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details in the normal order hamiltonian

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Raul QUINTERO MONSEBAIZ 2022-10-13 10:46:52 +02:00
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CCvsMBPT.tex
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@ -292,7 +292,7 @@ To be more specific, restricting ourselves to CCD, \ie, $\hT = \hT_2$, the eleme
\begin{equation} \begin{equation}
\mel*{ \Psi_{i}^{a} }{ \bHN }{ \Psi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi} \mel*{ \Psi_{i}^{a} }{ \bHN }{ \Psi_{j}^{b} } = \cF_{ab} \delta_{ij} - \cF_{ij} \delta_{ab} + \cW_{jabi}
\end{equation} \end{equation}
where $\bHN = e^{-\hT} \hH e^{\hT} - E_\text{CCD}$ is the normal-ordered similarity-transformed Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are where \textcolor{blue}{ $\bHN = e^{-\hT} \hH_{N} e^{\hT} $} is the normal-ordered similarity-transformed Hamiltonian, $\Psi_{i}^{a}$ are singly-excited determinants, the one-body terms are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:cFab} \label{eq:cFab}
@ -503,9 +503,9 @@ where $\be{}{\GW}$ is a diagonal matrix collecting the quasiparticle energies, t
\end{subequations} \end{subequations}
and the corresponding coupling blocks read and the corresponding coupling blocks read
\begin{align} \begin{align}
V^\text{2h1p}_{p,klc} & = \ERI{pc}{kl} V^\text{2h1p}_{p,klc} & =\textcolor{blue}{\sqrt{2}}\ERI{pc}{kl}
& &
V^\text{2p1h}_{p,kcd} & = \ERI{pk}{dc} V^\text{2p1h}_{p,kcd} & =\textcolor{blue}{\sqrt{2}} \ERI{pk}{dc}
\end{align} \end{align}
Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021} Going beyond the Tamm-Dancoff approximation is possible, but more cumbersome. \cite{Bintrim_2021}