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changes to the UMP hubbard dimer convergence

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Hugh Burton 2020-11-24 15:13:47 +00:00
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Manuscript/EPAWTFT.tex
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@ -27,7 +27,7 @@
%\newcommand{\latin}[1]{\textit{#1}} %\newcommand{\latin}[1]{\textit{#1}}
\newcommand{\ie}{\latin{i.e.}} \newcommand{\ie}{\latin{i.e.}}
\newcommand{\eg}{\latin{e.g.}} \newcommand{\eg}{\latin{e.g.}}
\newcommand{\etal}{\latin{et.\ al}} \newcommand{\etal}{\textit{et.\ al}}
\newcommand{\mc}{\multicolumn} \newcommand{\mc}{\multicolumn}
\newcommand{\fnm}{\footnotemark} \newcommand{\fnm}{\footnotemark}
@ -731,7 +731,7 @@ Early implementations were restricted to the fourth-order MP4 approach that was
to offer state-of-the-art quantitative accuracy.\cite{Pople_1978,Krishnan_1980} to offer state-of-the-art quantitative accuracy.\cite{Pople_1978,Krishnan_1980}
However, it was quickly realised that the MP series often demonstrated very slow, oscillatory, However, it was quickly realised that the MP series often demonstrated very slow, oscillatory,
or erratic convergence, with the UMP series showing particularly slow convergence.\cite{Laidig_1985,Knowles_1985,Handy_1985} or erratic convergence, with the UMP series showing particularly slow convergence.\cite{Laidig_1985,Knowles_1985,Handy_1985}
For example, RMP5 is worse than RMP4 for predicting the the homolytic barrier fission of \ce{He2^2+} using a minimal basis set, For example, RMP5 is worse than RMP4 for predicting the homolytic barrier fission of \ce{He2^2+} using a minimal basis set,
while the UMP series monotonically converges but becomes increasingly slow beyond UMP5.\cite{Gill_1986} while the UMP series monotonically converges but becomes increasingly slow beyond UMP5.\cite{Gill_1986}
The first examples of divergent MP series were observed in the heavy-atom \ce{N2} and \ce{F2} The first examples of divergent MP series were observed in the heavy-atom \ce{N2} and \ce{F2}
diatomics, where low-order RMP and UMP expansions give qualitatively wrong binding curves.\cite{Laidig_1987} diatomics, where low-order RMP and UMP expansions give qualitatively wrong binding curves.\cite{Laidig_1987}
@ -756,19 +756,21 @@ in the reference wave function.\cite{Nobes_1987}
Using the UHF framework allows} the singlet ground state wave function to mix with triplet wave functions, Using the UHF framework allows} the singlet ground state wave function to mix with triplet wave functions,
leading to \hugh{spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.} leading to \hugh{spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.}
\hugh{The link between slow UMP convergence and this spin-contamination was first systematically investigated} \hugh{The link between slow UMP convergence and this spin-contamination was first systematically investigated}
by Gill and \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988} by Gill \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988}
\hugh{In this work, the authors compared the UMP series with the exact RHF- and UHF-based FCI expansions \hugh{In this work, the authors compared the UMP series with the exact RHF- and UHF-based FCI expansions
and identified that the slow UMP convergence arises from its failure to correctly predict the amplitude of the and identified that the slow UMP convergence arises from its failure to correctly predict the amplitude of the
low-lying double excitation. low-lying double excitation.
This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference This erroneous description of the double excitation amplitude has the same origin as the spin-contamination in the reference
UHF wave function, creating the first direct link between spin-contamination and slow UMP convergence.\cite{Gill_1988} UHF wave function, creating the first direct link between spin-contamination and slow UMP convergence.\cite{Gill_1988}
Lepetit \etal\ later analysed the difference between the unrestricted MP and EN partitionings and argued Lepetit \etal\ later analysed the difference between perturbation convergence using the unrestricted MP
that the slow UMP convergence for stretched molecules arises from (i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2}) and EN partitionings.\cite{Lepetit_1988}
They argued that the slow UMP convergence for stretched molecules arises from
(i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2})
tends to a constant value instead of vanishing, and (ii) the slow convergence of contributions from the tends to a constant value instead of vanishing, and (ii) the slow convergence of contributions from the
singly-excited configurations that strongly couple to the doubly-excited configurations and first singly-excited configurations that strongly couple to the doubly-excited configurations and first
appear at fourth-order.\cite{Lepetit_1988} appear at fourth-order.\cite{Lepetit_1988}
Drawing these ideas together, we believe that slow UMP convergence occurs because the single excitations must focus on removing Drawing these ideas together, we believe that slow UMP convergence occurs because the single excitations must focus on removing
spin-contamination from the reference wave function rather than fine-tuning the amplitudes of the higher spin-contamination from the reference wave function, limiting their ability to fine-tune the amplitudes of the higher
excitations that capture the correlation energy. excitations that capture the correlation energy.
} }
@ -834,9 +836,9 @@ excitations that capture the correlation energy.
\label{fig:RMP}} \label{fig:RMP}}
\end{figure*} \end{figure*}
\hugh{The behaviour of the RMP and UMP series observed in \ce{H2} can again be analytically illustrated by considering \hugh{The behaviour of the RMP and UMP series observed in \ce{H2} can also be illustrated by considering
the Hubbard dimer with a complex-valued perturbation strength. the analytic Hubbard dimer with a complex-valued perturbation strength.
In this system, the stretching of a chemical bond is directly mirrored by an increase in the on-site repulsion $U$. In this system, the stretching of a chemical bond is directly mirrored by an increase in the electron correlation $U/t$.
} }
Using the ground-state RHF reference orbitals leads to the \hugh{parametrised} RMP Hamiltonian Using the ground-state RHF reference orbitals leads to the \hugh{parametrised} RMP Hamiltonian
\begin{widetext} \begin{widetext}
@ -915,7 +917,7 @@ Using the ground-state UHF reference orbitals in the Hubbard dimer yields the \h
While there is a closed-form expression for the ground-state energy, it is cumbersome and we eschew reporting it. While there is a closed-form expression for the ground-state energy, it is cumbersome and we eschew reporting it.
Instead, the radius of convergence of the UMP series can be obtained numerically as a function of $U/t$, as shown Instead, the radius of convergence of the UMP series can be obtained numerically as a function of $U/t$, as shown
in Fig.~\ref{fig:RadConv}. in Fig.~\ref{fig:RadConv}.
These numerical values reveal that the UMP ground state series has $\rc > 1$ for all $U/t$ and must always converge. These numerical values reveal that the UMP ground-state series has $\rc > 1$ for all $U/t$ and must always converge.
However, in the strong correlation limit (large $U$), this radius of convergence tends to unity, indicating that However, in the strong correlation limit (large $U$), this radius of convergence tends to unity, indicating that
the corresponding UMP series becomes increasingly slow. the corresponding UMP series becomes increasingly slow.
Furthermore, the doubly-excited state using the ground-state UHF orbitals has $\rc < 1$ for almost any value Furthermore, the doubly-excited state using the ground-state UHF orbitals has $\rc < 1$ for almost any value
@ -928,22 +930,34 @@ in the complex $\lambda$-plane.
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, while RMP becomes divergent for $U=7t$.
The 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 1 (Fig.~\ref{fig:RadConv}). for larger $U/t$ as the radius of convergence becomes increasingly close to 1 (Fig.~\ref{fig:RadConv}).
% EFFECT OF SYMMETRY BREAKING % EFFECT OF SYMMETRY BREAKING
Allowing the UHF orbitals to break the molecular symmetry introduces new couplings between electronic states As the UHF orbitals break the molecular symmetry, new coupling terms emerge between the electronic states that
and fundamentally changes the structure of the EPs in the complex $\lambda$-plane. cause fundamental changes to the structure of EPs in the complex $\lambda$-plane.
For example, while the RMP energy shows only one EP between the ground state and For example, while the RMP energy shows only one EP between the ground state and
the doubly-excited state (Fig.~\ref{fig:RMP}), the UMP energy has two EPs: one connecting the ground state with the the doubly-excited state (Fig.~\ref{fig:RMP}), the UMP energy has two EPs: one connecting the ground state with the
first-excited \titou{open-shell singlet?}, and the other connecting the \titou{open-shell singlet?} to the singly-excited open-shell singlet, and the other connecting this single excitation to the
doubly-excited second excitation (Fig.~\ref{fig:UMP}). doubly-excited second excitation (Fig.~\ref{fig:UMP}).
While this symmetry-breaking makes the ground-state UMP series convergent by moving the corresponding \hugh{%
EP outside the unit cylinder, this process also moves the excited-state EP within the unit cylinder This new ground-state EP always appears outside the unit cylinder and guarantees convergence of the ground-state energy.
and thus causes a deterioration in the convergence of the excited-state UMP series. However, the excited-state EP is moved within} the unit cylinder and causes the
Furthermore, since the UHF ground-state energy is already an improved approximation of the exact energy, the ground-state convergence of the excited-state UMP series to deteriorate.
UMP energy surface is relatively flat and the majority of the UMP expansion is concerned with removing \hugh{Our interpretation of this effect is that the symmetry-broken orbital optimisation has redistributed the strong
spin-contamination from the wave function. coupling between the ground- and doubly-excited states into weaker couplings between all states, and has thus
sacrificed convergence of the excited-state series so that the chance of ground-state convergence can be maximised.}
Since the UHF ground state already provides a good approximation to the exact energy, the ground-state sheet of
the UMP energy is relatively flat and the corresponding EP in the Hubbard dimer always lies outside the unit cylinder.
\hugh{The slow convergence observed in \ce{H2} can then be seen as this EP moves ever closer to one at larger $U/t$ values.}
Furthermore, the majority of the UMP expansion in this regime is concerned with removing spin-contamination from the wave
function \hugh{rather than improving the energy.
It is well-known that the spin-projection needed to remove spin-contamination can require non-linear combinations
of highly-excited determinants,\cite{Lowdin_1955c} and thus it is not surprising that this process proceeds
very slowly as the perturbation order is increased.
}
%The convergence of the UMP as a function of the ratio $U/t$ is shown in Fig.~\ref{subfig:UMP_cvg} for two specific values: the first ($U = 3t$) is well within the RMP convergence region, while the second ($U = 7t$) falls outside. %The convergence of the UMP as a function of the ratio $U/t$ is shown in Fig.~\ref{subfig:UMP_cvg} for two specific values: the first ($U = 3t$) is well within the RMP convergence region, while the second ($U = 7t$) falls outside.
%Note that in the case of UMP, there are now two pairs of EPs as the open-shell singlet now couples strongly with both the ground and doubly-excited states. %Note that in the case of UMP, there are now two pairs of EPs as the open-shell singlet now couples strongly with both the ground and doubly-excited states.