pq,rs}^{\la}(\yo)
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= \ERI{pq}{rs}
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+ 2 \sum_m \violet{\ERI{pq}{m}^{\lambda}} \violet{\ERI{rs}{m}^{\lambda}}
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\qty[ \frac{1}{\yo - \orange{\Om{m}{\lambda,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\lambda,\RPA}} - i \eta} ]
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+ 2 \sum_m \violet{\ERI{pq}{m}^{\la}} \violet{\ERI{rs}{m}^{\la}}
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\qty[ \frac{1}{\yo - \orange{\Om{m}{\la,\RPA}} + i \eta} - \frac{1}{\yo + \orange{\Om{m}{\la,\RPA}} - i \eta} ]
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\end{equation}
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\begin{equation}
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\violet{\ERI{pq}{m}^{\lambda}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\lambda,\RPA}+\bY{m}{\lambda,\RPA}})_{ia}
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\violet{\ERI{pq}{m}^{\la}} = \sum_{ia} \ERI{pq}{ia} (\orange{\bX{m}{\la,\RPA}+\bY{m}{\la,\RPA}})_{ia}
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\end{equation}
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\end{block}
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\end{frame}
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@ -1416,14 +1429,14 @@ decoration={snake,
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\begin{block}{ACFDT correlation energy from BSE}
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\begin{algorithmic}
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\Procedure{ACFDT for BSE}{}
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\State Compute $GW$ quasiparticle energies $\beGW$ and interaction kernel $\bK{}{}$
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\State Get Gauss-Legendre weights and roots $\{w_k,\lambda_k\}_{1\le k \le N_\text{grid}}$
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\State $\Ec \gets 0$
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\State Compute $GW$ quasiparticle energies $\blue{\beGW}$ and interaction kernel $\bK{}{}$
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\State Get Gauss-Legendre weights and roots $\{\purple{w_k},\violet{\lambda_k}\}_{1\le k \le N_\text{grid}}$
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\State $\green{\Ec} \gets 0$
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\For{$k=1,\ldots,N_\text{grid}$}
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\State Compute $W^{\lambda_k}$
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\State Perform BSE calculation at $\lambda = \lambda_k$ to get $\bX{}{\lambda_k}$ and $\bY{}{\lambda_k}$
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\State Form two-particle density matrix $\bP{}{\lambda_k}$
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\State $\Ec \gets \Ec + w_k \Tr( \bK{}{} \bP{}{\lambda_k})$
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\State Compute static screening elements $\highlight{W}_{pq,rs}^{\violet{\lambda_k}}$
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\State Perform BSE calculation at $\la = \violet{\lambda_k}$ to get $\bX{}{\violet{\lambda_k}}$ and $\bY{}{\violet{\lambda_k}}$
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\State Form two-particle density matrix $\bP{}{\violet{\lambda_k}}$
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\State $\green{\Ec} \gets \green{\Ec} + \purple{w_k} \Tr( \bK{}{} \bP{}{\violet{\lambda_k}})$
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\EndFor
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\EndProcedure
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\end{algorithmic}
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@ -1440,15 +1453,15 @@ decoration={snake,
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%-----------------------------------------------------
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%-----------------------------------------------------
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\begin{frame}{Useful papers}
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\begin{frame}{Useful papers/programs}
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\begin{itemize}
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\item \red{molGW:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
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\item \red{mol$GW$:} Bruneval et al. Comp. Phys. Comm. 208 (2016) 149
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\bigskip
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\item \green{Turbomole:} van Setten et al. JCTC 9 (2013) 232; Kaplan et al. JCTC 12 (2016) 2528
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\bigskip
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\item \violet{Fiesta:} Blase et al. Chem. Soc. Rev. 47 (2018) 1022
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\bigskip
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\item \purple{FHI-AIMS:} Caruso et al. 86 (2012) 081102
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\item \purple{FHI-AIMS:} Caruso et al. PRB 86 (2012) 081102
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\bigskip
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\item \orange{Review:}
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\begin{itemize}
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@ -1459,7 +1472,8 @@ decoration={snake,
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\item Blase et al. JPCL 11 (2020) 7371
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\end{itemize}
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\bigskip
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\item \red{GW100:} Data set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665
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\item \red{$GW$100:} IPs for a set of 100 molecules. van Setten et al. JCTC 11 (2015) 5665 (\url{http://gw100.wordpress.com})
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\end{itemize}
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\end{frame}
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%-----------------------------------------------------
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