diff --git a/sfBSE.tex b/sfBSE.tex index 8966e7d..1e1ae9a 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -40,6 +40,7 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \alert{Here comes the introduction.} Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript. +In the following, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin?orbit interaction. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Unrestricted $GW$ formalism} @@ -57,42 +58,47 @@ In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{ %================================ \subsection{The dynamical screening} %================================ -The one-body Green's function is defined as +The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. +The spin-$\sig$ component of the one-body Green's function reads \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} -Based on this Green's function, one can easily compute the non-interacting polarizability +where $\eta$ is a positive infinitesimal. +Based on it, one can easily compute the non-interacting polarizability (which is a sum over spins) \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} -and subseauently the dielectric function +and subsequently the dielectric function \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} +where $\delta(\br_1 - \br_2)$ is the Dirac functions. 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} - -Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations. -In the orbital basis, the spectral representation of $W(\omega)$ read +which is spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates. +Therefore, within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations. +In the orbital basis, the spectral representation of $W(\omega)$ reads \begin{multline} +\label{eq:W} 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 - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ] \end{multline} -where the two-electron integrals are +where the bare two-electron integrals are \cite{Gill_1994} \begin{equation} +\label{eq:sERI} \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 \end{equation} - +and the screened two-electron integrals (or spectral weights) are explicitly given by \begin{equation} \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} \end{equation} - +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 the following linear response system \begin{equation} \label{eq:LR-RPA} \begin{pmatrix} @@ -109,10 +115,10 @@ where the two-electron integrals are \begin{pmatrix} \bX{m}{\spc,\RPA} \\ \bY{m}{\spc,\RPA} \\ - \end{pmatrix}, + \end{pmatrix} \end{equation} -The spin structure of these matrices are general and reads +The spin structure of matrices $\bA{}{}$ and $\bB{}{}$ is general \begin{align} \label{eq:LR-RPA-AB} \bA{}{\spc} & = \begin{pmatrix} @@ -136,7 +142,8 @@ The spin structure of these matrices are general and reads \bB{}{\dwup,\updw} & \bO \\ \end{pmatrix} \end{align} -with +and does not only apply to the RPA but also to RPAx (\ie, RPA with exchange), BSE and TD-DFT. +At the RPA level, the matrix elements of $\bA{}{}$ and $\bB{}{}$ are \begin{subequations} \begin{align} \label{eq:LR_RPA-A} @@ -146,7 +153,7 @@ with \B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \ERI{i_\sig a_\tau}{j_\sigp b_\taup} \end{align} \end{subequations} -from which we obtain, at the RPA level, the following expressions +from which we obtain the following expressions \begin{subequations} \begin{align} \label{eq:LR_RPA-Asc} @@ -171,7 +178,7 @@ for the spin-flip excitations. %================================ \subsection{The $GW$ self-energy} %================================ -Within the acclaimed $GW$ approximation, the exchange-correlation part of the self-energy is defined as +Within the acclaimed $GW$ approximation, the exchange-correlation (xc) part of the self-energy \begin{equation} \begin{split} \Sig{}^{\text{xc},\sig}(\br_1,\br_2;\omega) @@ -180,7 +187,7 @@ Within the acclaimed $GW$ approximation, the exchange-correlation part of the se & = \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} \end{equation} -and the spectral representation of its exchange and correlation part read +is, like the one-body Green's function, spin-diagonal, and its spectral representation read \begin{gather} \SigX{p_\sig q_\sig} = - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig} @@ -192,16 +199,23 @@ and the spectral representation of its exchange and correlation part read & + \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} \end{split} \end{gather} - -The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation +which has been split in exchange (x) and correlation (c) parts for convenience. +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 \begin{equation} - \omega = \eHF{p\sigma} + \SigC{p\sigma}(\omega) + \omega = \e{p_\sig}{} - V_{p_\sigma}^{\xc} + \SigX{p\sigma} + \SigC{p\sigma}(\omega) \end{equation} +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. %================================ \subsection{The Bethe-Salpeter equation formalism} %================================ - +Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy. +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 \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) @@ -212,7 +226,15 @@ The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-depen \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} +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} +is the non-interacting counterpart of the two-particle correlation function $L$. +Within the $GW$ approximation, the BSE kernel is \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}} @@ -415,7 +437,7 @@ and \subsection{Oscillator strengths} \label{sec:os} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -For the spin-conserved transition, the transition dipole moment in the $x$ direction is +For the spin-conserved transition, 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} @@ -449,7 +471,7 @@ where \end{equation} is the overlap between spin-up and spin-down orbitals. -The explicit expressions of $\Delta \expval{S^2}_m^{\spc}$ and $\Delta \expval{S^2}_m^{\spf}$ are given in Ref.~\onlinecite{Li_2010}. +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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%