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sfBSE.rty
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sfBSE.rty
@ -93,9 +93,12 @@
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\Sig}[1]{\Sigma_{#1}}
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\newcommand{\SigGW}[1]{\Sigma^{GW}_{#1}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\Z}[1]{Z_{#1}}
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\newcommand{\MO}[1]{\phi_{#1}}
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\newcommand{\ERI}[2]{(#1|#2)}
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\newcommand{\rbra}[1]{(#1|}
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\newcommand{\rket}[1]{|#1)}
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\newcommand{\sERI}[2]{[#1|#2]}
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%% bold in Table
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@ -108,10 +111,6 @@
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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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@ -157,6 +156,15 @@
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\newcommand{\si}{\sigma}
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\newcommand{\sip}{\sigma'}
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\newcommand{\up}{\downarrow}
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\newcommand{\dw}{\uparrow}
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\newcommand{\upup}{\uparrow\uparrow}
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\newcommand{\updw}{\uparrow\downarrow}
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\newcommand{\dwup}{\downarrow\uparrow}
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\newcommand{\dwdw}{\downarrow\downarrow}
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\newcommand{\spc}{\text{sc}}
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\newcommand{\spf}{\text{sf}}
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% addresses
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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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100
sfBSE.tex
100
sfBSE.tex
@ -45,63 +45,127 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th
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\section{Unrestricted $GW$ formalism}
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\label{sec:UGW}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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Let us consider an electronic system consisting of $n = n_\up + n_\dw$ electrons (where $n_\up$ and $n_\dw$ are the number of spin-up and spin-down electrons respectively) and $N$ one-electron basis functions.
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The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
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The number of spin-conserved single excitations is then $S^\spc = S_{\up\up}^\spc + S_{\dw\dw}^\spc = O_\up V_\up + O_\dw V_\dw$, while the number of spin-flip excitations is $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
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Let us denote as $\MO{p\si}$ the $p$th orbital of spin $\sigma$ (where $\sigma = \up$ or $\dw$).
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In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations.
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It is important to understand that, in a spin-conserved excitation the hole orbital $\MO{i\si}$ and particle orbital $\MO{a\si}$ have the same spin $\si$.
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A bra and ket composed by these two orbitals will be denoted as $\rbra{ia\si}$ and $\rket{ia\si}$.
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In a spin-flip excitation, the hole and the particle states have different spin.
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%================================
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\subsection{The dynamical screening}
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%================================
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Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
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The matrix elements of $W(\omega)$ read
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\begin{multline}
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W_{pq\si,rs\sip}(\omega) = \ERI{pq\si}{rs\sip}
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+ \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m}
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\\
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+ \sum_m \sERI{pq\si}{m}\sERI{rs\sip}{m} \qty[ \frac{1}{\omega - \OmRPA{m} + i \eta} - \frac{1}{\omega + \OmRPA{m} - i \eta} ]
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\times \qty[ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
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\end{multline}
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where the two-electron integrals are
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\begin{equation}
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\ERI{pq\si}{rs\sip} = \iint \MO{p\si}(\br) \MO{q\si}(\br) \frac{1}{\abs{\br - \br'}} \MO{r\sip}(\br') \MO{s\sip}(\br') d\br d\br'
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\end{equation}
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\begin{equation}
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\sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\RPA}+\bY{m}{\RPA})_{ia\sip}
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\sERI{pq\si}{m} = \sum_{ia\sip} \ERI{pq\si}{rs\sip} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{ia\sip}
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\end{equation}
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\begin{equation}
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\label{eq:LR-RPA}
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\begin{pmatrix}
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\bA{\RPA} & \bB{\RPA} \\
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-\bB{\RPA} & -\bA{\RPA} \\
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\bA{\spc,\RPA} & \bB{\spc,\RPA} \\
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-\bB{\spc,\RPA} & -\bA{\spc,\RPA} \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{m}{\RPA} \\
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\bY{m}{\RPA} \\
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\bX{m}{\spc,\RPA} \\
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\bY{m}{\spc,\RPA} \\
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\end{pmatrix}
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=
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\OmRPA{m}
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\Om{m}{\spc,\RPA}
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\begin{pmatrix}
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\bX{m}{\RPA} \\
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\bY{m}{\RPA} \\
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\bX{m}{\spc,\RPA} \\
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\bY{m}{\spc,\RPA} \\
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\end{pmatrix},
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\end{equation}
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\begin{align}
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\label{eq:LR-RPA-AB}
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\bA{\spc} & = \begin{pmatrix}
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\bA{\upup,\upup} & \bA{\upup,\dwdw} \\
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\bA{\dwdw,\upup} & \bA{\dwdw,\dwdw} \\
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\end{pmatrix}
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&
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\bB{\spc} & = \begin{pmatrix}
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\bB{\upup,\upup} & \bB{\upup,\dwdw} \\
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\bB{\dwdw,\upup} & \bB{\dwdw,\dwdw} \\
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\end{pmatrix}
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\end{align}
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\begin{align}
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\label{eq:LR-RPA-AB}
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\bA{\spf} & = \begin{pmatrix}
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\bA{\updw,\updw} & \bO \\
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\bO & \bA{\dwup,\dwup} \\
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\end{pmatrix}
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&
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\bB{\spf} & = \begin{pmatrix}
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\bO & \bB{\updw,\dwup} \\
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\bB{\dwup,\updw} & \bO \\
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\end{pmatrix}
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\end{align}
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with
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-A}
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\A{ia\si,jb\sip}{\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a} - \e{i}) + 2 \ERI{ia\si}{jb\sip},
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\label{eq:LR_RPA-Asc}
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\A{ia\si,jb\sip}{\spc,\RPA} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\e{a\si} - \e{i\si}) + \ERI{ia\si}{jb\sip},
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\\
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\label{eq:LR_RPA-B}
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\B{ia\si,jb\sip}{\RPA} & = 2 \ERI{ia\si}{bj\sip},
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\label{eq:LR_RPA-Bsc}
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\B{ia\si,jb\sip}{\spc,\RPA} & = 0,
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\end{align}
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\end{subequations}
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%================================
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\subsection{The $GW$ self-energy}
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%================================
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\begin{equation}
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\begin{split}
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\SigC{pq\si}(\omega)
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& = \sum_i \sum_m \frac{\sERI{pi\si}{m} \sERI{qi\si}{m}}{\omega - \e{i\si} + \Om{m}{\spc,\RPA} - i \eta}
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\\
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& + \sum_a \sum_m \frac{\sERI{pa\si}{m} \sERI{qa\si}{m}}{\omega - \e{a\si} - \Om{m}{\spc,\RPA} + i \eta}
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\end{split}
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\end{equation}
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The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
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\begin{equation}
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\omega = \eHF{p\sigma} + \SigGW{p\sigma}(\omega)
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\omega = \eHF{p\sigma} + \SigC{p\sigma}(\omega)
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\end{equation}
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%================================
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\subsection{The Bethe-Salpeter formalism}
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%================================
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-Asc}
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\A{ia\si,jb\sip}{\spc,\BSE} & = \delta_{ij} \delta_{ab} \delta_{\si\sip} (\eGW{a\si} - \eGW{i\si}) + \ERI{ia\si}{jb\sip} - W_{ij}
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\\
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\label{eq:LR_BSE-Bsc}
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\B{ia\si,jb\sip}{\spc,\BSE} & = 0,
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\end{align}
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\end{subequations}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Computational details}
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\section{Computational details}
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\label{sec:compdet}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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