xavier EOD 17th Apr
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@ -207,11 +207,11 @@ In its press release announcing the attribution of the 2013 Nobel prize in Chemi
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The study of neutral electronic excitations in condensed matter systems, from molecules to extended solids, has witnessed the development of a large number of such approximate methods with numerous applications to a large variety of fields, from the prediction of the colour of precious metals and stones for jewellery, to the understanding, \eg, of the basic principles behind photovoltaics, photocatalysis or DNA damage under irradiation in the context of biology. The present Perspective aims at describing the current status and upcoming challenges for the Bethe-Salpeter equation (BSE) formalism that, while sharing many features with time-dependent density functional theory (TD-DFT), including computational cost scaling with system size, relies on a different formalism, with specific difficulties but also potential solutions to known difficulties.
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The Bethe-Salpeter equation 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} to which belong as well the algebraic diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads:
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The Bethe-Salpeter equation 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 e.g. the algebraic diagrammatic construction (ADC) techniques in quantum chemistry. \cite{Dreuw_2015} While the density and density matrix stand as the basic variables in DFT and Hartree-Fock, Green's function MBPT takes the one-body Green's function as the central quantity. The (time-ordered) one-body Green's function reads:
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\begin{equation}
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G(xt,x't') = -i \langle N | T \left[ {\hat \psi}(xt) {\hat \psi}^{\dagger}(x't') \right] | N \rangle
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\end{equation}
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where $| N \rangle $ is the N-electron ground-state wavefunction. The operators ${\hat \psi}(xt)$ and ${\hat \psi}^{\dagger}(x't')$ remove/add an electron in space-spin-time positions (xt) and (x't'), while $T$ is the time-ordering operator. For (t>t') the one-body Green's function provides the amplitude of probability of finding bla bla \\
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where $| N \rangle $ is the N-electron ground-state wavefunction. The operators ${\hat \psi}(xt)$ and ${\hat \psi}^{\dagger}(x't')$ remove/add an electron in space-spin-time positions (xt) and (x't'), while $T$ is the time-ordering operator. For (t>t') the one-body Green's function provides the amplitude of probability of finding, on top of the ground-state Fermi sea, an electron in (xt) that was previously introduced in (x't'), while for (t<t') it is the propagation of a hole that is monitored. \\
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\noindent{\textbf{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 :
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\begin{equation}
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@ -232,7 +232,8 @@ which resembles formally the Kohn-Sham equation with the difference that the sel
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\noindent{\textbf{The $GW$ self-energy.}} While the presented equations are formally exact, it remains to provide an expression for the exchange-correlation 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 follows the path of linear response by considering the variation of $G$ with respect to an external perturbation. The obtained equation, when compared with the equation for the time-evolution of $G$ (Eqn.~\ref{Gmotion}), leads to a formal expression for the self-energy :
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\noindent{\textbf{The $GW$ self-energy.}} While the presented equations are formally exact, it remains to provide an expression for the exchange-correlation 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
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\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. The obtained equation, when compared with the equation for the time-evolution of $G$ (Eqn.~\ref{Gmotion}), leads to a formal expression for the self-energy :
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\begin{equation}
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\Sigma(1,2) = i \int d34 \; G(1,4) W(3,1^{+}) \Gamma(42,3)
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\end{equation}
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@ -246,9 +247,9 @@ with $\chi_0$ the well-known independent electron susceptibility and $v$ the b
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\varepsilon_n^{\GW} = \varepsilon_n^{\KS} +
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\langle \phi_n^{\KS} | \Sigma^{\GW}(\varepsilon_n^{\GW}) -V^{\XC} | \phi_n^{\KS} \rangle
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\end{equation}
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Such an approach, where input Kohn-Sham energies are corrected to yield better electronic energy levels, is labeled the single-shot, or perturbative, $G_0W_0$ technique. This simple scheme was used in the early $GW$ studies of extended semiconductors and insulators, [REFS]
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surfaces [REFs], and 2D systems, [REFS] allowing to dramatically reduced the errors associated with Kohn-Sham eigenvalues in conjunction with the common LDA approximation.
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In particular, the well-known ``band gap" problem, [REFS] namely the underestimation of the occupied to unoccupied bands energy gap at the LDA Kohn-Sham level, was dramatically reduced, bringing the agreement with experiment to within a very few tenths of an eV [REFS] with an $\mathcal{O}(N^4)$ computational cost scaling (see below). As 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. [REFS] However, remaining a low order perturbative approach starting with mono-determinental mean-field solutions, it is not intended to explore strongly correlated systems. [REFS to Hubbard cluster and discuss bubbles in a Note ???] \\
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Such an approach, where input Kohn-Sham energies are corrected to yield better electronic energy levels, is labeled 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}
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surfaces, \cite{Northrup_1991,Blase_1994,Rohlfing_1995} and 2D systems, \cite{Blase_1995} allowing to dramatically reduced the errors associated with Kohn-Sham eigenvalues in conjunction with common local or gradient-corrected approximations to the exchange-correlation potential.
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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 Kohn-Sham level, was dramatically reduced, bringing the agreement with experiment to within a very few tenths of an eV [REFS] with an $\mathcal{O}(N^4)$ computational cost scaling (see below). As 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 mono-determinental mean-field solutions, it is not intended to explore strongly correlated systems. \cite{Verdozzi_1995} \\
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\noindent {\textbf{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:
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\begin{equation}
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@ -321,6 +322,7 @@ with $\eta=2,0$ for singlets/triplets and:
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W_{ai,bj} = \int d{\bf r} d{\bf r}'
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\phi_i({\bf r}) \phi_j({\bf r}) W({\bf r},{\bf r}'; \omega=0)
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\phi_a({\bf r}') \phi_b({\bf r}')
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\label{Wmatel}
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\end{equation}
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where we notice that the 2 occupied (virtual) eigenstates are taken at the same space position, in contrast with the
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$(ai|bj)$ bare Coulomb term. As compared to TD-DFT :
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@ -337,39 +339,29 @@ the use of the BSE formalism in condensed-matter physics emerged in the 60s at t
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Three decades latter, the first \textit{ab initio} implementations, starting with small clusters \cite{Onida_1995,Rohlfing_1998} and extended semiconductors and wide-gap insulators, \cite{Albrecht_1997,Benedict_1998,Rohlfing_1999}
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paved the way to the popularization in the solid-state physics community of the BSE formalism.
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Following early applications to periodic polymers and molecules, [REFS] the BSE formalism gained much momentum in the quantum chemistry community with in particular several benchmarks \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same running parameters (geometries, basis sets) than the available reference higher-level calculations such as CC3. [REFS] Such comparisons were grounded in the development of codes replacing the planewave solid-state physics paradigm by well documented correlation-consistent Gaussian basis sets, together with adequate auxiliary bases when resolution-of-the-identity techniques were used. [REFS]
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Following early applications to periodic polymers and molecules, [REFS] the BSE formalism gained much momentum in the quantum chemistry community with in particular several benchmarks \cite{Korbel_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017,Krause_2017,Gui_2018} on large molecular systems performed with the very same running parameters (geometries, basis sets) than the available reference higher-level calculations such as CC3. Such comparisons were grounded in the development of codes replacing the planewave solid-state physics paradigm by well documented correlation-consistent Gaussian basis sets, together with adequate auxiliary bases when resolution-of-the-identity techniques were used. [REFS]
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An important conclusion drawn from these calculations was that the quality of the BSE excitation energies are strongly correlated to the deviation of the preceding $GW$ HOMO-LUMO gap with the experimental (IP-AE) photoemission gap. Standard $G_0W_0$ calculations starting with Kohn-Sham eigenstates generated with (semi)local functionals yield much larger HOMO-LUMO gaps than the input Kohn-Sham one, but still too small as compared to the experimental (AE-IP) value. Such an underestimation of the (IP-AE) gap leads to a similar underestimation of the lowest optical excitation energies.
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Such a residual HOMO-LUMO gap problem can be significantly improved by adopting exchange-correlation functionals that yield a much improved HOMO-LUMO gap as a starting point for the $GW$ correction. Obviously, optimally tuned functionals such that the Kohn-Sham HOMO-LUMO gap matches the $\Delta$SCF (AE-IP) value, yields excellent $GW$ gaps and much improved resulting BSE excitations. Alternatively, self-consistent schemes, where ...
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Such a residual HOMO-LUMO gap problem can be significantly improved by adopting exchange-correlation functionals with a tuned amount of exact exchange that yield a much improved Kohn-Sham HOMO-LUMO gap as a starting point for the $GW$ correction. \cite{Bruneval_2013,Rangel_2016,Knight_2016} Alternatively, self-consistent schemes, where corrected eigenvalues, and possibly orbitals, are reinjected in the construction of $G$ and $W$, have been shown to lead to a significant improvement of the quasiparticle energies in the case of molecular systems, with the advantage of significantly removing the dependence on the starting point functional. \cite{Rangel_2016,Kaplan_2016,Caruso_2016} As a result, BSE excitation singlet energies starting from such improved quasiparticle energies were found to be in much better agreement with reference calculations such as CC3. For sake of illustration, an average 0.2 eV error was found for the well-known Thiel set comprising more than a hundred representative singlet excitations from a large variety of representative molecules.
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\cite{Jacquemin_2015a,Bruneval_2015,Gui_2018,Krause_2017} This is equivalent to the best TD-DFT results obtained by scanning a large variety of global hybrid functionals with varying fraction of exact exchange.
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A very remarkable success of the BSE formalism lies in the description of the charge-transfer (CT) excitations, a notoriously difficult problem for TD-DFT calculations adopting standard functionals. \cite{Dreuw_2004}
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Such a difficulty can be ascribed to the lack of long-range electron-hole interaction with local XC functionals. Such a problem can be cured within TD-DFT by including a long-range component in the kernel through an exact exchange contribution, a solution that explains in particular the success of range-separated hybrids for the description of CT excitations. \cite{Stein_2009} The analysis of the screened Coulomb potential matrix elements in the BSE kernel (see Eqn.~\ref{Wmatel}) reveals that such long-range (non-local) electron-hole interactions are properly described, including in environments (solvents, molecular solid, etc.) where screening reduces the long-range electron-hole interactions. The success of the BSE formalism to treat CT excitations has been demonstrated in several studies, \cite{Blase_2011b,Baumeier_2012,Duchemin_2012,Sharifzadeh_2013,Cudazzo_2010,Cudazzo_2013} opening the way to important applications such as doping, photovoltaics or photocatalysis in organic systems. We now leave the description of successes to discuss difficulties and Perspectives.\\
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\noindent {\textbf{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-Hartree-Fock. As such, searching iteratively for, typically, the lowest eigenstates presents the same computational cost within BSE and TD-DFT. The main bottleneck resides in the preceding calculations of the $GW$ quasiparticle energies. Within a planewave approach, or using resolution-of-the-identity techniques combined with localized basis sets, $GW$ calculations scale as $\mathcal{O}(N^4)$ with system size. Such a cost is mainly associated with calculating the free-electron susceptibility $\chi_0(\omega)$ at various frequencies with its entangled summations over occupied and virtual states. Pooling empty states with common energy denominators, \cite{Bruneval_2008}
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or replacing the sum over unoccupied states by iterative techniques \cite{Umari_2010,Giustino_2010} as already done in TDDFT, \cite{Walker_2006} are efficient techniques that do not change however the scaling with system size.
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Another approach, that has regained recently much interest, lies in the so-called space-time approach by Rojas and coworkers. \cite{Rojas_1995} Borrowing the idea of Laplace transform formulations, already used in quantum chemistry perturbation theories, \cite{Almlof_1991,Haser_1992} combined with a real-space grid formulation, the susceptibility can be factorized so as to decoupled summations over occupied and virtual states:
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\begin{equation}
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\chi_0({\bf r},{\bf r}'; i\tau) =
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\left( \sum_i \phi_i({\bf r}') {\tilde{\phi}}_{i}({\bf r}) \right)
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\left( \sum_a \phi_a({\bf r}') {\tilde{\phi}}_{a}({\bf r}) \right)
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\end{equation}
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with $\tilde{\phi}_{i}({\bf r}) = {\phi_ai}({\bf r}) e^{ \varepsilon_a \tau} $ and $\tilde{\phi}_{a}({\bf r}) = {\phi_a}({\bf r}) e^{ - \varepsilon_a \tau} $, taking the zero of energy at the chemical potential and with real orbitals. Such an approach leads to cubic scaling algorithms, independently of any arguments exploiting localization or sparcity in the limit of large systems. Such an approach has been recently adapted to cubic scaling RPA calculations \cite{} and is now blooming in quantum chemistry thanks to the concept of Interpolative Separable Density Fitting (ISDF) that allows decoupling occupied and virtual orbitals entangled in standard resolution of identity density- or coulomb fitting coefficients.
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Combined further with a stochastic treatment of virtual space sampling, impressive linear scaling formalisms could be established, paving the way to applying the $GW$/BSE formalism to systems comprising several hundred atoms on standard size laboratory clusters.
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too small by up to an eV in the case of small
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\vskip 5cm
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interest from the quantum chemistry community BSE formalism
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chemistry oriented reviews with, e.g., the language of localized basis and resolution-of-the-identity techniques, \cite{Ren_2012} or applications related to organic molecular systems, photoelectrochemistry, etc. \cite{Ping_2013,Leng_2016,Blase_2018}
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large molecular benchmarks with comparisons to TD-DFT and higher level wavefunctions techniques such as CC3
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Reinjecting the corrected eigenvalues into the construction of $G$ and $W$ leads to a partially self-consistent scheme where eigenvalues are updated, while keeping input one-body orbitals frozen to their Kohn-Sham ansatz. Such a simple self-consistent scheme is labelled ev$GW$, [REFS] as a simple alternative to more involved self-consistent schemes where both eigenvalues and eigenstates are updated. [REFS] \\
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charge transfer
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classification into local-, Rydberg-, or charge transfer-type
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\cite{Hirose_2017} ad developed extensively in to the TD-DFT community.
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\noindent {\textbf{The computational challenge.}} \\
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pooling empty state with common energy denominator techniques, \cite{Bruneval_2008}
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replacing the sum over unoccupied states by iterative techniques \cite{Umari_2010,Giustino_2010} already known in TDDFT. \cite{Walker_2006} Such techniques proved to be very efficient in the case in particular of planewave basis sets that generate very large number of empty states.
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%%
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Another approach, that proved very fruitful, lies in the so-called space-time approach by Rojas and coworkers, \cite{Rojas_1995} that borrows the idea of Laplace transform formulation, already used in quantum chemistry perturbation theories, \cite{Almlof_1991,Haser_1992} with the further concept of separability
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\noindent {\textbf{The challenge of Analytic gradients.}} \\
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@ -184,7 +184,37 @@
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Bdsk-Url-1 = {http://link.aps.org/doi/10.1103/PhysRevLett.45.290},
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Bdsk-Url-2 = {http://dx.doi.org/10.1103/PhysRevLett.45.290}}
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@article{Hybetsen_1986,
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@article{Perdew_1983,
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title = {Physical Content of the Exact Kohn-Sham Orbital Energies: Band Gaps and Derivative Discontinuities},
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Author = {Hybertsen, Mark S. and Louie, Steven G.},
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Publisher = {American Physical Society},
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Title = {Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies},
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Title = {Self-energy operators and exchange-correlation potentials in semiconductors},
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Title = {Self-Energy Operators and Exchange-Correlation Potentials in Semiconductors},
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Title = {Precise quasiparticle energies and Hartree-Fock bands of semiconductors and insulators},
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@article{Rohlfing_1995,
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title = {Efficient Scheme for GW Quasiparticle Band-Structure Calculations with Aapplications to Bulk Si and to the Si(001)-(2\ifmmode\times\else\texttimes\fi{}1) Surface},
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}
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@article{Verdozzi_1995,
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||||
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|
||||
}
|
||||
@article{Campillo_1999,
|
||||
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|
||||
publisher = {American Physical Society},
|
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|
||||
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|
||||
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|
||||
|
||||
@article{Onida_2002,
|
||||
Author = {Onida, Giovanni and Reining, Lucia and Rubio, Angel},
|
||||
Date-Modified = {2020-04-16 22:28:27 +0200},
|
||||
@ -1128,3 +1247,144 @@
|
||||
Year = {2010},
|
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|
||||
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|
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|
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|
||||
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|
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Reference in New Issue
Block a user