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Tab for Be

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Pierre-Francois Loos 2020-10-30 13:12:56 +01:00
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sfBSE.tex
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@ -129,7 +129,7 @@ where the expressions of the matrix elements of $\bA{}{}$ and $\bB{}{}$ are spec
The spin structure of these matrices, though, is general The spin structure of these matrices, though, is general
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LR-RPA-AB} \label{eq:LR-RPA-AB-sc}
\bA{}{\spc} & = \begin{pmatrix} \bA{}{\spc} & = \begin{pmatrix}
\bA{}{\upup,\upup} & \bA{}{\upup,\dwdw} \\ \bA{}{\upup,\upup} & \bA{}{\upup,\dwdw} \\
\bA{}{\dwdw,\upup} & \bA{}{\dwdw,\dwdw} \\ \bA{}{\dwdw,\upup} & \bA{}{\dwdw,\dwdw} \\
@ -140,7 +140,7 @@ The spin structure of these matrices, though, is general
\bB{}{\dwdw,\upup} & \bB{}{\dwdw,\dwdw} \\ \bB{}{\dwdw,\upup} & \bB{}{\dwdw,\dwdw} \\
\end{pmatrix} \end{pmatrix}
\\ \\
\label{eq:LR-RPA-AB} \label{eq:LR-RPA-AB-sf}
\bA{}{\spf} & = \begin{pmatrix} \bA{}{\spf} & = \begin{pmatrix}
\bA{}{\updw,\updw} & \bO \\ \bA{}{\updw,\updw} & \bO \\
\bO & \bA{}{\dwup,\dwup} \\ \bO & \bA{}{\dwup,\dwup} \\
@ -215,6 +215,7 @@ Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchan
\end{split} \end{split}
\end{equation} \end{equation}
is, like the one-body Green's function, spin-diagonal, and its spectral representation reads is, like the one-body Green's function, spin-diagonal, and its spectral representation reads
\begin{subequations}
\begin{gather} \begin{gather}
\Sig{p_\sig q_\sig}{\x} \Sig{p_\sig q_\sig}{\x}
= - \sum_{i} \ERI{p_\sig i_\sig}{i_\sig q_\sig} = - \sum_{i} \ERI{p_\sig i_\sig}{i_\sig q_\sig}
@ -226,6 +227,7 @@ is, like the one-body Green's function, spin-diagonal, and its spectral represen
& + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta} & + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta}
\end{split} \end{split}
\end{gather} \end{gather}
\end{subequations}
where the self-energy has been split in its exchange (x) and correlation (c) contributions. where the self-energy has been split in its exchange (x) and correlation (c) contributions.
The Dyson equation linking the Green's function and the self-energy holds separately for each spin component The Dyson equation linking the Green's function and the self-energy holds separately for each spin component
\begin{equation} \begin{equation}
@ -550,9 +552,26 @@ All the static and dynamic BSE calculations have been performed with the softwar
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% TABLE I %%% %%% TABLE I %%%
%\begin{table} \begin{squeezetable}
% \begin{table*}
%\end{table} \caption{
Spin-flip excitations of \ce{Be} obtained for various methods with the 6-31G basis.
The $GW$ calculations are performed with a HF starting point.
\label{tab:Be}}
\begin{ruledtabular}
\begin{tabular}{lcccccccccccc}
State & TD-BLYP & TD-B3LYP & TD-BHHLYP & CIS
& BSE@{\GOWO} & BSE@{\evGW} & dBSE@{\GOWO} & dBSE@{\evGW}
& ADC(2) & ADC(2)-x & ADC(3) & FCI \\
\hline
$^3P(1s^22s2p)$ & & & & 2.111 & & & & & & & & 2.862 \\
$^1P(1s^22s2p)$ & & & & 6.036 & & & & & & & & 6.577 \\
$^3P(1s^22p^2)$ & & & & 7.480 & & & & & & & & 7.669 \\
$^1P(1s^22p^2)$ & & & & 8.945 & & & & & & & & 8.624 \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\end{squeezetable}
%%% %%% %%% %%% %%% %%% %%% %%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion} \section{Conclusion}