diff --git a/Manuscript/BSE_JPCL.tex b/Manuscript/BSE_JPCL.tex index 021d849..1f7b294 100644 --- a/Manuscript/BSE_JPCL.tex +++ b/Manuscript/BSE_JPCL.tex @@ -207,11 +207,11 @@ In its press release announcing the attribution of the 2013 Nobel prize in Chemi The study of neutral electronic excitations in condensed matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the prediction of the colour of precious metals and stones for jewellery, to the understanding, \eg, of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. The present Perspective aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism that, while sharing many features with time-dependent density functional theory (TD-DFT), including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known difficulties. -The Bethe-Salpeter equation formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} to which belong as well the algebraic diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads: +The Bethe-Salpeter equation formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} belongs to the family of Green's function many-body perturbation theories (MBPT) \cite{Hedin_1965,Aryasetiawan_1998,Onida_2002,Reining_2017,ReiningBook} together with e.g. the algebraic diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads: \begin{equation} G(xt,x't') = -i \langle N | T \left[ {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't') \right] | N \rangle \end{equation} -where $| N \rangle $ is the N-electron ground-state wavefunction. The operators ${\hat \psi}(xt)$ and ${\hat \psi}^{\dagger}(x't')$ remove/add an electron in space-spin-time positions (xt) and (x't'), while $T$ is the time-ordering operator. For (t>t') the one-body Green's function provides the amplitude of probability of finding bla bla \\ +where $| N \rangle $ is the N-electron ground-state wavefunction. The operators ${\hat \psi}(xt)$ and ${\hat \psi}^{\dagger}(x't')$ remove/add an electron in space-spin-time positions (xt) and (x't'), while $T$ is the time-ordering operator. For (t>t') the one-body Green's function provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in (xt) that was previously introduced in (x't'), while for (t