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BSEdyn.tex
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@ -299,7 +299,7 @@ We present, in a second step, the perturbative implementation of the dynamical c
\subsection{General dynamical BSE} \subsection{General dynamical BSE}
%================================= %=================================
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, The two-body 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} \begin{equation}
iL(1,2; 1',2') = \pdv{G(1,1')}{U(2',2)}, iL(1,2; 1',2') = \pdv{G(1,1')}{U(2',2)},
\end{equation} \end{equation}
@ -344,7 +344,7 @@ is the BSE kernel that takes into account the self-consistent variation of the H
[where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the xc self-energy $ \Sigma_\text{xc}$ with respect to the variation of $G$. [where $\delta$ is Dirac's delta function and $v$ is the bare Coulomb operator] and the xc self-energy $ \Sigma_\text{xc}$ with respect to the variation of $G$.
In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add (respectively) an electron 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. In Eqs.~\eqref{eq:G1} and \eqref{eq:G2}, the field operators $\Hat{\psi}(\bx t)$ and $\Hat{\psi}^{\dagger}(\bx't')$ remove and add (respectively) an electron 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 $L_0$ and $L$ [see Eqs.~\eqref{eq:L0} and \eqref{eq:L}] over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$). The resolution of the dynamical BSE starts with the expansion of $L_0$ and $L$ [see Eqs.~\eqref{eq:L0} and \eqref{eq:L}] over the complete orthonormalized set of $N$-electron excited states $\ket{N,s}$ (with $\ket{N,0} \equiv \ket{N}$). \cite{Strinati_1988}
In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, and using the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, \eg, In the optical limit of instantaneous electron-hole creation and destruction, imposing $t_{2'} = t_2^+$ and $t_{1'} = t_1^+$, and using the relation between the field operators in their time-dependent (Heisenberg) and time-independent (Schr\"{o}dinger) representations, \eg,
\begin{equation} \label{Eisenberg} \begin{equation} \label{Eisenberg}
\hpsi(1) = e^{ i \hH t_1 } \hpsi(\bx_1) e^{-i \hH t_1 }, \hpsi(1) = e^{ i \hH t_1 } \hpsi(\bx_1) e^{-i \hH t_1 },
@ -368,7 +368,7 @@ where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\end{subequations} \end{subequations}
The $\Om{s}{}$'s are the neutral excitation energies of interest. The $\Om{s}{}$'s are the neutral excitation energies of interest.
Picking up the $e^{+i \Om{s}{} t_2 }$ component of $L(1,2; 1',2')$ 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 seeking the $e^{-i \Om{s}{} t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads Picking up the $e^{+i \Om{s}{} t_2 }$ component of both $L(1,2; 1',2')$ and $L(6,2;5,2')$, simplifying further by $\tchi_s(\bx_2,\bx_{2'})$ on both sides of the BSE [see Eq.~\eqref{eq:BSE}], we seek the $e^{-i \Om{s}{} t_1 }$ Fourier component associated with the right-hand side of a modified dynamical BSE, which reads
\begin{multline} \label{eq:BSE_2} \begin{multline} \label{eq:BSE_2}
\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Om{s}{} t_1 } \mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Om{s}{} t_1 }
\theta ( \tau_{12} ) \theta ( \tau_{12} )
@ -401,9 +401,11 @@ Projecting the Fourier component $L_0(\bx_1,4;\bx_{1'},3; \omega_1 = \Om{s}{} )
\\ \\
= =
\frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Om{s}{} t^{34} }} { \Om{s}{} - ( \e{a} - \e{i} ) + i \eta } \frac{ \MO{a}^*(\bx_3) \MO{i}(\bx_4) e^{i \Om{s}{} t^{34} }} { \Om{s}{} - ( \e{a} - \e{i} ) + i \eta }
\qty[ \theta( \tau_{34} ) e^{i ( \e{i} + \frac{\Om{s}{}}{2}) \tau_{34} } + \theta( - \tau_{34} ) e^{i (\e{a} - \frac{\Om{s}{}}{2}) \tau_{34} } ]. \qty[ \theta( \tau_{34} ) e^{i \qty( \e{i} + \frac{\Om{s}{}}{2}) \tau_{34} } + \theta( - \tau_{34} ) e^{i \qty(\e{a} - \frac{\Om{s}{}}{2}) \tau_{34} } ].
\end{multline} \end{multline}
with $t^{34} = (t_3 + t_4)/2$ and $\tau_{34} = t_3 -t_4$.More details are provided in the Appendix. 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 $t^{34} = (t_3 + t_4)/2$ and $\tau_{34} = t_3 -t_4$.
More details are provided in the Appendix.
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.
This is done by expanding the field operators over a complete orbital basis of creation/destruction operators. This is done by expanding the field operators over a complete orbital basis of creation/destruction operators.
For example, we have (see derivation in the Appendix) For example, we have (see derivation in the Appendix)
\begin{multline} \label{eq:spectral65} \begin{multline} \label{eq:spectral65}
@ -412,9 +414,9 @@ For example, we have (see derivation in the Appendix)
= - \qty( e^{ -i \Omega_s t^{65} } ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5) = - \qty( e^{ -i \Omega_s t^{65} } ) \sum_{pq} \MO{p}(\bx_6) \MO{q}^*(\bx_5)
\mel{N}{\ha_q^{\dagger} \ha_p}{N,s} \mel{N}{\ha_q^{\dagger} \ha_p}{N,s}
\\ \\
\times \qty[ \theta( \tau_{65} ) e^{- i ( \e{p} - \frac{\Om{s}{}}{2} ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i ( \e{q} + \frac{\Om{s}{}}{2}) \tau_{65} } ], \times \qty[ \theta( \tau_{65} ) e^{- i \qty( \e{p} - \frac{\Om{s}{}}{2} ) \tau_{65} } + \theta( - \tau_{65} ) e^{ - i \qty( \e{q} + \frac{\Om{s}{}}{2}) \tau_{65} } ],
\end{multline} \end{multline}
with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$. The $\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}$ are the unknown particle-hole amplitudes in the molecular orbitals product basis. with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$. The $\mel{N}{\ha_q^{\dagger} \ha_p}{N,s}$ are the unknown particle-hole amplitudes.% in the orbital product basis.
%================================ %================================
@ -432,7 +434,7 @@ leads to the following simplified BSE kernel
where $W$ is the dynamically-screened Coulomb operator. where $W$ is the dynamically-screened Coulomb operator.
The $GW$ quasiparticle energies $\eGW{p}$ are usually good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}. The $GW$ quasiparticle energies $\eGW{p}$ are usually good approximations to the removal/addition energies $\e{p}$ introduced in Eq.~\eqref{eq:G-Lehman}.
Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE (dBSE): Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW} into Eq.~\eqref{eq:BSE_2}, and projecting onto $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$, one gets after a few tedious manipulations (see {\SI}) the dynamical BSE:
\begin{equation} \label{eq:BSE-final} \begin{equation} \label{eq:BSE-final}
\begin{split} \begin{split}
( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia,s} ( \eGW{a} - \eGW{i} - \Om{s}{} ) X_{ia,s}
@ -442,7 +444,7 @@ Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW}
\end{split} \end{split}
\end{equation} \end{equation}
with $X_{jb,s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ and $Y_{jb,s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$, and where $\kappa = 2 $ or $0$ for singlet and triplet excited states (respectively). with $X_{jb,s} = \mel{N}{\ha_j^{\dagger} \ha_b}{N,s}$ and $Y_{jb,s} = \mel{N}{\ha_b^{\dagger} \ha_j}{N,s}$, and where $\kappa = 2 $ or $0$ for singlet and triplet excited states (respectively).
Neglecting the anti-resonant terms, $Y_{jb,s}$, in the dBSE, which are (usually) much smaller than their resonant counterparts, $X_{jb,s}$, leads to the well-known Tamm-Dancoff approximation (TDA). Neglecting the anti-resonant terms, $Y_{jb,s}$, in the dynamical BSE, which are (usually) much smaller than their resonant counterparts, $X_{jb,s}$, leads to the well-known Tamm-Dancoff approximation (TDA).
In Eq.~\eqref{eq:BSE-final}, In Eq.~\eqref{eq:BSE-final},
\begin{equation} \begin{equation}
\ERI{ia}{jb} = \iint d\br d\br' \, \MO{i}^*(\br) \MO{a}(\br) v(\br -\br') \MO{j}^*(\br') \MO{b}(\br'), \ERI{ia}{jb} = \iint d\br d\br' \, \MO{i}^*(\br) \MO{a}(\br) v(\br -\br') \MO{j}^*(\br') \MO{b}(\br'),
@ -460,8 +462,7 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
= \iint d\br d\br' \, \MO{i}(\br) \MO{j}^*(\br) W(\br ,\br'; \omega) \MO{a}^*(\br') \MO{b}(\br'). = \iint d\br d\br' \, \MO{i}(\br) \MO{j}^*(\br) W(\br ,\br'; \omega) \MO{a}^*(\br') \MO{b}(\br').
\end{equation} \end{equation}
\xavier{\sout{ A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting $\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s}$ and $L_0(\bx_1,4;\bx_{1'},3; \Om{s}{})$ onto $\MO{i}^*(\bx_1) \MO{a}(\bx_{1'})$ instead of %\xavier{\sout{ A second coupled equation for the $(X_{ia}^{s}, Y_{ia}^{s} )$ vector can be obtained by projecting $\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s}$ and $L_0(\bx_1,4;\bx_{1'},3; \Om{s}{})$ onto $\MO{i}^*(\bx_1) \MO{a}(\bx_{1'})$ instead of $\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$. } }
$\MO{a}^*(\bx_1) \MO{i}(\bx_{1'})$. } }
%================================ %================================
@ -522,8 +523,8 @@ The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
\\ \\
\times \qty[ \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ]. \times \qty[ \frac{1}{\Om{ib}{s} - \Om{m}{\RPA} + i\eta} + \frac{1}{\Om{ja}{s} - \Om{m}{\RPA} + i\eta} ].
\end{multline} \end{multline}
\xavier{One can verify that in the static limit, that can be obtained with \titou{One can verify that in the static limit, that can be obtained with
$\Om{m}{\RPA} \rightarrow +\infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to the static ${W}_{ij,ab}$ ones, and the dBSE formalism recovers the form of the standard BSE formalism.} $\Om{m}{\RPA} \rightarrow +\infty$, the matrix elements $\widetilde{W}_{ij,ab}$ correctly reduce to the static ${W}_{ij,ab}$ ones, and the dynamical BSE formalism recovers the form of the standard BSE formalism.}
Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{s} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{s}{})$ has no resonances. Due to excitonic effects, the lowest BSE excitation energy, $\Om{1}{}$, stands lower than the lowest RPA excitation energy, $\Om{1}{\RPA}$, so that, $\Om{ib}{s} - \Om{m}{\RPA} < 0 $ and $\widetilde{W}_{ij,ab}(\Om{s}{})$ has no resonances.
Furthermore, $\Om{ib}{s}$ and $\Om{ja}{s}$ are necessarily negative quantities for in-gap low-lying BSE excitations. Furthermore, $\Om{ib}{s}$ and $\Om{ja}{s}$ are necessarily negative quantities for in-gap low-lying BSE excitations.
Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$. Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$.