added notes and updated numbers
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56
BSEdyn.tex
56
BSEdyn.tex
@ -531,7 +531,7 @@ From a more practical point of view, Eq.~\eqref{eq:BSE-final} can be recast as a
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\begin{pmatrix}
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\bA{}(\Om{s}{}) & \bB{}(\Om{s}{})
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\\
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-\bB{}(\Om{s}{}) & -\bA{}(\Om{s}{})
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-\bB{}(-\Om{s}{}) & -\bA{}(-\Om{s}{})
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\\
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\end{pmatrix}
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\cdot
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@ -565,8 +565,8 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
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\begin{multline}
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\label{eq:LR-PT}
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\begin{pmatrix}
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\bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\
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-\bB{}(\Om{s}{}) & -\bA{}(\Om{s}{}) \\
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\bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\
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-\bB{}(-\Om{s}{}) & -\bA{}(-\Om{s}{}) \\
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\end{pmatrix}
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\\
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=
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@ -578,8 +578,8 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
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\end{pmatrix}
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+
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\begin{pmatrix}
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\bA{(1)}(\Om{s}{}) & \bB{(1)}(\Om{s}{}) \\
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-\bB{(1)}(\Om{s}{}) & -\bA{(1)}(\Om{s}{}) \\
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\bA{(1)}(\Om{s}{}) & \bB{(1)}(\Om{s}{}) \\
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-\bB{(1)}(-\Om{s}{}) & -\bA{(1)}(-\Om{s}{}) \\
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\end{pmatrix},
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\end{multline}
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with
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@ -662,7 +662,7 @@ Thanks to first-order perturbation theory, the first-order correction to the $s$
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\cdot
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\begin{pmatrix}
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\bA{(1)}(\Om{s}{(0)}) & \bB{(1)}(\Om{s}{(0)}) \\
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-\bB{(1)}(\Om{s}{(0)}) & -\bA{(1)}(\Om{s}{(0)}) \\
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-\bB{(1)}(-\Om{s}{(0)}) & -\bA{(1)}(-\Om{s}{(0)}) \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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@ -742,18 +742,18 @@ All the static and dynamic BSE calculations have been performed with the softwar
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& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
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& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
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\hline
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$^1\Pi_g(n \ra \pis)$ & Val. & 9.90 & -0.32 & -0.31 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & -0.42 \\
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$^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
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$^1\Delta_u(\pi \ra \pis)$ & Val. & 10.37 & -0.31 & -0.31 & 10.27 & -0.39 & -0.40 & 10.34 & -0.41 & -0.40 \\
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$^1\Sigma_g^+$(R) & Ryd. & 15.67 & -0.17 & -0.12 & 15.04 & -0.21 & -0.10 & 14.72 & -0.21 & -0.16 \\
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$^1\Pi_u$(R) & Ryd. & 15.00 & -0.21 & -0.21 & 14.75 & -0.27 & -0.26 & 14.72 & -0.29 & -0.26 \\
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$^1\Sigma_u^+$(R) & Ryd. & 22.88\fnm[1] & -0.15 & -0.21 & 19.03 & -0.08 & -0.06 & 16.78 & -0.06 & -0.07 \\
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$^1\Pi_u$(R) & Ryd. & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.13 & 16.93 & -0.09 & -0.09 \\
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$^1\Pi_g(n \ra \pis)$ & Val. & 9.90 & -0.32 & -0.33 & 9.92 & -0.40 & -0.42 & 10.01 & -0.42 & - \\
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$^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.31 & 9.61 & -0.42 & -0.36 & 9.69 & -0.44 & - \\
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$^1\Delta_u(\pi \ra \pis)$ & Val. & 10.37 & -0.31 & -0.32 & 10.27 & -0.39 & -0.42 & 10.34 & -0.41 & - \\
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$^1\Sigma_g^+$(R) & Ryd. & 15.67 & -0.17 & -0.59 & 15.04 & -0.21 & -0.46 & 14.72 & -0.21 & - \\
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$^1\Pi_u$(R) & Ryd. & 15.00 & -0.21 & -0.26 & 14.75 & -0.27 & -0.30 & 14.72 & -0.29 & - \\
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$^1\Sigma_u^+$(R) & Ryd. & 22.88\fnm[1] & -0.15 & -0.20 & 19.03 & -0.08 & -0.07 & 16.78 & -0.06 & - \\
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$^1\Pi_u$(R) & Ryd. & 23.62\fnm[1] & -0.11 & -0.10 & 19.15 & -0.11 & -0.10 & 16.93 & -0.09 & - \\
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\\
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$^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.39 & -0.48 & -0.63 & 7.46 & -0.59 & -0.56 & 7.59 & -0.62 & -0.60 \\
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$^3\Pi_g(n \ra \pis)$ & Val. & 8.07 & -0.42 & -0.44 & 8.14 & -0.52 & -0.50 & 8.24 & -0.54 & -0.51 \\
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$^3\Delta_u(\pi \ra \pis)$ & Val. & 8.56 & -0.41 & -0.46 & 8.52 & -0.52 & -0.50 & 8.62 & -0.55 & -0.51 \\
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$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.34 & 9.61 & -0.42 & -0.40 & 9.69 & -0.44 & -0.44 \\
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$^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.39 & -0.48 & -0.69 & 7.46 & -0.59 & -0.41 & 7.59 & -0.62 & - \\
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$^3\Pi_g(n \ra \pis)$ & Val. & 8.07 & -0.42 & -0.43 & 8.14 & -0.52 & -0.48 & 8.24 & -0.54 & - \\
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$^3\Delta_u(\pi \ra \pis)$ & Val. & 8.56 & -0.41 & -0.40 & 8.52 & -0.52 & -0.40 & 8.62 & -0.55 & - \\
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$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.70 & -0.33 & -0.31 & 9.61 & -0.42 & -0.36 & 9.69 & -0.44 & - \\
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\hline
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\\
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& & \mc{3}{c}{aug-cc-pVDZ ($\Eg^{\GW} = 19.49$ eV)}
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@ -764,18 +764,18 @@ All the static and dynamic BSE calculations have been performed with the softwar
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& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$}
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& \tabc{$\Om{s}{\stat}$} & \tabc{$\Delta\Om{s}{\dyn}$(dTDA)} & \tabc{$\Delta\Om{s}{\dyn}$} \\
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\hline
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$^1\Pi_g(n \ra \pis)$ & Val. & 10.18 & -0.41 & -0.43 & 10.42 & -0.42 & -0.40 & 10.52 & -0.43 & -0.40 \\
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$^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.44 & 10.11 & -0.45 & -0.45 & 10.20 & -0.45 & -0.45 \\
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$^1\Delta_u(\pi \ra \pis)$ & Val. & 10.57 & -0.41 & -0.40 & 10.75 & -0.42 & -0.41 & 10.85 & -0.42 & -0.42 \\
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$^1\Sigma_g^+$ & Ryd. & 13.72 & -0.04 & -0.04 & 13.60 & -0.03 & -0.03 & 13.55 & -0.02 & -0.02 \\
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$^1\Pi_u$ & Ryd. & 14.07 & -0.05 & -0.05 & 13.98 & -0.04 & -0.04 & 13.96 & -0.03 & -0.04 \\
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$^1\Sigma_u^+$ & Ryd. & 13.80 & -0.08 & -0.08 & 13.98 & -0.07 & -0.08 & 14.08 & -0.06 & -0.06 \\
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$^1\Pi_u$ & Ryd. & 14.22 & -0.04 & -0.03 & 14.24 & -0.03 & -0.03 & 14.26 & -0.03 & -0.02 \\
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$^1\Pi_g(n \ra \pis)$ & Val. & 10.18 & -0.41 & -0.43 & 10.42 & -0.42 & -0.41 & 10.52 & -0.43 & - \\
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$^1\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.41 & 10.11 & -0.45 & -0.42 & 10.20 & -0.45 & - \\
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$^1\Delta_u(\pi \ra \pis)$ & Val. & 10.57 & -0.41 & -0.41 & 10.75 & -0.42 & -0.45 & 10.85 & -0.42 & - \\
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$^1\Sigma_g^+$ & Ryd. & 13.72 & -0.04 & -0.05 & 13.60 & -0.03 & -0.03 & 13.55 & -0.02 & - \\
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$^1\Pi_u$ & Ryd. & 14.07 & -0.05 & -0.05 & 13.98 & -0.04 & -0.04 & 13.96 & -0.03 & - \\
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$^1\Sigma_u^+$ & Ryd. & 13.80 & -0.08 & -0.10 & 13.98 & -0.07 & -0.10 & 14.08 & -0.06 & - \\
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$^1\Pi_u$ & Ryd. & 14.22 & -0.04 & -0.03 & 14.24 & -0.03 & -0.03 & 14.26 & -0.03 & - \\
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\\
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$^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.75 & -0.63 & -1.32 & 8.02 & -0.64 & -0.60 & 8.12 & -0.64 & -0.66 \\
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$^3\Pi_g(n \ra \pis)$ & Val. & 8.42 & -0.54 & -0.53 & 8.66 & -0.56 & -0.79 & 8.75 & -0.56 & -0.50 \\
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$^3\Delta_u(\pi \ra \pis)$ & Val. & 8.86 & -0.54 & -0.55 & 9.04 & -0.56 & -0.59 & 9.14 & -0.56 & -0.60 \\
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$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.45 & 10.11 & -0.45 & -0.45 & 10.20 & -0.45 & -0.45 \\
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$^3\Sigma_u^+(\pi \ra \pis)$ & Val. & 7.75 & -0.63 & -2.42 & 8.02 & -0.64 & -0.45 & 8.12 & -0.64 & - \\
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$^3\Pi_g(n \ra \pis)$ & Val. & 8.42 & -0.54 & -0.50 & 8.66 & -0.56 & -0.79 & 8.75 & -0.56 & - \\
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$^3\Delta_u(\pi \ra \pis)$ & Val. & 8.86 & -0.54 & -0.47 & 9.04 & -0.56 & -0.52 & 9.14 & -0.56 & - \\
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$^3\Sigma_u^-(\pi \ra \pis)$ & Val. & 9.95 & -0.44 & -0.41 & 10.11 & -0.45 & -0.42 & 10.20 & -0.45 & - \\
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\end{tabular}
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\end{ruledtabular}
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\fnt[1]{Excitation energy larger than the fundamental gap.}
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467
Notes/BSEdyn-notes.tex
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467
Notes/BSEdyn-notes.tex
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@ -0,0 +1,467 @@
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\documentclass{article}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig}
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\usepackage[version=4]{mhchem}
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%\usepackage[utf8]{inputenc}
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%\usepackage[T1]{fontenc}
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%\usepackage{txfonts}
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\usepackage[
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colorlinks=true,
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citecolor=blue,
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breaklinks=true
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]{hyperref}
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\urlstyle{same}
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\newcommand{\ie}{\textit{i.e.}}
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\newcommand{\eg}{\textit{e.g.}}
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\newcommand{\alert}[1]{\textcolor{red}{#1}}
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\definecolor{darkgreen}{HTML}{009900}
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\usepackage[normalem]{ulem}
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\newcommand{\titou}[1]{\textcolor{red}{#1}}
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\newcommand{\trashPFL}[1]{\textcolor{red}{\sout{#1}}}
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\newcommand{\PFL}[1]{\titou{(\underline{\bf PFL}: #1)}}
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\newcommand{\mc}{\multicolumn}
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\newcommand{\fnm}{\footnotemark}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\tabc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\SI}{\textcolor{blue}{supplementary material}}
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\newcommand{\QP}{\textsc{quantum package}}
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\newcommand{\T}[1]{#1^{\intercal}}
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% coordinates
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\newcommand{\br}{\mathbf{r}}
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\newcommand{\dbr}{d\br}
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% methods
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\newcommand{\evGW}{ev$GW$}
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\newcommand{\qsGW}{qs$GW$}
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\newcommand{\GOWO}{$G_0W_0$}
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\xc}{\text{xc}}
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\newcommand{\Ha}{\text{H}}
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\newcommand{\co}{\text{x}}
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%
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\newcommand{\Norb}{N_\text{orb}}
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\newcommand{\Nocc}{O}
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\newcommand{\Nvir}{V}
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\newcommand{\IS}{\lambda}
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% operators
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\newcommand{\hH}{\Hat{H}}
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% methods
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\newcommand{\KS}{\text{KS}}
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\newcommand{\HF}{\text{HF}}
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\newcommand{\RPA}{\text{RPA}}
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\newcommand{\BSE}{\text{BSE}}
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\newcommand{\TDABSE}{\text{BSE(TDA)}}
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\newcommand{\dBSE}{\text{dBSE}}
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\newcommand{\TDAdBSE}{\text{dBSE(TDA)}}
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\newcommand{\GW}{GW}
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\newcommand{\stat}{\text{stat}}
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\newcommand{\dyn}{\text{dyn}}
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\newcommand{\TDA}{\text{TDA}}
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% energies
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\newcommand{\Enuc}{E^\text{nuc}}
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\newcommand{\Ec}{E_\text{c}}
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\newcommand{\EHF}{E^\text{HF}}
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\newcommand{\EBSE}{E^\text{BSE}}
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\newcommand{\EcRPA}{E_\text{c}^\text{RPA}}
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\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
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% orbital energies
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\newcommand{\e}[1]{\eps_{#1}}
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\newcommand{\eHF}[1]{\eps^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\eps^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\eps^\text{QP}_{#1}}
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\newcommand{\eGW}[1]{\eps^{GW}_{#1}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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% Matrix elements
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\newcommand{\Sig}[1]{\Sigma_{#1}}
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\ERI}[2]{(#1|#2)}
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\newcommand{\sERI}[2]{[#1|#2]}
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% excitation energies
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\newcommand{\OmRPA}[1]{\Omega_{#1}^{\text{RPA}}}
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\newcommand{\OmRPAx}[1]{\Omega_{#1}^{\text{RPAx}}}
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\newcommand{\OmBSE}[1]{\Omega_{#1}^{\text{BSE}}}
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\newcommand{\spinup}{\downarrow}
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\newcommand{\spindw}{\uparrow}
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\newcommand{\singlet}{\uparrow\downarrow}
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\newcommand{\triplet}{\uparrow\uparrow}
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% Matrices
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\newcommand{\bO}{\mathbf{0}}
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\newcommand{\bH}{\mathbf{H}}
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\newcommand{\bV}{\mathbf{V}}
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\newcommand{\bI}{\mathbf{1}}
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\newcommand{\bb}{\mathbf{b}}
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\newcommand{\bA}{\mathbf{A}}
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\newcommand{\bB}{\mathbf{B}}
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\newcommand{\bx}{\mathbf{x}}
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% units
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\newcommand{\IneV}[1]{#1 eV}
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\newcommand{\InAU}[1]{#1 a.u.}
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\newcommand{\InAA}[1]{#1 \AA}
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\newcommand{\kcal}{kcal/mol}
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\DeclareMathOperator*{\argmax}{argmax}
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\DeclareMathOperator*{\argmin}{argmin}
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% orbitals, gaps, etc
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\newcommand{\updw}{\uparrow\downarrow}
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\newcommand{\upup}{\uparrow\uparrow}
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\newcommand{\eps}{\varepsilon}
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\newcommand{\IP}{I}
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\newcommand{\EA}{A}
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\newcommand{\HOMO}{\text{HOMO}}
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\newcommand{\LUMO}{\text{LUMO}}
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\newcommand{\Eg}{E_\text{g}}
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\newcommand{\EgFun}{\Eg^\text{fund}}
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\newcommand{\EgOpt}{\Eg^\text{opt}}
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\newcommand{\EB}{E_B}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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\title{Notes on the Dynamical Bethe-Salpeter Equation}
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\author{Pierre-Fran\c{c}ois Loos}
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\begin{document}
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\maketitle
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{The concept of dynamical quantities}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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As a chemist, it is maybe difficult to understand the concept of dynamical properties, the motivation behind their introduction, and their actual usefulness.
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Here, we will try to give a pedagogical example showing the importance of dynamical quantities and their main purposes \cite{ReiningBook}.
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To do so, let us consider the usual chemical scenario where one wants to get the neutral excitations of a given system.
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In most cases, this can be done by solving a set of linear equations of the form
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\begin{equation}
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\label{eq:lin_sys}
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\bA \bx = \omega \bx
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\end{equation}
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where $\omega$ is one of the neutral excitation energies of the system associated with the transition vector $\bx$.
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If we assume that the operator $\bA$ has a matrix representation of size $K \times K$, this \textit{linear} set of equations yields $K$ excitation energies.
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However, in practice, $K$ might be very large, and it might therefore be practically useful to recast this system as two smaller coupled systems, such that
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\begin{equation}
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\label{eq:lin_sys_split}
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\begin{pmatrix}
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\bA_1 & \T{\bb} \\
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\bb & \bA_2 \\
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\end{pmatrix}
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\begin{pmatrix}
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\bx_1 \\
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\bx_2 \\
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\end{pmatrix}
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= \omega
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\begin{pmatrix}
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\bx_1 \\
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\bx_2 \\
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\end{pmatrix}
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\end{equation}
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where the blocks $\bA_1$ and $\bA_2$, of sizes $K_1 \times K_1$ and $K_2 \times K_2$ (with $K_1 + K_2 = K$), can be associated with, for example, the single and double excitations of the system.
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Note that this \textit{exact} decomposition does not alter, in any case, the values of the excitation energies, not their eigenvectors.
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Solving separately each row of the system \eqref{eq:lin_sys_split} yields
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\begin{subequations}
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\begin{gather}
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\label{eq:row1}
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\bA_1 \bx_1 + \T{\bb} \bx_2 = \omega \bx_1
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\\
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\label{eq:row2}
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\bx_2 = (\omega \bI - \bA_2)^{-1} \bb \bx_1
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\end{gather}
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\end{subequations}
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Substituting Eq.~\eqref{eq:row2} into Eq.~\eqref{eq:row1} yields the following effective \textit{non-linear}, frequency-dependent operator
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\begin{equation}
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\label{eq:non_lin_sys}
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\Tilde{\bA}_1(\omega) \bx_1 = \omega \bx_1
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\end{equation}
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with
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\begin{equation}
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\Tilde{\bA}_1(\omega) = \bA_1 + \T{\bb} (\omega \bI - \bA_2)^{-1} \bb
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\end{equation}
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which has, by construction, exactly the same solutions than the linear system \eqref{eq:lin_sys} but a smaller dimension.
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For example, an operator $\Tilde{\bA}_1(\omega)$ built in the basis of single excitations can potentially provide excitation energies for double excitations thanks to its frequency-dependent nature, the information from the double excitations being ``folded'' into $\Tilde{\bA}_1(\omega)$ via Eq.~\eqref{eq:row2} \cite{ReiningBook}.
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|
||||
How have we been able to reduce the dimension of the problem while keeping the same number of solutions?
|
||||
To do so, we have transformed a linear operator $\bA$ into a non-linear operator $\Tilde{\bA}_1(\omega)$ by making it frequency dependent.
|
||||
In other words, we have sacrificed the linearity of the system in order to obtain a new, non-linear systems of equations of smaller dimension.
|
||||
This procedure converting degrees of freedom into frequency or energy dependence is very general and can be applied in various contexts.
|
||||
Thanks to its non-linearity, Eq.~\eqref{eq:non_lin_sys} can produce more solutions than its actual dimension.
|
||||
However, because there is no free lunch, this non-linear system is obviously harder to solve than its corresponding linear analogue given by Eq.~\eqref{eq:lin_sys}.
|
||||
Nonetheless, approximations can be now applied to Eq.~\eqref{eq:non_lin_sys} in order to solve it efficiently.
|
||||
|
||||
One of these approximations is the so-called \textit{static} approximation, which corresponds to fix the frequency to a particular value.
|
||||
For example, as commonly done within the Bethe-Salpeter formalism, $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
|
||||
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
|
||||
This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$.
|
||||
Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{A two-level model}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
|
||||
Let us consider a two-level quantum system made of two orbitals \cite{Romaniello_2009b}.
|
||||
We will label these two orbitals as valence ($v$) and conduction ($c$) orbitals with respective one-electron energies $\e{v}$ and $\e{c}$.
|
||||
In a more quantum chemical language, these correspond to the HOMO and LUMO orbitals (respectively).
|
||||
The ground state has a one-electron configuration $v\bar{v}$, while the doubly-excited state has a configuration $c\bar{c}$.
|
||||
There is then only one single excitation which corresponds to the transition $v \to c$.
|
||||
As usual, this can produce a singlet singly-excited state of configuration $(v\bar{c} + c\bar{v})/\sqrt{2}$, and a triplet singly-excited state of configuration $(v\bar{c} - c\bar{v})/\sqrt{2}$ \cite{SzaboBook}.
|
||||
|
||||
Within many-body perturbation theory (MBPT), one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
|
||||
Assuming that the dynamically-screened Coulomb potential has been calculated at the random-phase approximation (RPA) level and within the Tamm-Dancoff approximation (TDA), the expression of the $\GW$ quasiparticle energy is
|
||||
\begin{equation}
|
||||
\e{p}^{\GW} = \e{p} + Z_{p} \Sig{p}(\e{p})
|
||||
\end{equation}
|
||||
where $p = v$ or $c$,
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
\label{eq:Sigv}
|
||||
\Sig{v}(\omega) & = \frac{2 \ERI{vv}{vc}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{vc}{cv}^2}{\omega - \e{c} + \Omega}
|
||||
\\
|
||||
\label{eq:Sigc}
|
||||
\Sig{c}(\omega) & = \frac{2 \ERI{vc}{cv}^2}{\omega - \e{v} + \Omega} + \frac{2 \ERI{vc}{cc}^2}{\omega - \e{c} + \Omega}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
are the correlation parts of the self-energy associated with the valence of conduction orbitals,
|
||||
\begin{equation}
|
||||
Z_{p} = \qty( 1 - \left. \pdv{\Sig{p}(\omega)}{\omega} \right|_{\omega = \e{p}} )^{-1}
|
||||
\end{equation}
|
||||
is the renormalization factor, and
|
||||
\begin{equation}
|
||||
\ERI{pq}{rs} = \iint p(\br) q(\br) \frac{1}{\abs{\br - \br'}} r(\br') s(\br') d\br d\br'
|
||||
\end{equation}
|
||||
are the usual (bare) two-electron integrals.
|
||||
In Eqs.~\eqref{eq:Sigv} and \eqref{eq:Sigc}, $\Omega = \Delta\e{} + 2 \ERI{vc}{vc}$ is the sole (singlet) RPA excitation energy of the system, with $\Delta\e{} = \eGW{c} - \eGW{v}$.
|
||||
|
||||
One can now build the dynamical Bethe-Salpeter equation (dBSE) Hamiltonian, which reads
|
||||
\begin{equation} \label{eq:HBSE}
|
||||
\bH^{\dBSE}(\omega) =
|
||||
\begin{pmatrix}
|
||||
R(\omega) & C(\omega)
|
||||
\\
|
||||
-C(-\omega) & -R(-\omega)
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
R(\omega) & = \Delta\e{} + 2 \sigma \ERI{vc}{cv} - W_R(\omega)
|
||||
\\
|
||||
C(\omega) & = 2 \sigma \ERI{vc}{cv} - W_C(\omega)
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
($\sigma = 1$ for singlets and $\sigma = 0$ for triplets) and
|
||||
\begin{subequations}
|
||||
\begin{align}
|
||||
W_R(\omega) & = \ERI{vv}{cc} + \frac{4 \ERI{vv}{vc} \ERI{vc}{cc}}{\omega - \Omega - \Delta\e{}}
|
||||
\\
|
||||
W_C(\omega) & = \ERI{vc}{cv} + \frac{4 \ERI{vc}{cv}^2}{\omega - \Omega}
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
are the elements of the dynamically-screened Coulomb potential for the resonant and coupling blocks of the dBSE Hamiltonian.
|
||||
It can be easily shown that solving the equation
|
||||
\begin{equation}
|
||||
\det[\bH^{\dBSE}(\omega) - \omega \bI] = 0
|
||||
\end{equation}
|
||||
yields 6 solutions (per spin manifold): 3 pairs of frequencies opposite in sign, which corresponds to the 3 resonant states and the 3 anti-resonant states.
|
||||
As mentioned in Ref.~\cite{Romaniello_2009b}, spurious solutions appears due to the approximate nature of the dBSE kernel.
|
||||
Indeed, diagonalizing the exact Hamiltonian would produce two singlet solutions corresponding to the singly- and doubly-excited states, while there is only one triplet state (see discussion earlier in the section).
|
||||
Therefore, there is one spurious solution for the singlet manifold and two spurious solution for the triplet manifold.
|
||||
|
||||
Within the static approximation, the BSE Hamiltonian is
|
||||
\begin{equation}
|
||||
\bH^{\BSE} =
|
||||
\begin{pmatrix}
|
||||
R^{\stat} & C^{\stat}
|
||||
\\
|
||||
-C^{\stat} & -R^{\stat}
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
with
|
||||
\begin{align}
|
||||
R^{\stat} & = R(\omega = \Delta\e{}) = \Delta\e{} + 2 \sigma \ERI{vc}{vc} - W_R(\omega = \Delta\e{})
|
||||
\\
|
||||
C^{\stat} & = C(\omega = 0) = 2 \sigma \ERI{vc}{vc} - W_C(\omega = 0)
|
||||
\end{align}
|
||||
In the static approximation, only one pair of solutions (per spin manifold) is obtained by diagonalizing $\bH^{\BSE}$.
|
||||
There are, like in the dynamical case, opposite in sign.
|
||||
Therefore, the static BSE Hamiltonian does not produce spurious excitations but misses the (singlet) double excitation.
|
||||
|
||||
Enforcing the TDA, which corresponds to neglecting the coupling term between the resonant and anti-resonant part of the BSE Hamiltonian, \ie, $C(\omega) = 0$, allows to remove some of these spurious excitations.
|
||||
In this case, the excitation energies are obtained by solving the simple equation $R(\omega) - \omega = 0$, which yields two solutions for each spin manifold.
|
||||
There is thus only one spurious excitation in the triplet manifold, the two solutions of the singlet manifold corresponding to the single and double excitations.
|
||||
|
||||
Another way to access dynamical effects while staying in the static framework is to use perturbation theory.
|
||||
To do so, one must decompose the dBSE Hamiltonian into a (zeroth-order) static part and a dynamical perturbation, such that
|
||||
\begin{equation}
|
||||
\bH^{\dBSE}(\omega) = \underbrace{\bH^{\BSE}}_{\bH^{(0)}} + \underbrace{\qty[ \bH^{\dBSE}(\omega) - \bH^{\BSE} ]}_{\bH^{(1)}}
|
||||
\end{equation}
|
||||
Thanks to (renormalized) first-order perturbation theory, one gets
|
||||
\begin{equation}
|
||||
\omega_{1,\sigma}^{\BSE1} = \omega_{1,\sigma}^{\BSE} + Z_{1} \T{\bV} \cdot \qty[ \bH^{\dBSE}(\omega = \omega_{1,\sigma}^{\BSE}) - \bH^{\BSE} ] \cdot \bV
|
||||
\end{equation}
|
||||
where
|
||||
\begin{equation}
|
||||
\bV =
|
||||
\begin{pmatrix}
|
||||
X \\ Y
|
||||
\end{pmatrix}
|
||||
\end{equation}
|
||||
are the eigenvectors of $\bH^{\BSE}$, and
|
||||
\begin{equation}
|
||||
Z_{1} = \qty{ 1 - \T{\bV} \cdot \left. \pdv{\bH^{\dBSE}(\omega)}{\omega} \right|_{\omega = \omega_{1,\sigma}^{\BSE}} \cdot \bV }^{-1}
|
||||
\end{equation}
|
||||
This corresponds to a dynamical correction to the static excitations, and the TDA can be applied to the dynamical correction, a scheme we label as dTDA in the following.
|
||||
|
||||
We now take a numerical example by considering the singlet ground state of the \ce{He} atom in the 6-31G basis set.
|
||||
This system contains two orbitals and the numerical values of the various quantities defined above are
|
||||
\begin{align}
|
||||
\e{v} & = -0.914\,127
|
||||
&
|
||||
\e{c} & = + 1.399\,859
|
||||
\\
|
||||
\ERI{vv}{cc} & = 0.858\,133
|
||||
&
|
||||
\ERI{vc}{cv} & = 0.227\,670
|
||||
\\
|
||||
\ERI{vv}{vc} & = 0.255\,554
|
||||
&
|
||||
\ERI{vc}{cc} & = 0.316\,490
|
||||
\end{align}
|
||||
which yields
|
||||
\begin{align}
|
||||
\Omega & = 2.769\,327
|
||||
&
|
||||
\eGW{v} & = -0.863\,700
|
||||
&
|
||||
\eGW{c} & = +1.373\,640
|
||||
\end{align}
|
||||
|
||||
%%% FIGURE 1 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{dBSE}
|
||||
\caption{
|
||||
$\det[\bH^{\dBSE}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds.
|
||||
\label{fig:dBSE}
|
||||
}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
Figure \ref{fig:dBSE} shows the three resonant solutions (for the singlet and triplet spin manifold) of the dynamical BSE Hamiltonian $\bH(\omega)$ defined in Eq.~\eqref{eq:HBSE}, the curve being invariant with respect to the transformation $\omega \to - \omega$ (electron-hole symmetry).
|
||||
Numerically, we find
|
||||
\begin{align}
|
||||
\omega_{1,\updw}^{\dBSE} & = 1.90527
|
||||
&
|
||||
\omega_{2,\updw}^{\dBSE} & = 2.78377
|
||||
&
|
||||
\omega_{3,\updw}^{\dBSE} & = 4.90134
|
||||
\end{align}
|
||||
for the singlet states, and
|
||||
\begin{align}
|
||||
\omega_{1,\upup}^{\dBSE} & = 1.46636
|
||||
&
|
||||
\omega_{2,\upup}^{\dBSE} & = 2.76178
|
||||
&
|
||||
\omega_{3,\upup}^{\dBSE} & = 4.91545
|
||||
\end{align}
|
||||
for the triplet states.
|
||||
it is interesting to mention that, around $\omega = \omega_1^{\sigma}$ ($\sigma =$ $\updw$ or $\upup$), the slope of the curves depicted in Fig.~\ref{fig:dBSE} is small, while the two other solutions, $\omega_2^{\sigma}$ and $\omega_3^{\sigma}$, stem from poles and consequently the slope is very large around these frequency values.
|
||||
|
||||
Diagonalizing the static BSE Hamiltonian yields the following singlet and triplet excitation energies:
|
||||
\begin{align}
|
||||
\omega_{1,\updw}^{\BSE} & = 1.92778
|
||||
&
|
||||
\omega_{1,\upup}^{\BSE} & = 1.48821
|
||||
\end{align}
|
||||
which shows that the physical single excitation stemming from the dynamical BSE Hamiltonian is the lowest one for each spin manifold, \ie, $\omega_1^{\updw}$ and $\omega_1^{\upup}$.
|
||||
|
||||
%%% FIGURE 2 %%%
|
||||
\begin{figure}
|
||||
\includegraphics[width=\linewidth]{dBSE-TDA}
|
||||
\caption{
|
||||
$\det[\bH^{\TDAdBSE}(\omega) - \omega \bI]$ as a function of $\omega$ for both the singlet (red) and triplet (blue) manifolds within the TDA.
|
||||
\label{fig:dBSE-TDA}
|
||||
}
|
||||
\end{figure}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
Figure \ref{fig:dBSE-TDA} shows the same curves as Fig.~\ref{fig:dBSE} but in the TDA.
|
||||
As one can see, the spurious solution $\omega_2^{\sigma}$ has disappeared, and two pairs of solutions remain for each spin manifold.
|
||||
Numerically, we have
|
||||
\begin{align}
|
||||
\omega_{1,\updw}^{\TDAdBSE} & = 1.94005
|
||||
&
|
||||
\omega_{3,\updw}^{\TDAdBSE} & = 4.90117
|
||||
\end{align}
|
||||
for the singlet states, and
|
||||
\begin{align}
|
||||
\omega_{1,\upup}^{\TDAdBSE} & = 1.47070
|
||||
&
|
||||
\omega_{3,\upup}^{\TDAdBSE} & = 4.91517
|
||||
\end{align}
|
||||
while the static values are
|
||||
\begin{align}
|
||||
\omega_{1,\updw}^{\TDABSE} & = 1.95137
|
||||
&
|
||||
\omega_{1,\upup}^{\TDABSE} & = 1.49603
|
||||
\end{align}
|
||||
|
||||
It is now instructive to provide the exact results, \ie, the excitation energies obtained by diagonalizing the exact Hamiltonian in the same basis set.
|
||||
A quick configuration interaction with singles and doubles (CISD) calculation provide the following excitation energies:
|
||||
\begin{align}
|
||||
\omega_{1}^{\updw} & = 1.92145
|
||||
&
|
||||
\omega_{1}^{\upup} & = 1.47085
|
||||
&
|
||||
\omega_{3}^{\updw} & = 3.47880
|
||||
\end{align}
|
||||
This evidences that BSE reproduces qualitatively well the singlet and triplet single excitations, but quite badly the double excitation which is off by more than 1 hartree.
|
||||
All these numerical results are gathered in Table \ref{tab:Ex}.
|
||||
|
||||
The perturbatively-corrected values are also reported, which shows that this scheme is very efficient at reproducing the dynamical value.
|
||||
Note that, although the BSE1(dTDA) value is further from the dBSE value than BSE1, it is quite close to the exact excitation energy.
|
||||
|
||||
%%% TABLE I %%%
|
||||
\begin{table}
|
||||
\caption{Singlet and triplet excitation energies at various levels of theory.
|
||||
\label{tab:Ex}
|
||||
}
|
||||
\begin{center}
|
||||
\small
|
||||
\begin{tabular}{|c|ccccccc|c|}
|
||||
\hline
|
||||
Singlets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1$ & 1.92778 & 1.90022 & 1.91554 & 1.90527 & 1.95137 & 1.94004 & 1.94005 & 1.92145 \\
|
||||
$\omega_2$ & & & & 2.78377 & & & & \\
|
||||
$\omega_3$ & & & & 4.90134 & & & 4.90117 & 3.47880 \\
|
||||
\hline
|
||||
Triplets & BSE & BSE1 & BSE1(dTDA) & dBSE & BSE(TDA) & BSE1(TDA) & dBSE(TDA) & Exact \\
|
||||
\hline
|
||||
$\omega_1$ & 1.48821 & 1.46860 & 1.46260 & 1.46636 & 1.49603 & 1.47070 & 1.47070 & 1.47085 \\
|
||||
$\omega_2$ & & & & 2.76178 & & & & \\
|
||||
$\omega_3$ & & & & 4.91545 & & & 4.91517 & \\
|
||||
\hline
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{table}
|
||||
%%% %%% %%% %%%
|
||||
|
||||
% BIBLIOGRAPHY
|
||||
\bibliographystyle{unsrt}
|
||||
\bibliography{../BSEdyn}
|
||||
|
||||
\end{document}
|
||||
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Notes/dBSE.pdf
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Reference in New Issue
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