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Pierre-Francois Loos 2022-02-22 15:36:18 +01:00
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@ -343,7 +343,7 @@ Similarly notations will be employed for the $(\Ne\pm1)$-electron configurations
In Fig.~\ref{fig:H2}, we report the variation of the quasiparticle energies of the four orbitals as functions of the internuclear distance $\RHH$. In Fig.~\ref{fig:H2}, we report the variation of the quasiparticle energies of the four orbitals as functions of the internuclear distance $\RHH$.
One can easily diagnose two problematic regions showing obvious discontinuities around $\RHH = \SI{1.2}{\angstrom}$ for the LUMO$+1$ ($p = 3$) and $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ ($p = 4$). One can easily diagnose two problematic regions showing obvious discontinuities around $\RHH = \SI{1.2}{\angstrom}$ for the LUMO$+1$ ($p = 3$) and $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ ($p = 4$).
As thoroughly explained in Ref.~\onlinecite{Veril_2018}, if one relies on the linearization of quasiparticle equation \eqref{eq:qp_eq} to compute the quasiparticle energies, \ie, $\eps{p}{\GW} = \eps{p}{\HF} + Z_{p} \SigC{p}(\eps{p}{\HF})$, these discontinuities are transformed into irregularities as the renormalization factor cancels out the singularities of the self-energy. As thoroughly explained in Ref.~\onlinecite{Veril_2018}, if one relies on the linearization of the quasiparticle equation \eqref{eq:qp_eq} to compute the quasiparticle energies, \ie, $\eps{p}{\GW} = \eps{p}{\HF} + Z_{p} \SigC{p}(\eps{p}{\HF})$, these discontinuities are transformed into irregularities as the renormalization factor cancels out the singularities of the self-energy.
Figure \ref{fig:H2_zoom} shows the evolution of the quasiparticle energy, the energetically close-by satellites and their corresponding weights as functions of $\RHH$. Figure \ref{fig:H2_zoom} shows the evolution of the quasiparticle energy, the energetically close-by satellites and their corresponding weights as functions of $\RHH$.
Let us first look more closely at the region around $\RHH = \SI{1.2}{\angstrom}$ involving the LUMO$+1$ (left panel of Fig.~\ref{fig:H2_zoom}). Let us first look more closely at the region around $\RHH = \SI{1.2}{\angstrom}$ involving the LUMO$+1$ (left panel of Fig.~\ref{fig:H2_zoom}).
@ -351,7 +351,7 @@ As one can see, an avoided crossing is formed between two solutions of the quasi
Inspection of their corresponding eigenvectors reveals that the $(\Ne+1)$-electron determinants principally involved are the reference 1p determinant $\ket*{1\Bar{1}3}$ and an excited $(\Ne+1)$-electron determinant of configuration $\ket*{12\Bar{2}}$ that becomes lower in energy than the reference determinant for $\RHH > \SI{1.2}{\angstrom}$. Inspection of their corresponding eigenvectors reveals that the $(\Ne+1)$-electron determinants principally involved are the reference 1p determinant $\ket*{1\Bar{1}3}$ and an excited $(\Ne+1)$-electron determinant of configuration $\ket*{12\Bar{2}}$ that becomes lower in energy than the reference determinant for $\RHH > \SI{1.2}{\angstrom}$.
By construction, the quasiparticle solution diabatically follows the reference determinant $\ket*{1\Bar{1}3}$ through the avoided crossing (thick lines in Fig.~\ref{fig:H2_zoom}) which is precisely the origin of the energetic discontinuity. By construction, the quasiparticle solution diabatically follows the reference determinant $\ket*{1\Bar{1}3}$ through the avoided crossing (thick lines in Fig.~\ref{fig:H2_zoom}) which is precisely the origin of the energetic discontinuity.
As similar scenario is at play in the region around $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ (right panel of Fig.~\ref{fig:H2_zoom}) but it now involves three solutions ($s = 5$, $s = 6$, and $s = 7$). A similar scenario is at play in the region around $\RHH = \SI{0.5}{\angstrom}$ for the LUMO$+2$ (right panel of Fig.~\ref{fig:H2_zoom}) but it now involves three solutions ($s = 5$, $s = 6$, and $s = 7$).
The electronic configurations of the Slater determinant involved are the $\ket*{1\Bar{1}4}$ reference determinant as well as two external determinants of configuration $\ket*{1\Bar{?}?}$ and $\ket*{1\Bar{?}?}$. The electronic configurations of the Slater determinant involved are 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 create two discontinuities in the energy surface (see Fig.~\ref{fig:H2}). These forms two avoided crossings in rapid successions, which create two discontinuities in the energy surface (see Fig.~\ref{fig:H2}).
In this region, although the ground-state wave function is well described by the $\Ne$-electron HF determinant, a situation that can be safely labeled as single-reference, one can see that the $(\Ne+1)$-electron state involves three Slater determinants and can then be labeled as a multi-reference (or strongly-correlated) situation with near-degenerate electronic configurations. In this region, although the ground-state wave function is well described by the $\Ne$-electron HF determinant, a situation that can be safely labeled as single-reference, one can see that the $(\Ne+1)$-electron state involves three Slater determinants and can then be labeled as a multi-reference (or strongly-correlated) situation with near-degenerate electronic configurations.
@ -413,7 +413,7 @@ Our investigations have shown that the following regularizer
\begin{equation} \begin{equation}
f_\eta(\Delta) = \frac{1-e^{-2\Delta^2/\eta^2}}{\Delta} f_\eta(\Delta) = \frac{1-e^{-2\Delta^2/\eta^2}}{\Delta}
\end{equation} \end{equation}
derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Wegner_1994,Glazek_1994,White_2002,Evangelista_2014} is particularly convenient and effective in the present context. derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Wegner_1994,Glazek_1994,White_2002} by Evangelista \cite{Evangelista_2014} is particularly convenient and effective in the present context.
Increasing $\eta$ gradually integrates out states with denominators $\Delta$ larger than $\eta$ while the states with $\Delta \ll \eta$ are not decoupled from the reference space, hence avoiding intruder state problems. \cite{Li_2019a} Increasing $\eta$ gradually integrates out states with denominators $\Delta$ larger than $\eta$ while the states with $\Delta \ll \eta$ are not decoupled from the reference space, hence avoiding intruder state problems. \cite{Li_2019a}
Of course, by construction, we have Of course, by construction, we have
\begin{equation} \begin{equation}
@ -423,7 +423,7 @@ Of course, by construction, we have
Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasiparticle energies in the two regions of interest for $\eta = 0.1$, $1$, and $10$. Figure \ref{fig:H2reg_zoom} compares the non-regularized and regularized quasiparticle energies in the two regions of interest for $\eta = 0.1$, $1$, and $10$.
It clearly shows how the regularization of the $GW$ self-energy diabatically linked the two solutions to get rid of the discontinuities. It clearly shows how the regularization of the $GW$ self-energy diabatically linked the two solutions to get rid of the discontinuities.
However, this diabatization is more or less accurate depending on the value of $\eta$. However, this diabatization is more or less accurate depending on the value of $\eta$.
For $\eta = 10$, the value is clearly too large which induces a large difference between the two sets of quasiparticle energies (purple curves). For $\eta = 10$, the value is clearly too large inducing a large difference between the two sets of quasiparticle energies (purple curves).
For $\eta = 0.1$, we have the opposite scenario where the value is too small and some irregularities remain (green curves). For $\eta = 0.1$, we have the opposite scenario where the value is too small and some irregularities remain (green curves).
We have found that $\eta = 1$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions. We have found that $\eta = 1$ is a good compromise that does not alter significantly the quasiparticle energies while providing a smooth transition between the two solutions.