minor corrections

RapportStage/Rapport.tex

@@ -173,20 +173,30 @@ More importantly here, although EPs usually lie off the real axis, these singula
 \section{Perturbation theory}
 %============================================================%
 
+Within (time-independent) Rayleigh-Schr\"odinger perturbation theory, the Schr\"odinger equation 
+\begin{equation} \label{eq:SchrEq}
+	\bH \Psi = E \Psi
+\end{equation} 
+is recast as 
+\begin{equation} \label{eq:SchrEq-PT}
+	\bH(\lambda) \Psi(\lambda) = (\bH^{(0)} + \lambda \bV ) \Psi(\lambda) = E(\lambda) \Psi(\lambda),
+\end{equation}
+where $\bH^{(0)}$ is the zeroth-order Hamiltonian and $\bV = \bH - \bH^{(0)}$ is the so-called perturbation.
+The ``physical'' system of interest is recovered by setting the coupling parameter $\lambda$ to unity.
+This decomposition is obviously non-unique and motivated by several factors as discussed below.
 
-Within perturbation theory, the Schr\"odinger equation is usually rewritten as 
-\begin{equation}
-	\bH(\lambda) \Psi = (\bH^{(0)} + \lambda \bV ) \Psi(\lambda) = E(\lambda) \Psi,
-\end{equation}
-with
-\begin{equation}
-\bV=\bH - \bH^{(0)}
-\end{equation}
-The energy can then be written as a power series of $\lambda$
-\begin{equation}
+Accordingly to Eq.~\eqref{eq:SchrEq-PT}, the energy can then be written as a power series of $\lambda$
+\begin{equation} \label{eq:Elambda}
 	E(\lambda) = \sum_{k=0}^\infty \lambda^k E^{(k)}
 \end{equation}
-where $\lambda$ is a coupling parameter set equal to 1 at the end of the calculation. However it is not guaranteed that the series $E(\lambda)$ has a radius of convergence $\abs{\lambda_0} < 1$. It means that the series is divergent for the physical system at $\lambda=1$. One can prove that $\abs{\lambda_0}$ can be obtained by extending $\lambda$ in the complex plane and looking for the singularities of $E(\lambda)$. This is due to the following theorem \cite{Goodson_2012}: The Taylor series about a point $z_0$ of a function over the complex $z$ plane will converge at a value $z_1$ if the function is non-singular at all values of $z$ in the circular region centered at $z_0$ with radius $\abs{z_1 − z_0}$. If the function has a singular point $z_s$ such that $\abs{z_s − z_0} < \abs{z_1 − z_0}$, then the series will diverge when evaluated at $z_1$. This theorem means that the radius of convergence of the perturbation series is equal to distance to the origin of the closest singularity of $E(\lambda)$.  
+However it is not guaranteed that the series \eqref{eq:Elambda} has a radius of convergence $\abs{\lambda_0} < 1$. 
+In other words, the series might well be divergent for the physical system at $\lambda = 1$. 
+One can prove that the actual value of the radius of convergence $\abs{\lambda_0}$ can be obtained by looking for the singularities of $E(\lambda)$ in the complex $\lambda$ plane.
+This is due to the following theorem \cite{Goodson_2012}: 
+\begin{quote}
+	\textit{``The Taylor series about a point $z_0$ of a function over the complex $z$ plane will converge at a value $z_1$ if the function is non-singular at all values of $z$ in the circular region centered at $z_0$ with radius $\abs{z_1 − z_0}$. If the function has a singular point $z_s$ such that $\abs{z_s − z_0} < \abs{z_1 − z_0}$, then the series will diverge when evaluated at $z_1$.''}
+\end{quote}
+This theorem means that the radius of convergence of the perturbation series is equal to the distance to the origin of the closest singularity of $E(\lambda)$.  
 
 The discovery of a partitioning of the Hamiltonian that allowed chemists to recover a part of the correlation energy (i.e. the difference between the exact energy and the Hartree-Fock energy) using perturbation theory has been a major step in the development of post-Hartree-Fock methods. This case of the Rayleigh-Schrödinger perturbation theory is called the M{\o}ller-Plesset perturbation theory \cite{Moller_1934}. In the MPPT the unperturbed Hamiltonian is the sum of the $n$ mono-electronic Fock operators which are the sum of the one-electron core Hamiltonian $h(i)$, the Coulomb $J_j(i)$ and Exchange $K_j(i)$ operators.
  
@@ -198,13 +208,13 @@ H_0= \sum\limits_{i=1}^{n} f(i)
 f(i) = h(i) + \sum\limits_{j=1,j \neq i}^{n} \left[J_j(i) - K_j(i)\right]
 \end{equation}
 
-In Hartree-Fock theory the exact wave function is approximated as a Slater-determinant (which is an anti-symmetric combination of mono-electronic orbitals) and those wave functions are eigenvectors of the Fock operators. In the perturbation theory the energy is a power series of $\lambda$ and the physical energy is obtained by taking $\lambda$ equal to 1. We will refer to the energy up to the n-th order as the MPn energy. The MP0 energy overestimates the energy by double counting the electron-electron interaction, the MP1 corrects this effect and the MP1 energy is equal to the Hartree-Fock energy. The MP2 energy starts to recover a part of the correlation energy.
+In Hartree-Fock theory the exact wave function is approximated as a Slater-determinant (which is an anti-symmetric combination of mono-electronic orbitals) and those wave functions are eigenvectors of the Fock operators. In the perturbation theory the energy is a power series of $\lambda$ and the physical energy is obtained by taking $\lambda$ equal to 1. We will refer to the energy up to the $n$-th order as the MP$n$ energy. The MP0 energy overestimates the energy by double counting the electron-electron interaction, the MP1 corrects this effect and the MP1 energy is equal to the Hartree-Fock energy. The MP2 energy starts to recover a part of the correlation energy.
 
 \begin{equation}
-E_{\text{MP\textsubscript{n}}}= \sum_{k=0}^n E^{(k)}
+E_{\text{MP$_{n}$}}= \sum_{k=0}^n E^{(k)}
 \end{equation}
 
-But as mentioned before \textit{a priori} there are no reasons that this power series is always convergent for $\lambda$=1 when n goes to infinity. In fact, it is known that when the Hartree-Fock wave function is a bad approximation of the exact wave function, for example for multi-reference states, the M{\o}ller-Plesset will give bad results\cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. A smart way to investigate the convergence properties of the MP series is to transform the coupling parameter $\lambda$ into a complex variable. By doing so the Hamiltonian and the energy become functions of this variable. The energy becomes a multivalued function on $n$ Riemann sheets. As mentioned above by searching the singularities of the function $E(\lambda)$ we can get information on the convergence properties of the MPPT. Those singularities of the energy are exactly the exceptional points connecting the electronic states mentioned in the introduction. The direct computation of the terms of the series is quite easy up to the 4th order and the 5th and 6th order can be obtained at high cost. But to deeply understand the behavior of the MP series and how it is connected to the singularities, we need to have access to high order terms of the series. For small systems we can have access to the whole series using Full Configuration Interaction. If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI basis set we get the exact energies (in this finite basis set) and expanding in $\lambda$ allows to to get the M{\o}ller-Plesset perturbation series at every order.
+But as mentioned before \textit{a priori} there are no reasons that this power series is always convergent for $\lambda$=1 when $n$ goes to infinity. In fact, it is known that when the Hartree-Fock wave function is a bad approximation of the exact wave function, for example for multi-reference states, the M{\o}ller-Plesset will give bad results\cite{Gill_1986, Gill_1988, Handy_1985, Lepetit_1988}. A smart way to investigate the convergence properties of the MP series is to transform the coupling parameter $\lambda$ into a complex variable. By doing so the Hamiltonian and the energy become functions of this variable. The energy becomes a multivalued function on $n$ Riemann sheets. As mentioned above by searching the singularities of the function $E(\lambda)$ we can get information on the convergence properties of the MPPT. Those singularities of the energy are exactly the exceptional points connecting the electronic states mentioned in the introduction. The direct computation of the terms of the series is quite easy up to the 4th order and the 5th and 6th order can be obtained at high cost. But to deeply understand the behavior of the MP series and how it is connected to the singularities, we need to have access to high order terms of the series. For small systems we can have access to the whole series using Full Configuration Interaction. If the Hamiltonian $H(\lambda)$ is diagonalized in the FCI basis set we get the exact energies (in this finite basis set) and expanding in $\lambda$ allows to to get the M{\o}ller-Plesset perturbation series at every order.
 
 %============================================================%
 \section{Historical overview}
@@ -317,7 +327,7 @@ Singularity $\alpha$ and quantum phase transition ?
 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}{\vb{r}_{12}}
+\widehat{H} = -\frac{\grad_1^2 + \grad_2^2}{2} + \frac{1}{\vb{r}_{12}}
 \end{equation}
 
 The laplacian operators are the kinetic operators for each electrons and $\vb{r}_{12}^{-1}$ 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 variables that may influence the physics of EPs: