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adding more references

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Pierre-Francois Loos 2020-08-28 15:45:25 +02:00
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%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-08-28 14:26:14 +0200
%% Created for Pierre-Francois Loos at 2020-08-28 15:45:04 +0200
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@ -173,7 +173,7 @@ For example, assuming that $\bA_2$ is a diagonal matrix is of common practice (s
Another of these approximations is the so-called \textit{static} approximation, where one sets the frequency to a particular value.
For example, as commonly done within the Bethe-Salpeter equation (BSE) formalism of many-body perturbation theory (MBPT), \cite{Strinati_1988} $\Tilde{\bA}_1(\omega) = \Tilde{\bA}_1 \equiv \Tilde{\bA}_1(\omega = 0)$.
In such a way, the operator $\Tilde{\bA}_1$ is made linear again by removing its frequency-dependent nature.
A similar example in the context of time-dependent density-functional theory (TDDFT) \cite{Runge_1984} is provided by the ubiquitous adiabatic approximation, \cite{Tozer_2000} which neglects all memory effects by making static the exchange-correlation (xc) kernel (\ie, frequency independent). \cite{Maitra_2016}
A similar example in the context of time-dependent density-functional theory (TDDFT) \cite{Runge_1984} is provided by the ubiquitous adiabatic approximation, \cite{Tozer_2000} which neglects all memory effects by making static the exchange-correlation (xc) kernel (\ie, frequency independent). \cite{Maitra_2012,Maitra_2016,Elliott_2011}
These approximations come with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $N$ to $N_1$.
Coming back to our example, in the static (or adiabatic) approximation, the operator $\Tilde{\bA}_1$ built in the single-excitation basis cannot provide double excitations anymore, and the $N_1$ excitation energies are associated with single excitations.
All additional solutions associated with higher excitations have been forever lost.
@ -384,7 +384,7 @@ Very recently, Loos and Blase have applied the dynamical correction to the BSE b
They compiled a comprehensive set of vertical transitions in prototypical molecules, providing benchmark data and showing that the dynamical corrections can be sizable and improve the static BSE excitations considerably. \cite{Loos_2020e}
Let us stress that, in all these studies, the TDA is applied to the dynamical correction (\ie, only the diagonal part of the BSE Hamiltonian is made frequency-dependent) and we shall do the same here.
Within the so-called $GW$ approximation of MBPT, \cite{Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook,Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals.
Within the so-called $GW$ approximation of MBPT, \cite{Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook,Golze_2019} one can easily compute the quasiparticle energies associated with the valence and conduction orbitals. \cite{Hybertsen_1985a,Hybertsen_1986,vanSetten_2013,Bruneval_2016}
Assuming that $W$ has been calculated at the random-phase approximation (RPA) level and within the TDA, the expression of the $\GW$ quasiparticle energy is simply \cite{Veril_2018}
\begin{equation}
\e{p}^{\GW} = \e{p} + Z_{p}^{\GW} \SigGW{p}(\e{p})
@ -642,7 +642,7 @@ and
C_{\dBSE2}^{\co,\upup} = \frac{\ERI{cc}{cc} \ERI{vc}{cv} + \ERI{vv}{vv} \ERI{vc}{cv} }{2 \Delta\eGF{}}
\end{gather}
\end{subequations}
As mentioned in Ref.~\onlinecite{Rebolini_2016}, the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
As mentioned in Ref.~\onlinecite{Rebolini_2016}, the BSE2 kernel has some similarities with the second-order polarization-propagator approximation \cite{Oddershede_1977,Nielsen_1980} (SOPPA) and second RPA kernels. \cite{Wambach_1988,Huix-Rotllant_2011,Huix-Rotllant_PhD,Sangalli_2011}
Unlike the dBSE Hamiltonian [see Eq.~\eqref{eq:HBSE}], the BSE2 dynamical kernel is spin aware with distinct expressions for singlets and triplets. \cite{Rebolini_PhD}
Like in dBSE, dBSE2 generates the right number of excitations for the singlet manifold (see Fig.~\ref{fig:BSE2}).