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Pierre-Francois Loos 2021-01-30 15:08:22 +01:00
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17
Manuscript/EPAWTFT.tex
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@ -392,6 +392,7 @@ These degeneracies can be conical intersections between two states with differen
for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the for real values of $\lambda$,\cite{Yarkony_1996} or EPs between two states with the
same symmetry for complex values of $\lambda$. same symmetry for complex values of $\lambda$.
\titou{Although the complex component of the energy is linked to lifetime in non-Hermitian quantum mechanics, \cite{MoiseyevBook} its meaning is unclear to us in the present context.}
%============================================================% %============================================================%
\subsection{Rayleigh--Schr\"odinger Perturbation Theory} \subsection{Rayleigh--Schr\"odinger Perturbation Theory}
@ -783,7 +784,7 @@ diatomics, where low-order RMP and UMP expansions give qualitatively wrong bindi
The divergence of RMP expansions for stretched bonds can be easily understood from two perspectives.\cite{Gill_1988a} The divergence of RMP expansions for stretched bonds can be easily understood from two perspectives.\cite{Gill_1988a}
Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the
RHF wave function no longer provides a qualitatively correct reference for the perturbation expansion. RHF wave function no longer provides a qualitatively correct reference for the perturbation expansion.
Secondly, the energy gap between the bonding and anti-bonding orbitals associated with the stretch becomes Secondly, the energy gap between the \titou{occupied and unoccupied} orbitals associated with the stretch becomes
increasingly small at larger bond lengths, leading to a divergence, for example, in the MP2 correction \eqref{eq:EMP2}. increasingly small at larger bond lengths, leading to a divergence, for example, in the MP2 correction \eqref{eq:EMP2}.
In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains
qualitatively correct at large bond lengths and the orbital degeneracy is avoided. qualitatively correct at large bond lengths and the orbital degeneracy is avoided.
@ -891,12 +892,12 @@ The Taylor expansion of the RMP energy can then be evaluated to obtain the $k$th
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% RADIUS OF CONVERGENCE PLOTS % RADIUS OF CONVERGENCE PLOTS
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The RMP series is convergent for $U = 3.5\,t$ with $\rc > 1$, as illustrated for the individual terms at each The RMP series is convergent \titou{at $\lambda = 1$} for $U = 3.5\,t$ with $\rc > 1$, as illustrated for the individual terms at each
perturbation order in Fig.~\ref{subfig:RMP_cvg}. perturbation order in Fig.~\ref{subfig:RMP_cvg}.
In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent. In contrast, for $U = 4.5t$ one finds $\rc < 1$, and the RMP series becomes divergent \titou{at $\lambda = 1$}.
The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and The corresponding Riemann surfaces for $U = 3.5\,t$ and $4.5\,t$ are shown in Figs.~\ref{subfig:RMP_3.5} and
\ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot) and the radius of convergence indicated \ref{subfig:RMP_4.5}, respectively, with the single EP at $\lep$ (black dot) and \titou{the radius of convergence indicated
by the vertical cylinder of unit radius. by the vertical cylinder of unit radius (TODO)}.
For the divergent case, $\lep$ lies inside this cylinder of convergence, while in the convergent case $\lep$ lies For the divergent case, $\lep$ lies inside this cylinder of convergence, while in the convergent case $\lep$ lies
outside this cylinder. outside this cylinder.
In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour In both cases, the EP connects the ground state with the doubly-excited state, and thus the convergence behaviour
@ -966,7 +967,7 @@ The convergence behaviour can be further elucidated by considering the full stru
in the complex $\lambda$-plane (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}). in the complex $\lambda$-plane (see Figs.~\ref{subfig:UMP_3} and \ref{subfig:UMP_7}).
These Riemann surfaces are illustrated for $U = 3t$ and $7t$ alongside the perturbation terms at each order These Riemann surfaces are illustrated for $U = 3t$ and $7t$ alongside the perturbation terms at each order
in Fig.~\ref{subfig:UMP_cvg}. in Fig.~\ref{subfig:UMP_cvg}.
At $U = 3t$, the RMP series is convergent, while RMP becomes divergent for $U=7t$. At $U = 3t$, the RMP series is convergent \titou{at $\lambda = 1$}, while RMP becomes divergent \titou{at $\lambda = 1$} for $U=7t$.
The ground-state UMP expansion is convergent in both cases, although the rate of convergence is significantly slower The ground-state UMP expansion is convergent in both cases, although the rate of convergence is significantly slower
for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}). for larger $U/t$ as the radius of convergence becomes increasingly close to one (Fig.~\ref{fig:RadConv}).
@ -1073,7 +1074,7 @@ The authors first considered molecules with low-lying doubly-excited states with
and spin symmetry as the ground state. \cite{Olsen_2000} and spin symmetry as the ground state. \cite{Olsen_2000}
In these systems, the exact wave function has a non-negligible contribution from the doubly-excited states, In these systems, the exact wave function has a non-negligible contribution from the doubly-excited states,
and thus the low-lying excited states are likely to become intruder states. and thus the low-lying excited states are likely to become intruder states.
For \ce{CH_2} in a diffuse, yet rather small basis set, the series is convergent at least up to the 50th order, and For \ce{CH_2} in a diffuse, yet rather small basis set, the series is convergent \titou{at $\lambda = 1$} at least up to the 50th order, and
the dominant singularity lies close (but outside) the unit circle, causing slow convergence of the series. the dominant singularity lies close (but outside) the unit circle, causing slow convergence of the series.
These intruder-state effects are analogous to the EP that dictates the convergence behaviour of These intruder-state effects are analogous to the EP that dictates the convergence behaviour of
the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}). the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}).
@ -1278,7 +1279,7 @@ The RMP critical point then corresponds to the intersection $E_{-} = E_{+}$, giv
\lc = 1 - \frac{\epsilon}{U}. \lc = 1 - \frac{\epsilon}{U}.
\end{equation} \end{equation}
Clearly the radius of convergence $\rc = \abs{\lc}$ is controlled directly by the ratio $\epsilon / U$, Clearly the radius of convergence $\rc = \abs{\lc}$ is controlled directly by the ratio $\epsilon / U$,
with a convergent RMP series occurring for $\epsilon > 2 U$. with a convergent RMP series \titou{at $\lambda = 1$} occurring for $\epsilon > 2 U$.
The on-site repulsion $U$ controls the strength of the HF potential localised around the ``atomic site'', with a The on-site repulsion $U$ controls the strength of the HF potential localised around the ``atomic site'', with a
stronger repulsion encouraging the electrons to be ionised at a less negative value of $\lambda$. stronger repulsion encouraging the electrons to be ionised at a less negative value of $\lambda$.
Large $U$ can be physically interpreted as strong electron repulsion effects in electron dense molecules. Large $U$ can be physically interpreted as strong electron repulsion effects in electron dense molecules.

14
Response_Letter/Response_Letter.tex
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@ -72,9 +72,11 @@ We look forward to hearing from you.
This is hinted at in the introduction but was the first question that entered my mind.} This is hinted at in the introduction but was the first question that entered my mind.}
\\ \\
\alert{In most figures, we indeed plot the real component of the energy. \alert{In most figures, we indeed plot the real component of the energy.
By going complex, the energies lost their meaning as there's no relation of order for complex numbers. By going complex, the energies lost their meaning as there is no relation of order for complex numbers.
In non-Hermitian processes like resonances or scattering, the imaginary part of the energy is indeed linked to lifetimes [see the book of Moiseyev]. In non-Hermitian processes like resonances or scattering, the imaginary part of the energy is indeed linked to lifetimes [see the book of Moiseyev].
However, the meaning of the imaginary part of the energy in the present context is unclear and we prefer not to speculate on this.} This is mention in the introduction section.
However, the meaning of the imaginary part of the energy in the present context is unclear and we prefer not to speculate on this.
We have added a comment on this point in the revised version of the manuscript.}
\item \item
{IIe.\\ {IIe.\\
@ -94,7 +96,7 @@ We look forward to hearing from you.
Also, in the general case (either RHF or UHF), the bonding orbitals for the two sigma values are occupied and the two antibonding orbitals are unoccupied. Also, in the general case (either RHF or UHF), the bonding orbitals for the two sigma values are occupied and the two antibonding orbitals are unoccupied.
Bonding and antibonding notation is not so common in the condensed matter literature and it may be missed that these correspond to occupied and empty single-particle states.} Bonding and antibonding notation is not so common in the condensed matter literature and it may be missed that these correspond to occupied and empty single-particle states.}
\\ \\
\alert{We now mention that Eqs. 20, 21a, and 21b are valid for both the RHF and UHF cases. \alert{We now mention that Eqs. 20, 21a, and 21b are valid for both RHF and UHF.
We have also changed the notations for the occupied and empty single-particle states.} We have also changed the notations for the occupied and empty single-particle states.}
\item \item
@ -106,7 +108,7 @@ We look forward to hearing from you.
Certain values away from $\lambda=1$ can also have physical meaning, like the autoionization process discussed later.} Certain values away from $\lambda=1$ can also have physical meaning, like the autoionization process discussed later.}
\\ \\
\alert{Only the $\lambda = 0$ (non-interacting system in some cases) and $\lambda = 1$ (fully interacting system) are physical systems as mentioned in the manuscript. \alert{Only the $\lambda = 0$ (non-interacting system in some cases) and $\lambda = 1$ (fully interacting system) are physical systems as mentioned in the manuscript.
However, it can be useful to study the systems at different $\lambda$ values.} However, it can be useful to study the systems at different $\lambda$ values as we do in Sec.~II.F.}
\item \item
{IIIc.\\ {IIIc.\\
@ -116,7 +118,7 @@ We look forward to hearing from you.
I first thought the cylinder was marking the radius of convergence. It is meant to show the position of the EPs and indicate if the radius of convergence is $>1$ or $<1$. I first thought the cylinder was marking the radius of convergence. It is meant to show the position of the EPs and indicate if the radius of convergence is $>1$ or $<1$.
The language could be improved.} The language could be improved.}
\\ \\
\alert{The language has been improved.} \alert{The language has been improved and we have mentioned in several places that we are talking about a convergent/divergent MP series at $\lambda = 1$.}
\item \item
{IIId.\\ {IIId.\\
@ -124,7 +126,7 @@ We look forward to hearing from you.
I raise this point because the journal is JPCM, and these terms are not common in condensed matter literature. I raise this point because the journal is JPCM, and these terms are not common in condensed matter literature.
Are intruder states equivalent to states creating exceptional points that ruin the series convergence?} Are intruder states equivalent to states creating exceptional points that ruin the series convergence?}
\\ \\
\alert{The classification of front-door and back-door intruder states have been clarified. \alert{We believe that the classification of front-door and back-door intruder states has been clearly defined in the original manuscript:
"Following this theory, a singularity in the unit circle is designated as an intruder state, with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative)."} "Following this theory, a singularity in the unit circle is designated as an intruder state, with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative)."}
\item \item