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sfBSE.bib
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@ -1,7 +1,7 @@
%% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-10-25 13:01:52 +0100
%% Created for Pierre-Francois Loos at 2020-10-25 21:09:09 +0100
%% Saved with string encoding Unicode (UTF-8)
@ -17,7 +17,8 @@
Pages = {4326},
Title = {Spin-Flip Methods in Quantum Chemistry},
Volume = {22},
Year = {2020}}
Year = {2020},
Bdsk-Url-1 = {https://doi.org/10.1039/c9cp06507e}}
@article{Zhang_2004,
Author = {Zhang, Fan and Burke, Kieron},

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sfBSE.tex
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@ -40,7 +40,7 @@
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\alert{Here comes the introduction.}
Unless otherwise stated, atomic units are used, and we assume real quantities throughout this manuscript.
In the following, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin?orbit interaction.
In the following, we consider systems with collinear spins and a spin-independent hamiltonian without contributions such as spin-orbit interaction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Unrestricted $GW$ formalism}
@ -58,15 +58,15 @@ In a spin-flip excitation, the hole and particle states, $\MO{i_\sig}$ and $\MO{
%================================
\subsection{The dynamical screening}
%================================
The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system.
The spin-$\sig$ component of the one-body Green's function reads
The pillar of Green's function many-body perturbation theory is the (time-ordered) one-body Green's function, which has poles at the charged excitations (i.e., ionization potentials and electron affinities) of the system. \cite{ReiningBook}
The spin-$\sig$ component of the one-body Green's function reads \cite{ReiningBook,Bruneval_2016a}
\begin{equation}
G^{\sig}(\br_1,\br_2;\omega)
= \sum_i \frac{\MO{i_\sig}(\br_1) \MO{i_\sig}(\br_2)}{\omega - \e{i_\sig}{} - i\eta}
+ \sum_a \frac{\MO{a_\sig}(\br_1) \MO{a_\sig}(\br_2)}{\omega - \e{a_\sig}{} + i\eta}
\end{equation}
where $\eta$ is a positive infinitesimal.
Based on it, one can easily compute the non-interacting polarizability (which is a sum over spins)
Based on the spin-up and spin-down components of $G$, one can easily compute the non-interacting polarizability (which is a sum over spins)
\begin{equation}
\chi_0(\br_1,\br_2;\omega) = - \frac{i}{2\pi} \sum_\sig \int G^{\sig}(\br_1,\br_2;\omega+\omega') G^{\sig}(\br_1,\br_2;\omega') d\omega'
\end{equation}
@ -74,14 +74,15 @@ and subsequently the dielectric function
\begin{equation}
\epsilon(\br_1,\br-2;\omega) = \delta(\br_1 - \br_2) - \int \frac{\chi_0(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation}
where $\delta(\br_1 - \br_2)$ is the Dirac functions.
where $\delta(\br_1 - \br_2)$ is the Dirac function.
Based on this latter ingredient, one can access the dynamically-screened Coulomb potential
\begin{equation}
W(\br_1,\br_2;\omega) = \int \frac{\epsilon^{-1}(\br_1,\br_3;\omega) }{\abs{\br_2 - \br_3}} d\br_3
\end{equation}
which is spin independent as the bare Coulomb interaction $\abs{\br_1 - \br_2}^{-1}$ does not depend on spin coordinates.
Therefore, within the $GW$ formalism, the dynamical screening $W(\omega)$ is computed at the RPA level using the spin-conserved neutral excitations.
In the orbital basis, the spectral representation of $W(\omega)$ reads
Within the $GW$ formalism, the is computed at the RPA level by considering only the manifold of the spin-conserved neutral excitation.
In the orbital basis, the spectral representation of $W$ reads
\begin{multline}
\label{eq:W}
W_{p_\sig q_\sig,r_\sigp s_\sigp}(\omega) = \ERI{p_\sig q_\sig}{r_\sigp s_\sigp}
@ -98,27 +99,27 @@ and the screened two-electron integrals (or spectral weights) are explicitly giv
\begin{equation}
\ERI{p_\sig q_\sig}{m} = \sum_{ia\sigp} \ERI{p_\sig q_\sig}{r_\sigp s_\sigp} (\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})_{i_\sigp a_\sigp}
\end{equation}
In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors $(\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})$ are obtained by solving the following linear response system
In Eqs.~\eqref{eq:W} and \eqref{eq:sERI}, the RPA spin-conserved neutral excitations $\Om{m}{\spc,\RPA}$ and their corresponding eigenvectors $(\bX{m}{\spc,\RPA}+\bY{m}{\spc,\RPA})$ are obtained by solving a linear response system of the form
\begin{equation}
\label{eq:LR-RPA}
\begin{pmatrix}
\bA{}{\spc,\RPA} & \bB{}{\spc,\RPA} \\
-\bB{}{\spc,\RPA} & -\bA{}{\spc,\RPA} \\
\bA{}{} & \bB{}{} \\
-\bB{}{} & -\bA{}{} \\
\end{pmatrix}
\cdot
\begin{pmatrix}
\bX{m}{\spc,\RPA} \\
\bY{m}{\spc,\RPA} \\
\bX{m}{} \\
\bY{m}{} \\
\end{pmatrix}
=
\Om{m}{\spc,\RPA}
\Om{m}{}
\begin{pmatrix}
\bX{m}{\spc,\RPA} \\
\bY{m}{\spc,\RPA} \\
\bX{m}{} \\
\bY{m}{} \\
\end{pmatrix}
\end{equation}
The spin structure of matrices $\bA{}{}$ and $\bB{}{}$ is general
where the expressions of the matrix elements of $\bA{}{}$ and $\bB{}{}$ are specific of the method and of the spin manifold.
The spin structure of these matrices, though, is general
\begin{align}
\label{eq:LR-RPA-AB}
\bA{}{\spc} & = \begin{pmatrix}
@ -142,7 +143,19 @@ The spin structure of matrices $\bA{}{}$ and $\bB{}{}$ is general
\bB{}{\dwup,\updw} & \bO \\
\end{pmatrix}
\end{align}
and does not only apply to the RPA but also to RPAx (\ie, RPA with exchange), BSE and TD-DFT.
In the absence of instabilities, the linear eigenvalue problem \eqref{eq:LR-RPA} has particle-hole symmetry which means that the eigenvalues are obtained by pairs $\pm \Om{m}{}$.
In such a case, $(\bA{}{}-\bB{}{})^{1/2}$ is positive definite, and Eq.~\eqref{eq:LR-RPA} can be recast as a Hermitian problem of half the dimension
\begin{equation}
\label{eq:small-LR}
(\bA{}{} - \bB{}{})^{1/2} \cdot (\bA{}{} + \bB{}{}) \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{} = \bOm{2} \cdot \bZ{}{}
\end{equation}
where the excitation amplitudes are
\begin{equation}
\bX{}{} + \bY{}{} = \bOm{-1/2} \cdot (\bA{}{} - \bB{}{})^{1/2} \cdot \bZ{}{}
\end{equation}
Within the Tamm-Dancoff approximation (TDA), the coupling terms between the resonant and anti-resonant parts, $\bA{}{}$ and $-\bA{}{}$, are neglected, which consist in setting $\bB{}{} = \bO$.
In such a case, Eq.~\eqref{eq:LR-RPA} reduces to $\bA{}{} \cdot \bX{m}{} = \Om{m}{} \bX{m}{}$.
At the RPA level, the matrix elements of $\bA{}{}$ and $\bB{}{}$ are
\begin{subequations}
\begin{align}
@ -187,7 +200,7 @@ Within the acclaimed $GW$ approximation, \cite{Hedin_1965,Golze_2019} the exchan
& = \frac{i}{2\pi} \int G^{\sig}(\br_1,\br_2;\omega+\omega') W(\br_1,\br_2;\omega') e^{i \eta \omega'} d\omega'
\end{split}
\end{equation}
is, like the one-body Green's function, spin-diagonal, and its spectral representation read
is, like the one-body Green's function, spin-diagonal, and its spectral representation reads
\begin{gather}
\SigX{p_\sig q_\sig}
= - \frac{1}{2} \sum_{i\sigp} \ERI{p_\sig i_\sigp}{i_\sigp q_\sig}
@ -199,7 +212,7 @@ is, like the one-body Green's function, spin-diagonal, and its spectral represen
& + \sum_{am} \frac{\ERI{p_\sig a_\sig}{m} \ERI{q_\sig a_\sig}{m}}{\omega - \e{a_\sig} - \Om{m}{\spc,\RPA} + i \eta}
\end{split}
\end{gather}
which has been split in exchange (x) and correlation (c) parts for convenience.
which has been split in its exchange (x) and correlation (c) contributions.
The Dyson equation linking the Green's function and the self-energy holds separately for each spin component, and the quasiparticle energies $\eGW{p_\sig}$ are obtained by solving the frequency-dependent quasiparticle equation
\begin{equation}
\omega = \e{p_\sig}{} - V_{p_\sigma}^{\xc} + \SigX{p\sigma} + \SigC{p\sigma}(\omega)
@ -209,6 +222,7 @@ with
V_{p_\sigma}^{\xc} = \int \MO{p_\sig}(\br) v^{\xc}(\br) \MO{p_\sig}(\br) d\br
\end{equation}
where $v^{\xc}(\br)$ the Kohn-Sham exchange-correlation.
\alert{Adding the Dyson equation? Introduce linearization of the quasiparticle equation and different degree of self-consistency.}
%================================
\subsection{The Bethe-Salpeter equation formalism}
@ -242,9 +256,9 @@ Within the $GW$ approximation, the BSE kernel is
- \delta_{\sig\sigp} W(\br_3,\br_4;\omega) \delta(\br_3 - \br_6) \delta(\br_4 - \br_6)
\end{multline}
where, as usual, we have not considered the higher-order terms in $W$ by neglecting the derivative $\partial W/\partial G$. \cite{Hanke_1980, Strinati_1982, Strinati_1984, Strinati_1988}
Within the static approximation, the spin-conserved and spin-flip optical excitation at the BSE level are obtained by solving a similar linear response problem
Within the static approximation which consists in neglecting the frequency dependence of the dynamically-screened Coulomb potential, the spin-conserved and spin-flip optical excitation at the BSE level are obtained by solving a similar linear response problem
\begin{equation}
\label{eq:LR-RPA}
\label{eq:LR-BSE}
\begin{pmatrix}
\bA{}{\BSE} & \bB{}{\BSE} \\
-\bB{}{\BSE} & -\bA{}{\BSE} \\
@ -498,6 +512,23 @@ As explained in Ref.~\onlinecite{Casanova_2020}, there are two sources of spin c
\section{Computational details}
\label{sec:compdet}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the systems under investigation here, we consider either an open-shell doublet or triplet reference state.
We then adopt the unrestricted formalism throughout this work.
The $GW$ calculations performed to obtain the screened Coulomb operator and the quasiparticle energies are done using a (unrestricted) UHF starting point.
Perturbative $GW$ (or {\GOWO}) \cite{Hybertsen_1985a,Hybertsen_1986,vanSetten_2013} quasiparticle energies are employed as starting points to compute the BSE neutral excitations.
These quasiparticle energies are obtained by linearizing the frequency-dependent quasiparticle equation, and the entire set of orbitals is corrected.
Further details about our implementation of {\GOWO} can be found in Refs.~\onlinecite{Loos_2018b,Veril_2018,Loos_2020,Loos_2020e}.
%Note that, for the present (small) molecular systems, {\GOWO}@UHF and ev$GW$@UHF yield similar quasiparticle energies and fundamental gap.
%Moreover, {\GOWO} allows to avoid rather laborious iterations as well as the significant additional computational effort of ev$GW$.
%In the present study, the zeroth-order Hamiltonian [see Eq.~\eqref{eq:LR-PT}] is always the ``full'' BSE static Hamiltonian, \ie, without TDA.
The dynamical correction is computed in the TDA throughout.
As one-electron basis sets, we employ the Dunning families cc-pVXZ and aug-cc-pVXZ (X = D, T, and Q) defined with cartesian Gaussian functions.
Finally, the infinitesimal $\eta$ is set to $100$ meV for all calculations.
%It is important to mention that the small molecular systems considered here are particularly challenging for the BSE formalism, \cite{Hirose_2015,Loos_2018b} which is known to work best for larger systems where the amount of screening is more important. \cite{Jacquemin_2017b,Rangel_2017}
%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_1995a} CCSD, \cite{Purvis_1982} and CC3 \cite{Christiansen_1995b} excitation energies are also extracted.
%Various statistical quantities are reported in the following: the mean signed error (MSE), mean absolute error (MAE), root-mean-square error (RMSE), and the maximum positive [Max($+$)] and maximum negative [Max($-$)] errors.
All the static and dynamic BSE calculations have been performed with the software \texttt{QuAcK}, \cite{QuAcK} freely available on \texttt{github}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Results}