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Manuscript/BSE_JPCL.tex
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@ -93,6 +93,8 @@
\newcommand{\EgOpt}{\Eg^\text{opt}}
\newcommand{\EB}{E_B}
\newcommand{\Nel}{N}
\newcommand{\Norb}{N_\text{orb}}
% units
\newcommand{\IneV}[1]{#1 eV}
@ -238,9 +240,9 @@ 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 , originally developed by Schirmer and Trofimov, \cite{Schirmer_1982,Schirmer_1991,Schirmer_2004d,Schirmer_2018} in quantum chemistry. \cite{Dreuw_2015}
While the one-body 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},
G(\bx t,\bx't') = -i \mel{\Nel}{T \qty[ \Hat{\psi}(\bx t) \Hat{\psi}^{\dagger}(\bx't') ]}{\Nel},
\end{equation}
where $\ket{N}$ is the $N$-electron ground-state wave function.
where $\ket{\Nel}$ is the $\Nel$-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 $t > t'$, $G$ 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.\\
@ -251,10 +253,10 @@ A central property of the one-body Green's function is that its frequency-depend
\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 $\varepsilon_s < \mu$.
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.
where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\varepsilon_s = E_s^{\Nel+1} - E_0^{\Nel}$ for $\varepsilon_s > \mu$, and $\varepsilon_s = E_0^{\Nel} - E_s^{\Nel-1}$ for $\varepsilon_s < \mu$.
Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ 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 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 $\Nel$-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}
@ -322,7 +324,7 @@ Here, I would prefer to introduce the frequency-dependent quasiparticle 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.
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).
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 a computational cost scaling quartically with the number of basis functions (see below).
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.
Further improvements may be obtained via self-consistency of the Hedin's equations (see Fig.~\ref{fig:pentagon}).
@ -453,16 +455,16 @@ An important conclusion drawn from these calculations was that the quality of th
\end{equation}
with the experimental (photoemission) fundamental gap \cite{Bredas_2014}
\begin{equation}
\EgFun = I^N - A^N,
\EgFun = I^\Nel - A^\Nel,
\end{equation}
where $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N+1} - E_0^N$ are the ionization potential and the electron affinity of the $N$-electron system (see Fig.~\ref{fig:gaps}).
where $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system (see Fig.~\ref{fig:gaps}).
%%% FIG 2 %%%
\begin{figure*}
\includegraphics[width=0.7\linewidth]{gaps}
\caption{
Definition of the optical gap $\EgOpt$ and fundamental gap $\EgFun$.
$\EB$ is the excitonic effect, while $I^N$ and $A^N$ are the ionization potential and the electron affinity of the $N$-electron system.
$\EB$ is the excitonic effect, while $I^\Nel$ and $A^\Nel$ are the ionization potential and the electron affinity of the $\Nel$-electron system.
$\Eg^{\KS}$ and $\Eg^{\GW}$ are the KS and $GW$ HOMO-LUMO gaps.
See main text for the definition of the other quantities
\label{fig:gaps}}
@ -479,7 +481,7 @@ but still too small as compared to the experimental value, \ie,
\end{equation}
Such an underestimation of the fundamental gap leads to a similar underestimation of the optical gap $\EgOpt$, \ie, the lowest optical excitation energy.
\begin{equation}
\EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB,
\EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB,
\end{equation}
where $\EB$ is the excitonic effect, that is, the stabilization implied by the attraction of the excited electron and its hole left behind (see Fig.~\ref{fig:gaps}).
Because of this, we have $\EgOpt < \EgFun$.
@ -516,18 +518,20 @@ We now leave the description of successes to discuss difficulties and Perspectiv
\subsection{The computational challenge}
%==========================================
As emphasized above, the BSE eigenvalue equation in the occupied-to-virtual product space is formally equivalent to that of TD-DFT or TD-HF. \cite{Dreuw_2005}
Searching iteratively for the lowest eigenstates presents the same $\order*{N^4}$ 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 $\order*{N^4}$ operations with standard RI techniques.
At the price of sacrificing the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\order*{N^3}$ operations using iterative techniques. \cite{Ljungberg_2015}
Searching iteratively for the lowest eigenstates presents the same $\order*{\Norb^4}$ matrix-vector multiplication computational cost within BSE and TD-DFT (where $\Norb$ is the number of orbitals).
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.
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}
In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculations that scale as $\order{N^4}$ 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 $GW$ calculations that scale 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.
%%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.
While the sparsity of ..., \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}
While the sparsity of ..., \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}
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 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. \\
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}
These ongoing developments pave the way to applying the $GW$@BSE formalism to systems comprising several hundred atoms on standard laboratory clusters.
\\
%==========================================
\subsection{The triplet instability challenge}
@ -542,7 +546,7 @@ 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 analytical gradients}
\subsection{The challenge of analytical nuclear 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
%
@ -552,46 +556,54 @@ The features of ground- and excited-state potential energy surfaces (PES) are cr
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.
For the last two decades, TD-DFT 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}
In this seminal work devoted to small molecules (\ce{CO} and \ce{NH3}), only the BSE excitation energy gradients were calculated, while computing the KS-LDA forces as its ground-state contribution.
\\
%==========================================
\subsection{Unphysical discontinuities}
\subsection{The challenge of the ground-state energy}
%==========================================
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_2009a,Romaniello_2012,DiSabatino_2015,DiSabatino_2016,Tarantino_2017,DiSabatino_2019}
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.
In contrast to TD-DFT which relies on KS-DFT 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}
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.
A promising route, which closely follows 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{Furche_2005,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 yields extremely accurate PES around equilibrium, and can even compete with high-order coupled cluster methods in terms of absolute ground-state energies and equilibrium distances. \cite{Loos_2020}
Their accuracy near the dissociation limit remains an open question. \cite{Caruso_2013,Olsen_2014,Colonna_2014,Hellgren_2015,Holzer_2018}
Indeed, in the largest available benchmark study \cite{Holzer_2018} encompassing the total energies of the atoms \ce{H}--\ce{Ne}, the atomization energies of the 26 small molecules forming the HEAT test set, and the bond lengths and harmonic vibrational frequencies of $3d$ transition-metal monoxides, the BSE correlation energy, as evaluated within the ACFDT framework, \cite{Furche_2005} was mostly discarded from the set of tested techniques due to instabilities (negative frequency modes in the BSE polarization propagator) and replaced by an approximate (RPAsX) approach where the screened-Coulomb potential matrix elements was removed from the resonant electron-hole contribution. \cite{Maggio_2016,Holzer_2018}
Moreover, we also observe \cite{Loos_2020} that, in some cases, unphysical irregularities on the ground-state PES due to the appearance of discontinuities as a function of the bond length for some of the $GW$ quasiparticle energies.
Such an unphysical behavior stems from defining the quasiparticle energy as the solution of the quasiparticle equation with the largest spectral weight in cases where several solutions can be found.
This shortcoming has been thoroughly described in several previous studies.\cite{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020}
\\
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).
%==========================================
%\subsection{Unphysical discontinuities}
%==========================================
%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_2009a,Romaniello_2012,DiSabatino_2015,DiSabatino_2016,Tarantino_2017,DiSabatino_2019}
%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).
%==========================================
\subsection{The double excitation challenge}

26
Manuscript/xbbib.bib
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@ -1,13 +1,37 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-05-16 09:03:07 +0200
%% Created for Pierre-Francois Loos at 2020-05-18 09:58:45 +0200
%% Saved with string encoding Unicode (UTF-8)
@article{Colonna_2014,
Author = {Nicola Colonna and Maria Hellgren and Stefano {de Gironcoli}},
Date-Added = {2020-05-18 09:58:44 +0200},
Date-Modified = {2020-05-18 09:58:44 +0200},
Doi = {10.1103/PhysRevB.90.125150},
Journal = {Phys. Rev. B},
Pages = {125150},
Title = {Correlation Energy Within Exact-Exchange Adiabatic Connection Fluctuation-Dissipation Theory: Systematic Development and Simple Approximations},
Volume = {90},
Year = {2014},
Bdsk-Url-1 = {https://doi.org/10.1103/PhysRevB.90.125150}}
@article{Furche_2005,
Author = {Filipp Furche and Troy {Van Voorhis}},
Date-Added = {2020-05-18 09:58:37 +0200},
Date-Modified = {2020-05-18 09:58:37 +0200},
Doi = {10.1063/1.1884112},
Journal = {J. Chem. Phys.},
Pages = {164106},
Title = {Fluctuation-Dissipation Theorem Density-Functional Theory},
Volume = {122},
Year = {2005},
Bdsk-Url-1 = {https://doi.org/10.1063/1.1884112}}
@article{DiSabatino_2019,
Author = {S. {Di Sabatino} and J. A. Berger and P. Romaniello},
Date-Added = {2020-05-16 09:03:04 +0200},