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HF/Loos-TCCM_HF.tex
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@ -149,7 +149,7 @@
\end{frame}
%-----------------------------------------------------
\section{Introduction}
\section{The electronic problem}
%-----------------------------------------------------
%-----------------------------------------------------
\subsection{Motivations}
@ -203,6 +203,7 @@
\begin{columns}
\begin{column}{0.4\textwidth}
\begin{block}{In atomic units ($m = e = \hbar = 1$)}
\begin{subequations}
\begin{align}
& \green{\cT_\text{n}} = - \sum_{A=1}^{M} \frac{\nabla_A^2}{2 M_A}
\\
@ -214,11 +215,12 @@
\\
& \violet{\cV_\text{nn}} = \sum_{A<B}^{M} \frac{Z_A Z_B}{R_{AB}}
\end{align}
\end{subequations}
\end{block}
\end{column}
\begin{column}{0.6\textwidth}
\begin{itemize}
\item $\nabla^2$ is the \alert{Laplace operator} (or Laplacian)
\item $\nabla^2$ is the \green{Laplace operator} (or Laplacian)
\bigskip
\item $M_A$ is the \orange{mass} of nucleus $A$
\bigskip
@ -274,7 +276,9 @@
\end{block}
\pause
\begin{block}{Solution:}
Let's try $\Psi = \Psi_A \Psi_B$. Then,
Let's try $\Psi = \Psi_A \Psi_B$ and see if we're lucky.
\\
Then,
\begin{equation*}
\begin{split}
\hH \Psi
@ -296,22 +300,24 @@
%-----------------------------------------------------
\begin{frame}{Spin of the electron}
We are interested by \red{electrons} which are \blue{fermions} $\Rightarrow$ \green{Pauli exclusion principle} (cf next slide)
\begin{block}{Spin functions}
We are interested by \green{electrons} which are \green{fermions} $\Rightarrow$ \green{Pauli exclusion principle} (cf next slide)
\begin{block}{Spin functions: $\ket{\sigma} = \ket{s,m_s} \quad s^2 \ket{s,m_s} = s(s+1) \ket{s,m_s} \quad s_z \ket{s,m_s} = m_s \ket{s,m_s}$}
\centering
$ \ket{\alpha} = $ \violet{spin-up} electron and $\ket{\beta}$ = \violet{spin-down} electron\\
$ \ket{\red{\alpha}} = \ket{\frac{1}{2},\frac{1}{2}}$ \red{spin-up} electron
$\qquad$
$\ket{\blue{\beta}}= \ket{\frac{1}{2},-\frac{1}{2}}$ = \blue{spin-down} electron\\
\begin{align}
\int \alpha^*(\omega) \beta(\omega) d\omega & = \int \beta^*(\omega) \alpha(\omega) d\omega & = 0
\int \red{\alpha}^*(\omega) \blue{\beta}(\omega) d\omega & = \int \blue{\beta}^*(\omega) \red{\alpha}(\omega) d\omega & = 0
& &
\int \alpha^*(\omega) \alpha(\omega) d\omega & = \int \beta^*(\omega) \beta(\omega) d\omega & = 1
\int \red{\alpha}^*(\omega) \red{\alpha}(\omega) d\omega & = \int \blue{\beta}^*(\omega) \blue{\beta}(\omega) d\omega & = 1
\\
\braket{\alpha}{\beta} & = \braket{\beta}{\alpha} & = 0
\braket{\red{\alpha}}{\blue{\beta}} & = \braket{\blue{\beta}}{\red{\alpha}} & = 0
& &
\braket{\alpha}{\alpha} & = \braket{\beta}{\beta} & = 1
\braket{\red{\alpha}}{\red{\alpha}} & = \braket{\blue{\beta}}{\blue{\beta}} & = 1
\end{align}
\end{block}
\bigskip
The composite variable $\bx$ combines \orange{spin ($\omega$)} and \alert{spatial ($\br$)} coordinates: $\boxed{\bx = (\br,\omega)}$
The \violet{composite variable $\bx$} combines \orange{spin ($\omega$)} and \red{spatial ($\br$)} coordinates: $\boxed{\violet{\bx} = (\orange{\omega},\red{\br})}$
\begin{block}{Antisymmetry principle}
\begin{equation}
\cH_\text{elec} \Phi(\bx_1,\bx_2,\ldots,\bx_N) = \cE_\text{elec} \Phi(\bx_1,\bx_2,\ldots,\bx_N)
@ -446,12 +452,12 @@
\begin{itemize}
\item $\{ \phi_{\mu} | i=1,\ldots,K \}$ are basis functions or \alert{atomic orbitals (AOs)}
\item $\{ \chi_i | i=1,\ldots,2K \}$ are the \orange{spin orbitals}
\item $\{ \psi_i | i=1,\ldots,K \}$ are the \violet{spatial orbitals} or \alert{molecular orbitals (MOs)}
\item $\{ \psi_i | i=1,\ldots,K \}$ are the \violet{spatial orbitals} or \violet{molecular orbitals (MOs)}
\bigskip
\item With $K$ AOs, we can create $K$ \violet{spatial orbitals} and $2K$ \orange{spin orbitals}
\item When a system has \orange{$2$ electrons in each orbital}, it is called a \alert{closed-shell} system, otherwise it is called a \alert{open-shell} system
\item The \violet{spatial orbitals} and the \orange{spin orbitals} \alert{form orthogonal bases}
\item \blue{The AOs are, a priori, not orthogonal to each other}
\item With $K$ AOs, one can create $K$ \violet{spatial orbitals} and $2K$ \orange{spin orbitals}
\item For the ground state, the first $N$ \orange{spin orbitals} are \underline{occupied} and the last $2K-N$ are \underline{vacant (unoccupied)}
\item When a system has \blue{$2$ electrons in each orbital}, it is called a \blue{closed-shell} system, otherwise it is called a \blue{open-shell} system
\item For the ground state of a closed shell, the first $N/2$ \violet{spatial orbitals} are \underline{doubly-occupied} and the last $K-N/2$ are \underline{vacant (unoccupied)}
\bigskip
\item The MOs are build by \green{linear combination of AOs (LCAO)}
\item The coefficient $C_{\mu i}$ are determined via the \alert{HF equations} based on \violet{variational principle}
@ -465,55 +471,23 @@
\end{center}
\end{frame}
%-----------------------------------------------------
\subsection{Variational principle}
%-----------------------------------------------------
\begin{frame}{Interlude 1: The variational principle}
\begin{block}{Problem}
\violet{\textit{``Let's suppose we know all the functions such as $\hH \varphi_i = E_i \varphi_i$, with $E_0 < E_1 < \ldots $ and $\braket{ \varphi_i }{ \varphi_j } = \delta_{ij}$.
Show that, for any normalized $\Psi$, we have $ E = \mel{ \Psi }{ \hH }{ \Psi } \ge E_0$''}}
\pause
\\
\bigskip
\end{block}
\begin{block}{Solution}
We expand $\Psi$ in a \alert{clever basis}
\begin{equation*}
\Psi = \sum_{i=1}^\infty c_i \,\varphi_i
\qq{with}
\sum_{i=1}^\infty c_i^2 = 1
\end{equation*}
\begin{equation*}
\begin{split}
E & = \mel{ \Psi }{ \hH }{ \Psi }
= \mel{ \sum_i c_i \varphi_i }{ \hH }{ \sum_j c_j \varphi_j }
= \sum_{ij} c_i c_j \mel{ \varphi_i }{ \hH }{ \varphi_j }
\\
& = \sum_{ij} c_i c_j E_j \braket{ \varphi_i }{ \varphi_j }
= \sum_{ij} c_i c_j E_j \delta_{ij}
= \sum_{i} c_i^2 E_i \ge E_0 \sum_{i} c_i^2 = E_0
\end{split}
\end{equation*}
\end{block}
\end{frame}
%-----------------------------------------------------
\subsection{Excited determinants}
%-----------------------------------------------------
\begin{frame}{Interlude 2: Excited determinants}
\begin{frame}{Excited determinants}
\begin{block}{Reference determinant}
\begin{equation}
\qq*{\green{The electrons are in the $N$ lowest orbitals (Aufbau principle):}} \ket{\Psi_0} = \ket{\chi_1 \ldots \chi_a \chi_b \ldots \chi_N}
\qq*{\green{The electrons are in the $N$ lowest orbitals (Aufbau principle):}} \ket{\Psi_0} = \ket{\chi_1 \ldots \chi_{\green{a}} \chi_{\green{b}} \ldots \chi_N}
\end{equation}
\end{block}
\begin{block}{Singly-excited determinants}
\begin{equation}
\qq*{\orange{Electron in $a$ promoted in $r$:}} \ket{\Psi_a^r} = \ket{\chi_1 \ldots \chi_r \chi_b \ldots \chi_N}
\qq*{Electron in $\green{a}$ promoted in $\orange{r}$:} \ket{\Psi_{\green{a}}^{\orange{r}}} = \ket{\chi_1 \ldots \chi_{\orange{r}} \chi_{\green{b}} \ldots \chi_N}
\end{equation}
\end{block}
\begin{block}{Doubly-excited determinants}
\begin{equation}
\qq*{\red{Electrons in $a$ and $b$ promoted in $r$ and $s$:}} \ket{\Psi_{ab}^{rs}} = \ket{\chi_1 \ldots \chi_r \chi_s \ldots \chi_N}
\qq*{Electrons in $\green{a}$ and $\green{b}$ promoted in $\red{r}$ and $\red{s}$:} \ket{\Psi_{\green{ab}}^{\red{rs}}} = \ket{\chi_1 \ldots \chi_{\red{r}} \chi_{\red{s}} \ldots \chi_N}
\end{equation}
\end{block}
\begin{center}
@ -525,6 +499,7 @@
\end{center}
\end{frame}
%-----------------------------------------------------
\section{HF approximation}
%-----------------------------------------------------
@ -726,7 +701,7 @@
%-----------------------------------------------------
\begin{frame}{The Hartree-Fock energy: examples}
\begin{block}{Problem: Normalization of the HF wave function}
\textit{\violet{``Show that the HF wave function built with two spin orbitals $\chi_1$ and $\chi_2$ is normalized''}}
\textit{\violet{``Show that the HF wave function built with two (normalized) spin orbitals $\chi_1$ and $\chi_2$ is normalized''}}
\end{block}
\pause
\begin{block}{Solution}
@ -1100,6 +1075,37 @@ These orbitals are called \orange{canonical molecular orbitals} (= eigenvectors
\end{block}
\end{frame}
%-----------------------------------------------------
\subsection{Variational principle}
%-----------------------------------------------------
\begin{frame}{The variational principle}
\begin{block}{Problem}
\violet{\textit{``Let's suppose we know all the functions such as $\hH \varphi_i = E_i \varphi_i$, with $E_0 < E_1 < \ldots $ and $\braket{ \varphi_i }{ \varphi_j } = \delta_{ij}$.
Show that, for any normalized $\Psi$, we have $ E = \mel{ \Psi }{ \hH }{ \Psi } \ge E_0$''}}
\pause
\end{block}
\begin{block}{Solution}
We expand $\Psi$ in a \alert{clever basis}
\begin{equation*}
\Psi = \sum_{i=1}^\infty c_i \,\varphi_i
\qq{with}
\sum_{i=1}^\infty c_i^2 = 1
\end{equation*}
\pause
\begin{equation*}
\begin{split}
E & = \mel{ \Psi }{ \hH }{ \Psi }
= \mel{ \sum_i c_i \varphi_i }{ \hH }{ \sum_j c_j \varphi_j }
= \sum_{ij} c_i c_j \mel{ \varphi_i }{ \hH }{ \varphi_j }
\\
& = \sum_{ij} c_i c_j E_j \braket{ \varphi_i }{ \varphi_j }
= \sum_{ij} c_i c_j E_j \delta_{ij}
= \sum_{i} c_i^2 E_i \ge E_0 \sum_{i} c_i^2 = E_0
\end{split}
\end{equation*}
\end{block}
\end{frame}
%-----------------------------------------------------
\subsection{Koopmans}
%-----------------------------------------------------