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\citation{Moller_1934}
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\citation{Moller_1934}
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\citation{SzaboBook}
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\citation{SzaboBook}
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\citation{Gill_1994}
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\citation{Gill_1994}
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\citation{Gill_1986,Gill_1988,Handy_1985,Lepetit_1988,Leininger_2000}
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\citation{JensenBook}
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\citation{JensenBook}
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\citation{Seidl_2018}
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\citation{Sergeev_2005}
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\citation{Sergeev_2005}
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\citation{Stillinger_2000}
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\citation{Olsen_2000}
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\citation{Olsen_2000}
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\citation{Cejnar_2009,Sachdev_2011}
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\citation{Cejnar_2009,Sachdev_2011}
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\bibcite{Leininger_2000}{{47}{2000}{{Leininger\ \emph {et~al.}}}{{Leininger, Allen, Schaefer,\ and\ Sherrill}}}
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\bibcite{JensenBook}{{48}{2017}{{Jensen}}{{}}}
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@article{Leininger_2000,
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author = {Leininger,Matthew L. and Allen,Wesley D. and Schaefer,Henry F. and Sherrill,C. David},
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date-added = {2020-11-12 14:50:57 +0100},
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date-modified = {2020-11-12 14:51:07 +0100},
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doi = {10.1063/1.481764},
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eprint = {https://doi.org/10.1063/1.481764},
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journal = {The Journal of Chemical Physics},
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number = {21},
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pages = {9213-9222},
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title = {Is Mo/ller--Plesset perturbation theory a convergent ab initio method?},
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url = {https://doi.org/10.1063/1.481764},
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volume = {112},
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year = {2000},
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@article{Nesbet_1955,
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@article{Nesbet_1955,
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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. },
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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. },
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author = {Nesbet, R. K. and Hartree, Douglas Rayner},
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author = {Nesbet, R. K. and Hartree, Douglas Rayner},
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year = {2019},
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year = {2019},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.5110554}}
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@article{Leininger_2000,
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author = {Leininger, Matthew L. and Allen, Wesley D. and Schaefer, Henry F. and Sherrill, C. David},
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date = {2000-05-17},
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doi = {10.1063/1.481764},
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journal = {J. Chem. Phys.},
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number = {21},
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pages = {9213--9222},
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title = {Is {M{\o}ller-Plesset} perturbation theory a convergent ab initio method?},
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volume = {112},
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Bdsk-Url-1 = {https://doi.org/10.1063/1.481764}}
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@article{Moller_1934,
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@article{Moller_1934,
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author = {M{\o}ller, Chr. and Plesset, M. S.},
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author = {M{\o}ller, Chr. and Plesset, M. S.},
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date = {1934-10-01},
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date = {1934-10-01},
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\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\documentclass[aps,prb,reprint,noshowkeys,superscriptaddress]{revtex4-1}
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\usepackage{graphicx,dcolumn,bm,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,longtable,txfonts,mhchem}
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\usepackage{graphicx,tabularx,array,booktabs,longtable,dcolumn,xcolor,microtype,multirow,amscd,amsmath,amssymb,amsfonts,physics,siunitx,mhchem}
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\usepackage[utf8]{inputenc}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage[T1]{fontenc}
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\usepackage{txfonts}
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\usepackage{txfonts}
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@ -27,7 +26,6 @@
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\fnt}{\footnotetext}
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\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\mcc}[1]{\multicolumn{1}{c}{#1}}
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\newcommand{\mr}{\multirow}
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\newcommand{\mr}{\multirow}
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\newcommand{\SI}{\textcolor{blue}{supporting information}}
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% operators
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% operators
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\newcommand{\bH}{\mathbf{H}}
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\newcommand{\bH}{\mathbf{H}}
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@ -302,10 +300,10 @@ with $\mel{pq}{}{rs} = \braket{pq}{rs} - \braket{pq}{sr}$, and where
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\begin{equation}
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\begin{equation}
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\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}}
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\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}}
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\end{equation}
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\end{equation}
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are two-electron integral in the spin-orbital basis. \cite{Gill_1994}
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are two-electron integrals in the spin-orbital basis. \cite{Gill_1994}
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As mentioned earlier, there is, \textit{a priori}, no guarantee that the MP$m$ series converges to the exact energy when $m \to \infty$.
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As mentioned earlier, there is, \textit{a priori}, no guarantee that the MP$m$ series converges to the exact energy when $m \to \infty$.
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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}
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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}
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A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable.
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A convenient way to investigate the convergence properties of the MP series is to analytically continue the coupling parameter $\lambda$ into the complex variable.
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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).
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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).
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As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series.
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As mentioned above, by searching the singularities of the function $E(\lambda)$, one can get information on the convergence properties of the MP series.
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@ -364,11 +362,16 @@ The discovery of this divergent behavior is worrying as in order to get meaningf
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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
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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
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\begin{equation}
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\begin{equation}
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\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},
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\underbrace{\mqty(\alpha & \delta \\ \delta & \beta)}_{\bH} = \underbrace{\mqty(\alpha & 0 \\ 0 & \beta + \gamma )}_{\bH^{(0)}} + \underbrace{\mqty( 0 & \delta \\ \delta & - \gamma)}_{\bV},
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\end{equation}
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\end{equation}
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where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ and the second matrix in the right-hand-side $\bV$ is the perturbation.
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where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ and the second matrix in the right-hand-side $\bV$ is the perturbation.
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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).
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In Ref.~\onlinecite{Olsen_2019}, the simple two-state model proposed by Olsen \textit{et al.} is generalized to a non-symmetric Hamiltonian
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\begin{equation}
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|
\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}.
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\end{equation}
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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.
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It is worth noting that only cases where $\text{sgn}(\delta_1) = - \text{sgn}(\delta_2)$ leads to new forms of perturbation expansions.
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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.
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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.
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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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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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