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Manuscript/BSE_JPCL.tex
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@ -180,8 +180,9 @@
\date{\today}
\begin{tocentry}
\vspace{1cm}
\centering
\includegraphics[width=0.9\textwidth]{../TOC/TOC}
\includegraphics[width=\textwidth]{../TOC/TOC}
\end{tocentry}
@ -257,7 +258,7 @@ where $\mu$ is the chemical potential, $\eta$ is a positive infinitesimal, $\var
Here, $E_s^{\Nel}$ is the total energy of the $s$\textsuperscript{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 poles of the Green's function $\lbrace \varepsilon_s \rbrace$ are proper addition/removal 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.
Contrary to standard $\Delta$SCF techniques, the knowledge of $G$ provides the full ionization spectrum, as measured by direct and inverse photoemission spectroscopy, 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}
@ -282,7 +283,7 @@ The knowledge of $\Sigma$ allows to access the true addition/removal energies, n
While the equations reported above are formally exact, it remains to provide an expression for the xc self-energy operator $\Sigma$.
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,Reining_2017,Golze_2019} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}).
\cite{Onida_2002,Golze_2019} follows the path of linear response by considering the variation of $G$ with respect to an external perturbation (see Fig.~\ref{fig:pentagon}).
The resulting 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),
@ -342,18 +343,16 @@ In particular, the well-known ``band gap" problem, \cite{Perdew_1983,Sham_1983}
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 Hedin's equations (see Fig.~\ref{fig:pentagon}).
There exists two main types of self-consistent $GW$ methods:
i) \textit{``eigenvalue-only quasiparticle''} $GW$ (ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011}
where the quasiparticle 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 quasiparticle energies and the corresponding orbitals.
i) \textit{``eigenvalue-only quasiparticle''} $GW$ (ev$GW$), \cite{Hybertsen_1986,Blase_2011} where the quasiparticle energies are updated at each iteration, and
ii) \textit{``quasiparticle self-consistent''} $GW$ (qs$GW$), \cite{vanSchilfgaarde_2006} where one updates both the quasiparticle 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_2013,Caruso_2013a,Koval_2014,Wilhelm_2018,Loos_2018}
Indeed, there is a long-standing debate about the importance of partial and full self-consistency in $GW$. \cite{Stan_2006,Rostgaard_2010,Caruso_2013a,Koval_2014}
In some situations, it has been found that self-consistency can worsen spectral properties compared to the simpler $G_0W_0$ method.\cite{deGroot_1995,Schone_1998,Ku_2002,Friedrich_2006,Kutepov_2016,Kutepov_2017,Loos_2018}
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 famous example has been provided by the calculations performed on the uniform electron gas. \cite{Holm_1998}
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_2013,Caruso_2013a,Koval_2014,Hung_2016,Blase_2018,Jacquemin_2017}
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_2013a,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}
@ -362,8 +361,8 @@ 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 $\rho(1)$ with respect to an external local perturbation $U(1)$, the BSE formalism considers a generalized 4-points susceptibility, or two-particle correlation function, that monitors the variation of the one-body Green's function $G(1,1')$ with respect to a non-local external perturbation $U(2,2')$: \cite{Strinati_1988}
Like TD-DFT, BSE deals with the calculations of optical (or neutral) excitations, as measured by optical (\eg, absorption) spectroscopy,
However, while TD-DFT starts with the variation of the charge density $\rho(1)$ with respect to an external local perturbation $U(1)$, the BSE formalism considers a generalized 4-points susceptibility, or two-particle correlation function, that monitors the variation of the one-body Green's function $G(1,1')$ with respect to a non-local external perturbation $U(2,2')$: \cite{Strinati_1988}
\begin{equation}
\chi(1,2) \stackrel{\DFT}{=} \pdv{\rho(1)}{U(2)}
\quad \rightarrow \quad
@ -576,8 +575,6 @@ Exploiting further the locality of the atomic orbital basis, the BSE absorption
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 $GW$ calculation that scales 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, for example, the overlap matrix in the atomic orbital basis allows to reduce the scaling in the large size limit, \cite{Foerster_2011,Wilhelm_2018} efficient real-space grids 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}
@ -588,9 +585,8 @@ These ongoing developments pave the way to applying the $GW$@BSE formalism to sy
%==========================================
\subsection{The triplet instability challenge}
%==========================================
\hl{Je ne pige pas la premiere phrase qui semble melanger differents concepts. A divisier ?}
The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels, contaminating as well TD-DFT calculations with popular range-separated hybrids that generally contains a large fraction of exact exchange in the long-range.
The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission or thermally activated delayed fluorescence (TADF).
From a more theoretical point of view, triplet instabilities are intimately linked to the stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and KS \cite{Bauernschmitt_1996} levels, hampering the applicability of TD-DFT for popular range-separated hybrids containing a large fraction of long-range exact exchange.
While TD-DFT with range-separated hybrids can benefit from tuning the range-separation parameter(s) as a mean to act on the triplet instability, \cite{Sears_2011} BSE calculations do not offer this pragmatic way-out since the screened Coulomb potential that builds the kernel does not offer any parameter to tune.
Benchmark calculations \cite{Jacquemin_2017b,Rangel_2017} clearly concluded that triplets are notably too low in energy within BSE and that the use of the Tamm-Dancoff approximation was able to partly reduce this error.
@ -601,14 +597,11 @@ An alternative cure was offered by hybridizing TD-DFT and BSE, that is, by addin
%==========================================
\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
%
%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}\\
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{Olivucci_2010}
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{Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Roos_1996,Angeli_2001}) cannot be afforded.
From a theoretical point of view, the accurate prediction of excited electronic states remains a challenge, \cite{Loos_2020a} especially for large systems where state-of-the-art computational techniques (such as multiconfigurational methods \cite{Roos_1996}) cannot be afforded.
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.
@ -636,38 +629,16 @@ Such an unphysical behavior stems from defining the quasiparticle energy as the
We refer the interested reader to Refs.~\citenum{vanSetten_2015,Maggio_2017,Loos_2018,Veril_2018,Duchemin_2020} for detailed discussions.
\\
%==========================================
%\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{Beyond the static approximation}
%==========================================
Going beyond the static approximation is a difficult challenge which has been, nonetheless, embraced by several groups.\cite{Strinati_1988,Rohlfing_2000,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Zhang_2013,Rebolini_2016,Olevano_2019}
As mentioned earlier in this \textit{Perspective}, most of BSE calculations are performed within the so-called static approximation, which substitutes the dynamically-screened (\ie, frequency-dependent) Coulomb potential $W(\omega)$ by its static limit $W(\omega = 0)$ [see Eq.~\eqref{eq:Wmatel}].
It is important to mention that diagonalizing the BSE Hamiltonian in the static approximation corresponds to solving a \textit{linear} eigenvalue problem in the space of single excitations, while it is, in its dynamical form, a non-linear eigenvalue problem (in the same space) which is much harder to solve from a numerical point of view.
In complete analogy with the ubiquitous adiabatic approximation in TD-DFT, one key consequence of the static approximation is that double (and higher) excitations are completely absent from the BSE spectrum, which obviously hampers the applicability of BSE as double excitation may play, indirectly, a key role in photochemistry mechanisms.
In complete analogy with the ubiquitous adiabatic approximation in TD-DFT, one key consequence of the static approximation is that double (and higher) excitations are completely absent from the BSE optical spectrum, which obviously hampers the applicability of BSE as double excitation may play, indirectly, a key role in photochemistry mechanisms.
Higher excitations would be explicitly present in the BSE Hamiltonian by ``unfolding'' the dynamical BSE kernel, and one would recover a linear eigenvalue problem with, nonetheless, a much larger dimension.
Corrections to take into account the dynamical nature of the screening may or may not recover these multiple excitations.
However, dynamical corrections permit, in any case, to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations (and, in particular, non-interacting double excitations).
However, dynamical corrections permit, in any case, to recover, for transitions with a dominant single-excitation character, additional relaxation effects coming from higher excitations.
From a more practical point of view, dynamical effects have been found to affect the positions and widths of core-exciton resonances in semiconductors, \cite{Strinati_1982,Strinati_1984} rare gas solids, and transition metals. \cite{Ankudinov_2003}
Thanks to first-order perturbation theory, Rohlfing and coworkers have developed an efficient way of taking into account the dynamical effects via a plasmon-pole approximation combined with TDA. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b}
@ -676,8 +647,6 @@ Studying PYP, retinal and GFP chromophore models, Ma \textit{et al.}~found that
Zhang \textit{et al.}~have studied the frequency-dependent second-order Bethe-Salpeter kernel and they have observed an appreciable improvement over configuration interaction with singles (CIS), time-dependent Hartree-Fock (TDHF), and adiabatic TD-DFT results. \cite{Zhang_2013}
Rebolini and Toulouse have performed a similar investigation in a range-separated context, and they have reported a modest improvement over its static counterpart. \cite{Rebolini_2016}
In these two latter studies, they also followed a (non-self-consistent) perturbative approach within TDA with a renormalization of the first-order perturbative correction.
%Finally, let us also mentioned the work of Romaniello and coworkers, \cite{Romaniello_2009b,Sangalli_2011} in which the authors genuinely accessed additional excitations by solving the non-linear, frequency-dependent eigenvalue problem.
%However, it is based on a rather simple model (the Hubbard dimer) which permits to analytically solve the dynamical equations.
\\
%==========================================
@ -749,16 +718,6 @@ In these two latter studies, they also followed a (non-self-consistent) perturba
%This approximation comes with a heavy price as the number of solutions provided by the system of equations \eqref{eq:non_lin_sys} has now been reduced from $K$ to $K_1$.
%Coming back to our example, in the static approximation, the operator $\Tilde{\bA}_1$ built in the basis of single excitations cannot provide double excitations anymore, and the only $K_1$ excitation energies are associated with single excitations.\\
%==========================================
%\subsection{Core-level spectroscopy}
%==========================================
%XANES, \cite{Olovsson_2009,Vinson_2011}
%diabatization and conical intersections \cite{Kaczmarski_2010}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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@ -10,20 +10,20 @@
\begin{tikzpicture}
\begin{scope}[very thick,
node distance=5cm,on grid,>=stealth',
block1/.style={rectangle,draw,fill=red!20},
block2/.style={rectangle,draw,fill=orange!20},
comp1/.style={rectangle,draw,fill=green!40},
comp2/.style={circle,draw,fill=green!40},
comp3/.style={rectangle,fill=white!40}]
theo1/.style={rectangle,draw,fill=red!20},
theo2/.style={rectangle,draw,fill=orange!20},
theo3/.style={rectangle,draw,fill=green!40},
exp1/.style={rectangle,draw,fill=cyan!40},
exp2/.style={rectangle,draw,fill=violet!40}]
\node [block1, text width=7cm, align=center] (KS)
\node [theo1, text width=7cm, align=center] (KS)
{\textbf{\LARGE Kohn-Sham DFT}
$$
\qty[ -\frac{\nabla^2}{2} + v_\text{ext} + V^{\text{Hxc}} ] \phi_p^{\text{KS}} = \varepsilon^{\text{KS}}_p \phi_p^{\text{KS}}
$$
};
\node [block2, text width=7cm, align=center] (GW) [below=of KS, yshift=2cm]
\node [theo2, text width=7cm, align=center] (GW) [below=of KS, yshift=2cm]
{\textbf{\LARGE $GW$ approximation}
$$
\varepsilon_p^{GW} = \varepsilon_p^{\text{KS}} +
@ -32,7 +32,7 @@
};
\node [comp1, text width=7cm, align=center] (BSE) [below=of GW, yshift=2cm]
\node [theo3, text width=7cm, align=center] (BSE) [below=of GW, yshift=2cm]
{\textbf{\LARGE Bethe-Salpeter equation}
$$
\begin{pmatrix}
@ -53,21 +53,18 @@
};
% \node [comp2, align=center] (phys) [left=of BSE, xshift=-2cm]
% {\LARGE Molecules};
\node [exp1, align=center] (photo) [right=of GW, xshift=3cm]
{\LARGE (Inverse) \\ \LARGE photoemission \\ \LARGE spectroscopy};
% \node [comp2, align=center] (chem) [below=of BSE, xshift=0cm]
% {\LARGE Materials};
\node [exp2, align=center] (abs) [right=of BSE, xshift=3cm]
{\LARGE Optical \\ \LARGE spectroscopy};
% \node [comp2, align=center] (bio) [right=of BSE, xshift=2cm]
% {\LARGE Clusters};
\path
(KS) edge [->,color=black] node [right,black] {\LARGE Fundamental gap} (GW)
(GW) edge [->,color=black] node [right,black] {\LARGE Excitonic effect} (BSE)
% (BSE) edge [->,color=black] node [above,black] {} (phys)
% (BSE) edge [->,color=black] node [above,black] {} (chem)
% (BSE) edge [->,color=black] node [above,black] {} (bio)
(photo) edge [<->,color=black] node [above,black] {Ionization potentials} node [below,black] {Electron affinities} (GW)
(abs) edge [<->,color=black] node [above,black] {Optical excitations} (BSE)
;