\section{Theoretical Model}
\label{sec:section2}
\subsection{Setup}\label{sec:section2.1}
In this section, I establish the theoretical framework for the model developed throughout this paper. Fix a non-atomic continuum of male and female agents and consider the dynamic two-sided market formed by the SBDP, which agents can join to search for potential romantic partners.
For ease of exposition, I assume that this market is heteronormative such that male agents search exclusively for female agents and vice-versa.
Time is discrete and indexed by $t=0, 1, 2, ...$ over an infinite horizon. Each period, masses $\lambda_m, \lambda_w>0$ of new men and women enter the platform, where agents are then paired and presented a candidate partner from the opposite side of the market.

We model agents with heterogeneous preferences (capturing the notion that `beauty lies in the eye of the beholder') and thus, after being paired, each agent observes an \textit{idiosyncratic attractiveness value} $\theta \in \Theta := [0,1]$ for their candidate.
These values are i.i.d according to a pair of absolutely continuous CDF's, $F_m$ and $F_w$, with corresponding PDF's $f_m,f_w$.
Female agents draw male candidate values from $F_m$ and vice versa but, importantly, the value man $i$ draws for woman $j$ does not necessarily equal the value that $j$ draws for $i$ and, for simplicity, these are modelled as independent from one another.

After observing their candidate's attractiveness, agents then choose to swipe left (dislike) or right (like) on them, yielding an action space $\mathcal{A}=\{ \text{Swipe Left}, \text{Swipe Right}\}$.
If either agent swipes left, then they both receive a payoff of zero; however, if both agents swipe right, they are said to have \textit{matched} and both receive a matching payoff $u(\theta)$, where $u(\cdot)$ is a continuous, strictly increasing function that satisfies $u(0) = 0$.
This last property stems from the fact that, in most SBDPs, users are allowed to unmatch with each other, and therefore matching with even the least attractive individual on the other side of the market is weakly preferred to not matching.

After payoffs have been received, agents are paired with different candidates and the stage interaction is repeated. Given the continuum of agents, I assume that interactions take place \textit{anonymously} in the style of \cite{jovanovic1988anonymous}. Furthermore, to the agents' knowledge, pairings are determined in an unknown manner (since SBDPs are generally secretive regarding the algorithms they use), effectively making their problem one of uniform random search.

\begin{figure}[ht]
    \centering
    \caption{Sequence of events within each time period}
    \vspace{20pt}
        \begin{tikzpicture}
            % draw horizontal line
        \draw[thick, -Triangle] (0,0) -- (\ImageWidth,0); %node[font=\scriptsize,below left=3pt and -8pt]{$t+1$};

        % draw vertical lines
        \foreach \x in {1,2,...,6}
        \draw (\x*2 cm,4pt) -- (\x*2 cm,-4pt);

        \foreach \x/\descr in {2/t, 4/\text{Arrivals}, 6/\text{Pairings}, 8/\text{Game Play}, 10/\text{Departures}, 12/t+1}
        \node[font=\scriptsize, text height=1.75ex,
        text depth=.5ex] at (\x,-.3) {$\descr$};
    \end{tikzpicture}
    \label{fig:timeline}
\end{figure}

Since swiping right in the above stage game is weakly dominant for all agents, one would not expect the repeated interaction to be massively revealing of real-life preferences, where initiating a romanic encounter is often perceived by humans as a costly endeavour \citep{dawkins2017selfish}. This becomes problematic since the main selling point of SBDPs is a reduction in searching costs for individuals seeking romantic encounters, and this is only accomplished if matches have a high likelihood of resulting in real-life romantic attraction. Because of this, SBDPs like Tinder place a cap on the total number of right swipes for each user, thus enabling this as a form of costly signalling. I refer to the total number of right-swipes a user has left as its \textit{budget}, $b$, which evolves dynamically according to the law of motion:
\begin{equation*}
  b_{t+1}= b_{t} - \mathbbm{1}\{a_t=\text{Swipe Right}\}
\end{equation*}

The budget sets for men and women are thus defined by $\mathcal{B}_{s}=\{b \in \mathbb{Z} : 1\leq b \leq B_s\}$, for each sex $s=m,w$, with budget caps $B_m$ and $B_w$ determined exogenously. Importantly, agents depart from the platform in one of two ways: they can leave \textit{endogenously}, if they expend their swiping budget, or \textit{exogenously} with probability $(1-\delta)$ in each time period. This admits to the interpretation of a geometrically distributed lifetime, such that agents discount future payments by a factor of $\delta\in(0,1)$.

One final remark for this setup is that, in a continuum market with anonymous interactions, the mean-field assumption established in \autoref{sec:section2.3} effectively restricts focus onto the set of (pure) symmetric stationary strategies. This is argued in more detail in the following sections but, for now, denote these strategies by functions $\mu: \Theta \times\mathcal{B}_m\rightarrow \mathcal{A}$ for men and $\omega:\Theta \times\mathcal{B}_w\rightarrow \mathcal{A}$ for women.

\subsection{The Dating Market}\label{sec:section2.2}
With the above framework in place, I now outline the platform state variables that make up the SBDP market.
Let $N_{mt}(b), N_{wt}(b)$ denote the mass of male and female agents (respectively) with a budget of $b\in\mathcal{B}$ in a given time period $t$.
%Additionally, with a slight abuse of notation, let $N_{mt}, N_{wt}$ denote the overall mass of male and female agents (respectively) in the platform, i.e. $\sum_{b\in\mathcal{B}_m}N_{mt}(b)$ and $\sum_{b\in\mathcal{B}_w}N_{wt}(b)$.
Since gender imbalances can leave some agents in the long side of the market unpaired, a pairings process must also be determined.
Given the automated nature of SBDPs, I assume an efficient matching technology and model pairings as a Bernoulli process parametrised by market tightness; thus, the probability of being paired with a candidate is defined for both sides as:
\begin{equation*}
    \tau_{mt}:=\min \left\{\frac{\sum_{b\in\mathcal{B}_w}N_{wt}(b)}{\sum_{b\in\mathcal{B}_m}N_{mt}(b)} , 1 \right\},
    \quad \tau_{wt}:= \left(\frac{\sum_{b\in\mathcal{B}_m}N_{mt}(b)}{\sum_{b\in\mathcal{B}_w}N_{wt}(b)} \right) \tau_{mt}
\end{equation*}

From the above, the platform state in time period $t$ can be defined as $\Psi_t=(N_{mt},N_{wt})$.
For most of this paper, I focus on characterising user behaviour and its resulting implications in a stationary setting (which is denoted by omitted time subscripts), although some discussion of coupled strategy and market dynamics is provided in \autoref{sec:section4}.
As a necessary requirement, the market steady state $\Psi_t=\Psi_{t+1}=...=\Psi$ must satisfy the balanced flow conditions \footnote{Formally, these conditions rely on the exact law of large numbers, which was rigorously developed in discrete-time settings by \cite{duffie2018dynamic}, but a technical discussion of this lies outside the scope of this paper.} for our continuum model; these are presented below for the female agents, but they apply analogously to the male side of the market.
Firstly, the entry flow of agents into the platform must equal the departure flow:
\begin{equation}\label{eq:ss1}
    \lambda_w\;=\; \underbrace{ (1-\delta)\sum_{b\in\mathcal{B}_w}N_{wt}(b)}_{\text{Exogenous Outflow}} \;+\; \underbrace{N_w(1) \delta \tau_w\int_{\Theta}\omega(\theta,1)\,dF_{m}(\theta)}_{\text{Endogenous Outflow}}
\end{equation}

Secondly, for both sides, the flow of agents into any particular budget level must equal the outflow of agents from that same level. Thus, for all $b\in\mathcal{B}_w$:
\begin{equation}\label{eq:ss2}
    \underbrace{N_w(b+1) \delta \tau_w \int_{\Theta} \omega(\theta,b+1)\,dF_{m}(\theta)}_{\text{Inflow into $b$}} \;=\; \underbrace{N_w(b) \Big[ (1-\delta) \;+\; \delta \tau_w\int_{\Theta} \omega(\theta,b)\,dF_{m}(\theta)\Big]}_{\text{Outflow from $b$}}
\end{equation}

Finally, the entry flow of agents into the platform must equal the outflow from the top budget level, hence:
\begin{equation}\label{eq:ss3}
    \lambda_w \;=\; \underbrace{N_w(B_w) \Big[ (1-\delta) \;+\; \tau_w \delta \int_{\Theta} \omega(\theta,B_w)\,dF_{m}(\theta) \Big]}_{\text{Outflow from $B_w$}}
\end{equation}

Importantly, the above conditions take the strategy profile $(\mu,\omega)$ as exogenously fixed. Over the next section, these are endogenously derived by characterising the agents’ best-responses in an SDBP setting, which themselves depend on the market steady state $\Psi$.

\subsection{The Search Problem}\label{sec:section2.3}
With the model framework outlined above, I now present the decision problem faced by female agents in the market given some platform steady state $\Psi$, with analogous results and implications for the male side.
Consider now a woman $i$ who is paired with a man $j$ in the platform. Since $j$'s swiping behaviour will depend on his own budget, which is unknown to woman $i$, then (under strict rationality) she would have to compute her expected payoff conditional on her beliefs for the market history.

This behaviour is both unreasonable and intractable in an SBDP setting, as explained in \autoref{sec:section1.1}; instead, given a stationary platform and a continuum of anonymous agents, it is reasonable to conjecture that the average swiping rate for men, $\overline\mu$, is also stationary, and that any individual agent's actions have a negligible effect on the platform state dynamics\footnote{I also assume that $\overline\mu>0$ to prune out degenerate equilibria where one side never swipes right.}.
As such, the expected ex-interim payoff for woman $i$ is the following:
\begin{equation*}
    \begin{aligned}
        U(\theta, a)&=\Big(\mathbbm{1}\{a=\text{Swipe Right}\}\Big)\,\overline{\mu}u(\theta)%, \\[8pt] \quad \text{where}\quad \overline{\mu} &= \sum_{b\in \mathcal{B}_m}\int_{\Theta}\frac{{N}_m(b)}{N_m}  \, \mu(\theta',b)\,dF_w(\theta')
    \end{aligned}
\end{equation*}

The above imposes a mean-field assumption, such that woman $i$ accounts for $j$'s behaviour only through $\overline\mu$ conditional on the platform state $\Psi$.
This modelling choice, which has been employed by \cite{immorlica2021designing} and \cite{iyer2014mean} among others, simplifies the full dynamic game by collapsing it onto a pair of Markov Decision Processes (MDPs), one for each side of the market, such that strategy $\omega$ is a best-response for women iff it is an optimal policy for the corresponding MDP.
Additionally, this also condenses $i$'s state of payoff-dependent variables to include only the value $\theta$ that she observes for $j$ and her own budget $b$.
Therefore, since all agents in the same side solve the same MDP, which depends only on their individual state, this effectively justifies the restricted focus on symmetric stationary strategies, as established in \autoref{sec:section2.1}.

Let the unrealized jump times of the pairing process for woman $i$ be indexed by $k$.
Given that, at the time of pairing, this women has a budget of $b$ right swipes left, she then solves the constrained MDP presented below, captured by the value function $V_w(\theta,b)$:
\begin{equation*}
    \begin{aligned}
        V_w(\theta,b)=\max_{\{a_k\}^\infty_{k=0}} \quad & \mathbb{E}_{\theta}\left[\sum^\infty_{k=0} \delta^{k} U(\theta_k, a_k) \;|\; \theta_0=\theta, b_0=b\right]\\
        \textrm{s.t.} \quad & b_{k+1} = b_k - \mathbbm{1}\{a_k=\text{Swipe Right}\} \\
        & b_k\in \mathcal{B}_w \cup \{0\},\\
        & a_k\in \mathcal{A}
    \end{aligned}
\end{equation*}

Importantly, the first two constraints, along with the exogenous departure process, make this problem non-trivial: by limiting woman $i$'s right-swiping budget, the platform imposes an opportunity cost for swiping right on $j$ and foregoing potential future matches with more attractive men, whilst the exogenous departure process removes the possibility of simply waiting around to swipe right on the top-$B_w$ most attractive men in the platform.
By standard dynamic programming arguments, this problem can be captured by two Bellman equations; one for when $j$ is paired and another for when she isn't:
\begin{align}
    \begin{split}
        V^{P}_w(\theta,b) &=\max \Big\{ \, \overline{\mu}\, u(\theta) \;+\; \delta \tau_w \,\mathbb{E}_\theta \Big[V^P_w(\theta', b-1)\Big] \;+\; \delta (1-\tau)V^{NP}_w(b-1) \, ,\\[6pt]
        & \quad\quad\quad\quad\;\delta \tau_w \, \mathbb{E}_\theta\Big[ V^P_w(\theta', b) \Big] \;+\; \delta (1-\tau_w) V^{NP}_w(b) \, \Big\}
    \end{split}\\[10pt]
    \begin{split}
        V^{NP}_w(b) &= {}\delta \tau_w \,\mathbb{E}_\theta \Big[ V^P_w(\theta', b)\Big] \,+\, \delta (1-\tau_w) V^{NP}_w(b)
    \end{split}
\end{align}

With some straightforward algebra, the above two equations can be merged into the full Bellman equation below. Note that, by imposing the swiping budget constraint from the above MDP, it must be the case that $V_w(\theta, 0)=0$ for all $\theta \in \Theta$, since agents who expend their budget must leave the platform and can't accumulate any additional payoffs:
\begin{equation}\label{eq:full bellman}
    \begin{aligned}
        V_w(\theta,b) \;=\;&\max\left\{\,\overline{\mu} \, u(\theta) +\alpha \,\mathbb{E}_\theta \Big[V_w(\theta', b-1)\Big]\,,\; \alpha\,\mathbb{E}_\theta \Big[ V_w(\theta', b)\Big]\,\right\}
    \end{aligned}
\end{equation}

Here, $\alpha$ is the effective discount rate accounting for the exogenous possibilities of both departures and pairings, defined as:
\begin{equation*}
\alpha:=\frac{\tau_w\delta}{1-\delta(1-\tau_w)}.
\end{equation*}

Upon inspection, it is clear that the value function is of a piecewise nature over $\Theta$. This is formally stated below (with the corresponding derivation included in \autoref{appx: b}):
\begin{proposition}\label{prop:piecewiseV}
Fix some $b\in\mathcal{B}_w$. Then there exists some unique reservation value $\widetilde{\omega}_b\in \Theta$ such that $V_w(\theta,b)$ admits the following piecewise form over $\Theta$:
\begin{equation*}
    \begin{split}
        V_w(\theta,b)=\begin{cases}
            \overline\mu u(\theta) +\alpha \,\mathbb{E}_{\theta}\Big[V_w(\theta', b-1)\Big],& \theta \geq \widetilde \omega_b \\[10pt]
            \alpha \,\mathbb{E}_{\theta}\Big[V_w(\theta', b)\Big],& \theta\leq\widetilde \omega_b
        \end{cases}\\[8pt]
    \end{split}
\end{equation*}
\end{proposition}

In the result above, all reservation values $\{\widetilde\omega\}_{b\in \mathcal{B}_w}$ are such that woman $i$ is indifferent between swiping left or right. Therefore, an optimal policy for woman $i$'s MDP involves swiping right for partners who exceed the reservation value for her current budget:
\begin{corollary}\label{cor:optpolicy}
    The following threshold policy $\widetilde\omega$, parametrised by $\{\widetilde\omega\}_{b\in \mathcal{B}_w}$, attains $V_w(\theta,b)$:
    \begin{equation*}
        \begin{split}
            \widetilde\omega(\theta,b)&=\begin{cases}
                \text{Swipe Right},\quad \theta\geq \widetilde{\omega}_b \\
                \text{Swipe Left}, \quad\theta< \widetilde\omega _b
            \end{cases}
        \end{split}
    \end{equation*}
\end{corollary}

Using both of these results, I now derive the following explicit characterisation for reservation values $\{\widetilde\omega_b\}_{b\in \mathcal{B}_w}$ (with a corresponding proof included in \autoref{appx: b}).
\begin{proposition}\label{prop:recurrence relation}
The set of reservation values for women, $\{\widetilde\omega_b\}_{b\in \mathcal{B}_w}$, uniquely satisfies the recurrence relation and initial condition below, over the budget set $\mathcal{B}_w$:
\begin{equation}\label{eq:recurrence relation}
    \begin{aligned}
        u(\widetilde \omega_b) \;=\; \alpha u(\widetilde \omega_b) F_m(\widetilde \omega_b) \;+\; \alpha u(\widetilde \omega_{b-1}) \Big[1- F_m(\widetilde \omega_{b-1})\Big] \;+\; \int^{\widetilde \omega_{b-1}}_{\widetilde \omega_b} \alpha u(\theta')\,dF_m(\theta')
    \end{aligned}
\end{equation}
\begin{equation}\label{eq:initial condition}
    u(\widetilde\omega_1) \;=\; \alpha u(\widetilde\omega_1)F(\widetilde\omega_1) \;+\; \alpha \int^1_{\widetilde\omega_1}u(\theta')dF(\theta')
\end{equation}
\end{proposition}

This explicit characterisation allows for an improved computation of $\{\widetilde\omega_b\}_{b\in \mathcal{B}_w}$ over traditional numerical methods such as value iteration \citep{bellman2015applied}, which has a runtime complexity of $\mathcal{O}(|\mathcal{B}_w\times\Theta|^2|\mathcal{A}|)$ and can become expensive for large budget sets or fine discretisations of $\Theta$. Using the recurrence relation in \autoref{prop:recurrence relation}, the optimal policy was computed for an arbitrary set of exogenous parameters, with results in \autoref{fig:swiping-rule} showing a clear cut-off rule for swiping right.
These cut-off values are decreasing in the agent's budget, which captures the notion that an agent's current swipe is more valuable than all preceding ones given the increasing opportunity cost.

\begin{figure}[ht]
    \centering
    \caption{The Optimal Swiping Rule}
    \includegraphics{swiping-rule.png}
    \label{fig:swiping-rule}
\end{figure}


\documentclass{memoir}
\usepackage{notestemplate}

%\logo{~/School-Work/Auxiliary-Files/resources/png/logo.png}
%\institute{Rice University}
%\faculty{Faculty of Whatever Sciences}
%\department{Department of Mathematics}
%\title{Class Notes}
%\subtitle{Based on MATH xxx}
%\author{\textit{Author}\\Gabriel \textsc{Gress}}
%\supervisor{Linus \textsc{Torvalds}}
%\context{Well, I was bored...}
%\date{\today}

\begin{document}

% \maketitle

% Notes taken on whenever

\section{Algebraic Extensions}
\label{sec:algebraic_extensions}

\begin{thm}[Tower Theorem]
	Let \(F \hookrightarrow E \hookrightarrow K\) be a composition of field extensions. Then \([K:F] = [K:E] [E:F]\).
\end{thm}
One can show this via vector space arguments (look at the bases of the spaces).
\begin{cor}
	If \(K / F\) is a finite extension, and \(E\) is a subfield of \(K\) containing \(F\), then \([E:F] \mid [K:F] \).
\end{cor}

\begin{exmp}
	Let
\begin{align*}
	K &= \Q(\sqrt[6]{2} )\\
	E &= \Q(\sqrt{2} )\\
	F &= \Q
\end{align*} It follow directly from previous work that \([\Q(\sqrt[6]{2}) : \Q] = 6\) and \([\Q(\sqrt{2} ):\Q] = 2\). As for \(K / E\), the minimal polynomial is \(x^3-\sqrt{2} \), which gives \([\Q(\sqrt[6]{2} ): \Q(\sqrt{2} )] = 3\), which corresponds to what the tower theorem gives us.
\end{exmp}

\begin{defn}
	An extension \(K / F\) is called \textbf{finitely generated} if there exist elements \(\alpha_1, \alpha_2,\ldots,\alpha_n\) such that
	\begin{align*}
		K = F(\alpha_1, \alpha_2, \ldots, \alpha_n) \quad \text{for }n<\infty
	\end{align*}
	Such an extension can be obtained recursively via simple extensions.
\end{defn}
	We have that \(F(\alpha ,\beta ) = (F(\alpha ))(\beta )\), hence the definition above is consistent.
\begin{exmp}
	\begin{itemize}
		\item \(\Q(\sqrt[6]{2} ,\sqrt{2} ) = \left( \Q(\sqrt[6]{2} ) \right) (\sqrt{2} ) = \Q(\sqrt[6]{2} )\) because \(\sqrt{2} = (\sqrt[6]{2} )^3\).
		\item One can check that \(\Q(\sqrt{2} ,\sqrt{3} )\) is a proper field extension for both \(\Q(\sqrt{2} )\) and \(\Q(\sqrt{3} )\).
	\end{itemize}
\end{exmp}
\begin{thm}
	\(K / F\) is finite if and only if \(K\) is generated by a finite number of algebraic elements over \(F\).
\end{thm}
We denote by \(\overline{\Q}\) the subfield of \(\C\) generated by all algebraic elements of \(\C\) over \(\Q\). \(\overline{\Q}\) is an infinite algebraic extension of \(\Q\), and referred to as the \textbf{field of algebraic numbers}. \\

\begin{thm}
	If \(E / F\) and \(K / E\) are algebraic, then \(K / F\) is algebraic.
\end{thm}
\end{document}

Abstract Algebra - Introductory/Algebra II/Notes/source/Lecture17 - TowerThm.tex


\section*{Research question and Methodology}

Language models can be used for a variety of purposes, such as: speech recognition,
spelling correction, grammar correction and automatic translation.
All these applications have the task of assigning a probability to a
sequence of words, based on the number of times they appear in one or more
documents. As briefly mentioned in the introductory chapter, the purpose
of the following project is to verify the existence of a further method, which
makes use of the concept of language model, specifically an \emph{n-grams} model,
capable of expanding a query. Achieving this goal means solving the problem
of mismatch between the terms present in the query and those present in a
corpus of documents. The probability of generating a new \emph{q} query given the
estimate of a Language Model for a \emph{D} document can only occur through a
ranking of relevant documents. If the corpus of documents is large, thinking
of generating \emph{n} Language models, with \emph{n} equal to the number of documents,
turns out to be a computationally expensive operation. This paper has used
a useful approach to be able to generate a first ranking of documents ordered
by relevance with the query \emph{q}. The technique applied is the \emph{tf-idf} recovery
model. The adoption of this method of weighing the terms, as well as being
widely used in the state of the art, produces excellent results. The introduction
of the \emph{LM}, and of other semantic analysis techniques, made it possible
to outperform the performance of the \emph{tf-idf} baseline, generating ranking,
starting from the latter, of documents more pertinent to the query \emph{q} \cite{09}. In
the following paper, tests are carried out which certify the veracity of this
thesis. It should be noted that the generation of the ranking, obtained from
the weights of the \emph{tf-idf} method, is obtained using the well-known \emph{cosine
similarity} metric between the weight vectors. To comply with the set objective,
the position of the relevant target document, that is the document
that the user is searching for, was kept track. This was possible because a
query was chosen, among those available, that was close to the title of
this document. The score assigned to the target document will represent the
minimum \emph{threshold} to be able to form the new ranking of documents. This step
is fundamental as, by setting a higher threshold, the target document would
be lost in subsequent calculations. By setting a lower threshold, however,
those documents that represent noise will be taken into account. From this
ranking, the \emph{LMs} for each document will be calculated \cite{10}, generating \emph{n} LMs,
with \emph{n} equal to the number of relevant documents, based on the terms in the
query. It should be noted that, in order to carry out this step, the concept
of \emph{skip-gram} has been applied to the query, with step \emph{s} equal to two. The
creation of each LM is possible only after using one of the existing smoothing
techniques. To prevent a linguistic model from assigning zero probability to
an invisible event, i.e. when a term present in the query is not present in
an LM, we should eliminate some probability mass from some more frequent
events and give it to events that we haven't never seen. There
are a variety of methods for smoothing, some of these are: \emph{Laplace (add-
one) smoothing}, \emph{Linear Interpolation smoothing}, \emph{add-k smoothing}, \emph{back-off
smoothing} and \emph{Kneser-Ney smoothing}. Among these, the following paper
has experimented the use of the first two smoothing methods, each of which
will produce different results useful for achieving the final goal. The core of
the algorithm lies in being able to derive the best ranking of relevant documents,
using the initial query, through an iterative process, as s changes.
This variation will lead to the generation of several LMs, each with step \emph{s},
with $s=\{2,3,\ldots,10\}$. At each iteration, a new ranking of documents will be generated, thanks to the calculation of the Maximum Likelihood Estimation \emph{MLE}, between the query \emph{q} and each document \emph{d}(\ref{MLE}).
\begin{eqnarray}\label{MLE}
    P(q|d) & \approx & P(q|M_d) \nonumber \\
           & \approx & \prod_i^n{P(w_i|w_{i-1})} \nonumber \\
           & \approx & \frac{count(w_i,w_{i-1})}{\sum_{j=1}^n count(w_j,w_{i-1})} \nonumber \\
           & = & \frac{count(w_i, w_{i-1})}{count(w_{i-1})}
\end{eqnarray}
Among all these nine rankings, after being sorted, the one that will have the
target document in the highest position, close to the first, will be chosen.
Also thanks to this, it will be possible to determine the lambda parameters
present in the interpolation smoothing technique. By focusing on the latter
technique, it was necessary to implement the concept suggested by \cite{11} as
regards the calculation of linear interpolation. Instead of adding one to
the probability calculation, as is well known in Laplace's smoothing (\ref{Laplace}), linear
interpolation, recursively, calculates the MLE of order two (\emph{bi-grams}) (\ref{LinInt}), up to
the order zero (\emph{zero-grams})(\ref{zero-grams}).
\begin{equation}\label{Laplace}
    P(w_i|w_{i-1}) = \frac{count(w_{i-1})+1}{count(w_{i-1})+|V|}
\end{equation}
\begin{equation}\label{LinInt}
    P(w_i|w_{i-1}) = \lambda P(q|M_d) + (1-\lambda)P(q|M_c)
\end{equation}
\begin{equation}\label{zero-grams}
    P(w_i) = \lambda \frac{1}{|V|} + (1-\lambda)P(w_i)
\end{equation}

where:
\begin{itemize}
    \item $\sum_i \lambda_i = 1$
    \item $M_d$: represents the language model of the single document;
    \item $M_c$: represents the language model of the entire collection of documents;
    \item $|V|$: represents the number of unique words within the corpus of documents.
\end{itemize}
It is always good to specify that, like the iterative process of steps \emph{s}, both $\lambda$
parameters also follow the same reasoning. The idea is to assign a range of
numbers $\lambda = \{0.1, 0.2,\ldots,1\}$ to both values. In both smoothing techniques,
the \emph{perplexity} level existing between the query word set $W = \{w_1, w_2,\ldots,w_N\}$ and the language model of
the target document present in the ranking of relevant documents returned
by the tf-idf model will be calculated(\ref{perplexity}) \cite{12}.
\begin{equation}\label{perplexity}
    pp(W) = \sqrt[N]{\prod_{i=1}^N\frac{1}{P(w_i|w_{i-1})}}
\end{equation}
After obtaining the best ranking, the next step is based on building a \emph{term-
term matrix} \cite{12}, where the terms in question are both those of the query and
those belonging to the language model of the entire collection of relevant
documents present in the ranking. Within this matrix, the numbers of co-occurrences
between all terms will be reported. On this type of matrix it is
possible to apply the calculation of \emph{Positive Pointwise Mutual Information
(PPMI)}. PPMI draws on the intuition that the best way to weigh the association
between two words is to ask how much more the two words co-occur in
our corpus than we would have a priori expected them to appear by chance.
This measure derives from the calculation of the standard PMI which represents
a measure of frequency between two events \emph{x} and \emph{y}, compared to what
we would expect if they were independent (\ref{pmi}).
\begin{equation}\label{pmi}
    \emph{pmi(x,y)} = \log_2\frac{P(x,y)}{P(x)P(y)}
\end{equation}
The ratio gives us an estimate of how much more the two words co-occur
than we expect by chance. PMI values can be positive, negative or infinite.
Negative values, which imply that events occur less often than we would
expect by chance, tend to be unreliable when we have documents consisting
of few terms, as in our case. To solve this problem, the calculation of the
PPMI is used which replaces negative values with zero(\ref{ppmi}).
\begin{equation}\label{ppmi}
    PPMI(x,y) = \max(\log_2\frac{P(x,y)}{P(x)P(y)}, 0)
\end{equation}
But the question is: why is the PPMI calculation used? The adoption of
this is useful for being able to calculate the similarity between words, i.e.
their synonymy, search for paraphrases, keep track of the change in meaning
of words and to automatically discover the meanings of words in different
corpora. To find the words most similar to those in the query, the cosine
similarity is calculated on the first ten word vectors that have the highest
positive PPMI values. Eventually, each token in the query will have a maximum
of ten expansion terms. Thanks to this, we are already seeing how
query expansion can be done. Moving on, there is a problem to solve before
we can calculate the similarity of the cosine: the \emph{high dimensionality}
of the matrix. In order to obtain a good similarity, relatively low at the
computational level, another concept has been implemented: \emph{Singular Value
Decomposition (SVD)}. The idea of applying SVD on a term-term matrix was
proposed by \cite{13}. Switching from sparse to dense vectors allows for better
similarity comparisons. The SVD allows to decompose the term-term matrix
(A), of dimensions $txd$, into three matrices \cite{14} (\ref{decomposition}):
\begin{equation}\label{decomposition}
    A = USV^T
\end{equation}
Where:
\begin{enumerate}
    \item \emph{U}: matrix of dimension $txm$ where the columns represent the left
    singular vectors of matrix A;
    \item \emph{S}: diagonal matrix of dimension $mxm$, containing the singular values
    of matrix A;
    \item \emph{$V^T$}: transposed matrix, of dimensions $mxd$, where the columns represent
    the right singular vectors of matrix A.
\end{enumerate}
Thanks to the product of the matrix \emph{U} with the matrix \emph{S}, a new matrix
\emph{$\mathcal{D}$} (\ref{matrixD}), of dimension $txm$, is formed containing all the terms that will be
compared, through cosine similarity, with the query terms present in the
matrix \emph{$\mathcal{T}$} (\ref{matrixT}) given by the product of \emph{S} and $V^T$ .
\begin{equation}\label{matrixD}
    \mathcal{D} = U*S
\end{equation}
\begin{equation}\label{matrixT}
    \mathcal{T} = S*V^T
\end{equation}
Having all the possible words available to be added to the query tokens, the
time has come to generate all the possible queries. The number of queries
produced will depend on the number of words present in the initial query and
on any words obtained from the calculation of the cosine similarity. Having
set a maximum limit of ten words per token, the total of the quries generated
is given by the product (\ref{possQueries}) of the number of possible pairs of terms:
\begin{equation}\label{possQueries}
    \#Queries = \prod_i^q (\#sim\_words_i)
\end{equation}
where \emph{i} represents the single word taken in the query and \emph{q} represents the
total of the words present in the starting query. Being new queries, these
will be subjected to the same procedure as the original query, that is, new
LMs will be produced and with them new rankings of documents that will
contain the target document in the first positions. Finally, the best query,
or queries, will be chosen based on the level of perplexity reached.

chapters/Research question and methodology.tex


%\documentclass[UTF8]{ctexart} % use larger type; default would be 10pt
\documentclass[a4paper]{article}
\usepackage{../mqc}

\title{\textbf{Modern Quantum Chemistry, Szabo \& Ostlund}\\HW}
\author{wsr
\vspace{5pt}\\
}
\date{\today} % Activate to display a given date or no date (if empty),
         % otherwise the current date is printed

\begin{document}
% \boldmath

\maketitle

\tableofcontents

\newpage

\setcounter{section}{2}
\section{The Hartree-Fock Approximation}
\subsection{The HF Equations}
\subsubsection{The Coulomb and Exchange Operators}
\subsubsection{The Fock Operator}
\ex{3.1}
\begin{align}
\Braket{\chi_i | \hat{f} | \chi_j} &= \Braket{ \chi_i(1) | h(1) + \sum_{b} [\mathscr{J}_b(1) - \mathscr{K}_b(1)] | \chi_j(1)} \notag\\
&= [i|h|j] + \sum_{b\neq j}\qty[\Braket{\chi_i(1) \chi_b(2) | \dfrac{1}{r_{12}} | \chi_b(2)\chi_j(1)} - \Braket{\chi_i(1)\chi_b(2) | \dfrac{1}{r_{12}} | \chi_b(1)\chi_j(2)}] \notag\\
&= [i|h|j] + \sum_{b\neq j}\qty([ij|bb] - [ib|bj])
\end{align}
Since
\begin{equation}\label{key}
[ij|jj] - [ij|jj] = 0
\end{equation}
we have
\begin{align}
\Braket{\chi_i | \hat{f} | \chi_j}
&= \Braket{i|h|j} + \sum_b\qty(\Braket{ib|jb} - \Braket{ib|bj}) \notag\\
&= \Braket{i|h|j} + \sum_b\Braket{ib||jb}
\end{align}

\subsection{Derivation of the HF Equations}
\subsubsection{Functional Variation}
\subsubsection{Minimization of the Energy of a Single Determinant}
\ex{3.2}
Take the complex conjugate of
\begin{equation}\label{key}
\mathscr{L}[\{\chi_\alpha\}] = E_0[\{\chi_\alpha\}] - \sum_a^N\sum_b^N \varepsilon_{ba}([a|b] - \delta_{ab})
\end{equation}
we have
\begin{equation}\label{key}
\mathscr{L}[\{\chi_\alpha\}]^* = E_0[\{\chi_\alpha\}]^* - \sum_a^N\sum_b^N \varepsilon_{ba}^*([a|b]^* - \delta_{ab}^*)
\end{equation}
i.e.
\begin{equation}\label{key}
\mathscr{L}[\{\chi_\alpha\}] = E_0[\{\chi_\alpha\}] - \sum_a^N\sum_b^N \varepsilon_{ba}^*([b|a] - \delta_{ab})
\end{equation}
thus
\begin{equation}\label{key}
\sum_a^N\sum_b^N \varepsilon_{ba}([a|b] - \delta_{ab}) = \sum_a^N\sum_b^N \varepsilon_{ba}^*([b|a] - \delta_{ab}) = \sum_b^N\sum_a^N \varepsilon_{ab}^*([a|b] - \delta_{ba})
\end{equation}
$ \therefore $
\begin{equation}\label{key}
\varepsilon_{ba} = \varepsilon_{ab}^*
\end{equation}

\ex{3.3}
$ \because $
\begin{align}
[\delta\chi_a | h | \chi_a] &= [\chi_a | h | \delta\chi_a]^*\\
[\chi_a\delta\chi_a | \chi_b\chi_b] &= [\delta\chi_a\chi_a | \chi_b\chi_b]^*\\
[\chi_a\chi_a | \chi_b\delta\chi_b] &= [\chi_a\chi_a | \delta\chi_b\chi_b]^*\\
[\chi_a\chi_b | \chi_b\delta\chi_a] &= [\chi_b\delta\chi_a | \chi_a\chi_b] = [\delta\chi_a\chi_b | \chi_b\chi_a]^*\\
[\chi_a\chi_b | \delta\chi_b\chi_a] &= [\delta\chi_b\chi_a | \chi_a\chi_b] = [\chi_a\delta\chi_b | \chi_b\chi_a]^*
\end{align}
$ \therefore $
\begin{align}
\delta E_0 &= \sum_a^N[\delta\chi_a | h | \chi_a]
+ \dfrac{1}{2}\sum_a^N\sum_b^N \qty([\delta\chi_a\chi_a | \chi_b\chi_b] + [\chi_a\chi_a | \delta\chi_b\chi_b]) \notag\\
&\quad{} - \dfrac{1}{2}\sum_a^N\sum_b^N \qty([\delta\chi_a\chi_b | \chi_b\chi_a] + [\chi_a\chi_b | \delta\chi_b\chi_a]) + \text{complex conjugates}
\end{align}
while
\begin{align}
\sum_a^N\sum_b^N [\chi_a\chi_a | \delta\chi_b\chi_b] &= \sum_b^N\sum_a^N [\chi_b\chi_b | \delta\chi_a\chi_a] = \sum_a^N\sum_b^N [\delta\chi_a\chi_a | \chi_b\chi_b]\\
\sum_a^N\sum_b^N [\chi_a\chi_b | \delta\chi_b\chi_a] &= \sum_b^N\sum_a^N [\chi_b\chi_a | \delta\chi_a\chi_b] = \sum_a^N\sum_b^N [\delta\chi_a\chi_b | \chi_b\chi_a]
\end{align}
thus
\begin{equation}\label{key}
\delta E_0 = \sum_a^N[\delta\chi_a | h | \chi_a]
+ \sum_a^N\sum_b^N \qty([\delta\chi_a\chi_a | \chi_b\chi_b] - [\delta\chi_a\chi_b | \chi_b\chi_a]) + \text{complex conjugates}
\end{equation}

\subsubsection{The Canonical HF Equations}

\subsection{Interpretation of Solutions to the HF Equations}
\subsubsection{Orbital Energies and Koopmans' Theorem}
\ex{3.4}
\begin{align}
f_{ij} = \Braket{\chi_i | f | \chi_j} = \Braket{i|h|j} + \sum_b\Braket{ib||jb}
\end{align}
\begin{align}
f_{ji}^* &= \Braket{\chi_j | f | \chi_i}^* = \Braket{j|h|i}^* + \sum_b\Braket{jb||ib}^*\notag\\
&= \Braket{i|h|j} + \sum_b\Braket{ib||jb}\notag\\
&= f_{ij}
\end{align}
thus the Fock operator is Hermitian.

\ex{3.5}
\begin{align}
\text{IP} &= ^{N-2}E - E_0 \notag\\
&= \sum_{a\neq c,d}\Braket{a|h|a} + \dfrac{1}{2}\sum_{a\neq c,d}\sum_{b\neq c,d} \Braket{ab||ab}  - \qty[\sum_a\Braket{a|h|a} + \dfrac{1}{2}\sum_a\sum_b \Braket{ab||ab} ] \notag\\
&= -\Braket{c|h|c} - \Braket{d|h|d} -  \dfrac{1}{2}\sum_{a\neq c,d}\Braket{ac||ac}  -  \dfrac{1}{2}\sum_{a\neq c,d}\Braket{ad||ad} - \dfrac{1}{2}\sum_{b\neq c,d} \Braket{cb||cb} - \dfrac{1}{2}\sum_{b\neq c,d} \Braket{db||db}  - \Braket{cd||cd} \notag\\
&= -\Braket{c|h|c} - \Braket{d|h|d} -  \sum_{a\neq c,d}\Braket{ac||ac}  -  \sum_{a\neq c,d}\Braket{ad||ad}   - \Braket{cd||cd} \notag\\
&= -\Braket{c|h|c} - \Braket{d|h|d} - \qty( \sum_{a\neq c}\Braket{ac||ac} - \Braket{dc||dc}) -  \qty(\sum_{a\neq d}\Braket{ad||ad} - \Braket{cd||cd})  - \Braket{cd||cd} \notag\\
&= -\varepsilon_c - \varepsilon_d + \Braket{cd|cd} - \Braket{cd|dc}
\end{align}

\ex{3.6}
\begin{align}
^N E_0 - ^{N+1}E^r &= \sum_a\Braket{a|h|a} + \dfrac{1}{2}\sum_a\sum_b \Braket{ab||ab} \notag\\
&{}\quad - \qty[\sum_a\Braket{a|h|a} + \Braket{r|h|r} + \dfrac{1}{2}\sum_a\sum_b \Braket{ab||ab} + \dfrac{1}{2}\sum_b \Braket{rb||rb} + \dfrac{1}{2}\sum_a \Braket{ar||ar}] \notag\\
&= - \Braket{r|h|r} - \dfrac{1}{2}\sum_b \Braket{rb||rb} - \dfrac{1}{2}\sum_b \Braket{br||br} \notag\\
&= - \Braket{r|h|r} - \sum_b \Braket{rb||rb}
\end{align}

\subsubsection{Brillouin's Theorem}
\subsubsection{The HF Hamiltonian}
\ex{3.7}
Suppose $ \mathscr{H}_0 $ commutes with $ \mathscr{P}_n $,
\begin{align}\label{key}
\mathscr{H}_0\ket{\Psi_0} &= \mathscr{H}_0 \dfrac{1}{\sqrt{N!}} \sum_n^{N!}(-1)^{p_n} \mathscr{P}_n \qty{\sum_i^N  f(i)\chi_j(1)\cdots\chi_k(N)} \notag\\
&= \dfrac{1}{\sqrt{N!}} \sum_n^{N!}(-1)^{p_n} \mathscr{P}_n \qty{ (\varepsilon_j + \cdots + \varepsilon_k) \chi_j(1)\cdots\chi_k(N)} \notag\\
&= \sum_a \varepsilon_a
\end{align}
Now we show $ \mathscr{H}_0 $ commutes with $ \mathscr{P}_n $, for example, $ \mathscr{P}_{ab} $
\begin{equation}\label{key}
\mathscr{P}_{ab}\mathscr{H}_0 = \mathscr{P}_{ab}(\cdots + f(a) + \cdots + f(b) + \cdots) = (\cdots + f(b) + \cdots + f(a) + \cdots)\mathscr{P}_{ab} = \mathscr{H}_0 \mathscr{P}_{ab}
\end{equation}

\ex{3.8}
\begin{equation}\label{key}
\mathscr{V} = \sum_i^N \sum_{j>i}^N \mathscr{O}_2 - \sum_i^N\sum_b^N [\mathscr{G}_b(i) - \mathscr{K}_b(i)]
\end{equation}
thus
\begin{align}
\Braket{\Psi_0 | \mathscr{V} | \Psi_0} &= \sum_i^N \sum_{j>i}^N \Braket{\Psi_0 | \mathscr{O}_2 | \Psi_0} - \sum_i^N\sum_b^N [\Braket{\Psi_0 | \mathscr{G}_b(i) - \mathscr{K}_b(i)| \Psi_0}] \notag\\
&= \dfrac{1}{2}\sum_a^N \sum_b^N \Braket{ab||ab} - \sum_i^N\sum_b^N [\Braket{ib|ib} - \Braket{ib|bi}] \notag\\
&= -\dfrac{1}{2}\sum_a^N \sum_b^N \Braket{ab||ab}
\end{align}

\subsection{Restricted Closed-shell HF: The Roothaan Equations}
\subsubsection{Closed-shell HF: Restricted Spin Orbitals}
\ex{3.9}
\begin{align}
\varepsilon_i %&= \Braket{\chi_i | h | \chi_i} + \sum_b^N \Braket{\chi_i\chi_b || \chi_i\chi_b} \notag\\
&= (i|h|i) + \sum_b^N(\Braket{ib|ib} - \Braket{ib|bi}) \notag\\
&= (i|h|i) + \sum_c^{N/2}(\Braket{ic|ic} - \Braket{ic|ci}) + \sum_{\bar{c}}^{N/2}(\Braket{i\bar{c}|i\bar{c}} - \Braket{i\bar{c}|\bar{c}i})
\end{align}
Assume $ \chi_j $ has $ \alpha $ spin, since assuming $ \alpha $ or $ \beta $ is identical
\begin{align}
\varepsilon_i &= (i|h|i) + \sum_c^{N/2}\qty[ (ic|ic)\Braket{\alpha|\alpha}\Braket{\alpha|\alpha}
- (ic|ci)\Braket{\alpha|\alpha}\Braket{\alpha|\alpha}]
+ \sum_c^{N/2}\qty[(ic|ic)\Braket{\alpha|\alpha}\Braket{\beta|\beta}
- (ic|ci)\Braket{\alpha|\beta}\Braket{\beta|\alpha}] \notag\\
&= (i|h|i) + \sum_c^{N/2}\qty[ 2(ic|ic) - (ic|ci)] \notag\\
&= (i|h|i) + \sum_n^{N/2} (2J_{ib} - K_{ib})
\end{align}

\subsubsection{Introduction of a Basis: The Roothaan Equations}
\ex{3.10}
\begin{align}
(\vb{C}^\dagger \vb{S} \vb{C})_{\mu\nu} &= \sum_i\sum_j C^\dagger_{\mu i} S_{ij} C_{j\nu} \notag\\
&= \sum_i\sum_j C^*_{i\mu} \Braket{\phi_i| \phi_j} C_{j\nu} \notag\\
&= \Braket{\phi_\mu | \phi_\nu} \notag\\
&= \delta_{\mu\nu}
\end{align}
thus
\begin{equation}\label{key}
\vb{C}^\dagger \vb{S} \vb{C} = \iden
\end{equation}

\subsubsection{The Charge Density}
\ex{3.11}
\begin{align}
\rho(\vb{r}) &= \Braket{\Psi_0 | \hat{\rho}(\vb{r}) | \Psi_0} \notag\\
&= \sum_i^N \dfrac{1}{N!} \sum_I^{N!}\sum_J^{N!} (-1)^{p_I} (-1)^{p_J} \int \dd \vb{x}_1\cdots \dd \vb{x}_N \hsP_I\{\chi_1(1)\cdots\chi_N(N)\}^* \delta(\vb{r}_i - \vb{r}) \hsP_J\{\chi_1(1)\cdots\chi_N(N)\}
\end{align}
Since $ \{\chi_m\} $ are orthogonal,
\begin{align}
\rho(\vb{r})
&= \sum_i^N \dfrac{1}{N!} \sum_I^{N!} \int \dd \vb{x}_1\cdots \dd \vb{x}_N \hsP_I\{\chi_1(1)\cdots\chi_N(N)\}^* \delta(\vb{r}_i - \vb{r}) \hsP_I\{\chi_1(1)\cdots\chi_N(N)\} \notag\\
&= \sum_i^N \dfrac{1}{N!} (N-1)!\sum_s^N\int \dd \vb{x}_i \chi_s^*(\vb{x}_i) \delta(\vb{r}_i - \vb{r}) \chi_s(\vb{x}_i) \notag\\
&= \sum_i^N \dfrac{1}{N}\cdot 2\sum_s^{N/2}\int \dd \vb{r}_i \phi_s(\vb{r}_i) \delta(\vb{r}_i - \vb{r}) \phi_s(\vb{r}_i) \notag\\
&= \sum_i^N \dfrac{2}{N} \sum_s^{N/2} \phi_s(\vb{r}) \phi_s(\vb{r}) \notag\\
&= N \dfrac{2}{N} \sum_s^{N/2} \phi_s(\vb{r}) \phi_s(\vb{r}) \notag\\
&= 2 \sum_s^{N/2} \phi_s(\vb{r}) \phi_s(\vb{r})
\end{align}

\ex{3.12}
From Ex 3.10, we have
\begin{equation}\label{key}
\vb{C}^\dagger \vb{S} \vb{C} = \iden
\end{equation}
i.e.
\begin{equation}\label{key}
\sum_i^K\sum_j^K C^*_{i\mu} S_{ij} C_{j\nu} = \delta_{\mu\nu}
\end{equation}
thus
\begin{align}
(\vb{P}\vb{S}\vb{P})_{\mu\sigma} &= \sum_\nu^K\sum_\lambda^K P_{\mu\nu} S_{\nu\lambda} P_{\lambda\sigma} \notag\\
&= 4\sum_\nu^K\sum_\lambda^K \sum_a^{N/2}C_{\mu a}C_{\nu a}^* S_{\nu\lambda} \sum_b^{N/2}C_{\lambda b}C_{\sigma b}^* \notag\\
&= 4\sum_a^{N/2}\sum_b^{N/2} C_{\mu a}\qty(\sum_\nu^K\sum_\lambda^K C_{\nu a}^* S_{\nu\lambda} C_{\lambda b}) C_{\sigma b}^* \notag\\
&= 4\sum_a^{N/2}\sum_b^{N/2}  C_{\mu a} \delta_{ab} C_{\sigma b}^* \notag\\
&= 4\sum_a^{N/2} C_{\mu a} C_{\sigma a}^* \notag\\
&= 2 P_{\mu\sigma}
\end{align}
thus
\begin{equation}\label{key}
\vb{P}\vb{S}\vb{P} = 2\vb{P}
\end{equation}

\ex{3.13}
Eq. 3.122 shows
\begin{equation}\label{key}
f(\vb{r}_1) = h(\vb{r}_1) + \sum_a^{N/2}\int \dd\vb{r}_2 \psi_a^*(\vb{r}_2) (2-\hsP_{12}) r_{12}^{-1} \psi_a(\vb{r}_2)
\end{equation}
thus
\begin{align}
f(\vb{r}_1) &= h(\vb{r}_1)
+ \sum_a^{N/2}\int \dd\vb{r}_2
\sum_\sigma C_{\sigma a}^* \phi_\sigma^*(\vb{r}_2) (2-\hsP_{12}) r_{12}^{-1} \sum_\lambda C_{\lambda a} \phi_\lambda(\vb{r}_2) \notag\\
&= h(\vb{r}_1)
+ \sum_\sigma\sum_\lambda \qty(\sum_a^{N/2} C_{\sigma a}^*C_{\lambda a} )
\int \dd\vb{r}_2 \phi_\sigma^*(\vb{r}_2) (2-\hsP_{12}) r_{12}^{-1}  \phi_\lambda(\vb{r}_2) \notag\\
&= h(\vb{r}_1)
+ \dfrac{1}{2} \sum_{\sigma,\lambda} P_{\lambda\sigma}
\int \dd\vb{r}_2 \phi_\sigma^*(\vb{r}_2) (2-\hsP_{12}) r_{12}^{-1}  \phi_\lambda(\vb{r}_2)
\end{align}

\subsubsection{Expression for the Fock Matrix}
\ex{3.14}
In expression $ (\mu\nu|\lambda\sigma) $, there are three interchangeable pairs, i.e. $\mu\leftrightarrow\nu $, $ \lambda\leftrightarrow\sigma $, and $ \mu\nu\leftrightarrow\lambda\sigma $. Thus $ (\mu\nu|\lambda\sigma) $ has an 8-fold symmetry. Similarly, %$ ,  $ have 4-fold symmetry,
$ (\mu\mu|\lambda\sigma), (\mu\nu|\mu\lambda), (\mu\nu|\mu\nu), (\mu\mu|\sigma\sigma) $ has 2-fold symmetry, and $ (\mu\mu|\mu\nu), (\mu\mu|\mu\mu) $ has 1-fold symmetry.\\
Therefore, the number of unique 2e integrals is
\begin{table}[H]
	\centering
	\begin{tabular}{ccc}
		\hline
		expression & number & $ K=100 $\\ \hline
		 $ (\mu\nu|\lambda\sigma) $ & $ K(K-1)(K-2)(K-3)/8 $ & 11763675 \\
		 $ (\mu\mu|\lambda\sigma)$ & $ K(K-1)(K-2)/2 $& 485100\\
		 $ (\mu\nu|\mu\lambda)$ & $ K(K-1)(K-2)/2 $& 485100\\
		 $ (\mu\nu|\mu\nu) $ & $ K(K-1)/2 $ & 4950\\
		 $ (\mu\mu|\sigma\sigma) $ & $ K(K-1)/2 $ & 4950\\
		 $ (\mu\mu|\mu\nu) $ & $ K(K-1) $ & 9900\\
		 $ (\mu\mu|\mu\mu) $ & $ K $ & 100\\
		 \hline
	\end{tabular}
\end{table}
thus the total number is $ \num{12753775} $.

\subsubsection{Orthogonalization of the Basis}
\ex{3.15}
$ \because $
\begin{equation}\label{key}
\vb{U}^\dagger \vb{S} \vb{U} = \vb{s}
\end{equation}
$ \therefore $
\begin{equation}\label{key}
\vb{S} \vb{U} = \vb{U} \vb{s}
\end{equation}
i.e.
\begin{equation}\label{key}
\sum_\nu S_{\mu\nu} U_{\nu i} = U_{\mu i} s_i
\end{equation}
thus
\begin{equation}\label{key}
\sum_\mu U_{\mu i}^* \sum_\nu S_{\mu\nu} U_{\nu i} = \sum_\mu U_{\mu i}^* U_{\mu i} s_i
\end{equation}
\begin{equation}\label{key}
\sum_\mu \sum_\nu U_{\mu i}^* \Braket{\phi_\mu| \phi_\nu} U_{\nu i} = s_i \sum_\mu \abs{ U_{\mu i}}^2
\end{equation}
Suppose
\begin{equation}\label{key}
\phi'_i = \sum_\nu U_{\nu i}\phi_\nu
\end{equation}
thus
\begin{equation}\label{key}
\Braket{\phi'_i | \phi'_i} = s_i \sum_\mu \abs{ U_{\mu i}}^2
\end{equation}
$ \because $
\begin{equation}\label{key}
\Braket{\phi'_i | \phi'_i} > 0 \qquad \abs{ U_{\mu i}}^2 > 0
\end{equation}
$ \therefore $
\begin{equation}\label{key}
s_i > 0
\end{equation}

\ex{3.16}
\subex{$ \bullet $} (3.174)\\
Since ($ \phi,\phi',\psi $ are row vectors)
%\begin{equation}\label{key}
%\bm\phi' = \bm\phi \vb{X}
%\end{equation}
\begin{equation}\label{key}
\bm\psi = \bm\phi \vb{C}
\end{equation}
\begin{equation}\label{key}
\bm\psi = \bm\phi' \vb{C}' = \bm\phi \vb{X} \vb{C}'
\end{equation}
we have
\begin{equation}\label{key}
\vb{C} = \vb{X} \vb{C}'
\end{equation}
i.e.
\begin{equation}\label{key}
\vb{C}' = \vb{X}^{-1} \vb{C}
\end{equation}

\subex{$ \bullet $} (3.177)
\begin{align}
F'_{\mu\nu} &= \Braket{\phi'_\mu | f | \phi'_\nu} \notag\\
&= \Braket{\sum_i \phi_i X_{i\mu} | f | \sum_j \phi_j X_{j\nu}} \notag\\
&= \sum_i\sum_j X^*_{i\mu} X_{j\nu} \Braket{\phi_i | f | \phi_j} \notag\\
&= \sum_i\sum_j X^*_{i\mu} F_{ij} X_{j\nu}
\end{align}
i.e.
\begin{equation}\label{key}
\vb{F}' = \vb{X}^\dagger \vb{F} \vb{X}
\end{equation}

\subsubsection{The SCF Procedure}

\subsubsection{Expectation Values and Population Analysis}
\ex{3.17}
From (3.148) in the textbook, we get
\begin{equation}\label{key}
F_{\mu\nu} = H_{\mu\nu}^{\core} + G_{\mu\nu} = H_{\mu\nu}^{\core} + \sum_a^{N/2}[ 2(\mu\nu|aa) - (\mu a| a\nu)]
\end{equation}
thus
\begin{align}
E_0 &= \sum_a^{N/2} [2 h_{aa} + \sum_b^{N/2} (2 J_{ab} - K_{ab})] \notag\\
&= 2 \sum_a^{N/2} (a|h|a) + \sum_a^{N/2}\sum_b^{N/2} [2 (aa|bb) - (ab|ba)] \notag\\
&= 2 \sum_a^{N/2} \sum_\mu\sum_\nu C_{\mu a}^* C_{\nu a} (\mu| h |\nu)
+ \sum_a^{N/2}\sum_b^{N/2} \qty[2\sum_\mu\sum_\nu C_{\mu a}^* C_{\nu a} (\mu\nu|bb) - \sum_\mu\sum_\nu C_{\mu a}^* C_{\nu a} (\mu b| b\nu)] \notag\\
&= \sum_\mu\sum_\nu P_{\nu\mu} H_{\mu\nu}^{\core} + \dfrac{1}{2}\sum_b^{N/2}\sum_\mu\sum_\nu[2 P_{\nu\mu}(\mu\nu|bb) - P_{\nu\mu}(\mu b|b\nu)] \notag\\
&= \sum_\mu\sum_\nu P_{\nu\mu} [H_{\mu\nu}^{\core} + \dfrac{1}{2}G_{\mu\nu}] \notag\\
&= \dfrac{1}{2}\sum_\mu\sum_\nu P_{\nu\mu} [H_{\mu\nu}^{\core} + F_{\mu\nu}] \notag\\
\end{align}

\ex{3.18}
For symmetrically orthogonalized basis,
\begin{equation}\label{key}
\vb{C}' = \vb{S}^{1/2}\vb{C}
\end{equation}
thus
\begin{align}
P'_{\mu\nu} &=  2\sum_a^{N/2} C'_{\mu a}C'^*_{\nu a} \notag\\
&=  2\sum_a^{N/2}  \sum_i S^{1/2}_{\mu i} C_{i a} \sum_j S^{1/2*}_{\nu j} C^*_{ja}  \notag\\
&=  \sum_i\sum_j  S^{1/2}_{\mu i} \qty( 2\sum_a^{N/2} C_{i a} C^*_{ja} ) S^{1/2*}_{\nu j}  \notag\\
&= \sum_i\sum_j  S^{1/2}_{\mu i} P_{ij} S^{1/2*}_{\nu j}  \notag\\
&= \sum_i\sum_j  S^{1/2}_{\mu i} P_{ij} S^{1/2}_{j\nu}
\end{align}
i.e.
\begin{equation}\label{key}
\vb{P}' = \vb{S}^{1/2} \vb{P} \vb{S}^{1/2}
\end{equation}
thus
\begin{equation}\label{key}
\sum_\mu (\vb{S}^{1/2} \vb{P} \vb{S}^{1/2})_{\mu\mu} = \sum_\mu \vb{P}'_{\mu\mu}
\end{equation}

\subsection{Model Calculations on \ce{H_2} and \ce{HeH^+}}
\subsubsection{The $ 1s $ Minimal STO-3G Basis Set}
\ex{3.19}
\begin{align}
\phi_{1s}^{\GF}(\alpha,\vb{r}-\vb{R}_A)\phi_{1s}^{\GF}(\alpha,\vb{r}-\vb{R}_B) &= \qty(\dfrac{2\alpha}{\pi})^{3/4} \e^{-\alpha\abs{\vb{r}-\vb{R}_A}^2} \qty(\dfrac{2\beta}{\pi})^{3/4} \e^{-\beta\abs{\vb{r}-\vb{R}_B}^2} \notag\\
&= \qty(\dfrac{2\alpha}{\pi})^{3/4} \qty(\dfrac{2\beta}{\pi})^{3/4} \e^{-\alpha\abs{\vb{r}-\vb{R}_A}^2 - \beta\abs{\vb{r}-\vb{R}_B}^2} \notag\\
&= \qty(\dfrac{2\alpha}{\pi})^{3/4} \qty(\dfrac{2\beta}{\pi})^{3/4} \exp(-\qty[(\alpha+\beta)\abs{\vb{r}}^2
	- 2\vb{r}\cdot(\alpha\vb{R}_A + \beta\vb{R}_B)
	+ \alpha\abs{\vb{R}_A}^2 +\beta\abs{\vb{R}_B}^2])
\end{align}
Let
\begin{equation}\label{key}
p = \alpha + \beta \qquad \vb{R}_P = \dfrac{\alpha\vb{R}_A + \beta\vb{R}_B} {\alpha+\beta}
\end{equation}
we have
\begin{align}
\phi_{1s}^{\GF}(\alpha,\vb{r}-\vb{R}_A)\phi_{1s}^{\GF}(\alpha,\vb{r}-\vb{R}_B)
&= \qty(\dfrac{2\alpha}{\pi}\dfrac{\beta}{\pi})^{3/4}
\exp(-\qty[p\abs{\vb{r}}^2
	- 2\vb{r}\cdot(p\vb{R}_P)
	+ \alpha\abs{\vb{R}_A}^2 +\beta\abs{\vb{R}_B}^2]) \notag\\
&= \qty(\dfrac{2\alpha}{\pi}\dfrac{2\beta}{\pi})^{3/4}
\exp(-\qty[p\abs{\vb{r} - \vb{R}_P}^2
	- p\abs{\vb{R}_P}^2
	+ \alpha\abs{\vb{R}_A}^2 +\beta\abs{\vb{R}_B}^2]) \notag\\
&= \qty(\dfrac{2\alpha\beta/p}{\pi})^{3/4} \qty(\dfrac{2p}{\pi})^{3/4}
\e^{-p\abs{\vb{r} - \vb{R}_P}^2 }
\exp(p\abs{\vb{R}_P}^2 - \alpha\abs{\vb{R}_A}^2 - \beta\abs{\vb{R}_B}^2)
\end{align}
Let
\begin{equation}\label{key}
\phi_{1s}^{\GF}(\alpha,\vb{r}-\vb{R}_A)\phi_{1s}^{\GF}(\alpha,\vb{r}-\vb{R}_B) = K_{AB} \qty(\dfrac{2p}{\pi})^{3/4}
\e^{-p\abs{\vb{r} - \vb{R}_P}^2 }
\end{equation}
thus
\begin{align}
K_{AB} &= \qty(\dfrac{2\alpha\beta/p}{\pi})^{3/4}
\exp(p\abs{\vb{R}_P}^2 - \alpha\abs{\vb{R}_A}^2 -\beta\abs{\vb{R}_B}^2) \notag\\
&= \qty(\dfrac{2\alpha\beta/p}{\pi})^{3/4}
\exp(\dfrac{1}{p}(
\alpha^2\abs{\vb{R}_A}^2 +\beta^2\abs{\vb{R}_B}^2 + 2\alpha\beta\vb{R}_A\cdot\vb{R}_B )
- \alpha\abs{\vb{R}_A}^2 -\beta\abs{\vb{R}_B}^2) \notag\\
&= \qty(\dfrac{2\alpha\beta/p}{\pi})^{3/4}
\exp(\dfrac{1}{p} \qty(\alpha^2\abs{\vb{R}_A}^2 +\beta^2\abs{\vb{R}_B}^2 + 2\alpha\beta\vb{R}_A\cdot\vb{R}_B
- p\alpha\abs{\vb{R}_A}^2 -p\beta\abs{\vb{R}_B}^2)) \notag\\
&= \qty(\dfrac{2\alpha\beta/p}{\pi})^{3/4}
\exp(\dfrac{1}{p} \qty(-\alpha\beta\abs{\vb{R}_A}^2 -\alpha\beta\abs{\vb{R}_B}^2 + 2\alpha\beta\vb{R}_A\cdot\vb{R}_B )) \notag\\
&= \qty(\dfrac{2\alpha\beta}{p\pi})^{3/4}
\exp(-\dfrac{\alpha\beta}{p} \abs{\vb{R}_A - \vb{R}_B}^2 )
\end{align}

\ex{3.20}
At $ r=0 $,
\begin{align}
\phi_{1s}^{\text{CGF}}(\zeta=1.0, \text{STO-1G}) &= \num{0.267656} \\
\phi_{1s}^{\text{CGF}}(\zeta=1.0, \text{STO-2G}) &= \num{0.389383} \\
\phi_{1s}^{\text{CGF}}(\zeta=1.0, \text{STO-3G}) &= \num{0.454986}
\end{align}
while
\begin{equation}\label{key}
\phi_{1s}^{\text{SF}}(\zeta=1.0) = \dfrac{1}{\sqrt{\pi}} = \num{0.56419}
\end{equation}

\subsubsection{STO-3G \ce{H_2}}
\ex{3.21}
\begin{equation}\label{key}
\phi_{1s}^{\text{CGF}}(\zeta=1.0, \text{STO-1G}) = \phi_{1s}^{\text{GF}}(\num{0.270950})
\end{equation}
Since $ \alpha = \alpha_{(\zeta=1.0)}\cross \zeta^2 $,
\begin{equation}\label{key}
\phi_{1s}^{\text{CGF}}(\zeta=1.24, \text{STO-1G}) = \phi_{1s}^{\text{GF}}(\num{0.416613})
\end{equation}
thus
\begin{align}
S_{12} &= K_{AB} \qty(\dfrac{2\cdot 2\alpha}{\pi})^{3/4} \int \dd\vb{r} \e^{-2\alpha\abs{\vb{r}-\vb{R}_P}^2} \notag\\
&= \qty(\dfrac{2\alpha}{2\pi})^{3/4} \e^{-\tfrac{\alpha}{2}R^2} \qty(\dfrac{2\cdot 2\alpha}{\pi})^{3/4} \int \dd\vb{r} \e^{-2\alpha\abs{\vb{r}-\vb{R}_A}^2} \notag\\
&= \qty(\dfrac{2\alpha}{\pi})^{3/2} \e^{-\tfrac{\alpha}{2}R^2} 4\pi \int \dd r r^2 \e^{-2\alpha r^2} \notag\\
&= \qty(\dfrac{2\alpha}{\pi})^{3/2} \e^{-\tfrac{\alpha}{2}R^2} 4\pi \dfrac{\sqrt{\pi}}{8\sqrt{2} \alpha^{3/2}} \notag\\
&= \e^{-\tfrac{\alpha}{2}R^2}
\end{align}
At $ R = 1.4, \alpha = \num{0.416613} $,
\begin{equation}\label{key}
S_{12} = \num{0.6648}
\end{equation}

\ex{3.22}
Let
\begin{equation}\label{key}
\psi_1 = c_1 (\phi_1 + \phi_2) \qquad \psi_2 = c_2 (\phi_1 - \phi_2)
\end{equation}
\begin{align}
1 &= \Braket{\phi_1|\psi_1} = c_1^2 (S_{11} + S_{12} + S_{21} + S_{22}) \notag\\
&= c_1^2 (2 + 2S_{12})
\end{align}
$ \therefore $
\begin{equation}\label{key}
c_1 = [2(1 + S_{12})]^{-1/2}
\end{equation}
\begin{align}
1 &= \Braket{\phi_2|\psi_2} = c_2^2 (S_{11} - S_{12} - S_{21} + S_{22}) \notag\\
&= c_2^2 (2 - 2S_{12})
\end{align}
$ \therefore $
\begin{equation}\label{key}
c_2 = [2(1 - S_{12})]^{-1/2}
\end{equation}

\ex{3.23}
Suppose
\begin{equation}\label{key}
\psi_1 = c_1 (\phi_1 + \phi_2) \qquad \psi_2 = c_2 (\phi_1 - \phi_2)
\end{equation}
thus
\begin{equation}\label{key}
\vb{H}^{\core} \vb{C} = \vb{S} \vb{C} \bm\varepsilon
\end{equation}
\begin{equation}\label{key}
\mqty(H_{11}^{\core} & H_{12}^{\core}\\ H_{21}^{\core} & H_{22}^{\core}) \mqty(c_1 & c_2\\ c_1 & -c_2) = \mqty(S_{11} & S_{12}\\ S_{21} & S_{22}) \mqty(c_1 & c_2\\ c_1 & -c_2) \mqty(\varepsilon_1 & 0\\ 0 &\varepsilon_2 )
\end{equation}
\begin{equation}\label{key}
\mqty((H_{11}^{\core} + H_{12}^{\core}) c_1 & (H_{11}^{\core} - H_{12}^{\core}) c_2 \\
(H_{21}^{\core} + H_{22}^{\core}) c_1 & (H_{21}^{\core} - H_{22}^{\core})c_2)
= \mqty((S_{11} + S_{12}) c_1\varepsilon_1 & (S_{11} - S_{12}) c_2\varepsilon_2\\
(S_{21} + S_{22}) c_1\varepsilon_1 & (S_{21} - S_{22}) c_2\varepsilon_2)
\end{equation}
$ \therefore $
\begin{equation}\label{key}
\left\{ \mqty{\varepsilon_1 = (H_{11}^{\core} + H_{12}^{\core})/(1 + S_{12}) \\
\varepsilon_2 = (H_{11}^{\core} - H_{12}^{\core})/(1 - S_{12})}
\right.
\end{equation}
\begin{align}
\varepsilon_1 = (-1.1204-0.9584)/(1+0.6593) = -1.2528\\
\varepsilon_2 = (-1.1204+0.9584)/(1-0.6593) = -0.4755
\end{align}

\ex{3.24}
\begin{equation}\label{key}
P_{\mu\nu} = 2\sum_a^{N/2} C_{\mu a}C_{\nu a}^* = 2 C_{\mu 1}C_{\nu 1}^*
\end{equation}
$ \therefore $
\begin{align}
\vb{P} &= 2 \mqty( C_{11}C_{11}^* & C_{11}C_{21}^*\\
                   C_{21}C_{11}^* & C_{21}C_{21}^*) \notag\\
&= 2 \mqty([2(1+S_{12})]^{-1/2}[2(1+S_{12})]^{-1/2} & [2(1+S_{12})]^{-1/2}[2(1+S_{12})]^{-1/2} \\ [2(1+S_{12})]^{-1/2}[2(1+S_{12})]^{-1/2} & [2(1+S_{12})]^{-1/2}[2(1+S_{12})]^{-1/2} ) \notag\\
&= (1+S_{12})^{-1} \mqty(1 & 1 \\ 1 & 1)
\end{align}
For $ \ce{H_2^+} $,
\begin{equation}\label{key}
\vb{P}_{\ce{H_2^+}} = \dfrac{1}{2} (1+S_{12})^{-1} \mqty(1 & 1 \\ 1 & 1)
\end{equation}

\ex{3.25}
\begin{align}
F_{\mu\nu} &= H^{\core}_{\mu\nu} + \sum_{\lambda\sigma} P_{\lambda\sigma} \qty[(\mu\nu|\sigma\lambda) - \dfrac{1}{2}(\mu\lambda|\sigma\nu)] \notag\\
&= H^{\core}_{\mu\nu} + (1+S_{12})^{-1} \sum_{\lambda\sigma} \qty[(\mu\nu|\sigma\lambda) - \dfrac{1}{2}(\mu\lambda|\sigma\nu)] %\notag\\
%&= H^{\core}_{\mu\nu} + (1+S_{12})^{-1}
%\qty[(\mu\nu|11) - \dfrac{1}{2}(\mu 1|1 \nu)
%+ (\mu\nu|21) - \dfrac{1}{2}(\mu 1|2 \nu)
%+ (\mu\nu|12) - \dfrac{1}{2}(\mu 2|1 \nu)
%+ (\mu\nu|22) - \dfrac{1}{2}(\mu 2|2 \nu)] \notag\\
\end{align}
$ \therefore $
\begin{align}
F_{11} &= H^{\core}_{11} + (1+S_{12})^{-1}
\qty[(11|11) - \dfrac{1}{2}(1 1|1 1)
+ (11|21) - \dfrac{1}{2}(1 1|2 1)
+ (11|12) - \dfrac{1}{2}(1 2|1 1)
+ (11|22) - \dfrac{1}{2}(1 2|2 1)] \notag\\
&= H^{\core}_{11} + (1+S_{12})^{-1}
\qty[\dfrac{1}{2}(1 1|1 1) + (1 1|2 1) + (11|22) - \dfrac{1}{2}(1 2|2 1)]
\end{align}
\iffalse
\begin{align}
F_{22} &= H^{\core}_{22} + (1+S_{12})^{-1}
\qty[(22|11) - \dfrac{1}{2}(2 1|1 2)
+ (22|21) - \dfrac{1}{2}(2 1|2 2)
+ (22|12) - \dfrac{1}{2}(2 2|1 2)
+ (22|22) - \dfrac{1}{2}(2 2|2 2)] \notag\\
&= H^{\core}_{22} + (1+S_{12})^{-1}
\qty[(22|11) - \dfrac{1}{2}(2 1|1 2)
+ (22|21)
+ \dfrac{1}{2}(2 2|2 2)] \notag\\
&= H^{\core}_{11} + (1+S_{12})^{-1}
\qty[(11|22) - \dfrac{1}{2}(1 2|2 1)
+ (11|12)
+ \dfrac{1}{2}(11|11)]
\end{align}
\fi
\begin{align}
F_{11} = F_{22} &= -1.1204 + (1 + 0.6593)^{-1} \qty(\dfrac{1}{2}\times 0.7746 + 0.4441 + 0.5697 -\dfrac{1}{2}\times 0.2970) \notag\\
&= -0.3655
\end{align}

\begin{align}
F_{12} &= H^{\core}_{12} + (1+S_{12})^{-1}
\qty[(12|11) - \dfrac{1}{2}(1 1|1 2)
+ (12|21) - \dfrac{1}{2}(1 1|22)
+ (12|12) - \dfrac{1}{2}(1 2|1 2)
+ (12|22) - \dfrac{1}{2}(1 2|2 2)] \notag\\
&= H^{\core}_{12} + (1+S_{12})^{-1}
\qty[ (1 1|1 2)
 - \dfrac{1}{2}(1 1|22)
+  \dfrac{3}{2}(1 2|1 2)]
\end{align}
\begin{align}
F_{12} = F_{21} &= -0.9584 + (1 + 0.6593)^{-1} \qty(0.4441 - \dfrac{1}{2}\times 0.5697 + \dfrac{3}{2}\times 0.2970) \notag\\
&= -0.5939
\end{align}

\ex{3.26}
Similar to the procedure in Ex 3.23, we get
\begin{align}
\varepsilon_1 &= \dfrac{F_{11} + F_{12}}{1 + S_{12}} = \dfrac{-0.3655 -0.5939}{1 + 0.6593} = -0.5782 \\
\varepsilon_2 &= \dfrac{F_{11} - F_{12}}{1 - S_{12}} = \dfrac{-0.3655 +0.5939}{1 - 0.6593} = 0.6703
\end{align}

\ex{3.27}
\begin{align}
E_0 &= \sum_{\mu\nu} \dfrac{1}{2}P_{\nu\mu} (H^{\core}_{\mu\nu} + F_{\mu\nu}) \notag\\
&= \dfrac{1}{2}\dfrac{1}{1 + S_{12}} \qty(H^{\core}_{11} + F_{11} + H^{\core}_{12} + F_{12} + H^{\core}_{21} + F_{21} + H^{\core}_{22} + F_{22}) \notag\\
&= \dfrac{H^{\core}_{11} + F_{11} + H^{\core}_{12} + F_{12}}{1 + S_{12}} \notag\\
&= \dfrac{-1.1204 -0.3655 - 0.9584 - 0.5939}{1 + 0.6593} \notag\\
&= -1.8310
\end{align}
\begin{equation}\label{key}
E_{tot} = E_0 + \dfrac{1}{R} = -1.1167
\end{equation}

\subsubsection{An SCF Calculation on STO-3G \ce{HeH^+}}
\ex{3.28}
\begin{align}
\vb{X}^\dagger_{\text{Schmidt}} \vb{S} \vb{X}_{\text{Schmidt}} &= \mqty(1 & 0 \\ -S_{12}/\sqrt{1-S_{12}^2} & 1/\sqrt{1-S_{12}^2})
\mqty(1 & S_{12} \\ S_{12} & 1)
\mqty(1 & -S_{12}/\sqrt{1-S_{12}^2}\\ 0 & 1/\sqrt{1-S_{12}^2}) \notag\\
&= \mqty(1 & S_{12} \\ 0 & \sqrt{1-S_{12}^2})
\mqty(1 & -S_{12}/\sqrt{1-S_{12}^2}\\ 0 & 1/\sqrt{1-S_{12}^2}) \notag\\
&= \mqty(1 & 0\\ 0 & 1)
\end{align}
thus the Schmidt transformation produces orthonormal basis.

\ex{3.29}
\begin{align}
E_0(R\ra \infty) &= \dfrac{1}{2}\sum_\mu\sum_\nu P_{\nu\mu}(R\ra \infty) [2H_{\mu\nu}^{\core} + G_{\mu\nu}]
\end{align}
where
\begin{equation}\label{key}
P_{\nu\mu}(R\ra \infty) = \mqty( 2 & 0\\ 0 & 0)
\end{equation}
\begin{align}\label{key}
G_{\mu\nu} &= \sum_\lambda\sum_\sigma P_{\lambda\sigma}(R\ra\infty) \qty[(\mu\nu|\sigma\lambda) - \dfrac{1}{2}(\mu\lambda|\sigma\nu)] \notag\\
&= 2\qty[(\mu\nu|\phi_1\phi_1) - \dfrac{1}{2}(\mu\phi_1|\phi_1\nu)]
\end{align}
thus
\begin{align}
E_0(R\ra \infty) &= \dfrac{1}{2}\sum_\mu\sum_\nu P_{\nu\mu}(R\ra \infty) [2H_{\mu\nu}^{\core} + G_{\mu\nu}] \notag\\
&= \dfrac{1}{2}\times 2[2H_{11}^{\core} + G_{11}] \notag\\
&= 2(T_{11} + V^1_{11}) + 2\qty[(\phi_1\phi_1|\phi_1\phi_1) - \dfrac{1}{2}(\phi_1\phi_1|\phi_1\phi_1)] \notag\\
&= 2T_{11} + 2V^1_{11} + (\phi_1\phi_1|\phi_1\phi_1)
\end{align}

\subsection{Polyatomic Basis Sets}
\subsubsection{Contracted Gaussian Functions}
\subsubsection{Minimal Basis Sets: STO-3G}
\subsubsection{Double Zeta Basis Sets: 4-31G}
\ex{3.30}
The outer basis function
\begin{equation}\label{key}
\phi''_{1s}(\vb{r}) = g_{1s}(0.298073, \vb{r})
\end{equation}
The inner basis function
\begin{equation}\label{key}
\phi'_{1s}(\vb{r}) = N[0.46954 g_{1s}(1.242567, \vb{r})
+ 0.15457 g_{1s}(5.782948, \vb{r}) + 0.02373 g_{1s}(38.47497, \vb{r})]
\end{equation}
Renormalize it, we get
\begin{equation}\label{key}
N = 1.689
\end{equation}
thus
\begin{equation}\label{key}
\phi'_{1s}(\vb{r}) = 0.79330 g_{1s}(1.242567, \vb{r})
+ 0.26115 g_{1s}(5.782948, \vb{r}) + 0.04009 g_{1s}(38.47497, \vb{r})
\end{equation}

\subsubsection{Polarized Basis Sets: 6-31G* and 6-31G**}
\ex{3.31}
~\\
\begin{table}[H]
	\centering
	\begin{tabular}{cccc}
		\hline
		 & C & H & total\\ \hline
		STO-3G & 5 & 1 & 36\\
		4-31G & 9 & 2 & 66\\
		6-31G* (Cartesian) & 15 & 2 & 102\\
		6-31G** (Cartesian) & 15 & 5 & 120\\
		\hline
	\end{tabular}
\end{table}

\subsection{Some Illustrative Closed-shell Calculations}
\subsubsection{Total Energies}
\ex{3.32}~\\
Reaction I
\begin{table}[H]
	\begin{tabular}{ccc|c}
		\hline
		basis & $ \Delta E /  $ a.u.& $ \Delta E /(\si{kcal/mol}) $& \\
		\hline
		STO-3G   & -0.061 & -38.28 &\\
		4-31G    & -0.069 & -43.30 &\\
		6-31G*   & -0.045 & -28.24 & exoergic\\
		6-31G**  & -0.055 &	-34.51 & \\
		HF-limit & -0.051 & -32.00 &\\
		\hline
	\end{tabular}
\end{table}
Reaction II
\begin{table}[H]
	\begin{tabular}{ccc|c}
		\hline
		basis & $ \Delta E /  $ a.u.& $ \Delta E /(\si{kcal/mol}) $ &\\
		\hline
		STO-3G   &  0.186 &	116.72 & endoergic	\\ \cline{4-4}
		4-31G    & -0.114 &	-71.54 & \multirow{4}{*}{exoergic}\\
		6-31G*   & -0.088 &	-55.22 &\\
		6-31G**  & -0.095 &	-59.61 &\\
		HF-limit & -0.097 &	-60.87 &\\
		\hline
	\end{tabular}
\end{table}
The contribution of zero-point vibrations to the energy change of reaction I would be $ \SI{-0.37}{kcal/mol} $, to the energy change of reaction II would be $ \SI{17.78}{kcal/mol} $. Thus the effect of zero-point vibrations should not be ignored.

\subsubsection{Ionization Potentials}

\subsubsection{Equilibrium Geometries}

\subsubsection{Population Analysis and Dipole Moments}

\subsection{Unrestricted Open-shell HF: The Pople-Nesbet Equations}
\subsubsection{Open-shell HF: Unrestricted Spin Orbitals}
\ex{3.33}
\begin{align}
f^\alpha(1) &= \int\dd\omega_1 \alpha^*(\omega_1)
\qty[h(1) + \sum_a\int\dd\vb{x}_2 \chi_a^*(2)r_{12}^{-1}(1-\hsP_{12})\chi_a(2)] \alpha(\omega_1) \notag\\
&= h(1) + \sum_a^{N_\alpha}
\qty[ \int\dd\omega_1 \alpha^*(\omega_1) \int\dd\vb{x}_2 \chi_a^*(2)r_{12}^{-1}\chi_a(2) \alpha(\omega_1)
 - \int\dd\omega_1 \alpha^*(\omega_1) \int\dd\vb{x}_2 \chi_a^*(2)r_{12}^{-1}\chi_a(1) \alpha(\omega_2) ] \notag\\
&\quad {} + \sum_a^{N_\beta}
\qty[ \int\dd\omega_1 \alpha^*(\omega_1) \int\dd\vb{x}_2 \chi_a^*(2)r_{12}^{-1}\chi_a(2) \alpha(\omega_1)
-  \int\dd\omega_1 \alpha^*(\omega_1) \int\dd\vb{x}_2 \chi_a^*(2)r_{12}^{-1}\chi_a(1) \alpha(\omega_2) ]  \notag\\
&= h(1) + \sum_a^{N_\alpha}
\qty[\int\dd\vb{r}_2 \psi_a^{\alpha*}(\vb{r}_2) r_{12}^{-1} \psi_a^\alpha(\vb{r}_2)
- \int\dd\vb{r}_2\int\dd\omega_2\int\dd\omega_1
\alpha^*(\omega_1)\alpha^*(\omega_2) \psi_a^{\alpha*}(\vb{r}_2) r_{12}^{-1} \psi_a^{\alpha}(\vb{r}_1) \alpha(\omega_1) \alpha(\omega_2) ] \notag\\
&\quad {} + \sum_a^{N_\beta}\qty[\int\dd\vb{r}_2 \psi_a^{\beta*}(\vb{r}_2) r_{12}^{-1} \psi_a^\beta(\vb{r}_2)
- \int\dd\vb{r}_2\int\dd\omega_2\int\dd\omega_1
\alpha^*(\omega_1)\beta^*(\omega_2) \psi_a^{\beta*}(\vb{r}_2) r_{12}^{-1} \psi_a^{\beta}(\vb{r}_1) \beta(\omega_1) \alpha(\omega_2) ]  \notag\\
&= h(1) + \sum_a^{N_\alpha}
\qty[\int\dd\vb{r}_2 \psi_a^{\alpha*}(\vb{r}_2) r_{12}^{-1} \psi_a^\alpha(\vb{r}_2)
- \int\dd\vb{r}_2 \psi_a^{\alpha*}(\vb{r}_2) r_{12}^{-1} \psi_a^{\alpha}(\vb{r}_1) ]
+ \sum_a^{N_\beta} \qty[\int\dd\vb{r}_2 \psi_a^{\beta*}(\vb{r}_2)r_{12}^{-1} \psi_a^\beta(\vb{r}_2) - 0 ]  \notag\\
&= h(1) + \sum_a^{N_\alpha}\qty[J_a^\alpha(1) - K_a^{\alpha}(1) ] + \sum_a^{N_\beta} J_a^\beta(1)
\end{align}

\ex{3.34}
\begin{align}
E_0 &= \sum_a h_{aa} + \dfrac{1}{2}\sum_a^{N_\alpha}\sum_b^{N_\alpha} (J_{ab}^{\alpha\alpha} - K_{ab}^{\alpha\alpha})
%+ \dfrac{1}{2}\sum_a^{N_\beta}\sum_b^{N_\beta} (J_{ab}^{\beta\beta} - K_{ab}^{\beta\beta})
+ \sum_a^{N_\alpha}\sum_b^{N_\beta} J_{ab}^{\alpha\beta} \notag\\
&= h_{11}^\alpha + h_{22}^\alpha + h_{11}^\alpha + J_{12}^{\alpha\alpha} - K_{12}^{\alpha\alpha} + J_{11}^{\alpha\beta} + J_{21}^{\alpha\beta}
\end{align}

\ex{3.35}
\begin{align}
\varepsilon_i^\alpha &= (\psi_i^\alpha(1) | h(1) + \sum_a^{N_\alpha}\qty[J_a^\alpha(1) - K_a^{\alpha}(1) ] + \sum_a^{N_\beta} J_a^\beta(1) | \psi_i^\alpha(1)) \notag\\
&= h_{ii}^\alpha + \sum_a^{N_\alpha}\qty[J_{ia}^{\alpha\alpha} - K_{ia}^{\alpha\alpha}] + \sum_a^{N_\beta} J_{ia}^{\alpha\beta}
\end{align}
\begin{align}
\varepsilon_i^\beta &= (\psi_i^\beta(1) | h(1) + \sum_a^{N_\alpha}\qty[J_a^\beta(1) - K_a^{\beta}(1) ] + \sum_a^{N_\beta} J_a^\alpha(1) | \psi_i^\beta(1)) \notag\\
&= h_{ii}^\beta + \sum_a^{N_\alpha}\qty[J_{ia}^{\beta\beta} - K_{ia}^{\beta\beta}] + \sum_a^{N_\beta} J_{ia}^{\beta\alpha}
\end{align}
Since
\begin{equation}\label{key}
E_0 = \sum_a h_{aa} + \dfrac{1}{2}\sum_a^{N_\alpha}\sum_b^{N_\alpha} (J_{ab}^{\alpha\alpha} - K_{ab}^{\alpha\alpha}) + \dfrac{1}{2}\sum_a^{N_\beta}\sum_b^{N_\beta} (J_{ab}^{\beta\beta} - K_{ab}^{\beta\beta})
+ \sum_a^{N_\alpha}\sum_b^{N_\beta} J_{ab}^{\alpha\beta}
\end{equation}
we have
\begin{align}
E_0 = \sum_i^{N_\alpha} \varepsilon_i^\alpha
+ \sum_i^{N_\beta} \varepsilon_i^\beta
- \dfrac{1}{2}\sum_i^{N_\alpha}\sum_a^{N_\alpha} (J_{ia}^{\alpha\alpha} - K_{ia}^{\alpha\alpha})
- \dfrac{1}{2}\sum_i^{N_\beta}\sum_a^{N_\beta} (J_{ia}^{\beta\beta} - K_{ia}^{\beta\beta})
- \sum_i^{N_\beta}\sum_a^{N_\alpha} J_{ia}^{\beta\alpha}
\end{align}

\subsubsection{Introduction of a Basis: The Pople-Nesbet Equations}

\subsubsection{Unrestricted Density Matrices}
\ex{3.36}
\begin{align}
\int\dd\vb{r} \rho^S(\vb{r}) &= \int\dd\vb{r} \qty[\rho^\alpha(\vb{r}) - \rho^\beta(\vb{r})] \\
&= N_\alpha - N_\beta
\end{align}
Since
\begin{equation}\label{key}
\ev{\hsS_z} = \dfrac{1}{2}(N_\alpha - N_\beta)
\end{equation}
we get
\begin{equation}\label{key}
\int\dd\vb{r} \rho^S(\vb{r}) = 2\ev{\hsS_z}
\end{equation}

\ex{3.37}
\begin{align}
\rho^\alpha(\vb{r}) &= \sum_a^{N_\alpha}  \psi_a^{\alpha*}(\vb{r})\psi_a^\alpha(\vb{r}) \notag\\
&=  \sum_a^{N_\alpha} \sum_\nu C^{\alpha*}_{\nu a}\phi_\nu^*(\vb{r})
\sum_\mu C_{\mu a}^\alpha\phi_\mu(\vb{r}) \notag\\
&= \sum_\nu\sum_\mu \qty[\sum_a^{N_\alpha} C^{\alpha*}_{\nu a}C_{\mu a}^\alpha ] \phi_\nu^*(\vb{r})\phi_\mu(\vb{r}) \notag\\
\end{align}
Let
\begin{equation}\label{key}
P^\alpha_{\mu\nu} = \sum_a^{N_\alpha} C^{\alpha*}_{\nu a}C_{\mu a}^\alpha
\end{equation}
thus
\begin{equation}\label{key}
\rho^\alpha(\vb{r}) = \sum_\nu\sum_\mu P^\alpha_{\mu\nu} \phi_\mu(\vb{r}) \phi_\nu^*(\vb{r})
\end{equation}
The formulation for $ \beta $ spin is similar.

\ex{3.38}
\begin{align}
\ev{\mathscr{O}_1} &= \sum_i^N \Braket{\chi_i | h | \chi_i} \notag\\
&= \sum_i^{N_\alpha} (\psi_i^\alpha | h | \psi_i^\alpha) + \sum_i^{N_\beta} (\psi_i^\beta | h | \psi_i^\beta) \notag\\
&= \sum_i^{N_\alpha} \sum_\nu\sum_\mu C_{\nu a}^{\alpha*} (\phi_\nu | h | \phi_\mu) C_{\mu a}^\alpha
+ \sum_i^{N_\beta} \sum_\nu\sum_\mu C_{\nu a}^{\beta*} (\phi_\nu | h | \phi_\mu) C_{\mu a}^\beta \notag\\
&= \sum_\nu\sum_\mu P_{\mu\nu}^\alpha (\phi_\nu | h | \phi_\mu)
+ \sum_\nu\sum_\mu P_{\mu\nu}^\beta (\phi_\nu | h | \phi_\mu)  \notag\\
&= \sum_\nu\sum_\mu P_{\mu\nu}^T (\phi_\nu | h | \phi_\mu)
\end{align}

\ex{3.39}
\begin{align}
\ev{\hat\rho^S} &= \Braket{\Psi_0 | \hat\rho^S | \Psi_0} \notag\\
&= \dfrac{1}{N!}\sum_{i,j}^{N!} (-1)^{p_i}(-1)^{p_j} \int\dd\vb{x}_1\cdots\dd\vb{x}_N \hsP_i\{\chi_1(1)\cdots\chi_N(N)\} \sum_m^N 2\delta(\vb{r}_m - \vb{R})s_z(m) \hsP_j\{\chi_1(1)\cdots\chi_N(N)\} \notag\\
&= \dfrac{2}{N!}\sum_{i}^{N!} \sum_m^N  \int\dd\vb{x}_1\cdots\dd\vb{x}_N \hsP_i\{\chi_1(1)\cdots\chi_N(N)\} \delta(\vb{r}_m - \vb{R})s_z(m) \hsP_i\{\chi_1(1)\cdots\chi_N(N)\} \notag\\
&= \dfrac{2}{N}\sum_{s}^{N} \sum_m^N  \int\dd\vb{x}_m \chi_s^*(m) \delta(\vb{r}_m - \vb{R})s_z(m) \chi_s(m) \notag\\
&= \dfrac{2}{N}\sum_m^N  \qty[ \sum_{s}^{N_\alpha} \int\dd\vb{r}_m \psi_s^{\alpha*}(m) \delta(\vb{r}_m - \vb{R})s_z(m) \psi_s^\alpha(m)
+ \sum_{s}^{N_\beta} \int\dd\vb{r}_m \psi_s^{\beta*}(m) \delta(\vb{r}_m - \vb{R})s_z(m) \psi_s^\beta(m) ] \notag\\
&= \dfrac{2}{N}\sum_m^N  \qty[ \dfrac{1}{2}\sum_{s}^{N_\alpha} \int\dd\vb{r}_m \psi_s^{\alpha*}(m) \delta(\vb{r}_m - \vb{R}) \psi_s^\alpha(m)
- \dfrac{1}{2}\sum_{s}^{N_\beta} \int\dd\vb{r}_m \psi_s^{\beta*}(m) \delta(\vb{r}_m - \vb{R}) \psi_s^\beta(m) ] \notag\\
&= \dfrac{2}{N}N  \qty[ \dfrac{1}{2}\sum_{s}^{N_\alpha} \psi_s^{\alpha*}(\vb{R})\psi_s^\alpha(\vb{R})
- \dfrac{1}{2}\sum_{s}^{N_\beta} \psi_s^{\beta*}(\vb{R})\psi_s^\beta(\vb{R}) ] \notag\\
&= 2 \qty[ \dfrac{1}{2} \rho^\alpha(\vb{R})
- \dfrac{1}{2}\rho^\beta(\vb{R}) ] \notag\\
&= \rho^S(\vb{R})
\end{align}
where
\begin{align}
\rho^S(\vb{R}) &= \sum_\nu\sum_\mu P^S_{\nu\mu} \phi_\mu^*(\vb{R})\phi_\nu(\vb{R}) \notag\\
&= \sum_\nu (\vb{P}^S\vb{A})_{\nu\nu} \notag\\
&= \tr(\vb{P}^S\vb{A})
\end{align}

\subsubsection{Expression for the Fock Matrices}

\subsubsection{Solution of the Unrestricted SCF Equations}
\ex{3.40}
\begin{align}
E_0 &= \sum_a^{N_\alpha} h_{aa}^\alpha + \sum_a^{N_\beta} h_{aa}^\beta + \dfrac{1}{2}\sum_a^{N_\alpha}\sum_b^{N_\alpha} (J_{ab}^{\alpha\alpha} - K_{ab}^{\alpha\alpha}) + \dfrac{1}{2}\sum_a^{N_\beta}\sum_b^{N_\beta} (J_{ab}^{\beta\beta} - K_{ab}^{\beta\beta})
+ \sum_a^{N_\alpha}\sum_b^{N_\beta} J_{ab}^{\alpha\beta} \notag\\
&= \sum_\mu\sum_\nu P^\alpha_{\nu\mu}H^{\core}_{\mu\nu} + \sum_\mu\sum_\nu P^\beta_{\nu\mu}H^{\core}_{\mu\nu} + \dfrac{1}{2}\sum_\mu\sum_\nu\sum_\lambda\sum_\sigma P^\alpha_{\nu\mu}P^\alpha_{\sigma\lambda} [(\mu\nu|\lambda\sigma) - (\mu\sigma|\lambda\nu)] \notag\\
&\quad{} + \dfrac{1}{2}\sum_\mu\sum_\nu\sum_\lambda\sum_\sigma P^\beta_{\nu\mu}P^\beta_{\sigma\lambda} [(\mu\nu|\lambda\sigma) - (\mu\sigma|\lambda\nu)] + \sum_\mu\sum_\nu\sum_\lambda\sum_\sigma P^\alpha_{\nu\mu}P^\beta_{\sigma\lambda} (\mu\nu|\lambda\sigma) \notag\\
&= \sum_\mu\sum_\nu P^\alpha_{\nu\mu} \qty{H^{\core}_{\mu\nu} + \dfrac{1}{2}\sum_\lambda\sum_\sigma P^\alpha_{\sigma\lambda} [(\mu\nu|\lambda\sigma) - (\mu\sigma|\lambda\nu)]
+ \dfrac{1}{2}\sum_\lambda\sum_\sigma P^\beta_{\sigma\lambda} (\mu\nu|\lambda\sigma)}      \notag\\
&\quad{} + \sum_\mu\sum_\nu P^\beta_{\nu\mu} \qty{ H^{\core}_{\mu\nu} + \dfrac{1}{2}\sum_\lambda\sum_\sigma P^\beta_{\sigma\lambda} [(\mu\nu|\lambda\sigma) - (\mu\sigma|\lambda\nu)]
+ \dfrac{1}{2}\sum_\lambda\sum_\sigma P^\alpha_{\sigma\lambda}(\mu\nu|\lambda\sigma)} \notag\\
&= \sum_\mu\sum_\nu P^\alpha_{\nu\mu} \qty{H^{\core}_{\mu\nu} + \dfrac{1}{2}\sum_\lambda\sum_\sigma [P^T_{\sigma\lambda} (\mu\nu|\lambda\sigma) - P^\alpha_{\sigma\lambda}(\mu\sigma|\lambda\nu)] }      \notag\\
&\quad{} + \sum_\mu\sum_\nu P^\beta_{\nu\mu} \qty{H^{\core}_{\mu\nu} + \dfrac{1}{2}\sum_\lambda\sum_\sigma [P^T_{\sigma\lambda} (\mu\nu|\lambda\sigma) - P^\beta_{\sigma\lambda}(\mu\sigma|\lambda\nu)] } \notag\\
\end{align}
while
\begin{align}
F^\alpha_{\mu\nu} &= H^{\core}_{\mu\nu} + \sum_\lambda\sum_\sigma \qty[ P^T_{\lambda\sigma}(\mu\nu|\sigma\lambda) - P^\alpha_{\lambda\sigma}(\mu\lambda|\sigma\nu)] \\
F^\beta_{\mu\nu} &= H^{\core}_{\mu\nu} + \sum_\lambda\sum_\sigma \qty[ P^T_{\lambda\sigma}(\mu\nu|\sigma\lambda) - P^\beta_{\lambda\sigma}(\mu\lambda|\sigma\nu)]
\end{align}
thus
\begin{align}
E_0 &= \sum_\mu\sum_\nu \qty{P^\alpha_{\nu\mu} \qty[\dfrac{1}{2} H^{\core}_{\mu\nu} + \dfrac{1}{2} F^\alpha_{\mu\nu} ]
+ P^\beta_{\nu\mu} \qty[\dfrac{1}{2} H^{\core}_{\mu\nu} + \dfrac{1}{2} F^\beta_{\mu\nu} ]} \notag\\
&= \dfrac{1}{2} \sum_\mu\sum_\nu \qty[P^T_{\nu\mu} H^{\core}_{\mu\nu} + P^\alpha_{\nu\mu}F^\alpha_{\mu\nu}  + P^\beta_{\nu\mu}F^\beta_{\mu\nu} ]
\end{align}

\subsubsection{Illustrative Unrestricted Calculations}
\ex{3.41}
\begin{align}
\ev{\hsS^2} &= \Braket{c_1 {^2\Psi} + c_2 {^4\Psi} | \hsS^2 | c_1 {^2\Psi} + c_2 {^4\Psi}} \notag\\
&= \dfrac{3}{4}c_1^2 + \dfrac{15}{4}c_2^2 \notag\\
&= \dfrac{3}{4}(1 - c_2^2) + \dfrac{15}{4}c_2^2 \notag\\
&= \dfrac{3}{4} + 3c_2^2
\end{align}
thus
\linespread{1.2}
\begin{table}[H]
	\centering
	\begin{tabular}{cccc}
		\hline
		basis & $ \ev{\hsS^2} $ & $ c_2 $ & contamination/\% \\ \hline
		STO-3G & 0.7652 & 0.07118 & 0.5067\\
		4-31G  & 0.7622 & 0.06377 & 0.4067\\
		6-31G* & 0.7618 & 0.06272 & 0.3933\\
		6-31G**& 0.7614 & 0.06164 & 0.3800\\
		\hline
	\end{tabular}
\end{table}
\linespread{1.0}

\subsubsection{The Dissociation Problem and Its Unrestricted Solution}
\ex{3.42}
\begin{align}
\Braket{\psi_1^\alpha|\psi_1^\alpha} &= \Braket{\cos\theta\psi_1 + \sin\theta\psi_2 | \cos\theta\psi_1 + \sin\theta\psi_2} = \cos\theta^2 + \sin\theta^2 = 1 \\
\Braket{\psi_2^\alpha|\psi_2^\alpha} &= \Braket{-\sin\theta\psi_1 + \cos\theta\psi_2 | -\sin\theta\psi_1 + \cos\theta\psi_2} = \sin\theta^2 + \cos\theta^2 = 1 \\
\Braket{\psi_1^\alpha|\psi_2^\alpha} &= \Braket{\cos\theta\psi_1 + \sin\theta\psi_2 | -\sin\theta\psi_1 + \cos\theta\psi_2} = -\cos\theta\sin\theta + \sin\theta\cos\theta = 0
\end{align}
thus $ \{\psi_1^\alpha, \psi_2^\alpha\} $ is orthonormal.\\
Similarly conclusion can be derived for $ \beta $ orbitals.

\ex{3.43}
For $ R = 1.4 $,
\begin{align}
\eta &= \dfrac{h_{22} - h_{11} + J_{22} - J_{12} + 2K_{12}}{J_{11} + J_{22} - 2J_{12} + 4K_{12}} \notag\\
&= \dfrac{(\varepsilon_2 − 2J_{12} + K_{12} ) - (\varepsilon_1 - J_{11}) + J_{22} - J_{12} + 2K_{12}}{J_{11} + J_{22} - 2J_{12} + 4K_{12}} \notag\\
&= \dfrac{\varepsilon_2 - \varepsilon_1 + J_{11} + J_{22} − 3J_{12} + 3K_{12}   }{J_{11} + J_{22} - 2J_{12} + 4K_{12}} \notag\\
&= \dfrac{0.6703 + 0.5782 + 0.6746 + 0.6975 - 3\times 0.6636 + 3\times 0.1813}{0.6746 + 0.6975 - 2\times 0.6636 + 4\times 0.1813}
notag\\
&= 1.524 > 1
\end{align}
thus unrestricted solution does not exist for this case.\\
For $ R = 4.0 $,
\begin{align}
\eta %&= \dfrac{h_{22} - h_{11} + J_{22} - J_{12} + 2K_{12}}{J_{11} + J_{22} - 2J_{12} + 4K_{12}} \notag\\
%&= \dfrac{(\varepsilon_2 − 2J_{12} + K_{12} ) - (\varepsilon_1 - J_{11}) + J_{22} - J_{12} + 2K_{12}}{J_{11} + J_{22} - 2J_{12} + 4K_{12}} \notag\\
&= \dfrac{\varepsilon_2 - \varepsilon_1 + J_{11} + J_{22} − 3J_{12} + 3K_{12}   }{J_{11} + J_{22} - 2J_{12} + 4K_{12}} \notag\\
&= \dfrac{0.0916 + 0.2542 + 0.5026 + 0.5259 - 3\times 0.5121 + 3\times 0.2651}{ 0.5026 + 0.5259 - 2\times 0.5121 + 4\times 0.2651}
notag\\
&= 0.5948
\end{align}
\begin{equation}\label{key}
\theta = \arccos(\sqrt{\eta}) = 0.6900 = 39.53\textdegree
\end{equation}

\ex{3.44}
\begin{align}
\lim\limits_{R\ra\infty} \ket{\Psi_0} &= \dfrac{1}{2}\qty[\ket{\psi_1\bar{\psi}_1} - \ket{\psi_2\bar{\psi}_2} - {\sqrt{2}}\ket{^3\Psi_1^2}] \notag\\
&= \dfrac{1}{2}\qty[\ket{\psi_1\bar{\psi}_1} - \ket{\psi_2\bar{\psi}_2} - \qty(\ket{\psi_1\bar{\psi}_2} - \ket{\psi_2\bar{\psi}_1})] \notag\\
&= \dfrac{1}{2}\qty[\dfrac{1}{2}\ket{(\phi_1 + \phi_2)(\bar\phi_1 + \bar\phi_2)}
- \dfrac{1}{2}\ket{(\phi_1 - \phi_2)(\bar\phi_1 - \bar\phi_2)}
- \qty(\dfrac{1}{2}\ket{(\phi_1 + \phi_2)(\bar\phi_1 - \bar\phi_2)}
- \dfrac{1}{2}\ket{(\phi_1 - \phi_2)(\bar\phi_1 + \bar\phi_2)})] \notag\\
&=\dfrac{1}{4}\qty(\ket{\phi_1\bar\phi_1} + \ket{\phi_1\bar\phi_2}
+ \ket{\phi_2\bar\phi_1} + \ket{\phi_2\bar\phi_2} ) - \dfrac{1}{4}\qty(\ket{\phi_1\bar\phi_1} - \ket{\phi_1\bar\phi_2}
- \ket{\phi_2\bar\phi_1} + \ket{\phi_2\bar\phi_2} ) \notag\\
&\quad{} - \dfrac{1}{4}\qty[\qty(\ket{\phi_1\bar\phi_1} - \ket{\phi_1\bar\phi_2}
+ \ket{\phi_2\bar\phi_1} - \ket{\phi_2\bar\phi_2} )
- \qty(\ket{\phi_1\bar\phi_1} + \ket{\phi_1\bar\phi_2}
- \ket{\phi_2\bar\phi_1} - \ket{\phi_2\bar\phi_2} )] \notag\\
&= \dfrac{1}{2}\qty(\ket{\phi_1\bar\phi_2}  + \ket{\phi_2\bar\phi_1} )
- \dfrac{1}{4}\qty(- 2\ket{\phi_1\bar\phi_2} + 2\ket{\phi_2\bar\phi_1}) \notag\\
&= \ket{\phi_1\bar\phi_2}
\end{align}

\end{document}