clean up BSE theory
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BSEdyn.tex
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BSEdyn.tex
@ -80,14 +80,14 @@
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\newcommand{\EcBSE}{E_\text{c}^\text{BSE}}
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% orbital energies
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\newcommand{\e}[1]{\epsilon_{#1}}
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\newcommand{\eHF}[1]{\epsilon^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\epsilon^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\epsilon^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\epsilon^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\epsilon^{GW}_{#1}}
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\newcommand{\eevGW}[1]{\epsilon^\text{\evGW}_{#1}}
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\newcommand{\eGnWn}[2]{\epsilon^\text{\GnWn{#2}}_{#1}}
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\newcommand{\e}[1]{\eps_{#1}}
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\newcommand{\eHF}[1]{\eps^\text{HF}_{#1}}
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\newcommand{\eKS}[1]{\eps^\text{KS}_{#1}}
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\newcommand{\eQP}[1]{\eps^\text{QP}_{#1}}
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\newcommand{\eGOWO}[1]{\eps^\text{\GOWO}_{#1}}
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\newcommand{\eGW}[1]{\eps^{GW}_{#1}}
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\newcommand{\eevGW}[1]{\eps^\text{\evGW}_{#1}}
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\newcommand{\eGnWn}[2]{\eps^\text{\GnWn{#2}}_{#1}}
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\newcommand{\Om}[2]{\Omega_{#1}^{#2}}
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\newcommand{\tOm}[2]{\Tilde{\Omega}_{#1}^{#2}}
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@ -180,14 +180,18 @@
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\newcommand{\pis}{\pi^*}
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\newcommand{\ra}{\rightarrow}
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\newcommand\vari{{\varepsilon}_i}
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\newcommand\vara{{\varepsilon}_a}
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\newcommand\varj{{\varepsilon}_j}
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\newcommand\varb{{\varepsilon}_b}
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\newcommand\varn{{\varepsilon}_n}
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\newcommand\varm{{\varepsilon}_m}
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\newcommand\vari{{\eps}_i}
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\newcommand\vara{{\eps}_a}
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\newcommand\varj{{\eps}_j}
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\newcommand\varb{{\eps}_b}
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\newcommand\varn{{\eps}_n}
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\newcommand\varm{{\eps}_m}
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\newcommand\Oms{{\Omega}_s}
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\newcommand\hOms{\frac{{\Omega}_s}{2}}
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\newcommand{\hpsi}{\Hat{\psi}}
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\newcommand{\ha}{\Hat{a}}
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\newcommand{\tchi}{\Tilde{\chi}}
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\newcommand{\bx}{\mathbf{x}}
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\newcommand{\NEEL}{Universit\'e Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France}
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\newcommand{\LCPQ}{Laboratoire de Chimie et Physique Quantiques (UMR 5626), Universit\'e de Toulouse, CNRS, UPS, France}
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@ -227,11 +231,11 @@ In recent years, it has been shown to be a valuable tool for computational theor
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Taking the optical gap (\ie, the lowest optical excitation energy) as an example, BSE builds on top of a $GW$ calculation by adding up excitonic effects $\EB$ to the $GW$ HOMO-LUMO gap
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\begin{equation}
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\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \varepsilon_{\HOMO}^{\GW},
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\Eg^{\GW} = \eps_{\LUMO}^{\GW} - \eps_{\HOMO}^{\GW},
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\end{equation}
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which is itself a corrected version of the Kohn-Sham (KS) gap
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\begin{equation}
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\Eg^{\KS} = \eps_{\LUMO}^{\KS} - \varepsilon_{\HOMO}^{\KS} \ll \Eg^{\GW} \approx \EgFun,
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\Eg^{\KS} = \eps_{\LUMO}^{\KS} - \eps_{\HOMO}^{\KS} \ll \Eg^{\GW} \approx \EgFun,
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\end{equation}
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in order to approximate the optical gap
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\begin{equation}
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@ -272,100 +276,123 @@ Unless otherwise stated, atomic units are used.
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\label{sec:theory}
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%%%%%%%%%%%%%%%%%%%%%%%%
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In this Section, we describe the theoretical foundations leading to the dynamical Bethe-Salpeter equation, following the seminal work by Strinati, \cite{Strinati_1988} presenting in a second step the perturbative implementation \cite{Rohlfing_2000,Ma_2009a,Ma_2009b} of the dynamical correction as compared to the standard adiabatic calculations. More details of the derivation are provided in ...
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In this Section, following the seminal work by Strinati, \cite{Strinati_1988} we describe, first, the theoretical foundations leading to the dynamical Bethe-Salpeter equation.
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We present, in a second step, the perturbative implementation \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} of the dynamical correction as compared to the standard static approximation.
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More details of the derivation are provided as {\SI}.
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%================================
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\subsection{General dynamical BSE theory}
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%=================================
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The resolution \cite{Strinati_1988} of the Bethe-Salpeter equation:
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\begin{align*}
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L(1,2; & 1',2') = L_0(1,2;1',2') + \\
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&+ \int d3456 \;
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L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2')
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\end{align*}
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with:
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\begin{align*}
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iL(1,2; 1',2') &= -G_2(1,2;1',2') + G(1,1')G(2,2') \\
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i^2 G_2(1,2;1',2') &= \langle N | T {\hat \psi}(1) {\hat \psi}(2) {\hat \psi}^{\dagger}(2') {\hat \psi}^{\dagger}(1') | N \rangle
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\end{align*}
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where e.g. $1 = (x_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set $ |N,s \rangle $ of the N-electron system excited states, with $| N \rangle = | N,0 \rangle$ the ground-state. In the optical limit of instantaneous electron-hole creation and destruction, imposing
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$t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, with e.g. $t_2^+ = t_2 + 0^+$ where $0^+$ is a small positive infinitesimal, one obtains:
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\begin{align*}
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iL(1,2;1',2') &= \theta(\tau_{12}) \sum_{s > 0} \chi_s(x_1,x_{1'}) {\tilde \chi}_s(x_2,x_{2'})
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e^{ -i \Oms \tau_{12} } \\
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&- \theta(-\tau_{12}) \sum_{s > 0} \chi_s(x_2,x_{2'}) {\tilde \chi}_s(x_1,x_{1'})
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e^{ + i \Oms \tau_{12} }
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\end{align*}
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with $\tau_{12} = t_1 - t_2$ and
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\begin{align*}
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\chi_s(x_1,x_{1'}) = \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle \\
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{\tilde \chi}_s(x_2,x_{2'}) = \langle N,s | T {\hat \psi}(x_2) {\hat \psi}^{\dagger}(x_{2'}) | N \rangle
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\end{align*}
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The $\Oms$ are the neutral excitation energies of interest. Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by ${\tilde \chi}_s(x_2,x_{2'})$ on both side of the Bethe-Salpeter equation, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the BSE. For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system, the $L_0(1,2;1',2')$ term cannot contribute since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionisation potential.
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The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables:
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\begin{align*}
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[iL_0]( \omega_1 ) = \int \frac{ d \omega }{ 2\pi } \; G(\omega - \frac{\omega_1}{2} ) G( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
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\end{align*}
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with $\tau_{34} = t_3 - t_4$ and
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$t^{34} = (t_3 + t_4)/2$. Plugging now the 1-body Green's function Lehman representation, e.g.
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$$
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G(x_1,x_3 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_3) } { \omega - \varepsilon_n + i \eta \text{sgn} \times (\varepsilon_n - \mu) }
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$$
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and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component
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\begin{align*}
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\int dx_1 dx_{1'} \; & \phi_a^*(x_1) \phi_i(x_{1'}) L_0(x_1,3;x_{1'},4; \Oms) = e^{i \Oms t^{34} } \times \\
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& \frac{ \phi_a^*(x_3) \phi_i(x_4) } { \Oms - ( \vara - \vari ) + i \eta }
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\left[ \theta( \tau ) e^{i ( \vari + \hOms) \tau }
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+ \theta( - \tau ) e^{i (\vara - \hOms \tau } \right]
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\end{align*}
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with $\tau = \tau_{34}$. Adopting now the $GW$ approximation for the exchange-correlation self-energy leads to a simplification of the BSE kernel:
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$$
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\Xi(3,5;4,6) = v(3,6) \delta(34) \delta(56) - W(3^+,4) \delta(36) \delta(45)
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$$
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We further obtain the needed spectral representation of
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$\langle N | T {\hat \psi}(3) {\hat \psi}^{\dagger}(4) | N,s \rangle$
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expanding the field operators over a complete orbital basis creation/destruction operators:
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\begin{align*}
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\langle N | T {\hat \psi}(3) & {\hat \psi}^{\dagger}(4) | N,s \rangle = - \Big( e^{ -i \Omega_s t^{34} } \Big) \sum_{mn} \phi_m(x_3) \phi_n^*(x_4) \times \nonumber \\
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\times & \langle N | {\hat a}_n^{\dagger} {\hat a}_m | N,s \rangle \;\Big[ \theta( \tau ) e^{- i ( \varepsilon_m - \hOms ) \tau } + \theta( -\tau ) e^{ - i ( \varepsilon_n + \hOms) \tau } \Big]
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\end{align*}
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with $\tau = \tau_{34}$ and where the $ \lbrace \varepsilon_{n/m} \rbrace$ are proper addition/removal energies such that e.g.
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$$
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e^{ i H \tau } {\hat a}_m^{\dagger} | N \rangle = e^{ i (E_0^N + \varepsilon_m ) \tau } {\hat a} _m^{\dagger} | N \rangle
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$$
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The $GW$ quasiparticle energies $ \varepsilon_{n/m}^{GW}$ are good approximations to such removal/addition energies. Selecting (n,m)=(j,b) yields the largest components
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$A_{jb}^{s} = \langle N | {\hat a}_j^{\dagger} {\hat a}_b | N,s \rangle $, while (n,m)=(b,j) yields much weaker
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$B_{jb}^{s} = \langle N | {\hat a}_b^{\dagger} {\hat a}_j | N,s \rangle $ contributions. We used chemist notations with (i,j) indexing occupied orbitals and (a,b) virtual ones. Neglecting the $B_{jb}^{s}$ weights leads to the Tamm Dancoff approximation (TDA). Working out the same expansion for $ \langle N | T {\hat \psi}(5) {\hat \psi}^{\dagger}(5) | N,s \rangle$ and $ \langle N | T {\hat \psi}(x_1) {\hat \psi}^{\dagger}(x_{1'}) | N,s \rangle$, and projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$,
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one obtains after a few tedious manipulations (see Supplemental Information) the dynamical Bethe-Salpeter equation (dBSE) :
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\begin{align}
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( \varepsilon_a - \varepsilon_i - \Omega_s ) A_{ia}^{s}
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&+ \sum_{jb} \Big( v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) \Big) A_{jb}^{s} \\
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&+ \sum_{bj} \Big( v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) \Big) B_{jb}^{s}
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= 0
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\end{align}
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with an effective dynamically screened Coulomb potential (see Pina eq. 24):
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\begin{align}
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\widetilde{W}_{ij,ab}(\Oms) &= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega) \times \\
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\hskip 1cm &\times \left[ \frac{1}{ \Omega_{ib}^s - \omega +i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } \right] \nonumber
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\end{align}
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where $\; \Omega_{ib}^s = \Oms - ( \varb - \vari )$ and $\; \Omega_{ja}^s = \Oms - ( \vara - \varj ).$
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In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
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\begin{align*}
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W_{ij,ab}(\omega) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
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& \times \Big( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } \Big)
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\end{align*}
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($\Omega_m^{RPA} > 0 $) so that
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The resolution \cite{Strinati_1988} of the Bethe-Salpeter equation
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\begin{multline}
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L(1,2; 1',2')
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= L_0(1,2;1',2')
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\\
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+ \int d3456 \; L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2'),
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\end{multline}
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with
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\begin{align}
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\widetilde{W}_{ij,ab}( \Oms ) &= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m] \times \\
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& \times \left[ \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta } + \frac{ 1}{ \Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta }
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\right] \nonumber
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iL(1,2; 1',2')
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& = - G_2(1,2;1',2') + G(1,1') G(2,2'),
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\\
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i^2 G_2(1,2;1',2')
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& = \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N},
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\end{align}
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where, \eg, $1 = (\bx_1,t_1)$ a space-spin plus time variable, starts with the expansion of the 2-body Green's function $G_2$ and response function $L$ over the complete orthonormalized set $\ket{N,s}$ of the $N$-electron system excited states, with $\ket{N} = \ket{N,0}$ the ground-state.
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In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, with, \eg, $t_2^+ = t_2 + 0^+$ where $0^+$ is a small positive infinitesimal, one gets
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\begin{equation}
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\begin{split}
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iL(1,2;1',2')
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& = \theta(+\tau_{12}) \sum_{s > 0} \chi_s(x_1,x_{1'}) \tchi_s(x_2,x_{2'}) e^{ - i \Oms \tau_{12} }
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\\
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& - \theta(-\tau_{12}) \sum_{s > 0} \chi_s(x_2,x_{2'}) \tchi_s(x_1,x_{1'}) e^{ + i \Oms \tau_{12} },
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\end{split}
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\end{equation}
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with $\tau_{12} = t_1 - t_2$ and
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\begin{align}
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\chi_s(x_1,x_{1'}) & = \mel{N}{T \hpsi(x_1) \hpsi^{\dagger}(x_{1'})}{N,s}
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\\
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\tchi_s(x_2,x_{2'}) & = \mel{N,s}{T \hpsi(x_2) \hpsi^{\dagger}(x_{2'})}{N}
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\end{align}
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The $\Oms$'s are the neutral excitation energies of interest.
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Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(x_2,x_{2'})$ on both side of the BSE, we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the BSE.
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For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system, the $L_0(1,2;1',2')$ term cannot contribute since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization potential.
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The Fourier components with respect to time $t_1$ of $iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1')$ reads, dropping the (space/spin)-variables:
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\begin{align}
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[iL_0]( \omega_1 ) = \int \frac{d\omega}{2\pi} \; G\qty(\omega - \frac{\omega_1}{2} ) G\qty( {\omega} + \frac{\omega_1}{2} ) e^{ i \omega \tau_{34} } e^{ i \omega_1 t^{34} }
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\end{align}
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with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$.
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Plugging now the 1-body Green's function Lehman representation, \eg,
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\begin{equation}
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G(x_1,x_2 ; \omega) = \sum_n \frac{ \phi_n(x_1) \phi_n^*(x_2) } { \omega - \eps_n + i \eta \text{sgn} \times (\eps_n - \mu) }
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\end {equation}
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and projecting on $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains the $\omega_1= \Oms$ component
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\begin{multline}
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\int dx_1 dx_{1'} \; \phi_a^*(x_1) \phi_i(x_{1'}) L_0(x_1,3;x_{1'},4; \Oms)
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\\
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= e^{i \Oms t^{34} }
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\frac{ \phi_a^*(x_3) \phi_i(x_4) } { \Oms - ( \vara - \vari ) + i \eta }
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\times \qty[ \theta( \tau ) e^{i ( \vari + \hOms) \tau } + \theta( - \tau ) e^{i (\vara - \hOms \tau } ]
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\end{multline}
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with $\tau = \tau_{34}$. Adopting now the $GW$ approximation for the exchange-correlation self-energy leads to a simplification of the BSE kernel:
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\begin{equation}
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\Xi(3,5;4,6) = v(3,6) \delta(34) \delta(56) - W(3^+,4) \delta(36) \delta(45)
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\end{equation}
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We further obtain the needed spectral representation of $\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}$ expanding the field operators over a complete orbital basis creation/destruction operators:
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\begin{multline}
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\mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s}
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\\
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= - \qty( e^{ -i \Omega_s t^{34} } ) \sum_{mn} \phi_m(x_3) \phi_n^*(x_4)
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\mel{N}{\ha_n^{\dagger} \ha_m}{N,s}
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\\
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\times \qty[ \theta( \tau ) e^{- i ( \eps_m - \hOms ) \tau } + \theta( -\tau ) e^{ - i ( \eps_n + \hOms) \tau } ]
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\end{multline}
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with $\tau = \tau_{34}$ and where the $ \lbrace \eps_{n/m} \rbrace$ are proper addition/removal energies such that
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\begin{equation}
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e^{i H \tau} \ha_m^{\dagger} \ket{N} = e^{ i (E_0^N + \eps_m ) \tau } \ha _m^{\dagger} \ket{N}
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\end{equation}
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The $GW$ quasiparticle energies $\eps{n/m}^{GW}$ are good approximations to such removal/addition energies.
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Selecting $(n,m)=(j,b)$ yields the largest components
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$A_{jb}^{s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$, while $(n,m)=(b,j)$ yields much weaker
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$B_{jb}^{s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$ contributions. We used chemist notations with $(i,j)$ indexing occupied orbitals and $(a,b)$ virtual ones.
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Neglecting the $B_{jb}^{s}$ weights leads to the Tamm-Dancoff approximation (TDA).
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Working out the same expansion for $\mel{N}{T \hpsi(5) \hpsi^{\dagger}(5)}{N,s}$ and $\mel{N}{T \hpsi(x_1) \hpsi^{\dagger}(x_{1'})}{N,s}$, and projecting onto $\phi_a^*(x_1) \phi_i(x_{1'})$, one obtains after a few tedious manipulations (see {\SI}) the dynamical Bethe-Salpeter equation (dBSE) :
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\begin{equation}
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\begin{split}
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( \eps_a - \eps_i - \Omega_s ) A_{ia}^{s}
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& + \sum_{jb} \qty[ v_{ai,bj} - \widetilde{W}_{ij,ab}(\Oms) ] A_{jb}^{s} \\
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& + \sum_{jb} \qty[ v_{ai,jb} - \widetilde{W}_{ib,aj}(\Oms) ] B_{jb}^{s}
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= 0
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\end{split}
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\end{equation}
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with an effective dynamically screened Coulomb potential (see Pina eq. 24):
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\begin{multline}
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\widetilde{W}_{ij,ab}(\Oms)
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= \frac{ i }{ 2 \pi} \int d\omega \; e^{-i \omega 0^+ } W_{ij,ab}(\omega)
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\\
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\times \qty[ \frac{1}{ \Omega_{ib}^s - \omega + i \eta } + \frac{1}{ \Omega_{ja}^{s} + \omega + i\eta } ]
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\end{multline}
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where $\Omega_{ib}^s = \Oms - ( \varb - \vari )$ and $\Omega_{ja}^s = \Oms - ( \vara - \varj )$.
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In the present study, we use the exact spectral representation of $W(\omega)$ at the RPA level:
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\begin{multline}
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W_{ij,ab}(\omega)
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= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m]
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\\
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||||
\times \qty( \frac{1}{ \omega-\Omega_m^{RPA} + i\eta } - \frac{1}{ \omega + \Omega_m^{RPA} - i\eta } )
|
||||
\end{multline}
|
||||
($\Omega_m^{RPA} > 0 $) so that
|
||||
\begin{multline}
|
||||
\widetilde{W}_{ij,ab}( \Oms )
|
||||
= (ij|ab) + 2 \sum_m^{OV} [ij|m] [ab|m]
|
||||
\\
|
||||
\times \qty( \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA} + i\eta} + \frac{1}{\Omega_{ja}^{s} - \Omega_m^{RPA} + i\eta} )
|
||||
\end{multline}
|
||||
\textcolor{red}{Due to excitonic effects, the lowest BSE ${\Omega}_1$ excitation energy stands lower than the lowest $\Omega_m^{RPA}$ excitation energy, so that
|
||||
e.g. $( \Omega_{ib}^{s} - \Omega_m^{RPA} )$ is strictly negative and $\widetilde{W}_{ij,ab}( \Oms )$ presents no resonances. Further, $\Omega_{ib}^{s}$ and $\Omega_{ja}^{s}$ are necessarily negative for in-gap low lying BSE excitations, such that e.g.
|
||||
$$
|
||||
\left| \frac{ 1 }{ \Omega_{ib}^{s} - \Omega_m^{RPA} } \right|
|
||||
< \frac{1}{ \Omega_m^{RPA} }
|
||||
\abs{ \frac{1}{\Omega_{ib}^{s} - \Omega_m^{RPA}} } < \frac{1}{ \Omega_m^{RPA}}
|
||||
$$
|
||||
This leads to reduced electron-hole screening, namely larger electron-hole stabilising binding energy, as compared to the standard adiabatic BSE, leading to smaller (red-shifted) excitation energies. }
|
||||
|
||||
@ -738,6 +765,7 @@ All the BSE calculations have been performed with our locally developed $GW$ sof
|
||||
& 5.81 & 5.73 & 6.30 & 5.72 & 6.26
|
||||
& 5.63 & 5.85 & 6.22 & 5.94 & 6.33 \\
|
||||
\\
|
||||
%T2: check state ordering in BSE calculation
|
||||
\ce{C2H4} & $^1B_{3u}(\pi \ra 3s)$ & 11.49 & 7.64 & 7.62 & -0.03 & 1.004
|
||||
& 7.35 & 7.34 & 7.42 & 7.29 & 7.35
|
||||
& 6.63 & 6.88 & 6.94 & 6.93 & 7.57 \\
|
||||
|
||||
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Reference in New Issue
Block a user