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@ -14,7 +14,6 @@
\usepackage{amsmath}
\usepackage{newtxtext,newtxmath}
\usepackage{pifont}
\usepackage{graphicx}
\usepackage{dcolumn}
@ -237,27 +236,25 @@ The present \textit{Perspective} aims at describing the current status and upcom
%%%%%%%%%%%%%%%%%%%%%%
The BSE 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, for example, the algebraic-diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015}
While the density stands as the basic variable in density-functional theory DFT, \cite{Hohenberg_1964,Kohn_1965,ParrBook} Green's function MBPT takes the one-body Green's function as the central quantity.
The (time-ordered) one-body Green's function reads
While the density stands as the basic variable in density-functional theory DFT, \cite{Hohenberg_1964,Kohn_1965,ParrBook} the pillar of Green's function MBPT is the (time-ordered) one-body Green's function
\begin{equation}
G(\bx t,\bx't') = -i \mel{N}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{N},
\end{equation}
where $\ket{N}$ is the $N$-electron ground-state wave function.
The operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while \titou{$T$ 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 ($\bx t$) that was previously introduced in ($\bx't'$), while for ($t < t'$) the propagation of a hole is monitored.\\
For $t > t'$, $G$ provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in ($\bx t$) that was previously introduced in ($\bx't'$), while for $t < t'$ the propagation of a hole is monitored.\\
%===================================
\subsection{Charged excitations}
%===================================
A central property of the one-body Green's function is that its spectral representation presents poles at the charged excitation energies of the system
A central property of the one-body Green's function is that its frequency-dependent (\ie, dynamical) spectral representation has poles at the charged excitation energies of the system
\begin{equation}\label{eq:spectralG}
G(\bx,\bx'; \omega ) = \sum_s \frac{ f_s(\bx) f^*_s(\bx') }{ \omega - \varepsilon_s + i \eta \, \text{sgn}(\varepsilon_s - \mu ) },
\end{equation}
where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{N+1} - E_0^{N}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{N} - E_s^{N-1}$ for $\varepsilon_s < \mu$,
where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{N+1} - E_0^{N}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{N} - E_s^{N-1}$ for $\varepsilon_s < \mu$.
Here, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ corresponds to its ground-state energy.
The $f_s$'s are the so-called Lehmann amplitudes that reduce to one-body orbitals in the case of single-determinant many-body wave functions (see below).
Unlike Kohn-Sham (KS) eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are thus the proper \titou{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 orbitals.
Unlike Kohn-Sham (KS) eigenvalues, the Green's function poles $\lbrace \varepsilon_s \rbrace$ are proper \titou{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 orbitals.
Using the equation-of-motion formalism for the creation/destruction operators, it can be shown formally that $G$ verifies
\begin{equation}\label{eq:Gmotion}
@ -265,14 +262,15 @@ Using the equation-of-motion formalism for the creation/destruction operators, i
= \delta(1,2),
\end{equation}
where we introduce the usual composite index, \eg, $1 \equiv (\bx_1,t_1)$.
Here, $h$ is the \titou{one-body Hartree Hamiltonian} and $\Sigma$ the so-called exchange-correlation (xc) self-energy operator.
Here, $h$ is the \titou{one-body Hartree Hamiltonian} and $\Sigma$ is the so-called exchange-correlation (xc) self-energy operator.
Using the spectral representation of $G$ [see Eq.~\eqref{eq:spectralG}], one obtains the familiar eigenvalue equation, \ie,
\begin{equation}
h(\br) f_s(\br) + \int d\br' \, \Sigma(\br,\br'; \varepsilon_s ) f_s(\br) = \varepsilon_s f_s(\br),
\end{equation}
which resembles formally the KS equation with the difference that the self-energy $\Sigma$ is non-local, energy dependent and non-hermitian.
The knowledge of $\Sigma$ allows thus to obtain the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.
which resembles formally the KS equation with the difference that the self-energy $\Sigma$ is non-local, energy-dependent and non-hermitian.
The knowledge of $\Sigma$ allows to obtain the true addition/removal energies, namely the entire spectrum of occupied and virtual electronic energy levels, at the cost of solving a generalized one-body eigenvalue equation.
\titou{[INTRODUCE QUASIPARTICLES and OTHER solutions ??]}
\titou{The spin variable has disappear. How do we deal with this?}
\\
%===================================
@ -283,29 +281,52 @@ While the equations reported above are formally exact, it remains to provide an
This is where Green's function practical theories differ.
Developed by Lars Hedin in 1965 with application to the interacting homogeneous electron gas, \cite{Hedin_1965} the $GW$ approximation
\cite{Aryasetiawan_1998,Farid_1999,Onida_2002,Ping_2013,Leng_2016,Golze_2019rev} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}).
The obtained equation, when compared with the equation for the time-evolution of $G$ [Eq.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy
\begin{equation}
The obtained equation, when compared with the equation for the time-evolution of $G$ [see Eq.~\eqref{eq:Gmotion}], leads to a formal expression for the self-energy
\begin{equation}\label{eq:Sig}
\Sigma(1,2) = i \int d34 \, G(1,4) W(3,1^{+}) \Gamma(42,3),
\end{equation}
where $W$ is the dynamically-screened Coulomb potential and $\Gamma$ is a ``vertex" function that can be written as $\Gamma(12,3) = \delta(12)\delta(13) + \order{W}$, where $\order{W}$ means a corrective term with leading linear order in terms of $W$.
The neglect of the vertex leads to the so-called $GW$ approximation for $\Sigma$ that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with
\begin{align}
W(1,2) & = v(1,2) + \int d34 \, v(1,2) \chi_0(3,4) W(4,2),
The neglect of the vertex leads to the so-called $GW$ approximation of the self-energy
\begin{equation}\label{eq:SigGW}
\Sigma^{\GW}(1,2) = i \, G(1,2) W(2,1^{+}),
\end{equation}
that can be regarded as the lowest-order perturbation in terms of the screened Coulomb potential $W$ with
\begin{gather}
W(1,2) = v(1,2) + \int d34 \, v(1,2) \chi_0(3,4) W(4,2),
\\
\chi_0(1,2) & = -i \int d34 \, G(2,3) G(4,2),
\end{align}
\chi_0(1,2) = -i \int d34 \, G(2,3) G(4,2),
\end{gather}
where $\chi_0$ is the independent electron susceptibility and $v$ the bare Coulomb potential.
In practice, the input $G$ and $\chi_0$ needed to build $\Sigma$ are taken to be the ``best'' Green's function and susceptibility that can be easily calculated, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.
\titou{In practice, the input $G$ and $\chi_0$ needed to build $\Sigma^{\GW}$ are taken to be the ``best'' Green's function and susceptibility that can be easily calculated, namely the KS or Hartree-Fock (HF) ones where the $\lbrace \varepsilon_p, f_p \rbrace$ of Eq.~\eqref{eq:spectralG} are taken to be KS (or HF) eigenstates.}
Taking then $( \Sigma^{\GW}-V^{\XC} )$ as a correction to the KS xc potential $V^{\XC}$, a first-order correction to the input KS energies $\lbrace \varepsilon_p^{\KS} \rbrace$ is obtained as follows:
\begin{equation}
\varepsilon_p^{\GW} = \varepsilon_p^{\KS} +
\mel{\phi_p^{\KS}}{\Sigma^{\GW}(\varepsilon_p^{\GW}) - V^{\XC}}{\phi_p^{\KS}}.
\end{equation}
Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique.
\titou{T2: Shall we introduce the renormalization factor?}
Such an approach, where input KS energies are corrected to yield better electronic energy levels, is labeled as the single-shot, or perturbative, $G_0W_0$ technique.
This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, \cite{Strinati_1980,Hybertsen_1986,Godby_1988,Linden_1988}
surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with KS eigenvalues in conjunction with common local or gradient-corrected approximations to the xc potential.
In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983} namely the underestimation of the occupied to unoccupied bands energy gap at the LDA KS level, was dramatically reduced, bringing the agreement with experiment to within a few tenths of an eV [REFS] with an $\mathcal{O}(N^4)$ computational scaling (see below).
Further improvements may be obtained via partial self-consistency.
There exist two main types of partially self-consistent $GW$ methods:
i) \textit{``eigenvalue-only quasiparticle''} $GW$ (ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011}
where the quasiparticle (QP) energies are updated at each iteration, and
ii) \textit{``quasiparticle self-consistent''} $GW$ (qs$GW$), \cite{Faleev_2004,vanSchilfgaarde_2006,Kotani_2007,Ke_2011,Kaplan_2016}
where one updates both the QP energies and the corresponding orbitals.
Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
However, self-consistency does not always improve things, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006, Stan_2009, Rostgaard_2010, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Wilhelm_2018}
In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.
A famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998, Holm_1999,Holm_2000, Garcia-Gonzalez_2001}
This was further evidenced in real extended systems by several authors. \cite{Schone_1998, Ku_2002, Kutepov_2016, Kutepov_2017}
However, other approximations may have caused such deterioration, \eg, pseudo-potentials \cite{deGroot_1995} or finite-basis set effects. \cite{Friedrich_2006}
These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent $GW$ methods have shown great promise. \cite{Ke_2011,Blase_2011,Faber_2011,Caruso_2012,Caruso_2013,Caruso_2013a,Caruso_2013b,Koval_2014,Hung_2016,Blase_2018,Jacquemin_2017}
Another important feature compared to other perturbative techniques, the $GW$ formalism can tackle finite size and periodic systems, and does not present any divergence in the limit of zero gap (metallic) systems. \cite{Campillo_1999}
However, remaining a low order perturbative approach starting with single-determinant mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995}\\
@ -323,18 +344,18 @@ However, remaining a low order perturbative approach starting with single-determ
\subsection{Neutral excitations}
%===================================
While TD-DFT starts with the variation of the charge density $\rho$ with respect to an external local perturbation, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation:
While TD-DFT starts with the variation of the charge density $\rho$ with respect to an external local perturbation \titou{$U(1)$}, the BSE formalism considers a generalized 4-points susceptibility that monitors the variation of the Green's function with respect to a non-local external perturbation \titou{U($1,2$)}:
\begin{equation}
\chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
\quad \rightarrow \quad
L(12,34) \stackrel{\BSE}{=} -i \pdv{G(1,2)}{U(3,4)}.
\end{equation}
%with the relation $\chi(1,2) = L(11,22)$ since $\rho(1) = -iG(1,1^{+})$, as a first bridge between the TD-DFT and BSE worlds.
The equation of motion for $G$ [Eq.~\ref{eq:Gmotion}] can be reformulated in the form of a Dyson equation
The equation of motion for $G$ [see Eq.~\ref{eq:Gmotion}] can be reformulated in the form of a Dyson equation
\begin{equation}
G = G_0 + G_0 \Sigma G,
\end{equation}
that relates the full (interacting) Green's function, $G$, to its Hartree version, $G_0$, obtained by replacing the $\lbrace \varepsilon_s, f_s \rbrace$ by the Hartree eigenvalues and eigenfunctions.
that relates the full (interacting) Green's function, $G$, to its Hartree version, $G_0$, obtained by replacing the $\lbrace \varepsilon_p, f_p \rbrace$ by the Hartree eigenvalues and eigenfunctions.
The derivative with respect to $U$ of the Dyson equation yields
\begin{multline}\label{eq:DysonL}
L(12,34) = L_0(12,34)
@ -346,7 +367,6 @@ where $L_0 = \partial G_0 / \partial U$ is the Hartree 4-point susceptibility an
\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) + \pdv{\Sigma(5,6)}{G(7,8)}
\end{equation}
is the so-called BSE kernel.
This equation can be compared to its TD-DFT analog
\begin{equation}
\chi(1,2) = \chi_0(1,2) + \int d34 \, \chi_0(1,3) \Xi^{\DFT}(3,4) \chi(4,2),
@ -356,12 +376,12 @@ where
\Xi^{\DFT}(3,4) = v(3,4) + \pdv{V^{\XC}(3)}{\rho(4)}
\end{equation}
is the TD-DFT kernel.
Plugging now the $GW$ self-energy, in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel
Plugging now the $GW$ self-energy [see Eq.~\eqref{eq:SigGW}], in a scheme that we label the BSE@$GW$ approach, leads to an approximation to the BSE kernel
\begin{equation}\label{eq:BSEkernel}
\Xi^{\BSE}(5,6,7,8) = v(5,7) \delta(56) \delta(78) -W(5,6) \delta(57) \delta(68 ),
\end{equation}
where it is traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$.
Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces thus the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied orbitals and $(a,b)$ are unoccupied orbitals), leads to an eigenvalue problem similar to the so-called Casida's equations in TD-DFT:
where it is traditional to neglect the derivative $( \partial W / \partial G)$ that introduces again higher orders in $W$. \cite{Hanke_1980,Strinati_1982,Strinati_1984}
Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential, that replaces the static DFT xc kernel, and expressing Eq.~\eqref{eq:DysonL} in the standard product space $\lbrace \phi_i(\br) \phi_a(\br') \rbrace$ [where $(i,j)$ are occupied orbitals and $(a,b)$ are unoccupied orbitals), leads to an eigenvalue problem similar to the so-called Casida's equations in TD-DFT:
\begin{equation}
\begin{pmatrix}
R & C
@ -381,14 +401,15 @@ Taking the static limit, \ie, $W(\omega=0)$, for the screened Coulomb potential,
Y^m
\end{pmatrix},
\end{equation}
with electron-hole (e-h) eigenstates written as
with electron-hole ($eh$) eigenstates written as
\begin{equation}
\psi_{m}^{eh}(\br_e,\br_h)
= \sum_{ia} \qty[ X_{ia}^{m} \phi_i(\br_h) \phi_a(\br_e)
+ Y_{ia}^{m} \phi_i(\br_e) \phi_a(\br_h) ],
\end{equation}
where $\lambda$ index the electronic excitations.
where $m$ indexes the electronic excitations.
The $\lbrace \phi_{i/a} \rbrace$ are the input (KS) eigenstates used to build the $GW$ self-energy.
\titou{T2: this is only true in the case of $G_0W_0$.}
The resonant and anti-resonant parts of the BSE Hamiltonian read
\begin{gather}
R_{ai,bj} = \qty( \varepsilon_a^{\GW} - \varepsilon_i^{\GW} ) \delta_{ij} \delta_{ab} + \eta (ai|bj) - W_{ai,bj},
@ -408,40 +429,19 @@ As compared to TD-DFT,
\item the $GW$ quasiparticle energies $\lbrace \varepsilon_{i/a}^{\GW} \rbrace$ replace the KS eigenvalues
\item the non-local screened Coulomb matrix elements replaces the DFT xc kernel.
\end{itemize}
\titou{T2: would it be useful to say that there is 100\% exact exchange in BSE@$GW$?}
We emphasise that these equations can be solved at exactly the same cost as the standard TD-DFT equations once the quasiparticle energies and screened Coulomb potential $W$ are inherited from preceding $GW$ calculations. This defines the standard (static) BSE@$GW$ scheme that we discuss in this \textit{Perspective}, emphasizing its pros and cons. \\
%===================================
\subsection{Practical considerations}
%\subsection{Practical considerations}
%===================================
From a practical point of view, it is important to understand that, to compute the BSE neutral excitations, one must perform, first, several calculations.
First, a KS-DFT (or HF) calculation has to be performed in order to get orbitals and their corresponding energies.
Then, these are used as input variables for the $GW$ calculation, whose main purpose is to correct these quantities.
Depending on the level of self-consistency, only the eigenvalues or both the eigenvalues and the orbitals are updated.
In the case of a $G_0W_0$ calculation, a single, perturbative correction is applied to the orbital energies only.
The partially self-consistent ev$GW$ scheme update
The simplest and most popular variant is perturbative GW, or $G_0W_0$, \cite{Hybertsen_1985, Hybertsen_1986} which has been widely used in the literature to study solids, atoms and molecules. \cite{Bruneval_2012, Bruneval_2013, vanSetten_2015, vanSetten_2018}
Although $G_0W_0$ provides accurate results (at least for weakly/moderately correlated systems), it is strongly starting-point dependent due to its perturbative nature, and violates some important conservation laws, such as the conservation of energy, momentum and particle number. \cite{Martin_1959, Baym_1961, Baym_1962}
Improvements may be obtained via partial self-consistency while the conservation laws are satisfied at full self-consistency.
However, things are not that simple, as self-consistency and vertex corrections are known to cancel to some extent. \cite{ReiningBook}
Indeed, there is a long-standing debate about the importance of partial and full self-consistency in GW. \cite{Stan_2006, Stan_2009, Rostgaard_2010, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Wilhelm_2018}
In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.
A famous example has been provided by the calculations performed on the uniform electron gas, \cite{Holm_1998, Holm_1999,Holm_2000, Garcia-Gonzalez_2001}
a paradigm central to many areas of physics and chemistry. \cite{Loos_2016}
This was further evidenced in real extended systems by several authors. \cite{Schone_1998, Ku_2002, Kutepov_2016, Kutepov_2017}
However, other approximations may have caused such deterioration, \eg, pseudo-potentials \cite{deGroot_1995} or finite-basis set effects. \cite{Friedrich_2006}
These studies have cast doubt on the importance of self-consistent schemes within $GW$, at least for solid-state calculations.
For finite systems such as atoms and molecules, the situation is less controversial, and partially or fully self-consistent GW methods have shown great promise. \cite{Ke_2011, Blase_2011, Faber_2011, Caruso_2012, Caruso_2013, Caruso_2013a, Caruso_2013b, Koval_2014, Hung_2016, Blase_2018, Jacquemin_2017}
There exist two main types of partially self-consistent GW methods:
i) \textit{``eigenvalue-only quasiparticle''} GW (ev$GW$), \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011}
where the quasiparticle (QP) energies are updated at each iteration, and
ii) \textit{``quasiparticle self-consistent''} GW (qs$GW$), \cite{Faleev_2004, vanSchilfgaarde_2006, Kotani_2007, Ke_2011, Kaplan_2016}
where one updates both the QP energies and the corresponding orbitals.
Note that a starting point dependence remains in ev$GW$ as the orbitals are not self-consistently optimized in this case.
\\
%From a practical point of view, it is important to understand that, to compute the BSE neutral excitations, one must perform, beforehand, several calculations.
%First, a KS-DFT (or HF) calculation has to be performed in order to get orbitals and their corresponding energies.
%Then, these quantities are used as input variables for the $GW$ calculation, whose main purpose is to correct these quantities.
%Depending on the level of self-consistency, only the orbital energies or both the orbitals and their energies are corrected.
%The $GW$ calculation being performed, the BSE excitation energies can then be computed using as input the $GW$ quasiparticle energies and the screened Coulomb operator $W$.
%\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Historical overview}
@ -548,16 +548,55 @@ benchmarks \cite{Jacquemin_2017b,Rangel_2017}
a first cure was offered by hybridizing TD-DFT and BSE, namely adding to the BSE kernel the correlation part of the underlying DFT functional used to build the susceptibility and resulting screened Coulomb potential $W$. \cite{Holzer_2018b}\\
%==========================================
\subsection{The challenge of analytic gradients}
\subsection{The challenge of analytical gradients}
%==========================================
An additional issue concerns the formalism taken to calculate the ground-state energy for a given atomic configuration. Since the BSE formalism presented so far concerns the calculation of the electronic excitations, namely the difference of energy between the GS and the ES, gradients of the ES absolute energy require
%An additional issue concerns the formalism taken to calculate the ground-state energy for a given atomic configuration. Since the BSE formalism presented so far concerns the calculation of the electronic excitations, namely the difference of energy between the GS and the ES, gradients of the ES absolute energy require
%
%This points to another direction for the BSE formalism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}\\
This points to another direction for the BSE formlism, namely the calculation of GS total energy with the correlation energy calculated at the BSE level. Such a task was performed by several groups using in particular the adiabatic connection fluctuation-dissipation theorem (ACFDT), focusing in particular on small dimers. \cite{Olsen_2014,Holzer_2018b,Li_2020,Loos_2020}\\
The features of ground- and excited-state potential energy surfaces (PES) are critical for the faithful description and a deeper understanding of photochemical and photophysical processes. \cite{Bernardi_1996,Olivucci_2010,Robb_2007}
For example, chemoluminescence, fluorescence and other related processes are associated with geometric relaxation of excited states, and structural changes upon electronic excitation. \cite{Navizet_2011}
Reliable predictions of these mechanisms which have attracted much experimental and theoretical interest lately require exploring the ground- and excited-state PES.
From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Gonzales_2012, Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Andersson_1990,Andersson_1992,Roos_1996,Angeli_2001}) cannot be afforded.
For the last two decades, time-dependent density-functional theory (TD-DFT) \cite{Casida_1995} has been the go-to method to compute absorption and emission spectra in large molecular systems.
In TD-DFT, the PES for the excited states can be easily and efficiently obtained as a function of the molecular geometry by simply adding the ground-state DFT energy to the excitation energy of the selected state.
One of the strongest assets of TD-DFT is the availability of first- and second-order analytic nuclear gradients (\ie, the first derivatives of the excited-state energy with respect to the nuclear displacements), which enables the exploration of excited-state PES.\cite{Furche_2002}
A significant limitation of the BSE formalism, as compared to TD-DFT, lies in the lack of analytical nuclear gradients for both the ground and excited states, preventing efficient studies of excited-state processes.
While calculations of the $GW$ quasiparticle energy ionic gradients is becoming increasingly popular,
\cite{Lazzeri_2008,Faber_2011b,Yin_2013,Faber_2015,Montserrat_2016,Zhenglu_2019} only one pioneering study of the excited-state BSE gradients has been published so far. \cite{Beigi_2003}
In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the Kohn-Sham (KS) LDA forces as its ground-state contribution.
In contrast to TD-DFT which relies on KS-DFT \cite{Hohenberg_1964,Kohn_1965,ParrBook} as its ground-state analog, the ground-state BSE energy is not a well-defined quantity, and no clear consensus has been found regarding its formal definition.
Consequently, the BSE ground-state formalism remains in its infancy with very few available studies for atomic and molecular systems. \cite{Olsen_2014,Holzer_2018,Li_2019,Li_2020,Loos_2020}
A promising route, which closely follows random-phase approximation (RPA)-type formalisms, \cite{Furche_2008,Toulouse_2009,Toulouse_2010,Angyan_2011,Ren_2012} is to calculated the ground-state BSE energy within the adiabatic-connection fluctuation-dissipation theorem (ACFDT) framework. \cite{Olsen_2014,Maggio_2016,Holzer_2018,Loos_2020}
Thanks to comparisons with both similar and state-of-art computational approaches, we have recently showed that the ACFDT@BSE@$GW$ approach is surprisingly accurate, and can even compete with high-order coupled cluster (CC) methods in terms of absolute ground-state energies and equilibrium distances. \cite{Loos_2020}
However, we also observe that, in some cases, unphysical irregularities on the ground-state PES, which are due to the appearance of a satellite resonance with a weight similar to that of the $GW$ quasiparticle peak. \cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020,Loos_2020}
\\
%==========================================
\subsection{Unphysical discontinuities}
%==========================================
\titou{T2: talking about multiple solution issues.}
The GW approximation of many-body perturbation theory has been highly successful at predicting the electronic properties of solids and molecules. \cite{Onida_2002, Aryasetiawan_1998, Reining_2017}
However, it is also known to be inadequate to model strongly correlated systems. \cite{Romaniello_2009, Romaniello_2012, DiSabatino_2015, DiSabatino_2016, Tarantino_2017}
Here, we have found severe shortcomings of two widely-used variants of $GW$ in the weakly correlated regime.
We report unphysical irregularities and discontinuities in some key experimentally-measurable quantities computed within the $GW$ approximation
of many-body perturbation theory applied to molecular systems.
In particular, we show that the solution obtained with partially self-consistent GW schemes depends on the algorithm one uses to solve self-consistently the quasi-particle (QP) equation.
The main observation of the present study is that each branch of the self-energy is associated with a distinct QP solution, and that each switch between solutions implies a significant discontinuity in the quasiparticle energy as a function of the internuclear distance.
Moreover, we clearly observe ``ripple'' effects, i.e., a discontinuity in one of the QP energies induces (smaller) discontinuities in the other QP energies.
Going from one branch to another implies a transfer of weight between two solutions of the QP equation.
The case of occupied, virtual and frontier orbitals are separately discussed on distinct diatomics.
In particular, we show that multisolution behavior in frontier orbitals is more likely if the HOMO-LUMO gap is small.
We have evidenced that one can hit multiple solution issues within $G_0W_0$ and ev$GW$ due to the location of the QP solution near poles of the self-energy.
Within linearized $G_0W_0$, this implies irregularities in key experimentally-measurable quantities of simple diatomics, while, at the partially self-consistent ev$GW$ level, discontinues arise.
Because the RPA correlation energy \cite{Casida_1995, Dahlen_2006, Furche_2008, Bruneval_2016} and the Bethe-Salpeter excitation energies \cite{Strinati_1988, Leng_2016, Blase_2018} directly dependent on the QP energies, these types of discontinuities are also present in these quantities, hence in the energy surfaces of ground and excited states.
In a recent article, \cite{Loos_2018} while studying a model two-electron system, we have observed that, within partially self-consistent $GW$ (such as ev$GW$ and qs$GW$), one can observe, in the weakly correlated regime, (unphysical) discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, HOMO-LUMO gap, total and correlation energies, as well as vertical excitation energies).
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Year = {2007},
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Date-Added = {2020-05-15 23:23:07 +0200},
Date-Modified = {2020-05-15 23:23:07 +0200},
Doi = {10.1039/cs9962500321},
Issn = {0306-0012, 1460-4744},
Journal = {Chem. Soc. Rev.},
Language = {en},
Number = {5},
Pages = {321},
Title = {Potential Energy Surface Crossings in Organic Photochemistry},
Volume = {25},
Year = {1996},
Bdsk-Url-1 = {https://doi.org/10.1039/cs9962500321}}
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Date-Added = {2020-05-15 22:38:14 +0200},
Date-Modified = {2020-05-15 22:38:14 +0200},
Doi = {10.1103/PhysRevB.21.4656},
Journal = {Phys. Rev. B},
Pages = {4656},
Title = {Many-Particle Effects in the Optical Spectrum of a Semiconductor},
Volume = {21},
Year = {1980},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.21.4656}}
@article{Martin_1959,
Author = {Martin, Paul C. and Schwinger, Julian},
Date-Added = {2020-05-15 15:13:03 +0200},