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regularizer

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Pierre-Francois Loos 2022-02-21 15:06:05 +01:00
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@ -295,7 +295,7 @@ where $\bI$ is the identity matrix.
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.
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 eigenvalues 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 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.
@ -359,8 +359,8 @@ Our investigation has shown that the following regularizer
\begin{equation}
f_\eta(\Delta) = \frac{1-e^{-2\Delta^2/\eta^2}}{\Delta}
\end{equation}
derived from the (second-order) perturbative analysis of the similarity renormalization group equations \cite{Evangelista_2014,Li_2019a} is particularly convenient and effective in the present context.
%Decreasing $\eta$ gradually decouples states with small denominator in the self-energy (which is exactly our purpose).
derived from the (second-order) perturbative analysis of the similarity renormalization group equations \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 (which is exactly our purpose), hence avoiding intruder state problems. \cite{Li_2019a}
Of course, by construction, we have
\begin{equation}
\lim_{\eta\to0} \rSigC{p}(\omega;\eta) = \SigC{p}(\omega)