moving Hubbard MP section
This commit is contained in:
parent
ff78e32218
commit
b9dccf97cd
@ -6,12 +6,13 @@
|
||||
\BOOKMARK [1][-]{section*.7}{Perturbation theory}{section*.2}% 6
|
||||
\BOOKMARK [2][-]{section*.8}{Rayleigh-Schr\366dinger perturbation theory}{section*.7}% 7
|
||||
\BOOKMARK [2][-]{section*.9}{The Hartree-Fock Hamiltonian}{section*.7}% 8
|
||||
\BOOKMARK [2][-]{section*.11}{M\370ller-Plesset perturbation theory}{section*.7}% 9
|
||||
\BOOKMARK [1][-]{section*.12}{Historical overview}{section*.2}% 10
|
||||
\BOOKMARK [2][-]{section*.13}{Behavior of the M\370ller-Plesset series}{section*.12}% 11
|
||||
\BOOKMARK [2][-]{section*.16}{Insights from a two-state model}{section*.12}% 12
|
||||
\BOOKMARK [2][-]{section*.17}{The singularity structure}{section*.12}% 13
|
||||
\BOOKMARK [2][-]{section*.18}{The physics of quantum phase transitions}{section*.12}% 14
|
||||
\BOOKMARK [1][-]{section*.19}{Conclusion}{section*.2}% 15
|
||||
\BOOKMARK [1][-]{section*.20}{Acknowledgments}{section*.2}% 16
|
||||
\BOOKMARK [1][-]{section*.21}{References}{section*.2}% 17
|
||||
\BOOKMARK [2][-]{section*.10}{Complex adiabatic connection}{section*.7}% 9
|
||||
\BOOKMARK [2][-]{section*.12}{M\370ller-Plesset perturbation theory}{section*.7}% 10
|
||||
\BOOKMARK [1][-]{section*.14}{Historical overview}{section*.2}% 11
|
||||
\BOOKMARK [2][-]{section*.15}{Behavior of the M\370ller-Plesset series}{section*.14}% 12
|
||||
\BOOKMARK [2][-]{section*.17}{Insights from a two-state model}{section*.14}% 13
|
||||
\BOOKMARK [2][-]{section*.18}{The singularity structure}{section*.14}% 14
|
||||
\BOOKMARK [2][-]{section*.19}{The physics of quantum phase transitions}{section*.14}% 15
|
||||
\BOOKMARK [1][-]{section*.20}{Conclusion}{section*.2}% 16
|
||||
\BOOKMARK [1][-]{section*.21}{Acknowledgments}{section*.2}% 17
|
||||
\BOOKMARK [1][-]{section*.22}{References}{section*.2}% 18
|
||||
|
||||
@ -512,6 +512,10 @@ by swapping $\ta^{\text{UHF}}$ and $\tb^{\text{UHF}}$ in Eqs.~\eqref{eq:ta_uhf}
|
||||
Note that the RHF wave function remains a genuine solution of the HF equations for $U \ge 2t$, but corresponds to a saddle point
|
||||
of the HF energy rather than a minimum.
|
||||
|
||||
%============================================================%
|
||||
\subsection{Complex adiabatic connection}
|
||||
%============================================================%
|
||||
|
||||
\titou{In a recent paper, \cite{Burton_2019} using holomorphic Hartree-Fock (h-HF) \cite{Hiscock_2014,Burton_2018,Burton_2016} as an analytic continuation of conventional HF theory, we have demonstrated, on a simple model, that one can interconvert states of different symmetries and natures by following well-defined contours in the complex $\lambda$-plane, where $\lambda$ is the strength of the electron-electron interaction (see Fig.~\ref{fig:iAC}).
|
||||
In particular, by slowly varying $\lambda$ in a similar (yet different) manner to an adiabatic connection in density-functional theory, \cite{Langreth_1975,Gunnarsson_1976,Zhang_2004} one can ``morph'' a ground-state wave function into an excited-state wave function via a stationary path of HF solutions. \cite{Seidl_2018}
|
||||
In such a way, we could obtain a doubly-excited state wave function starting from the ground state wave function, a process which is not as easy as one might think. \cite{Gilbert_2008,Thom_2008,Shea_2018}
|
||||
@ -573,12 +577,71 @@ In order to better understand the behavior of the MP series and how it is connec
|
||||
For small systems, one can access the whole terms of the series using full configuration interaction (FCI).
|
||||
If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI space, one gets the exact energies (in this finite Hilbert space) and the Taylor expansion with respect to $\lambda$ allows to access the MP perturbation series at any order.
|
||||
|
||||
Let us illustrate the behaviour of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}).
|
||||
Within the RMP partition technique, we have
|
||||
\begin{equation}
|
||||
\label{eq:H_RMP}
|
||||
\bH_\text{RMP} =
|
||||
\begin{pmatrix}
|
||||
-2t + U - \lambda U/2 & 0 & 0 & \lambda U/2 \\
|
||||
0 & U - \lambda U/2 & \lambda U/2 & 0 \\
|
||||
0 & \lambda U/2 & U - \lambda U/2 & 0 \\
|
||||
\lambda U/2 & 0 & 0 & 2t + U - \lambda U/2 \\
|
||||
\end{pmatrix},
|
||||
\end{equation}
|
||||
which yields the ground-state energy
|
||||
\begin{equation}
|
||||
\label{eq:E0MP}
|
||||
E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}.
|
||||
\end{equation}
|
||||
From this expression, it is easy to identify that the radius of convergence is defined by the EPs at $\lambda = \pm i 4t / U$ (which are actually located at the same position than the exact EPs discussed in Sec.~\ref{sec:example}).
|
||||
Equation \eqref{eq:E0MP} can be Taylor expanded in terms of $\lambda$ to obtain the $n$th-order MP correction
|
||||
\begin{equation}
|
||||
E_\text{MP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2),
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
E_{-}(\lambda) = \sum_{k=0}^\infty E_\text{MP}^{(k)} \lambda^k.
|
||||
\end{equation}
|
||||
We illustrate this by plotting the energy expansions at orders of $\lambda$ just below and above the radius of convergence in Fig.~\ref{fig:RMP}.
|
||||
|
||||
At the UMP level now, we have
|
||||
\begin{widetext}
|
||||
\begin{equation}
|
||||
\label{eq:H_UMP}
|
||||
\bH_\text{UMP} =
|
||||
\begin{pmatrix}
|
||||
-2t^2 \lambda/U & 0 & 0 & +2t^2 \lambda/U \\
|
||||
0 & U - 2t^2 \lambda/U & +2t^2\lambda/U & +2t \sqrt{U^2 - (2t)^2} \lambda/U \\
|
||||
0 & +2t^2\lambda/U & U - 2t^2 \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U \\
|
||||
+2t^2 \lambda/U & +2t \sqrt{U^2 - (2t)^2} \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U & 2U(1-\lambda) + 6t^2\lambda/U \\
|
||||
\end{pmatrix},
|
||||
\end{equation}
|
||||
\end{widetext}
|
||||
A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting its expression.
|
||||
The convergence of the UMP as a function of the ratio $U/t$ is depicted in Fig.~\ref{fig:UMP}.
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure*}
|
||||
\centering
|
||||
\includegraphics[height=0.23\textwidth]{fig3a}
|
||||
\hspace{0.05\textwidth}
|
||||
\includegraphics[height=0.23\textwidth]{fig3b}
|
||||
\hspace{0.05\textwidth}
|
||||
\includegraphics[height=0.23\textwidth]{fig3c}
|
||||
\caption{
|
||||
Convergence of the URMP series as a function of the perturbation order $n$ energies for the Hubbard dimer at $U/t = 3$ and $7$.
|
||||
The Riemann surfaces of the FCI energies are also represented for these two values of $U/t$.
|
||||
\label{fig:UMP}}
|
||||
\end{figure*}
|
||||
|
||||
Obviously, although practically convenient for electronic structure calculations, the MP partitioning is not the only possibility, and alternative partitioning have been proposed in the literature:
|
||||
i) the Epstein-Nesbet (EN) partitioning which consists in taking the diagonal elements of $\hH$ as the zeroth-order Hamiltonian. \cite{Nesbet_1955,Epstein_1926}
|
||||
Hence, the off-diagonal elements of $\hH$ are the perturbation operator,
|
||||
ii) the weak correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\hH^{(0)}$ and the two-electron part is the perturbation operator $\hV$, and
|
||||
iii) the strong coupling partitioning where the two operators are inverted compared to the weak correlation partitioning. \cite{Seidl_2018}
|
||||
|
||||
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
\section{Historical overview}
|
||||
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
||||
@ -640,64 +703,6 @@ The authors illustrate this protocol on the dissociation curve of \ce{LiH} and t
|
||||
\label{fig:RMP}}
|
||||
\end{figure*}
|
||||
|
||||
Let us illustrate the behaviour of the RMP and UMP series on the Hubbard dimer (see Fig.~\ref{fig:RMP}).
|
||||
Within the RMP partition technique, we have
|
||||
\begin{equation}
|
||||
\label{eq:H_RMP}
|
||||
\bH_\text{RMP} =
|
||||
\begin{pmatrix}
|
||||
-2t + U - \lambda U/2 & 0 & 0 & \lambda U/2 \\
|
||||
0 & U - \lambda U/2 & \lambda U/2 & 0 \\
|
||||
0 & \lambda U/2 & U - \lambda U/2 & 0 \\
|
||||
\lambda U/2 & 0 & 0 & 2t + U - \lambda U/2 \\
|
||||
\end{pmatrix},
|
||||
\end{equation}
|
||||
which yields the ground-state energy
|
||||
\begin{equation}
|
||||
\label{eq:E0MP}
|
||||
E_{-}(\lambda) = U - \frac{\lambda U}{2} - \frac{1}{2} \sqrt{(4t)^2 + \lambda ^2 U^2}.
|
||||
\end{equation}
|
||||
From this expression, it is easy to identify that the radius of convergence is defined by the EPs at $\lambda = \pm i 4t / U$ (which are actually located at the same position than the exact EPs discussed in Sec.~\ref{sec:example}).
|
||||
Equation \eqref{eq:E0MP} can be Taylor expanded in terms of $\lambda$ to obtain the $n$th-order MP correction
|
||||
\begin{equation}
|
||||
E_\text{MP}^{(k)} = U \delta_{0,k} - \frac{1}{2} \frac{U^k}{(4t)^{k-1}} \mqty( 1/2 \\ k/2),
|
||||
\end{equation}
|
||||
with
|
||||
\begin{equation}
|
||||
E_{-}(\lambda) = \sum_{k=0}^\infty E_\text{MP}^{(k)} \lambda^k.
|
||||
\end{equation}
|
||||
We illustrate this by plotting the energy expansions at orders of $\lambda$ just below and above the radius of convergence in Fig.~\ref{fig:RMP}.
|
||||
|
||||
At the UMP level now, we have
|
||||
\begin{widetext}
|
||||
\begin{equation}
|
||||
\label{eq:H_UMP}
|
||||
\bH_\text{UMP} =
|
||||
\begin{pmatrix}
|
||||
-2t^2 \lambda/U & 0 & 0 & +2t^2 \lambda/U \\
|
||||
0 & U - 2t^2 \lambda/U & +2t^2\lambda/U & +2t \sqrt{U^2 - (2t)^2} \lambda/U \\
|
||||
0 & +2t^2\lambda/U & U - 2t^2 \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U \\
|
||||
+2t^2 \lambda/U & +2t \sqrt{U^2 - (2t)^2} \lambda/U & -2t \sqrt{U^2 - (2t)^2} \lambda/U & 2U(1-\lambda) + 6t^2\lambda/U \\
|
||||
\end{pmatrix},
|
||||
\end{equation}
|
||||
\end{widetext}
|
||||
A closed-form expression for the ground-state energy can be obtained but it is cumbersome, so we eschew reporting its expression.
|
||||
The convergence of the UMP as a function of the ratio $U/t$ is depicted in Fig.~\ref{fig:UMP}.
|
||||
|
||||
%%% FIG 3 %%%
|
||||
\begin{figure*}
|
||||
\centering
|
||||
\includegraphics[height=0.23\textwidth]{fig3a}
|
||||
\hspace{0.05\textwidth}
|
||||
\includegraphics[height=0.23\textwidth]{fig3b}
|
||||
\hspace{0.05\textwidth}
|
||||
\includegraphics[height=0.23\textwidth]{fig3c}
|
||||
\caption{
|
||||
Convergence of the URMP series as a function of the perturbation order $n$ energies for the Hubbard dimer at $U/t = 3$ and $7$.
|
||||
The Riemann surfaces of the FCI energies are also represented for these two values of $U/t$.
|
||||
\label{fig:UMP}}
|
||||
\end{figure*}
|
||||
|
||||
%==========================================%
|
||||
\subsection{Insights from a two-state model}
|
||||
%==========================================%
|
||||
|
||||
Loading…
Reference in New Issue
Block a user