From 92735a3ee232ac01aec2eacf1a5b593b7a428925 Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Sun, 25 Oct 2020 21:29:40 +0100 Subject: [PATCH] saving work in GW --- sfBSE.bib | 5 ++-- sfBSE.tex | 75 +++++++++++++++++++++++++++++++++++++++---------------- 2 files changed, 56 insertions(+), 24 deletions(-) diff --git a/sfBSE.bib b/sfBSE.bib index 854437b..e50c071 100644 --- a/sfBSE.bib +++ b/sfBSE.bib @@ -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}, diff --git a/sfBSE.tex b/sfBSE.tex index e000fdb..4584900 100644 --- a/sfBSE.tex +++ b/sfBSE.tex @@ -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}