saving work in BSE section
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sfBSE.tex
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sfBSE.tex
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@ -47,10 +47,10 @@ Unless otherwise stated, atomic units are used, and we assume real quantities th
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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, 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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The number of spin-conserved (sc) 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 $\MO{p_\sig}(\br)$ 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 a spin-flip (sf) 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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@ -66,7 +66,11 @@ The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBo
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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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where $\eta$ is a positive infinitesimal.
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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)
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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 $\e{p_\sig}{}$.
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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 $\e{p_\sig}^{\KS}$.
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Within self-consistent schemes, these quantities can be replaced by quasiparticle energies and orbitals evaluated within the $GW$ approximation (see below).
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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)
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\begin{equation}
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\label{eq:chi0}
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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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@ -76,7 +80,7 @@ and subsequently the dielectric function
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\label{eq:eps}
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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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where $\delta(\br)$ is the Dirac delta 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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\label{eq:W}
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@ -102,7 +106,7 @@ and the screened two-electron integrals (or spectral weights) are explicitly giv
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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_spectral} 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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In Eqs.~\eqref{eq:W_spectral} and \eqref{eq:sERI}, the spin-conserved RPA 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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@ -149,7 +153,7 @@ The spin structure of these matrices, though, is general
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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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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 its original dimension
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\begin{equation}
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\label{eq:small-LR}
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(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{} = \bOm{2} \cdot \bZ{}{}
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@ -159,10 +163,12 @@ where the excitation amplitudes are
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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 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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In such a case, Eq.~\eqref{eq:LR-RPA} reduces to a 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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Note that, for spin-flip excitations, it is quite common to enforce the TDA especially when one considers a triplet reference as the first ``excited-state'' is usually the ground state of the closed-shell system (hence, corresponding to a negative excitation energy).
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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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@ -220,7 +226,7 @@ is, like the one-body Green's function, spin-diagonal, and its spectral represen
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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 the self-energy has been split in its exchange (x) and correlation (c) contributions.
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where 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{equation}
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\label{eq:Dyson_G}
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@ -231,7 +237,7 @@ The Dyson equation linking the Green's function and the self-energy holds separa
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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{split}
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\end{equation}
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where $G_{\KS}^{\sig}$ is the Kohn-Sham Green's function built with Kohn-Sham orbitals $\MO{p_\sig}^{\KS}(\br)$ and one-electron energies $\e{p_\sig}^{\KS}$ according to Eq.~\eqref{eq:G} and $v^{\xc}(\br)$ is the Kohn-Sham local exchange-correlation potential.
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where $v^{\xc}(\br)$ is the KS (local) exchange-correlation potential.
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The target quantities here are the quasiparticle energies $\eGW{p_\sig}$, \ie, the poles of $G$ [see Eq.~\eqref{eq:G}], which correspond to well-defined addition/removal energies (unlike the KS orbital energies).
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Because the exchange-correlation part of the self-energy is, itself, constructed with the Green's function [see Eq.~\eqref{eq:Sig}], the present process is, by nature, self-consistent.
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The same comment applies to the dynamically-screened Coulomb potential $W$ entering the definition of $\Sig{}{\xc}$ [see Eq.~\eqref{eq:Sig}] which is also constructed from $G$ [see Eqs.~\eqref{eq:chi0}, \eqref{eq:eps}, and \eqref{eq:W}].
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@ -245,25 +251,25 @@ In its simplest perturbative (\ie, one-shot) version, known as {\GOWO}, a single
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\label{eq:QP-eq}
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\omega = \e{p_\sig}{} + \Sig{p_\sig}{\xc}(\omega) - V_{p_\sig}^{\xc}
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\end{equation}
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where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and its offspring quantities have been constructed at the Kohn-Sham level, and
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where $\Sig{p_\sig}{\xc}(\omega) \equiv \Sig{p_\sig p_\sig}{\xc}(\omega)$ and its offspring quantities have been constructed at the KS level, and
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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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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
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\begin{equation}
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\eGOWO{p_\sig}
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= \e{p_\sig}^{\KS} + Z_{p_\sig} [\Sig{p_\sig p_\sig}{\xc}(\e{p_\sig}^{\KS}) - V_{p_\sig}^{\xc} ]
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= \e{p_\sig}^{\KS} + Z_{p_\sig} [\Sig{p_\sig}{\xc}(\e{p_\sig}^{\KS}) - V_{p_\sig}^{\xc} ]
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\end{equation}
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where
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\begin{equation}
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Z_{p_\sig} = \qty[ 1 - \left. \pdv{\Sig{p_\sig p_\sig}{\xc}(\omega)}{\omega} \right|_{\omega = \e{p_\sig}^{\KS}} ]^{-1}
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Z_{p_\sig} = \qty[ 1 - \left. \pdv{\Sig{p_\sig}{\xc}(\omega)}{\omega} \right|_{\omega = \e{p_\sig}^{\KS}} ]^{-1}
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\end{equation}
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is a renormalization factor which also represents the spectral weight of the quasiparticle solution.
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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.
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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 Kohn-Sham level).
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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).
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Finally, within the quasiparticle self-consistent $GW$ scheme (qs$GW$), both the one-electron energies and the orbitals are updated until convergence is reached.
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Finally, within the quasiparticle self-consistent $GW$ (qs$GW$) scheme, both the one-electron energies and the orbitals are updated until convergence is reached.
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These are obtained via the diagonalization of an effective Fock matrix which includes explicitly a frequency-independent and hermitian self-energy defined as
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\begin{equation}
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\Tilde{\Sigma}_{p_\sig q_\sig}^{\xc} = \frac{1}{2} \qty[ \Sig{p_\sig q_\sig}{\xc}(\e{p_\sig}{}) + \Sig{q_\sig p_\sig}{\xc}(\e{p_\sig}{}) ]
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@ -274,6 +280,9 @@ These are obtained via the diagonalization of an effective Fock matrix which inc
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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. \cite{Salpeter_1951,Strinati_1988}
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Using the BSE formalism, one can access the spin-conserved and spin-flip excitations.
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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.
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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.
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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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@ -301,7 +310,7 @@ Within the $GW$ approximation, the BSE kernel is
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- \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6)
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\end{multline}
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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}
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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
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Within the static approximation which consists in neglecting the frequency dependence of the dynamically-screened Coulomb potential, the spin-conserved and spin-flip BSE optical excitations are obtained by solving the usual Casida-like linear response (eigen)problem:
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\begin{equation}
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\label{eq:LR-BSE}
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\begin{pmatrix}
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@ -320,7 +329,7 @@ Within the static approximation which consists in neglecting the frequency depen
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\bY{m}{\BSE} \\
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\end{pmatrix}
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\end{equation}
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Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega = 0)$, we have
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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
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-A}
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@ -330,7 +339,7 @@ Defining $W^{\stat}_{p_\sig q_\sig,r_\sigp s_\sigp} = W_{p_\sig q_\sig,r_\sigp s
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\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}
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\end{align}
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\end{subequations}
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from which we obtain, at the BSE level, the following expressions
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from which we obtain, at the BSE level, the following expressions for the spin-conserved and spin-flip excitations:
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\begin{subequations}
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\begin{align}
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\label{eq:LR_BSE-Asc}
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@ -338,11 +347,7 @@ from which we obtain, at the BSE level, the following expressions
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\\
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\label{eq:LR_BSE-Bsc}
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\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}
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\end{align}
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\end{subequations}
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for the spin-conserved excitations and
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\begin{subequations}
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\begin{align}
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\\
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\label{eq:LR_BSE-Asf}
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\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}
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\\
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@ -350,7 +355,12 @@ for the spin-conserved excitations and
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\B{i_\sig a_\bsig,j_\bsig b_\sig}{\spf,\BSE} & = - W^{\stat}_{i_\sig b_\sig,j_\bsig a_\bsig}
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\end{align}
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\end{subequations}
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for the spin-flip excitations.
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At this stage, it is of particular interest to discuss the form of the spin-flip matrix elements defined in Eqs.~\eqref{eq:LR_BSE-Asf} and \eqref{eq:LR_BSE-Bsf}.
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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.
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This is also the case at the TD-DFT level when one relies on (semi-)local functionals.
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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.
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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}.
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%================================
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\subsection{Dynamical correction}
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@ -511,7 +521,6 @@ and
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\Om{m}{\dBSE} = \Om{m}{(0)} + Z_{m} \Om{m}{(1)}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Oscillator strengths}
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\label{sec:os}
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@ -528,7 +537,7 @@ and the total oscillator strength is given by
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\begin{equation}
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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 ]
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\end{equation}
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For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix elements $(i_\sig|x|a_\bsig)$ vanish.
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For spin-flip transitions, we have $f_{m}^{\spf} = 0$ as the transition matrix elements $(i_\sig|x|a_\bsig)$ vanish via integration over the spin coordinate.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Spin contamination}
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