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Pierre-Francois Loos 2020-06-06 14:12:42 +02:00
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BSEdyn.tex
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@ -363,7 +363,7 @@ where $\tau_{12} = t_1 - t_2$, $\theta$ is the Heaviside step function, and
\end{subequations}
The $\Om{s}{}$'s are the neutral excitation energies of interest.
Picking up the $e^{+i \Om{s}{} t_2 }$ 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 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 $L(1,2; 1',2')$ 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
\begin{multline} \label{eq:BSE_2}
\mel{N}{T [ \hpsi(\bx_1) \hpsi^{\dagger}(\bx_{1}') ] } {N,s} e^{ - i \Om{s}{} t_1 }
\theta ( \tau_{12} )
@ -373,10 +373,10 @@ Picking up the $e^{+i \Om{s}{} t_2 }$ component in $L(1,2; 1',2')$ and $L(6,2;5,
\times \mel{N}{T [\hpsi(6) \hpsi^{\dagger}(5)] }{N,s}
\theta [\min(t_5,t_6) - t_2].
\end{multline}
For the lowest neutral excitation energies falling in the fundamental gap of the system (\ie, $\Om{s}{} < \EgFun$ due to excitonic effects), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Om{s}{} t_1 }$ response since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
For the lowest neutral excitation energies falling in the fundamental gap of the system (\ie, $\Om{s}{} < \EgFun$ due to excitonic effects), $L_0(1,2;1',2')$ cannot contribute to the $e^{-i \Om{s}{} t_1 }$ response term since its lowest excitation energy is precisely the fundamental gap [see Eq.~\eqref{eq:Egfun}].
Consequently, special care has to be taken for high-lying excited states (like core or Rydberg excitations) where additional terms have to be taken into account (see Refs.~\onlinecite{Strinati_1982,Strinati_1984}).
Dropping the (space/spin) variables, the Fourier components with respect to $t_1$ of $L_0(1,4;1',3)$ reads
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} )
@ -418,7 +418,7 @@ with $t^{65} = (t_5 + t_6)/2$ and $\tau_{65} = t_6 -t_5$.
\subsection{Dynamical BSE within the $GW$ approximation}
%=================================
Adopting now the $GW$ approximation for the xc self-energy, \ie,
Adopting now the $GW$ approximation \cite{Hedin_1965} for the xc self-energy, \ie,
\begin{equation}
\Sigma_\text{xc}^{\GW}(1,2) = i G(1,2) W(1^+,2),
\end{equation}
@ -444,7 +444,7 @@ Substituting Eqs.~\eqref{eq:iL0bis}, \eqref{eq:spectral65}, and \eqref{eq:Xi_GW}
\end{split}
\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).
Neglecting the anti-resonant terms, $Y_{jb,s}$, in the dBSE, which are 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 dBSE, 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},
\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'),
@ -471,7 +471,7 @@ is an effective dynamically-screened Coulomb potential, \cite{Romaniello_2009b}
In the present study, we consider the exact spectral representation of $W$ at the RPA level:
\begin{multline}
\label{eq:W}
\label{eq:W-RPA}
W_{ij,ab}(\omega)
= \ERI{ij}{ab} + 2 \sum_m \sERI{ij}{m} \sERI{ab}{m}
\\
@ -483,7 +483,7 @@ where $m$ labels single excitations, and
\sERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} (\bX{m}{\RPA} + \bY{m}{\RPA})_{ia}
\end{equation}
are the spectral weights.
In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are RPA neutral excitations and their corresponding transition vectors computed by solving the (static) linear response problem
In Eqs.~\eqref{eq:W-RPA} and \eqref{eq:sERI}, $\OmRPA{m}{}$ and $(\bX{m}{\RPA} + \bY{m}{\RPA})$ are RPA neutral excitations and their corresponding transition vectors computed by solving the (static) linear response problem
\begin{equation}
\label{eq:LR-RPA}
\begin{pmatrix}
@ -525,17 +525,17 @@ The analysis of the poles of the integrand in Eq.~\eqref{eq:wtilde} yields
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.
Thus, we have $\abs*{\Omega_{ib}^{s} - \Om{m}{\RPA}} > \Omega_m^{\RPA}$.
As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole binding energy, as compared to the standard static BSE, and yields smaller (red-shifted) excitation energies.
As a consequence, we observe a reduction of the electron-hole screening, \ie, an enhancement of electron-hole binding energy, as compared to the standard static BSE, and consequently smaller (red-shifted) excitation energies.
%================================
\subsection{Perturbative dynamical correction}
%=================================
From a more practical point of view, to compute the BSE excitation energies of a closed-shell system, one must solve the following (non-linear) dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988}
From a more practical point of view, Eq.~\eqref{eq:BSE-final} can be recast as an non-linear eigenvalue problem and, to compute the BSE excitation energies of a closed-shell system, one must solve the following dynamical (\ie, frequency-dependent) response problem \cite{Strinati_1988}
\begin{equation}
\label{eq:LR-dyn}
\begin{pmatrix}
\bA{}(\Om{s}{}) & \bB{}(\Om{s}{})
\bA{}(\Om{s}{}) & \bB{}(\Om{s}{})
\\
-\bB{}(\Om{s}{}) & -\bA{}(\Om{s}{})
\\
@ -618,10 +618,12 @@ and
\B{ia,jb}{(1)}(\Om{s}{}) & = - \tW{ib,aj}{}(\Om{s}{}) + \W{ib,aj}{\text{stat}},
\end{align}
\end{subequations}
where we have defined the static version of the screened Coulomb potential
where we have defined the static version of the screened Coulomb potential [see Eq.~\eqref{eq:W-RPA}]
\begin{equation}
\label{eq:Wstat}
\W{ij,ab}{\text{stat}} = W_{ij,ab}(\omega = 0) = \ERI{ij}{ab} - 4 \sum_m \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}.
\W{ij,ab}{\text{stat}}
\equiv W_{ij,ab}(\omega = 0)
= \ERI{ij}{ab} - 4 \sum_m \frac{\sERI{ij}{m} \sERI{ab}{m}}{\OmRPA{m}{} - i \eta}.
\end{equation}
According to perturbation theory, the $s$th BSE excitation energy and its corresponding eigenvector can then expanded as
\begin{subequations}
@ -716,7 +718,7 @@ The static BSE Hamiltonian is computed once during the static BSE calculation an
Searching iteratively for the lowest eigenstates, via Davidson's algorithm for instance, can be performed in $\order*{\Norb^4}$ computational cost.
Constructing the static and dynamic BSE Hamiltonians is much more expensive as it requires the complete diagonalization of the $(\Nocc \Nvir \times \Nocc \Nvir)$ RPA linear response matrix [see Eq.~\eqref{eq:LR-RPA}], which corresponds to a $\order*{\Nocc^3 \Nvir^3} = \order*{\Norb^6}$ computational cost.
Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is the computational bottleneck in our current implementation.
Although it might be reduced to $\order*{\Norb^4}$ operations with standard resolution-of-the-identity techniques, \cite{Duchemin_2019,Duchemin_2020} this step is the computational bottleneck in the current implementation.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details}
@ -731,8 +733,8 @@ Further details about our implementation of {\GOWO} can be found in Refs.~\onlin
As one-electron basis sets, we employ the augmented Dunning family (aug-cc-pVXZ) defined with cartesian Gaussian functions.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
For comparison purposes, we employ the theoretical best estimates and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2 \cite{Christiansen_1995}, CCSD, \cite{Purvis_1982} and CC3, \cite{Christiansen_1995b} excitation energies are also extracted.
All the BSE calculations have been performed with the $GW$ software, \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}, where the present perturbative correction has been implemented.
For comparison purposes, we employ the theoretical best estimates (TBEs) and geometries of Refs.~\onlinecite{Loos_2018a,Loos_2019,Loos_2020b} from which CIS(D), \cite{Head-Gordon_1994,Head-Gordon_1995} ADC(2), \cite{Trofimov_1997,Dreuw_2015} CC2 \cite{Christiansen_1995}, and CCSD \cite{Purvis_1982} excitation energies are also extracted.
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}, where the present perturbative correction has been implemented.
%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results and Discussion}