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RapportStage/Rapport.tex

@@ -461,16 +461,18 @@ When one relies on MP perturbation theory (and more generally on any perturbativ
 %In other words, each time a higher-order term is computed, one would like to obtained an overall result closer to the exact energy. 
 In other words, one would like a monotonic convergence of the MP series. Assuming this, the only limiting process to get the exact correlation energy (in a finite basis set) would be our ability to compute the terms of this perturbation series.
 Unfortunately this is not as easy as one might think because i) the terms of the perturbative series become rapidly computationally cumbersome, and ii) erratic behavior of the perturbative coefficients are not uncommon. For example, in the late 80's, Gill and Radom reported deceptive and slow convergences in stretch systems \cite{Gill_1986, Gill_1988} (see also Refs.~\cite{Handy_1985, Lepetit_1988}). 
-In Fig.~\ref{fig:RUMP_Gill}, which has been extracted from Ref.~\cite{Gill_1986}, one can see that the restricted MP series is convergent, yet oscillating which is far from being convenient if one is only able to compute the first few terms of the expansion (for example here RMP5 is worse than RMP4). 
-On the other hand, the unrestricted MP series is monotonically convergent (except for the first few orders) but very slowly so one cannot use it in practice for systems where one can only compute the first terms.
+In Fig.~\ref{fig:RUMP_Gill}, which has been extracted from Ref.~\cite{Gill_1986}, one can see that the restricted MP \titou{(RMP)} series is convergent, yet oscillating which is far from being convenient if one is only able to compute the first few terms of the expansion (for example here RMP5 is worse than RMP4). 
+On the other hand, the unrestricted MP series \titou{(UMP)} is monotonically convergent (except for the first few orders) but very slowly. 
+Thus, one cannot practically use it for systems where only the first terms can be computed.
 \titou{There is a problem here as one has not introduce restricted and unrestricted formalisms.}
 
 When a bond is stretched, in most cases the exact wave function becomes more and more multi-reference. Yet the HF wave function is restricted to be single Slater determinant.
-Thus, it is inappropriate to model (even qualitatively) stretched system. Nevertheless, the HF wave function can undergo a symmetry breaking to lower its energy by sacrificing one of the symmetry of the exact wave function during the process (see for example the case of \ce{H_2} in Ref.~\cite{SzaboBook}). \antoine{It could be expected that  the restricted MP series has inappropriate convergence properties as a restricted HF Slater determinant is a poor approximation of the exact wave function for stretched system. However even in the unrestricted formalism the series does not give accurate results at low orders. Whereas the unrestricted formalism allows a better description of a stretched system because the spatial orbitals of electrons $\alpha$ and $\beta$ are not restricted to be the same \cite{Fukutome_1981}.} Even with this improvement of the zeroth-order wave function the series does not have the smooth and rapidly converging behavior wanted. 
+Thus, it is inappropriate to model (even qualitatively) stretched system. Nevertheless, the HF wave function can undergo a symmetry breaking to lower its energy by sacrificing one of the symmetry of the exact wave function during the process (see for example the case of \ce{H_2} in Ref.~\cite{SzaboBook}). 
+One could then potentially claim that the RMP series exhibits deceptive convergence properties as the RHF Slater determinant is a poor approximation of the exact wave function for stretched system. However, even in the unrestricted formalism which clearly represents a better description of a stretched system, the UMP series does not have the smooth and rapidly converging behavior that one would wish for. 
 
 \begin{table}[h!]
     \centering
-    \caption{\centering Percentage of electron correlation energy recovered and $\expval{S^2}$ for the \ce{H2} molecule as a function of bond length (r,\si{\angstrom}) in the STO-3G basis set (taken from \cite{Gill_1988}).}
+    \caption{Percentage of electron correlation energy recovered and $\expval{S^2}$ for the \ce{H2} molecule as a function of bond length (r,\si{\angstrom}) in the STO-3G basis set (taken from \cite{Gill_1988}).}
     \begin{tabular}{ccccccc}
 \hline
 \hline
@@ -486,13 +488,14 @@ Thus, it is inappropriate to model (even qualitatively) stretched system. Nevert
     \label{tab:SpinContamination}
 \end{table}
 
-In the unrestricted framework the singlet ground state wave function is allowed to mix with triplet wave function which leads to spin contamination. Gill \textit{et al.}~highlighted the link between slow convergence of the unrestricted MP series and spin contamination of the wave function as shown in Table \ref{tab:SpinContamination} in the example of \ce{H_2} in a minimal basis \cite{Gill_1988}. 
-Handy and coworkers exhibited the same behavior of the series (oscillating and monotonically slowly) in stretched \ce{H_2O} and \ce{NH_2} systems \cite{Handy_1985}. Lepetit et al.~analyzed the difference between the MP and EN partitioning for the unrestricted HF reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the singly and doubly excited configurations. Moreover, the MP and EN numerators in Eqs.~\eqref{eq:EMP2} and \eqref{eq:EEN2} are the same and they vanish when the bond length $r$ goes to infinity. Yet the MP denominators tends towards a constant when $r \to \infty$ so the terms vanish, whereas the EN denominators tends to zero which improves the convergence but can also make the series diverge.
-Cremer and He analyzed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and regrouped all of these systems in two classes. The class A systems where one observes a monotonic convergence to the FCI energy and the class B for which convergence is erratic after initial oscillations. The sample of systems contains stretched molecules as well as molecules at their equilibrium geometry for various basis sets. They highlighted that \cite{Cremer_1996}
+In the unrestricted framework the singlet ground state wave function is allowed to mix with triplet wave function, leading to the so-called spin contamination issue. Gill \textit{et al.}~highlighted the link between slow convergence of the UMP series and spin contamination, as shown in Table \ref{tab:SpinContamination} for the example of \ce{H2} in a minimal basis \cite{Gill_1988}. 
+Handy and coworkers reported the same behavior of the series (oscillating and \titou{slowly monotonically convergent}) in stretched \ce{H2O} and \ce{NH2} systems \cite{Handy_1985}. Lepetit et al.~analyzed the difference between the MP and EN partitioning for the UHF reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the singly- and doubly-excited configurations. 
+%Moreover, the MP and EN numerators in Eqs.~\eqref{eq:EMP2} and \eqref{eq:EEN2} are the same and they vanish when the bond length $r$ goes to infinity. Yet the MP denominators tends towards a constant when $r \to \infty$ so the terms vanish, whereas the EN denominators tends to zero which improves the convergence but can also make the series diverge.
+Cremer and He analyzed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and grouped them in two classes: i) the class A systems where one observes a monotonic convergence to the FCI energy, and ii) the class B for which convergence is erratic after initial oscillations. The sample of systems contains stretched molecules as well as molecules at their equilibrium geometry for various basis sets. They highlighted that \cite{Cremer_1996}
 \begin{quote}
 	\textit{``Class A systems are characterized 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.''}
 \end{quote}
-They proved that using different extrapolation formulas of the first terms of the MP series for class A and class B systems, this improves the precision of those formulas. The mean deviation from FCI correlation energies is $0.3$ millihartree whereas with the formula that do not distinguish the system it is 12 millihartree. Even if there were still shaded areas and that this classification was incomplete, this work showed that understanding the origin of the different modes of convergence would lead to a more rationalized use of the MP perturbation theory and to more accurate correlation energies.
+\titou{They proved that using different extrapolation formulas of the first terms of the MP series for class A and class B systems, this improves the precision of those formulas.} The mean deviation from FCI correlation energies is $0.3$ millihartree whereas with the formula that do not distinguish the system it is 12 millihartree. Even if there were still shaded areas and that this classification was incomplete, this work showed that understanding the origin of the different modes of convergence would lead to a more rationalized use of the MP perturbation theory and to more accurate correlation energies.
 
 \subsection{Cases of divergence}
 
@@ -555,7 +558,7 @@ It seems like our understanding of the physics of spatial and/or spin symmetry b
 Simple systems that are analytically solvable (or at least quasi-exactly solvable) are of great importance in theoretical chemistry. Those systems are very useful benchmarks to test new methods as they are mathematically easy but retain much of the key physics. To investigate the physics of EPs we use one such system named spherium model. It consists of two electrons confined to the surface of a sphere interacting through the long-range Coulomb potential. Thus the Hamiltonian is:
 
 \begin{equation}
-\widehat{H} = -\frac{\grad_1^2 + \grad_2^2}{2} + \frac{1}{r_{12}}
+	\hH = -\frac{\grad_1^2 + \grad_2^2}{2} + \frac{1}{r_{12}}
 \end{equation}
 
 The laplacian operators are the kinetic operators for each electrons and $r_{12}^{-1} = \abs{\vb{r}_1 - \vb{r}_2}$ is the Coulomb operator. The radius R of the sphere dictates the correlation regime, i.e., weak correlation regime at small $R$ where the kinetic energy dominates, or strong correlation regime where the electron repulsion term drives the physics. We will use this model to try to rationalize the effects of the parameters that may influence the physics of EPs: