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BSEdyn.tex
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BSEdyn.tex
@ -180,8 +180,6 @@
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\newcommand{\pis}{\pi^*}
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\newcommand{\ra}{\rightarrow}
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\newcommand\Oms{{\Omega}_s}
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\newcommand\hOms{\frac{{\Omega}_s}{2}}
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\newcommand{\hpsi}{\Hat{\psi}}
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\newcommand{\ha}{\Hat{a}}
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\newcommand{\tchi}{\Tilde{\chi}}
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@ -279,7 +277,7 @@ Additional details about this derivation are provided as {\SI}.
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We present, in a second step, the perturbative implementation of the dynamical correction as compared to the standard static approximation.
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%================================
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\subsection{General dynamical BSE theory}
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\subsection{General dynamical BSE}
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%=================================
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The two-particle correlation function $L(1,2; 1',2')$ --- a central quantity in the BSE formalism --- relates the variation of the one-body Green's function $G(1,1')$ with respect to an external non-local perturbation $U(2',2)$, \ie,
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@ -332,9 +330,9 @@ In the optical limit of instantaneous electron-hole creation and destruction, im
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\begin{equation}
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\begin{split}
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iL(1,2; 1',2')
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& = \theta(+\tau_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Oms \tau_{12} }
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& = \theta(+\tau_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Om{s}{} \tau_{12} }
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\\
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& - \theta(-\tau_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms \tau_{12} },
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& - \theta(-\tau_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Om{s}{} \tau_{12} },
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\end{split}
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\end{equation}
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where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
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@ -345,15 +343,15 @@ where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
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\tchi_s(\bx_1,\bx_{2}) & = \mel{N,s}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})] }{N}.
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\end{align}
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\end{subequations}
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The $\Oms$'s are the neutral excitation energies of interest. We have used the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, e.g.
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The $\Om{s}{}$'s are the neutral excitation energies of interest. We have used the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, e.g.
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$$
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\hpsi(1) = e^{ i {\hat H} t_1 } \hpsi(\bx_1) e^{-i {\hat H} t_1 }
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$$
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with $\hat H$ the exact many-body Hamiltonian.
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Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE [see Eq.~\eqref{eq:BSE}], we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads
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Picking up the $e^{+i \Om{s}{} t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE [see Eq.~\eqref{eq:BSE}], we are left with the search of the $e^{-i \Om{s}{} t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads
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\begin{multline} \label{eq:BSE_2}
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\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Oms t_1 }
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\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Om{s}{} t_1 }
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\theta ( \tau_{12} )
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\\
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= \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6)
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@ -361,7 +359,7 @@ Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$
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\times \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
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\theta [\min(t_5,t_6) - t_2].
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\end{multline}
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For the lowest neutral excitation energies falling in the fundamental gap of the system (\ie, $\Oms < \EgFun$ due to excitonic effects), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
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For the lowest neutral excitation energies falling in the fundamental gap of the system (\ie, $\Om{s}{} < \EgFun$ due to excitonic effects), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Om{s}{} t_1 }$ response since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
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Consequently, special care has to be taken for high-lying excited states (like core or Rydberg excitations) where additional terms have to be taken into account (see Refs.~\onlinecite{Strinati_1982,Strinati_1984}).
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Dropping the (space/spin) variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads
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@ -384,13 +382,13 @@ The $\e{p}$'s in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies (\ie, proper
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%$\hH$ being the exact many-body Hamiltonian.
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In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
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%\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)}
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Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Oms )$ onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$ yields
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Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Om{s}{} )$ onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$ yields
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\begin{multline} \label{eq:iL0bis}
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\iint d\bx_1 d\bx_{1'} \, \MO{a}^*(\bx_1) \MO{i}(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Oms)
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\iint d\bx_1 d\bx_{1'} \, \MO{a}^*(\bx_1) \MO{i}(\bx_{1'}) L_0(\bx_1,4;\bx_{1'},3; \Om{s}{})
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\\
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=
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\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta }
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\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \hOms) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \hOms ) \tau_{34} } ].
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\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Om{s}{} t^{34} }} { \Om{s}{} - ( \e{a} - \e{i} ) + i \eta }
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\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \frac{\Om{s}{}}{2}) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \frac{\Om{s}{}}{2}) \tau_{34} } ].
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\end{multline}
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% and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively.
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As a final step, we express the terms $\mel{N}{T [\hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')] }{N,s}$ and $\mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}$ from Eq.~\eqref{eq:BSE_2} in the standard electron-hole product (or single-excitation) space.
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@ -403,11 +401,15 @@ For example, we have
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= - \qty( e^{ -i \Omega_s t^{65} } ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5)
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\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
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\\
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\times \qty[ \theta( \tau_{65} ) e^{- i ( \e{p} - \hOms ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i ( \e{q} + \hOms) \tau_{65} } ]
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\times \qty[ \theta( \tau_{65} ) e^{- i ( \e{p} - \frac{\Om{s}{}}{2} ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i ( \e{q} + \frac{\Om{s}{}}{2}) \tau_{65} } ]
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\end{multline}
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with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
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%with a similar expression for $\mel{N}{T [\hpsi(\bx_3) \hpsi^{\dagger}(\bx_4)] }{N,s}$.
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%================================
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\subsection{Dynamical BSE within the $GW$ approximation}
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%=================================
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Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie,
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\begin{equation}
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\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
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@ -427,9 +429,9 @@ The $GW$ quasiparticle energies $\eGW{p}$ are good approximations to the removal
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Substituting Eqs.~\eqref{eq:iL0bis},\eqref{eq:spectral65},\eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE):
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\begin{equation} \label{eq:BSE-final}
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\begin{split}
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( \eGW{a} - \eGW{i} - \Oms ) X_{ia}^{s}
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& + \sum_{jb} \qty[ \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Oms) ] X_{jb}^{s} \\
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& + \sum_{jb} \qty[ \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Oms) ] Y_{jb}^{s}
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( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia}^{s}
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& + \sum_{jb} \qty[ \ERI{ia}{jb} - \widetilde{W}_{ij,ab}(\Om{s}{}) ] X_{jb}^{s} \\
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& + \sum_{jb} \qty[ \ERI{ia}{bj} - \widetilde{W}_{ib,aj}(\Om{s}{}) ] Y_{jb}^{s}
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= 0,
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\end{split}
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\end{equation}
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@ -441,12 +443,12 @@ In Eq.~\eqref{eq:BSE-final},
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\end{equation}
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are the bare two-electron integrals in the spatial orbital basis $\lbrace \MO{p}(\br{}) \rbrace$, and
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\begin{multline}
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\widetilde{W}_{ij,ab}(\Oms)
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\widetilde{W}_{ij,ab}(\Om{s}{})
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= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega)
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\\
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\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ],
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\end{multline}
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is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{s} = \Oms - ( \eGW{q} - \eGW{p} )$ and
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is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b} where $\Om{pq}{s} = \Om{s}{} - ( \eGW{q} - \eGW{p} )$ and
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\begin{equation}
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W_{pq,rs}({\omega})
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= \iint d\br d\br' \, \MO{p}(\br) \MO{q}^*(\br) W(\br ,\br'; \omega) \MO{r}^*(\br') \MO{s}(\br').
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@ -454,13 +456,18 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
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\xavier{A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting now onto the $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1'})}{N,s}$ left-hand side and right-hand-side of the BSE, leading to : }
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%================================
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\subsection{Dynamical screening}
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%=================================
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In the present study, we consider the exact spectral representation of $W(\omega)$ at the random-phase approximation (RPA) level, which reads
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\begin{multline}
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\label{eq:W}
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W_{ij,ab}(\omega)
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= \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m}
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\\
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\times \qty[ \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } ]
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\times \qty[ \frac{1}{ \omega-\Om{m}{\RPA} + i\eta } - \frac{1}{ \omega + \Om{m}{\RPA} - i\eta } ],
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\end{multline}
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where
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\begin{equation}
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@ -498,20 +505,19 @@ with
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\end{align}
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\end{subequations}
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where the $\e{p}$'s are taken as the HF orbital energies in the case of $G_0W_0$ or as the $GW$ quasiparticle energies in the case of self-consistent schemes such as ev$GW$.
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The RPA matrices $\bA{\RPA}$ and $\bB{\RPA}$ in Eq.~\eqref{eq:LR-RPA} are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}$, and $\bY{m}{}$ are (eigen)vectors of length $\Nocc \Nvir$.
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Because $\Om{m}{\RPA} > 0$, we have
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\begin{multline}
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\widetilde{W}_{ij,ab}( \Oms )
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= \ERI{ij}{ab} + 2 \sum_m^{OV} \sERI{ij}{m} \sERI{ab}{m}
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\widetilde{W}_{ij,ab}( \Om{s}{} )
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= \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m}
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\\
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\times \qty( \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} )
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\times \qty[ \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ].
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\end{multline}
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\titou{Due to excitonic effects, the lowest BSE excitation energy, ${\Omega}_1$, stands lower than the lowest RPA excitation energy, $\Omega_m^{RPA}$, so that
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e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and the $\widetilde{W}_{ij,ab}( \Oms )$ present no resonances.
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Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that e.g.
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$$
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\abs{ \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA}} } < \frac{1}{ \Omega_m^{RPA}}
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$$
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This leads to reduced electron-hole screening, namely larger electron-hole stabilising binding energy, as compared to the standard adiabatic BSE, leading to smaller (red-shifted) excitation energies. }
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Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{s} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{s}{})$ has no resonances.
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Furthermore, $\Om{ib}{s}$ and $\Om{ja}{s}$ are necessarily negative quantities for in-gap low-lying BSE excitations.
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Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$.
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As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole stabilizing binding energy, as compared to the standard static BSE, and yields smaller (red-shifted) excitation energies.
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%================================
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\subsection{Perturbative dynamical correction}
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@ -521,36 +527,35 @@ For a closed-shell system in a finite basis, to compute the BSE excitation energ
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\begin{equation}
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\label{eq:LR-dyn}
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\begin{pmatrix}
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\bA{}(\omega) & \bB{}(\omega) \\
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-\bB{}(\titou{-}\omega) & -\bA{}(\titou{-}\omega) \\
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\bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\
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-\bB{}(\titou{-}\Om{s}{}) & -\bA{}(\titou{-}\Om{s}{}) \\
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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\bX{m}{}(\omega) \\
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\bY{m}{}(\omega) \\
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\bX{s}{} \\
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\bY{s}{} \\
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\end{pmatrix}
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=
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\omega
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\Om{s}{}
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\begin{pmatrix}
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\bX{m}{}(\omega) \\
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\bY{m}{}(\omega) \\
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\bX{s}{} \\
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\bY{s}{} \\
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\end{pmatrix},
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\end{equation}
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where the dynamical matrices $\bA{}(\omega)$ and $\bB{}(\omega)$ are of size $\Nocc \Nvir \times \Nocc \Nvir$, where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$ is the total number of spatial orbitals), respectively, and $\bX{m}{}(\omega)$, and $\bY{m}{}(\omega)$ are (eigen)vectors of length $\Nocc \Nvir$.
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In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
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Note that, due to its non-linear nature, Eq.~\eqref{eq:LR-dyn} may provide more than one solution for each value of $m$. \cite{Romaniello_2009b,Sangalli_2011,Martin_2016}
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where the dynamical matrices $\bA{}$ and $\bB{}$, as well as $\bX{s}{}$, and $\bY{s}{}$, have the same size as their RPA counterparts.
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Note that, due to its non-linear nature, Eq.~\eqref{eq:LR-dyn} may provide more than one solution for each value of $s$. \cite{Romaniello_2009b,Sangalli_2011,Martin_2016}
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The BSE matrix elements read
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\begin{subequations}
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\begin{align}
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\label{eq:BSE-Adyn}
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\A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \sigma \ERI{ia}{jb} - \tW{ij,ab}{}(\omega),
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\A{ia,jb}{}(\Om{s}{}) & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + 2 \sigma \ERI{ia}{jb} - \tW{ij,ab}{}(\Om{s}{}),
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\\
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\label{eq:BSE-Bdyn}
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\B{ia,jb}{}(\omega) & = 2 \sigma \ERI{ia}{bj} - \tW{ib,aj}{}(\omega),
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\B{ia,jb}{}(\Om{s}{}) & = 2 \sigma \ERI{ia}{bj} - \tW{ib,aj}{}(\Om{s}{}),
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\end{align}
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\end{subequations}
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where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies, and $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively).
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where $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively).
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%\begin{equation}
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% \ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
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%\end{equation}
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@ -566,8 +571,8 @@ 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{}(\omega) & \bB{}(\omega) \\
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-\bB{}(\titou{-}\omega) & -\bA{}(\titou{-}\omega) \\
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\bA{}(\Om{s}{}) & \bB{}(\Om{s}{}) \\
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-\bB{}(\titou{-}\Om{s}{}) & -\bA{}(\titou{-}\Om{s}{}) \\
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\end{pmatrix}
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\\
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=
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@ -577,8 +582,8 @@ 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)}(\omega) & \bB{(1)}(\omega) \\
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-\bB{(1)}(\titou{-}\omega) & -\bA{(1)}(\titou{-}\omega) \\
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\bA{(1)}(\Om{s}{}) & \bB{(1)}(\Om{s}{}) \\
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-\bB{(1)}(\titou{-}\Om{s}{}) & -\bA{(1)}(\titou{-}\Om{s}{}) \\
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\end{pmatrix}
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\end{multline}
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with
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@ -595,35 +600,35 @@ and
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\begin{subequations}
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\begin{align}
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\label{eq:BSE-A1}
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\A{ia,jb}{(1)}(\omega) & = - \tW{ij,ab}{}(\omega) + \W{ij,ab}{\text{stat}},
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\A{ia,jb}{(1)}(\Om{s}{}) & = - \tW{ij,ab}{}(\Om{s}{}) + \W{ij,ab}{\text{stat}},
|
||||
\\
|
||||
\label{eq:BSE-B1}
|
||||
\B{ia,jb}{(1)}(\omega) & = - \tW{ib,aj}{}(\omega) + \W{ib,aj}{\text{stat}},
|
||||
\B{ia,jb}{(1)}(\Om{s}{}) & = - \tW{ib,aj}{}(\Om{s}{}) + \W{ib,aj}{\text{stat}},
|
||||
\end{align}
|
||||
\end{subequations}
|
||||
where we have defined the static version of the screened Coulomb potential
|
||||
\begin{equation}
|
||||
\label{eq:Wstat}
|
||||
\W{ij,ab}{\text{stat}} = W_{ij,ab}(\omega = 0) = \ERI{ij}{ab} - 4 \sum_m^{\Nocc \Nvir} \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}.
|
||||
\W{ij,ab}{\text{stat}} = W_{ij,ab}(\omega = 0) = \ERI{ij}{ab} - 4 \sum_m \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}.
|
||||
\end{equation}
|
||||
According to perturbation theory, the $m$th BSE excitation energy and its corresponding eigenvector can then decomposed as
|
||||
According to perturbation theory, the $s$th BSE excitation energy and its corresponding eigenvector can then decomposed as
|
||||
\begin{subequations}
|
||||
\begin{gather}
|
||||
\Om{m}{} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots,
|
||||
\Om{s}{} = \Om{s}{(0)} + \Om{s}{(1)} + \ldots,
|
||||
\\
|
||||
\begin{pmatrix}
|
||||
\bX{m}{} \\
|
||||
\bY{m}{} \\
|
||||
\bX{s}{} \\
|
||||
\bY{s}{} \\
|
||||
\end{pmatrix}
|
||||
=
|
||||
\begin{pmatrix}
|
||||
\bX{m}{(0)} \\
|
||||
\bY{m}{(0)} \\
|
||||
\bX{s}{(0)} \\
|
||||
\bY{s}{(0)} \\
|
||||
\end{pmatrix}
|
||||
+
|
||||
\begin{pmatrix}
|
||||
\bX{m}{(1)} \\
|
||||
\bY{m}{(1)} \\
|
||||
\bX{s}{(1)} \\
|
||||
\bY{s}{(1)} \\
|
||||
\end{pmatrix}
|
||||
+ \ldots.
|
||||
\end{gather}
|
||||
@ -637,50 +642,50 @@ Solving the zeroth-order static problem yields
|
||||
\end{pmatrix}
|
||||
\cdot
|
||||
\begin{pmatrix}
|
||||
\bX{m}{(0)} \\
|
||||
\bY{m}{(0)} \\
|
||||
\bX{s}{(0)} \\
|
||||
\bY{s}{(0)} \\
|
||||
\end{pmatrix}
|
||||
=
|
||||
\Om{m}{(0)}
|
||||
\Om{s}{(0)}
|
||||
\begin{pmatrix}
|
||||
\bX{m}{(0)} \\
|
||||
\bY{m}{(0)} \\
|
||||
\bX{s}{(0)} \\
|
||||
\bY{s}{(0)} \\
|
||||
\end{pmatrix},
|
||||
\end{equation}
|
||||
and, thanks to first-order perturbation theory, the first-order correction to the $m$th excitation energy is
|
||||
and, thanks to first-order perturbation theory, the first-order correction to the $s$th excitation energy is
|
||||
\begin{equation}
|
||||
\label{eq:Om1}
|
||||
\Om{m}{(1)} =
|
||||
\Om{s}{(1)} =
|
||||
\T{\begin{pmatrix}
|
||||
\bX{m}{(0)} \\
|
||||
\bY{m}{(0)} \\
|
||||
\bX{s}{(0)} \\
|
||||
\bY{s}{(0)} \\
|
||||
\end{pmatrix}}
|
||||
\cdot
|
||||
\begin{pmatrix}
|
||||
\bA{(1)}(\Om{m}{(0)}) & \bB{(1)}(\Om{m}{(0)}) \\
|
||||
-\bB{(1)}(\titou{-}\Om{m}{(0)}) & -\bA{(1)}(\titou{-}\Om{m}{(0)}) \\
|
||||
\bA{(1)}(\Om{s}{(0)}) & \bB{(1)}(\Om{s}{(0)}) \\
|
||||
-\bB{(1)}(\titou{-}\Om{s}{(0)}) & -\bA{(1)}(\titou{-}\Om{s}{(0)}) \\
|
||||
\end{pmatrix}
|
||||
\cdot
|
||||
\begin{pmatrix}
|
||||
\bX{m}{(0)} \\
|
||||
\bY{m}{(0)} \\
|
||||
\bX{s}{(0)} \\
|
||||
\bY{s}{(0)} \\
|
||||
\end{pmatrix}.
|
||||
\end{equation}
|
||||
From a practical point of view, if one enforces the TDA, we obtain the very simple expression
|
||||
\begin{equation}
|
||||
\label{eq:Om1-TDA}
|
||||
\Om{m}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}.
|
||||
\Om{s}{(1)} = \T{(\bX{s}{(0)})} \cdot \bA{(1)}(\Om{s}{(0)}) \cdot \bX{s}{(0)}.
|
||||
\end{equation}
|
||||
This correction can be renormalized by computing, at basically no extra cost, the renormalization factor which reads, in the TDA,
|
||||
\begin{equation}
|
||||
\label{eq:Z}
|
||||
Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}.
|
||||
Z_{s} = \qty[ 1 - \T{(\bX{s}{(0)})} \cdot \left. \pdv{\bA{(1)}(\Om{s}{})}{\Om{s}{}} \right|_{\Om{s}{} = \Om{s}{(0)}} \cdot \bX{s}{(0)} ]^{-1}.
|
||||
\end{equation}
|
||||
This finally yields
|
||||
\begin{equation}
|
||||
\Om{m}{\text{dyn}} = \Om{m}{\text{stat}} + \Delta\Om{m}{\text{dyn}} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}.
|
||||
\Om{s}{\text{dyn}} = \Om{s}{\text{stat}} + \Delta\Om{s}{\text{dyn}} = \Om{s}{(0)} + Z_{s} \Om{s}{(1)}.
|
||||
\end{equation}
|
||||
with $\Om{m}{\text{stat}} \equiv \Om{m}{(0)}$ and $\Delta\Om{m}{\text{dyn}} = Z_{m} \Om{m}{(1)}$.
|
||||
with $\Om{s}{\text{stat}} \equiv \Om{s}{(0)}$ and $\Delta\Om{s}{\text{dyn}} = Z_{s} \Om{s}{(1)}$.
|
||||
This is our final expression.
|
||||
|
||||
%%% FIG 1 %%%
|
||||
|
||||
Loading…
Reference in New Issue
Block a user