Updates from Overleaf

Manuscript/BSE_JPCL.tex

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 %%% INTRODUCTION %%%
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-In its press release announcing the attribution of the 2013 Nobel prize in Chemistry to Martin Karplus, Michael Levitt and Arieh Warshel, the Royal Swedish Academy of Sciences concluded that ``Today the computer is just as important a tool for chemists as the test tube. Simulations are so realistic that they predict the outcome of traditional experiments." Martin Karplus Nobel lecture moderated this bold statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are "much too complicated to be soluble", urging the scientist to develop "approximate practical methods." This is where the methodology community stands, attempting to develop robust approximations to study with increasing accuracy the properties of complex systems. The study of charged or neutral electronic excitations in condensed matter systems, from molecules to extended solids, has witnessed the development of a large number of such `àpproximate" methods
+In its press release announcing the attribution of the 2013 Nobel prize in Chemistry to  Karplus,  Levitt and  Warshel, the Royal Swedish Academy of Sciences concluded that ``Today the computer is just as important a tool for chemists as the test tube. Simulations are so realistic that they predict the outcome of traditional experiments." Martin Karplus Nobel lecture moderated this bold statement, introducing his presentation by a 1929 quote from Dirac emphasizing that laws of quantum mechanics are "much too complicated to be soluble", urging the scientist to develop "approximate practical methods." This is where the methodology community stands, attempting to develop robust approximations to study with increasing accuracy the properties of complex systems. 
+
+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 e.g. 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  belongs to the family of  Green's function many-body perturbation theories (MBPT). While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT take the one-body and two-body Green's function as central quantities. The (time-ordered) one-body Green's function reads:
+\begin{equation}
+    G(xt,x't') = -i \langle N | T {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't')} | 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 
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+A central property of the one-body Green's function is that its spectral representation presents poles at the charged excitation of the system :
+\begin{equation}
+    G(x,x'; \omega ) = \sum_n \frac{ f_s(x) f^*_s(x') }{ \omega - \varepsilon_s + i \eta \times \text{sgn}(\varepsilon_s - \mu})
+\end{equation}
+where $\varepsilon_s = E_s(N+1) - E_0(N)$ for $\varepsilon_s > \mu$ ($\mu$ chemical potential, $\eta$ small positive infinitesimal) and $\varepsilon_s = E_0(N) - E_s(N-1)$ for $\varepsilon_s < \mu$. The quantities $E_s(N+1)$ and $E_s(N-1)$ are the total energy of the s-th excited state of the (N+1) and (N-1)-electron systems, while $E_0(N)$ is the N-electron ground-state energy. Contrary to the Kohn-Sham eigenvalues, the Green's function poles $\varepsilon_s$ are the proper charging energies of the N-electron system, leading to well-defined ionization potentials and electronic affinities. Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission, not only that associated with frontier orbtials. The  $\labrace f_s \rbrace$ are called the Lehmann amplitudes.  
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