update

RapportStage/Rapport.tex

@@ -224,7 +224,7 @@ which correspond to square-root singularities in the complex-$\lambda$ plane (se
 These two $\lambda$ values, given by Eq.~\eqref{eq:lambda_EP}, are the so-called EPs and one can clearly see that they connect the ground and excited states.
 Starting from $\lambda_\text{EP}$, two square-root branch cuts run on the imaginary axis towards $\pm i \infty$.
 In the real $\lambda$ axis, the point for which the states are the closest ($\lambda = 0$) is called an avoided crossing and this occurs at $\lambda = \Re(\lambda_\text{EP})$.
-The ``shape'' of the avoided crossing in linked to the magnitude of $\Im(\lambda_\text{EP})$: the smaller $\Im(\lambda_\text{EP})$, the sharper the avoided crossing is.
+The ``shape'' of the avoided crossing is linked to the magnitude of $\Im(\lambda_\text{EP})$: the smaller $\Im(\lambda_\text{EP})$, the sharper the avoided crossing is.
 
 Around $\lambda = \lambda_\text{EP}$, Eq.~\eqref{eq:E_2x2} behaves as \cite{MoiseyevBook}
 \begin{equation}
@@ -241,6 +241,36 @@ and we have
 	E_{\pm}(4\pi) & = E_{\pm}(0).
 \end{align}
 This evidences that encircling non-Hermitian degeneracies at EPs leads to an interconversion of states, and two loops around the EP are necessary to recover the initial energy.
+
+The eigenvectors associated to the energies \eqref{eq:E_2x2} are
+\begin{equation}
+\phi_{\pm}=\begin{pmatrix}
+\frac{1}{2\lambda}(\epsilon_1 - \epsilon_2 \pm \sqrt{(\epsilon_1 - \epsilon_2)^2 + 4\lambda^2}) \\ 1
+\end{pmatrix},
+\end{equation}
+and for $\lambda=\lambda_\text{EP}$ they become
+\begin{equation}
+\phi_{\pm}=\begin{pmatrix}
+\mp i \\ 1
+\end{pmatrix},
+\end{equation}
+which are clearly self-orthogonal. The equation (7) can be rewrite as
+\begin{equation}
+\phi_{\pm}=\begin{pmatrix}
+\frac{1}{\lambda}(\frac{\epsilon_1 - \epsilon_2}{2} \pm \sqrt{2\lambda_\text{EP}} \sqrt{\lambda - \lambda_\text{EP}}) \\
+\end{pmatrix},
+\end{equation}
+we can see that if we normalise them they will behave as
+$(\lambda - \lambda_\text{EP})^{-1/4}$ resulting in the following pattern when looping around one EP:
+\begin{align}
+	\phi_{\pm}(2\pi) & = \phi_{\mp}(0),
+	&
+	\phi_{\pm}(4\pi) & = -\phi_{\pm}(0) \\
+	\phi_{\pm}(6\pi) & = -\phi_{\mp}(0),
+	&
+	\phi_{\pm}(8\pi) & = \phi_{\pm}(0),
+\end{align}
+showing that 4 loops around the EP are necessary to recover the initial state. We can also that looping around the other way round leads to a different pattern.
 \titou{Maybe you should add a few equations here to highlight the self-orthogonality process. 
 What do you think?
 You could also show that the behaviour of the wave function when one follows the complex contour around the EP.}
@@ -327,9 +357,29 @@ But as mentioned before \textit{a priori} there are no reasons that $E_{\text{MP
 
 \subsection{Alternative partitioning}\label{sec:AlterPart}
 
-The M{\o}ller-Plesset partitioning is not the only one possible in electronic structure theory. An other possibility, even more natural than the M{\o}ller-Plesset one, is to take the diagonal elements of $\bH$ as the zeroth-order Hamiltonian. Hence, the off-diagonal elements of $\bH(\lambda)$ are the perturbation operator. 
+The M{\o}ller-Plesset partitioning is not the only one possible in electronic structure theory. An other possibility, even more natural than the M{\o}ller-Plesset one, is to take the diagonal elements of $\bH$ as the zeroth-order Hamiltonian. Hence, the off-diagonal elements of $\bH(\lambda)$ are the perturbation operator. This partitioning leads to the Epstein-Nesbet (EN) perturbation theory. The zeroth-order eigenstates for this partitioning are Slater determinants as for the M{\o}ller-Plesset partitioning. The expression of the second order correction to the energy is given for both M{\o}ller-Plesset and Epstein-Nesbet. The energies at the MP denominator are the orbitals energies whereas in the EN case it is the excitation energies. The i,j indices represent the occupied orbitals and r,s the virtual orbitals of the basis sets.
 
-The electronic Hamiltonian can be separated in the kinetic part and the potential part. We can use this to consider to other partitioning
+\begin{equation}\label{eq:EMP2}
+E_{\text{MP2}}=\sum\limits_{\substack{i<j \\ r<s}}^{n}\frac{\abs{\bra{ij}\hspace{1pt}\ket{rs}}^2}{\epsilon_i + \epsilon_j - \epsilon_r - \epsilon_s}
+\end{equation}
+\begin{equation}\label{eq:EEN2}
+E_{\text{EN2}}=\sum\limits_{\substack{i<j \\ r<s}}^{n}\frac{\abs{\bra{ij}\hspace{1pt}\ket{rs}}^2}{\epsilon_i + \epsilon_j - \epsilon_r - \epsilon_s - (J_{ij} + J_{rs} - J_{ir} - J_{is} - J_{jr} + J_{js})}
+\end{equation}
+where $J_{ij}$ is the matrix element of the Coulomb operator \eqref{eq:CoulOp} and with
+\begin{equation}
+\bra{ij}\hspace{1pt}\ket{rs}=\bra{ij}\ket{rs} - \bra{ij}\ket{sr}
+\end{equation}
+where $\bra{ij}\ket{rs}$ is the two-electron integral
+\begin{equation}
+\bra{ij}\ket{rs}=\int \dd\vb{x}_1\dd\vb{x}_2\chi_i^*(\vb{x}_1)\chi_j^*(\vb{x}_2)r_{12}^{-1}\chi_r(\vb{x}_1)\chi_s(\vb{x}_2)
+\end{equation}
+
+Additionally, we will consider two other partitioning. The electronic Hamiltonian can be separated in a one-electron part and in a two-electron part as seen previously. We can use this separation to create two other partitioning:
+
+\begin{itemize}
+\item The Weak Correlation partitioning in which the one-electron part is consider as the unperturbed Hamiltonian $\bH^{(0)}$ and the two-electron part is the perturbation operator $\bV$.
+\item The Strong Coupling partitioning where the two operators are inverted compared to the weak correlation partitioning.
+\end{itemize}
 
 %============================================================%
 \section{Historical overview}
@@ -368,8 +418,7 @@ When a bond is stretched the exact wave function becomes more and more multi-ref
 \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 et al.~highlighted the link between the slow convergence of the unrestricted MP series and the spin contamination of the wave function as shown in the \autoref{tab:SpinContamination} in the example of \ce{H_2} in a minimal basis \cite{Gill_1988}. 
-Handy and co-workers exhibited the same behaviors 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 M{\o}ller-Plesset and Epstein-Nesbet partitioning for the unrestricted Hartree-Fock reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the singly with the doubly excited configuration. Moreover the MP denominators tends towards a constant so each contribution become very small when the bond is stretched. \antoine{Rephrase this sentence}
+Handy and co-workers exhibited the same behaviors 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 M{\o}ller-Plesset and Epstein-Nesbet partitioning for the unrestricted Hartree-Fock reference \cite{Lepetit_1988}. They concluded that the slow convergence is due to the coupling of the singly with the doubly excited configuration. Moreover the MP and EN numerators in Eq. \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 $r\rightarrow\infty$ so the terms vanish. Where as the EN denominators tends to 0 which improve the convergence but can also make diverge the series.
-
 Cremer and He analyzed 29 atomic and molecular systems at the FCI level \cite{Cremer_1996} and regrouped all the systems in two classes. The class A systems which have a monotonic convergence to the FCI value and the class B which converge erratically after initial oscillations. The sample of systems contains stretched molecules and also some at equilibrium geometry, there are also some systems in 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}
@@ -503,7 +552,7 @@ In addition, there is also the well-known symmetry-broken UHF solution. For $R>3
 E_{\text{sb-UHF}}=-\frac{75}{112R^3}+\frac{25}{28R^2}+\frac{59}{84R}
 \end{equation}
 
-The exact solution for the ground state is a singlet so this wave function does not have the true symmetry\antoine{add an equation that shows this}.  However this solution gives more accurate results for the energy at large R as shown in \autoref{tab:ERHFvsEUHF}. In fact at the Coulson-Fischer point, it becomes more efficient to minimize the Coulomb repulsion than the kinetic energy in order to minimize the total energy. Thus the wave function break the spin symmetry because it allows a more efficient minimization of the Coulomb repulsion. This type of symmetry breaking is called a spin density wave because the system oscillates between the two symmetry-broken configurations \cite{GiulianiBook}.
+The exact solution for the ground state is a singlet so this wave function does not have the true symmetry. Indeed, the spherical harmonics are eigenvectors of $S^2$ but the symmetry-broken solution is a linear combination of the two eigenvectors and is not an eigenvector of $S^2$. However this solution gives more accurate results for the energy at large R as shown in \autoref{tab:ERHFvsEUHF}. In fact at the Coulson-Fischer point, it becomes more efficient to minimize the Coulomb repulsion than the kinetic energy in order to minimize the total energy. Thus the wave function break the spin symmetry because it allows a more efficient minimization of the Coulomb repulsion. This type of symmetry breaking is called a spin density wave because the system oscillates between the two symmetry-broken configurations \cite{GiulianiBook}.
 
 \begin{table}[h!]
 \centering