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19
BSEdyn.bib
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@ -1,13 +1,26 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-05-19 14:00:59 +0200
%% Created for Pierre-Francois Loos at 2020-05-19 16:23:54 +0200
%% Saved with string encoding Unicode (UTF-8)
@misc{QuAcK,
Author = {P. F. Loos},
Date-Added = {2020-05-19 16:22:58 +0200},
Date-Modified = {2020-05-19 16:22:58 +0200},
Doi = {10.5281/zenodo.3745928},
Note = {\url{https://github.com/pfloos/QuAcK}},
Publisher = {Zenodo},
Title = {{{QuAcK: a software for emerging quantum electronic structure methods}}},
Url = {https://github.com/pfloos/QuAcK},
Year = {2019},
Bdsk-Url-1 = {https://github.com/LCPQ/quantum_package},
Bdsk-Url-2 = {http://dx.doi.org/10.5281/zenodo.200970}}
@article{Loos_2018a,
Author = {P. F. Loos and A. Scemama and A. Blondel and Y. Garniron and M. Caffarel and D. Jacquemin},
Date-Added = {2020-05-19 14:00:54 +0200},
@ -1536,10 +1549,10 @@
Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.92.075422},
Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.92.075422}}
@article{Loos_2018,
@article{Loos_2018b,
Author = {P. F. Loos and P. Romaniello and J. A. Berger},
Date-Added = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-18 21:40:28 +0200},
Date-Modified = {2020-05-19 16:23:52 +0200},
Doi = {10.1021/acs.jctc.8b00260},
Journal = {J. Chem. Theory Comput.},
Pages = {3071--3082},

61
BSEdyn.tex
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@ -414,14 +414,13 @@ The BSE matrix elements read
\begin{subequations}
\begin{align}
\label{eq:BSE-Adyn}
\A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{}(\omega),
\A{ia,jb}{}(\omega) & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \sigma \ERI{ia}{jb} - \W{ij,ab}{}(\omega),
\\
\label{eq:BSE-Bdyn}
\B{ia,jb}{}(\omega) & = 2 \ERI{ia}{bj} - \W{ib,aj}{}(\omega),
\B{ia,jb}{}(\omega) & = 2 \sigma \ERI{ia}{bj} - \W{ib,aj}{}(\omega),
\end{align}
\end{subequations}
\titou{singlet and triplet}
where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies,
where $\eGW{ia} = \eGW{a} - \eGW{i}$ are occupied-to-virtual differences of $GW$ quasiparticle energies, $\sigma = 1 $ or $0$ for singlet and triplet excited states (respectively),
\begin{equation}
\ERI{pq}{rs} = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}'
\end{equation}
@ -440,7 +439,7 @@ where $\eta$ is a positive infinitesimal, and
are the spectral weights.
In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are direct (\ie, without exchange) RPA neutral excitations and their corresponding transition vectors computed by solving the (static) linear response problem
\begin{equation}
\label{eq:LR-stat}
\label{eq:LR-RPA}
\begin{pmatrix}
\bA{\RPA} & \bB{\RPA} \\
-\bB{\RPA} & -\bA{\RPA} \\
@ -487,14 +486,14 @@ Now, let us decompose, using basic perturbation theory, the non-linear eigenprob
-\bB{(1)}(\omega) & -\bA{(1)}(\omega) \\
\end{pmatrix}
\end{equation}
where
with
\begin{subequations}
\begin{align}
\label{eq:BSE-A0}
\A{ia,jb}{(0)} & = \delta_{ij} \delta_{ab} \eGW{ia} + 2 \ERI{ia}{jb} - \W{ij,ab}{\text{stat}},
\\
\label{eq:BSE-B0}
\B{ia,jb}{(0)} & = 2 \ERI{ia}{bj} - \W{ib,aj}{\text{stat}},
\B{ia,jb}{(0)} & = 2 \ERI{ia}{bj} - \W{ib,aj}{\text{stat}}.
\end{align}
\end{subequations}
and
@ -507,15 +506,15 @@ and
\B{ia,jb}{(1)}(\omega) & = - \W{ib,aj}{}(\omega) + \W{ib,aj}{\text{stat}},
\end{align}
\end{subequations}
The static version of the screened Coulomb potential reads
where we have defined the static version of the screened Coulomb potential
\begin{equation}
\label{eq:Wstat}
\W{ij,ab}{\text{stat}} = \ERI{ij}{ab} - 4 \sum_m^{\Nocc \Nvir} \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}.
\end{equation}
The $m$th BSE excitation energy and its corresponding eigenvector can then decomposed as
According to perturbation theory, the $m$th BSE excitation energy and its corresponding eigenvector can then decomposed as
\begin{subequations}
\begin{gather}
\Om{m}{} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots
\Om{m}{} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots,
\\
\begin{pmatrix}
\bX{m}{} \\
@ -531,11 +530,12 @@ The $m$th BSE excitation energy and its corresponding eigenvector can then decom
\bX{m}{(1)} \\
\bY{m}{(1)} \\
\end{pmatrix}
+ \ldots
+ \ldots.
\end{gather}
\end{subequations}
Solving the zeroth-order static problem yields
\begin{equation}
\label{eq:LR-BSE-stat}
\begin{pmatrix}
\bA{(0)} & \bB{(0)} \\
-\bB{(0)} & -\bA{(0)} \\
@ -552,8 +552,9 @@ Solving the zeroth-order static problem yields
\bY{m}{(0)} \\
\end{pmatrix},
\end{equation}
Thanks to first-order perturbation theory, the first-order correction to the $m$th excitation energy is
and, thanks to first-order perturbation theory, the first-order correction to the $m$th excitation energy is
\begin{equation}
\label{eq:Om1}
\Om{m}{(1)} =
\T{\begin{pmatrix}
\bX{m}{(0)} \\
@ -570,14 +571,15 @@ Thanks to first-order perturbation theory, the first-order correction to the $m$
\bY{m}{(0)} \\
\end{pmatrix}.
\end{equation}
From a practical point of view, if one enforces the Tamm-Dancoff approximation (TDA), we obtain the very simple expression
From a practical point of view, if one enforces the TDA, we obtain the very simple expression
\begin{equation}
\label{eq:Om1-TDA}
\Om{m}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)}.
\end{equation}
This correction can be renormalized by computing, at basically no extra cost, the renormalization factor
\begin{equation}
Z_{m} = \qty[ \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1}.
\label{eq:Z}
Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1},
\end{equation}
which finally yields
\begin{equation}
@ -594,31 +596,35 @@ This is our final expression.
%\end{figure}
%%% %%% %%%
In terms of computational consideration, because Eq.~\eqref{eq:EcBSE} requires the entire BSE singlet excitation spectrum, we perform a complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
In terms of computational cost, if one decides to compute the dynamical correction of the $M$ lowest excitation energies, one must perform, first, a conventional (static) BSE calculation and extract the $M$ lowest eigenvalues and their corresponding eigenvectors [see Eq.~\eqref{eq:LR-BSE-stat}].
These are then used to compute the first-order correction from Eq.~\eqref{eq:Om1} or Eq.~\eqref{eq:Om1-TDA}, which also require to construct and evaluate the dynamical part of the BSE Hamiltonian for each excitation one wants to dynamically correct.
The static BSE Hamiltonian is computed once during the static BSE calculation and does not dependent on the targeted excitation.
Searching iteratively for the lowest eigenstates, via the Davidson algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost.
Constructing the static and dynamic BSE Hamiltonians is much more expensive as it requires the complete diagonalization of the $(\Nocc \Nvir \times \Nocc \Nvir)$ RPA linear response matrix [see Eq.~\eqref{eq:LR-RPA}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is, by far, the computational bottleneck in our current implementation.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%
The restricted HF formalism has been systematically employed in the present study.
All the $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (restricted) HF starting point.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent ev$GW$ calculations are employed as starting points to compute the BSE neutral excitations.
Within $GW$, all orbitals are corrected.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent ev$GW$ \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011}calculations are employed as starting points to compute the BSE neutral excitations.
For both {\GOWO} and {\evGW}, the entire set of orbitals are corrected.
In the case of {\GOWO}, the quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation.
For ev$GW$, the convergence creiterion has been set to $10^{-5}$.
Further details about our implementation of {\GOWO} and evGW can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
For ev$GW$, the convergence criterion has been set to $10^{-5}$.
Further details about our implementation of {\GOWO} and {\evGW} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018}.
As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
For comparison purposes, we employ the reference excitation energies and geometries from Ref.~\onlinecite{Loos_2018a} also computed the PES at the second-order M{\o}ller-Plesset perturbation theory (MP2), as well as with various increasingly accurate CC methods, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3. \cite{Christiansen_1995b}
All the other calculations have been performed with our locally developed $GW$ software. \cite{Loos_2018b,Veril_2018}
As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions.
For comparison purposes, we employ the theoretical best estimates and geometries of Ref.~\onlinecite{Loos_2018a} from which coupled cluster (CC) excitation energies, namely, CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3, \cite{Christiansen_1995b} are also extracted.
All the BSE calculations have been performed with our locally developed $GW$ software, \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}, where the present perturbative correction has been implemented.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
\label{sec:conclusion}
\section{Results and Discussion}
\label{sec:resdis}
%%%%%%%%%%%%%%%%%%%%%%%%
This is the conclusion
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
@ -629,6 +635,7 @@ This is the conclusion
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements{
%%%%%%%%%%%%%%%%%%%%%%%%
%PFL thanks the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481) for financial support.
This work was performed using HPC resources from GENCI-TGCC (Grant No.~2019-A0060801738) and CALMIP (Toulouse) under allocation 2020-18005.
Funding from the \textit{``Centre National de la Recherche Scientifique''} is acknowledged.
This work has also been supported through the EUR grant NanoX ANR-17-EURE-0009 in the framework of the \textit{``Programme des Investissements d'Avenir''.}}