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Manuscript/EPAWTFT.aux
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@ -55,7 +55,7 @@
\citation{Moller_1934} \citation{Moller_1934}
\citation{SzaboBook} \citation{SzaboBook}
\citation{Gill_1994} \citation{Gill_1994}
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@ -89,10 +89,10 @@
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@ -148,34 +148,35 @@
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@ -6,7 +6,7 @@
%Control: page (0) single %Control: page (0) single
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@ -521,6 +521,19 @@
{journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume} {journal} {\bibinfo {journal} {J. Chem. Phys.}\ }\textbf {\bibinfo {volume}
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@ -1,13 +1,28 @@
%% This BibTeX bibliography file was created using BibDesk. %% This BibTeX bibliography file was created using BibDesk.
%% http://bibdesk.sourceforge.net/ %% http://bibdesk.sourceforge.net/
%% Created for Pierre-Francois Loos at 2020-11-12 10:02:40 +0100 %% Created for Pierre-Francois Loos at 2020-11-12 14:54:07 +0100
%% Saved with string encoding Unicode (UTF-8) %% Saved with string encoding Unicode (UTF-8)
@article{Leininger_2000,
author = {Leininger,Matthew L. and Allen,Wesley D. and Schaefer,Henry F. and Sherrill,C. David},
date-added = {2020-11-12 14:50:57 +0100},
date-modified = {2020-11-12 14:51:07 +0100},
doi = {10.1063/1.481764},
eprint = {https://doi.org/10.1063/1.481764},
journal = {The Journal of Chemical Physics},
number = {21},
pages = {9213-9222},
title = {Is Mo/ller--Plesset perturbation theory a convergent ab initio method?},
url = {https://doi.org/10.1063/1.481764},
volume = {112},
year = {2000},
Bdsk-Url-1 = {https://doi.org/10.1063/1.481764}}
@article{Nesbet_1955, @article{Nesbet_1955,
abstract = { A systematic method is developed for estimating or calculating corrections for configuration interaction in atomic, molecular and nuclear wave-function calculations. Solutions of the Hartree-Fock equations for a single Slater determinant or approximate Hartree-Fock solutions obtained by Roothaan's iterative procedure have special properties which are used to simplify the matrix of the many-particle Hamiltonian. A restricted self-consistent field method is proposed for treating states of low symmetry. This method avoids the off-diagonal Lagrange multipliers encountered in previous methods and is adapted to configuration interaction calculations. }, abstract = { A systematic method is developed for estimating or calculating corrections for configuration interaction in atomic, molecular and nuclear wave-function calculations. Solutions of the Hartree-Fock equations for a single Slater determinant or approximate Hartree-Fock solutions obtained by Roothaan's iterative procedure have special properties which are used to simplify the matrix of the many-particle Hamiltonian. A restricted self-consistent field method is proposed for treating states of low symmetry. This method avoids the off-diagonal Lagrange multipliers encountered in previous methods and is adapted to configuration interaction calculations. },
author = {Nesbet, R. K. and Hartree, Douglas Rayner}, author = {Nesbet, R. K. and Hartree, Douglas Rayner},
@ -240,17 +255,6 @@
year = {2019}, year = {2019},
Bdsk-Url-1 = {https://doi.org/10.1063/1.5110554}} Bdsk-Url-1 = {https://doi.org/10.1063/1.5110554}}
@article{Leininger_2000,
author = {Leininger, Matthew L. and Allen, Wesley D. and Schaefer, Henry F. and Sherrill, C. David},
date = {2000-05-17},
doi = {10.1063/1.481764},
journal = {J. Chem. Phys.},
number = {21},
pages = {9213--9222},
title = {Is {M{\o}ller-Plesset} perturbation theory a convergent ab initio method?},
volume = {112},
Bdsk-Url-1 = {https://doi.org/10.1063/1.481764}}
@article{Moller_1934, @article{Moller_1934,
author = {M{\o}ller, Chr. and Plesset, M. S.}, author = {M{\o}ller, Chr. and Plesset, M. S.},
date = {1934-10-01}, date = {1934-10-01},

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@ -1,6 +1,5 @@
\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1} \documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,txfonts,mhchem} \usepackage{graphicx,tabularx,array,booktabs,longtable,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx,mhchem}
\usepackage[utf8]{inputenc} \usepackage[utf8]{inputenc}
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@ -27,7 +26,6 @@
\newcommand{\fnt}{\footnotetext} \newcommand{\fnt}{\footnotetext}
\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}} \newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
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% operators % operators
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@ -302,10 +300,10 @@ with $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$, and where
\begin{equation} \begin{equation}
\braket{pq}{rs} = \iint \dd\vb{x}_1\dd\vb{x}_2\frac{\phi_p(\vb{x}_1)\phi_q(\vb{x}_2)\phi_r(\vb{x}_1)\phi_s(\vb{x}_2)}{\abs{\vb{r}_1 - \vb{r}_2}} \braket{pq}{rs} = \iint \dd\vb{x}_1\dd\vb{x}_2\frac{\phi_p(\vb{x}_1)\phi_q(\vb{x}_2)\phi_r(\vb{x}_1)\phi_s(\vb{x}_2)}{\abs{\vb{r}_1 - \vb{r}_2}}
\end{equation} \end{equation}
are two-electron integral in the spin-orbital basis. \cite{Gill_1994} are two-electron integrals in the spin-orbital basis. \cite{Gill_1994}
As mentioned earlier, there is, \textit{a priori}, no guarantee that the MP$m$ series converges to the exact energy when $m \to \infty$. As mentioned earlier, there is, \textit{a priori}, no guarantee that the MP$m$ series converges to the exact energy when $m \to \infty$.
In fact, it is known that when the HF wave function is a poor approximation to the exact wave function, for example in multi-reference systems, the MP method yields deceptive results. \cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988} In fact, it is known that when the HF wave function is a poor approximation to the exact wave function, for example in multi-reference systems, the MP method yields deceptive results. \cite{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000}
A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable. A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable.
By doing so, the Hamiltonian and the energy become complex-valued functions of $\lambda$, and the energy becomes a multivalued function on $K$ Riemann sheets (where $K$ is the number of basis functions). By doing so, the Hamiltonian and the energy become complex-valued functions of $\lambda$, and the energy becomes a multivalued function on $K$ Riemann sheets (where $K$ is the number of basis functions).
As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series. As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series.
@ -364,11 +362,16 @@ The discovery of this divergent behavior is worrying as in order to get meaningf
A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative). Their method consists in performing a scan of the real axis to detect the avoided crossing responsible for the pair of dominant singularities. Then, by modeling this avoided crossing via a two-state Hamiltonian one can get an approximation of the dominant conjugate pair of singularities by finding the EPs of the following $2\times2$ matrix A singularity in the unit circle is designated as an intruder state, more precisely as a front-door (respectively back-door) intruder state if the real part of the singularity is positive (respectively negative). Their method consists in performing a scan of the real axis to detect the avoided crossing responsible for the pair of dominant singularities. Then, by modeling this avoided crossing via a two-state Hamiltonian one can get an approximation of the dominant conjugate pair of singularities by finding the EPs of the following $2\times2$ matrix
\begin{equation} \begin{equation}
\underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha + \alpha_s & 0 \\ 0 & \beta + \beta_s )}_{\bH^{(0)}} + \underbrace{\mqty(- \alpha_s & \delta \\ \delta & - \beta_s)}_{\bV}, \underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta \\ \delta & - \gamma)}_{\bV},
\end{equation} \end{equation}
where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ and the second matrix in the right-hand-side $\bV$ is the perturbation. where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ and the second matrix in the right-hand-side $\bV$ is the perturbation.
In Ref.~\onlinecite{Olsen_2019}, the simple $2 \times 2$ model proposed by Olsen \textit{et al.} is generalized to a non-symmetric (i.e, non-Hermitian) Hamiltonian and various choice of perturbation (not only the MP partioning). In Ref.~\onlinecite{Olsen_2019}, the simple two-state model proposed by Olsen \textit{et al.} is generalized to a non-symmetric Hamiltonian
\begin{equation}
\underbrace{\mqty(\alpha & \delta_1 \\ \delta_2 & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta_2 \\ \delta_1 & - \gamma)}_{\bV}.
\end{equation}
allowing an analysis of various choice of perturbation (not only the MP partioning) such as coupled cluster perturbation expansions and other non-Hermitian perturbation methods.
It is worth noting that only cases where $\text{sgn}(\delta_1) = - \text{sgn}(\delta_2)$ leads to new forms of perturbation expansions.
Interestingly, they showed that the convergence pattern of a given perturbation method can be characterized by its archetype which defines the overall ``shape'' of the energy convergence and can be subdivided in five classes for Hermitian Hamiltonians (zigzag, interspersed zigzag, triadic, ripples, and geometric), while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians. Interestingly, they showed that the convergence pattern of a given perturbation method can be characterized by its archetype which defines the overall ``shape'' of the energy convergence and can be subdivided in five classes for Hermitian Hamiltonians (zigzag, interspersed zigzag, triadic, ripples, and geometric), while two additional archetypes (zigzag-geometric and convex-geometric) are observed in non-Hermitian Hamiltonians.
Other features characterizing the convergence behavior of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern. Other features characterizing the convergence behavior of a perturbation method are its rate of convergence, its length of recurring period, and its sign pattern.

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References/Olsen_2019.pdf
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