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sfBSE.tex
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sfBSE.tex
@ -40,6 +40,7 @@
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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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@ -57,42 +58,47 @@ In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{
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%================================
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\subsection{The dynamical screening}
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%================================
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The one-body Green's function is defined as
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The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system.
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The spin-$\sig$ component of the one-body Green's function reads
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\begin{equation}
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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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\end{equation}
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Based on this Green's function, one can easily compute the non-interacting polarizability
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where $\eta$ is a positive infinitesimal.
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Based on it, one can easily compute the non-interacting polarizability (which is a sum over spins)
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\begin{equation}
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\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'
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\end{equation}
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and subseauently the dielectric function
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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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\end{equation}
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where $\delta(\br_1 - \br_2)$ is the Dirac functions.
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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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Within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
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In the orbital basis, the spectral representation of $W(\omega)$ read
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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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Therefore, within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
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In the orbital basis, the spectral representation of $W(\omega)$ reads
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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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+ \sum_m \ERI{p_\sig q_\sig}{m}\ERI{r_\sigp s_\sigp}{m}
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\\
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\times \qty[ \frac{1}{\omega - \Om{m}{\spc,\RPA} + i \eta} - \frac{1}{\omega + \Om{m}{\spc,\RPA} - i \eta} ]
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\end{multline}
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where the two-electron integrals are
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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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\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 the following linear response system
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\begin{equation}
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\label{eq:LR-RPA}
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\begin{pmatrix}
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@ -109,10 +115,10 @@ where the two-electron integrals are
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\begin{pmatrix}
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\bX{m}{\spc,\RPA} \\
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\bY{m}{\spc,\RPA} \\
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\end{pmatrix},
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\end{pmatrix}
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\end{equation}
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The spin structure of these matrices are general and reads
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The spin structure of matrices $\bA{}{}$ and $\bB{}{}$ is general
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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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@ -136,7 +142,8 @@ The spin structure of these matrices are general and reads
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\bB{}{\dwup,\updw} & \bO \\
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\end{pmatrix}
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\end{align}
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with
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and does not only apply to the RPA but also to RPAx (\ie, RPA with exchange), BSE and TD-DFT.
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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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\label{eq:LR_RPA-A}
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@ -146,7 +153,7 @@ with
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\B{i_\sig a_\tau,j_\sigp b_\taup}{\RPA} & = \ERI{i_\sig a_\tau}{j_\sigp b_\taup}
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\end{align}
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\end{subequations}
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from which we obtain, at the RPA level, the following expressions
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from which we obtain the following expressions
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\begin{subequations}
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\begin{align}
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\label{eq:LR_RPA-Asc}
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@ -171,7 +178,7 @@ for the spin-flip excitations.
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%================================
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\subsection{The $GW$ self-energy}
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%================================
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Within the acclaimed $GW$ approximation, the exchange-correlation part of the self-energy is defined as
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Within the acclaimed $GW$ approximation, 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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@ -180,7 +187,7 @@ Within the acclaimed $GW$ approximation, the exchange-correlation part of the se
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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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and the spectral representation of its exchange and correlation part read
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is, like the one-body Green's function, spin-diagonal, and its spectral representation read
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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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@ -192,16 +199,23 @@ and the spectral representation of its exchange and correlation part read
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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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The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-dependent quasiparticle equation
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which has been split in exchange (x) and correlation (c) parts for convenience.
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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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\begin{equation}
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\omega = \eHF{p\sigma} + \SigC{p\sigma}(\omega)
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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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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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%================================
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\subsection{The Bethe-Salpeter equation formalism}
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%================================
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Like its TD-DFT cousin, BSE deals with the calculation of (neutral) optical excitations as measured by absorption spectroscopy.
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Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
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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
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\begin{multline}
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L^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
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= L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
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@ -212,7 +226,15 @@ The quasiparticle energies $\eGW{p}$ are obtained by solving the frequency-depen
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\times L^{\sig\sigp}(\br_6,\br_2;\br_5,\br_2';\omega)
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d\br_3 d\br_4 d\br_5 d\br_6
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\end{multline}
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where
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\begin{multline}
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L_{0}^{\sig\sigp}(\br_1,\br_2;\br_1',\br_2';\omega)
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\\
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= \frac{1}{2\pi} \int G^{\sig}(\br_1,\br_2';\omega+\omega') G^{\sig}(\br_1',\br_2;\omega') d\omega'
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\end{multline}
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is the non-interacting counterpart of the two-particle correlation function $L$.
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Within the $GW$ approximation, the BSE kernel is
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\begin{multline}
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i \Xi^{\sig\sigp}(\br_3,\br_5;\br_4,\br_6;\omega)
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= \frac{\delta(\br_3 - \br_4) \delta(\br_5 - \br_6) }{\abs{\br_3-\br_6}}
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@ -415,7 +437,7 @@ and
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\subsection{Oscillator strengths}
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\label{sec:os}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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For the spin-conserved transition, the transition dipole moment in the $x$ direction is
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For the spin-conserved transition, the $x$ component of the transition dipole moment is
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\begin{equation}
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\mu_{x,m}^{\spc} = \sum_{ia\sig} (i_\sig|x|a_\sig)(\bX{m}{\spc}+\bY{m}{\spc})_{i_\sig a_\sig}
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\end{equation}
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@ -449,7 +471,7 @@ where
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\end{equation}
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is the overlap between spin-up and spin-down orbitals.
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The explicit expressions of $\Delta \expval{S^2}_m^{\spc}$ and $\Delta \expval{S^2}_m^{\spf}$ are given in Ref.~\onlinecite{Li_2010}.
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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.
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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.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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