reviewing Hugh stuff. Corrected a few typos
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@ -259,9 +259,9 @@ Analytically solvable models are essential in theoretical chemistry and physics
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easily tested while retaining the key physical phenomena.
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Using the (localised) site basis, the Hilbert space of the Hubbard dimer comprises the four configurations
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\begin{align}
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\begin{align*}
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& \ket{\Lup \Ldown} & & \ket{\Lup\Rdown} & & \ket{\Rup\Ldown} & & \ket{\Rup\Rdown}
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\end{align}
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\end{align*}
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where $\Lsi$ ($\Rsi$) denotes an electron with spin $\sigma$ on the left (right) site.
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The exact, or full configuration interaction (FCI), Hamiltonian is then
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\begin{equation}
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@ -504,11 +504,13 @@ E_\text{HF}(\ta, \tb) = -t\, \qty( \sin \ta + \sin \tb ) + \frac{U}{2} \qty( 1 +
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\end{equation}
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where we have introduced bonding $\mathcal{B}^{\sigma}$ and anti-bonding $\mathcal{A}^{\sigma}$ molecular orbitals for
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the spin-$\sigma$ electrons as
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\begin{subequations}
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\begin{align}
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\mathcal{B}^{\sigma} & = \hphantom{-} \cos(\frac{\theta_\sigma}{2}) \Lsi + \sin(\frac{\theta_\sigma}{2}) \Rsi,
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\\
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\mathcal{A}^{\sigma} & = - \sin(\frac{\theta_\sigma}{2}) \Lsi + \cos(\frac{\theta_\sigma}{2}) \Rsi
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\end{align}
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\end{subequations}
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In the weak correlation regime $0 \le U \le 2t$, the angles which minimise the HF energy,
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\ie, $\pdv*{E_\text{HF}}{\theta_\sigma} = 0$, are
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\begin{equation}
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@ -556,6 +558,7 @@ Note that the RHF wave function remains a genuine solution of the HF equations f
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of the HF energy rather than a minimum.
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This critical point is analogous to the infamous Coulson--Fischer point identified in the hydrogen dimer.\cite{Coulson_1949}
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For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become
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\begin{subequations}
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\begin{align}
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\ta^\text{UHF} & = \arctan (-\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\label{eq:ta_uhf}
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@ -563,6 +566,7 @@ For $U \ge 2t$, the optimal orbital rotation angles for the UHF orbitals become
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\tb^\text{UHF} & = \arctan (+\frac{2t}{\sqrt{U^2 - 4t^2}}),
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\label{eq:tb_uhf}
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\end{align}
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\end{subequations}
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with the corresponding UHF ground-state energy (Fig.~\ref{fig:HF_real})
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\begin{equation}
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E_\text{UHF} \equiv E_\text{HF}(\ta^\text{UHF}, \tb^\text{UHF}) = - \frac{2t^2}{U}.
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@ -981,8 +985,8 @@ sacrificed convergence of the excited-state series so that the ground-state conv
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Since the UHF ground state already provides a good approximation to the exact energy, the ground-state sheet of
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the UMP energy is relatively flat and the corresponding EP in the Hubbard dimer always lies outside the unit cylinder.
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\hugh{The slow convergence observed in stretched \ce{H2}\cite{Gill_1988} can then be seen as this EP
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moves increasingly close to the unit cylinder at large $U/t$ and $\rc$ approaches one (from above).}
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The slow convergence observed in stretched \ce{H2}\cite{Gill_1988} can then be seen as this EP
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moves increasingly close to the unit cylinder at large $U/t$ and $\rc$ approaches one (from above).
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Furthermore, the majority of the UMP expansion in this regime is concerned with removing spin-contamination from the wave
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function rather than improving the energy.
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It is well-known that the spin-projection needed to remove spin-contamination can require non-linear combinations
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@ -1010,16 +1014,16 @@ very slowly as the perturbation order is increased.
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%==========================================%
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% CREMER AND HE
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\hugh{As computational implementations of higher-order MP terms improved, the systematic investigation
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of convergence behaviour in a broader class of molecules became possible.}
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Cremer and He \hugh{introduced an efficient MP6 approach and used it to analyse the RMP convergence of}
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As computational implementations of higher-order MP terms improved, the systematic investigation
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of convergence behaviour in a broader class of molecules became possible.
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Cremer and He introduced an efficient MP6 approach and used it to analyse the RMP convergence of
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29 atomic and molecular systems with respect to the FCI energy.\cite{Cremer_1996}
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They established two general classes: ``class A'' systems that exhibit monotonic convergence;
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and ``class B'' systems for which convergence is erratic after initial oscillations.
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%Their system set contains stretched molecules as well as molecules at their equilibrium geometry for various basis sets.
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\hugh{By analysing the different cluster contributions to the MP energy terms, they proposed that
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By analysing the different cluster contributions to the MP energy terms, they proposed that
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class A systems generally include well-separated and weakly correlated electron pairs, while class B systems
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are characterised by dense electron clustering in one or more spatial regions.}\cite{Cremer_1996}
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are characterised by dense electron clustering in one or more spatial regions.\cite{Cremer_1996}
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%\textit{``Class A systems are characterised by electronic structures with well-separated electron pairs while class B systems are characterized by electronic structures with electron clustering in one or more regions.''}
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%Moreover, they analysed the contribution of the triple (T) excitations to the MP4, MP5 and MP6 energies next to the single, double and quadruple (SDQ) excitations contribution.
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In class A systems, they showed that the majority of the correlation energy arises from pair correlation,
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@ -1028,9 +1032,9 @@ On the other hand, triple excitations have an important contribution in class B
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orbital relaxation, and these contributions lead to oscillations of the total correlation energy.
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%This observation on the contribution to the MP$n$ energy corroborates the electronic structure discussed above.
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\hugh{%
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Using these classifications, Cremer and He then introduced simple extrapolation formulas for estimating the
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exact correlation energy $\Delta E$ using terms up to MP6\cite{Cremer_1996}
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\begin{subequations}
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\begin{align}
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\Delta E_{\text{A}}
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&= \Emp^{(2)} + \Emp^{(3)} + \Emp^{(4)}
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@ -1039,6 +1043,7 @@ exact correlation energy $\Delta E$ using terms up to MP6\cite{Cremer_1996}
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\Delta E_{\text{B}}
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&= \Emp^{(2)} + \Emp^{(3)} + \qty(\Emp^{(4)} + \Emp^{(5)}) \exp(\Emp^{(6)} / \Emp^{(5)}).
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\end{align}
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\end{subequations}
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%As one can only compute the first terms of the MP series, a smart way of getting more accurate results is to use extrapolation formula, \ie, estimating the limit of the series with only few terms.
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%Cremer and He proved that using specific extrapolation formulas of the MP series for class A and class B systems improves the precision of the results compared to the formula used without resorting to classes. \cite{Cremer_1996}
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These class-specific formulas reduced the mean absolute error from the FCI correlation energy by a
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@ -1046,7 +1051,6 @@ factor of four compared to previous class-independent extrapolations,
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highlighting how one can leverage a deeper understanding of MP convergence to improve estimates of
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the correlation energy at lower computational costs.
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In Section~\ref{sec:Resummation}, we consider more advanced extrapolation routines that take account of EPs in the complex $\lambda$-plane.
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}
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%The mean absolute deviation taking the FCI correlation energies as reference is $0.3$ millihartree with the class-specific formula whereas the deviation increases to 12 millihartree using the general formula.
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%Even if there were still shaded areas in their analysis and that their classification was incomplete, the work of Ref.~\onlinecite{Cremer_1996} clearly evidenced that understanding the origin of the different modes of convergence could potentially lead to a more rationalised use of MP perturbation theory and, hence, to more accurate correlation energy estimates.
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@ -1055,9 +1059,9 @@ They showed that the series could be divergent even in systems that were conside
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such as \ce{Ne} or the \ce{HF} molecule. \cite{Olsen_1996, Christiansen_1996}
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Cremer and He had already studied these two systems and classified them as \textit{class B} systems.\cite{Cremer_1996}
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However, Olsen and co-workers performed their analysis in larger basis sets containing diffuse functions,
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finding that the correspoding MP series becomes divergent at (very) high order.
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The discovery of this divergent behaviour is particularly worrying as large basis sets \trashHB{(as close to
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the complete basis set as possible)} are required to get meaningful and accurate energies.\cite{Loos_2019d,Giner_2019}
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finding that the corresponding MP series becomes divergent at (very) high order.
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The discovery of this divergent behaviour is particularly worrying as large basis sets
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are required to get meaningful and accurate energies.\cite{Loos_2019d,Giner_2019}
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Furthermore, diffuse functions are particularly important for anions and/or Rydberg excited states, where the wave function
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is inherently more diffuse than the ground state.\cite{Loos_2018a,Loos_2020a}
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@ -1068,8 +1072,7 @@ Their analysis is based on Darboux's theorem: \cite{Goodson_2011}
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\begin{quote}
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\textit{``In the limit of large order, the series coefficients become equivalent to the Taylor series coefficients of the singularity closest to the origin. ''}
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\end{quote}
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\trashHB{Following the result of this theorem, the convergence patterns of the MP series can be explained by looking at the dominant singularity.}
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\hugh{Following this theory,} a singularity in the unit circle is designated as an intruder state,
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Following this theory, a singularity in the unit circle is designated as an intruder state,
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with a front-door (or back-door) intruder state if the real part of the singularity is positive (or negative).
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Using their observations in Ref.~\onlinecite{Olsen_1996}, Olsen and collaborators proposed
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@ -1084,7 +1087,7 @@ the dominant singularities as the EPs of the $2\times2$ matrix
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+ \underbrace{\mqty( -\alpha_{\text{s}} & \delta \\ \delta & - \beta_{\text{s}})}_{\bV},
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\end{equation}
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where the diagonal matrix is the unperturbed Hamiltonian matrix $\bH^{(0)}$ with level shifts
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$\alpha_{\text{s}}$ and $\beta_{\text{s}}$, \hugh{and $\bV$ represents the perturbation.}
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$\alpha_{\text{s}}$ and $\beta_{\text{s}}$, and $\bV$ represents the perturbation.
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The authors first considered molecules with low-lying doubly-excited states with the same spatial
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and spin symmetry as the ground state. \cite{Olsen_2000}
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@ -1092,14 +1095,14 @@ In these systems, the exact wave function has a non-negligible contribution from
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and thus the low-lying excited states are likely to become intruder states.
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For \ce{CH_2} in a diffuse, yet rather small basis set, the series is convergent at least up to the 50th order, and
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the dominant singularity lies close (but outside) the unit circle, causing slow convergence of the series.
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\hugh{These intruder-state effects are analogous to the EP that dictates the convergence behaviour of
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the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}).}
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These intruder-state effects are analogous to the EP that dictates the convergence behaviour of
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the RMP series for the Hubbard dimer (Fig.~\ref{fig:RMP}).
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Furthermore, the authors demonstrated that the divergence for \ce{Ne} is due to a back-door intruder state
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that arise when the ground state undergos sharp avoided crossings with highly diffuse excited states.
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that arise when the ground state undergoes sharp avoided crossings with highly diffuse excited states.
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%They used their two-state model on this avoided crossings and the model was actually predicting the divergence of the series.
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%They concluded that the divergence of the series was due to the interaction with a highly diffuse excited state.
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\hugh{This divergence is related to a more fundamental critical point in the MP energy surface that we will
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discuss in Section~\ref{sec:MP_critical_point}.}
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This divergence is related to a more fundamental critical point in the MP energy surface that we will
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discuss in Section~\ref{sec:MP_critical_point}.
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Finally, Ref.~\onlinecite{Olsen_1996} proved that the extrapolation formulas of Cremer and He \cite{Cremer_1996}
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are not mathematically motivated when considering the complex singularities causing the divergence, and therefore
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@ -1130,14 +1133,14 @@ The three remaining Hermitian archetypes seem to be rarely observed in MP pertur
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In contrast, the non-Hermitian coupled cluster perturbation theory,%
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\cite{Pawlowski_2019a,Pawlowski_2019b,Pawlowski_2019c,Pawlowski_2019d,Pawlowski_2019e} exhibits a range of archetypes
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including the interspersed zigzag, triadic, ripple, geometric, and zigzag-geometric forms.
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\hugh{This analysis highlights the importance of the primary critical point in controlling the high-order convergergence,
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regardless of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000}}
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This analysis highlights the importance of the primary critical point in controlling the high-order convergence,
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regardless of whether this point is inside or outside the complex unit circle. \cite{Handy_1985,Olsen_2000}
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%=======================================
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\subsection{The singularity structure}
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\label{sec:MP_critical_point}
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%=======================================
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In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analyzed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and collaborators were trying to find the dominant singularity causing the divergence. \cite{Olsen_1996,Olsen_2000,Olsen_2019}
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In the 2000's, Sergeev and Goodson \cite{Sergeev_2005, Sergeev_2006} analysed this problem from a more mathematical point of view by looking at the whole singularity structure where Olsen and collaborators were trying to find the dominant singularity causing the divergence. \cite{Olsen_1996,Olsen_2000,Olsen_2019}
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They regrouped singularities in two classes: i) $\alpha$ singularities which have ``large'' imaginary parts, and ii) $\beta$ singularities which have very small imaginary parts.
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Singularities of $\alpha$-type are related to large avoided crossing between the ground and low-lying excited states, whereas $\beta$ singularities come from a sharp avoided crossing between the ground state and a highly diffuse state.
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They succeeded to explain the divergence of the series caused by $\beta$ singularities following previous work by Stillinger. \cite{Stillinger_2000}
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