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@ -26,7 +26,8 @@
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\newcommand{\Hxc}{\text{Hxc}}
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\newcommand{\xc}{\text{xc}}
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\newcommand{\Ha}{\text{H}}
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\newcommand{\co}{\text{x}}
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\newcommand{\co}{\text{c}}
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\newcommand{\x}{\text{x}}
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%
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\newcommand{\Norb}{N_\text{orb}}
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@ -92,7 +93,7 @@
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\newcommand{\tW}[2]{\widetilde{W}_{#1}^{#2}}
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\newcommand{\Wc}[1]{W^\text{c}_{#1}}
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\newcommand{\vc}[1]{v_{#1}}
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\newcommand{\Sig}[1]{\Sigma_{#1}}
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\newcommand{\Sig}[2]{\Sigma_{#1}^{#2}}
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\newcommand{\SigC}[1]{\Sigma^\text{c}_{#1}}
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\newcommand{\SigX}[1]{\Sigma^\text{x}_{#1}}
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\newcommand{\SigXC}[1]{\Sigma^\text{xc}_{#1}}
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60
sfBSE.tex
60
sfBSE.tex
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@ -40,20 +40,19 @@
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\alert{Here comes the introduction.}
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Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
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In the following, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Unrestricted $GW$ formalism}
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\label{sec:UGW}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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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.
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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.
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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.
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The number of spin-up and spin-down occupied orbitals are $O_\up = n_\up$ and $O_\dw = n_\dw$, respectively, and, assuming no linear dependencies in the one-electron basis set, there is $V_\up = N - O_\up$ and $V_\dw = N - O_\dw$ spin-up and spin-down virtual (\ie, unoccupied) orbitals.
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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 $S^\spf = S_{\up\dw}^\spf + S_{\dw\up}^\spf = O_\up V_\dw + O_\dw V_\up$.
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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.
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In the present context these orbitals can originate from a HF or KS calculation.
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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.
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Let us denote as $\MO{p_\sig}(\br)$ the $p$th (spin)orbital of spin $\sig$ (where $\sig =$ $\up$ or $\dw$) and $\e{p_\sig}{}$ its one-electron energy.
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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 $\sig$.
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In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{a_\bsig}$, have opposite spins, $\sig$ and $\bsig$.
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In the following, we assume real quantities throughout this manuscript, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, $p$, $q$, $r$, and $s$ indicate arbitrary orbitals, and $m$ labels single excitations.
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Moreover, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.
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%================================
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\subsection{The dynamical screening}
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@ -61,6 +60,7 @@ In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{
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The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook}
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The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBook,Bruneval_2016a}
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\begin{equation}
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\label{eq:G}
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G^{\sig}(\br_1,\br_2;\omega)
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= \sum_i \frac{\MO{i_\sig}(\br_1) \MO{i_\sig}(\br_2)}{\omega - \e{i_\sig}{} - i\eta}
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+ \sum_a \frac{\MO{a_\sig}(\br_1) \MO{a_\sig}(\br_2)}{\omega - \e{a_\sig}{} + i\eta}
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@ -72,17 +72,17 @@ Based on the spin-up and spin-down components of $G$, one can easily compute the
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\end{equation}
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and subsequently the dielectric function
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\begin{equation}
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\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
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\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
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\end{equation}
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where $\delta(\br_1 - \br_2)$ is the Dirac function.
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Based on this latter ingredient, one can access the dynamically-screened Coulomb potential
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\begin{equation}
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W(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
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\end{equation}
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which is spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates.
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which is naturally spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates.
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Within the $GW$ formalism, the is computed at the RPA level by considering only the manifold of the spin-conserved neutral excitation.
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In the orbital basis, the spectral representation of $W$ reads
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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.
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In the orbital basis, the spectral representation of $W$ is
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\begin{multline}
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\label{eq:W}
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W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
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@ -92,14 +92,14 @@ In the orbital basis, the spectral representation of $W$ reads
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\end{multline}
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where the bare two-electron integrals are \cite{Gill_1994}
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\begin{equation}
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\label{eq:sERI}
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\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
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\end{equation}
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and the screened two-electron integrals (or spectral weights) are explicitly given by
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\begin{equation}
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\label{eq:sERI}
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\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}
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\end{equation}
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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
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In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors, $\bX{m}{\spc,\RPA}$ and $\bY{m}{\spc,\RPA}$, are obtained by solving a linear response system of the form
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\begin{equation}
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\label{eq:LR-RPA}
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\begin{pmatrix}
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@ -120,6 +120,7 @@ In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitat
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\end{equation}
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where the expressions of the matrix elements of $\bA{}{}$ and $\bB{}{}$ are specific of the method and of the spin manifold.
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The spin structure of these matrices, though, is general
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\begin{subequations}
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\begin{align}
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\label{eq:LR-RPA-AB}
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\bA{}{\spc} & = \begin{pmatrix}
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@ -143,6 +144,7 @@ The spin structure of these matrices, though, is general
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\bB{}{\dwup,\updw} & \bO \\
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\end{pmatrix}
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\end{align}
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\end{subequations}
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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 $\pm \Om{m}{}$.
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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
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\begin{equation}
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@ -153,9 +155,11 @@ where the excitation amplitudes are
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\begin{equation}
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\bX{}{} + \bY{}{} = \bOm{-1/2} \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{}
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\end{equation}
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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 $\bB{}{} = \bO$.
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In such a case, Eq.~\eqref{eq:LR-RPA} reduces to $\bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{}$.
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Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $\bA{}{}$ and $-\bA{}{}$, are neglected, which consists in setting $\bB{}{} = \bO$.
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In such a case, Eq.~\eqref{eq:LR-RPA} reduces to straightforward Hermitian problem of the form:
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\begin{equation}
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\bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{}
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\end{equation}
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At the RPA level, the matrix elements of $\bA{}{}$ and $\bB{}{}$ are
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\begin{subequations}
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\begin{align}
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@ -194,26 +198,35 @@ for the spin-flip excitations.
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Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchange-correlation (xc) part of the self-energy
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\begin{equation}
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\begin{split}
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\Sig{}^{\text{xc},\sig}(\br_1,\br_2;\omega)
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& = \Sig{}^{\text{x},\sig}(\br_1,\br_2) + \Sig{}^{\text{c},\sig}(\br_1,\br_2;\omega)
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\Sig{}{\xc,\sig}(\br_1,\br_2;\omega)
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& = \Sig{}{\x,\sig}(\br_1,\br_2) + \Sig{}{\co,\sig}(\br_1,\br_2;\omega)
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\\
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& = \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'
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\end{split}
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\end{equation}
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is, like the one-body Green's function, spin-diagonal, and its spectral representation reads
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\begin{gather}
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\SigX{p_\sig q_\sig}
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= - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig}
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\Sig{p_\sig q_\sig}{\x}
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= - \sum_{i} \ERI{p_\sig i_\sig}{i_\sig q_\sig}
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\\
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\begin{split}
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\SigC{p_\sig q_\sig}(\omega)
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\Sig{p_\sig q_\sig}{\co}(\omega)
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& = \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}
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\\
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& + \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}
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\end{split}
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\end{gather}
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which has been split in its exchange (x) and correlation (c) contributions.
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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
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which the self-energy has been split in its exchange (x) and correlation (c) contributions.
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The Dyson equation linking the Green's function and the self-energy holds separately for each spin component
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\begin{multline}
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\qty[ G^{\sig}(\br_1,\br_2;\omega) ]^{-1}
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= \qty[ G_{\KS}^{\sig}(\br_1,\br_2;\omega) ]^{-1}
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\\
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+ \Sig{}{\xc,\sig}(\br_1,\br_2;\omega) - v^{\xc}(\br_1) \delta(\br_1 - \br_2)
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\end{multline}
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where $G_{\KS}^{\sig}$ is the Kohn-Sham Green's function built with Kohn-Sham orbitals and one-electron energies according to Eq.~\eqref{eq:G} and $v^{\xc}(\br)$ is the Kohn-Sham local exchange-correlation potential.
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The quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation
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\begin{equation}
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\omega = \e{p_\sig}{} - V_{p_\sigma}^{\xc} + \SigX{p\sigma} + \SigC{p\sigma}(\omega)
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\end{equation}
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@ -221,8 +234,7 @@ with
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\begin{equation}
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V_{p_\sigma}^{\xc} = \int \MO{p_\sig}(\br) v^{\xc}(\br) \MO{p_\sig}(\br) d\br
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\end{equation}
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where $v^{\xc}(\br)$ the Kohn-Sham exchange-correlation.
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\alert{Adding the Dyson equation? Introduce linearization of the quasiparticle equation and different degree of self-consistency.}
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\alert{Introduce linearization of the quasiparticle equation and different degree of self-consistency.}
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%================================
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\subsection{The Bethe-Salpeter equation formalism}
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