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corrected errors from Wednesday lecture

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Pierre-Francois Loos 2021-09-01 18:45:10 +02:00
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HF/Loos-TCCM_HF.tex
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@ -330,23 +330,23 @@
\begin{frame}{Antisymmetry} \begin{frame}{Antisymmetry}
\begin{block}{Problem:} \begin{block}{Problem:}
\violet{\textit{``Show that, for a system of two fermions, the wave function vanishes when they are at the same point in space''}} \violet{\textit{``Show that, for a system of two fermions, the wave function vanishes when they are at the same point in spin-space''}}
\end{block} \end{block}
\pause \pause
\begin{block}{Solution} \begin{block}{Solution}
\blue{Indistinguishable} particles means \blue{Indistinguishable} particles means
\begin{equation} \begin{equation}
\boxed{\abs{\Psi(x_1,x_2)}^2 = \abs{\Psi(x_2,x_1)}^2 \Rightarrow \Psi(x_1,x_2) = \alert{\pm} \Psi(x_2,x_1)} \boxed{\abs{\Psi(\bx_1,\bx_2)}^2 = \abs{\Psi(\bx_2,\bx_1)}^2 \Rightarrow \Psi(\bx_1,\bx_2) = \alert{\pm} \Psi(\bx_2,\bx_1)}
\end{equation} \end{equation}
\\ \\
\bigskip \bigskip
\pause \pause
\violet{Bosons} mean \violet{$\Psi(x_1,x_2) = \Psi(x_2,x_1)$} and \orange{Fermions} mean \orange{$\Psi(x_1,x_2) = -\Psi(x_2,x_1)$} \violet{Bosons} mean \violet{$\Psi(\bx_1,\bx_2) = \Psi(\bx_2,\bx_1)$} and \orange{Fermions} mean \orange{$\Psi(\bx_1,\bx_2) = -\Psi(\bx_2,\bx_1)$}
\\ \\
\bigskip \bigskip
Let's put them at the same spot, i.e. \blue{$x = x_1 = x_2$} Let's put them at the same spot, i.e. \blue{$\bx = \bx_1 = \bx_2$}
\begin{equation} \begin{equation}
\text{\violet{For Fermions, }} \Psi(x,x) = - \Psi(x,x) \quad \Rightarrow \quad \boxed{\Psi(x,x) = 0} \text{\violet{For Fermions, }} \Psi(\bx,\bx) = - \Psi(\bx,\bx) \qq{$\Rightarrow$} \boxed{\Psi(\bx,\bx) = 0}
\end{equation} \end{equation}
\alert{The wave function vanishes! $\quad \Rightarrow \quad$ This is called the Fermi hole!} \alert{The wave function vanishes! $\quad \Rightarrow \quad$ This is called the Fermi hole!}
\end{block} \end{block}
@ -354,29 +354,29 @@
\begin{frame}{Antisymmetry (Take 2)} \begin{frame}{Antisymmetry (Take 2)}
\begin{block}{Problem:} \begin{block}{Problem:}
\violet{\textit{``Given two one-electron functions $\chi_1(x)$ and $\chi_2(x)$, could you construct a two-electron (fermionic) wave function $\Psi(x_1,x_2)$?''}} \violet{\textit{``Given two one-electron functions $\chi_1(\bx)$ and $\chi_2(\bx)$, could you construct a two-electron (fermionic) wave function $\Psi(\bx_1,\bx_2)$?''}}
\end{block} \end{block}
\pause \pause
\begin{block}{Solution} \begin{block}{Solution}
A possible solution is A possible solution is
\begin{equation} \begin{equation}
\Psi(x_1,x_2) = \chi_1(x_1) \chi_2(x_2) - \chi_1(x_2) \chi_2(x_1) \Psi(\bx_1,\bx_2) = \chi_1(\bx_1) \chi_2(\bx_2) - \chi_1(\bx_2) \chi_2(\bx_1)
\end{equation} \end{equation}
This has been popularized by \blue{Slater}: This has been popularized by \blue{Slater}:
\begin{equation} \begin{equation}
\boxed{ \boxed{
\Psi(x_1,x_2) = \Psi(\bx_1,\bx_2) =
\begin{vmatrix} \begin{vmatrix}
\chi_1(x_1) & \chi_2(x_1) \\ \chi_1(\bx_1) & \chi_2(\bx_1) \\
\chi_1(x_2) & \chi_2(x_2) \\ \chi_1(\bx_2) & \chi_2(\bx_2) \\
\end{vmatrix} \end{vmatrix}
= \chi_1(x_1) \chi_2(x_2) - \chi_1(x_2) \chi_2(x_1) = \chi_1(\bx_1) \chi_2(\bx_2) - \chi_1(\bx_2) \chi_2(\bx_1)
} }
\end{equation} \end{equation}
\alert{This is called a Slater determinant!} \alert{This is called a Slater determinant!}
\\ \\
\bigskip \bigskip
A wave function of the form \green{$\Psi(x_1,x_2) = \chi_1(x_1) \chi_2(x_2)$} is called a \green{Hartree product} A wave function of the form \green{$\Psi(\bx_1,\bx_2) = \chi_1(\bx_1) \chi_2(\bx_2)$} is called a \green{Hartree product}
\end{block} \end{block}
\end{frame} \end{frame}
@ -562,34 +562,34 @@
\begin{itemize} \begin{itemize}
\item \orange{Coulomb operator} \item \orange{Coulomb operator}
\begin{equation} \begin{equation}
\cJ_{\green{i}}(\blue{1}) \ket{ \chi_{\purple{j}}(\red{2}) } \cJ_{\green{j}}(\blue{1}) \ket{ \chi_{\purple{i}}(\blue{1}) }
= \mel{ \chi_{\green{i}}(\blue{1}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\green{i}}(\blue{1}) } \ket{ \chi_{\purple{j}}(\red{2}) } = \mel{ \chi_{\green{j}}(\red{2}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\green{j}}(\red{2}) } \ket{ \chi_{\purple{i}}(\blue{1}) }
= \qty[ \int d\blue{\bx_1} \chi_{\green{i}}^*(\blue{\bx_1}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{i}}(\blue{\bx_1}) ] \ket{ \chi_{\purple{j}}(\red{\bx_2}) } = \qty[ \int d\red{\bx_2} \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{j}}(\red{\bx_2}) ] \ket{ \chi_{\purple{i}}(\blue{\bx_1}) }
\end{equation} \end{equation}
\item \orange{Coulomb matrix elements} \item \orange{Coulomb matrix elements}
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\cJ_{\green{i}\purple{j}} \cJ_{\purple{i}\green{j}}
& = \mel{\chi_{\purple{j}}(\red{2})}{\cJ_{\green{i}}(\blue{1})}{\chi_{\purple{j}}(\red{2})} & = \mel{\chi_{\purple{i}}(\blue{1})}{\cJ_{\green{j}}(\blue{1})}{\chi_{\purple{i}}(\blue{1})}
= \mel{ \chi_{\green{i}}(\blue{1}) \chi_{\purple{j}}(\red{2}) }{r_{\blue{1}\red{2}}^{-1}}{ \chi_{\green{i}}(\blue{1}) \chi_{\purple{j}}(\red{2})} = \mel{ \chi_{\purple{i}}(\blue{1}) \chi_{\green{j}}(\red{2}) }{r_{\blue{1}\red{2}}^{-1}}{ \chi_{\purple{i}}(\blue{1}) \chi_{\green{j}}(\red{2})}
\\ \\
& = \iint \chi_{\green{i}}^*(\blue{\bx_1}) \chi_{\purple{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{i}}(\blue{\bx_1}) \chi_{\green{i}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2} & = \iint \chi_{\purple{i}}^*(\blue{\bx_1}) \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{i}}(\blue{\bx_1}) \chi_{\green{j}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2}
\end{split} \end{split}
\end{equation} \end{equation}
\item \violet{(non-local) Exchange operator} \item \violet{(non-local) Exchange operator}
\begin{equation} \begin{equation}
\cK_{\green{i}}(\blue{1}) \ket{ \chi_{\purple{j}}(\red{2}) } \cK_{\green{j}}(\blue{1}) \ket{ \chi_{\purple{i}}(\blue{1}) }
= \mel{ \chi_{\green{i}}(\blue{1}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\purple{j}}(\blue{1}) } \ket{ \chi_{\green{i}}(\red{2}) } = \mel{ \chi_{\green{j}}(\red{2}) }{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\purple{i}}(\red{2}) } \ket{ \chi_{\green{j}}(\blue{1}) }
= \qty[ \int \chi_{\green{i}}^*(\blue{\bx_1}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{j}}(\blue{\bx_1}) d\blue{\bx_1} ] \ket{ \chi_{\green{i}}(\red{\bx_2}) } = \qty[ \int d\red{\bx_2} \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{i}}(\red{\bx_2}) ] \ket{ \chi_{\green{j}}(\blue{\bx_2}) }
\end{equation} \end{equation}
\item \violet{Exchange matrix elements} \item \violet{Exchange matrix elements}
\begin{equation} \begin{equation}
\begin{split} \begin{split}
\cK_{\green{i}\purple{j}} \cK_{\green{i}\purple{j}}
& = \mel{\chi_{\purple{j}}(\red{2})}{\cK_{\green{i}}(\blue{1})}{\chi_{\purple{j}}(\red{2})} & = \mel{\chi_{\purple{i}}(\blue{1})}{\cK_{\green{j}}(\blue{1})}{\chi_{\purple{i}}(\blue{1})}
= \mel{ \chi_{\green{i}}(\blue{1}) \chi_{\purple{j}}(\red{2})}{ r_{\blue{1}\red{2}}^{-1} }{ \chi_{\purple{j}}(\blue{1}) \chi_{\green{i}}(\red{2}) } = \mel{ \chi_{\purple{i}}(\blue{1}) \chi_{\green{j}}(\red{2}) }{r_{\blue{1}\red{2}}^{-1}}{ \chi_{\green{j}}(\blue{1}) \chi_{\purple{i}}(\red{2})}
\\ \\
& = \iint \chi_{\green{i}}^*(\blue{\bx_1}) \chi_{\purple{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\purple{j}}(\blue{\bx_1}) \chi_{\green{i}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2} & = \iint \chi_{\purple{i}}^*(\blue{\bx_1}) \chi_{\green{j}}^*(\red{\bx_2}) r_{\blue{1}\red{2}}^{-1} \chi_{\green{j}}(\blue{\bx_1}) \chi_{\purple{i}}(\red{\bx_2}) d\blue{\bx_1} d\red{\bx_2}
\end{split} \end{split}
\end{equation} \end{equation}
\end{itemize} \end{itemize}
@ -634,19 +634,27 @@
\end{frame} \end{frame}
\begin{frame}{Permutation symmetry} \begin{frame}{Permutation symmetry}
\begin{block}{Permutation symmetry in physicist notation} \begin{block}{Permutation symmetry in physicts' notations}
\begin{equation} \begin{equation}
\braket{ij}{kl} \braket{ij}{kl}
= \braket{\chi_i \chi_j}{\chi_k \chi_l} = \braket{\chi_i \chi_j}{\chi_k \chi_l}
= \iint \chi_i^*(\bx_1) \chi_j^*(\bx_2) \frac{1}{r_{12}} \chi_k(\bx_1) \chi_l(\bx_2) d\bx_1 d\bx_2 = \iint \chi_i^*(\bx_1) \chi_j^*(\bx_2) \frac{1}{r_{12}} \chi_k(\bx_1) \chi_l(\bx_2) d\bx_1 d\bx_2
\end{equation} \end{equation}
\begin{align} \begin{equation}
\qq*{\red{Complex-valued integrals:}} & \qq*{\red{Complex-valued integrals:}}
\braket{ij}{kl} = \braket{ji}{lk} = \braket{kl}{ij}^* \braket{ij}{kl} = \braket{ji}{lk} = \braket{kl}{ij}^* = \braket{lk}{ji}^*
\\ \end{equation}
\qq*{\green{Real-valued integrals:}} & \end{block}
\braket{ij}{kl} = \braket{ji}{kl} = \braket{ij}{lk} = \braket{ji}{lk} = \braket{kl}{ij} = \braket{lk}{ij} = \braket{kl}{ji} = \braket{lk}{ji} \begin{block}{Permutation symmetry in chemists' notations}
\end{align} \begin{equation}
[ij|kl]
= [\chi_i \chi_j|\chi_k \chi_l]
= \iint \chi_i^*(\bx_1) \chi_j(\bx_1) \frac{1}{r_{12}} \chi_k^*(\bx_2) \chi_l(\bx_2) d\bx_1 d\bx_2
\end{equation}
\begin{equation}
\qq*{\green{Real-valued integrals:}}
[ij|kl] = [ji|kl] = [ij|lk] = [ji|lk] = [kl|ij] = [lk|ij] = [kl|ji] = [lk|ji]
\end{equation}
\end{block} \end{block}
\end{frame} \end{frame}
@ -664,7 +672,7 @@
\end{block} \end{block}
\begin{block}{\orange{Case 2 = differ by one spinorbital}: $\ket{K} = \ket{\ldots m n \ldots}$ and $\ket{L} = \ket{\ldots p n \ldots}$} \begin{block}{\orange{Case 2 = differ by one spinorbital}: $\ket{K} = \ket{\ldots m n \ldots}$ and $\ket{L} = \ket{\ldots p n \ldots}$}
\begin{equation} \begin{equation}
\mel{K}{\cO_1}{L} = \sum_m \mel{m}{h}{p} \mel{K}{\cO_1}{L} = \mel{m}{h}{p}
\end{equation} \end{equation}
\end{block} \end{block}
\begin{block}{\red{Case 3 = differ by two spinorbitals}: $\ket{K} = \ket{\ldots m n \ldots}$ and $\ket{L} = \ket{\ldots p q \ldots}$} \begin{block}{\red{Case 3 = differ by two spinorbitals}: $\ket{K} = \ket{\ldots m n \ldots}$ and $\ket{L} = \ket{\ldots p q \ldots}$}