diff --git a/BSEdyn.bib b/BSEdyn.bib index d7b9829..b503245 100644 --- a/BSEdyn.bib +++ b/BSEdyn.bib @@ -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}, diff --git a/BSEdyn.tex b/BSEdyn.tex index ef5fde3..37c7f31 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -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''.}}