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91
Manuscript/MRGW.bib
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@ -1,13 +1,79 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-02-24 15:41:54 +0100
%% Created for Pierre-Francois Loos at 2022-05-03 10:33:26 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Loos_2022,
author = {Loos,Pierre-Fran{\c c}ois and Romaniello,Pina},
date-added = {2022-04-27 14:39:38 +0200},
date-modified = {2022-04-27 14:39:53 +0200},
doi = {10.1063/5.0088364},
journal = {J. Chem. Phys.},
number = {16},
pages = {164101},
title = {Static and dynamic Bethe--Salpeter equations in the T-matrix approximation},
volume = {156},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1063/5.0088364}}
@article{Pokhilko_2021a,
author = {Pokhilko,Pavel and Zgid,Dominika},
date-added = {2022-04-24 15:40:03 +0200},
date-modified = {2022-04-24 15:40:15 +0200},
doi = {10.1063/5.0055191},
journal = {J. Chem. Phys.},
number = {2},
pages = {024101},
title = {Interpretation of multiple solutions in fully iterative GF2 and GW schemes using local analysis of two-particle density matrices},
volume = {155},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1063/5.0055191}}
@article{Pokhilko_2021b,
author = {Pokhilko,Pavel and Iskakov,Sergei and Yeh,Chia-Nan and Zgid,Dominika},
date-added = {2022-04-24 15:38:35 +0200},
date-modified = {2022-04-24 15:39:02 +0200},
doi = {10.1063/5.0054661},
journal = {J. Chem. Phys.},
number = {2},
pages = {024119},
title = {Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green's function methods: Theory and application to GW and GF2},
volume = {155},
year = {2021},
bdsk-url-1 = {https://doi.org/10.1063/5.0054661}}
@article{Pokhilko_2022,
author = {Pokhilko,Pavel and Yeh,Chia-Nan and Zgid,Dominika},
date-added = {2022-04-24 15:35:30 +0200},
date-modified = {2022-04-24 15:35:48 +0200},
doi = {10.1063/5.0082586},
journal = {J. Chem. Phys.},
number = {9},
pages = {094101},
title = {Iterative subspace algorithms for finite-temperature solution of Dyson equation},
volume = {156},
year = {2022},
bdsk-url-1 = {https://doi.org/10.1063/5.0082586}}
@article{Hedin_1999,
abstract = {The GW approximation (GWA) extends the well-known Hartree-Fock approximation (HFA) for the self-energy (exchange potential), by replacing the bare Coulomb potential v by the dynamically screened potential W, e.g. Vex = iGv is replaced by GW = iGW. Here G is the one-electron Green's function. The GWA like the HFA is self-consistent, which allows for solutions beyond perturbation theory, like say spin-density waves. In a first approximation, iGW is a sum of a statically screened exchange potential plus a Coulomb hole (equal to the electrostatic energy associated with the charge pushed away around a given electron). The Coulomb hole part is larger in magnitude, but the two parts give comparable contributions to the dispersion of the quasi-particle energy. The GWA can be said to describe an electronic polaron (an electron surrounded by an electronic polarization cloud), which has great similarities to the ordinary polaron (an electron surrounded by a cloud of phonons). The dynamical screening adds new crucial features beyond the HFA. With the GWA not only bandstructures but also spectral functions can be calculated, as well as charge densities, momentum distributions, and total energies. We will discuss the ideas behind the GWA, and generalizations which are necessary to improve on the rather poor GWA satellite structures in the spectral functions. We will further extend the GWA approach to fully describe spectroscopies like photoemission, x-ray absorption, and electron scattering. Finally we will comment on the relation between the GWA and theories for strongly correlated electronic systems. In collecting the material for this review, a number of new results and perspectives became apparent, which have not been published elsewhere.},
author = {Lars Hedin},
date-added = {2022-04-21 13:21:07 +0200},
date-modified = {2022-04-21 13:21:24 +0200},
doi = {10.1088/0953-8984/11/42/201},
journal = {J. Phys. Condens. Matter},
number = {42},
pages = {R489--R528},
title = {On correlation effects in electron spectroscopies and the$\less$i$\greater${GW}$\less$/i$\greater$approximation},
volume = {11},
year = 1999,
bdsk-url-1 = {https://doi.org/10.1088/0953-8984/11/42/201}}
@article{Marie_2021,
abstract = {We explore the non-Hermitian extension of quantum chemistry in the complex plane and its link with perturbation theory. We observe that the physics of a quantum system is intimately connected to the position of complex-valued energy singularities, known as exceptional points. After presenting the fundamental concepts of non-Hermitian quantum chemistry in the complex plane, including the mean-field Hartree--Fock approximation and Rayleigh--Schr{\"o}dinger perturbation theory, we provide a historical overview of the various research activities that have been performed on the physics of singularities. In particular, we highlight seminal work on the convergence behaviour of perturbative series obtained within M{\o}ller--Plesset perturbation theory, and its links with quantum phase transitions. We also discuss several resummation techniques (such as Pad{\'e} and quadratic approximants) that can improve the overall accuracy of the M{\o}ller--Plesset perturbative series in both convergent and divergent cases. Each of these points is illustrated using the Hubbard dimer at half filling, which proves to be a versatile model for understanding the subtlety of analytically-continued perturbation theory in the complex plane.},
author = {Antoine Marie and Hugh G A Burton and Pierre-Fran{\c{c}}ois Loos},
@ -48,16 +114,6 @@
year = {2013},
bdsk-url-1 = {https://doi.org/10.1063/1.4851816}}
@misc{Loos_2022,
archiveprefix = {arXiv},
author = {Pierre-Fran{\c c}ois Loos and Pina Romaniello},
date-added = {2022-02-22 14:34:24 +0100},
date-modified = {2022-02-22 14:34:28 +0100},
eprint = {2202.07936},
primaryclass = {physics.chem-ph},
title = {Static and Dynamic Bethe-Salpeter Equations in the $T$-Matrix Approximation},
year = {2022}}
@article{Backhouse_2020b,
author = {Backhouse, Oliver J. and Booth, George H.},
date-added = {2022-02-22 13:31:29 +0100},
@ -232,17 +288,6 @@
year = {2020},
bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.9b03423}}
@article{Hedin_1999,
author = {Lars Hedin},
date-added = {2022-02-19 13:51:59 +0100},
date-modified = {2022-02-19 13:51:59 +0100},
journal = {J Phys.: Cond. Mat.},
number = {42},
pages = {R489-R528},
title = {On correlation effects in electron spectroscopies and the GW approximation},
volume = {11},
year = {1999}}
@article{Bruneval_2006,
author = {Bruneval, Fabien and Vast, Nathalie and Reining, Lucia},
date-added = {2022-02-19 13:51:49 +0100},

35
Manuscript/MRGW.tex
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@ -185,7 +185,7 @@ with the following sum rules:
&
\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{}
\end{align}
Here, we $p,q,r$ indicate arbitrary (\ie, occupied or unoccupied) orbitals, $i,j,k,l$ are occupied orbitals, while $a,b,c,d$ are unoccupied (virtual) orbitals.
As shown recently, the quasiparticle equation \eqref{eq:qp_eq} can be recast as a linear eigensystem with exactly the same solution:
\begin{equation}
\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
@ -242,7 +242,8 @@ with
\end{equation}
In the presence of intruder states, it might be interesting to move additional electronic configurations in the reference space.
Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$)
Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $qia$ the additional 2h1p ($qia = ija$) or 2p1h ($qia = iab$) that one wants to consider explicitly in the model space.
Equation \label{eq:Hp} can then be written exactly as
\begin{equation}
\label{eq:Hp_qia}
\bH^{(p,qia)} =
@ -257,18 +258,22 @@ Let us label as $p$ the reference 1h ($p = i$) or 1p ($p = a$) determinant and $
\end{pmatrix}
\end{equation}
with new blocks defined as
\begin{subequations}
\begin{gather}
V_{p,qia} = V_{qia,p} = \sqrt{2} \ERI{pq}{ia}
\\
\eps{qia}{} \equiv C_{qia,qia} = \text{sgn}(\eps{q}{} - \mu) \qty[ \qty(\eps{q}{} + \eps{a}{} - \eps{i}{} ) + 2 \ERI{ia}{ia} ]
\eps{qia}{} = \text{sgn}(\eps{q}{} - \mu) \qty[ \qty(\eps{q}{} + \eps{a}{} - \eps{i}{} ) + 2 \ERI{ia}{ia} ]
\\
C_{qia,KLC}^\text{2h1p} = - 2 \ERI{ia}{CL} \delta_{qK}
\\
C_{qia,KCD}^\text{2p1h} = + 2 \ERI{ia}{KC} \delta_{qD}
\\
V_{p,qia} = V_{qia,p} = \sqrt{2} \ERI{pq}{ia}
\end{gather}
The expressions of $\bC{p}{\text{2h1p}}$, $\bC{p}{\text{2p1h}}$, $\bV{}{\text{2h1p}}$, and $\bV{}{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has remove the contribution from the 2h1p or 2p1h configuration $qia$.
While $\eps{p}{\HF}$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps{qia}{} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration.
Therefore, when $\eps{p}{\HF}$ and $\eps{qia}{}$ becomes of similar mangitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems.
\end{subequations}
where $\text{sgn}$ is the sign function and $\mu$ is the chemical potential.
The expressions of $\bC{p}{\text{2h1p}}$, $\bC{p}{\text{2p1h}}$, $\bV{}{\text{2h1p}}$, and $\bV{}{\text{2p1h}}$ remain identical to the ones given in Eqs.~\eqref{eq:C2h1p}, \eqref{eq:C2p1h}, \eqref{eq:V2h1p}, and \eqref{eq:V2p1h} but one has to remove the contribution from the 2h1p or 2p1h configuration $qia$.
While $\eps{p}{}$ represents the relative energy (with respect to the $N$-electron HF reference determinant) of the 1h or 1p configuration, $\eps{qia}{} \equiv C_{qia,qia}$ is the relative energy of the 2h1p or 2p1h configuration with respect to the $N$-electron HF reference determinant.
Therefore, when $\eps{p}{}$ and $\eps{qia}{}$ becomes of similar mangitude, one might want to move the 2h1p or 2p1h configuration from the external to the internal space in order to avoid intruder state problems.
Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-energy matrix
\begin{equation}
@ -281,7 +286,7 @@ Downfolding Eq.~\eqref{eq:Hp_qia} yields the following frequency-dependent self-
\\
\end{pmatrix}
\end{equation}
with
with the dynamical self-energies
\begin{subequations}
\begin{gather}
\begin{split}
@ -315,6 +320,18 @@ with
\end{subequations}
Of course, the present procedure can be generalized to any number of states.
Solving
\begin{equation}
\bH^{(p,qia)} \cdot \bc{}{(p,qia,s)} = \eps{p,s}{\GW} \bc{}{(p,qia,s)}
\end{equation}
Because both the 1h or 1p configuration $p$ and the 2h1p or 2p1h configuration $qia$ are in the internal space, we have a new definition of the spectral weight:
\begin{equation}
\label{eq:Z_proj}
Z_{p,qia,s} = \qty[ c_{1}^{(p,qia,s)} ]^{2} + \qty[ c_{2}^{(p,qia,s)} ]^{2}
\end{equation}
Without doubt, the present procedure has similarities with the dressed time-dependent density-functional theory method developed by Maitra and coworkers, \cite{Cave_2004,Maitra_2004} where one doubly-excited configuration is included in the space of single excitations, hence resulting in a dynamical kernel.
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\section{Conclusion}
%%%%%%%%%%%%%%%%%%%%%%