From 4039ca433ef0fe236a43a69970e441234a13f75a Mon Sep 17 00:00:00 2001 From: Pierre-Francois Loos Date: Fri, 27 Aug 2021 17:19:26 +0200 Subject: [PATCH] HF lecture --- BIO_RESUME_TCCM_IIC2021_LOOS.DOC | Bin 0 -> 43520 bytes HF/Loos-TCCM_HF.tex | 1773 ++++++++++++++++++++++++++++++ HF/fig/CNRS.png | Bin 0 -> 123758 bytes HF/fig/Cramer.jpg | Bin 0 -> 31143 bytes HF/fig/ERC.jpg | Bin 0 -> 57629 bytes HF/fig/GS.png | Bin 0 -> 60232 bytes HF/fig/H2.png | Bin 0 -> 33927 bytes HF/fig/HF_det.png | Bin 0 -> 106479 bytes HF/fig/Helgaker.jpg | Bin 0 -> 103839 bytes HF/fig/Jensen.jpg | Bin 0 -> 30315 bytes HF/fig/LCPQ.pdf | Bin 0 -> 15994 bytes HF/fig/RHF_UHF.png | Bin 0 -> 19960 bytes HF/fig/SBG.png | Bin 0 -> 128554 bytes HF/fig/Szabo.jpg | Bin 0 -> 3172373 bytes HF/fig/Szabo_Ostlund.pdf | Bin 0 -> 3182374 bytes HF/fig/UPS.pdf | Bin 0 -> 235750 bytes HF/fig/amazon.png | Bin 0 -> 733206 bytes HF/fig/coord.png | Bin 0 -> 128850 bytes HF/fig/double.png | Bin 0 -> 55579 bytes HF/fig/ortho.png | Bin 0 -> 62548 bytes HF/fig/single.png | Bin 0 -> 49895 bytes HF_Abstract_TCCM_IIC2021.DOC | Bin 0 -> 27648 bytes PFLoos.png | Bin 0 -> 1141815 bytes 23 files changed, 1773 insertions(+) create mode 100644 BIO_RESUME_TCCM_IIC2021_LOOS.DOC create mode 100644 HF/Loos-TCCM_HF.tex create mode 100755 HF/fig/CNRS.png create mode 100644 HF/fig/Cramer.jpg create mode 100644 HF/fig/ERC.jpg create mode 100644 HF/fig/GS.png create mode 100644 HF/fig/H2.png create mode 100644 HF/fig/HF_det.png create mode 100644 HF/fig/Helgaker.jpg create mode 100644 HF/fig/Jensen.jpg create mode 100644 HF/fig/LCPQ.pdf create mode 100644 HF/fig/RHF_UHF.png create mode 100644 HF/fig/SBG.png create mode 100644 HF/fig/Szabo.jpg create mode 100644 HF/fig/Szabo_Ostlund.pdf create mode 100644 HF/fig/UPS.pdf create mode 100644 HF/fig/amazon.png create mode 100644 HF/fig/coord.png create mode 100644 HF/fig/double.png create mode 100644 HF/fig/ortho.png create mode 100644 HF/fig/single.png create mode 100644 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+\usepackage{amsmath,amssymb,amsfonts,pgfpages,graphicx,subfigure,xcolor,bm,multirow,microtype,wasysym,tabularx,amscd,pgfgantt,mhchem,physics,libertine,mathpazo} +\usetheme{Warsaw} +\usecolortheme{seahorse} + +\usepackage{hyperref} +\hypersetup{ + colorlinks=true, + linkcolor=cyan, + filecolor=magenta, + urlcolor=cyan, + citecolor=purple +} +\urlstyle{same} + +% *********** +% * PACKAGE * +% *********** + +% energies +\newcommand{\Ec}{E_\text{c}} +\newcommand{\EHF}{E_\text{HF}} + +% bold symbols +\newcommand{\br}{\boldsymbol{r}} +\newcommand{\bx}{\boldsymbol{x}} +\newcommand{\bs}{\boldsymbol{s}} +\newcommand{\bR}{\boldsymbol{R}} +\newcommand{\bo}{\boldsymbol{o}} +\newcommand{\bA}{\boldsymbol{A}} +\newcommand{\bB}{\boldsymbol{B}} +\newcommand{\bC}{\boldsymbol{C}} +\newcommand{\bE}{\boldsymbol{E}} +\newcommand{\bG}{\boldsymbol{G}} +\newcommand{\bJ}{\boldsymbol{J}} +\newcommand{\bK}{\boldsymbol{K}} +\newcommand{\bP}{\boldsymbol{P}} +\newcommand{\bI}{\boldsymbol{I}} +\newcommand{\bU}{\boldsymbol{U}} +\newcommand{\bO}{\boldsymbol{O}} +\newcommand{\bS}{\boldsymbol{S}} +\newcommand{\bT}{\boldsymbol{T}} +\newcommand{\bF}{\boldsymbol{F}} +\newcommand{\bH}{\boldsymbol{H}} +\newcommand{\bV}{\boldsymbol{V}} +\newcommand{\bX}{\boldsymbol{X}} + +% hat symbols +\newcommand{\hH}{\Hat{H}} +\newcommand{\hh}{\Hat{h}} +\newcommand{\hT}{\Hat{T}} +\newcommand{\hV}{\Hat{V}} +\newcommand{\hO}{\Hat{O}} + +% curly symbols +\newcommand{\cO}{\mathcal{O}} +\newcommand{\cJ}{\mathcal{J}} +\newcommand{\cK}{\mathcal{K}} +\newcommand{\cH}{\mathcal{H}} +\newcommand{\cT}{\mathcal{T}} +\newcommand{\cV}{\mathcal{V}} +\newcommand{\cE}{\mathcal{E}} +\newcommand{\cP}{\mathcal{P}} + +% colors and others +\newcommand{\mc}{\multicolumn} +\definecolor{darkgreen}{RGB}{0, 180, 0} +\newcommand{\purple}[1]{\textcolor{purple}{#1}} +\newcommand{\red}[1]{\textcolor{red}{#1}} +\newcommand{\orange}[1]{\textcolor{orange}{#1}} +\newcommand{\green}[1]{\textcolor{darkgreen}{#1}} +\newcommand{\blue}[1]{\textcolor{blue}{#1}} +\newcommand{\violet}[1]{\textcolor{violet}{#1}} +\newcommand{\pub}[1]{\small \textcolor{purple}{#1}} + +% shortcuts +\newcommand{\si}{\sigma} +\newcommand{\la}{\lambda} + +\newcommand{\mycirc}[1][black]{\Large\textcolor{#1}{\ensuremath\bullet}} + +% ************* +% * HEAD DATA * +% ************* + \title[The HF approximation]{ + \LARGE The Hartree--Fock Approximation + } + \author[PF Loos]{Pierre-Fran\c{c}ois LOOS} + \date{TCCM 2021} + \institute[CNRS@LCPQ]{ + Laboratoire de Chimie et Physique Quantiques (UMR 5626),\\ + Universit\'e de Toulouse, CNRS, UPS, Toulouse, France. + } + \titlegraphic{ + \vspace{0.2\textheight} + \includegraphics[height=0.05\textwidth]{fig/UPS} + \hspace{0.2\textwidth} + \includegraphics[height=0.05\textwidth]{fig/ERC} + \hspace{0.2\textwidth} + \includegraphics[height=0.05\textwidth]{fig/LCPQ} + \hspace{0.2\textwidth} + \includegraphics[height=0.05\textwidth]{fig/CNRS} + } + +\begin{document} + +%----------------------------------------------------- +%%% TITLE %%% +%----------------------------------------------------- +\begin{frame} + \titlepage +\end{frame} + +%%% SLIDE X %%% +\begin{frame}{How to perform a HF calculation in practice?} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{The SCF algorithm for Hartree-Fock (HF) calculations (p.~146)} + \begin{enumerate} + \item \orange{Specify molecule} $\{\bR_A\}$ and $\{Z_A\}$ and \violet{basis set} $\{\phi_\mu\}$ + \item Calculate integrals $S_{\mu \nu}$, $H_{\mu \nu}$ and $\braket{\mu \nu}{\lambda \sigma}$ + \item Diagonalize $\bS$ and compute $\bX = \bS^{-1/2}$ + \item Obtain \alert{guess density matrix} for $\bP$ + \begin{enumerate} + \item[1.] Calculate $\bJ$ and $\bK$, then $\bF = \bH + \bJ + \bK$ + \item[2.] Compute $\bF' = \bX^\dag \cdot \bF \cdot \bX$ + \item[3.] Diagonalize $\bF'$ to obtain $\bC'$ and $\bE$ + \item[4.] Calculate $\bC= \bX \cdot \bC'$ + \item[5.] Form a \blue{new density matrix} $\bP = \bC \cdot \bC^\dag$ + \item[6.] \alert{Am I converged?} If not go back to 1. + \end{enumerate} + \item Calculate stuff that you want, like $\EHF$ for example + \end{enumerate} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/Szabo} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{Szabo's and Ostlund's book} + \begin{center} + \includegraphics[width=\textwidth]{fig/amazon} + \end{center} +\end{frame} + +%----------------------------------------------------- +\section{Introduction} +%----------------------------------------------------- +%----------------------------------------------------- +\subsection{Motivations} +%----------------------------------------------------- +\begin{frame}{Motivations \& Assumptions} + \begin{itemize} + \item We consider the \violet{time-independent} Schr\"odinger equation + \bigskip + \item HF is an \alert{ab initio method}, i.e., there's no parameter + \bigskip + \item We don't care about \violet{relativistic effects} + \bigskip + \item HF is an \blue{independent-particle model} + \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} + +%----------------------------------------------------- +\subsection{Born-Oppenheimer} +%----------------------------------------------------- +\begin{frame}{The Hamiltonian} + In the \alert{Schr\"odinger equation} + \begin{equation} + \cH \Phi(\{\br_i\},\{\bR_A\}) = \cE \Phi(\{\br_i\},\{\bR_A\}) + \end{equation} + the \violet{total Hamiltonian} is + \begin{equation} + \boxed{ + \cH = \green{\cT_\text{n}} + \blue{\cT_\text{e}} + \orange{\cV_\text{ne}} + \alert{\cV_\text{ee}} + \violet{\cV_\text{nn}} + } + \end{equation} + \begin{block}{What are all these terms?} + \begin{itemize} + \item \green{$\cT_\text{n}$} is the \green{kinetic energy of the nuclei} + \bigskip + \item \blue{$\cT_\text{e}$} is the \blue{kinetic energy of the electrons} + \bigskip + \item \orange{$\cV_\text{ne}$} is the \orange{Coulomb attraction between nuclei and electrons} + \bigskip + \item \alert{$\cV_\text{ee}$} is the \alert{Coulomb repulsion between electrons} + \bigskip + \item \violet{$\cV_\text{nn}$} is the \violet{Coulomb repulsion between nuclei} + \end{itemize} + \end{block} +\end{frame} + +\begin{frame}{The Hamiltonian (Take 2)} + \begin{columns} + \begin{column}{0.5\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} + \\ + & \blue{\cT_\text{e}} = - \sum_{i=1}^{N} \frac{\nabla_i^2}{2} + \\ + & \orange{\cV_\text{ne}} = - \sum_{A=1}^{M} \sum_{i=1}^N \frac{Z_A}{r_{iA}} + \\ + & \alert{\cV_\text{ee}} = \sum_{i0} + \underbrace{(\alert{J_{12}} - \orange{J_{11}})}_{<0} - \alert{K_{12}} + \end{split} + \end{equation*} + \end{block} +\end{frame} + +\begin{frame}{HF Energy of Atoms} + \begin{block}{Problem: HF energy of the \ce{Li} atom} + \violet{``Find the HF energy of the \ce{Li} atom in terms of the spatial MOs''} + \end{block} + \pause + \begin{block}{Solution:} + $$ \chi_1 = \alpha \, \psi_1 \qquad \chi_2 = \beta \, \psi_1 + \qquad \chi_3 = \alpha \, \psi_2 \qquad \chi_4 = \beta \, \psi_2$$ + \begin{equation*} + E_\text{HF} = 2 h_1 + h_2 + J_{11} + 2J_{12} - K_{12} + \end{equation*} + \end{block} + \pause + \begin{block}{Problem: HF energy of the \ce{B} atom} + \violet{``Find the HF energy of the \ce{B} atom' in terms of the spatial MOs'} + \end{block} + \pause + \begin{block}{Solution:} + $$ E_\text{HF} =2h_1 + 2h_2 + h_3 + J_{11} + 4J_{12} + J_{22} - 2K_{12} + 2J_{13} + 2J_{23} - K_{13} - K_{23}$$ + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{Spin to spatial} +%----------------------------------------------------- +\begin{frame}{From spin to spatial orbitals} + \begin{columns} + \begin{column}{0.6\textwidth} + \begin{block}{Two-electron example: \ce{H2} in minimal basis} + In the spin orbital basis, we have + \begin{equation*} + \begin{split} + \EHF + & = \mel{\chi_1}{h}{\chi_1} + \mel{\chi_2}{h}{\chi_2} + + \braket{\chi_1 \chi_2}{\chi_1 \chi_2} - \braket{\chi_1 \chi_2}{\chi_2 \chi_1} + \\ + & = [\chi_1|h|\chi_1] + [\chi_2|h|\chi_2] + + [\chi_1 \chi_1|\chi_2 \chi_2] - [\chi_1 \chi_2|\chi_2 \chi_1] + \end{split} + \end{equation*} + Spin to spatial transformation: + \begin{align*} + \chi_1(\bx) & \equiv \psi_1(\bx) = \psi_1(\br) \alpha(\omega) + \\ + \chi_2(\bx) & \equiv \Bar{\psi}_1(\bx) = \psi_1(\br) \beta(\omega) + \end{align*} + \begin{equation*} + \EHF + = [\psi_1|h|\psi_1] + [\Bar{\psi}_1|h|\Bar{\psi}_1] + + [\psi_1 \psi_1 | \Bar{\psi}_1 \Bar{\psi}_1] - [\psi_1 \Bar{\psi_1} | \Bar{\psi}_1 \psi_1] + \end{equation*} + Therefore, in the spatial orbital basis, we have + \begin{equation*} + \EHF = 2(\psi_1|h|\psi_1) + (\psi_1 \psi_1|\psi_1 \psi_1) = 2(1|h|1) + (11|11) + \end{equation*} + \end{block} + \end{column} + \begin{column}{0.4\textwidth} + \begin{center} + \includegraphics[width=\textwidth]{fig/H2} + \end{center} + \end{column} + \end{columns} +\end{frame} + +\begin{frame}{From spin to spatial orbitals (Take 2)} + \begin{block}{One-electron terms} + \begin{equation*} + \begin{split} + [\chi_1|h|\chi_1] + & = \int \chi_1^*(\bx) h(\br) \chi_1(\bx) d\bx + \\ + & = \int \alpha^*(\omega) \psi_1^*(\br) h(\br) \alpha(\omega) \psi_1(\br) d\omega d\br + \\ + & = \underbrace{\qty[ \int \alpha^*(\omega) \alpha(\omega) d\omega]}_{=1} + \underbrace{\qty[ \int \psi_1^*(\br) h(\br) \psi_1(\br) d\br ]}_{(\psi_1|h|\psi_1)} + \end{split} + \end{equation*} + \begin{equation*} + \begin{split} + [\chi_2|h|\chi_2] + & = \int \chi_2^*(\bx) h(\br) \chi_2(\bx) d\bx + \\ + & = \int \beta^*(\omega) \psi_1^*(\br) h(\br) \beta(\omega) \psi_1(\br) d\omega d\br + \\ + & = \underbrace{\qty[ \int \beta^*(\omega) \beta(\omega) d\omega]}_{=1} + \underbrace{\qty[ \int \psi_1^*(\br) h(\br) \psi_1(\br) d\br ]}_{(\psi_1|h|\psi_1)} + \end{split} + \end{equation*} + \end{block} +\end{frame} + +\begin{frame}{From spin to spatial orbitals (Take 3)} + \begin{block}{Two-electron terms} + \begin{equation*} + \begin{split} + [\chi_1 \chi_1|\chi_2 \chi_2] + & = \iint \chi_1^*(\bx_1) \chi_1(\bx_1) r_{12}^{-1} \chi_2^*(\bx_2) \chi_2(\bx_2) d\bx_1 d\bx_2 + \\ + & = \iint \alpha^*(\omega_1) \psi_1^*(\br_1) \alpha(\omega_1) \psi_1(\br_1) r_{12}^{-1} \beta^*(\omega_2) \psi_1^*(\br_2) \beta(\omega_2) \psi_1(\br_2) d\omega_1 d\br_1 d\omega_2 d\br_2 + \\ + & = \underbrace{\qty[ \int \alpha^*(\omega_1) \alpha(\omega_1) d\omega_1]}_{=1} + \underbrace{\qty[ \int \beta^*(\omega_2) \beta(\omega_2) d\omega_2]}_{=1} + \underbrace{\qty[ \iint \psi_1^*(\br_1) \psi_1(\br_1) r_{12}^{-1} \psi_1^*(\br_2) \psi_1(\br_2) d\br_1 d\br_2 ]}_{(\psi_1 \psi_1|\psi_1 \psi_1)} + \end{split} + \end{equation*} + \begin{equation*} + \begin{split} + [\chi_1 \chi_2|\chi_2 \chi_1] + & = \iint \chi_1^*(\bx_1) \chi_2(\bx_1) r_{12}^{-1} \chi_2^*(\bx_2) \chi_1(\bx_2) d\bx_1 d\bx_2 + \\ + & = \iint \alpha^*(\omega_1) \psi_1^*(\br_1) \beta(\omega_1) \psi_1(\br_1) r_{12}^{-1} \beta^*(\omega_2) \psi_1^*(\br_2) \alpha(\omega_2) \psi_1(\br_2) d\omega_1 d\br_1 d\omega_2 d\br_2 + \\ + & = \underbrace{\qty[ \int \alpha^*(\omega_1) \beta(\omega_1) d\omega_1]}_{=0} + \underbrace{\qty[ \int \beta^*(\omega_2) \alpha(\omega_2) d\omega_2]}_{=0} + \underbrace{\qty[ \iint \psi_1^*(\br_1) \psi_1(\br_1) r_{12}^{-1} \psi_1^*(\br_2) \psi_1(\br_2) d\br_1 d\br_2 ]}_{(\psi_1 \psi_1|\psi_1 \psi_1)} + \end{split} + \end{equation*} + \end{block} +\end{frame} + + + +\begin{frame}{From spin to spatial orbitals (Take 4)} + \begin{block}{General expression} + \begin{equation} + \EHF + = \sum_a^N [a|h|a] + \frac{1}{2} \sum_a^N \sum_b^N \qty( [aa|bb] - [ab|ba] ) + = 2 \sum_a^{N/2} (a|h|a) + \sum_a^{N/2} \sum_b^{N/2} \qty[ 2(aa|bb) - (ab|ba) ] + \end{equation} + \end{block} + \begin{block}{One- and two-electron terms} + \small + \begin{equation} + \sum_a^N [a|h|a] = \sum_a^{N/2} [a|h|a] + \sum_a^{N/2} [\Bar{a}|h|\Bar{a}] = 2 \sum_a^{N/2} [a|h|a] + \end{equation} + \begin{equation} + \begin{split} + \frac{1}{2} \sum_a^N \sum_b^N \qty( [aa|bb] - [ab|ba] ) + & = \frac{1}{2} \Bigg\{ + \sum_a^{N/2} \sum_b^{N/2} \qty( [aa|bb] - [ab|ba] ) + + \sum_a^{N/2} \sum_b^{N/2} \qty( [aa|\Bar{b}\Bar{b}] - [a\Bar{b}|\Bar{b}a] ) + \\ + & \qquad + \sum_a^{N/2} \sum_b^{N/2} \qty( [\Bar{a}\Bar{a}|bb] - [\Bar{a}b|b\Bar{a}] ) + + \sum_a^{N/2} \sum_b^{N/2} \qty( [\Bar{a}\Bar{a}|\Bar{b}\Bar{b}] - [\Bar{a}\Bar{b}|\Bar{b}\Bar{a}] ) + \Bigg\} + \\ + & = \sum_a^{N/2} \sum_b^{N/2} \qty[ 2(aa|bb) - (ab|ba) ] + \end{split} + \end{equation} + \end{block} +\end{frame} + + +%----------------------------------------------------- +\subsection{Fock matrix} +%----------------------------------------------------- +\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}}} + \end{equation*} + For a \orange{closed-shell system} (i.e. two electrons in each orbital) + \begin{equation*} + f(1) = h(1) + \sum_a^{N/2} [2 J_a(1) - K_a(1)] \quad \text{\alert{(closed shell)}} + \end{equation*} +These orbitals are called \orange{canonical molecular orbitals} (= eigenvectors): + \begin{equation*} + \boxed{f(1)\,\psi_i(1) = \varepsilon_i \, \psi_i(1)} + \end{equation*} + and $\varepsilon_i$ are called the \violet{MO energies} (= eigenvalues) +\end{frame} + +\begin{frame}{Fock matrix elements in the MO basis} + \begin{block}{Problem:} + \violet{`` Find the expression of the matrix elements $f_{ij} = \mel{\chi_i}{f}{\chi_j}$''} + \end{block} + \pause + \begin{block}{Solution:} + \begin{equation*} + \begin{split} + \mel{\chi_i}{f}{\chi_j} + & = \mel{\chi_i}{h + \sum_a \qty( \cJ_a - \cK_a)}{\chi_j} + \\ + & = \mel{\chi_i}{h}{\chi_j} + \sum_a \qty( \mel{\chi_i}{\cJ_a}{\chi_j} - \mel{\chi_i}{\cK_a}{\chi_j} ) + \\ + & = \mel{i}{h}{j} + \sum_a \qty[ \braket{ia}{ja} - \braket{ia}{aj} ] + \\ + & = \mel{i}{h}{j} + \sum_a \mel{ia}{}{ja} + \end{split} + \end{equation*} + \end{block} +\end{frame} + +\begin{frame}{MO energies in the MO basis} + \begin{block}{Problem:} + \violet{`` Deduce the expression of $\varepsilon_i$''} + \end{block} + \pause + \begin{block}{Solution:} + \begin{align*} + f \ket{\chi_i} = \varepsilon_i \ket{\chi_i} + & \Rightarrow \quad \mel{\chi_i}{f}{\chi_i} = \varepsilon_i \braket{\chi_i}{\chi_i} = \varepsilon_i + \\ + & \Rightarrow \quad \varepsilon_i = \mel{i}{h}{i} + \sum_a \qty[ \braket{ia}{ia} - \braket{ia}{ai} ] + \\ + & \Rightarrow \quad \varepsilon_i = \mel{i}{h}{i} + \sum_a \mel{ia}{}{ia} + \end{align*} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{Koopmans} +%----------------------------------------------------- +\begin{frame}{Koopmans' theorem} + \begin{block}{Ground-state energy of the $N$-electron system} + \begin{equation} + {}^{N} E_0 = \sum_{a} h_a + \frac{1}{2} \sum_{ab} \mel{ab}{}{ab} + \end{equation} + \end{block} + \begin{block}{Energy of the $(N-1)$-electron system (cation)} + \begin{equation} + {}^{N-1}E_c = \sum_{a \neq c} h_a + \frac{1}{2} \sum_{a \neq c} \sum_{b \neq c} \mel{ab}{}{ab} + \end{equation} + \end{block} + \begin{block}{Ionization potential (IP)} + \begin{equation} + \begin{split} + \text{IP} + & = {}^{N-1}E_c - {}^{N} E_0 + \\ + & = - \mel{c}{h}{c} - \frac{1}{2} \sum_{a} \mel{ac}{}{ac} - \frac{1}{2} \sum_{b} \mel{cb}{}{cb} + \\ + & = - \mel{c}{h}{c} - \sum_{a} \mel{ac}{}{ac} + = - \varepsilon_c + \end{split} + \end{equation} + \end{block} +\end{frame} + +\begin{frame}{Koopmans' theorem for electron affinity (EA)} + \begin{block}{Problem:} + \violet{``Show that Koopmans' theorem applies to electron affinities''} + \end{block} + \pause + \begin{block}{Solution:} + \begin{equation} + \begin{split} + \text{EA} + & = {}^{N}E_0 - {}^{N+1} E^r + \\ + & = - \mel{r}{h}{r} - \sum_{a} \mel{ra}{}{ra} + \\ + & = - \varepsilon_r + \end{split} + \end{equation} + + \end{block} +\end{frame} + +%----------------------------------------------------- +\section{Roothaan-Hall equations} +%----------------------------------------------------- +%----------------------------------------------------- +\subsection{Basis set approximation} +%----------------------------------------------------- +\begin{frame}{Roothaan-Hall equations: introduction of a basis} + \begin{block}{Expansion in a basis} + $$ \psi_i(\br) = \sum_\mu^K C_{\mu i} \phi_{\mu}(\br) + \qquad \equiv \qquad + \ket{i} = \sum_\mu^K C_{\mu i} \ket{\mu}$$ + \alert{\bf $K$ AOs gives $K$ MOs:} + \blue{$N/2$ are occupied MOs} and \orange{$K-N/2$ are vacant/virtual MOs} + \end{block} + \begin{block}{Roothaan-Hall equations} + \begin{equation*} + \begin{split} + f \ket{i} = \varepsilon_i \ket{i} + & \quad \Rightarrow \quad + f \sum_{\nu} C_{\nu i} \ket{\nu} = \varepsilon_i \sum_{\nu} C_{\nu i} \ket{\nu} + \\ + & \quad \Rightarrow \quad \mel{ \mu }{ f \sum_{\nu} C_{\nu i} }{ \nu } = \varepsilon_i \mel{ \mu }{ \sum_{\nu} C_{\nu i} }{ \nu } + \\ + & \quad \Rightarrow \quad \sum_{\nu} C_{\nu i} \mel{ \mu }{ f }{ \nu } = \sum_{\nu} C_{\nu i} \varepsilon_i \braket{ \mu }{ \nu } + \\ + & \quad \Rightarrow \quad + \boxed{\alert{\sum_{\nu} F_{\mu \nu} C_{\nu i} = \sum_{\nu} S_{\mu \nu} C_{\nu i} \varepsilon_i }} + \end{split} + \end{equation*} + \end{block} +\end{frame} + +\begin{frame}{Introduction of a basis (Take 2)} + \begin{block}{Matrix form of the Roothaan-Hall equations} + \begin{align} + \bF \cdot \bC & = \bS \cdot \bC \cdot \bE + & + & \Leftrightarrow + & + \bF^\prime \cdot \bC^\prime & = \bC^\prime \cdot \bE + \\ + \bF' & = \bX^\dag \cdot \bF \cdot \bX + & + \bC & = \bX \cdot \bC' + & + \bX^\dag \cdot \bS \cdot \bX & = \bI + \end{align} + \begin{itemize} + \item \violet{Fock matrix} $ F_{\mu\nu} = \mel{ \mu }{ f }{ \nu }$ and \blue{Overlap matrix} $S_{\mu\nu} = \braket{ \mu }{ \nu }$ + \item We need to determine the \orange{coefficient matrix} $\bC$ and the \purple{orbital energies} $\bE$ + \end{itemize} + \begin{align} + \bC & = + \begin{pmatrix} + C_{11} & C_{12} & \cdots & C_{1K} \\ + C_{21} & C_{22} & \cdots & C_{2K} \\ + \vdots & \vdots & \ddots & \vdots \\ + C_{K1} & C_{K2} & \cdots & C_{KK} \\ + \end{pmatrix} + & + \bE & = + \begin{pmatrix} + \varepsilon_{1} & 0 & \cdots & 0 \\ + 0 & \varepsilon_{2} & \cdots & 0 \\ + \vdots & \vdots & \ddots & \vdots \\ + 0 & 0 & \cdots & \varepsilon_{K} \\ + \end{pmatrix} + \end{align} + \end{block} + \begin{block}{Self-consistent field (SCF) procedure} + \begin{equation} + \bF(\bC) \cdot \bC + = \bS \cdot \bC \cdot \bE + \qq{\alert{\bf How do we solve these HF equations?}} + \end{equation} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{Fock matrix} +%----------------------------------------------------- +\begin{frame}{Expression of the Fock matrix} + \begin{block}{Problem:} + \violet{\textit{``Find the expression of the Fock matrix in terms of the one- and two-electron integrals''}} + \end{block} + \pause + \begin{block}{Solution:} + \begin{equation*} + \begin{split} + F_{\mu \nu} + & = \mel{\mu}{h + \sum_a^N (\cJ_a - \cK_a)}{\nu} = H_{\mu \nu} + \sum_a^N \mel{\mu}{\cJ_a - \cK_a}{\nu} + \\ + & = H_{\mu \nu} + \sum_a^N (\mel{ \mu \chi_a }{ r_{12}^{-1} }{ \nu \chi_a } - \mel{ \mu \chi_a }{ r_{12}^{-1} }{ \chi_a \nu }) + \\ + & = H_{\mu \nu} + \sum_a^N \sum_{\lambda \sigma} C_{\lambda a} C_{\sigma a} (\mel{ \mu \lambda }{ r_{12}^{-1} }{ \nu \sigma } - \mel{ \mu \lambda }{ r_{12}^{-1} }{ \sigma \nu }) + \\ + & = H_{\mu \nu} + \sum_{\lambda \sigma} \alert{P_{\lambda \sigma}} (\braket{ \mu \lambda }{ \nu \sigma } - \braket{ \mu \lambda }{ \sigma \nu }) + = H_{\mu \nu} + \sum_{\lambda \sigma} \alert{P_{\lambda \sigma}} \mel{ \mu \lambda }{}{ \nu \sigma } + = H_{\mu \nu} + G_{\mu \nu} + \end{split} + \end{equation*} + \begin{equation*} + F_{\mu \nu} + = H_{\mu \nu} + \sum_{\lambda \sigma} P_{\lambda \sigma} (\langle \mu \lambda | \nu \sigma \rangle - \frac{1}{2} \langle \mu \lambda | \sigma \nu \rangle) + \quad + \text{\alert{(closed shell)}} + \end{equation*} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{Density matrix \& Integrals} +%----------------------------------------------------- +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame}{One- and two-electron integrals (Appendix A)} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{One-electron integrals: overlap \& core Hamiltonian} + \begin{equation} + S_{\mu\nu} + = \braket{\mu}{\nu} + = \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} + \end{equation} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/SBG} + \end{column} + \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} + \end{equation} + \begin{equation} + ( \mu \blue{\nu} \orange{||} \lambda \red{\sigma} ) = ( \mu \blue{\nu} | \lambda \red{\sigma} ) - ( \mu \red{\sigma} | \lambda \blue{\nu} ) + \end{equation} + \end{block} + \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} + \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 + \end{equation} + \end{block} +\end{frame} + + +\begin{frame}{Computation of the Fock matrix and energy} + \begin{block}{Density matrix (closed-shell system)} + \begin{equation} + P_{\red{\mu \nu}} = 2 \sum_{a}^{N/2} C_{\red{\mu} a} C_{\red{\nu} a} + \qqtext{or} + \boxed{\bP = 2 \, \bC \cdot \bC^{\dag}} + \end{equation} + \end{block} + \begin{block}{Fock matrix in the AO basis (closed-shell system)} + \begin{equation} + F_{\red{\mu\nu}} + = H_{\red{\mu\nu}} + + \underbrace{\sum_{\blue{\la \si}} P_{\blue{\la\si}} (\red{\mu\nu}|\blue{\la\si})}_{J_{\red{\mu \nu}} = \text{ Coulomb}} + \underbrace{ - \frac{1}{2} \sum_{\blue{\la \si}} P_{\blue{\la\si}} (\red{\mu}\blue{\si}|\blue{\la}\red{\nu})}_{K_{\red{\mu \nu}} = \text{ exchange}} + \end{equation} + \end{block} + \begin{block}{HF energy in the AO basis (closed-shell system)} + \begin{equation} + E_\text{HF} = \sum_{\red{\mu \nu}} P_{\red{\mu \nu}} H_{\red{\mu \nu}} + + \frac{1}{2} \sum_{\red{\mu \nu} \blue{\la\si}} P_{\red{\mu \nu}} \qty[ (\red{\mu \nu} | \blue{\lambda \sigma}) - \frac{1}{2} (\red{\mu} \blue{\sigma} | \red{\lambda} \blue{\nu}) ] P_{\blue{\lambda\sigma}} + \qqtext{or} + \boxed{E_\text{HF} = \frac{1}{2} \text{Tr}{\qty[\bP \cdot (\bH + \bF)]}} + \end{equation} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{HF energy} +%----------------------------------------------------- +\begin{frame}{Expression of the HF energy } + \begin{block}{Problem:} + \violet{\textit{``Find the expression of the HF energy in terms of the one- and two-electron integrals''}} + \end{block} + \pause + \begin{block}{Solution:} + \begin{equation*} + \begin{split} + E_\text{HF} + & = \sum_a^N h_a + \frac{1}{2} \sum_{ab}^N (\cJ_{ab} - \cK_{ab}) \quad \alert{\text{(cf few slides ago)}} + \\ + & = \sum_a^N \mel{ \sum_\mu C_{\mu a} \phi_\mu }{ h }{ \sum_\nu C_{\nu a} \phi_\nu } + + \frac{1}{2} \sum_{ab}^N \mel{ \qty(\sum_\mu C_{\mu a} \phi_\mu) \qty(\sum_\lambda C_{\lambda b} \phi_\lambda)} {} { \qty(\sum_\nu C_{\nu a} \phi_\nu) \qty(\sum_\sigma C_{\sigma b} \phi_\sigma)} + \\ + & = \sum_{\mu \nu} P_{\mu \nu} \qty[ H_{\mu \nu} + \frac{1}{2} \sum_{\lambda \sigma} P_{\lambda \sigma} \mel{ \mu \lambda }{}{ \nu \sigma } ] + \end{split} + \end{equation*} + \end{block} +\end{frame} + + +%----------------------------------------------------- +\subsection{SCF} +%----------------------------------------------------- +\begin{frame}{How to perform a HF calculation in practice?} + \begin{block}{The SCF algorithm} + \begin{enumerate} + \item \orange{Specify molecule} $\{\bR_A\}$ and $\{Z_A\}$ and \violet{basis set} $\{\phi_\mu\}$ + \item Calculate integrals $S_{\mu \nu}$, $H_{\mu \nu}$ and $\braket{ \mu \nu }{ \lambda \sigma }$ + \item Diagonalize $\bS$ and compute $\bX$ + \item Obtain \alert{guess density matrix} for $\bP$ + \begin{enumerate} + \item[1.] Calculate $\bG$ and then $\bF = \bH + \bG$ + \item[2.] Compute $\bF' = \bX^\dag \cdot \bF \cdot \bX$ + \item[3.] Diagonalize $\bF'$ to obtain $\bC'$ and $\bE$ + \item[4.] Calculate $\bC= \bX \cdot \bC'$ + \item[5.] Form a \blue{new density matrix} $\bP = \bC \cdot \bC^\dag$ + \item[6.] \alert{Am I converged?} If not go back to 1. + \end{enumerate} + \item Calculate stuff that you want, like $\EHF$ for example + \end{enumerate} + \end{block} +\end{frame} + +\begin{frame}{Orthogonalization matrix} + \red{\bf We are looking for a matrix in order to orthogonalize the AO basis, i.e.~$\bX^\dag \cdot \bS \cdot \bX = \bI$} + \\ + \bigskip + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{block}{Symmetric (or L\"owdin) orthogonalization} + \begin{equation} + \text{$\bX =\bS^{-1/2} = \bU \cdot \bs^{-1/2} \cdot \bU^\dag$ is one solution...} + \end{equation} + \purple{\bf Is it working?} + \begin{equation} + \bX^\dag \cdot \bS \cdot \bX + = \bS^{-1/2} \cdot \bS \cdot \bS^{-1/2} + = \bS^{-1/2} \cdot \bS \cdot \bS^{-1/2} + = \bI \quad \green{\checkmark} + \end{equation} + \end{block} + \begin{block}{Canonical orthogonalization} + \begin{equation} + \text{$\bX =\bU \cdot \bs^{-1/2}$ is another solution (when you have linear dependencies)...} + \end{equation} + \purple{\bf Is it working?} + \begin{equation} + \bX^\dag \cdot \bS \cdot \bX + = \bs^{-1/2} \cdot \underbrace{\bU^{\dag} \cdot \bS \cdot \bU}_{\bs} \cdot \bs^{-1/2} + = \bI \quad \green{\checkmark} + \end{equation} + \end{block} + \end{column} + \begin{column}{0.3\textwidth} + \includegraphics[width=\textwidth]{fig/ortho} + \end{column} + \end{columns} +\end{frame} + + +\begin{frame}{How to obtain a good guess for the MOs or density matrix?} + \begin{block}{Possible initial density matrix} + \begin{enumerate} + \bigskip + \item We can set \purple{$\bP = \mathbf{0}$ $\Rightarrow$ $\bF = \bH$} (\orange{core Hamiltonian approximation}):\\ + $\Rightarrow$ Usually a poor guess but easy to implement + \bigskip + \item Use \alert{EHT or semi-empirical methods}:\\ + $\Rightarrow$ Out of fashion + \bigskip + \item Using \violet{tabulated atomic densities}:\\ + $\Rightarrow$ ``SAD'' guess in QChem + \bigskip + \item \blue{Read the MOs of a previous calculation:}\\ + $\Rightarrow$ Very common and very useful + \bigskip + \end{enumerate} + \end{block} +\end{frame} + +\begin{frame}{How do I know I have converged (or not)?} + \begin{block}{Convergence in SCF calculations} + \begin{enumerate} + \bigskip + \item You can check the \orange{energy and/or the density matrix}:\\ + $\Rightarrow$ The energy/density \textbf{should not} change at convergence + \bigskip + \item You can check the commutator \alert{$\bF \cdot \bP \cdot \bS - \bS \cdot \bP \cdot \bF$}:\\ + $\Rightarrow$ At convergence, we have \alert{$\bF \cdot \bP \cdot \bS - \bS \cdot \bP \cdot \bF = \mathbf{0}$} + \bigskip + \item The \violet{DIIS (direct inversion in the iterative subspace) method} is usually used to speed up convergence:\\ + $\Rightarrow$ \blue{Extrapolation of the Fock matrix} using previous iterations + $$ \bF_{m+1} = \sum_{i=m-k}^{m} c_i \, \bF_i $$ + \bigskip + \end{enumerate} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{Properties} +%----------------------------------------------------- +\begin{frame}{Dipole moments} + \begin{block}{Classical vs Quantum} + \begin{equation} + \boldsymbol{\mu} = (\mu_x,\mu_y,\mu_z) + = \underbrace{\green{\sum_i q_i \br_i}}_{\text{\green{classical definition}}} + \end{equation} + \begin{equation} + \boldsymbol{\mu} = (\mu_x,\mu_y,\mu_z) + = \underbrace{\red{\mel{\Psi_0}{- \sum_i^N \br_i}{\Psi_0}}}_{\text{\red{electrons}}} + \underbrace{\blue{\sum_A^M Z_A \bR_A}}_{\text{\blue{nuclei}}} + = \red{- \sum_{\mu \nu} P_{\mu\nu} (\nu|\br|\mu)} + \blue{\sum_A^M Z_A \bR_A} + \end{equation} + \end{block} + \begin{block}{Vector components} + \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 + \end{equation} + \end{block} +\end{frame} + +\begin{frame}{Charge analysis} + \begin{block}{Electron density} + \begin{equation} + \rho(\br) = \sum_{\mu\nu} \phi_\mu(\br) P_{\mu\nu} \phi_\nu(\br) + \qq{with} + \int \rho(\br) d\br = N + \qq{$\Rightarrow$} + N = \sum_{\mu\nu} P_{\mu\nu} S_{\nu\mu} = \sum_\mu (\bP \cdot \bS)_{\mu\mu} = \Tr(\bP \cdot \bS) + \end{equation} + \end{block} + \begin{block}{Mulliken population analysis} + Assuming that the basis functions are atom-centered + \begin{equation} + \underbrace{\blue{q_A^\text{Mulliken}}}_{\text{net charge on $A$}} = Z_A - \sum_{\mu \in A} (\bP \cdot \bS)_{\mu\mu} + \end{equation} + \end{block} + \begin{block}{L{\"o}wdin population analysis} + Because $\Tr(\bA \cdot \bB) = \Tr(\bB \cdot \bA)$, we have, for any $\alpha$, + $N = \sum_{\mu} (\bS^{\alpha} \cdot \bP \cdot \bS^{1-\alpha})_{\mu\mu}$ + \begin{equation} + \qq*{For \red{$\alpha = 1/2$}, we get:} + N = \sum_{\mu} (\bS^{1/2} \cdot \bP \cdot \bS^{1/2})_{\mu\mu} + \qq{$\Rightarrow$} + \red{q_A^\text{L{\"o}wdin}} = Z_A - \sum_{\mu \in A} (\bS^{1/2} \cdot \bP \cdot \bS^{1/2})_{\mu\mu} + \end{equation} + \end{block} +\end{frame} + +%----------------------------------------------------- +\section{Unrestricted HF} +%----------------------------------------------------- +%----------------------------------------------------- +\subsection{UHF} +%----------------------------------------------------- +\begin{frame}{Unrestricted HF (UHF)} + \begin{block}{How to model open-shell systems?} + \begin{itemize} + \item RHF is made to describe \alert{closed-shell systems} and we have used \orange{restricted spin orbitals}: + \begin{equation*} + \chi_i^\text{RHF}(\bx) = + \begin{cases} + \alpha(\omega) \, \psi_i(\br) + \\ + \beta(\omega) \, \psi_i(\br) + \end{cases} + \end{equation*} + \item It does {\bf not} describe \alert{open-shell systems} + \item For open-shell systems we can use \violet{unrestricted spin orbitals} + \begin{equation*} + \chi_i^\text{UHF}(\bx) = + \begin{cases} + \alpha(\omega) \, \orange{\psi_i^\alpha(\br)} + \\ + \beta(\omega) \, \red{\psi_i^\beta(\br)} + \end{cases} + \end{equation*} \ + \item RHF = \orange{Restricted} Hartree-Fock $\leftrightarrow$ \blue{Roothaan-Hall equations} + \item UHF = \red{Unrestricted} Hartree-Fock $\leftrightarrow$ \violet{Pople-Nesbet equations} + \item \blue{Restricted Open-shell Hartree-Fock (ROHF)} do exist but we won't talk about it + \end{itemize} + \end{block} +\end{frame} + +\begin{frame}{RHF, ROHF and UHF} + \center + \includegraphics[width=0.4\textwidth]{fig/RHF_UHF} + \begin{itemize} + \item RHF = \orange{Restricted} Hartree-Fock + \item UHF = \red{Unrestricted} Hartree-Fock + \item ROHF = \blue{Restricted Open-shell} Hartree-Fock + \end{itemize} +\end{frame} + +%----------------------------------------------------- +%\subsection{UHF wave function} +%----------------------------------------------------- +%\begin{frame}{The UHF wave function} +% \begin{block}{Slater determinants for UHF} +% \small +% \begin{equation*} +% \Psi_\text{UHF}%(\br_1,\ldots,\br_N) +% = +% \underbrace{ +% \frac{1}{\sqrt{N^\alpha!}} +% \begin{vmatrix} +% \psi_1^\alpha(\br_1) & \cdots & \psi_{N^\alpha}^\alpha(\br_1) \\ +% \vdots & \ddots & \vdots \\ +% \psi_1^\alpha(\br_{N^\alpha}) & \cdots & \psi_{N^\alpha}^\alpha(\br_{N^\alpha}) \\ +% \end{vmatrix} +% }_{\orange{\Psi^\alpha(\br_1,\ldots,\br_{N^\alpha}) }} +% \underbrace{ +% \frac{1}{\sqrt{N^\beta!}} +% \begin{vmatrix} +% \psi_1^\beta(\br_{N^\alpha+1}) & \cdots & \psi_{N^\beta}^\beta(\br_{N^\alpha+1}) \\ +% \vdots & \ddots & \vdots \\ +% \psi_1^\beta(\br_{N}) & \cdots & \psi_{N^\beta}^\beta(\br_{N}) \\ +% \end{vmatrix} +% }_{\alert{\Psi^\beta(\br_{N^\alpha+1},\ldots,\br_{N}) }} +% \end{equation*} +% \normalsize +% \begin{itemize} +% \item The UHF wave function is \violet{a product of two determinants} +% \begin{itemize} +% \item One for the \orange{spin-up electrons} $\orange{\Psi^\alpha(\br_1,\ldots,\br_{N^\alpha}) }$ +% \item One for the \alert{spin-down electrons} $\alert{\Psi^\beta(\br_{N^\alpha+1},\ldots,\br_{N}) }$ +% \end{itemize} +% \item The \alert{Pauli exclusion principle} only requires the \blue{wave function to be antisymmetric wrt the exchange of two same-spin electrons} +% \end{itemize} +% \end{block} +%\end{frame} + +%----------------------------------------------------- +\subsection{UHF equations} +%----------------------------------------------------- +\begin{frame}{Unrestricted Hartree-Fock equations} + \begin{block}{UHF equations for unrestricted spin orbitals} + \bigskip + \violet{To minimize the UHF energy}, the unrestricted spin orbitals must be eigenvalues of the \blue{$\alpha$ and $\beta$ Fock operators}: + \begin{align} + & \boxed{ + \orange{f^\alpha(1) \, \psi_i^\alpha(1) = \varepsilon_i^\alpha \, \psi_j^\alpha(1)} + } + & + & \boxed{ + \alert{f^\beta(1) \, \psi_i^\beta(1) = \varepsilon_i^\beta \, \psi_j^\beta(1)} + } + \end{align} + where + \begin{align} + \orange{f^\alpha(1)} & = h(1) + \sum_{a}^{N^\alpha} [ \orange{J_a^\alpha(1) - K_a^\alpha(1)} ] + \sum_{a}^{N^\beta} \alert{J_a^\beta(1)} + \\ + \alert{f^\beta(1)} & = h(1) + \sum_{a}^{N^\beta} [ \alert{J_a^\beta(1) - K_a^\beta(1)} ] + \sum_{a}^{N^\alpha} \orange{J_a^\alpha(1)} + \end{align} + The \blue{Coulomb} and \violet{Exchange} operators are + \begin{align} + \blue{J_i^\sigma(1)} & = \int \psi_i^\sigma(2) r_{12}^{-1} \psi_i^\sigma(2) d\br_2 + & + \violet{K_i^\sigma(1)} \psi_j^\sigma(1) & = \qty[ \int \psi_i^\sigma(2) r_{12}^{-1} \psi_j^\sigma(2) d\br_2 ] \psi_i^\sigma(1) + \end{align} + \end{block} +\end{frame} + +\begin{frame}{Unrestricted Hartree-Fock equations (Take 2)} + \begin{block}{UHF energy} + \bigskip + The UHF energy is composed by three contributions: + \begin{equation} + E_\text{UHF} = \orange{E_\text{UHF}^{\alpha\alpha}} + \red{E_\text{UHF}^{\beta\beta}} + \violet{E_\text{UHF}^{\alpha\beta}} + \end{equation} + which yields + \begin{equation} + \small \boxed{ + E_\text{UHF} = + \orange{\sum_{a}^{N^\alpha} h_i^\alpha + + \frac{1}{2} \sum_{ab}^{N^\alpha} (J_{ab}^{\alpha\alpha} - K_{ab}^{\alpha\alpha})} + + \red{\sum_{a}^{N^\beta} h_a^\beta + + \frac{1}{2} \sum_{ab}^{N^\beta} (J_{ab}^{\beta\beta} - K_{ab}^{\beta\beta})} + + \violet{\sum_{a}^{N^\alpha} \sum_{b}^{N^\beta} J_{ab}^{\alpha\beta}} + } + \end{equation} + The matrix elements are given by + \begin{align} + h_i^\sigma & = \mel{ \psi_i^\sigma }{ h }{ \psi_i^\sigma } + & + J_{ij}^{\sigma\sigma'} & = \braket{ \psi_i^\sigma \psi_j^{\sigma'} }{ \psi_i^\sigma \psi_j^{\sigma'} } + & + K_{ij}^{\sigma\sigma} & = \braket{ \psi_i^\sigma \psi_j^\sigma }{ \psi_j^\sigma \psi_j^\sigma } + \end{align} + Note that \blue{$K_{ij}^{\alpha\beta} = 0$} $\Leftrightarrow$ \alert{there is no exchange between opposite-spin electrons} + \end{block} +\end{frame} + +\begin{frame}{UHF energy of the \ce{Li} atom} + \begin{block}{Problem} + \violet{\textit{``Write down the UHF energy of the doublet state of the lithium atom''}} + \end{block} + \pause + \begin{block}{Solution} +% The UHF wave function for the doublet state of \ce{Li} is +% \begin{equation*} +% \Psi_\text{UHF}(\br_1,\br_2,\br_3) +% = \frac{1}{\sqrt{2}} +% \begin{vmatrix} +% \psi_1^\alpha(\br_1) & \psi_2^\alpha(\br_1) \\ +% \psi_1^\alpha(\br_2) & \psi_2^\alpha(\br_2) \\ +% \end{vmatrix} +% \psi_1^\beta(\br_3) +% \end{equation*} +% while the corresponding energy is + \begin{equation*} + E_\text{UHF} = h_1^\alpha + h_1^\beta + h_2^\alpha + J_{12}^{\alpha\alpha} - K_{12}^{\alpha\alpha} + J_{11}^{\alpha\beta} + J_{21}^{\alpha\beta} + \end{equation*} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{Pople-Nesbet} +%----------------------------------------------------- +\begin{frame}{The Pople-Nesbet Equations} + \small + \begin{block}{Expansion of the unrestricted spin orbitals in a basis} + \begin{align} + \psi_i^\alpha (\br) & = \sum_{\mu=1}^K \blue{C_{\mu i}^\alpha} \, \phi_{\mu} (\br) + & + \psi_i^\beta (\br) & = \sum_{\mu=1}^K \violet{C_{\mu i}^\beta} \, \phi_{\mu} (\br) + \end{align} + \end{block} + \begin{block}{The Pople-Nesbet equations} + \begin{align} + \orange{\bF^\alpha} \cdot \blue{\bC^\alpha} & = \bS \cdot \blue{\bC^\alpha} \cdot \bE^\alpha + & + \red{\bF^\beta} \cdot \violet{\bC^\beta} & = \bS \cdot \violet{\bC^\beta} \cdot \bE^\beta + \end{align} + \begin{gather} + \orange{F_{\mu \nu}^\alpha} + = H_{\mu \nu} + + \sum_{\lambda \sigma} \blue{P_{\lambda \sigma}^\alpha} [ (\mu \nu | \sigma \lambda) - (\mu \lambda | \sigma \nu) ] + + \sum_{\lambda \sigma} \violet{P_{\lambda \sigma}^\beta} (\mu \nu | \sigma \lambda) + \\ + \red{F_{\mu \nu}^\beta} + = H_{\mu \nu} + + \sum_{\lambda \sigma} \violet{P_{\lambda \sigma}^\beta} [ (\mu \nu | \sigma \lambda) - (\mu \lambda | \sigma \nu) ] + + \sum_{\lambda \sigma} \blue{P_{\lambda \sigma}^\alpha} (\mu \nu | \sigma \lambda) + \end{gather} + $\orange{\bF^\alpha}$ and $\red{\bF^\beta}$ are both functions of $\blue{\bC^\alpha}$ and $\violet{\bC^\beta}$ + $\Rightarrow$ \alert{There's a coupling between $\alpha$ and $\beta$ MOs!} + \end{block} +\end{frame} + +\begin{frame}{Unrestricted Density Matrices} + \begin{block}{Spin-up and spin-down density matrices} + \begin{align} + &\boxed{ + \orange{P_{\mu \nu}^\alpha} = \sum_{a=1}^{N^\alpha} C_{\mu a}^\alpha C_{\nu a}^\alpha + \quad \Leftrightarrow \quad \orange{\bP^\alpha} + } + & + &\boxed{ + \alert{P_{\mu \nu}^\beta} = \sum_{ a=1}^{N^\beta} C_{\mu a}^\beta C_{\nu a}^\beta + \quad \Leftrightarrow \quad \alert{\bP^\beta} + } + \end{align} + \end{block} + \begin{block}{Properties of the density $(\sigma = \alpha \text{ or } \beta)$} + \begin{align} + \rho^\sigma(\br) & = \sum_{\mu \nu} \phi_{\mu}(\br) P_{\mu \nu}^\sigma \phi_{\nu}(\br) + & + \int \rho^\sigma(\br) d\br & = N^\sigma + \end{align} + \end{block} + \begin{block}{Total and Spin density matrices} + \begin{align} + \underbrace{\blue{\bP^\text{T}}}_{\blue{\text{Charge density}}} & = \orange{\bP^\alpha} + \alert{\bP^\beta} + & + \underbrace{\violet{\bP^\text{S}}}_{\violet{\text{Spin density}}} & = \orange{\bP^\alpha} - \alert{\bP^\beta} + \end{align} + \end{block} +\end{frame} + +%----------------------------------------------------- +\subsection{SCF for UHF} +%----------------------------------------------------- +\begin{frame}{How to perform a UHF calculation in practice?} + \begin{block}{The SCF algorithm} + \begin{enumerate} + \item \orange{Specify molecule} $\{\bR_A\}$ and $\{Z_A\}$ and \violet{basis set} $\{\phi_\mu\}$ \alert{(same as RHF)} + \item Calculate integrals $S_{\mu \nu}$, $H_{\mu \nu}$ and $\braket{ \mu \nu }{ \lambda \sigma }$ \alert{(same as RHF)} + \item Diagonalize $\bS$ and compute $\bX$ \alert{(same as RHF)} + \item Obtain \alert{guess density matrix} for $\bP^\alpha$ and $\bP^\beta$ + \begin{enumerate} + \item[1a.] Calculate $\bG^\alpha$ and then $\bF^\alpha = \bH + \bG^\alpha$ + \item[1b.] Calculate $\bG^\beta$ and then $\bF^\beta = \bH + \bG^\beta$ + \item[2.] Compute $(\bF^\alpha)' = \bX^\dag \cdot \bF^\alpha \cdot \bX$ and $(\bF^\beta)' = \bX^\dag \cdot \bF^\beta \cdot \bX$ + \item[3a.] Diagonalize $(\bF^\alpha)'$ to obtain $(\bC^\alpha)'$ and $\bE^\alpha$ + \item[3b.] Diagonalize $(\bF^\beta)'$ to obtain $(\bC^\beta)'$ and $\bE^\beta$ + \item[4.] Calculate $\bC^\alpha= \bX \cdot (\bC^\alpha)'$ and $\bC^\beta= \bX \cdot (\bC^\beta)'$ + \item[5.] Form the new \blue{new density matrix} $\bP^\alpha$ and $\bP^\beta$, and compute $\bP^\text{T} = \bP^\alpha + \bP^\beta$ + \item[6.] \alert{Am I converged?} If not go back to 1. + \end{enumerate} + \item Calculate stuff that you want, like $E_\text{UHF}$ for example + \end{enumerate} + \end{block} +\end{frame} + +%----------------------------------------------------- +\section{Books} +%----------------------------------------------------- +\begin{frame}{Good books} + \begin{columns} + \begin{column}{0.7\textwidth} + \begin{itemize} + \item Introduction to Computational Chemistry (Jensen) + \\ + \vspace{1cm} + \item Essentials of Computational Chemistry (Cramer) + \\ + \vspace{1cm} + \item Modern Quantum Chemistry (Szabo \& Ostlund) + \\ + \vspace{1cm} + \item Molecular Electronic Structure Theory (Helgaker, Jorgensen \& Olsen) + \\ + \vspace{1cm} + \end{itemize} + \end{column} + \begin{column}{0.3\textwidth} + \centering + \includegraphics[height=0.3\textwidth]{fig/Jensen} + \\ + \bigskip + \includegraphics[height=0.3\textwidth]{fig/Cramer} + \\ + \bigskip + \includegraphics[height=0.3\textwidth]{fig/Szabo} + \\ + \bigskip + \includegraphics[height=0.3\textwidth]{fig/Helgaker} + \end{column} + \end{columns} 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