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hermitian -> Hermitian

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Pierre-Francois Loos 2023-02-20 11:02:36 +01:00
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Manuscript/SRGGW.tex
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@ -80,7 +80,7 @@ The family of Green's function methods based on the $GW$ approximation has gaine
Despite this, self-consistent versions still pose challenges in terms of convergence. Despite this, self-consistent versions still pose challenges in terms of convergence.
A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem. A recent study \href{https://doi.org/10.1063/5.0089317}{[J. Chem. Phys. 156, 231101 (2022)]} has linked these convergence issues to the intruder-state problem.
In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods. In this work, a perturbative analysis of the similarity renormalization group (SRG) approach is performed on Green's function methods.
The SRG formalism enables us to derive, from first principles, the expression of a new, naturally hermitian form of the static self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations. The SRG formalism enables us to derive, from first principles, the expression of a new, naturally Hermitian form of the static self-energy that can be employed in quasiparticle self-consistent $GW$ (qs$GW$) calculations.
The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code. The resulting SRG-based regularized self-energy significantly accelerates the convergence of qs$GW$ calculations, slightly improves the overall accuracy, and is straightforward to implement in existing code.
\bigskip \bigskip
\begin{center} \begin{center}
@ -167,7 +167,7 @@ Unless otherwise stated, atomic units are used throughout.
\label{sec:gw} \label{sec:gw}
%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%
The central equation of many-body perturbation theory based on Hedin's equations is the so-called dynamical and non-hermitian quasiparticle equation which, within the $GW$ approximation, reads The central equation of many-body perturbation theory based on Hedin's equations is the so-called dynamical and non-Hermitian quasiparticle equation which, within the $GW$ approximation, reads
\begin{equation} \begin{equation}
\label{eq:quasipart_eq} \label{eq:quasipart_eq}
\qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx), \qty[ \bF + \bSig(\omega = \epsilon_p) ] \psi_p(\bx) = \epsilon_p \psi_p(\bx),
@ -178,7 +178,7 @@ Throughout the manuscript, the indices $p,q,r,s$ are general orbitals while $i,j
The indices $\mu$ and $\nu$ are composite indices, that is, $\nu=(ia)$, referring to neutral (single) excitations. The indices $\mu$ and $\nu$ are composite indices, that is, $\nu=(ia)$, referring to neutral (single) excitations.
The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects. The self-energy can be physically understood as a correction to the Hartree-Fock (HF) problem (represented by $\bF$) accounting for dynamical screening effects.
Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently but the dynamical and non-hermitian nature of $\bSig(\omega)$, as well as its functional form, makes it much more challenging to solve from a practical point of view. Similarly to the HF case, Eq.~\eqref{eq:quasipart_eq} has to be solved self-consistently but the dynamical and non-Hermitian nature of $\bSig(\omega)$, as well as its functional form, makes it much more challenging to solve from a practical point of view.
The matrix elements of $\bSig(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016} The matrix elements of $\bSig(\omega)$ have the following closed-form expression \cite{Hedin_1999,Tiago_2006,Bruneval_2012,vanSetten_2013,Bruneval_2016}
\begin{equation} \begin{equation}
@ -212,7 +212,7 @@ and where
are bare two-electron integrals in the spin-orbital basis. are bare two-electron integrals in the spin-orbital basis.
The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problem defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}. The diagonal matrix $\boldsymbol{\Omega}$ contains the positive eigenvalues of the RPA problem defined in Eq.~\eqref{eq:full_dRPA} and its elements $\Omega_\nu$ appear in Eq.~\eqref{eq:GW_selfenergy}.
In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$). In the Tamm-Dancoff approximation (TDA), one sets $\bB = \bO$ in Eq.~\eqref{eq:full_dRPA} which reduces to a Hermitian eigenvalue problem of the form $\bA \bX = \bX \bOm$ (hence $\bY=0$).
As mentioned above, because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task. As mentioned above, because of the frequency dependence of the self-energy, solving exactly the quasiparticle equation \eqref{eq:quasipart_eq} is a rather complicated task.
Hence, several approximate schemes have been developed to bypass full self-consistency. Hence, several approximate schemes have been developed to bypass full self-consistency.
@ -241,9 +241,9 @@ However, if one of the quasiparticle equations does not have a well-defined quas
Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012} Even at convergence, the starting point dependence is not totally removed as the quasiparticle energies still depend on the initial set of orbitals. \cite{Marom_2012}
In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$. In order to update both the orbitals and their corresponding energies, one must consider the off-diagonal elements in $\bSig(\omega)$.
To avoid solving the non-hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qsGW}$. To avoid solving the non-Hermitian and dynamic quasiparticle equation defined in Eq.~\eqref{eq:quasipart_eq}, one can resort to the qs$GW$ scheme in which $\bSig(\omega)$ is replaced by a static approximation $\bSig^{\qsGW}$.
Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qsGW}$. Then, the qs$GW$ equations are solved via a standard self-consistent field procedure similar to the HF algorithm where $\bF$ is replaced by $\bF + \bSig^{\qsGW}$.
Various choices for $\bSig^{\qsGW}$ are possible but the most popular is the following hermitian approximation Various choices for $\bSig^{\qsGW}$ are possible but the most popular is the following Hermitian approximation
\begin{equation} \begin{equation}
\label{eq:sym_qsgw} \label{eq:sym_qsgw}
\Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ]. \Sigma_{pq}^{\qsGW} = \frac{1}{2}\Re[\Sigma_{pq}(\epsilon_p) + \Sigma_{pq}(\epsilon_q) ].
@ -257,7 +257,7 @@ The corresponding matrix elements are
+ \eta^2} +\frac{\Delta_{qr}^{\nu}}{(\Delta_{qr}^{\nu})^2 + \eta^2} ] W_ {pr}^{\nu} W_{qr}^{\nu}. + \eta^2} +\frac{\Delta_{qr}^{\nu}}{(\Delta_{qr}^{\nu})^2 + \eta^2} ] W_ {pr}^{\nu} W_{qr}^{\nu}.
\end{equation} \end{equation}
with $\Delta_{pr}^{\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level). with $\Delta_{pr}^{\nu} = \epsilon_p - \epsilon_r - \sgn(\epsilon_r-\epsilon_F)\Omega_\nu$ (where $\epsilon_F$ is the energy of the Fermi level).
One of the main results of the present manuscript is the derivation, from first principles, of an alternative static hermitian form for the $GW$ self-energy. One of the main results of the present manuscript is the derivation, from first principles, of an alternative static Hermitian form for the $GW$ self-energy.
Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level. Once again, in cases where multiple solutions have large spectral weights, self-consistency can be difficult to reach at the qs$GW$ level.
Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy. Multiple solutions of Eq.~\eqref{eq:G0W0} arise due to the $\omega$ dependence of the self-energy.
@ -588,7 +588,7 @@ For a fixed value of the energy cutoff $\Lambda$, if $\abs*{\Delta_{pr}^{\nu}} \
\subsection{Alternative form of the static self-energy} \subsection{Alternative form of the static self-energy}
%///////////////////////////% %///////////////////////////%
Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and hermitian, the new alternative form of the self-energy reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{eq:sym_qsgw}. Because the large-$s$ limit of Eq.~\eqref{eq:GW_renorm} is purely static and Hermitian, the new alternative form of the self-energy reported in Eq.~\eqref{eq:static_F2} can be naturally used in qs$GW$ calculations to replace Eq.~\eqref{eq:sym_qsgw}.
Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible. Unfortunately, as we shall discuss further in Sec.~\ref{sec:results}, as $s\to\infty$, self-consistency is once again quite difficult to achieve, if not impossible.
However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm} (see cyan curve in Fig.~\ref{fig:flow}). However, one can define a more flexible new static self-energy, which will be referred to as SRG-qs$GW$ in the following, by discarding the dynamic part in Eq.~\eqref{eq:GW_renorm} (see cyan curve in Fig.~\ref{fig:flow}).
This yields a $s$-dependent static self-energy which matrix elements read This yields a $s$-dependent static self-energy which matrix elements read
@ -599,7 +599,7 @@ This yields a $s$-dependent static self-energy which matrix elements read
\\ \\
\times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s} ]. \times \qty[1 - e^{-\qty[(\Delta_{pr}^{\nu})^2 + (\Delta_{qr}^{\nu})^2 ] s} ].
\end{multline} \end{multline}
Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is straightforward to implement in existing code and is naturally hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization. Note that the static SRG-qs$GW$ approximation defined in Eq.~\eqref{eq:SRG_qsGW} is straightforward to implement in existing code and is naturally Hermitian as opposed to the usual case [see Eq.~\eqref{eq:sym_qsGW}] where it is enforced by brute-force symmetrization.
Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator. Another important difference is that the SRG regularizer is energy-dependent while the imaginary shift is the same for every self-energy denominator.
Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms. Yet, these approximations are closely related because, for $\eta=0$ and $s\to\infty$, they share the same diagonal terms.