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Pierre-Francois Loos 2020-12-01 16:32:09 +01:00
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@ -644,7 +644,6 @@ role of \textit{quasi}-EPs in determining the behaviour of the HF approximation.
\label{sec:MP} \label{sec:MP}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%=====================================================% %=====================================================%
\subsection{Background Theory} \subsection{Background Theory}
%=====================================================% %=====================================================%
@ -734,10 +733,9 @@ diatomics, where low-order RMP and UMP expansions give qualitatively wrong bindi
% SLOW UMP CONVERGENCE AND SPIN CONTAMINATION % SLOW UMP CONVERGENCE AND SPIN CONTAMINATION
The divergence of RMP expansions for stretched bonds can be easily understood from two perspectives.\cite{Gill_1988a} The divergence of RMP expansions for stretched bonds can be easily understood from two perspectives.\cite{Gill_1988a}
Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the Firstly, the exact wave function becomes increasingly multi-configurational as the bond is stretched, and the
HF wave function no longer provides a qualitatively correct reference for the perturbation expansion. \titou{R}HF wave function no longer provides a qualitatively correct reference for the perturbation expansion.
Secondly, the energy gap between the bonding and anitbonding orbitals associated with the stretch becomes Secondly, the energy gap between the bonding and antibonding orbitals associated with the stretch becomes
increasingly small at larger bond lengths, leading to a divergence in the Rayleigh--Schr\"odinger perturbation increasingly small at larger bond lengths, \titou{leading to a divergence, for example, in the second-order MP correction \eqref{eq:EMP2}.}
expansion Eq.~\eqref{eq:EMP2}.
In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains In contrast, the origin of slow UMP convergence is less obvious as the reference UHF energy remains
qualitatively correct at large bond lengths and the orbital degeneracy is avoided. qualitatively correct at large bond lengths and the orbital degeneracy is avoided.
Furthermore, this slow convergence can also be observed in molecules with a UHF ground state at the equilibrium Furthermore, this slow convergence can also be observed in molecules with a UHF ground state at the equilibrium
@ -748,14 +746,14 @@ Using the UHF framework allows the singlet ground state wave function to mix wit
leading to spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator. leading to spin contamination where the wave function is no longer an eigenfunction of the $\Hat{\cS}^2$ operator.
The link between slow UMP convergence and this spin-contamination was first systematically investigated The link between slow UMP convergence and this spin-contamination was first systematically investigated
by Gill \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988} by Gill \etal\ using the minimal basis \ce{H2} model.\cite{Gill_1988}
In this work, the authors compared the UMP series with the exact RHF- and UHF-based FCI expansions In this work, the authors compared \titou{the UMP series with the exact RHF- and UHF-based FCI expansions (T2: I don't understand this)}
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 CHAT % LEPETIT CHAT
Lepetit \etal\ later analysed the difference between perturbation convergence using the unrestricted MP Lepetit \etal\ later analysed the difference between perturbation convergence using the UMP
and EN partitionings. \cite{Lepetit_1988} and EN partitionings. \cite{Lepetit_1988}
They argued that the slow UMP convergence for stretched molecules arises from They argued that the slow UMP convergence for stretched molecules arises from
(i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2}) (i) the fact that the MP denominator (see Eq.~\ref{eq:EMP2})