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Pierre-Francois Loos 2020-02-11 16:20:32 +01:00
parent f3629cc974
commit 20aa963400
2 changed files with 4 additions and 4 deletions
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BSE-PES.tex
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@ -146,7 +146,7 @@
\newcommand{\bB}[1]{\mathbf{B}^{#1}}
\newcommand{\bX}[1]{\mathbf{X}^{#1}}
\newcommand{\bY}[1]{\mathbf{Y}^{#1}}
\newcommand{\bZ}[1]{\mathbf{Z}^{#1}}
\newcommand{\bZ}[2]{\mathbf{Z}_{#1}^{#2}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bP}[1]{\mathbf{P}^{#1}}
@ -302,14 +302,14 @@ In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$
In the absence of instabilities (\ie, when $\bA{\IS} - \bB{\IS}$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension
\begin{equation}
\label{eq:small-LR}
(\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{\IS}_m = (\Om{m}{\IS})^2 \bZ{\IS}_m,
(\bA{\IS} - \bB{\IS})^{1/2} (\bA{\IS} + \bB{\IS}) (\bA{\IS} - \bB{\IS})^{1/2} \bZ{m}{\IS} = (\Om{m}{\IS})^2 \bZ{m}{\IS},
\end{equation}
where the excitation amplitudes are
\begin{subequations}
\begin{align}
\bX{\IS} + \bY{\IS} = (\bOm{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{+1/2} \bZ{\IS},
(\bX{\IS} + \bY{\IS})_m = (\Om{m}{\IS})^{-1/2} (\bA{\IS} - \bB{\IS})^{+1/2} \bZ{m}{\IS},
\\
\bX{\IS} - \bY{\IS} = (\bOm{\IS})^{+1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{\IS}.
(\bX{\IS} - \bY{\IS})_m = (\Om{m}{\IS})^{+1/2} (\bA{\IS} - \bB{\IS})^{-1/2} \bZ{m}{\IS}.
\end{align}
\end{subequations}
Introducing the so-called Mulliken notation for the bare two-electron integrals

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Data/H2_GS_VDZ.pdf
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