2020-10-20 21:58:35 +02:00
\documentclass [aip,jcp,reprint,noshowkeys,superscriptaddress] { revtex4-1}
\usepackage { graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,wrapfig,txfonts}
\usepackage [version=4] { mhchem}
\usepackage [utf8] { inputenc}
\usepackage [T1] { fontenc}
\usepackage { txfonts}
\usepackage [
colorlinks=true,
citecolor=blue,
breaklinks=true
]{ hyperref}
\urlstyle { same}
\begin { document}
\title { Spin-Conserved and Spin-Flip Optical Excitations From the Bethe-Salpeter Equation Formalism}
\author { Enzo \surname { Monino} }
\affiliation { \LCPQ }
\author { Pierre-Fran\c { c} ois \surname { Loos} }
\email { loos@irsamc.ups-tlse.fr}
\affiliation { \LCPQ }
\begin { abstract}
\alert { Here comes the abstract.}
%\bigskip
%\begin{center}
% \boxed{\includegraphics[width=0.5\linewidth]{TOC}}
%\end{center}
%\bigskip
\end { abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Introduction}
\label { sec:intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\alert { Here comes the introduction.}
Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
2020-10-25 21:29:40 +01:00
In the following, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.
2020-10-20 21:58:35 +02:00
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Unrestricted $ GW $ formalism}
\label { sec:UGW}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2020-10-21 22:58:11 +02:00
Let us consider an electronic system consisting of $ n = n _ \up + n _ \dw $ electrons (where $ n _ \up $ and $ n _ \dw $ are the number of spin-up and spin-down electrons respectively) and $ N $ one-electron basis functions.
The number of spin-up and spin-down occupied orbitals are $ O _ \up = n _ \up $ and $ O _ \dw = n _ \dw $ , respectively, and there is $ V _ \up = N - O _ \up $ and $ V _ \dw = N - O _ \dw $ spin-up and spin-down virtual (\ie , unoccupied) orbitals.
The number of spin-conserved single excitations is then $ S ^ \spc = S _ { \up \up } ^ \spc + S _ { \dw \dw } ^ \spc = O _ \up V _ \up + O _ \dw V _ \dw $ , while the number of spin-flip excitations is $ S ^ \spf = S _ { \up \dw } ^ \spf + S _ { \dw \up } ^ \spf = O _ \up V _ \dw + O _ \dw V _ \up $ .
2020-10-23 15:13:15 +02:00
Let us denote as $ \MO { p _ \sig } $ the $ p $ th orbital of spin $ \sig $ (where $ \sig = $ $ \up $ or $ \dw $ ) and $ \e { p _ \sig } { } $ its one-electron energy.
In the present context these orbitals can originate from a HF or KS calculation.
2020-10-21 22:58:11 +02:00
In the following, $ i $ and $ j $ are occupied orbitals, $ a $ and $ b $ are unoccupied orbitals, $ p $ , $ q $ , $ r $ , and $ s $ indicate arbitrary orbitals, and $ m $ labels single excitations.
2020-10-22 12:40:48 +02:00
It is important to understand that, in a spin-conserved excitation the hole orbital $ \MO { i _ \sig } $ and particle orbital $ \MO { a _ \sig } $ have the same spin $ \sig $ .
2020-10-23 15:13:15 +02:00
In a spin-flip excitation, the hole and particle states, $ \MO { i _ \sig } $ and $ \MO { a _ \bsig } $ , have opposite spins, $ \sig $ and $ \bsig $ .
2020-10-21 22:58:11 +02:00
%================================
2020-10-20 21:58:35 +02:00
\subsection { The dynamical screening}
2020-10-21 22:58:11 +02:00
%================================
2020-10-25 21:29:40 +01:00
The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite { ReiningBook}
The spin-$ \sig $ component of the one-body Green's function reads \cite { ReiningBook,Bruneval_ 2016a}
2020-10-23 15:13:15 +02:00
\begin { equation}
G^ { \sig } (\br _ 1,\br _ 2;\omega )
= \sum _ i \frac { \MO { i_ \sig } (\br _ 1) \MO { i_ \sig } (\br _ 2)} { \omega - \e { i_ \sig } { } - i\eta }
+ \sum _ a \frac { \MO { a_ \sig } (\br _ 1) \MO { a_ \sig } (\br _ 2)} { \omega - \e { a_ \sig } { } + i\eta }
\end { equation}
2020-10-24 22:09:28 +02:00
where $ \eta $ is a positive infinitesimal.
2020-10-25 21:29:40 +01:00
Based on the spin-up and spin-down components of $ G $ , one can easily compute the non-interacting polarizability (which is a sum over spins)
2020-10-23 15:13:15 +02:00
\begin { equation}
\chi _ 0(\br _ 1,\br _ 2;\omega ) = - \frac { i} { 2\pi } \sum _ \sig \int G^ { \sig } (\br _ 1,\br _ 2;\omega +\omega ') G^ { \sig } (\br _ 1,\br _ 2;\omega ') d\omega '
\end { equation}
2020-10-24 22:09:28 +02:00
and subsequently the dielectric function
2020-10-23 15:13:15 +02:00
\begin { equation}
\epsilon (\br _ 1,\br -2;\omega ) = \delta (\br _ 1 - \br _ 2) - \int \frac { \chi _ 0(\br _ 1,\br _ 3;\omega ) } { \abs { \br _ 2 - \br _ 3} } d\br _ 3
\end { equation}
2020-10-25 21:29:40 +01:00
where $ \delta ( \br _ 1 - \br _ 2 ) $ is the Dirac function.
2020-10-23 15:13:15 +02:00
Based on this latter ingredient, one can access the dynamically-screened Coulomb potential
\begin { equation}
W(\br _ 1,\br _ 2;\omega ) = \int \frac { \epsilon ^ { -1} (\br _ 1,\br _ 3;\omega ) } { \abs { \br _ 2 - \br _ 3} } d\br _ 3
\end { equation}
2020-10-24 22:09:28 +02:00
which is spin independent as the bare Coulomb interaction $ \abs { \br _ 1 - \br _ 2 } ^ { - 1 } $ does not depend on spin coordinates.
2020-10-25 21:29:40 +01:00
Within the $ GW $ formalism, the is computed at the RPA level by considering only the manifold of the spin-conserved neutral excitation.
In the orbital basis, the spectral representation of $ W $ reads
2020-10-20 23:14:57 +02:00
\begin { multline}
2020-10-24 22:09:28 +02:00
\label { eq:W}
2020-10-22 12:40:48 +02:00
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}
2020-10-20 23:14:57 +02:00
\\
2020-10-21 22:58:11 +02:00
\times \qty [ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
2020-10-20 23:14:57 +02:00
\end { multline}
2020-10-24 22:09:28 +02:00
where the bare two-electron integrals are \cite { Gill_ 1994}
2020-10-20 23:14:57 +02:00
\begin { equation}
2020-10-24 22:09:28 +02:00
\label { eq:sERI}
2020-10-23 15:13:15 +02:00
\ERI { p_ \sig q_ \tau } { r_ \sigp s_ \taup } = \int \frac { \MO { p_ \sig } (\br _ 1) \MO { q_ \tau } (\br _ 1) \MO { r_ \sigp } (\br _ 2) \MO { s_ \taup } (\br _ 2)} { \abs { \br _ 1 - \br _ 2} } d\br _ 1 d\br _ 2
2020-10-20 23:14:57 +02:00
\end { equation}
2020-10-24 22:09:28 +02:00
and the screened two-electron integrals (or spectral weights) are explicitly given by
2020-10-20 23:14:57 +02:00
\begin { equation}
2020-10-22 12:40:48 +02:00
\ERI { p_ \sig q_ \sig } { m} = \sum _ { ia\sigp } \ERI { p_ \sig q_ \sig } { r_ \sigp s_ \sigp } (\bX { m} { \spc ,\RPA } +\bY { m} { \spc ,\RPA } )_ { i_ \sigp a_ \sigp }
2020-10-20 23:14:57 +02:00
\end { equation}
2020-10-25 21:29:40 +01:00
In Eqs.~\eqref { eq:W} and \eqref { eq:sERI} , the RPA spin-conserved neutral excitations $ \Om { m } { \spc , \RPA } $ and their corresponding eigenvectors $ ( \bX { m } { \spc , \RPA } + \bY { m } { \spc , \RPA } ) $ are obtained by solving a linear response system of the form
2020-10-20 23:14:57 +02:00
\begin { equation}
\label { eq:LR-RPA}
\begin { pmatrix}
2020-10-25 21:29:40 +01:00
\bA { } { } & \bB { } { } \\
-\bB { } { } & -\bA { } { } \\
2020-10-20 23:14:57 +02:00
\end { pmatrix}
\cdot
\begin { pmatrix}
2020-10-25 21:29:40 +01:00
\bX { m} { } \\
\bY { m} { } \\
2020-10-20 23:14:57 +02:00
\end { pmatrix}
=
2020-10-25 21:29:40 +01:00
\Om { m} { }
2020-10-20 23:14:57 +02:00
\begin { pmatrix}
2020-10-25 21:29:40 +01:00
\bX { m} { } \\
\bY { m} { } \\
2020-10-24 22:09:28 +02:00
\end { pmatrix}
2020-10-20 23:14:57 +02:00
\end { equation}
2020-10-25 21:29:40 +01:00
where the expressions of the matrix elements of $ \bA { } { } $ and $ \bB { } { } $ are specific of the method and of the spin manifold.
The spin structure of these matrices, though, is general
2020-10-21 22:58:11 +02:00
\begin { align}
\label { eq:LR-RPA-AB}
2020-10-22 12:40:48 +02:00
\bA { } { \spc } & = \begin { pmatrix}
2020-10-24 14:19:04 +02:00
\bA { } { \upup ,\upup } & \bA { } { \upup ,\dwdw } \\
\bA { } { \dwdw ,\upup } & \bA { } { \dwdw ,\dwdw } \\
2020-10-21 22:58:11 +02:00
\end { pmatrix}
&
2020-10-22 12:40:48 +02:00
\bB { } { \spc } & = \begin { pmatrix}
2020-10-24 14:19:04 +02:00
\bB { } { \upup ,\upup } & \bB { } { \upup ,\dwdw } \\
\bB { } { \dwdw ,\upup } & \bB { } { \dwdw ,\dwdw } \\
2020-10-21 22:58:11 +02:00
\end { pmatrix}
2020-10-22 12:40:48 +02:00
\\
2020-10-21 22:58:11 +02:00
\label { eq:LR-RPA-AB}
2020-10-22 12:40:48 +02:00
\bA { } { \spf } & = \begin { pmatrix}
2020-10-24 14:19:04 +02:00
\bA { } { \updw ,\updw } & \bO \\
\bO & \bA { } { \dwup ,\dwup } \\
2020-10-21 22:58:11 +02:00
\end { pmatrix}
&
2020-10-22 12:40:48 +02:00
\bB { } { \spf } & = \begin { pmatrix}
2020-10-24 14:19:04 +02:00
\bO & \bB { } { \updw ,\dwup } \\
\bB { } { \dwup ,\updw } & \bO \\
2020-10-21 22:58:11 +02:00
\end { pmatrix}
\end { align}
2020-10-25 21:29:40 +01:00
In the absence of instabilities, the linear eigenvalue problem \eqref { eq:LR-RPA} has particle-hole symmetry which means that the eigenvalues are obtained by pairs $ \pm \Om { m } { } $ .
In such a case, $ ( \bA { } { } - \bB { } { } ) ^ { 1 / 2 } $ is positive definite, and Eq.~\eqref { eq:LR-RPA} can be recast as a Hermitian problem of half the dimension
\begin { equation}
\label { eq:small-LR}
(\bA { } { } - \bB { } { } )^ { 1/2} \cdot (\bA { } { } + \bB { } { } ) \cdot (\bA { } { } - \bB { } { } )^ { 1/2} \cdot \bZ { } { } = \bOm { 2} \cdot \bZ { } { }
\end { equation}
where the excitation amplitudes are
\begin { equation}
\bX { } { } + \bY { } { } = \bOm { -1/2} \cdot (\bA { } { } - \bB { } { } )^ { 1/2} \cdot \bZ { } { }
\end { equation}
Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $ \bA { } { } $ and $ - \bA { } { } $ , are neglected, which consist in setting $ \bB { } { } = \bO $ .
In such a case, Eq.~\eqref { eq:LR-RPA} reduces to $ \bA { } { } \cdot \bX { m } { } = \Om { m } { } \bX { m } { } $ .
2020-10-24 22:09:28 +02:00
At the RPA level, the matrix elements of $ \bA { } { } $ and $ \bB { } { } $ are
2020-10-20 23:14:57 +02:00
\begin { subequations}
2020-10-22 12:40:48 +02:00
\begin { align}
\label { eq:LR_ RPA-A}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } & = \delta _ { ij} \delta _ { ab} \delta _ { \sig \sigp } \delta _ { \tau \taup } (\e { a_ \tau } - \e { i_ \sig } ) + \ERI { i_ \sig a_ \tau } { b_ \sigp j_ \taup }
\\
\label { eq:LR_ RPA-B}
\B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } & = \ERI { i_ \sig a_ \tau } { j_ \sigp b_ \taup }
\end { align}
\end { subequations}
2020-10-24 22:09:28 +02:00
from which we obtain the following expressions
2020-10-22 12:40:48 +02:00
\begin { subequations}
2020-10-20 23:14:57 +02:00
\begin { align}
2020-10-21 22:58:11 +02:00
\label { eq:LR_ RPA-Asc}
2020-10-22 12:40:48 +02:00
\A { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } & = \delta _ { ij} \delta _ { ab} \delta _ { \sig \sigp } (\e { a_ \sig } - \e { i_ \sig } ) + \ERI { i_ \sig a_ \sig } { b_ \sigp j_ \sigp }
2020-10-20 23:14:57 +02:00
\\
2020-10-21 22:58:11 +02:00
\label { eq:LR_ RPA-Bsc}
2020-10-22 12:40:48 +02:00
\B { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } & = \ERI { i_ \sig a_ \sig } { j_ \sigp b_ \sigp }
2020-10-20 23:14:57 +02:00
\end { align}
\end { subequations}
2020-10-22 12:40:48 +02:00
for the spin-conserved excitations and
\begin { subequations}
\begin { align}
\label { eq:LR_ RPA-Asf}
\A { i_ \sig a_ \bsig ,j_ \sig b_ \bsig } { \spf ,\RPA } & = \delta _ { ij} \delta _ { ab} (\e { a_ \bsig } - \e { i_ \sig } )
\\
\label { eq:LR_ RPA-Bsf}
\B { i_ \sig a_ \bsig ,j_ \bsig b_ \sig } { \spf ,\RPA } & = 0
\end { align}
\end { subequations}
for the spin-flip excitations.
2020-10-21 22:58:11 +02:00
%================================
2020-10-20 21:58:35 +02:00
\subsection { The $ GW $ self-energy}
2020-10-21 22:58:11 +02:00
%================================
2020-10-25 13:30:30 +01:00
Within the acclaimed $ GW $ approximation, \cite { Hedin_ 1965,Golze_ 2019} the exchange-correlation (xc) part of the self-energy
2020-10-23 15:13:15 +02:00
\begin { equation}
2020-10-24 14:19:04 +02:00
\begin { split}
\Sig { } ^ { \text { xc} ,\sig } (\br _ 1,\br _ 2;\omega )
& = \Sig { } ^ { \text { x} ,\sig } (\br _ 1,\br _ 2) + \Sig { } ^ { \text { c} ,\sig } (\br _ 1,\br _ 2;\omega )
\\
& = \frac { i} { 2\pi } \int G^ { \sig } (\br _ 1,\br _ 2;\omega +\omega ') W(\br _ 1,\br _ 2;\omega ') e^ { i \eta \omega '} d\omega '
\end { split}
2020-10-23 15:13:15 +02:00
\end { equation}
2020-10-25 21:29:40 +01:00
is, like the one-body Green's function, spin-diagonal, and its spectral representation reads
2020-10-24 14:19:04 +02:00
\begin { gather}
\SigX { p_ \sig q_ \sig }
2020-10-23 15:13:15 +02:00
= - \frac { 1} { 2} \sum _ { i\sigp } \ERI { p_ \sig i_ \sigp } { i_ \sigp q_ \sig }
2020-10-24 14:19:04 +02:00
\\
2020-10-21 22:58:11 +02:00
\begin { split}
2020-10-22 12:40:48 +02:00
\SigC { p_ \sig q_ \sig } (\omega )
2020-10-23 15:13:15 +02:00
& = \sum _ { im} \frac { \ERI { p_ \sig i_ \sig } { m} \ERI { q_ \sig i_ \sig } { m} } { \omega - \e { i_ \sig } + \Om { m} { \spc ,\RPA } - i \eta }
2020-10-21 22:58:11 +02:00
\\
2020-10-23 15:13:15 +02:00
& + \sum _ { am} \frac { \ERI { p_ \sig a_ \sig } { m} \ERI { q_ \sig a_ \sig } { m} } { \omega - \e { a_ \sig } - \Om { m} { \spc ,\RPA } + i \eta }
2020-10-21 22:58:11 +02:00
\end { split}
2020-10-24 14:19:04 +02:00
\end { gather}
2020-10-25 21:29:40 +01:00
which has been split in its exchange (x) and correlation (c) contributions.
2020-10-24 22:09:28 +02:00
The Dyson equation linking the Green's function and the self-energy holds separately for each spin component, and the quasiparticle energies $ \eGW { p _ \sig } $ are obtained by solving the frequency-dependent quasiparticle equation
2020-10-20 21:58:35 +02:00
\begin { equation}
2020-10-24 22:09:28 +02:00
\omega = \e { p_ \sig } { } - V_ { p_ \sigma } ^ { \xc } + \SigX { p\sigma } + \SigC { p\sigma } (\omega )
2020-10-20 21:58:35 +02:00
\end { equation}
2020-10-24 22:09:28 +02:00
with
\begin { equation}
V_ { p_ \sigma } ^ { \xc } = \int \MO { p_ \sig } (\br ) v^ { \xc } (\br ) \MO { p_ \sig } (\br ) d\br
\end { equation}
where $ v ^ { \xc } ( \br ) $ the Kohn-Sham exchange-correlation.
2020-10-25 21:29:40 +01:00
\alert { Adding the Dyson equation? Introduce linearization of the quasiparticle equation and different degree of self-consistency.}
2020-10-21 22:58:11 +02:00
%================================
2020-10-22 12:40:48 +02:00
\subsection { The Bethe-Salpeter equation formalism}
2020-10-21 22:58:11 +02:00
%================================
2020-10-25 13:30:30 +01:00
Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. \cite { Salpeter_ 1951,Strinati_ 1988}
2020-10-24 22:09:28 +02:00
Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
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
2020-10-23 15:13:15 +02:00
\begin { multline}
L^ { \sig \sigp } (\br _ 1,\br _ 2;\br _ 1',\br _ 2';\omega )
= L_ { 0} ^ { \sig \sigp } (\br _ 1,\br _ 2;\br _ 1',\br _ 2';\omega )
\\
+ \int L_ { 0} ^ { \sig \sigp } (\br _ 1,\br _ 4;\br _ 1',\br _ 3;\omega )
\Xi ^ { \sig \sigp } (\br _ 3,\br _ 5;\br _ 4,\br _ 6;\omega )
\\
\times L^ { \sig \sigp } (\br _ 6,\br _ 2;\br _ 5,\br _ 2';\omega )
d\br _ 3 d\br _ 4 d\br _ 5 d\br _ 6
\end { multline}
2020-10-24 22:09:28 +02:00
where
\begin { multline}
L_ { 0} ^ { \sig \sigp } (\br _ 1,\br _ 2;\br _ 1',\br _ 2';\omega )
\\
= \frac { 1} { 2\pi } \int G^ { \sig } (\br _ 1,\br _ 2';\omega +\omega ') G^ { \sig } (\br _ 1',\br _ 2;\omega ') d\omega '
\end { multline}
2020-10-25 13:30:30 +01:00
is the non-interacting analog of the two-particle correlation function $ L $ .
2020-10-23 15:13:15 +02:00
2020-10-24 22:09:28 +02:00
Within the $ GW $ approximation, the BSE kernel is
2020-10-23 15:13:15 +02:00
\begin { multline}
i \Xi ^ { \sig \sigp } (\br _ 3,\br _ 5;\br _ 4,\br _ 6;\omega )
= \frac { \delta (\br _ 3 - \br _ 4) \delta (\br _ 5 - \br _ 6) } { \abs { \br _ 3-\br _ 6} }
\\
- \delta _ { \sig \sigp } W(\br _ 3,\br _ 4;\omega ) \delta (\br _ 3 - \br _ 6) \delta (\br _ 4 - \br _ 6)
\end { multline}
2020-10-25 13:30:30 +01:00
where, as usual, we have not considered the higher-order terms in $ W $ by neglecting the derivative $ \partial W / \partial G $ . \cite { Hanke_ 1980, Strinati_ 1982, Strinati_ 1984, Strinati_ 1988}
2020-10-25 21:29:40 +01:00
Within the static approximation which consists in neglecting the frequency dependence of the dynamically-screened Coulomb potential, the spin-conserved and spin-flip optical excitation at the BSE level are obtained by solving a similar linear response problem
2020-10-25 13:30:30 +01:00
\begin { equation}
2020-10-25 21:29:40 +01:00
\label { eq:LR-BSE}
2020-10-25 13:30:30 +01:00
\begin { pmatrix}
\bA { } { \BSE } & \bB { } { \BSE } \\
-\bB { } { \BSE } & -\bA { } { \BSE } \\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { m} { \BSE } \\
\bY { m} { \BSE } \\
\end { pmatrix}
=
\Om { m} { \BSE }
\begin { pmatrix}
\bX { m} { \BSE } \\
\bY { m} { \BSE } \\
\end { pmatrix}
\end { equation}
2020-10-22 12:40:48 +02:00
Defining $ W ^ { \stat } _ { p _ \sig q _ \sig ,r _ \sigp s _ \sigp } = W _ { p _ \sig q _ \sig ,r _ \sigp s _ \sigp } ( \omega = 0 ) $ , we have
\begin { subequations}
\begin { align}
\label { eq:LR_ BSE-A}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \BSE } & = \A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } - \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau }
\\
\label { eq:LR_ BSE-B}
\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
2020-10-21 22:58:11 +02:00
\begin { subequations}
\begin { align}
\label { eq:LR_ BSE-Asc}
2020-10-22 12:40:48 +02:00
\A { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\BSE } & = \A { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } - \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig j_ \sigp ,b_ \sigp a_ \sig }
2020-10-21 22:58:11 +02:00
\\
\label { eq:LR_ BSE-Bsc}
2020-10-22 12:40:48 +02:00
\B { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\BSE } & = \B { i_ \sig a_ \sig ,j_ \sigp b_ \sigp } { \spc ,\RPA } - \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig b_ \sigp ,j_ \sigp a_ \sig }
\end { align}
\end { subequations}
for the spin-conserved excitations and
\begin { subequations}
\begin { align}
\label { eq:LR_ BSE-Asf}
\A { i_ \sig a_ \bsig ,j_ \sig b_ \bsig } { \spf ,\BSE } & = \A { i_ \sig a_ \bsig ,j_ \sig b_ \bsig } { \spf ,\RPA } - W^ { \stat } _ { i_ \sig j_ \sig ,b_ \bsig a_ \bsig }
\\
\label { eq:LR_ BSE-Bsf}
\B { i_ \sig a_ \bsig ,j_ \bsig b_ \sig } { \spf ,\BSE } & = - W^ { \stat } _ { i_ \sig b_ \sig ,j_ \bsig a_ \bsig }
2020-10-21 22:58:11 +02:00
\end { align}
\end { subequations}
2020-10-22 12:40:48 +02:00
for the spin-flip excitations.
2020-10-21 22:58:11 +02:00
2020-10-22 12:40:48 +02:00
%================================
2020-10-22 13:23:19 +02:00
\subsection { Dynamical correction}
2020-10-22 12:40:48 +02:00
%================================
2020-10-25 13:30:30 +01:00
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}
2020-10-22 13:23:19 +02:00
\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} ]
\end { multline}
\begin { equation}
\label { eq:LR-dyn}
\begin { pmatrix}
\bA { } { \dBSE } (\omega ) & \bB { } { \dBSE } (\omega )
\\
-\bB { } { \dBSE } (-\omega ) & -\bA { } { \dBSE } (-\omega )
\\
\end { pmatrix}
\cdot
\begin { pmatrix}
\bX { m} { \dBSE } \\
\bY { m} { \dBSE } \\
\end { pmatrix}
=
\Om { m} { \dBSE }
\begin { pmatrix}
\bX { m} { \dBSE } \\
\bY { m} { \dBSE } \\
\end { pmatrix}
\end { equation}
\begin { subequations}
\begin { align}
\label { eq:LR_ dBSE-A}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \dBSE } (\omega ) & = \A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \RPA } - \delta _ { \sig \sigp } \widetilde { W} _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau } (\omega )
\\
\label { eq:LR_ dBSE-B}
\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}
\begin { multline}
\label { eq:LR-PT}
\begin { pmatrix}
\bA { } { \dBSE } (\omega ) & \bB { } { \dBSE } (\omega ) \\
-\bB { } { \dBSE } (-\omega ) & -\bA { } { \dBSE } (-\omega ) \\
\end { pmatrix}
\\
=
\begin { pmatrix}
\bA { } { (0)} & \bB { } { (0)}
\\
-\bB { } { (0)} & -\bA { } { (0)}
\\
\end { pmatrix}
+
\begin { pmatrix}
\bA { } { (1)} (\omega ) & \bB { } { (1)} (\omega ) \\
-\bB { } { (1)} (-\omega ) & -\bA { } { (1)} (-\omega ) \\
\end { pmatrix}
\end { multline}
with
\begin { subequations}
\begin { align}
\label { eq:BSE-A0}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { (0)} & = \A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \BSE }
\\
\label { eq:BSE-B0}
\B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { (0)} & = \B { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { \BSE }
\end { align}
\end { subequations}
and
\begin { subequations}
\begin { align}
\label { eq:BSE-A1}
\A { i_ \sig a_ \tau ,j_ \sigp b_ \taup } { (1)} (\omega ) & = - \delta _ { \sig \sigp } \widetilde { W} _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau } (\omega ) + \delta _ { \sig \sigp } W^ { \stat } _ { i_ \sig j_ \sigp ,b_ \taup a_ \tau }
\\
\label { eq:BSE-B1}
\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}
\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}
\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}
\begin { equation}
\label { eq:Om1-TDA}
\Om { S} { (1)} = \T { (\bX { m} { (0)} )} \cdot \bA { } { (1)} (\Om { m} { (0)} ) \cdot \bX { m} { (0)}
\end { equation}
\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)}
\end { equation}
2020-10-24 14:19:04 +02:00
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection { Oscillator strengths}
\label { sec:os}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2020-10-24 22:09:28 +02:00
For the spin-conserved transition, the $ x $ component of the transition dipole moment is
2020-10-24 14:19:04 +02:00
\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
\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
\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}
For spin-flip transitions, we have $ f _ { m } ^ { \spf } = 0 $ as the transition matrix elements $ ( i _ \sig |x|a _ \bsig ) $ vanish.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection { Spin contamination}
\label { sec:spin}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin { equation}
\expval { S^ 2} _ m = \expval { S^ 2} _ 0 + \Delta \expval { S^ 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}
where
\begin { equation}
(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.
2020-10-24 22:09:28 +02:00
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.
2020-10-24 14:19:04 +02:00
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.
2020-10-22 13:23:19 +02:00
2020-10-20 21:58:35 +02:00
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2020-10-21 22:58:11 +02:00
\section { Computational details}
2020-10-20 21:58:35 +02:00
\label { sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
2020-10-25 21:29:40 +01:00
For the systems under investigation here, we consider either an open-shell doublet or triplet reference state.
We then adopt the unrestricted formalism throughout this work.
The $ GW $ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (unrestricted) UHF starting point.
Perturbative $ GW $ (or { \GOWO } ) \cite { Hybertsen_ 1985a,Hybertsen_ 1986,vanSetten_ 2013} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
Further details about our implementation of { \GOWO } can be found in Refs.~\onlinecite { Loos_ 2018b,Veril_ 2018,Loos_ 2020,Loos_ 2020e} .
%Note that, for the present (small) molecular systems, {\GOWO}@UHF and ev$GW$@UHF yield similar quasiparticle energies and fundamental gap.
%Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
%In the present study, the zeroth-order Hamiltonian [see Eq.~\eqref{eq:LR-PT}] is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
The dynamical correction is computed in the TDA throughout.
As one-electron basis sets, we employ the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.
Finally, the infinitesimal $ \eta $ is set to $ 100 $ meV for all calculations.
%It is important to mention that the small molecular systems considered here are particularly challenging for the BSE formalism, \cite{Hirose_2015,Loos_2018b} which is known to work best for larger systems where the amount of screening is more important. \cite{Jacquemin_2017b,Rangel_2017}
%For comparison purposes, we employ the theoretical best estimates (TBEs) and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2, \cite{Christiansen_1995a} CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} excitation energies are also extracted.
%Various statistical quantities are reported in the following: the mean signed error (MSE), mean absolute error (MAE), root-mean-square error (RMSE), and the maximum positive [Max($+$)] and maximum negative [Max($-$)] errors.
All the static and dynamic BSE calculations have been performed with the software \texttt { QuAcK} , \cite { QuAcK} freely available on \texttt { github} .
2020-10-20 21:58:35 +02:00
2020-10-24 14:19:04 +02:00
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Results}
\label { sec:res}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% TABLE I %%%
%\begin{table}
%
%\end{table}
%%% %%% %%% %%%
2020-10-20 21:58:35 +02:00
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section { Conclusion}
\label { sec:ccl}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgements {
We would like to thank Xavier Blase and Denis Jacquemin for insightful discussions.
2020-10-22 12:40:48 +02:00
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No.~863481).}
2020-10-20 21:58:35 +02:00
%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section * { Data availability statement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The data that supports the findings of this study are available within the article and its supplementary material.
%%%%%%%%%%%%%%%%%%%%%%%%
2020-10-25 13:30:30 +01:00
\bibliography { sfBSE}
2020-10-20 21:58:35 +02:00
%%%%%%%%%%%%%%%%%%%%%%%%
\end { document}