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Manuscript/ufGW.bib
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@ -1,13 +1,39 @@
%% This BibTeX bibliography file was created using BibDesk.
%% https://bibdesk.sourceforge.io/
%% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2022-02-19 13:53:00 +0100
%% Created for Pierre-Francois Loos at 2022-02-21 11:30:33 +0100
%% Saved with string encoding Unicode (UTF-8)
@article{Golze_2018,
author = {Golze, Dorothea and Wilhelm, Jan and van Setten, Michiel J. and Rinke, Patrick},
date-added = {2022-02-21 11:30:10 +0100},
date-modified = {2022-02-21 11:30:30 +0100},
doi = {10.1021/acs.jctc.8b00458},
journal = {J. Chem. Theory Comput.},
number = {9},
pages = {4856-4869},
title = {Core-Level Binding Energies from GW: An Efficient Full-Frequency Approach within a Localized Basis},
volume = {14},
year = {2018},
bdsk-url-1 = {https://doi.org/10.1021/acs.jctc.8b00458}}
@article{Golze_2020,
author = {Golze, Dorothea and Keller, Levi and Rinke, Patrick},
date-added = {2022-02-21 11:29:21 +0100},
date-modified = {2022-02-21 11:29:40 +0100},
doi = {10.1021/acs.jpclett.9b03423},
journal = {J. Phys. Chem. Lett.},
number = {5},
pages = {1840-1847},
title = {Accurate Absolute and Relative Core-Level Binding Energies from GW},
volume = {11},
year = {2020},
bdsk-url-1 = {https://doi.org/10.1021/acs.jpclett.9b03423}}
@article{Hedin_1999,
author = {Lars Hedin},
date-added = {2022-02-19 13:51:59 +0100},

61
Manuscript/ufGW.tex
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@ -174,6 +174,8 @@ Electron correlation is then explicitly incorporated into one-body quantities vi
In recent studies, \cite{Loos_2018b,Veril_2018,Berger_2021,DiSabatino_2021} we discovered that one can observe (unphysical) irregularities and/or discontinuities in the energy surfaces of several key quantities (ionization potential, electron affinity, fundamental gap, total and correlation energies, as well as vertical excitation energies) even in the weakly correlated regime.
These issues were discovered in Ref.~\onlinecite{Loos_2018b} while studying a model two-electron system \cite{Seidl_2007,Loos_2009a,Loos_2009c} and they were further investigated in Ref.~\onlinecite{Veril_2018}, where we provided additional evidences and explanations of these undesirable features in real molecular systems.
In particular, we showed that each branch of the self-energy $\Sigma$ is associated with a distinct quasiparticle solution, and that each switch between solutions implies a significant discontinuity in the quasiparticle energy due to the transfer of weight between two solutions of the quasiparticle equation. \cite{Veril_2018}
Multiple solution issues in $GW$ appears frequently, especially for orbitals that are energetically far from the Fermi level, such as in core ionized states. \cite{Golze_2018,Golze_2020}
It was shown that these problems could be alleviated by using a static Coulomb-hole plus screened-exchange (COHSEX) \cite{Hedin_1965,Hybertsen_1986,Hedin_1999,Bruneval_2006} self-energy \cite{Berger_2021} or by considering a fully self-consistent scheme. \cite{DiSabatino_2021}
However, none of these solutions is completely satisfying as a static approximation of the self-energy can induce significant loss in accuracy and fully self-consistent calculations can be quite challenging in terms of implementation and cost.
@ -200,7 +202,7 @@ where $\eps{p}{\HF}$ is the $p$th HF orbital energy and the correlation part of
+ \sum_{am} \frac{2\ERI{pa}{m}^2}{\omega - \eps{a}{\HF} - \Om{m}{\RPA}}
\end{equation}
is constituted by a hole (h) and a particle (p) term.
Within the Tamm-Dancoff approximation, the screened two-electron integrals are given by
Within the Tamm-Dancoff approximation (that we enforce here for the sake of simplicity), the screened two-electron integrals are given by
\begin{equation}
\ERI{pq}{m} = \sum_{ia} \ERI{pq}{ia} X_{ia,m}^\RPA
\end{equation}
@ -217,8 +219,10 @@ and
\ERI{pq}{ia} = \iint \MO{p}(\br_1) \MO{q}(\br_1) \frac{1}{\abs{\br_1 - \br_2}} \MO{i}(\br_2) \MO{a}(\br_2) d\br_1 \dbr_2
\end{equation}
are two-electron integrals over the HF (spatial) orbitals $\MO{p}(\br)$.
As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW}$ and their corresponding weight are given by the value of the following renormalization factor
As a non-linear equation, Eq.~\eqref{eq:qp_eq} has many solutions $\eps{p,s}{\GW}$ and their corresponding weights are given by the value of the following renormalization factor
\begin{equation}
\label{eq:Z}
0 \le Z_{p,s} = \qty[ 1 - \eval{\pdv{\SigC{p}(\omega)}{\omega}}_{\omega = \eps{p,s}{\GW}} ]^{-1} \le 1
\end{equation}
In a well-behaved case, one of the solution (the so-called quasiparticle) $\eps{p}{\GW} \equiv \eps{p,s=0}{\GW}$ has a large weight $Z_{p} \equiv Z_{p,s=0}$.
@ -228,15 +232,16 @@ Note that we have the following important conservation rules \cite{Martin_1959,B
&
\sum_{s} Z_{p,s} \eps{p,s}{\GW} & = \eps{p}{\HF}
\end{align}
which physically shows that the mean-field solution of unit weight is ``scattered'' by the effect of correlation in many solutions with smaller weights.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Upfolding: the linear $GW$ problem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The non-linear quasiparticle equation \eqref{eq:qp_eq} can be transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
The non-linear quasiparticle equation \eqref{eq:qp_eq} can be \textit{exactly} transformed into a larger linear problem via an upfolding process where the 2h1p and 2p1h sectors
are upfolded from the 1h and 1p sectors. \cite{Bintrim_2021a}
For each orbital $p$, this yields a linear eigenvalue problem of the form
\begin{equation}
\bH^{(p)} \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
\bH^{(p)} \cdot \bc{}{(p,s)} = \eps{p,s}{\GW} \bc{}{(p,s)}
\end{equation}
with
\begin{equation}
@ -264,13 +269,14 @@ and the corresponding coupling blocks read
\end{align}
The size of this eigenvalue problem is $1 + O^2 V + O V^2$ (where $O$ and $V$ are the number of occupied and virtual orbitals, respectively), and this eigenvalue problem has to be solved for each orbital that one wishes to correct.
Note, however, that the blocks $\bC{}{\text{2h1p}}$ and $\bC{}{\text{2p1h}}$ do not need to be recomputed for each orbital.
Because the renormalization factor corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference space, the weight of a solution $(p,s)$ is given by the the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie,
Because the renormalization factor \eqref{eq:Z} corresponds to the projection of the vector $\bc{}{(p,s)}$ onto the reference (or internal) space, the weight of a solution $(p,s)$ is given by the first coefficient of their corresponding eigenvector $\bc{}{(p,s)}$, \ie,
\begin{equation}
\label{eq:Z_proj}
Z_{p,s} = \qty[ c_{1}^{(p,s)} ]^{2}
\end{equation}
It is important to understand that diagonalizing $\bH^{(p)}$ in Eq.~\eqref{eq:Hp} is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}.
This can be further illustrate by expanding the secular equation associated with Eq.~\eqref{eq:Hp}
It is paramount to understand that diagonalizing $\bH^{(p)}$ [see Eq.~\eqref{eq:Hp}] is completely equivalent to solving the quasiparticle equation \eqref{eq:qp_eq}.
This can be further illustrated by expanding the secular equation associated with Eq.~\eqref{eq:Hp}
\begin{equation}
\det[ \bH^{(p)} - \omega \bI ] = 0
\end{equation}
@ -282,26 +288,23 @@ and comparing it with Eq.~\eqref{eq:qp_eq} by setting
+ \bV{p}{\text{2p1h}} \cdot \qty(\omega \bI - \bC{}{\text{2p1h}} )^{-1} \cdot \T{\qty(\bV{p}{\text{2p1h}})}
\end{multline}
where $\bI$ is the identity matrix.
The main mathematical difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the eigenvectors associated with each quasiparticle and satellites.
One can see this downfolding process as the construction of a frequency-dependent effective Hamiltonian where the reference (zeroth-order) space is composed by a single determinant of the 1h or 1p sector and the external (first-order) space by all the 2h1p and 2p1h configurations.
One can see this downfolding process as the construction of a frequency-dependent effective Hamiltonian where the internal space is composed by a single determinant of the 1h or 1p sector and the external (or outer) space by all the 2h1p and 2p1h configurations. \cite{Dvorak_2019a,Dvorak_2019b,Bintrim_2021a}
The main mathematical difference between the two approaches is that, by diagonalizing Eq.~\eqref{eq:Hp}, one has directly access to the internal and external components of the eigenvectors associated with each quasiparticle and satellite, and not only their projection in the reference space as shown by Eq.~\eqref{eq:Z_proj}.
The element $\eps{p}{\HF}$ of $\bH^{(p)}$ [see Eq.~\eqref{eq:Hp}] corresponds to the relative energy of the $(\Ne\pm1)$-electron reference determinant (compared to the $\Ne$-electron HF determinant) while the diagonal elements of the blocks $\bC{}{\text{2h1p}}$ and $\bC{}{\text{2p1h}}$ provide an estimate of the relative energy of the 2h1p and 2p1h determinants.
In some situations, one of these determinant from the outer space may become of similar energy than the reference determinant, a situation that one could label as intruder state problem.
Hence, the two diabatic electronic configurations may cross and form an avoided crossing.
As we shall see below, this is when discontinuities occur and is ubiquitous in molecular systems.
In some situations, one of these determinants from the external space may become of similar energy than the reference determinant.
Hence, these two diabatic electronic configurations may cross and form an avoided crossing, and this outer-space determinant may be labeled as an intruder state.
As we shall see below, discontinuities, which are ubiquitous in molecular systems, arise in such scenarios.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{An illustrative example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Multiple solution issues in $GW$ appears all the time, especially for orbitals that are far in energy from the Fermi level.
Therefore, such issues are ubiquitous when one wants to compute core ionized states for example.
In order to illustrate the appearance and the origin of these multiple solutions, we consider the hydrogen molecule in the 6-31G basis set which corresponds to a system with 2 electrons and 4 spatial orbitals (one occupied and three virtual).
This example was already considered in our previous work \cite{Veril_2018} but here we provide further insights on the origin of the appearances of these multiple solutions.
The algorithm diabatically follows the solution, while it should be adiabatic.
This is not the $\Ne$-electron situation where one has to check if it is multireference, but for the $\Ne\pm1$-electron situations.
In order to illustrate the appearance and the origin of these multiple solutions, we consider the hydrogen molecule in the 6-31G basis set which corresponds to a two-electron system with four spatial orbitals (one occupied and three virtuals).
This example was already considered in our previous work \cite{Veril_2018} but here we provide further insights on the origin of the appearances of these discontinuities.
We denote as $\ket*{1\Bar{1}}$ the $\Ne$-electron ground-state determinant where the orbital 1 is occupied by one spin-up and spin-down electron.
Similarly notations will be employed for the $(\Ne\pm1)$-electron configurations.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% FIGURE 1
@ -316,17 +319,27 @@ This is not the $\Ne$-electron situation where one has to check if it is multire
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Figure \ref{fig:H2} shows the evolution of the quasiparticle energies at the {\GOWO}@HF/6-31G level as a function on the internuclear distance $\RHH$.
As one can see there are two problematic regions showing obvious discontinuities around $\RHH = \SI{1}{\AA}$ and $\RHH = \SI{1}{\AA}$
Figure \ref{fig:H2} shows the evolution of the quasiparticle energy, energetically close-by satellites and their corresponding weights at the {\GOWO}@HF level as a function on the internuclear distance $\RHH$.
One can easily diagnose two problematic regions showing obvious discontinuities around $\RHH = \SI{0.5}{\angstrom}$ for the HOMO$+3$ ($p = 4$) and $\RHH = \SI{1.1}{\angstrom}$ for the HOMO$+2$ ($p = 3$).
Let us first look more closely at the region around $\RHH = \SI{1.1}{\angstrom}$.
As one can see, an avoided crossing is formed between two $(\Ne+1)$-electron.
Inspection of their corresponding eigenvectors reveals that the determinants involved are the reference 1p determinant $\ket*{1\Bar{1}3}$ and an excited $(\Ne+1)$-electron of configuration $\ket*{12\Bar{2}}$.
The algorithm diabatically follows the reference determinant $\ket*{1\Bar{1}3}$, while it should be adiabatically switching to the $\ket*{12\Bar{2}}$ determinant which becomes lower in energy for $\RHH > \SI{1.1}{\angstrom}$.
This is not the $\Ne$-electron situation where one has to check if it is multireference, but for the $\Ne\pm1$-electron situations.
As similar scenario is at the play in the region around $\RHH = \SI{0.5}{\angstrom}$ but it now involves 3 electronic configurations: the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration $\ket*{1\Bar{?}?}$ and $\ket*{1\Bar{?}?}$.
These forms two avoided crossings in rapid successions, which involves two discontinuties in the energy surface.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introducing regularized $GW$ methods}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
One direct consequences of such irregularities is the difficulty to converge (partially) self-consistent $GW$ methods as, depending on the iteration, the self-consistent procedure will jump erratically from one solution to the other (even if DIIS is employed).
One way to hamper such issues is to resort to regularization of the $GW$ self-energy.
One way to hamper such issues and to massively improve convergence is to resort to regularization of the $GW$ self-energy.
Of course, the way of regularizing the self-energy is not unique but here we consider three different ways directly imported from MP2 theory.
This helps greatly convergence for (partially) self-consistent $GW$ methods.
We show that it removes discontinuities, speed up convergence without altering the quasiparticle energy.
The regularized self-energy is
\begin{equation}