diff --git a/BSEdyn.tex b/BSEdyn.tex index 019dade..cc8fb72 100644 --- a/BSEdyn.tex +++ b/BSEdyn.tex @@ -52,7 +52,6 @@ \newcommand{\co}{\text{x}} % -\newcommand{\Nel}{N} \newcommand{\Norb}{N_\text{orb}} \newcommand{\Nocc}{O} \newcommand{\Nvir}{V} @@ -233,14 +232,14 @@ which is itself a corrected version of the Kohn-Sham (KS) gap \end{equation} in order to approximate the optical gap \begin{equation} - \EgOpt = E_1^{\Nel} - E_0^{\Nel} = \EgFun + \EB, + \EgOpt = E_1^{N} - E_0^{N} = \EgFun + \EB, \end{equation} where \begin{equation} \label{eq:Egfun} - \EgFun = I^\Nel - A^\Nel + \EgFun = I^N - A^N \end{equation} -is the the fundamental gap, \cite{Bredas_2014} $I^\Nel = E_0^{\Nel-1} - E_0^\Nel$ and $A^\Nel = E_0^{\Nel+1} - E_0^\Nel$ being the ionization potential and the electron affinity of the $\Nel$-electron system. -Here, $E_s^{\Nel}$ is the total energy of the $s$th excited state of the $\Nel$-electron system, and $E_0^\Nel$ corresponds to its ground-state energy. +is the the fundamental gap, \cite{Bredas_2014} $I^N = E_0^{N-1} - E_0^N$ and $A^N = E_0^{N+1} - E_0^N$ being the ionization potential and the electron affinity of the $N$-electron system. +Here, $E_s^{N}$ is the total energy of the $s$th excited state of the $N$-electron system, and $E_0^N$ corresponds to its ground-state energy. Because the excitonic effect corresponds physically to the stabilization implied by the attraction of the excited electron and its hole left behind, we have $\EgOpt < \EgFun$. Most of BSE implementations rely on the so-called static approximation, which approximates the dynamical (\ie, frequency-dependent) BSE kernel by its static limit. @@ -275,33 +274,36 @@ Unless otherwise stated, atomic units are used. %%%%%%%%%%%%%%%%%%%%%%%% In this Section, following Strinati's seminal work, \cite{Strinati_1988} we first derive in some details the theoretical foundations leading to the dynamical Bethe-Salpeter equation. -We present, in a second step, the perturbative implementation of the dynamical correction \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} as compared to the standard static approximation. +We present, in a second step, the perturbative implementation of the dynamical correction as compared to the standard static approximation. \cite{Rohlfing_2000,Ma_2009a,Ma_2009b,Baumeier_2012b} More details about this derivation are provided as {\SI}. %================================ \subsection{General dynamical BSE theory} %================================= -The two-particle correlation function $L(1,2; 1',2')$ central to the BSE formalism relates the variation of the one-particle Green's function $G(1,1')$ with respect to an external non-local perturbation $U(2',2)$, namely: -$$ -iL(1,2; 1',2') = \frac{ \partial G(1,1') }{ \partial U(2',2) } -$$ -where, \eg, $1 \equiv (\bx_1 t_1)$ is a space-spin plus time composite variable. The relation between $G$ and the charge density $\; \rho(1) = -i G(1,1^+)$ provides a direct connection with the density-density susceptibility at the core of TD-DFT with $\chi(1,2) = L(1,2;1^+,2^+)$. The notation $1^+$ means that the time $t_1$ is taken at $t_1^{+} = t_1 + 0^+$ where $0^+$ is a small positive infitesimal. This two-particle correlation function $L$ satisfies the self-consistent Bethe-Salpeter equation\cite{Strinati_1988} +The two-particle correlation function $L(1,2; 1',2')$ --- a central quantity in the BSE formalism --- relates the variation of the one-body Green's function $G(1,1')$ with respect to an external non-local perturbation $U(2',2)$, \ie, +\begin{equation} + iL(1,2; 1',2') = \pdv{G(1,1')}{U(2',2)} +\end{equation} +where, \eg, $1 \equiv (\bx_1 t_1)$ is a space-spin plus time composite variable. +The relation between $G$ and the charge density $\rho(1) = -i G(1,1^+)$ provides a direct connection with the density-density susceptibility $\chi(1,2) = L(1,2;1^+,2^+)$ at the core of TD-DFT. +(The notation $1^+$ means that the time $t_1$ is taken at $t_1^{+} = t_1 + 0^+$ where $0^+$ is a small positive infinitesimal.) +The two-body correlation function $L$ satisfies the self-consistent Bethe-Salpeter equation \cite{Strinati_1988} \begin{multline} \label{eq:BSE} - L(1,2; 1',2') - = L_0(1,2;1',2') - \\ - + \int d3456 \; L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2'), + L(1,2; 1',2') = L_0(1,2;1',2') + \\ + + \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6) L(6,2;5,2'), \end{multline} -where $\Xi$ is the BSE kernel +where \begin{equation} \Xi(3,5;4,6) = i \fdv{[v_\text{H}(3) \delta(3,4) + \Sigma_\text{xc}(3,4)]}{G(6,5)} \end{equation} -that takes into account the self-consistent variation of the Hartree potential +is the BSE kernel that takes into account the self-consistent variation of the Hartree potential \begin{equation} - v_\text{H}(1) = - i \int d2 v(1,2) G(2,2^+), + v_\text{H}(1) = - i \int d2 \, v(1,2) G(2,2^+), \end{equation} -[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the exchange-correlation self-energy $ \Sigma_\text{xc}$ with respect to the variation of the one-body Green's function $G$. $L$ and $L_0$ can be expressed as a function of the one-body and two-body ($G_2$) Green's functions: +[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the exchange-correlation self-energy $ \Sigma_\text{xc}$ with respect to the variation of $G$. +$L$ and $L_0$ can be expressed as a function of the one- and two-body ($G_2$) Green's functions as follows: \begin{gather} \label{eq:L0} iL_0(1, 4; 1', 3) = G(1, 3)G(4, 1'), @@ -312,19 +314,19 @@ that takes into account the self-consistent variation of the Hartree potential \label{eq:G2} i^2 G_2(1,2;1',2') = \mel{N}{T \hpsi(1) \hpsi(2) \hpsi^{\dagger}(2') \hpsi^{\dagger}(1')}{N}, \end{gather} -where the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ in Eq.~\eqref{eq:G2} remove and add an electron (respectively) in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator. +where the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add an electron (respectively) to the $N$-electron ground state $\ket{N}$ in space-spin-time positions ($\bx t$) and ($\bx't'$), while $T$ is the time-ordering operator. -The resolution of the dynamical BSE equation\cite{Strinati_1988} starts with the expansion of the two-body Green's function $G_2$ and the response function $L$ over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (where $\ket{N} \equiv \ket{N,0}$ corresponds to the ground state). +The resolution of the dynamical BSE equation \cite{Strinati_1988} starts with the expansion of $G_2$ and $L$ over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$). In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, one gets \begin{equation} \begin{split} iL(1,2; 1',2') - & = \theta(+\tau_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Oms \tau_{12} } + & = \theta(+t_{12}) \sum_{s > 0} \chi_s(\bx_1,\bx_{1'}) \tchi_s(\bx_2,\bx_{2'}) e^{ - i \Oms t_{12} } \\ - & - \theta(-\tau_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms \tau_{12} }, + & - \theta(-t_{12}) \sum_{s > 0} \chi_s(\bx_2,\bx_{2'}) \tchi_s(\bx_1,\bx_{1'}) e^{ + i \Oms t_{12} }, \end{split} \end{equation} -with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and +where $t_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and \begin{subequations} \begin{align} \chi_s(\bx_1,\bx_{2}) & = \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{2})}{N,s}, @@ -333,43 +335,44 @@ with $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and \end{align} \end{subequations} The $\Oms$'s are the neutral excitation energies of interest. -\titou{T2: shall we specify the physical meaning of $\chi_s$ and $\tchi_s$?} -Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$ functions, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ 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 modified dynamical Bethe-Salpeter equation: +Picking up the $e^{+i \Oms t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both side of the BSE [see Eq.~\eqref{eq:BSE}], we are left with the search of the $e^{-i \Oms t_1 }$ Fourier component associated with the right-hand side of the modified dynamical Bethe-Salpeter equation: \begin{multline} \label{eq:BSE_2} - \mel{N}{T \hpsi(\bx_1) &\hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 } - \theta ( \tau_{12} ) = \int d3456 \times + \mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')}{N,s} e^{ - i \Oms t_1 } + \theta ( t_{12} ) \\ - \times L_0(1,4;1',3) \Xi(3,5;4,6) - \mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s} - \theta (t^{56}_m - t_2) + = \int d3456 \, L_0(1,4;1',3) \Xi(3,5;4,6) + \\ + \times \mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s} + \theta [\min(t_5,t_6) - t_2]. \end{multline} -with $t^{56}_m = \min(t_5,t_6)$. \titou{For the lowest $\Oms$ excitation energies falling in the quasiparticle gap of the system} due to excitonic effects, $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response since its lowest excitation energy is precisely the quasiparticle gap, namely the difference between the electronic affinity and the ionization potential. -\titou{T2: I think we should specify at which level of theory this quasiparticle gap is computed. What do you think?} +For the lowest excitation energies falling in the fundamental gap of the system (\ie, $\Oms < \EgFun$), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Oms t_1 }$ response due to excitonic effects since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}]. +\titou{T2: Xavier, should we mention the consequences of this more explicitly?} -The Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads, dropping the (space/spin) variables +Dropping the (space/spin) variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads \begin{align} \label{eq:iL0} [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} } + e^{ i \omega t_{34} } e^{ i \omega_1 t^{34} } \end{align} -with $\tau_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$. +with $t_{34} = t_3 - t_4$ and $t^{34} = (t_3 + t_4)/2$. We now adopt the Lehman representation of the one-body Green's function in the quasiparticle approximation, \ie, -\begin{equation} +\begin{equation} \label{eq:G-Lehman} G(\bx_1,\bx_2 ; \omega) = \sum_p \frac{ \phi_p(\bx_1) \phi_p^*(\bx_2) } { \omega - \e{p} + i \eta \times \text{sgn} (\e{p} - \mu) } \end{equation} -where \titou{$\mu$ is the chemical potential}. -The set $\lbrace \e{p} \rbrace$ are quasiparticle energies and $\lbrace \phi_p \rbrace$ is their associated one-body \titou{(spin)} orbitals, \titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)} -After projecting onto \titou{$\phi_a^*(\bx_1) \phi_i(\bx_{1'})$}, one obtains the $\omega_1= \Oms$ component +where $\mu$ is the chemical potential. +The set $\lbrace \e{p} \rbrace$ in Eq.~\eqref{eq:G-Lehman} are quasiparticle energies and $\lbrace \phi_p \rbrace$ is their associated one-body (spin)orbitals. +In the following, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. +%\titou{namely $GW$ quasiparticle energies and input Hartree-Fock molecular orbitals in the present study. (T2: shall we really mention this here?)} +After projecting onto $\phi_a^*(\bx_1) \phi_i(\bx_{1'})$, one gets \begin{multline} \int d\bx_1 d\bx_{1'} \; \phi_a^*(\bx_1) \phi_i(\bx_{1'}) L_0(\bx_1,3;\bx_{1'},4; \Oms) - = e^{i \Oms t^{34} } \times \\ - \times - \frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) } { \Oms - ( \e{a} - \e{i} ) + i \eta } - \times \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ] + \\ + = + \frac{ \phi_a^*(\bx_3) \phi_i(\bx_4) e^{i \Oms t^{34} }} { \Oms - ( \e{a} - \e{i} ) + i \eta } + \qty[ \theta( \tau ) e^{i ( \e{i} + \hOms) \tau } + \theta( - \tau ) e^{i (\e{a} - \hOms \tau) } ] \end{multline} -with $\tau = \tau_{34}$ and where - $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively. +with $\tau = t_{34}$. % and $(i,j)$/$(a,b)$ index occupied/virtual orbitals, respectively. Adopting now the $GW$ approximation for the exchange-correlation self-energy, \ie, \begin{equation} \Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2), @@ -380,17 +383,15 @@ leads to the following simplified BSE kernel \end{equation} where $W$ is its dynamically-screened Coulomb operator. -As a final step, we express the $\mel{N}{T \hpsi(\bx_1) &\hpsi^{\dagger}(\bx_{1}')}{N,s}$ and $\mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s}$ weights present in Eq.~\ref{eq:BSE_2} in the standard -electron-hole product space, with -$(6,5) \rightarrow (5,5) \; \text{or} \; (3,4)$ when multiplied by $\delta(5,6)$ or $\delta(3,6) \delta(4,5)$, respectively. -This is done by expanding the field operators over a complete orbital basis creation/destruction operators, with e.g. \titou{(T2: I don't understand why we need this spectral representation)}: +As a final step, we express the terms $\mel{N}{T \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}')}{N,s}$ and $\mel{N}{T \hpsi(6) \hpsi^{\dagger}(5)}{N,s}$ from Eq.~\eqref{eq:BSE_2} in the standard electron-hole product (or single-excitation) space, with $(6,5) \rightarrow (5,5) \; \text{or} \; (3,4)$ when multiplied by $\delta(5,6)$ or $\delta(3,6) \delta(4,5)$, respectively. +This is done by expanding the field operators over a complete orbital basis creation/destruction operators, with \eg, \begin{multline} \mel{N}{T \hpsi(3) \hpsi^{\dagger}(4)}{N,s} \\ = - \qty( e^{ -i \Omega_s t^{34} } ) \sum_{pq} \phi_p(\bx_3) \phi_q^*(\bx_4) \mel{N}{\ha_q^{\dagger} \ha_p}{N,s} \\ - \times \qty[ \theta( \tau_{34} ) e^{- i ( \e{p} - \hOms ) \tau_{34} } + \theta( -\tau_{34} ) e^{ - i ( \e{q} + \hOms) \tau_{34} } ] + \times \qty[ \theta( t_{34} ) e^{- i ( \e{p} - \hOms ) t_{34} } + \theta( - t_{34} ) e^{ - i ( \e{q} + \hOms) t_{34} } ] \end{multline} where the $ \lbrace \eps_{p/q} \rbrace$ are proper addition/removal energies \titou{(T2: shall it be mentioned earlier around Eq. (14)?)} such that \begin{equation}