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Pierre-Francois Loos 2020-05-28 14:57:24 +02:00
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Manuscript/BSE_JPCL.tex
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@ -554,18 +554,18 @@ We now leave the description of successes to discuss difficulties and future dir
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As emphasized above, the BSE eigenvalue equation in the single-excitation space [see Eq.~\eqref{eq:BSE-eigen}] is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005} As emphasized above, the BSE eigenvalue equation in the single-excitation space [see Eq.~\eqref{eq:BSE-eigen}] is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
Searching iteratively for the lowest eigenstates exhibits the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT. Searching iteratively for the lowest eigenstates exhibits the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT.
Concerning the construction of the BSE Hamiltonian, it is no more expensive than building the TD-DFT one with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques. Concerning the construction of the BSE Hamiltonian, it is no more expensive than building its TD-DFT analogue with hybrid functionals, reducing again to $\order*{\Norb^4}$ operations with standard RI techniques.
At the price of sacrificing the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\order*{\Norb^3}$ operations using iterative techniques. \cite{Ljungberg_2015} At the price of sacrificing the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\order*{\Norb^3}$ operations using iterative techniques. \cite{Ljungberg_2015}
With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015} With the same restriction on the eigenvectors, a time-propagation approach, similar to that implemented for TD-DFT, \cite{Yabana_1996} combined with stochastic techniques to reduce the cost of building the BSE Hamiltonian matrix elements, allows quadratic scaling with systems size. \cite{Rabani_2015}
In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculations that scale as $\order{\Norb^4}$ with system size using plane-wave basis sets or RI techniques, but with a rather large prefactor. In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculation that scales as $\order{\Norb^4}$ with system size using plane-wave basis sets or RI techniques, but with a rather large prefactor.
%%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states. %%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states.
%%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019} %%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019}
The field of low-scaling $GW$ calculations is however witnessing significant advances. The field of low-scaling $GW$ calculations is however witnessing significant advances.
\xavier{While the sparsity of e.g. the overlap matrix in the atomic basis allows to reduce the scaling in the large size limit,} \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992} \xavier{While the sparsity of e.g. the overlap matrix in the atomic basis allows to reduce the scaling in the large size limit,} \cite{Foerster_2011,Wilhelm_2018} efficient real-space-grid and time techniques are blooming, \cite{Rojas_1995,Liu_2016} borrowing in particular the well-known Laplace transform approach used in quantum chemistry. \cite{Haser_1992}
Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017} Together with a stochastic sampling of virtual states, this family of techniques allow to set up linear scaling $GW$ calculations. \cite{Vlcek_2017}
The separability of occupied and virtual states summations lying at the heart of these approaches are now blooming in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in random-phase approximation (RPA) and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020} The separability of occupied and virtual states summations lying at the heart of these approaches are now spreading fast in quantum chemistry within the interpolative separable density fitting (ISDF) approach applied to calculating with cubic scaling the susceptibility needed in random-phase approximation (RPA) and $GW$ calculations. \cite{Lu_2017,Duchemin_2019,Gao_2020}
These ongoing developments pave the way to applying the $GW$@BSE formalism to systems comprising several hundred atoms on standard laboratory clusters. These ongoing developments pave the way to applying the $GW$@BSE formalism to systems containing several hundred atoms on standard laboratory clusters.
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@ -747,7 +747,8 @@ Here goes the conclusion.
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\section*{Acknowledgments} \section*{Acknowledgments}
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Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged. PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is also acknowledged.
This work has also been supported through the EUR Grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}. This work has also been supported through the EUR Grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''}.
DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support. DJ acknowledges the \textit{R\'egion des Pays de la Loire} for financial support.