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BSE-PES.bib
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%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-01-07 22:26:48 +0100 %% Created for Pierre-Francois Loos at 2020-01-25 14:38:41 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Angyan_2011,
Author = {J. G. Angyan and R.-F. Liu and J. Toulouse and G. Jansen},
Date-Added = {2020-01-25 14:20:51 +0100},
Date-Modified = {2020-01-25 14:23:02 +0100},
Doi = {10.1021/ct200501r},
Journal = {J. Chem. Theory Comput.},
Pages = {3116--3130},
Title = {Correlation Energy Expressions from the Adiabatic-Connection Fluctuation Dissipation Theorem Approach},
Volume = {7},
Year = {2011}}
@article{Ghosh_2018, @article{Ghosh_2018,
Author = {Ghosh, Soumen and Verma, Pragya and Cramer, Christopher J. and Gagliardi, Laura and Truhlar, Donald G.}, Author = {Ghosh, Soumen and Verma, Pragya and Cramer, Christopher J. and Gagliardi, Laura and Truhlar, Donald G.},
Date-Added = {2020-01-07 16:02:08 +0100}, Date-Added = {2020-01-07 16:02:08 +0100},

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BSE-PES.tex
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@ -54,6 +54,7 @@
\newcommand{\Norb}{N} \newcommand{\Norb}{N}
\newcommand{\Nocc}{O} \newcommand{\Nocc}{O}
\newcommand{\Nvir}{V} \newcommand{\Nvir}{V}
\newcommand{\IS}{\lambda}
% operators % operators
\newcommand{\hH}{\Hat{H}} \newcommand{\hH}{\Hat{H}}
@ -125,6 +126,8 @@
\newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}} \newcommand{\OmtBSE}[1]{{}^{3}\Omega^\text{BSE}_{#1}}
% Matrices % Matrices
\newcommand{\bO}{\mathbf{0}}
\newcommand{\bI}{\mathbf{1}}
\newcommand{\bvc}{\mathbf{v}} \newcommand{\bvc}{\mathbf{v}}
\newcommand{\bSig}{\mathbf{\Sigma}} \newcommand{\bSig}{\mathbf{\Sigma}}
\newcommand{\bSigX}{\mathbf{\Sigma}^\text{x}} \newcommand{\bSigX}{\mathbf{\Sigma}^\text{x}}
@ -144,6 +147,8 @@
\newcommand{\bX}{\mathbf{X}} \newcommand{\bX}{\mathbf{X}}
\newcommand{\bY}{\mathbf{Y}} \newcommand{\bY}{\mathbf{Y}}
\newcommand{\bZ}{\mathbf{Z}} \newcommand{\bZ}{\mathbf{Z}}
\newcommand{\bK}{\mathbf{K}}
\newcommand{\bP}{\mathbf{P}}
% units % units
\newcommand{\IneV}[1]{#1 eV} \newcommand{\IneV}[1]{#1 eV}
@ -281,7 +286,7 @@ Finally, the static approximation is enforced, \ie, $\W{}(1,2) = \W{}(\{\br{1},
\label{sec:BSE_basis} \label{sec:BSE_basis}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To compute the singlet and triplet BSE excitation energies in a finite basis within the static approximation, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016} For a closed-shell system, in order to compute the singlet BSE excitation energies within the static approximation in a finite basis, one must solve the following linear response problem \cite{Casida,Dreuw_2005,Martin_2016}
\begin{equation} \begin{equation}
\label{eq:LR} \label{eq:LR}
\begin{pmatrix} \begin{pmatrix}
@ -293,38 +298,42 @@ To compute the singlet and triplet BSE excitation energies in a finite basis wit
\bY_m \\ \bY_m \\
\end{pmatrix} \end{pmatrix}
= =
\OmBSE{m} \Om{m}
\begin{pmatrix} \begin{pmatrix}
\bX_m \\ \bX_m \\
\bY_m \\ \bY_m \\
\end{pmatrix}, \end{pmatrix},
\end{equation} \end{equation}
where $\OmBSE{m}$ is the $m$th excitation energy with eigenvector $\T{(\bX_m \, \bY_m)}$, and we have assumed real-valued orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$. where $\Om{m}$ is the $m$th excitation energy with eigenvector $\T{(\bX_m \, \bY_m)}$, and we have assumed real-valued spatial orbitals $\{\MO{p}(\br{})\}_{1 \le p \le \Norb}$.
The matrices $\bA$, $\bB$, $\bX$, and $\bY$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$), respectively. The matrices $\bA$, $\bB$, $\bX$, and $\bY$ are all of size $\Nocc \Nvir \times \Nocc \Nvir$ where $\Nocc$ and $\Nvir$ are the number of occupied and virtual orbitals (\ie, $\Norb = \Nocc + \Nvir$), respectively.
In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$, $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals. In the following, the index $m$ labels the $\Nocc \Nvir$ single excitations, $i$ and $j$ are occupied orbitals, $a$ and $b$ are unoccupied orbitals, while $p$, $q$, $r$, and $s$ indicate arbitrary orbitals.
In the absence of triplet instabilities, \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension In the absence of instabilities (\ie, $\bA - \bB$ is positive-definite), \cite{Dreuw_2005} Eq.~\eqref{eq:LR} is usually transformed into an Hermitian eigenvalue problem of smaller dimension
\begin{equation} \begin{equation}
\label{eq:small-LR} \label{eq:small-LR}
(\bA - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ, (\bA - \bB)^{1/2} (\bA + \bB) (\bA - \bB)^{1/2} \bZ = \bOm^2 \bZ,
\end{equation} \end{equation}
where the excitation amplitudes are where the excitation amplitudes are
\begin{equation} \begin{subequations}
\bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ. \begin{align}
\end{equation} \bX + \bY = \bOm^{-1/2} (\bA - \bB)^{1/2} \bZ,
The specific expression of the matrix elements of $\bA$ and $\bB$ are \\
\bX - \bY = \bOm^{1/2} (\bA - \bB)^{-1/2} \bZ.
\end{align}
\end{subequations}
In the case of BSE, the specific expression of the matrix elements are
\begin{subequations} \begin{subequations}
\begin{align} \begin{align}
\label{eq:LR_BSE} \label{eq:LR_BSE}
\A{ia,jb} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \W{ia,bj}(\omega = 0) - (ij|ba), \ABSE{ia,jb} & = \delta_{ij} \delta_{ab} (\eGW{a} - \eGW{i}) + \W{ia,bj}(\omega = 0) - (ia|jb),
\\ \\
\B{ia,jb} & = \W{ia,jb}(\omega = 0) - (ib|ja) , \BBSE{ia,jb} & = \W{ia,jb}(\omega = 0) - (ib|ja) ,
\end{align} \end{align}
\end{subequations} \end{subequations}
where $\eGW{p}$ are the {\GW} quasiparticle energies, where $\eGW{p}$ are the {\GW} quasiparticle energies,
\begin{multline} \begin{multline}
\label{eq:W} \label{eq:W}
\W{ia,jb}(\omega) = (1 + \delta_{\sigma \sigma^{\prime}}) (ia|jb) \W{ia,jb}(\omega) = 2 (ia|jb)
\\ \\
+ \sum_m^{\Nocc \Nvir} [ia|m] [jb|m] \qty(\frac{1}{\omega - \OmRPA{m} + i \eta} + \frac{1}{\omega + \OmRPA{m} - i \eta}) + \sum_m^{\Nocc \Nvir} [ia|m] [jb|m] \qty(\frac{1}{\omega - \OmRPA{m} + i \eta} + \frac{1}{\omega + \OmRPA{m} - i \eta})
\end{multline} \end{multline}
@ -336,56 +345,79 @@ are the screened two-electron integrals,
\begin{equation} \begin{equation}
(pq|rs) = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}', (pq|rs) = \iint \frac{\MO{p}(\br{}) \MO{q}(\br{}) \MO{r}(\br{}') \MO{s}(\br{}')}{\abs*{\br{} - \br{}'}} \dbr{} \dbr{}',
\end{equation} \end{equation}
are the bare two-electron integrals, $\delta_{pq}$ is the Kronecker delta, and are the bare two-electron integrals, and $\delta_{pq}$ is the Kronecker delta.
\begin{equation} In Eq.~\eqref{eq:W}, $\eta$ is a positive infinitesimal, and $\OmRPA{m}$ are the direct (\ie, without exchange) RPA neutral excitation energies computed during the {\GW} calculation by solving the linear eigenvalue problem \eqref{eq:LR} with matrix elements
\delta_{\sigma \sigma^{\prime}} = \begin{subequations}
\begin{cases} \begin{align}
0, & \sigma \neq \sigma^{\prime} \text{ (singlet manifold)}, \label{eq:LR_BSE}
\ARPA{ia,jb} & = \delta_{ij} \delta_{ab} (\eHF{a} - \eHF{i}) + (ia|bj),
\\ \\
1, & \sigma = \sigma^{\prime} \text{ (triplet manifold)}. \BRPA{ia,jb} & = (ia|jb),
\end{cases} \end{align}
\end{equation} \end{subequations}
In Eq.~\eqref{eq:W}, $\OmRPA{m}$ are the neutral direct (\ie, without exchange) dRPA excitation energies computed during the {\GW} calculation, and $\eta$ is a positive infimitesimal. and $\eHF{p}$ are the HF orbital energies.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Ground-state BSE energy} \subsection{Ground-state BSE energy}
\label{sec:BSE_energy} \label{sec:BSE_energy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The key quantity to define in the present context is the total ground-state BSE energy. The key quantity to define in the present context is the total ground-state BSE energy.
Although not unique, we propose to define it as Although this choice is not unique, \cite{Holzer_2018} we propose to define it as
\begin{equation} \begin{equation}
\label{eq:EtotBSE} \label{eq:EtotBSE}
\EBSE{m} = \Enuc + \EHF + \EcBSE \EBSE{m} = \Enuc + \EHF + \EcBSE
\end{equation} \end{equation}
where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), where $\Enuc$ and $\EHF$ are the nuclear repulsion energy and ground-state HF energy (respectively), and
\begin{equation} \begin{equation}
\label{eq:EcBSE} \label{eq:EcBSE}
\EcBSE = \frac{1}{2} \qty[ {\sum_m}' \OmBSE{m} - \Tr(\bA) ] \EcBSE = \frac{1}{2} \int_0^1 \Tr(\bK \bP_\IS) d\IS
\end{equation} \end{equation}
is the ground-state BSE correlation energy computed in the adiabatic connection framework is the ground-state BSE correlation energy computed in the adiabatic connection framework, where
\begin{equation}
\bK =
\begin{pmatrix}
\bA' & \bB \\
\bB & \bA' \\
\end{pmatrix}
\end{equation}
is the interaction kernel \cite{Angyan_2011, Holzer_2018} [with $\A{ia,jb}' = (ia|bj)$],
\begin{equation}
\bP_\lambda =
\begin{pmatrix}
\bY_\IS \T{\bY}_\IS & \bY_\IS \T{\bX}_\IS \\
\bX_\IS \T{\bY}_\IS & \bX_\IS \T{\bX}_\IS \\
\end{pmatrix}
-
\begin{pmatrix}
\bO & \bO \\
\bO & \bI \\
\end{pmatrix}
\end{equation}
is the correlation part of the two-electron density matrix at interaction strength $\IS$, $\Tr$ denotes the matrix trace.
Note that, it is unnecessary to compute the triplet contribution as it is strictly zero.
Note that the present formulation is different from the plasmon formulation. \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020} Note that the present formulation is different from the plasmon formulation. \cite{Schuck_Book, Gell-Mann_1957, Rowe_1968, Sawada_1957b, Li_2020}
Note that, at the dRPA level, the plasmon and adiabatic connection formulations are equivalent. \cite{Sawada_1957b, Fukuta_1964, Furche_2008} Note that, at the dRPA level, the plasmon and adiabatic connection formulations are equivalent. \cite{Sawada_1957b, Fukuta_1964, Furche_2008}
Howewer, this is not the case at the BSE level. However, this is not the case at the BSE level.
One of the undisputable advantage of the adiabtic connection formulation is that the triplet does not contribute. One of the indisputable advantage of the adiabatic connection formulation is that the triplet does not contribute.
Therefore, the triplet instabilities, some of the triplet excitation energies are negative, and must be discarded as the resonant-only part of the BSE excitonic Hamiltonian has to be considered. Therefore, the triplet instabilities, some of the triplet excitation energies are negative, and must be discarded as the resonant-only part of the BSE excitonic Hamiltonian has to be considered.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Computational details} \section{Computational details}
\label{sec:comp_details} \label{sec:comp_details}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we consider here. All the preliminary {\GW} calculations performed to obtain the screened Coulomb operator and the quasiparticle energies have been done using a (restricted) Hartree-Fock (HF) starting point, which is a very adequate choice in the case of the (small) systems that we have considered here.
Dunning's basis sets are defined in cartesian gaussians. Perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} calculations are employed as starting point to compute the BSE neutral excitations.
Both perturbative {\GW} (or {\GOWO}) \cite{Hybertsen_1985a, Hybertsen_1986} and partially self-consistent {\evGW} \cite{Hybertsen_1986, Shishkin_2007, Blase_2011, Faber_2011} calculations are employed as starting point to compute the BSE neutral excitations. These will be labeled as BSE@{\GOWO}.
These will be labeled as BSE@{\GOWO} and BSE@{\evGW}, respectively.
In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation. In the case of {\GOWO}, the quasiparticle energies have been obtained by linearizing the non-linear, frequency-dependent quasiparticle equation.
For {\evGW}, the quasiparticle energies are obtained self-consistently and we have used the DIIS convergence accelerator technique proposed by Pulay \cite{Pulay_1980,Pulay_1982} to avoid convergence issues. Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018,Veril_2018}.
Further details about our implementation of {\GOWO} and {\evGW} can be found in Refs.~\onlinecite{Loos_2018,Veril_2018}.
Finally, the infinitesimal $\eta$ has been set to zero for all calculations. Finally, the infinitesimal $\eta$ has been set to zero for all calculations.
\titou{For sake of comparison, no frozen core approximation. The numerical integration required to compute the correlation energy along the adiabatic path has been performed with a 21-point Gauss-Legendre quadrature.
The numerical integration required to compute the correlation energy along the adiabatic path has been computed with a 21-point Gauss-Legendre quadrature. } As one-electron basis sets, we employ the Dunning family (cc-pVXZ) defined with cartesian gaussian functions.
Unless, otherwise stated, the frozen-core approximation has not been enforced in our calculations in order to provide a fairer comparison between methods.
However, we have found that the conclusions drawn in the present study hold within the frozen-core approximation (see {\SI}).
Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both singlet and triplet), we perform a complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3}$ computational cost. Because Eq.~\eqref{eq:EcBSE} requires the entire BSE excitation spectrum (both singlet and triplet), we perform a complete diagonalization of the $\Nocc \Nvir \times \Nocc \Nvir$ BSE linear response matrix [see Eq.~\eqref{eq:small-LR}], which corresponds to a $\order{\Nocc^3 \Nvir^3} = \order{\Norb^6}$ computational cost.
This step is, by far, the computational bottleneck in our current implementation. This step is, by far, the computational bottleneck in our current implementation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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