T2 starts again
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@ -349,33 +349,17 @@ An important conclusion drawn from these calculations was that the quality of th
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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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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} Similar difficulties emerge as well in solid-state physics for semiconductors where extended Wannier excitons are characterized by weakly overlapping electrons and holes, causing a dramatic deficit of spectral weight at low energy. \cite{Botti_2004} These difficulties can be ascribed to the lack of long-range electron-hole interaction with local XC functionals that can be cured 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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\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. Searching iteratively for the lowest eigenstates presents the same $\mathcal{O}(N^4)$ 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 $\mathcal{O}(N^4)$ operations with standard resolution-of-identity techniques. At the price of sacrifying the knowledge of the eigenvectors, the BSE absorption spectrum can be known with $\mathcal{O}(N^3)$ 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}
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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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In practice, the main bottleneck for standard BSE calculations as compared to TD-DFT resides in the preceding $GW$ calculations that scale as $\mathcal{O}(N^4)$ with system size using PWs or RI techniques, but with a rather large prefactor.
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%%Such a cost is mainly associated with calculating the free-electron susceptibility with its entangled summations over occupied and virtual states.
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%%While attempts to bypass the $GW$ calculations are emerging, replacing quasiparticle energies by Kohn-Sham eigenvalues matching energy electron addition/removal, \cite{Elliott_2019}
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The field of low-scaling $GW$ calculations is however witnessing significant advances. While the sparcity 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}
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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 withing 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. \\
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\noindent {\textbf{The challenge of Analytic gradients.}} \\
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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
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This points to another direction for the BSE foramlism, 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}
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\noindent {\textbf{The Triplet Instability Challenge.}} \\
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The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
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\noindent {\textbf{The Triplet Instability Challenge.}} The analysis of the singlet-triplet splitting is central to numerous applications such as singlet fission, thermally activated delayed fluorescence (TADF) or
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stability analysis of restricted closed-shell solutions at the HF \cite{Seeger_1977} and TD-DFT \cite{Bauernschmitt_1996} levels.
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contaminating as well TD-DFT calculations with popular range-separated hybrids (RSH) that generally contains a large fraction of exact exchange in the long-range. \cite{Sears_2011}
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While TD-DFT with RSH can benefit from tuning the range-separation parameter 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.
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@ -384,6 +368,13 @@ benchmarks \cite{Jacquemin_2017b,Rangel_2017}
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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}
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\noindent {\textbf{The challenge of Analytic gradients.}} \\
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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
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This points to another direction for the BSE foramlism, 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}
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\noindent {\textbf{Dynamical kernels and multiple excitations.}} \\
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\cite{Zhang_2013}
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@ -1410,3 +1410,180 @@
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5044503}}
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}
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@article{Liu_2016,
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title = {Cubic Scaling $GW$: Towards Fast Quasiparticle Calculations},
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author = {Liu, Peitao and Kaltak, Merzuk and Klime\ifmmode \check{s}\else \v{s}\fi{}, Ji\ifmmode \check{r}\else \v{r}\fi{}\'{\i} and Kresse, Georg},
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}
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@article{Duchemin_2019,
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author = {Duchemin,Ivan and Blase,Xavier },
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title = {Separable resolution-of-the-identity with all-electron Gaussian bases: Application to cubic-scaling RPA},
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}
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@article{Lu_2017,
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title = "Cubic Scaling Algorithms for RPA Correlation Using Interpolative Separable Density Fitting",
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journal = "J. Comput. Phys.",
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volume = "351",
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pages = "187 - 202",
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year = "2017",
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url = "http://www.sciencedirect.com/science/article/pii/S002199911730671X",
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author = "Jianfeng Lu and Kyle Thicke",
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keywords = "Electronic structure theory, Density fitting, Random phase approximation, Fast algorithms, Contour integral",
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abstract = "We present a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in χ0 by use of Cauchy's integral formula. This introduces an additional integral to be carried out, for which we provide a geometrically convergent quadrature rule. Our scheme also uses the newly developed Interpolative Separable Density Fitting algorithm to further reduce the computational cost in a way analogous to that of the Resolution of Identity method."
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}
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}
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title = {Time-Dependent Stochastic Bethe-Salpeter Approach},
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}
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