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@ -308,36 +308,40 @@ This decomposition is obviously non-unique and motivated by several factors as d
Accordingly to Eq.~\eqref{eq:SchrEq-PT}, the energy can then be written as a power series of $\lambda$ Accordingly to Eq.~\eqref{eq:SchrEq-PT}, the energy can then be written as a power series of $\lambda$
\begin{equation} \label{eq:Elambda} \begin{equation} \label{eq:Elambda}
E(\lambda) = \sum_{k=0}^\infty \lambda^k E^{(k)} E(\lambda) = \sum_{k=0}^\infty \lambda^k E^{(k)}.
\end{equation} \end{equation}
However it is not guaranteed that the series \eqref{eq:Elambda} has a radius of convergence $\abs{\lambda_0} < 1$. 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 $\lambda = 1$. 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. 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}: This is due to the following theorem \cite{Goodson_2012}:
\begin{quote} \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 $z_1$.''} \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} \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 $E(\lambda)$. To illustrate this result we consider the simple function \eqref{eq:DivExample}. This function is smooth for $x \in \mathbb{R}$ and infinitely differentiable in $\mathbb{R}$. One would expect that the Taylor series of such a function would be convergent for all $x \in \mathbb{R}$, however this series is divergent for $x\geq1$. This is because the function has four singularities in the complex plane ($x = e^{i\pi/4}, e^{-i\pi/4}, e^{i3\pi/4}, e^{-i3\pi/4}$) with a modulus equal to 1. This simple example emphasizes the importance of the singularities in the complex plane to understand the convergence properties on the real axis. 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 $E(\lambda)$. To illustrate this result we consider the simple function \cite{BenderBook}
\begin{equation} \label{eq:DivExample} \begin{equation} \label{eq:DivExample}
f(x)=\frac{1}{1+x^4} f(x)=\frac{1}{1+x^4}
\end{equation} \end{equation}
This function is smooth for $x \in \mathbb{R}$ and infinitely differentiable in $\mathbb{R}$. One would expect that the Taylor series of such a function would be convergent $\forall x \in \mathbb{R}$. However this series is divergent for $x \ge 1$. This is because the function has four singularities in the complex plane ($x = e^{i\pi/4}$, $e^{-i\pi/4}$, $e^{i3\pi/4}$, and $e^{-i3\pi/4}$) with a modulus equal to $1$. This simple example emphasizes the importance of the singularities in the complex plane to understand the convergence properties on the real axis.
\subsection{The Hartree-Fock Hamiltonian} \subsection{The Hartree-Fock Hamiltonian}
In the Born-Oppenheimer approximation, the equation \eqref{eq:ExactHamiltonian} gives the exact electronic Hamiltonian for a chemical system with $n$ electrons and $N$ nuclei with respective charge $Z_k$. The first term is the kinetic energy of the electrons, the two following terms account respectively for the electron-nuclei attraction and the electron-electron repulsion. Within the Born-Oppenheimer approximation,
\begin{equation}\label{eq:ExactHamiltonian} \begin{equation}\label{eq:ExactHamiltonian}
\bH=\sum\limits_{i=1}^{n}\left[ -\frac{1}{2}\grad_i^2 - \sum\limits_{k=1}^{N} \frac{Z_k}{|\vb{r}_i-\vb{R}_k|}+\sum\limits_{i<j}^{n}\frac{1}{|\vb{r}_i-\vb{r}_j|}\right] \bH = - \sum_{i}^{n} \frac{1}{2}\grad_i^2 - \sum_{i}^{n} \sum_{A}^{N} \frac{Z_A}{|\vb{r}_i-\vb{R}_A|} + \sum_{i<j}^{n}\frac{1}{|\vb{r}_i-\vb{r}_j|}
\end{equation} \end{equation}
is the exact electronic Hamiltonian for a chemical system with $n$ electrons (where $\vb{r}_i$ is the position of the $i$th electron) and $N$ nuclei (where $\vb{R}_A$ is the position of the $A$th nucleus and $Z_A$ its charge). The first term is the kinetic energy of the electrons, the two following terms account respectively for the electron-nucleus attraction and the electron-electron repulsion.
Note that we use atomic units throughout unless otherwise stated.
In the Hartree-Fock (HF) approximation the wave function is approximated as a single Slater determinant (which is an anti-symmetric combination of $n$ one electron spin-orbitals). Rather than solving the equation \eqref{eq:SchrEq}, the Hartree-Fock theory uses the variational principle to find an approximation to $\Psi$. Hence the Slater determinants are not eigenfuctions of the exact Hamiltonian $\bH$. However, they are eigenfunctions of an approximated Hamiltonian $\bH^{\text{HF}}$, called the Hartree-Fock Hamiltonian, which is the sum of the one-electron Fock operators. In the Hartree-Fock (HF) approximation the wave function is approximated as a single Slater determinant (which is an anti-symmetric combination of $n$ one-electron spin-orbitals). Rather than solving Eq.~\eqref{eq:SchrEq}, HF theory uses the variational principle to find an approximation of $\Psi$ as a single Slater determinant. Hence a Slater determinant is not an eigenfunction of the exact Hamiltonian $\bH$. However, it is, by definition, an eigenfunction of the so-called (approximated) HF many-electron Hamiltonian, which is the sum of the one-electron Fock operators:
\begin{equation}\label{eq:HFHamiltonian} \begin{equation}\label{eq:HFHamiltonian}
\bH^{\text{HF}}= \sum\limits_{i=1}^{n} f(i) \bH^{\text{HF}}= \sum\limits_{i=1}^{n} f(\vb{r}_i)
\end{equation} \end{equation}
% Note that the eigenvalue of this Hamiltonian is not the HF energy but the sum of the eigenvalues
The eigenfunctions of $f(i)$ are the one-electron spin-orbitals $\phi_a(i)$ used to create the $n$-electron Slater determinant. The equation \eqref{eq:FockOp} gives the eigenvalue equation for the one-electron Fock operator associated with the electron 1. The one-electron core Hamiltonian $h(i)$ are the sum of the kinetic energy of the electron $i$ and the attraction of the nuclei on this electron. The two other terms are the the Coulomb $J_a(i)$ and Exchange $K_a(i)$ operators. Their action on spin-orbital (occupied or virtual) are given by the equation \eqref{eq:CoulOp} and \eqref{eq:ExcOp}. The integration is over the spatial and spin coordinates. \titou{I STOPPED HERE.}
The eigenfunctions of $f(\vb{r}_i)$ are the one-electron spin-orbitals $\phi_p(i)$ used to create the $n$-electron Slater determinant. Equation \eqref{eq:FockOp} gives the eigenvalue equation for the one-electron Fock operator associated with the electron $i$. The one-electron core Hamiltonian $h(\vb{r}_i)$ are the sum of the kinetic energy of the electron $i$ and the attraction of the nuclei on this electron. The two other terms are the the Coulomb $J_a(\vb{r}_i)$ and Exchange $K_a(\vb{r}_i)$ operators. Their action on spin-orbital (occupied or virtual) are given by Eqs.~\eqref{eq:CoulOp} and \eqref{eq:ExcOp}. The integration is over the spatial and spin coordinates.
\begin{equation}\label{eq:FockOp} \begin{equation}\label{eq:FockOp}
f(1)\phi_i(1) = \left[h(1) + \sum\limits_{j=1}^{n} J_j(1) - K_j(1)\right]\phi_i(1)=\epsilon_i\phi_i(1) f(1)\phi_i(1) = \left[h(1) + \sum\limits_{j=1}^{n} J_j(1) - K_j(1)\right]\phi_i(1)=\epsilon_i\phi_i(1)