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HF/Loos-TCCM_HF.tex
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@ -162,11 +162,10 @@
\bigskip
\item We don't care about \violet{relativistic effects}
\bigskip
\item HF is an \blue{independent-particle model}
\item HF is an \blue{independent-particle model}, i.e.,
the motion of one electron \violet{is considered to be independent of the dynamics of all other electrons}
\orange{$\Rightarrow$ interactions are taken into account in an average fashion}
\bigskip
\item the motion of one electron \violet{is considered to be independent of the dynamics of all other electrons}
\bigskip
\item \purple{HF is the starting point of pretty much anything!}
\end{itemize}
\end{frame}
@ -202,7 +201,7 @@
\begin{frame}{The Hamiltonian (Take 2)}
\begin{columns}
\begin{column}{0.5\textwidth}
\begin{column}{0.4\textwidth}
\begin{block}{In atomic units ($m = e = \hbar = 1$)}
\begin{align}
& \green{\cT_\text{n}} = - \sum_{A=1}^{M} \frac{\nabla_A^2}{2 M_A}
@ -217,7 +216,7 @@
\end{align}
\end{block}
\end{column}
\begin{column}{0.5\textwidth}
\begin{column}{0.6\textwidth}
\begin{itemize}
\item $\nabla^2$ is the \alert{Laplace operator} (or Laplacian)
\bigskip
@ -242,12 +241,12 @@
\end{frame}
\begin{frame}{The Born-Oppenheimer approximation}
\begin{block}{Electronic Hamiltonian}
The \alert{electronic Hamiltonian} is
\begin{block}{Born-Oppenheimer approximation = decoupling nuclei and electrons}
Because $M_A \gg 1$, the nuclear coordinates are ``parameters'' $\Rightarrow$ \blue{potential energy surface (PES)}
\begin{equation}
\cH_\text{elec} \Phi_\text{elec} = \cE_\text{elec} \Phi_\text{elec}
\Phi(\{\br_i\},\{\bR_A\}) = \Phi_\text{nucl}(\{\bR_A\}) \Phi_\text{elec}(\{\br_i\},\{\bR_A\})
\qq{with}
\boxed{\cH_\text{elec} = \blue{\cT_\text{e}} + \orange{\cV_\text{ne}} + \alert{\cV_\text{ee}}}
\cE_\text{tot} = \cE_\text{elec} + \sum_{A<B}^{M} \frac{Z_A Z_B}{R_{AB}}
\end{equation}
\end{block}
\begin{block}{Nuclear Hamiltonian}
@ -259,16 +258,39 @@
\end{equation}
It describes the vibration, rotation and translation of the molecules
\end{block}
\begin{block}{Born-Oppenheimer approximation = decoupling nuclei and electrons}
Because $M_A \gg 1$, the nuclear coordinates are ``parameters'' $\Rightarrow$ \blue{potential energy surface (PES)}
\begin{block}{Electronic Hamiltonian}
The \alert{electronic Hamiltonian} is
\begin{equation}
\Phi(\{\br_i\},\{\bR_A\}) = \Phi_\text{nucl}(\{\bR_A\}) \Phi_\text{elec}(\{\br_i\},\{\bR_A\})
\cH_\text{elec} \Phi_\text{elec} = \cE_\text{elec} \Phi_\text{elec}
\qq{with}
\cE_\text{tot} = \cE_\text{elec} + \sum_{A<B}^{M} \frac{Z_A Z_B}{R_{AB}}
\boxed{\cH_\text{elec} = \blue{\cT_\text{e}} + \orange{\cV_\text{ne}} + \alert{\cV_\text{ee}}}
\end{equation}
\end{block}
\end{frame}
\begin{frame}{Separability of the Schr\"odinger equation}
\begin{block}{Problem:}
\textit{\violet{``Assuming that $\hH = \hH_A + \hH_B$ with $\hH_A \Psi_A = E_A \Psi_A$ and $\hH_B \Psi_B = E_B \Psi_B$, find the expression of $\Psi$ and $E$ such that $\hH \Psi = E \Psi$''}}
\end{block}
\pause
\begin{block}{Solution:}
Let's try $\Psi = \Psi_A \Psi_B$. Then,
\begin{equation*}
\begin{split}
\hH \Psi
& = ( \hH_A + \hH_B) \Psi_A \Psi_B
\\
& = \hH_A \Psi_A \Psi_B + \hH_B \Psi_A \Psi_B
\\
& = E_A \Psi_A \Psi_B + E_B \Psi_A \Psi_B
\\
& = \underbrace{(E_A + E_B)}_{E} \underbrace{\Psi_A \Psi_B}_{\Psi}
\end{split}
\end{equation*}
\end{block}
\end{frame}
%-----------------------------------------------------
\subsection{Pauli}
%-----------------------------------------------------
@ -301,7 +323,7 @@
\end{frame}
\begin{frame}{Antisymmetry}
\begin{block}{Problem: Pauli Exclusion Principle}
\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''}}
\end{block}
\pause
@ -325,7 +347,7 @@
\end{frame}
\begin{frame}{Antisymmetry (Take 2)}
\begin{block}{Problem: Fermionic wave function}
\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 $\Psi(x_1,x_2)$?''}}
\end{block}
\pause
@ -481,17 +503,17 @@
\begin{frame}{Interlude 2: Excited determinants}
\begin{block}{Reference determinant}
\begin{equation}
\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_a \chi_b \ldots \chi_N}
\end{equation}
\end{block}
\begin{block}{Singly-excited determinants determinant}
\begin{block}{Singly-excited determinants}
\begin{equation}
\qq*{Electron in $a$ promoted in $r$:} \ket{\Psi_a^r} = \ket{\chi_1 \ldots \chi_r \chi_b \ldots \chi_N}
\qq*{\orange{Electron in $a$ promoted in $r$:}} \ket{\Psi_a^r} = \ket{\chi_1 \ldots \chi_r \chi_b \ldots \chi_N}
\end{equation}
\end{block}
\begin{block}{Doubly-excited determinants determinant}
\begin{block}{Doubly-excited determinants}
\begin{equation}
\qq*{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*{\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}
\end{equation}
\end{block}
\begin{center}
@ -503,48 +525,6 @@
\end{center}
\end{frame}
\begin{frame}{One-electron operators}
\begin{equation}
\boxed{\cO_1 = \sum_i^N h(i)}
\end{equation}
\begin{block}{\green{Case 1 = differ by zero spinorbital}: $\ket{K} = \ket{\ldots m n \ldots}$}
\begin{equation}
\mel{K}{\cO_1}{K} = \sum_m^N \mel{m}{h}{m}
\end{equation}
\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{equation}
\mel{K}{\cO_1}{L} = \sum_m \mel{m}{h}{p}
\end{equation}
\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{equation}
\mel{K}{\cO_1}{L} = 0
\end{equation}
\end{block}
\end{frame}
\begin{frame}{Two-electron operators}
\begin{equation}
\boxed{\cO_2 = \sum_{i<j}^N r_{ij}^{-1}}
\end{equation}
\begin{block}{\green{Case 1 = differ by zero spinorbital}: $\ket{K} = \ket{\ldots m n \ldots}$}
\begin{equation}
\mel{K}{\cO_2}{K} = \frac{1}{2} \sum_{mn}^N \mel{m}{h}{m} \mel{mn}{}{mn}
\end{equation}
\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{equation}
\mel{K}{\cO_2}{L} = \sum_n^N \mel{mn}{}{pn}
\end{equation}
\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{equation}
\mel{K}{\cO_2}{L} = \mel{mn}{}{pq}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
\section{HF approximation}
%-----------------------------------------------------
@ -686,15 +666,61 @@
= \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}
\begin{align}
\qq*{Complex-valued integrals:} &
\qq*{\red{Complex-valued integrals:}} &
\braket{ij}{kl} = \braket{ji}{lk} = \braket{kl}{ij}^*
\\
\qq*{Real-valued integrals:} &
\qq*{\green{Real-valued integrals:}} &
\braket{ij}{kl} = \braket{ji}{kl} = \braket{ij}{lk} = \braket{ji}{lk} = \braket{kl}{ij} = \braket{lk}{ij} = \braket{kl}{ji} = \braket{lk}{ji}
\end{align}
\end{block}
\end{frame}
%-----------------------------------------------------
\subsection{Slater-Condon}
%-----------------------------------------------------
\begin{frame}{Slater-Condon rules: One-electron operators}
\begin{equation}
\boxed{\cO_1 = \sum_i^N h(i)}
\end{equation}
\begin{block}{\green{Case 1 = differ by zero spinorbital}: $\ket{K} = \ket{\ldots m n \ldots}$}
\begin{equation}
\mel{K}{\cO_1}{K} = \sum_m^N \mel{m}{h}{m}
\end{equation}
\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{equation}
\mel{K}{\cO_1}{L} = \sum_m \mel{m}{h}{p}
\end{equation}
\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{equation}
\mel{K}{\cO_1}{L} = 0
\end{equation}
\end{block}
\end{frame}
\begin{frame}{Slater-Condon rules: Two-electron operators}
\begin{equation}
\boxed{\cO_2 = \sum_{i<j}^N r_{ij}^{-1}}
\end{equation}
\begin{block}{\green{Case 1 = differ by zero spinorbital}: $\ket{K} = \ket{\ldots m n \ldots}$}
\begin{equation}
\mel{K}{\cO_2}{K} = \frac{1}{2} \sum_{mn}^N \mel{mn}{}{mn}
\end{equation}
\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{equation}
\mel{K}{\cO_2}{L} = \sum_n^N \mel{mn}{}{pn}
\end{equation}
\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{equation}
\mel{K}{\cO_2}{L} = \mel{mn}{}{pq}
\end{equation}
\end{block}
\end{frame}
%-----------------------------------------------------
\subsection{Examples}
%-----------------------------------------------------
@ -1023,7 +1049,7 @@
\begin{frame}{The Fock matrix}
Using the \alert{variational principle}, one can show that, to minimise the energy, the MOs need to diagonalise the \alert{one-electron} \blue{Fock operator}
\begin{equation*}
\boxed{ f(1) = h(1) + \underbrace{\sum_a^N [\cJ_a(1) - \cK_a(1)]}_{\nu^\text{HF}(1) \text{ = Hartree-Fock potential}}}
\boxed{ f(1) = h(1) + \underbrace{\sum_a^N [\cJ_a(1) - \cK_a(1)]}_{\nu^\text{HF}(1) \text{ = \blue{Hartree-Fock potential}}}}
\end{equation*}
For a \orange{closed-shell system} (i.e. two electrons in each orbital)
\begin{equation*}
@ -1244,12 +1270,12 @@ These orbitals are called \orange{canonical molecular orbitals} (= eigenvectors
\begin{equation}
S_{\mu\nu}
= \braket{\mu}{\nu}
= \int \phi_\mu(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
= \int \phi_\mu^*(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
\end{equation}
\begin{equation}
H_{\mu\nu}
= \mel{\mu}{\hH^\text{c}}{\nu}
= \int \phi_\mu(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
= \int \phi_\mu^*(\orange{\br}) \hH^\text{c}(\orange{\br}) \phi_\nu(\orange{\br}) d\orange{\br}
\end{equation}
\end{block}
\end{column}
@ -1259,7 +1285,7 @@ These orbitals are called \orange{canonical molecular orbitals} (= eigenvectors
\end{columns} \begin{block}{Chemist/Mulliken notation for two-electron integrals}
\begin{equation}
( \mu \nu | \lambda \sigma )
= \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\alert{\br_1}) \frac{1}{r_{12}} \phi_\lambda(\blue{\br_2}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2}
= \iint \phi_\mu^*(\alert{\br_1}) \phi_\nu(\alert{\br_1}) \frac{1}{r_{12}} \phi_\lambda^*(\blue{\br_2}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2}
\end{equation}
\begin{equation}
( \mu \blue{\nu} \orange{||} \lambda \red{\sigma} ) = ( \mu \blue{\nu} | \lambda \red{\sigma} ) - ( \mu \red{\sigma} | \lambda \blue{\nu} )
@ -1268,7 +1294,7 @@ These orbitals are called \orange{canonical molecular orbitals} (= eigenvectors
\begin{block}{Physicist/Dirac notation for two-electron integrals}
\begin{equation}
\langle \mu \nu | \lambda \sigma \rangle
= \iint \phi_\mu(\alert{\br_1}) \phi_\nu(\blue{\br_2}) \frac{1}{r_{12}} \phi_\lambda(\alert{\br_1}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2}
= \iint \phi_\mu^*(\alert{\br_1}) \phi_\nu^*(\blue{\br_2}) \frac{1}{r_{12}} \phi_\lambda(\alert{\br_1}) \phi_\sigma(\blue{\br_2}) d\red{\br_1} d\blue{\br_2}
\end{equation}
\begin{equation}
\langle \mu \nu \orange{||} \blue{\lambda} \red{\sigma} \rangle = \langle \mu \nu | \blue{\lambda} \red{\sigma} \rangle - \langle \mu \nu | \red{\sigma} \blue{\lambda} \rangle
@ -1444,7 +1470,7 @@ These orbitals are called \orange{canonical molecular orbitals} (= eigenvectors
\begin{equation}
\mu_x = \red{- \sum_{\mu \nu} P_{\mu\nu} (\nu|x|\mu)} + \blue{\sum_A^M Z_A X_A}
\qq{with}
\underbrace{(\nu|x|\mu)}_{\text{one-electron integrals}} = \int \phi_\nu(\br) \,x\, \phi_\mu(\br) d\br
\underbrace{(\nu|x|\mu)}_{\text{one-electron integrals}} = \int \phi_\nu^*(\br) \,x\, \phi_\mu(\br) d\br
\end{equation}
\end{block}
\end{frame}