diff --git a/sfBSE.bib b/sfBSE.bib index e50c071..f06f3ed 100644 --- a/sfBSE.bib +++ b/sfBSE.bib @@ -1,13 +1,208 @@ %% This BibTeX bibliography file was created using BibDesk. %% http://bibdesk.sourceforge.net/ -%% Created for Pierre-Francois Loos at 2020-10-25 21:09:09 +0100 +%% Created for Pierre-Francois Loos at 2020-10-28 14:37:29 +0100 %% Saved with string encoding Unicode (UTF-8) +@article{Puschnig_2002, + Author = {Puschnig, Peter and Ambrosch-Draxl, Claudia}, + Date-Added = {2020-10-28 14:36:34 +0100}, + Date-Modified = {2020-10-28 14:36:34 +0100}, + Doi = {10.1103/PhysRevLett.89.056405}, + Issue = {5}, + Journal = {Phys. Rev. Lett.}, + Month = {Jul}, + Numpages = {4}, + Pages = {056405}, + Publisher = {American Physical Society}, + Title = {Suppression of Electron-Hole Correlations in 3D Polymer Materials}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.89.056405}, + Volume = {89}, + Year = {2002}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.89.056405}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.89.056405}} + +@article{Horst_1999, + Author = {van der Horst, J.-W. and Bobbert, P. A. and Michels, M. A. J. and Brocks, G. and Kelly, P. J.}, + Date-Added = {2020-10-28 14:34:58 +0100}, + Date-Modified = {2020-10-28 14:34:58 +0100}, + Doi = {10.1103/PhysRevLett.83.4413}, + Issue = {21}, + Journal = {Phys. Rev. Lett.}, + Month = {Nov}, + Numpages = {0}, + Pages = {4413--4416}, + Publisher = {American Physical Society}, + Title = {Ab Initio Calculation of the Electronic and Optical Excitations in Polythiophene: Effects of Intra- and Interchain Screening}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.83.4413}, + Volume = {83}, + Year = {1999}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.83.4413}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.83.4413}} + +@article{Rohlfing_1995, + Author = {Rohlfing, Michael and Kr{\"u}ger, Peter and Pollmann, Johannes}, + Date-Added = {2020-10-28 14:34:35 +0100}, + Date-Modified = {2020-10-28 14:34:35 +0100}, + Doi = {10.1103/PhysRevB.52.1905}, + Issue = {3}, + Journal = {Phys. Rev. B}, + Month = {Jul}, + Numpages = {0}, + Pages = {1905--1917}, + Publisher = {American Physical Society}, + 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}, + Url = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905}, + Volume = {52}, + Year = {1995}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.52.1905}} + +@article{Rohlfing_1998, + Author = {Rohlfing, Michael and Louie, Steven G.}, + Date-Added = {2020-10-28 14:34:35 +0100}, + Date-Modified = {2020-10-28 14:34:35 +0100}, + Doi = {10.1103/PhysRevLett.81.2312}, + Issue = {11}, + Journal = {Phys. Rev. Lett.}, + Month = {Sep}, + Numpages = {0}, + Pages = {2312--2315}, + Publisher = {American Physical Society}, + Title = {Electron-Hole Excitations in Semiconductors and Insulators}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.81.2312}, + Volume = {81}, + Year = {1998}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.81.2312}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.81.2312}} + +@article{Rohlfing_1999a, + Author = {Rohlfing, Michael and Louie, Steven G.}, + Date-Added = {2020-10-28 14:34:35 +0100}, + Date-Modified = {2020-10-28 14:34:35 +0100}, + Doi = {10.1103/PhysRevLett.82.1959}, + Issue = {9}, + Journal = {Phys. Rev. Lett.}, + Month = {Mar}, + Numpages = {0}, + Pages = {1959--1962}, + Publisher = {American Physical Society}, + Title = {Optical Excitations in Conjugated Polymers}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.82.1959}, + Volume = {82}, + Year = {1999}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.82.1959}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.82.1959}} + +@article{Rohlfing_1999b, + Author = {Rohlfing, Michael and Louie, Steven G.}, + Date-Added = {2020-10-28 14:34:35 +0100}, + Date-Modified = {2020-10-28 14:34:35 +0100}, + Doi = {10.1103/PhysRevLett.83.856}, + Issue = {4}, + Journal = {Phys. Rev. Lett.}, + Month = {Jul}, + Numpages = {0}, + Pages = {856--859}, + Publisher = {American Physical Society}, + Title = {Excitons and Optical Spectrum of the $\mathrm{Si}(111)\ensuremath{-}(2\ifmmode\times\else\texttimes\fi{}1)$ Surface}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.83.856}, + Volume = {83}, + Year = {1999}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.83.856}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.83.856}} + +@article{Rohlfing_2012, + Author = {Rohlfing, Michael}, + Date-Added = {2020-10-28 14:34:35 +0100}, + Date-Modified = {2020-10-28 14:34:35 +0100}, + Doi = {10.1103/PhysRevLett.108.087402}, + Issue = {8}, + Journal = {Phys. Rev. Lett.}, + Month = {Feb}, + Numpages = {5}, + Pages = {087402}, + Publisher = {American Physical Society}, + Title = {Redshift of Excitons in Carbon Nanotubes Caused by the Environment Polarizability}, + Url = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402}, + Volume = {108}, + Year = {2012}, + Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.108.087402}, + Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.108.087402}} + +@article{Li_2016, + Author = {Li, Zhendong and Liu, Wenjian}, + Date-Added = {2020-10-28 14:21:46 +0100}, + Date-Modified = {2020-10-28 14:21:52 +0100}, + Doi = {10.1021/acs.jctc.5b01158}, + Eprint = {https://doi.org/10.1021/acs.jctc.5b01158}, + Journal = {Journal of Chemical Theory and Computation}, + Note = {PMID: 26672389}, + Number = {1}, + Pages = {238-260}, + Title = {Critical Assessment of TD-DFT for Excited States of Open-Shell Systems: I. Doublet--Doublet Transitions}, + Url = {https://doi.org/10.1021/acs.jctc.5b01158}, + Volume = {12}, + Year = {2016}, + Bdsk-Url-1 = {https://doi.org/10.1021/acs.jctc.5b01158}} + +@article{Li_2010, + Author = {Li,Zhendong and Liu,Wenjian}, + Date-Added = {2020-10-28 14:19:49 +0100}, + Date-Modified = {2020-10-28 14:19:52 +0100}, + Doi = {10.1063/1.3463799}, + Eprint = {https://doi.org/10.1063/1.3463799}, + Journal = {The Journal of Chemical Physics}, + Number = {6}, + Pages = {064106}, + Title = {Spin-adapted open-shell random phase approximation and time-dependent density functional theory. I. Theory}, + Url = {https://doi.org/10.1063/1.3463799}, + Volume = {133}, + Year = {2010}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.3463799}} + +@article{Li_2011b, + Author = {Li,Zhendong and Liu,Wenjian}, + Date-Added = {2020-10-28 14:18:56 +0100}, + Date-Modified = {2020-10-28 14:19:00 +0100}, + Doi = {10.1063/1.3660688}, + Eprint = {https://doi.org/10.1063/1.3660688}, + Journal = {The Journal of Chemical Physics}, + Number = {19}, + Pages = {194106}, + Title = {Spin-adapted open-shell time-dependent density functional theory. III. An even better and simpler formulation}, + Url = {https://doi.org/10.1063/1.3660688}, + Volume = {135}, + Year = {2011}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.3660688}} + +@article{Li_2011a, + Author = {Li,Zhendong and Liu,Wenjian and Zhang,Yong and Suo,Bingbing}, + Date-Added = {2020-10-28 14:18:03 +0100}, + Date-Modified = {2020-10-28 14:18:28 +0100}, + Doi = {10.1063/1.3573374}, + Eprint = {https://doi.org/10.1063/1.3573374}, + Journal = {The Journal of Chemical Physics}, + Number = {13}, + Pages = {134101}, + Title = {Spin-adapted open-shell time-dependent density functional theory. II. Theory and pilot application}, + Url = {https://doi.org/10.1063/1.3573374}, + Volume = {134}, + Year = {2011}, + Bdsk-Url-1 = {https://doi.org/10.1063/1.3573374}} + +@article{Authier_2020, + Author = {J. Authier and P. F. Loos}, + Date-Added = {2020-10-28 11:21:07 +0100}, + Date-Modified = {2020-10-28 11:21:41 +0100}, + Journal = {J. Chem. Phys.}, + Title = {Dynamical Kernels for Optical Excitations}, + Year = {in press}} + @article{Casanova_2020, Author = {D. Casanova and A. I. Krylov}, Date-Added = {2020-10-25 13:00:27 +0100}, @@ -2293,24 +2488,6 @@ Year = {2007}, Bdsk-Url-1 = {https://doi.org/10.1002/9780470125922.ch2}} -@article{Rohlfing_1995, - Author = {Rohlfing, Michael and Kr{\"u}ger, Peter and Pollmann, Johannes}, - Date-Added = {2020-05-18 21:40:28 +0200}, - Date-Modified = {2020-05-18 21:40:28 +0200}, - Doi = {10.1103/PhysRevB.52.1905}, - Issue = {3}, - Journal = {Phys. Rev. B}, - Month = {Jul}, - Numpages = {0}, - Pages = {1905--1917}, - Publisher = {American Physical Society}, - 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}, - Url = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905}, - Volume = {52}, - Year = {1995}, - Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevB.52.1905}, - Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevB.52.1905}} - @article{Rohlfing_1999, Author = {Rohlfing, Michael and Louie, Steven G.}, Date-Added = {2020-05-18 21:40:28 +0200}, @@ -7294,20 +7471,6 @@ Year = {2006}, Bdsk-Url-1 = {https://doi.org/10.1080/00268970500416145}} -@article{Li_2011, - Author = {Xiangzhu Li and Josef Paldus}, - Date-Added = {2020-01-01 21:36:51 +0100}, - Date-Modified = {2020-01-01 21:36:52 +0100}, - Doi = {10.1063/1.3595513}, - Journal = {J. Chem. Phys.}, - Number = {21}, - Pages = {214118}, - Title = {Multi-Reference State-Universal Coupled-Cluster Approaches to Electronically Excited States}, - Url = {https://doi.org/10.1063/1.3595513}, - Volume = {134}, - Year = {2011}, - Bdsk-Url-1 = {https://doi.org/10.1063/1.3595513}} - @article{Li_2013, Author = {Li, Yan-Ni and Wang, Shengguang and Wang, Tao and Gao, Rui and Geng, Chun-Yu and Li, Yong-Wang and Wang, Jianguo and Jiao, Haijun}, Date-Added = {2020-01-01 21:36:51 +0100}, @@ -14408,22 +14571,6 @@ Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.80.4510}, Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.80.4510}} -@article{Rohlfing_1998, - Author = {Rohlfing, Michael and Louie, Steven G.}, - Doi = {10.1103/PhysRevLett.81.2312}, - Issue = {11}, - Journal = {Phys. Rev. Lett.}, - Month = {Sep}, - Numpages = {0}, - Pages = {2312--2315}, - Publisher = {American Physical Society}, - Title = {Electron-Hole Excitations in Semiconductors and Insulators}, - Url = {https://link.aps.org/doi/10.1103/PhysRevLett.81.2312}, - Volume = {81}, - Year = {1998}, - Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.81.2312}, - Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.81.2312}} - @article{vanderHorst_1999, Author = {van der Horst, J.-W. and Bobbert, P. A. and Michels, M. A. J. and Brocks, G. and Kelly, P. J.}, Doi = {10.1103/PhysRevLett.83.4413}, @@ -14440,22 +14587,6 @@ Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.83.4413}, Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.83.4413}} -@article{Pushchnig_2002, - Author = {Puschnig, Peter and Ambrosch-Draxl, Claudia}, - Doi = {10.1103/PhysRevLett.89.056405}, - Issue = {5}, - Journal = {Phys. Rev. Lett.}, - Month = {Jul}, - Numpages = {4}, - Pages = {056405}, - Publisher = {American Physical Society}, - Title = {Suppression of Electron-Hole Correlations in 3D Polymer Materials}, - Url = {https://link.aps.org/doi/10.1103/PhysRevLett.89.056405}, - Volume = {89}, - Year = {2002}, - Bdsk-Url-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.89.056405}, - Bdsk-Url-2 = {https://doi.org/10.1103/PhysRevLett.89.056405}} - @article{Tiago_2003, Author = {Tiago, Murilo L. and Northrup, John E. and Louie, Steven G.}, Date-Modified = {2020-02-05 20:45:49 +0100}, diff --git a/sfBSE.rty b/sfBSE.rty index 35374b7..4e50766 100644 --- a/sfBSE.rty +++ b/sfBSE.rty @@ -33,11 +33,10 @@ \newcommand{\Norb}{N_\text{orb}} \newcommand{\Nocc}{O} \newcommand{\Nvir}{V} -\newcommand{\IS}{\lambda} % operators \newcommand{\hH}{\Hat{H}} -\newcommand{\ha}{\Hat{a}} +\newcommand{\hS}{\Hat{S}} % methods \newcommand{\KS}{\text{KS}} diff --git a/sfBSE.tex b/sfBSE.tex index ef8563a..5deaa73 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -68,7 +68,7 @@ The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBo where $\eta$ is a positive infinitesimal. As readily seen in Eq.~\eqref{eq:G}, the Green's function can be evaluated at different levels of theory depending on the choice of orbitals and energies, $\MO{p_\sig}$ and $\e{p_\sig}{}$. For example, $G_{\KS}^{\sig}$ is the independent-particle Green's function built with KS orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$. -Within self-consistent schemes, these quantities can be replaced by quasiparticle energies and orbitals evaluated within the $GW$ approximation (see below). +Within self-consistent schemes, these quantities can be replaced by quasiparticle energies and orbitals evaluated within the $GW$ approximation (see below). \cite{Hedin_1965,Golze_2019} Based on the spin-up and spin-down components of $G$ defined in Eq.~\eqref{eq:G}, one can easily compute the non-interacting polarizability (which is a sum over spins) \begin{equation} @@ -88,7 +88,7 @@ Based on this latter ingredient, one can access the dynamically-screened Coulomb \end{equation} which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates. -Within the $GW$ formalism, the dynamical screening is computed at the random-phase approximation (RPA) level by considering only the manifold of the spin-conserved neutral excitations. +Within the $GW$ formalism, \cite{Hedin_1965,Onida_2002,Golze_2019} the dynamical screening is computed at the random-phase approximation (RPA) level by considering only the manifold of the spin-conserved neutral excitations. In the orbital basis, the spectral representation of $W$ is \begin{multline} \label{eq:W_spectral} @@ -246,7 +246,7 @@ The same comment applies to the dynamically-screened Coulomb potential $W$ enter \subsection{Level of self-consistency} %================================ This is where $GW$ schemes differ. -In its simplest perturbative (\ie, one-shot) version, known as {\GOWO}, a single iteration is performed, and the quasiparticle energies $\eGOWO{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation +In its simplest perturbative (\ie, one-shot) version, known as {\GOWO}, \cite{Strinati_1980,Hybertsen_1985a,Hybertsen_1986,Godby_1988,Linden_1988,Northrup_1991,Blase_1994,Rohlfing_1995,Shishkin_2007} a single iteration is performed, and the quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation \begin{equation} \label{eq:QP-eq} \omega = \e{p_\sig}{} + \Sig{p_\sig}{\xc}(\omega) - V_{p_\sig}^{\xc} @@ -257,17 +257,18 @@ where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and it \end{equation} Because, from a practical point of view, one is usually interested by the so-called quasiparticle solution (or peak), the quasiparticle equation \eqref{eq:QP-eq} is often linearized around $\omega = \e{p_\sig}^{\KS}$, yielding \begin{equation} - \eGOWO{p_\sig} + \eGW{p_\sig} = \e{p_\sig}^{\KS} + Z_{p_\sig} [\Sig{p_\sig}{\xc}(\e{p_\sig}^{\KS}) - V_{p_\sig}^{\xc} ] \end{equation} where \begin{equation} +\label{eq:Z_GW} Z_{p_\sig} = \qty[ 1 - \left. \pdv{\Sig{p_\sig}{\xc}(\omega)}{\omega} \right|_{\omega = \e{p_\sig}^{\KS}} ]^{-1} \end{equation} -is a renormalization factor which also represents the spectral weight of the quasiparticle solution. +is a renormalization factor (with $0 \le Z_{p_\sig} \le 1$) which also represents the spectral weight of the quasiparticle solution. In addition to the principal quasiparticle peak which, in a well-behaved case, contains most of the spectral weight, the frequency-dependent quasiparticle equation \eqref{eq:QP-eq} generates a finite number of satellite resonances with smaller weights. -Within the ``eigenvalue'' self-consistent $GW$ scheme (known as ev$GW$), several iterations are performed during which only the one-electron energies entering the definition of the Green's function [see Eq.~\eqref{eq:G}] are updated by the quasiparticle energies obtained at the previous iteration (the corresponding orbitals remain evaluated at the KS level). +Within the ``eigenvalue'' self-consistent $GW$ scheme (known as ev$GW$), \cite{Hybertsen_1986,Shishkin_2007,Blase_2011,Faber_2011,Rangel_2016,Kaplan_2016,Gui_2018} several iterations are performed during which only the one-electron energies entering the definition of the Green's function [see Eq.~\eqref{eq:G}] are updated by the quasiparticle energies obtained at the previous iteration (the corresponding orbitals remain evaluated at the KS level). Finally, within the quasiparticle self-consistent $GW$ (qs$GW$) scheme, both the one-electron energies and the orbitals are updated until convergence is reached. These are obtained via the diagonalization of an effective Fock matrix which includes explicitly a frequency-independent and hermitian self-energy defined as @@ -276,12 +277,17 @@ These are obtained via the diagonalization of an effective Fock matrix which inc \end{equation} %================================ -\subsection{The Bethe-Salpeter equation formalism} +\section{Unrestricted Bethe-Salpeter equation formalism} %================================ -Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \cite{Salpeter_1951,Strinati_1988} +Like its TD-DFT cousin, the BSE formalism \cite{Salpeter_1951,Strinati_1988,Albrecht_1998,Rohlfing_1998,Benedict_1998,vanderHorst_1999} deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \cite{Rohlfing_1999a,Horst_1999,Puschnig_2002,Tiago_2003,Boulanger_2014,Jacquemin_2015a,Bruneval_2015,Jacquemin_2015b,Hirose_2015,Jacquemin_2017a,Jacquemin_2017b,Rangel_2017,Krause_2017,Gui_2018} Using the BSE formalism, one can access the spin-conserved and spin-flip excitations. In a nutshell, BSE builds on top of a $GW$ calculation by adding up excitonic effects (\ie, the electron-hole binding energy) to the $GW$ fundamental gap which is itself a corrected version of the KS gap. -The purpose of the underlying $GW$ calculation is to provide quasiparticle energies and a dynamically-screened Coulomb potential that are used to build the BSE Hamiltonian from which the vertical excitations of the system are going to be extracted. +The purpose of the underlying $GW$ calculation is to provide quasiparticle energies and a dynamically-screened Coulomb potential that are used to build the BSE Hamiltonian from which the vertical excitations of the system are extracted. + +%================================ +\subsection{Static approximation} +%================================ + The Dyson equation that links the generalized four-point susceptibility $L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)$ and the BSE kernel $\Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)$ is \begin{multline} @@ -329,7 +335,7 @@ Within the static approximation which consists in neglecting the frequency depen \bY{m}{\BSE} \\ \end{pmatrix} \end{equation} -Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, the general expressions of the BSE matrix elements are +Defining the elements of the static screening as $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, the general expressions of the BSE matrix elements are \begin{subequations} \begin{align} \label{eq:LR_BSE-A} @@ -339,7 +345,7 @@ Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s \B{i_\sig a_\tau,j_\sigp b_\taup}{\BSE} & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau} \end{align} \end{subequations} -from which we obtain, at the BSE level, the following expressions for the spin-conserved and spin-flip excitations: +from which we obtain the following expressions for the spin-conserved and spin-flip BSE excitations: \begin{subequations} \begin{align} \label{eq:LR_BSE-Asc} @@ -360,26 +366,31 @@ At this stage, it is of particular interest to discuss the form of the spin-flip As readily seen from Eq.~\eqref{eq:LR_RPA-Asf}, at the RPA level, the spin-flip excitations are given by the difference of one-electron energies, hence missing out on key exchange and correlation effects. This is also the case at the TD-DFT level when one relies on (semi-)local functionals. This explains why most of spin-flip TD-DFT calculations are performed with hybrid functionals containing a substantial amount of Hartree-Fock exchange as only the exact exchange integral of the form $\ERI{i_\sig j_\sig}{b_\bsig a_\bsig}$ survive spin-symmetry requirements. -At the BSE level, these matrix elements are, of course, also present thanks to the contribution of $W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig}$ but it also includes correlation effects as evidenced in Eq.~\eqref{eq:W_spectral}. +At the BSE level, these matrix elements are, of course, also present thanks to the contribution of $W^{\stat}_{i_\sig j_\sig,b_\bsig a_\bsig}$ as evidenced in Eq.~\eqref{eq:W_spectral} but it also includes correlation effects. %================================ \subsection{Dynamical correction} %================================ -The dynamical correction to the static BSE kernel is defined in the Tamm-Dancoff approximation as \cite{Strinati_1988,Rohlfing_2000,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Loos_2020e} +In order to go beyond the ubiquitous static approximation of BSE \cite{Strinati_1988,Rohlfing_2000,Sottile_2003,Myohanen_2008,Ma_2009a,Ma_2009b,Romaniello_2009b,Sangalli_2011,Huix-Rotllant_2011,Sakkinen_2012,Zhang_2013,Rebolini_2016,Olevano_2019,Lettmann_2019} (which is somehow similar to the adiabatic approximation of TD-DFT \cite{Casida_2005,Huix-Rotllant_2011,Casida_2016,Maitra_2004,Cave_2004,Elliott_2011,Maitra_2012}), we have recently implemented, following Strinati's seminal work \cite{Strinati_1982,Strinati_1984,Strinati_1988} (see also the work of Romaniello \textit{et al.} \cite{Romaniello_2009b} and Sangalli \textit{et al.} \cite{Sangalli_2011}), a renormalized first-order perturbative correction in order to take into consideration the dynamical nature of the screened Coulomb potential $W$. \cite{Loos_2020e,Authier_2020} +This dynamical correction to the static BSE kernel (dubbed as dBSE in the following) does permit to recover additional relaxation effects coming from higher excitations. +Our implementation follows closely the work of Rohlfing and co-workers \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} in which they computed the dynamical correction in the TDA and plasmon-pole approximation. +However, our scheme goes beyond the plasmon-pole approximation as the spectral representation of the dynamically-screened Coulomb potential is computed exactly at the RPA level consistently with the underlying $GW$ calculation: \begin{multline} \widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} + \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m} \\ - \times \qty[ \frac{1}{\omega - (\e{s_\sigp}{} - \e{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} + \frac{1}{\omega - (\e{r_\sigp}{} - \e{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} ] + \times \Bigg[ \frac{1}{\omega - (\e{s_\sigp}{} - \e{q_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} + \\ + + \frac{1}{\omega - (\e{r_\sigp}{} - \e{p_\sig}{}) - \Om{m}{\spc,\RPA} + i \eta} \Bigg] \end{multline} - -\begin{equation} +The dBSE non-linear response problem is +\begin{multline} \label{eq:LR-dyn} \begin{pmatrix} - \bA{}{\dBSE}(\omega) & \bB{}{\dBSE}(\omega) + \bA{}{\dBSE}(\Om{m}{\dBSE}) & \bB{}{\dBSE}(\Om{m}{\dBSE}) \\ - -\bB{}{\dBSE}(-\omega) & -\bA{}{\dBSE}(-\omega) + -\bB{}{\dBSE}(-\Om{m}{\dBSE}) & -\bA{}{\dBSE}(-\Om{m}{\dBSE}) \\ \end{pmatrix} \cdot @@ -387,14 +398,15 @@ The dynamical correction to the static BSE kernel is defined in the Tamm-Dancoff \bX{m}{\dBSE} \\ \bY{m}{\dBSE} \\ \end{pmatrix} + \\ = \Om{m}{\dBSE} \begin{pmatrix} \bX{m}{\dBSE} \\ \bY{m}{\dBSE} \\ \end{pmatrix} -\end{equation} - +\end{multline} +where the dynamical matrices are generally defined as \begin{subequations} \begin{align} \label{eq:LR_dBSE-A} @@ -404,8 +416,8 @@ The dynamical correction to the static BSE kernel is defined in the Tamm-Dancoff \B{i_\sig a_\tau,j_\sigp b_\taup}{\dBSE}(\omega) & = \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega) \end{align} \end{subequations} - - +from which one can easily obtained the matrix elements for the spin-conserved and spin-flip manifolds similarly to Eqs.~\eqref{eq:LR_BSE-Asc}, \eqref{eq:LR_BSE-Bsc}, \eqref{eq:LR_BSE-Asf}, and \eqref{eq:LR_BSE-Bsf}. +Following Rayleigh-Schr\"odinger perturbation theory, we then decompose the non-linear eigenproblem \eqref{eq:LR-dyn} as a zeroth-order static (\ie, linear) reference and a first-order dynamic (\ie, non-linear) perturbation such that \begin{multline} \label{eq:LR-PT} \begin{pmatrix} @@ -446,94 +458,37 @@ and \B{i_\sig a_\tau,j_\sigp b_\taup}{(1)}(\omega) & = - \delta_{\sig \sigp} \widetilde{W}_{i_\sig b_\taup,j_\sigp a_\tau}(\omega) + \delta_{\sig \sigp} W^{\stat}_{i_\sig b_\taup,j_\sigp a_\tau} \end{align} \end{subequations} - -\begin{subequations} -\begin{gather} - \Om{m}{\dBSE} = \Om{m}{(0)} + \Om{m}{(1)} + \ldots - \\ - \begin{pmatrix} - \bX{m}{\dBSE} \\ - \bY{m}{\dBSE} \\ - \end{pmatrix} - = - \begin{pmatrix} - \bX{m}{(0)} \\ - \bY{m}{(0)} \\ - \end{pmatrix} - + - \begin{pmatrix} - \bX{m}{(1)} \\ - \bY{m}{(1)} \\ - \end{pmatrix} - + \ldots -\end{gather} -\end{subequations} - +The dBSE excitation energies are then obtained via \begin{equation} -\label{eq:LR-BSE-stat} - \begin{pmatrix} - \bA{}{(0)} & \bB{}{(0)} \\ - -\bB{}{(0)} & -\bA{}{(0)} \\ - \end{pmatrix} - \cdot - \begin{pmatrix} - \bX{S}{(0)} \\ - \bY{S}{(0)} \\ - \end{pmatrix} - = - \Om{m}{(0)} - \begin{pmatrix} - \bX{m}{(0)} \\ - \bY{m}{(0)} \\ - \end{pmatrix} + \Om{m}{\dBSE} = \Om{m}{\BSE} + \zeta_{m} \Om{m}{(1)} \end{equation} - -\begin{equation} -\label{eq:Om1} - \Om{m}{(1)} = - \T{\begin{pmatrix} - \bX{m}{(0)} \\ - \bY{m}{(0)} \\ - \end{pmatrix}} - \cdot - \begin{pmatrix} - \bA{}{(1)}(\Om{m}{(0)}) & \bB{}{(1)}(\Om{m}{(0)}) \\ - -\bB{}{(1)}(-\Om{m}{(0)}) & -\bA{}{(1)}(-\Om{m}{(0)}) \\ - \end{pmatrix} - \cdot - \begin{pmatrix} - \bX{m}{(0)} \\ - \bY{m}{(0)} \\ - \end{pmatrix} -\end{equation} - +where $\Om{m}{\BSE} \equiv \Om{m}{(0)}$ are the static (zeroth-order) BSE excitation energies obtained by solving Eq.~\eqref{eq:LR-BSE}, and \begin{equation} \label{eq:Om1-TDA} - \Om{S}{(1)} = \T{(\bX{m}{(0)})} \cdot \bA{}{(1)}(\Om{m}{(0)}) \cdot \bX{m}{(0)} + \Om{m}{(1)} = \T{(\bX{m}{\BSE})} \cdot \bA{}{(1)}(\Om{m}{\BSE}) \cdot \bX{m}{\BSE} \end{equation} - +are first-order corrections (with $\bX{m}{\BSE} \equiv \bX{m}{(0)}$) obtained within the dynamical TDA (dTDA) with the renormalization factor \begin{equation} \label{eq:Z} - Z_{m} = \qty[ 1 - \T{(\bX{m}{(0)})} \cdot \left. \pdv{\bA{}{(1)}(\Om{m}{})}{\Om{S}{}} \right|_{\Om{m}{} = \Om{m}{(0)}} \cdot \bX{m}{(0)} ]^{-1} -\end{equation} - -\begin{equation} - \Om{m}{\dBSE} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)} + \zeta_{m} = \qty[ 1 - \T{(\bX{m}{\BSE})} \cdot \left. \pdv{\bA{}{(1)}(\omega)}{\omega} \right|_{\omega = \Om{m}{\BSE}} \cdot \bX{m}{\BSE} ]^{-1} \end{equation} +which, unlike the $GW$ case [see Eq.~\eqref{eq:Z_GW}], is not restricted to be between $0$ and $1$. +In most cases, the value of $\zeta_{m}$ is close to unity which indicates that the perturbative expansion behaves nicely. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Oscillator strengths} \label{sec:os} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -For the spin-conserved transition, the $x$ component of the transition dipole moment is +For the spin-conserved transitions, the $x$ component of the transition dipole moment is \begin{equation} \mu_{x,m}^{\spc} = \sum_{ia\sig} (i_\sig|x|a_\sig)(\bX{m}{\spc}+\bY{m}{\spc})_{i_\sig a_\sig} \end{equation} -with +where \begin{equation} (p_\sig|x|q_\sigp) = \int \MO{p_\sig}(\br) \, x \, \MO{q_\sigp}(\br) d\br \end{equation} -and the total oscillator strength is given by +are one-electron integrals in the orbital basis. +The total oscillator strength is given by \begin{equation} f_{m}^{\spc} = \frac{2}{3} \Om{m}{\spc} \qty[ \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 + \qty(\mu_{x,m}^{\spc})^2 ] \end{equation} @@ -543,24 +498,29 @@ For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix e \subsection{Spin contamination} \label{sec:spin} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +One of the key issues of linear response formalism based on unrestricted references is spin contamination. +As nicely explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference configuration for which, for example, $\expval{\hS^2} > 2$ for a high-spin triplets, and ii) spin-contamination of the excited states due to spin incompleteness of the configuration interaction expansion. +The latter issue is an important source of spin contamination in the present context as BSE is limited to single excitations with respect to the reference configuration. +Specific schemes have been developed to palliate these shortcomings and we refer the interested reader to Ref.~\onlinecite{Casanova_2020} for a detailed discussion on this matter. +In order to monitor closely how contaminated are these states, we compute \begin{equation} - \expval{S^2}_m = \expval{S^2}_0 + \Delta \expval{S^2}_m + \expval{\hS^2}_m = \expval{\hS^2}_0 + \Delta \expval{\hS^2}_m \end{equation} - +where \begin{equation} - \expval{S^2}_{0} + \expval{\hS^2}_{0} = \frac{n_{\up} - n_{\dw}}{2} \qty( \frac{n_{\up} - n_{\dw}}{2} + 1 ) + n_{\dw} - \sum_p (p_{\up}|p_{\dw})^2 \end{equation} -where +is the expectation value of $\hS^2$ for the reference configuration, the first term correspoding to the exact value of $\expval{\hS^2}$, and \begin{equation} +\label{eq:OV} (p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br \end{equation} -is the overlap between spin-up and spin-down orbitals. +are overlap integrals between spin-up and spin-down orbitals. -The explicit expressions of $\Delta \expval{S^2}_m^{\spc}$ and $\Delta \expval{S^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2010} for spin-conserved and spin-flip excitations, and are functions of the $\bX{m}{}$ and $\bY{m}{}$ vectors and the orbital overlaps. -As explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin contamination: i) spin contamination of the reference, and ii) spin-contamination of the excited states due to the spin incompleteness. +For a given single excitation $m$, the explicit expressions of $\Delta \expval{\hS^2}_m^{\spc}$ and $\Delta \expval{\hS^2}_m^{\spf}$ can be found in the Appendix of Ref.~\onlinecite{Li_2011a} for spin-conserved and spin-flip excitations, and are functions of the $\bX{m}{}$ and $\bY{m}{}$ vectors and the orbital overlaps defined in Eq.~\eqref{eq:OV}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Computational details}