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sfBSE.bib
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@ -1,13 +1,208 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% 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) %% 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, @article{Casanova_2020,
Author = {D. Casanova and A. I. Krylov}, Author = {D. Casanova and A. I. Krylov},
Date-Added = {2020-10-25 13:00:27 +0100}, Date-Added = {2020-10-25 13:00:27 +0100},
@ -2293,24 +2488,6 @@
Year = {2007}, Year = {2007},
Bdsk-Url-1 = {https://doi.org/10.1002/9780470125922.ch2}} 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, @article{Rohlfing_1999,
Author = {Rohlfing, Michael and Louie, Steven G.}, Author = {Rohlfing, Michael and Louie, Steven G.},
Date-Added = {2020-05-18 21:40:28 +0200}, Date-Added = {2020-05-18 21:40:28 +0200},
@ -7294,20 +7471,6 @@
Year = {2006}, Year = {2006},
Bdsk-Url-1 = {https://doi.org/10.1080/00268970500416145}} 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, @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}, 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}, 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-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.80.4510},
Bdsk-Url-2 = {https://doi.org/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, @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.}, 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}, 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-1 = {https://link.aps.org/doi/10.1103/PhysRevLett.83.4413},
Bdsk-Url-2 = {https://doi.org/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, @article{Tiago_2003,
Author = {Tiago, Murilo L. and Northrup, John E. and Louie, Steven G.}, Author = {Tiago, Murilo L. and Northrup, John E. and Louie, Steven G.},
Date-Modified = {2020-02-05 20:45:49 +0100}, Date-Modified = {2020-02-05 20:45:49 +0100},

3
sfBSE.rty
View File

@ -33,11 +33,10 @@
\newcommand{\Norb}{N_\text{orb}} \newcommand{\Norb}{N_\text{orb}}
\newcommand{\Nocc}{O} \newcommand{\Nocc}{O}
\newcommand{\Nvir}{V} \newcommand{\Nvir}{V}
\newcommand{\IS}{\lambda}
% operators % operators
\newcommand{\hH}{\Hat{H}} \newcommand{\hH}{\Hat{H}}
\newcommand{\ha}{\Hat{a}} \newcommand{\hS}{\Hat{S}}
% methods % methods
\newcommand{\KS}{\text{KS}} \newcommand{\KS}{\text{KS}}

162
sfBSE.tex
View File

@ -68,7 +68,7 @@ The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBo
where $\eta$ is a positive infinitesimal. 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 $\e{p_\sig}{}$. 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 $\e{p_\sig}^{\KS}$. 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) 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} \begin{equation}
@ -88,7 +88,7 @@ Based on this latter ingredient, one can access the dynamically-screened Coulomb
\end{equation} \end{equation}
which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates. 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 In the orbital basis, the spectral representation of $W$ is
\begin{multline} \begin{multline}
\label{eq:W_spectral} \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} \subsection{Level of self-consistency}
%================================ %================================
This is where $GW$ schemes differ. 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} \begin{equation}
\label{eq:QP-eq} \label{eq:QP-eq}
\omega = \e{p_\sig}{} + \Sig{p_\sig}{\xc}(\omega) - V_{p_\sig}^{\xc} \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} \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 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} \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} ] = \e{p_\sig}^{\KS} + Z_{p_\sig} [\Sig{p_\sig}{\xc}(\e{p_\sig}^{\KS}) - V_{p_\sig}^{\xc} ]
\end{equation} \end{equation}
where where
\begin{equation} \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} Z_{p_\sig} = \qty[ 1 - \left. \pdv{\Sig{p_\sig}{\xc}(\omega)}{\omega} \right|_{\omega = \e{p_\sig}^{\KS}} ]^{-1}
\end{equation} \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. 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. 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 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} \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. 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. 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 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} \begin{multline}
@ -329,7 +335,7 @@ Within the static approximation which consists in neglecting the frequency depen
\bY{m}{\BSE} \\ \bY{m}{\BSE} \\
\end{pmatrix} \end{pmatrix}
\end{equation} \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{subequations}
\begin{align} \begin{align}
\label{eq:LR_BSE-A} \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} \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{align}
\end{subequations} \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{subequations}
\begin{align} \begin{align}
\label{eq:LR_BSE-Asc} \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. 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 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. 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} \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} \begin{multline}
\widetilde{W}_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} \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} + \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} \end{multline}
The dBSE non-linear response problem is
\begin{equation} \begin{multline}
\label{eq:LR-dyn} \label{eq:LR-dyn}
\begin{pmatrix} \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} \end{pmatrix}
\cdot \cdot
@ -387,14 +398,15 @@ The dynamical correction to the static BSE kernel is defined in the Tamm-Dancoff
\bX{m}{\dBSE} \\ \bX{m}{\dBSE} \\
\bY{m}{\dBSE} \\ \bY{m}{\dBSE} \\
\end{pmatrix} \end{pmatrix}
\\
= =
\Om{m}{\dBSE} \Om{m}{\dBSE}
\begin{pmatrix} \begin{pmatrix}
\bX{m}{\dBSE} \\ \bX{m}{\dBSE} \\
\bY{m}{\dBSE} \\ \bY{m}{\dBSE} \\
\end{pmatrix} \end{pmatrix}
\end{equation} \end{multline}
where the dynamical matrices are generally defined as
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LR_dBSE-A} \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) \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{align}
\end{subequations} \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} \begin{multline}
\label{eq:LR-PT} \label{eq:LR-PT}
\begin{pmatrix} \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} \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{align}
\end{subequations} \end{subequations}
The dBSE excitation energies are then obtained via
\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}
\begin{equation} \begin{equation}
\label{eq:LR-BSE-stat} \Om{m}{\dBSE} = \Om{m}{\BSE} + \zeta_{m} \Om{m}{(1)}
\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}
\end{equation} \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}
\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}
\begin{equation} \begin{equation}
\label{eq:Om1-TDA} \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} \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} \begin{equation}
\label{eq:Z} \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} \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}
\begin{equation}
\Om{m}{\dBSE} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}
\end{equation} \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} \subsection{Oscillator strengths}
\label{sec:os} \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} \begin{equation}
\mu_{x,m}^{\spc} = \sum_{ia\sig} (i_\sig|x|a_\sig)(\bX{m}{\spc}+\bY{m}{\spc})_{i_\sig a_\sig} \mu_{x,m}^{\spc} = \sum_{ia\sig} (i_\sig|x|a_\sig)(\bX{m}{\spc}+\bY{m}{\spc})_{i_\sig a_\sig}
\end{equation} \end{equation}
with where
\begin{equation} \begin{equation}
(p_\sig|x|q_\sigp) = \int \MO{p_\sig}(\br) \, x \, \MO{q_\sigp}(\br) d\br (p_\sig|x|q_\sigp) = \int \MO{p_\sig}(\br) \, x \, \MO{q_\sigp}(\br) d\br
\end{equation} \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} \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 ] 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} \end{equation}
@ -543,24 +498,29 @@ For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix e
\subsection{Spin contamination} \subsection{Spin contamination}
\label{sec:spin} \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} \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}
\begin{equation}
\expval{S^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} \end{equation}
where where
\begin{equation} \begin{equation}
\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}
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 (p_\sig|q_\sigp) = \int \MO{p_\sig}(\br) \MO{q_\sigp}(\br) d\br
\end{equation} \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. 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}.
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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details} \section{Computational details}