\par
\section{Driver programs for the {\tt Tree} object}
\label{section:Tree:drivers}
\par
%=======================================================================
\begin{enumerate}
%-----------------------------------------------------------------------
\item
\begin{verbatim}
drawTree msglvl msgFile inTreeFile inTagsFile outEPSfile
         heightflag coordflag radius bbox[4] frame[4] tagflag fontsize
\end{verbatim}
This driver program reads in a {\tt Tree} file and optionally a
tags {\tt IV} file and creates an EPS file with a simple
picture of a tree.
\par
\begin{itemize}
\item
The {\tt msglvl} parameter determines the amount of output.
\item
The {\tt msgFile} parameter determines the message file --- if {\tt
msgFile} is {\tt stdout}, then the output file is {\it stdout},
otherwise a file is opened with {\it append} status to receive any
output data.
\item
The {\tt inTreeFile} parameter is the input file for the {\tt Tree}
object. It must be of the form {\tt *.treef} or {\tt *.treeb}.
The {\tt Tree} object is read from the file via the
{\tt Tree\_readFromFile()} method.
\item
The {\tt inTagsFile} parameter is the input file for the {\tt IV}
vector object than holds the tags for the nodes.
It must be of the form {\tt *.ivf} or {\tt *.ivb} or {\tt none}.
The {\tt IV} object is read from the file via the
{\tt IV\_readFromFile()} method.
\item
The {\tt outEPSfile} parameter is name of the
encapsulated Postscript file to be written.
\item
The {\tt heightflag} parameter is {\tt 'D'} to use a depth metric,
(i.e., parent and child are in adjacent levels),
and {\tt 'H'} to use a height metric (i.e., a leaf is on the
outermost level).
\item
The {\tt coordflag} parameter is {\tt 'C'} to put the tree in a
Cartesian coordinate system and {\tt 'P'} for a polar coordinate
system.
\item
The {\tt radius} parameter is the radius of each node in the tree.
\item
The {\tt bbox} parameter a sequence of four numbers that form the
bounding box:
lower left $x$ value,
lower left $y$ value,
width and height.
\item
The {\tt frame} parameter a sequence of four numbers that form the
frame of the plot within the bounding box:
lower left $x$ value,
lower left $y$ value,
width and height.
\item
When {\tt tagflag = 1}, tags are drawn on the nodes.
If {\tt tagsFile} is {\tt NULL}, then node ids will be drawn on the
nodes.
Otherwise, node ids will be taken from the {\tt tagsIV} object.
\item
The {\tt fontsize} parameter is the size of the font to be used to
draw the node labels.
\end{itemize}
Use the {\tt doDraw} script file as an example.
Four plots of a tree for the {\tt R2D100} matrix ordered by nested
dissection are found below.
%-----------------------------------------------------------------------
\end{enumerate}
\par
\begin{figure}[htbp]
\caption{{\sc R2D100}: domain/separator tree.
         On the left {\tt heightflag = 'H'} and {\tt coordflag = 'C'},
         on the right {\tt heightflag = 'D'} and {\tt coordflag = 'C'}.}
\label{fig-R2D100-tree-HC}
\begin{center}
\mbox{
\psfig{file=../../Tree/doc/R2D100HC.eps,height=3.00in,width=3.00in}
}
\mbox{
\psfig{file=../../Tree/doc/R2D100DC.eps,height=3.00in,width=3.00in}
}
\end{center}
\end{figure}
\par

\begin{figure}[htbp]
\caption{{\sc R2D100}: domain/separator tree.
         On the left {\tt heightflag = 'H'} and {\tt coordflag = 'P'},
         on the right {\tt heightflag = 'D'} and {\tt coordflag = 'P'}.}
\label{fig-R2D100-tree-HP}
\begin{center}
\mbox{
\psfig{file=../../Tree/doc/R2D100HP.eps,height=3.00in,width=3.00in}
}
\mbox{
\psfig{file=../../Tree/doc/R2D100DP.eps,height=3.00in,width=3.00in}
}
\end{center}
\end{figure}


\documentclass[twocolumn]{article}

\usepackage{amsmath}
\usepackage{bm}
\usepackage{caption}
\usepackage{graphicx}
\usepackage{float}
\usepackage{subcaption}
\usepackage{gensymb}

% Make table names in captions boldfaced
\captionsetup[table]{labelfont=bf}
\captionsetup[figure]{labelfont=bf}

%opening
\title{Laboratory Observations of the Zeeman Effect}
\author{Jack Nelson,\footnote{With student Chongmo Tang as lab partner.}\\
	Physics Department,\\
	Occidental College, Los Angeles, CA}
\date{November 2016}

\begin{document}

\maketitle

\begin{abstract}
	We report our observations of the Zeeman effect of the 546.1 nm mercury spectral line in a 1T magnetic field.
	Using a polarizing filter, we were able to observe hyperfine structure in the spectrum from transitions with particular changes in the quantum number $M_j$, verifying through observation the properties anticipated by theory.
	We used measurements of the hyperfine structure of the ``Normal" Zeeman effect to calculate a value for the Bohr magneton, $\mu_B = 9.22\times10^{-24}\pm 0.06\times10^{-24} J\cdot T^{-1}$, which is consistent with the modern accepted value to within 0.63\% agreement.
\end{abstract}

\section{Introduction} \label{sec:Intro}
	We begin with an overview of the theory behind the Zeeman effect, then proceed into a discussion and derivation of the equation we used to calculate the value of the Bohr magneton from our measurements.

	\subsection{Theoretical Overview} \label{sec:Theory}
		The splitting of the spectra of mercury into hyperfine lines when an external magnetic field is applied is the result of the mercury atom's magnetic-dipole interaction with the external magnetic field.

		In its ground state, the mercury atom has 80 electrons in which the n = 1, 2, 3, 4, and 5 shells are filled, leaving two 6s valence electrons.
		The spectral emissions of mercury come from the transitions of these two valence electrons between different quantum states.
		Figure \ref{fig:MercuryEnergyLevelsDiagram} shows the different possible transitions of the mercury atom.
		Our observations will be limited to the $7^3S_1 \rightarrow 6^3P_2$ transition which yields the 546.1 nm green spectral line.

		When no external magnetic field is present, hyperfine structure in atomic spectra can be observed due to the spin-angular momentum coupling of electrons.
		See pp. 47 of \cite{melissinos_experiments_1966} for a more detailed discussion.
		In the presence of an external magnetic field, the interaction of the total angular momentum of the electrons with the magnetic field also splits the spectral lines into finer lines of slightly different wavelength.
		Figure \ref{fig:SpectralSplitting} shows how a single transition from the $l=2$ to $l=1$ states results in hyperfine structure in the presence of a magnetic field.

		\begin{figure}
			\centering
			\includegraphics[width=1.0\linewidth]{Images/SpectralSplitting}
			\caption{A single spectral line caused by an atomic energy transition will be split into multiple ``hyperfine" lines due to the interaction of the total angular momentum of the valence electrons with an external magnetic field. (a) shows the transition in the absence of an external magnetic field. The transition between the $l=2$ and the $l=1$ states ($d\rightarrow p$) produces a single spectral line. (b) shows how these formerly single energy-levels are split into multiple states in the presence of an external magnetic field. (c) shows the possible transitions between the new states, each of which produces a different spectral line with slightly different wavelength.\cite{melissinos_experiments_1966}}
			\label{fig:SpectralSplitting}
		\end{figure}

		The following discussion is a summary of the theory behind the Zeeman effect. For more detailed discussions and derivations, see \cite{melissinos_experiments_1966} and \cite{stoltenberg_zeeman_2007}.

		Hyperfine splitting in the presence of a magnetic field is due to the magnetic dipole interactions of an electron with the magnetic field.
		These interactions result in an added energy $\Delta E$ per electron for a quantum state that is given by
		\begin{equation}
			\Delta E = m\mu_0B
		\end{equation}
		where $m$ is the quantum number quantifying the projection of the total angular momentum along the z-axis of the atom, $\mu_0$ is the Bohr magneton, the fundamental magnetic moment of the electron, and $B$ is the applied magnetic field.

		To understand the origin of these interactions, consider a simple current loop such as the one shown in Figure \ref{fig:CurrentLoopDiagram}.
		The associated magnetic moment $\mathbf{\mu}$ of the loop has the magnitude:
		\begin{equation}
			\mu = IA
		\end{equation}
		Where $A=\pi a^2$ is the disk area enclosed by the loop.
		The direction of $\mathbf{\mu}$ is given by the right-hand rule.

		When an external magnetic field $\mathbf{B}$ is applied, the torque on the loop is given by
		\begin{equation}
			\boldsymbol{\tau} = \boldsymbol{\mu} \times \mathbf{B}
		\end{equation}
		The resulting potential energy of the current loop in the magnetic field is
		\begin{equation} \label{eq:MagneticEnergy}
			E_{B} = -\boldsymbol{\mu}\cdot \mathbf{B}
		\end{equation}
		That is, the magnetic field will do work on the current loop when its orientation is changed in the field.

		Thus, we can classically model the electron as a loop of current, as shown in Figure \ref{fig:CurrentLoopDiagram}, with magnetic moment given by
		\begin{equation}
			\boldsymbol{\mu_l} = -\left(\frac{e}{2m_e}\right) \mathbf{l}
		\end{equation}
		where $m_e$ is the mass of the electron, $e$ is the electron charge, and $\mathbf{l}$ is the orbital angular momentum.
		This is then conventionally stated as
		\begin{equation} \label{eq:MagneticMoment}
			\boldsymbol{\mu_l} = -g_l \left( \frac{e\hbar}{2m_e} \right) \left(\frac{\mathbf{l}}{\hbar}\right) = -g_l\mu_B \left(\frac{\mathbf{l}}{\hbar}\right)
		\end{equation}
		where $\mu_B$ is the Bohr magneton, and $g_l$ is a unitless ``g factor" which relates the magnitude of the magnetic moment in units of $\mu_B$ to the magnitude of the orbital angular momentum in units of $\hbar$.
		This ``g factor" will be important later.

		\begin{figure}
			\centering
			\includegraphics[width=0.7\linewidth]{Images/CurrentLoopDiagram}
			\caption{A loop of current about a vertical axis with magnetic moment $\mathbf{\mu}$.\cite{stoltenberg_zeeman_2007}}
			\label{fig:CurrentLoopDiagram}
		\end{figure}

		We use this classical model of the electron as a current loop to begin our theory of the Zeeman effect, but it doesn't get us far.
		For the purposes of simplicity, we will at first consider an atom with only one valence electron, such as sodium.
		Later, we will include the effect of multiple valence electrons, such as in mercury.
		\footnote{As such, we will use lower-case $lsmj$ to denote the quantum numbers of single electrons, while upper-case $LSMJ$ will be used for systems of more than one electron.}

		The magnetic torque on an atom is due to the coupled interaction of an electron's total angular momentum, $j$, with the magnetic field.
		The two forms of angular momentum in electrons - orbital angular momentum, $l$, and intrinsic spin momentum, $s$ - couple to form a measure of total angular momentum, $j$:

		\begin{equation}
		\mathbf{j} = \mathbf{l} + \mathbf{s}
		\label{eq:TotalAngularP}
		\end{equation}
		which has magnitude $\left|\mathbf{j}\right| = \sqrt{j\left(j+1\right)}\hbar$.
		Figure \ref{fig:TotalAngularMomentumDiagram} shows how these two quantities (vectors, actually) combine to form total angular momentum.

		The z-component of $j$, $j_z$, is also given by the sum of the z-components of $s$ and $l$
		\begin{equation}
			j_z = l_z + s_z = m_j\hbar
		\end{equation}
		where $m_j$ is the quantum number quantifying the projection of total angular momentum along the z-axis.

		\begin{figure}
			\centering
			\includegraphics[width=0.7\linewidth]{Images/TotalAngularMomentumDiagram}
			\caption{Electron spin angular momentum $\mathbf{s}$ and orbital angular momentum $\mathbf{l}$ sum to create total angular momentum $\mathbf{j}$.\cite{stoltenberg_zeeman_2007}}
			\label{fig:TotalAngularMomentumDiagram}
		\end{figure}

		Selection rules place restrictions on how the quantum numbers may change when a valence electron undergoes a transition.
		These selection rules are:
		\begin{enumerate}
			\item $\Delta s = 0$
			\item $\Delta l = 0, \pm1$
			\item $\Delta j = 0, \pm1$, but not $\Delta j = 0 \rightarrow \Delta j = 0$
			\item $\Delta m_j = 0, \pm1$, but not $m_j=0\rightarrow m_j=0$ if $\Delta j = 0$
		\end{enumerate}

		From equation (\ref{eq:MagneticEnergy}), the energy change due to a magnetic field is related to the magnetic moment of the electron and the magnitude of the field.
		Using equation (\ref{eq:MagneticMoment}), we can develop an expression for the quantum-mechanically accurate energy difference of the valence electron in the presence of the magnetic field:
		\begin{equation}
			\Delta E = E_B = g_jm_j\mu_BB
			\label{eq:EnergyShift}
		\end{equation}

		The g factor in equation (\ref{eq:EnergyShift}) is known as the Land\'{e} g factor, and is given by
		\begin{equation}
		g_L = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}
		\label{eq:LandeGFactor}
		\end{equation}

		The Lande\'{e} g factor is derived from considering the projections of the spin and orbital magnetic moments along the total angular momentum j.
		\cite{stoltenberg_zeeman_2007}

		When we consider multiple valence electrons, under LS or Russell-Saunders coupling, we can couple together the angular momenta for more than one electron.
		The individual orbital and spin angular momenta combine to form total angular momenta, thus $L$ and $S$ combine to form $J$ just as $l$ and $s$ combine to form $j$:
		\begin{equation}
			\mathbf{J} = \mathbf{L} + \mathbf{S}
		\end{equation}

		Figure \ref{fig:SpectralSplitting} shows that spectral splitting is dependent on the quantum number $m_j$.
		$m_j$ can take on $2l+1$ the values ranging from $l$ to $-l$.

		The polarization of the emitted light during the transition is also entirely determined by $\Delta m_j$.
		For $\Delta m_j = 0$, the emitted photons are polarized linearly and parallel to the field lines of the applied magnetic field $\mathbf{B}$.
		For $\Delta m_j=\pm1$, emitted photons are circularly polarized along the direction of $\mathbf{B}$ and appear linearly polarized perpendicular to $\mathbf{B}$ when viewed side-on to $\mathbf{B}$.
		\begin{figure}
			\centering
			\includegraphics[width=0.5\textwidth]{Images/MercuryEnergyLevelsDiagram.png}
			\caption{Energy level transitions of mercury between quantum states. For the purposes of our observations, we are interested in the $7^3S_1 \rightarrow 6^3P_2$ transition which produces 546.1 nm light. Figure 7.39 of \cite{melissinos_experiments_1966}}
			\label{fig:MercuryEnergyLevelsDiagram}
		\end{figure}

		The energy difference between the hyperfine spectra is caused by the differing values of $\Delta m_j$ and determined by measuring the relative differences in wavelength between the fine spectra.
		From the basic Planck relation we have
		\begin{equation}
			E = h\nu = \frac{hc}{\lambda}
		\end{equation}
		where $h$ is the Planck constant, $\nu$ is wave frequency, and $\lambda$ is wavelength.

		The relative energy difference of two wavelengths is then given by

		\begin{equation}
			\Delta E = M_j g_L B \mu_B
			\label{eq:EnergyShift}
		\end{equation}

		\subsection{Calculation of the Bohr Magneton} \label{subsec:BohrCalc}
			The fundamental unit of magnetic moment, the Bohr magneton, is given by

			\begin{equation}
				\mu_B = \frac{eh}{4\pi m_e}
			\end{equation}
			where $m_e$ is the electron mass, $e$ is the electron charge, and $h$ is Planck's constant.
			We are interested in measuring this fundamental quantity through observation.
			To do so, we will measure the magnitude of the hyperfine spectral line splitting in the presence of a magnetic field, and relate this splitting to the magnitude of the Bohr magneton.

			Consider the 546.1 nm mercury line in the presence of a magnetic field.
			As described in the previous section, the interaction of the spin-coupled total angular momentum of the valence electron will interact with the magnetic field to produce hyperfine structure with energy, and thus wavelength, dependent on the value of $\Delta M_j$ in the transition.

			For the case of $M_j = 0 \rightarrow M_j = 0$, the resulting spectral line will be the same as the un-split spectral line in the $B=0$ case with $\lambda = 546.1 nm$.
			In the $M_j = +1 \rightarrow M_j = +1$ and $M_j = -1 \rightarrow M_j = -1$ cases, however, the resulting hyperfine spectral line will be of slightly greater or less energy than the $B=0$ spectral line, thus resulting in a ``triplet" hyperfine structure in which the central 546.1 nm spectral line is surrounded on either side by two additional hyperfine spectral lines.

			We calculate the Land\'{e} g-factor for both the S and P states using equation (\ref{eq:LandeGFactor}).
			For the $^3S_1$ state, $L=0$ and $S=1$, thus $J = 1$, and $g_S = 2$.
			For the $^3P_2$ state, $L=1$ and $S=1$, thus $J=2$, and $g_P = 3/2$.

			Thus, using equation (\ref{eq:EnergyShift}), the energy shifts for the $M_j = \pm1$ cases for the $S$ and $P$ states in the presence of a magnetic field will be:
			\begin{equation}
				\Delta E_S = \pm 2B\mu_B
			\end{equation}
			\begin{equation}
				\Delta E_P = \pm \frac{3}{2}B\mu_B
			\end{equation}

			Thus, the photon energy for the $M_j = +1 \rightarrow M_j = +1$ transition will be
			\begin{equation}
				E = h\nu_+ = (E_S + \Delta E_S) - (E_P + \Delta E_P)
			\end{equation}
			and for the $M_j = -1 \rightarrow M_j = -1$ it will be
			\begin{equation}
				E = h\nu_- = (E_S - \Delta E_S) - (E_P - \Delta E_P)
			\end{equation}

			The difference in energy between the two hyperfine lines will then be
			\begin{equation}
				h(\nu_+ - \nu_-) = 2(\Delta E_S - \Delta E_P)
			\end{equation}
			Which is then related to the Bohr magneton by
			\begin{equation}
				2(\Delta E_S - \Delta E_P) = 2\left( 2B\mu_B - \frac{3}{2}B \mu_B \right) = B \mu_B
			\end{equation}
			Thus we have a relation for the difference between the inner and outer hyperfine spectra frequencies (and therefore their wavelengths), and the value of the Bohr magneton.
			\begin{equation}
				h(\nu_+ - \nu_-) = B\mu_B
			\end{equation}

			We convert these frequencies to wavelengths using the fundamental relation $f = c/\lambda$

			\begin{equation}
				h \left( \frac{c}{\lambda_+} - \frac{c}{\lambda_-} \right) = \frac{hc \left(\lambda_- - \lambda_+ \right)}{\lambda_- \lambda_+}
			\end{equation}
			\begin{equation}
				= \frac{hc\Delta\lambda}{\lambda^2} = B\mu_B
			\end{equation}
			where $\lambda_-\lambda_+ = \lambda^2$ to a good approximation because of the very small difference in wavelength between the hyperfine spectra.

			Rearranging, we have
			\begin{equation}
				\Delta\lambda = \frac{B\mu_B\lambda^2}{hc}
				\label{eq:WavelengthEq}
			\end{equation}

			We now have a relation between the change in wavelength between the outer hyperfine spectra and the Bohr magneton.
			We must now develop a method and theory of measuring these very slight differences in wavelength.
			This is accomplished using a Fabry-Perot interferometer to make these slight wavelength differences measurable.

			\subsubsection{Fabry-Perot Interferometer}
				In order to measure very slight wavelength differences, we employed a Fabry-Perot interferometer.
				The main mechanism of the interferometer is the etalon, which consists of two partially optically reflected glass plates very precisely aligned parallel to each other.

				\begin{figure}
					\centering
					\includegraphics[width=0.95\linewidth]{Images/EtalonDiagram}
					\caption{Cross-section of the Fabry-Perot etalon. Light entering the etalon undergoes multiple internal reflections before being reflected or transmitted externally.}
					\label{fig:Fabry-Perot}
				\end{figure}

				In the diagram shown in figure \ref{fig:Fabry-Perot}, the difference in path length between light rays 1 and 2 is given by
				\begin{equation}
					\Delta l = DA + AB - CB
				\end{equation}
				\begin{equation}
					= \frac{d}{Cos\theta} + \frac{d}{Cos\theta} - 2d(Sin\theta)Tan\theta
				\end{equation}
				\begin{equation}
					= \frac{2d}{Cos\theta} - \frac{2dSin^2\theta}{Cos\theta} =2dCos\theta
				\end{equation}
				Where $d$ is the gap distance of the etalon, approximately 1.995 mm.

				We can expand $Cos\theta$ in a Taylor series and only keep the first two terms because $\theta$ is very small
				\begin{equation}
					2dCos\theta = 2d\left(1-\frac{\theta^2}{2}\right)
				\end{equation}

				Thus, the total path length difference of the two rays is
				\begin{equation}
					\Delta l = 2d\left(1-\frac{\theta^2}{2}\right)
					\label{eq:EtalonPathLengthDiff}
				\end{equation}

				The two rays will be in phase and produce an interference maximum if and only if
				\begin{equation}
					\Delta l = (n+k)\lambda, k = 0, 1, 2, 3, ...
				\end{equation}

				Thus equation (\ref{eq:EtalonPathLengthDiff}) gives us
				\begin{equation}
					2d\left(1-\frac{\theta_k^2}{2}\right) = (n_k)\lambda, k = 0, 1, 2, 3,...
				\end{equation}
				where $\theta_k$ is the angle to the $k-th$ interference ring.
				See figure \ref{fig:B=0Pattern} for an example of the interference pattern seen through the Fabry-Perot etalon.

				Given a camera focal length f as shown in figre \ref{fig:CameraGeometry}, we can approximate the angle $\theta_k$ to be $\theta_k = R_k/f$, giving us
				\begin{equation}
					2d\left(1 - \frac{R_k^2}{2f^2}\right) = (n-k)\lambda, k=1,2,3,...
					\label{eq:Rk}
				\end{equation}

				We can get around not knowing the camera focal length using a measurement of the $B=0$ interference rings.
				For the $k=0$ ring, we have
				\begin{equation}
					2d\left(1-\frac{R_0^2}{2f^2}\right) = n\lambda
					\label{eq:R0}
				\end{equation}

				Subtracting equation (\ref{eq:Rk}) from (\ref{eq:R0}), we arrive at
				\begin{equation}
					\frac{d}{f^2}\lambda = \frac{k}{R_k^2 - R_0^2} = C_0
					\label{eq:ConstantEq}
				\end{equation}

				Thus by measuring the $R$ values for the interference rings with $B=0$, we can determine $C_0$.

				\begin{figure}
					\centering
					\includegraphics[width=0.95\linewidth]{Images/CameraGeometry}
					\caption{The camera lens geometry. Figure 4 from \cite{brolight_technology_co._ltd._zeeman_????}}
					\label{fig:CameraGeometry}
				\end{figure}

			\subsubsection{Final Derivation of the Bohr Magneton Equation}

				With equation (\ref{eq:Rk}), we have a general expression relating the interference rings' radial measurements to their wavelengths.
				Thus, we could measure the inner and outer rings of the hyperfine structure and take the difference to get the difference in wavelength between them.
				We thus arrive at
				\begin{equation}
					\left(\frac{d}{f^2}\right)\left(R_{k+}^2 - R_{k-}^2\right) = (n-k)(\lambda_- - \lambda_+)
				\end{equation}
				where $R_{k+}$ is the outer ring radius, $R_{k-}$ is the inner ring radius, $\lambda_+$ is the outer ring's wavelength, and $\lambda_-$ the inner ring's wavelength.

				Using equation (\ref{eq:WavelengthEq}) and the approximation $(n-k)\lambda = n\lambda = 2d$ and solving for the Bohr magneton, we have
				\begin{equation}
					\mu_B = \left( \frac{C_0 hc}{2dB} \right) \left( R_{k+}^2 - R_{k-}^2 \right)
					\label{eq:BohrEquation}
				\end{equation}

				We will use this equation in Section \ref{subsec:BohrCalc} to calculate an experimental value for the Bohr magneton.

\section{Methods and Procedures}
	The Pasco Scientific Model SE-9654 Zeeman Effect Experiment was used to make all observations.
	The experiment consisted of two large solenoids oriented along the same axis with an air gap between them in which a mercury lamp was placed.
	An optical rail was used to align a polarizing filter, a collimating lens, an interference filter, a Fabry-Perot interferometer, and a CMOS camera.
	The basic experimental technique consisted of imaging the interference pattern of the 546.1 nm mercury spectral line with the CMOS camera through the polarizer and Fabry-Perot interferometer.
	Figures \ref{fig:ContextView} and \ref{fig:SolenoidDetail} show the optics bench setup, including the solenoid electromagnet.

	\begin{figure*}
		\centering
		\begin{subfigure}{0.5\textwidth}
			\includegraphics[width = 0.98\textwidth]{Images/ExperimentOverview.jpg}
			\caption{}
			\label{subfig:Overview}

		\end{subfigure}%
		\begin{subfigure}{0.5\textwidth}
				\includegraphics[width = 0.98\textwidth]{Images/OpticsRail.jpg}
				\caption{}
				\label{subfig:OpticsRail}

		\end{subfigure}%
		\caption{Overview of the Experiment configuration. Figure \ref{subfig:Overview} shows the entire apparatus with the variable power supply for the solenoid and mercury lamp, the solenoid, the mercury lamp (hidden by the solenoid), the optical track, and the optical instruments. Figure \ref{subfig:OpticsRail} shows the optical instruments in detail.}
		\label{fig:ContextView}
	\end{figure*}

	\begin{figure*}
		\centering
		\begin{subfigure}{0.5\textwidth}
			\includegraphics[width = 0.98\textwidth]{Images/PerpendicularView.jpg}
			\caption{}
			\label{subfig:PerpView}
		\end{subfigure}%
		\begin{subfigure}{0.5\textwidth}
			\includegraphics[width = 0.98\textwidth]{Images/AxialView.jpg}
			\caption{}
			\label{subfig:AxialView}
		\end{subfigure}%
		\caption{Detail view of the solenoid with the mercury lamp source. Figure \ref{subfig:PerpView} shows the configuration with the magnetic field lines perpendicular to the light source. Figure \ref{subfig:AxialView} shows the axial field configuration looking down the sight-hole of the solenoid to the mercury lamp.}
		\label{fig:SolenoidDetail}
	\end{figure*}

	To study the Zeeman effect, observations of a mercury lamp were made first with no magnetic field and $90\deg$ polarization, then with an approximately $1T$ magnetic field.
	Observations with a magnetic field included:
	\begin{itemize}
		\item No polarization
		\item Field-parallel polarization
		\item Field-perpendicular polarization
		\item Axial B-field
	\end{itemize}

	Figure \ref{fig:FieldConfig} shows the perpendicular and axial configurations of the magnetic field used for the observations.

	The observations and results of each of these field configurations will be presented and discussed in Section \ref{sec:DataAndResults}.

	\begin{figure*}
		\centering
		\begin{subfigure}{0.5\textwidth}
			\includegraphics[width = 1.0\textwidth]{Images/FieldConfigA.jpg}
			\caption{Perpendicular field configuration.}
			\label{subfig:PerpFieldConfig}
		\end{subfigure}%
		\begin{subfigure}{0.5\textwidth}
			\centering
			\includegraphics[width=1.0\textwidth]{Images/FieldConfigB.jpg}
			\caption{Axial field configuration.}
			\label{subfig:AxialFieldConfig}
		\end{subfigure}%
		\caption{The two magnetic field configurations observed. In (\ref{subfig:PerpFieldConfig}), the magnetic field is oriented perpendicularly to the path of the observed light from the mercury lamp. $90\degree$ polarization is defined to be perpendicular to the field lines, while $0\degree$ polarization is parallel to the field lines. In configuration (\ref{subfig:AxialFieldConfig}), light from the lamp travels along the magnetic field lines down the center of one of the solenoids through a sight hole.}
		\label{fig:FieldConfig}
	\end{figure*}

\section{Data and Results} \label{sec:DataAndResults}

	 With no applied magnetic field, the 546.1nm mercury spectral line as viewed through the Fabry-Perot interferometer produces the interference pattern as shown in Figure \ref{fig:B=0Pattern}.
	 The $B=0$ interference pattern shows discrete rings, with the innermost ring denoted as the $k=0$ ring, the next innermost ring the $k=1$ ring, and so on.

	 \begin{figure*}
	 	\centering
	 	\includegraphics[width=0.65\linewidth]{Images/B=0Pattern}
	 	\caption{Interference pattern of the 546.1nm mercury line with no applied magnetic field. The interference lines are singular.}
	 	\label{fig:B=0Pattern}
	 \end{figure*}

	 Table \ref{tab:B0Data} shows the measured radii of the interference rings for the $k=1$ through $k=4$ rings.
	 The $k=0$ ring was not used in subsequent observations and calculations because it was not resolved enough to be measurable.

	 The ``bullseye'' $B=0$ pattern will be the standard interference pattern we will compare all subsequent patterns on.

	\subsection{Observations perpendicular to the Magnetic Field}
		The first observations were made such that the mercury lamp was viewed perpendicularly to the magnetic field, as is shown in Figure \ref{subfig:PerpFieldConfig}.

		When viewed without a polarizer, the interference pattern appears as in Figure \ref{fig:Run2_NoPolarizer_detail}.
		All nine of the transition spectra, corresponding to all nine possible unique $\Delta M_j$ transition values, can be seen.

		\begin{figure*}
			\centering
			\includegraphics[width=0.6\linewidth]{Images/Run2_NoPolarizer_detail}
			\caption{Mercury spectra in the presence of a magnetic field viewed without a polarizer. All spectra from all nine unique $\Delta M_j$ transitions can be seen.}
			\label{fig:Run2_NoPolarizer_detail}
		\end{figure*}

		From the theory discussed in section \ref{sec:Theory}, different values of $\Delta M_j$ in the mercury transition will produce different polarizations of the emitted light.
		Thus, we can use polarization to identify particular hyperfine lines which correspond to certain values of $\Delta M_j$.

		\subsubsection{$\mathbf{\Delta M_j = 0}$ Transition spectra}
			With a $1T$ magnetic field and the polarizer set parallel to the field lines, we observed the mercury spectra with polarization parallel to the applied magnetic field. A distinct triple ring structure can be seen Figure \ref{fig:Triplet90deg} in which each interference ring consists of a bright central ring surrounded by two fainter rings.
			This is what has been historically termed the ``Normal" Zeeman effect.

			\begin{figure*}
				\centering
				\includegraphics[width=0.6\linewidth]{Images/Run1_ParallelPolarizer}
				\caption{The ``Normal" Zeeman effect. Mercury interference pattern viewed perpendicular to the magnetic field lines through a polarizer oriented parallel to the field. The interference rings correspond to photons produced by $\Delta M_j = 0$ transitions polarized parallel to the magnetic field lines.}
				\label{fig:Triplet90deg}
			\end{figure*}

			Using PASCO Software, The radii of the inner and outer rings of the $k=1$ through $k=4$ interference rings were measured.
			These radii are given in Table \ref{tab:RadiusMeasurements} and will be used to calculate the value of the Bohr magneton in Section \ref{subsec:BohrMagneton}.
			\begin{table*}[]
				\centering
				\begin{tabular}{l|lll}
					k = 1 & & & \\
					Measurements & $R+$ (m) & $R-$ (m)  & $\Delta$R (m) \\ \hline
					1         & 0.4282 & 0.3893  & 0.0389 \\
					2         & 0.4278 & 0.3896  & 0.0382 \\
					3         & 0.4275 & 0.3887  & 0.0388 \\
					Mean      & $0.4278\pm0.4\times10^{-3}$       & $0.3892\pm0.5\times10^{-3}$        & $0.039\pm0.4\times10^{-3}$ \\ \hline
					k = 2 & & & \\
					Measurements & $R+$ (m) & $R-$ (m)  & $\Delta$R (m) \\ \hline
					1           & 0.5819 & 0.5539 & 0.0280 \\
					2           & 0.5815 & 0.5536 & 0.0279 \\
					3           & 0.5818 & 0.5536 & 0.0282 \\
					Mean        & $0.5817\pm0.2\times10^{-3}$ & $0.5537\pm0.2\times10^{-3}$ & $0.0280\pm0.2\times10^{-3}$ \\ \hline
					k = 3 & & & \\
					Measurements & $R+$ (m) & $R-$ (m)  & $\Delta$R (m) \\ \hline
					1    & 0.7035 & 0.679  & 0.0245 \\
					2    & 0.7033 & 0.6798 & 0.0235 \\
					3    & 0.7025 & 0.6805 & 0.0220 \\
					Mean & $0.7031\pm0.5\times10^{-3}$ & $0.6798\pm0.8\times10^{-3}$ & $0.0233\pm0.001$ \\ \hline
					k = 4 & & & \\
					Measurements & $R+$ (m) & $R-$ (m)  & $\Delta$R (m) \\ \hline
					1    & 0.8058 & 0.7861 & 0.02  \\
					2    & 0.8057 & 0.7865 & 0.02  \\
					3    & 0.8059 & 0.7857 & 0.02  \\
					Mean & $0.8058\pm0.1\times10^{-3}$ & $0.7861\pm0.4\times10^{-3}$ & $0.020\pm0.001$
				\end{tabular}

				\caption{Radius measurements of the hyperfine structure of the k=1 through k=4 interference rings with a 1T magnetic field and $\mathbf{90\degree}$ polarization.}
				\label{tab:RadiusMeasurements}
			\end{table*}

		\subsubsection{$\mathbf{\Delta M_j = \pm1}$ Transition Spectra}
			By orienting the polarizer perpendicular to the magnetic field lines, we were able to observe the spectra due to the circularly-polarized $\Delta M_j = \pm1$ transitions.
			The result is shown in figure \ref{fig:Run2_0Deg}, in which it can be seen that each interference ring is now missing the $\Delta M_j=0$ rings which have been filtered by the polarizer, while the outer rings due to the $\Delta M_j = \pm1$ transitions are visible.

		\begin{figure*}
			\centering
			\includegraphics[width=0.55\linewidth]{Images/Run2_PerpendicularPolarizer}
			\caption{Interference pattern viewed through a polarizer oriented perpendicular to the magnetic field lines. The $\Delta M_j = 0$ transition spectra polarized parallel to the magnetic field cannot be seen, leaving a large gap between the circularly-polarized $\Delta M_j = \pm1$ transition spectra.}
			\label{fig:Run2_0Deg}
		\end{figure*}

	\subsection{Spectra Along an Axial Magnetic Field}
		Our next observation was axial with the magnetic field, as shown in figures \ref{subfig:AxialFieldConfig} and \ref{subfig:AxialView}.
		The result is shown in figure \ref{fig:Run3_AxialField}.
		Regardless of whether a polarizer was used or not, and indeed regardless of the orientation of the polarizer, the pattern did not change.
		This agrees with the theory, as the linearly-polarized $\Delta M_j = 0$ transitions cannot propagate axially, as their field vectors $\mathbf{E}$ and $\mathbf{B}$ must be mutually perpendicular with their propagation direction.

		The circularly-polarized $\Delta M_j = \pm1$ transitions can be seen, however, as they are polarized circularly about the magnetic field lines.

		\begin{figure*}
			\centering
			\includegraphics[width=0.55\linewidth]{Images/Run3_AxialField}
			\caption{Interference pattern viewed axially along the magnetic field through the sight-hole in the electromagnet as shown in Figure \ref{subfig:AxialFieldConfig}. Only the circularly-polarized $\Delta M_j = \pm1$ transition spectra can be seen since photons from the $\Delta M_j = 0$ transition spectra cannot propagate in this direction because their $\mathbf{E}$ and $\mathbf{B}$ field vectors must be perpendicular to their propagation direction.}
			\label{fig:Run3_AxialField}
		\end{figure*}

	\subsection{Measurement of the Bohr Magneton} \label{subsec:BohrMagneton}
		Using the measurements of the split interference rings from the observation of the ``Normal" Zeeman effect with the $\Delta M_j = 0$ transition spectra, we were able to establish a value for the Bohr magneton.
		The inner and outer radius of each interference triplet was measured using PASCO software and is tabulated in Table \ref{tab:RadiusMeasurements}.

		To calculate the magnitude of the Bohr magneton, we use the theory from Section \ref{subsec:BohrCalc}.
		First, we must calculate the constant $C_0$ in equation (\ref{eq:ConstantEq}).
		This is accomplished using the un-split interference ring radii from the $B=0$ case.
		These measurements are given in Table \ref{tab:B0Data}.

		\begin{table}[h]
			\centering
			\begin{tabular}{l|l}
				k & Radius (m) \\ \hline
				4 & 0.7956 $\pm$ $0.1\times10^{-3}$    \\
				3 & 0.6911 $\pm$ $0.2\times10^{-3}$    \\
				2 & 0.5678 $\pm$ $0.1\times10^{-3}$    \\
				1 & 0.4096 $\pm$ $0.1\times10^{-3}$
			\end{tabular}
			\caption{Interference ring radii of the mercury 546.1 nm spectra with no magnetic field.}
			\label{tab:B0Data}
		\end{table}

		Using equation (\ref{eq:ConstantEq}), the value of $C_0$ for each interference ring $k=1$ through $k=4$ was calculated.
		The average of these values was then taken as the best value for $C_0$.
		Table \ref{tab:C0} lists the calculated values for each interference ring, as well as the overall mean and standard deviation.

		\begin{table}[]
			\centering
			\begin{tabular}{l|l}
				k & $C_0$ $(m^{-2})$ \\ \hline
				4 & 6.4523 \\
				3 & 6.4575 \\
				2 & 6.4633 \\
				1 & 6.4598 \\ \hline
				Mean & $6.458\pm0.005$
			\end{tabular}
			\caption{Measured values of the constant $C_0$ from the $B=0$ interference rings.}
			\label{tab:C0}
		\end{table}

		With a precise measured value for $C_0$, values for the Bohr magneton can then be calculated with equation (\ref{eq:BohrEquation}).
		The difference between inner and outer radii for each interference ring in the $B\neq 0$, parallel-field polarization case, which is related to the magnitude of the change in wavelength between the $\Delta M_j = \pm1$ transitions, is used to calculate the Bohr magneton.
		These values for $R_{k+}$ and $R_{k-}$ are given in Table \ref{tab:RadiusMeasurements}.

		The remaining constants in equation (\ref{eq:BohrEquation}) are Planck's constant, $h = 6.63\times10^{-34}$, the speed of light, $c = 2.99\times10^8 m/s$, the gap distance of the Fabry-Perot etalon, $d = 1.995\times10^{-3} m$, and the magnitude of the magnetic field at $5 A$, $B = 1.105 T$.

		The calculated values for the Bohr magneton for each interference ring as well as the mean and standard deviation are given in Table \ref{tab:BohrCalcs}.

		\begin{table}[]
			\centering
			\begin{tabular}{l|l}
				$k$  & $\mu_B \ (J\cdot T^{-1})$ \\ \hline
				1    & $9.16\times10^{-24}$   \\
				2    & $9.24\times10^{-24}$   \\
				3    & $9.36\times10^{-24}$   \\
				4    & $9.10\times10^{-24}$   \\ \hline
				Mean & $9.22\times10^{-24}\pm0.06\times10^{-24}$
			\end{tabular}
			\caption{Calculated values of the Bohr magneton $\mu_B$ for each interference ring, and the mean calculated value for all interference rings.}
			\label{tab:BohrCalcs}
		\end{table}

		Our calculated best value for the magnitude of the Bohr magneton is thus
		\begin{equation}
			\mu_B = 9.22\times10^{-24}\pm 0.06\times10^{-24} J\cdot T^{-1}
		\end{equation}

		For comparison, the SI value of the Bohr magneton is $9.274\times10^{-24} J\cdot T^{-1}$\cite{_codata_????}, which is within our margin of measured random error.
		Additionally, our measured value agrees with the accepted SI value to within 0.63\%.

\section{Conclusion} \label{sec:Conclusion}
	We observed the Zeeman  effect and verified its effect on the 546.1 nm mercury spectral line in the presence of a $\approx1T$ magnetic field.
	We verified that the hyperfine structure was produced by transitions of various changes of the quantum number $\Delta M_j$ which are constrained by the quantum selection rules.
	Using a polarizer, we showed that the different hyperfine spectra have either parallel or circular polarization along the magnetic field lines.
	We were able to separately observe light polarized parallel to the magnetic field lines created in $\Delta M_j = 0$ transitions, as well as axially circular-polarized light created in $\Delta M_j = \pm1$ transitions.
	Finally, we arrived at a measured value for the Bohr magneton of $\mu_B = 9.22\times10^{-24}\pm 0.06\times10^{-24} J\cdot T^{-1}$, which is consistent with the modern accepted SI value to within 0.63\%.
\bibliographystyle{ieeetr}
\bibliography{ZeemanPaper}

\end{document}


\chapter*{Abstract}

This thesis considers a Follow-the-Leader model where the velocity of a vehicle depends both on the distance to the vehicle ahead and the position of the vehicle. The main part of the thesis is a proof that the microscopic Follow-the-Model model converges in the many particle limit to the macroscopic Lighthill-Whitham-Richards model for traffic flow, with a space dependent flux.

\iffalse
The \texttt{ntnuthesis} document class is a customised version of the standard \LaTeX{} \texttt{report} document class. It can be used for theses at all levels – bachelor, master and PhD – and is available in English (British and American) and Norwegian (Bokmål and Nynorsk). This document is ment to serve (i) as a description of the document class, (ii) as an example of how to use it, and (iii) as a thesis template.
\fi


\documentclass{article}
%\usepackage{graphpap}
\begin{document}

\section{Basic}
Math: $a=b$
\begin{picture}(100,100)
\put(0,0){\line(1,0){100}}
\end{picture}
%\end{document}

\section{Pictures}
\paragraph{Lines:} using integers, scaling with floats, \LaTeX counters, TeX counters.
All should be the same.
\par
\begin{picture}(100,100)
\put(0,0){\line(1,0){100}}
\put(0,0){\line(1,1){50}}
\put(0,0){\line(0,1){100}}
\end{picture}
{\unitlength=100pt\relax
\begin{picture}(1.0,1.0)
\put(0,0){\line(1,0){1.0}}
\put(0,0){\line(1,1){0.5}}
\put(0,0){\line(0,1){1.0}}
\end{picture}
}

\newcounter{foo}
\setcounter{foo}{50}
\begin{picture}(100,100)
\put(0,0){\line(1,0){100}}
%\put(0,0){\line(1,1){100}}
\put(0,0){\line(1,1){\value{foo}}}
\put(0,0){\line(0,1){100}}
\end{picture}
\newcount\ffoo\ffoo=50\relax
\begin{picture}(100,100)
\put(0,0){\line(1,0){100}}
%\put(0,0){\line(1,1){100}}
\put(0,0){\line(1,1){\ffoo}}
\put(0,0){\line(0,1){100}}
\end{picture}
%\end{document}

\paragraph{Vectors:}
\begin{picture}(100,100)(-100,-100)
\put(0,0){\vector(-1, 0){100}}
\put(0,0){\vector(-1,-1){100}}
\put(0,0){\vector( 0,-1){100}}
\end{picture}

\paragraph{Boxes;}
\begin{picture}(100,100)
\put(0,0){\bf A}
\put(15,15){\framebox(15,15){A}}
\put(30,30){\framebox(15,15){\bf A}}
\put(45,45){\framebox(15,15){A}}
\put(60,20){\frame{\shortstack{A\\B\\\bf C\\D}}}
\put(75,20){\framebox(15,80){\shortstack{A\\\bf B\\C\\D}}}
\end{picture}

\paragraph{Circles:}
\begin{picture}(100,100)
\put(30,30){\circle{30}\vector(0,1){15}\circle*{5}}
\put(70,70){\oval(60,30)}
\end{picture}

\vskip 1em

\paragraph{Curves:}
\begin{picture}(100,100)
\qbezier(40,80)(80,120)(100,80)
\qbezier[50](40,89)(80,129)(100,89)
\end{picture}

\paragraph{Repeats:}
\begin{picture}(100,100)
\multiput(0,0)(10,10){11}{a}
\end{picture}

\begin{picture}(100,100)
\put(10,10){\circle{5}}
\put(20,20){\framebox(10,10){\circle{5}}}
\put(30,30){\framebox(10,10)[l]{\circle{5}}}
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\put(50,50){\framebox(10,10)[t]{\circle{5}}}
\put(60,60){\framebox(10,10)[b]{\circle{5}}}
\end{picture}

\end{document}


\section{Increase Attribute}\label{sec:increaseattribute}
\textbf{Cost:} 100 CP\\
\textbf{Requirements:} -\\
\textbf{Passive, Repeatable}\\
You increase one of your Attribute value by one, to a maximum of 12.\\
If you increase Strength, Intellect or Vitality, your Stamina, Mana or Health also increase by 1, plus 1 for each rank of ~\nameref{perk:increasestamina},~\nameref{perk:increasemana} or ~\nameref{perk:increasehealth} you have.
Each time you gain a higher rank for this perk, you increase an Attribute value by one point, to a maximum of 12.\\
\\
Rank Progression:\\
\\
\begin{minipage}{0.22\textwidth}
	\rowcolors{2}{lightgray}{white}
	\begin{tabular}{l | l}
		Rank & Cost\\
		II & 250\\
		III & 500\\
		IV & 750\\
		V & 1,000\\
		VI & 2,500\\
	\end{tabular}
\end{minipage}
\begin{minipage}{0.22\textwidth}
	\rowcolors{2}{lightgray}{white}
	\begin{tabular}{l | l}
		Rank & Cost\\ \hline
		VII & 5,000\\
		VIII & 7,500\\
		IX & 10,000\\
		X & 25,000\\
		XI & 75,000\\
	\end{tabular}
\end{minipage}
\begin{minipage}{0.22\textwidth}
	\rowcolors{2}{lightgray}{white}
	\begin{tabular}{l | l}
		Rank & Cost\\ \hline
		XII & 100,000\\
		XIII & 250,000\\
		XIV & 500,000\\
		XV & 750,000\\
		XVI & 1,000,000\\
	\end{tabular}
\end{minipage}
\begin{minipage}{0.22\textwidth}
	\rowcolors{2}{lightgray}{white}
	\begin{tabular}{l | l}
		Rank & Cost\\ \hline
		XVII & 2,500,000\\
		XVIII & 5,000,000\\
		XIX & 7,500,000\\
		XX & 10,000,000\\
		XXI & 25,000,000\\
	\end{tabular}
\end{minipage}


\section{From Passive to Active memory model (aka Threshold-based epidemic dynamics)}
Comment: Here goes a summary from the Threshold-based epidemic dynamics paper
\subsection{Problem description}
\subsection{Simple simulations}
\subsection{Overview of results taken from the paper}


\section{Literature Review}

Process calculi (or process algebras) is a family of algebraic approaches to study concurrent systems \cite{historyPA}.  A process calculus need to address following questions:
\begin{inparaenum}[1.]
  \item What are primitive components in a concurrent system?
  \item What kind of computations and process behaviours could be modelled in this calculus?
  \item What does equivalence between two processes mean in this calculus?
\end{inparaenum}
In the past 30 years, a number of process calculi have been proposed and studied.  Each calculus answers the first two questions from slightly different perspectives.  For the third question, it is worth to note that a particular calculus may have more than one interpretation for process equivalences on different levels.

This chapter will present selective results in concurrent theories and their implementations.  \S\ref{sec:PA} will use the $\pi$-calculus as an example to sketch research topics in the area of process calculi, which provides formal basis for the design of concurrent languages.  \S\ref{sec:join} and \S\ref{sec:ambient} will focus on the programmability of the join-calculus and the ambient-calculus.  Both sections will introduce how to the subject calculus addresses problems in distributed programming and how ideas in the subject calculus are applied in real programming languages.  Algebraic properties of the subject calculus, however, will not be investigated in depth in this short review.  Then, \S\ref{sec:Erlang} will introduce OTP design principles which make distributed Erlang programs more reliable and maintainable.  \S\ref{sec:blue} will present indirect and direct encodings for the $\lambda$-calculus in process calculi.  Lastly, \S\ref{sec:vanroy} will give a short survey on a general framework for implementing concurrent programming languages.

\input{pi}
\input{join}
\input{ambient}
\input{erlang}
\input{blue}
\input{vanroy}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % History
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{History of Artificial Life}

% % 2. History of AI and Computing, Turing, VonNue, etc %    History with ALife,
% Langton, GameOfLife % %      Charles Darwin % 1937 Alan Turing % 1945 John von
% Neumann %      Wolfram % 1970 Cellular automaton, John Conway % 1986
% Christopher Langton %      Marvin Minsky

History of artificial life.  artificial intelligence, cellular automata, sixty
years ago.

Charles Darwin, Alan Turning, John von Neumann, Wolfram, John McCarthy

If we really wanted to create a thorough simulation of life, we should model
everything in our Universe.  We should go back to the big bang.  We should go
back to early earth and model the earth formations.  We should look at the first
forms of cell life.  Model water and geodesic reactions.  We should look at
everything several billion years ago and then model as much as we can from that
point and model several billion years later.

Just the thought of several billion years ago and modeling chemical reactions
and water, sun, early life.  If our goal is to model 4 billion years of earth's
history with the goal of researching complex life forms.  Just consider the
complex life on earth today.  There are 6 billion humans alive today post 4
billion years since the earth's creation.  We estimate  that 100 billion humans
have lived.

That assumes that you only consider human life to be interesting.  What about
all the other forms of interesting life forms that have lived.  How many insect
have lived since early earth.  The numbers are infestimal.

Accurately modeling ... the Universe is complex.   Modeling the earth is equally
complex.

So, we have to focus on the more interesting aspects of early life and biology.

Let's look at organic material vs inorganic

DNA, Life, Death, Mutations, Evolution.  Blind Watchmaker

Conway's Game of Life cellular automaton is one of the most prominent examples
of cellular automata theory. The one dimensional program consists of a cell grid
typically with several dozen or more rows and similar number of columns. Each
cell on the grid has an on or off Boolean state. Every cell on the grid survives
or dies to the next generation depending on the game of life rules. If there are
too many neighbors surrounding a cell then the cell dies due to overcrowding. If
there is only one neighbor cell, the base cell dies due to under-population.
Activity on a particular cell is not interesting but when you run the entire
system for many generations, a group of patterns begin to form.

You may notice some common patterns in the figure. After so many iterations
through the game of life rules, only a few cells tend to stay alive. We started
with a large random number of alive cells and over time those cells died off. In
a controlled environment you may begin with carefully placed live cells and
monitor the patterns that emerge to model some other natural phenomena.

The name Stephan Wolfram has been mentioned several times in this post. He is
the founder of Wolfram|Research, his company is known for the popular
Mathematica software suite and Wolfram|Alpha knowledge engine. He did not
initially discover cellular automata but recently he has been a prominent figure
in its advocacy. He spent 10 years working on his book, A New Kind of Science.
In the 1300 page tome, he discusses how cellular automata can be applied to
every field of science from biology to physics. NKA is a detailed study of
cellular automata programs.

The diagram above depicts the rule 30 program (or rule 30 elementary cellular
automaton). There are 8 input states (2 ^ 3) and an output state of one or zero.
If you look at the diagram from left to right. The first sequence of blocks on
the left depict an input state of { 1 1 1 } with an output of 0. Given input of
cells { 1 1 1}, the output will be set to 0. Subsequently, the next set of
blocks consist of an input state of { 1 1 0 } with an output of 0.


\documentclass{lessonplan}

% Constants
\newcommand{\lessonNumber}{Infinity}
\title{The Infinity Lesson}
\author{\linkHome}
\date{}

% Buildup
\setLessonNumber{\lessonNumber}

%Document
\begin{document}

  \maketitle

  \section{Overview}
    \subsection{Summary}
      This lesson teaches students about infinite vs. finite, discrete vs. continuous, (self-similar) fractals,
      tessellations, and vector vs. pixel art.

      It will also give students the basis to understand sequences, series, mapping, and proof by contradiction.

      Students will be introduced to powerful design technology, get a chance to hone their fine motor skills, and
      even get to take home their creations.

      This lesson can, and should, be run differently every time depending on what students are curious and excited
      about, and what materials are available.
    \subsection{Objectives}
    \begin{itemize}
      \item Think about infinity (big infinity and little infinity) and how the real world is finite.
      \item Cut and fold paper into repeating patterns.
      \item (Intermediate) Think about continuity and how the real world is discrete.
      \item (Intermediate) Think about how pixels are discrete and vectors are continuous, and how technology can be
        used to model continuity with vectors. Make vector and pixel art.
      \item (Advanced) Reason about properties of little and big infinity.
      \item (Advanced) Reason about infinities of different sizes.
      \item (Cooperative) Create collaborative fractals and/or tessellations.
    \end{itemize}
    \subsection{Key Vocabulary Terms}
    \begin{itemize}
      \item infinite, infinitesimal, finite, fractal, tessellation, translation, rotation, reflection
      \item (Intermediate) discrete, continuous
      \item (Advanced) vector, pixel, (sequence/pattern/repetition) depth
    \end{itemize}
    \subsection{Materials}
      \begin{itemize}
        \item White letter paper -- 1 ream
        \item Color letter paper -- 1 ream, or enough sheets for each student and instructor to get 1
        \item Tissue paper -- 1 sheet
        \item Scissors -- a few pairs; if not enough for every student, they can share
        \item Tape -- 1 or 2 rolls, color/opacity doesn't matter
        \item Some sort of writing implement (pencils, pens, markers, etc.) for each student
        \item Origami paper -- at least 1 sheet per intermediate student and instructor
        \item Origami hydrangea fractals and herringbone tessellations of different depths, created by Verda, shown
          below for instructor reference during preparation for the lesson. The originals should be brought to the
          lesson, not just these images.
          \procedureImage[0.1]{example-hydrangea.jpg}
          \procedureImage[0.1]{example-herringbone.jpg}
        \item iPad (alternative: computer with Adobe Photoshop and Adobe Illustrator installed)
        \item (on iPad) Adobe Fresco document with the same layer in vector and pixel format, with large enough
          dimensions that the vector and pixel layers look the same (or as close as possible) at 100\% zoom.
          Alternative if no iPad: same layer in Photoshop and Illustrator, but one is pixel and one is vector.
        \item (on iPad) Adobe Fresco document with Sierpinski triangle. Alternative if no iPad: Illustrator document.
        \item (on iPad) Adobe Fresco document with triangle tessellation. Alternative if no iPad: Illustrator document.
        \item (on iPad or computer) YouTube Video 1:
          \url{https://www.youtube.com/watch?v=4YDHsMUQbVg}
        \item (on iPad or computer) YouTube Video 2:
          \url{https://www.youtube.com/watch?v=Ms655fMOxWE&list=PLUx6ByGW00PzyFMhIoL47_ZezTEQpquh1&index=1}
        \item (on iPad or computer) YouTube Video 3:
          \url{https://www.youtube.com/watch?v=7bFPoGRUJqc&list=PLUx6ByGW00PzyFMhIoL47_ZezTEQpquh1&index=2}
        \item (on iPad or computer) YouTube Video 4:
          \url{https://www.youtube.com/watch?v=U7TKtjQl1zk&list=PLUx6ByGW00PzyFMhIoL47_ZezTEQpquh1&index=3}
        \item (on iPad or computer) YouTube Video 5: \url{https://www.youtube.com/watch?v=RDN7JZpXor4}
        \item (on iPad or computer) YouTube Video 6: \url{https://www.youtube.com/watch?v=nw5RLvN7fYA}
      \end{itemize}
  \section{Procedure}
    \subsection{Setup (10 minutes)}
      Cut a few sheets of white letter paper into small (about 3x3in) pieces, so there are enough for each student and
      each instructor to have 3 pieces. Distribute 5 full sheets of white letter paper, 1 full sheet of color paper,
      and 3 small pieces of white letter paper to each student. Pass around writing implements, scissors, and tape.
    \subsection{Introduction (10 minutes)}
      ``Today we're going to talk about infinity.''

      Take a sheet of white letter paper and fold it in half.

      ``How many times can you fold a piece of paper?''

      Ask each student to take 1 sheet of their white letter paper and fold it as many times as they can.
      Go around and ask how many folds. Give them your answer -- should be around 7.

      Take the sheet of tissue paper and fold it as many times as you can. Should be more than 7. This shows how the
      thinner a paper is compared to its area, the more times it can be folded.

      Pick up a fresh piece of white letter paper.
      \subsubsection{Introduce little infinity}
        ``Imagine a piece of paper so thin that you could fold it a hundred times. Imagine a piece of paper so thin that
        you could fold it a million times. Now, imagine a piece of paper so thin that you could keep folding it forever.
        That's little infinity. We would say that that paper is infinitely thin, or that its thickness is infinitesimal,
        meaning infinitely small.''

        Pause for questions.
      \subsubsection{Introduce big infinity}
        ``Now let's think about the opposite, things that are really big. Imagine that we have a piece of paper that's
        this thick'' (hold up letter paper) ``but it's the size of this table. What about a piece of paper the size of
        this room? This city? This galaxy? The biggest thing you could possibly imagine? All of those papers would
        eventually run out of folding space, because even though they're really big, they still have limits. Now imagine
        a piece of paper that's this thick but goes on forever, without limits.'' (gesture to denote spreading out
        radially) ``That's big infinity. We would say that that paper is infinitely vast/wide, or that its area is
        infinite.''
      \subsubsection{Introduce finitude}
        ``If something isn't infinite or infinitesimal, it's finite. It has clearly defined limits. We can see the limits
        of this paper right here:'' (hold up letter paper and gesture to its edges and its thickness) ``Everything in the
        real world is finite -- if we want infinity, we look to the world of math. Sometimes it's really useful to think
        about, `what if we took this finite pattern and kept repeating it forever?'"
      \subsubsection{Introduce today's activities}
        ``Today we're going to make a few patterns and think about what it might look like if we kept repeating them
        forever. The two types of patterns we're going to make are called fractals and tessellations.''

        Show triangle translation tessellation in Fresco.

        ``Tessellations are a type of repeating pattern where elements of the pattern are next to each other.''

        Show rotation and reflection tessellations in Fresco.

        ``The elements might stay the same when they're repeated, or they might be rotated or flipped.''

        Show Sierpinski triangle in Fresco.

        ``Fractals are a type of repeating pattern where elements of the pattern are within each other. This particular
        fractal is called the Sierpinski triangle, named after a mathematician named Sierpinski.''

        ``Both fractals and tessellations can go on repeating forever, or we can cut them off at a limit we choose. If
        we cut them off at a limit, the number of repetitions is called the depth of the pattern. If we never cut it
        off, the depth would be big infinity.''
    \subsection{Demonstration and Guided Practice}
      It might be helpful to split up into two groups here -- beginner and intermediate -- so the beginner students can
      take their time and intermediate students can have a bit more time to work on the more complex art and math
      concepts in the Extensions section.
      \subsubsection{Fractals}
        Show them YouTube Video 1, demonstrating making a Sierpinski triangle with cuts in a piece of paper.
        Follow along yourself, and have the students do the same with their second sheet of white letter paper. Pause at
        every step so all students can catch up. At the very end, have the students attach their color paper to the back
        of their creations.
      \subsubsection{Tessellations}
        ``In tessellations, as opposed to fractals, elements of the pattern are repeated next to each other, not inside
        one another. The elements might stay the same when they're repeated, or they might be rotated or flipped.''

        Show triangle tessellation in Fresco.

        Show them YouTube Video 2, demonstrating making a tessellation by translation. Follow along yourself, and have
        the students do the same. Pause at every step so everyone can catch up. Label your creation ``TRANSLATION'' and
        have the students do the same. Trace tessellation element across a whole sheet of white letter paper.

        Show them YouTube Video 3, demonstrating making a tessellation by reflection. Follow along yourself, and have
        the students do the same. Pause at every step so everyone can catch up. Label your creation ``REFLECTION'' and
        have the students do the same. Trace tessellation element across a whole sheet of white letter paper.

        Show them YouTube Video 4, demonstrating making a tessellation by rotation. Follow along yourself, and have
        the students do the same. Pause at every step so everyone can catch up. Label your creation ``ROTATION'' and
        have the students do the same. Trace tessellation element across a whole sheet of white letter paper.
    \subsection{Extensions}
      \subsubsection{Collaborative fractal drawing}
        One student draw a big shape on a sheet of paper, another student draw another shape inside it, another draw
        a third inside the second, and then draw the first big shape again but smaller within the third shape. All
        students keep going with the repetition as small as they can.
      \subsubsection{Collaborative tessellation creation}
        Students form a group and make the biggest tessellation they can together! Make a lot of copies of the same
        tessellation element (by tracing and cutting copies of the original element), and set them up all over the
        room as far as they can go.
      \subsubsection{More complex origami fractals and tessellations}
        Show off Verda's example origami hydrangeas and herringbone tessellations, as well as YouTube videos
        demonstrating how to make them (YouTube Videos 5, 6); give students origami paper and let them try their
        hand at making the origami fractals/tessellations to whatever depth they want.
      \subsubsection{(Intermediate) Discrete vs. continuous}
      \subsubsection{(Intermediate) Vector vs. pixel art}
        Talk about how pixel art is discrete because pixels are distinct from each other and there is clear separation
        between them. When you zoom in enough, you can see where one pixel ends and the next one begins. Vector art,
        on the other hand, is continuous. You can keep zooming in (theoretically; Fresco actually limits how far you
        can zoom) and still see a smooth line.

        Let the kids play around with Fresco. Make sure they're being gentle with the iPad. Talk about how this really
        cool technology is created and maintained by software designers and developers. Answer questions about software
        design and development to the best of your ability.
      \subsubsection{(Advanced) Reasoning about infinity}
        ``This paper'' (show letter papers, unfolded and folded) ``runs out of folding space because each time it's
        folded, it gets thicker. With the infinitely thin paper, it would have to be the case that when it's folded,
        it stays infinitely thin.''

        Why?

        ``Imagine the infinitely thin paper got thicker when you folded it. It would eventually get as thick as this
        paper, and by that time, its area would be a lot smaller. That means it would eventually run out of space,
        and we wouldn't be able to fold it anymore. If we can't fold it anymore, that means it's not infinite. This
        tells us that infinite folds means the paper doesn't get any thicker with each fold.''
      \subsubsection{(Advanced) Reasoning about infinities of different sizes}
        Talk about the Infinite Hotel Paradox
        (\url{https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel}). Consider finitely many new
        guests or infinitely many new guests.

        Same situation as paper not getting any thicker with each fold: x = 2x if x is infinite or infinitesimal.
        Doesn't have to be 2: x = nx where n element of N, if x is infinite or infinitesimal.

        2 infinities are the same size if there's a 1:1 correspondence; if we can pair each element of Infinity 1 to
        a different element of Infinity 2, and there's none left over in Infinity 2. The set of natural numbers, or
        ``counting numbers'', is infinite -- you can keep adding 1 and getting a new number forever. We can use this
        infinite set to measure the size of other infinities. If we can count each element in another infinite set
        using the natural numbers, then it's the same size as N. If there are other numbers that can't be counted
        using N, then it's a bigger infinity than N.

        Draw infinities (N, Q, R) on paper to help kids visualize the process of counting them.
      \url{https://www.businessinsider.com/the-different-sizes-of-infinity-2013-11}
    \subsection{Conclusion and Evaluation}
      Give the kids a chance to show the fractals and tessellations that they created.
      \subsubsection{Exit Questions}
      \begin{itemize}
        \item What's another word to describe little infinity? (Hint: ``little infinity'' is a noun, but this word is an
          adjective)
        \item What's another word to describe big infinity?
        \item What do you call something that isn't infinitesimal or infinite?
        \item Out of infinite, infinitesimal, and finite, which one(s) exist(s) in the real world?
        \item What's the difference between a fractal and a tessellation?
        \item What is the depth of the fractal you created?
        \item (Intermediate) What's the difference between discrete and continuous?
        \item (Intermediate) What's the difference between vector and pixel art?
      \end{itemize}

\end{document}


\section{Introduction}
The use of high energy radiation has many wide spread uses. In particular,
the understanding of how radiation in the X ray regime interacts with matter
has applications in areas such as astronomy, medicine and in the study of the
structure of materials such as in the field of crystallography.

In order to be able to interpret results of experiments which make use of X rays
or other high energy photons we must understand how such photons interact with
matter. The most fundamental process that can be considered is the interaction
of a single photon with a single isolated atom. This can include the study of
coherent and incoherent scattering processes, photoionisation, photoabsorption
and pair production.

In particular, the theoretical calculation of atomic form factors allows the
determination of quantities such as cross sections and attenuation coefficients
which are valuable in obtaining information about the atom-photon interaction
process.

The study of atom-photon interactions through the calculation of atomic form
factors covering all elements has been the subject of decades of experimental
and theoretical research. This work includes the synthesis by Henke~\cite{Henke-Experimental}
of experimental and theoretical results, theoretical results using the
relativistic dipole approximation of Cromer and
Libermann~\cite{Cromer-1,Cromer-2,Cromer-3}, theoretical results of Kane et al.
based on S-matrix theory~\cite{Kane-Kissel-Pratt}, recent theoretical
tabulations by Chantler~\cite{Chantler-Tabulation}, the results of
Saloman, Hubbell and Scofield~\cite{Saloman-Hubbell-Scofield} and the results
of Creagh~\cite{Creagh-Hubbell,Creagh-McAuley}.

\section{Project Aims}
Major improvements on current results~\cite{Chantler-Book}
would also involve many years of additional research.
However, a detailed comprehension of the assumptions made in research to date
may be made by a detailed and critical study of form factor calculations for
hydrogen; in particular looking at the different assumptions and methods used to
compute atomic form factors.
This study of hydrogen allows us to show new results for:
\begin{enumerate}
    \item Analytic forms for the relativistic normal form factor for hydrogenic
          atoms.
    \item Numerical results for imaginary
          component of the anomalous form factor for hydrogen using a
          relativistic S-matrix approach.
    \item The angular dependence of the photionisation differential cross
          sections at selected energies.
    \item Semi-analytic results for hydrogenic bound-bound transition matrix elements.
\end{enumerate}


\section{Set Theory}
\label{sec:set-theory}


\subsection{Residual Supply Index (RSI)}


\subsubsection{1991: Jon Orwan and the Doppelgänger system}
\label{sec:orwant_doppelganger}

Doppelgänger is a user modelling system that performs inferences upon user data
and makes this information available to applications. The system allows users to
modify their models, it makes implicit generalizations about the data (which is
gathered through different channels) and provides an extensible architecture~\citep{orwant_doppelgangeruser_1991}.

% ``Figure 1 The DOPPELGÄNGER user modelling system gathers data about users from sensors, makes inferences on
% those data, and makes the results available to applications'' cogido de http://www.cs.ucf.edu/~dcm/Teaching/COT4810-Spring2011/Presentations/FrankHines-CheapUserModelingForAdaptiveSystems.pdf

User data is gathered through a continuously operative sensor network which
senses users' everyday activities (see Listing~\ref{lst:orwant_sensor_1}). Hence,
both long-term and short-term information about the user are stored. Sensors are
integrated in users' activities. However, Doppelgänger acknowledges the imperfection
of real world data. Data streams are usually incomplete or erroneous. The main
objective of the Doppelgänger system is to recover from these imperfections through
several learning techniques:

\begin{itemize}
  \item The Beta distribution~\citep{drake1967fundamentals}, which is used to
  determine the preference strength, probability of accuracy and the confidence
  of an estimation.
  \item Linear prediction, to predict a possible next event.
  \item Markov models, which represents user's behaviour through several states and
  all possible transitions between them with the corresponding probability.
\end{itemize}

The system maintains an accuracy estimation for each sensor, which helps the
system to decide a confidence metric for the gathered data.

\newpage

\begin{minted}[linenos=true, fontsize=\footnotesize, frame=lines]{vim}
(object orwant location (place 344) (time 779562701)
(id active-badge))
\end{minted}
\captionof{listing}{Message from a sensor to the server~\citep{orwant_heterogeneous_1994}.\label{lst:orwant_sensor_1}}

The user models are represented in SPONGE, a LISP based data structure manipulated with
C and Pearl programs \citep{orwant_heterogeneous_1994} (see Listing~\ref{lst:orwant_sensor_2}).
Models are stored as Unix directories consisting of domain submodels and conditional
submodels. The first one contains information about the user behaviour (i.e.,
location, preferences, and so forth); the second group contains triggering information
for deciding actions when certain situations are met. Besides, each user model
is conceptually represented as a point in a high dimensional space in which the
dimensions are determined by the number of sensors in the network.

% \InsertFig{orwant}{fig:orwant}{Orwant user model \citep{orwant_heterogeneous_1994}}{}{0.70}{}
%
% \InsertFig{orwant}{fig:orwant_sensors_apps}{The DOPPELGÄNGER user modelling system gathers data about users from sensors, makes inferences on
% those data, and makes the results available to applications. \citep{orwant_for_1996}}{}{0.70}{}

\begin{minted}[linenos=true, fontsize=\footnotesize, frame=lines]{vim}
(object orwant primary
  (object biographical_data
    (string_binding "true name" "Jon Orwant")
    (string_binding "e-mail address" orwant@media.mit.edu)
  ...)
  (object control
    (int_binding "doppelganger ID" 4))
...)
\end{minted}
\captionof{listing}{Orwant user model~\citep{orwant_heterogeneous_1994}.\label{lst:orwant_sensor_2}}

% \begin{figure}
% \centering
% \includegraphics[width=0.7\textwidth]{orwant_sensors_apps.pdf}
% \caption{The DOPPELGÄNGER user modelling system gathers data about users from sensors, makes inferences on
% those data, and makes the results available to applications~\citep{orwant_for_1996}.}
% \label{fig:orwant_sensors_apps}
% \end{figure}
% ----------------------------------------------------------------------


\documentclass{article}
\usepackage{amsmath,mathtools,amssymb}
\usepackage{simplewick}
\usepackage{graphicx}
\usepackage{booktabs}
\usepackage{blkarray}
\usepackage{gensymb}
\usepackage{verbatim}
\usepackage{mathrsfs}
\usepackage{bbm}
\usepackage{braket}
\usepackage{hyperref}
\usepackage{verbatim}
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\usepackage{xcolor}
\newcommand{\fctr}{\contraction}
\usepackage[margin=1.0in]{geometry}
\newcommand{\ol}{\overline}
\newcommand{\lp}{\left(}
\newcommand{\rp}{\right)}
\newcommand{\eps}{\varepsilon}
\newcommand{\lam}{\lambda}
\newcommand{\h}{\circ}
\newcommand{\p}{\bullet}
\newcommand{\mc}{\mathcal}
\newcommand{\mr}{\mathrm}
\newcommand{\cd}{\cdots}
\newcommand{\Ezero}{E^{(0)}}
\newcommand{\Rz}{\mathcal{R}_{0}}

\newcommand{\Phizero}{\Phi^{(0)}}
\newcommand{\Eone}{E^{(1)}}
\newcommand{\En}{E^{(n)}}
\newcommand{\Phione}{\Phi^{(1)}}
\newcommand{\Phin}{\Phi^{(n)}}

\newcommand{\Ecorr}{E_{\mathrm{corr}}}
\newcommand{\Ec}{E_{\mathrm{c}}}
\newcommand{\Vc}{V_{\mathrm{c}}}
\newcommand{\Hc}{H_{\mathrm{c}}}
\newcommand{\dg}{\ensuremath{^\dagger} }
\def\*#1{\mathbf{#1}}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}

\title{Lecture 6: Coupled Cluster Theory}
\date{April 13, 2020}
\begin{document}
\maketitle
\noindent

\section{Quantum Chemistry Wavefunctions}
In the last set of notes, we introduced the concept of a wave operator $\Omega$, which transforms
    our reference determinant $\Phi$ into the exact wavefunction $\Psi$
\[\ket{\Psi} = \Omega \ket{\Phi} \]
Different valid choices of the wave operator  $\Omega$ constitute different \textit{ansatz},
    which each arrive at valid form for the same exact wavefunction $\ket{\Psi}$.
In perturbation theory (PT), our wave operator is of the form,
$\Omega = P + \sum_{n=0}^{\infty} (\Rz V')^n \Rz V P$, so that our wavefunction is
\[\ket{\Psi_{\mathrm{PT}}} =  \ket{\Phi_0} + \sum_{n=0}^{\infty} (\Rz V')^n \Rz V \ket{\Phi} \]
In configuration interaction (CI), our wave operator (though we did not call it that at the time)
takes the form $\Omega = (1 + C_1 + C_2 + \cdots + C_n)$ so that
\[\ket{\Psi_{\mathrm{CI}}} = (1 + C_1 + C_2 + \cdots + C_n) \ket{\Phi} \]
In coupled cluster theory (CC), our wave operator, and therefore our wavefunction,
is an exponential form,
\[\ket{\Psi_{\mathrm{CC}}} = \exp(T_1 + T_2 + \cdots + T_n) \ket{\Phi} \]
Or simply,
\[\ket{\Psi_{\mathrm{CC}}} = \exp(T) \ket{\Phi} \]

Each of the wavefunctions, in their limit, approach the exact same wavefunction.
However, perturbation theory is an odd duck, since it always has an infinite number of
    contributions to the wavefunction and energy,
    whereas the CC and CI wavefunctions for a given $n$-electron system
    terminate at $n$ terms (you can't have a quadruple excitation in a 3-electron system).

\section{Coupled Cluster Theory}
Our coupled cluster wavefunction from before is
\[\ket{\Psi_{\mathrm{CC}}} = \exp(T) \ket{\Phi} \]
\[\ket{\Psi_{\mathrm{CC}}} = \exp(T_1 + T_2 + \cdots + T_n) \ket{\Phi} \]
where $T$ is the ``cluster operator'' which is a sum over
$k$-electron excitation operators
\[T = T_1 + T_2 + T_3 + \cdots \]
\[T_k = \left(\frac{1}{k!}\right)^2 t_{a_1 \cdots a_k}^{i_1 \cdots i_k } \tilde{a}_{i_1 \cdots i_k}^{a_1 \cdots a_k } \]
where $t_{a_1 \cdots a_k}^{i_1 \cdots i_k }$ are the coupled cluster \textit{amplitudes}.
It is worth briefly pointing out some buried complexity here. Noting that the function $e^T$ can be expanded according to it's Taylor series, the full form of the coupled cluster wavefunction
looks like the following,
\[ \ket{\Psi_{\mathrm{CC}}} = \left[ 1 + (T_1 + T_2 + T_3 + T_4 + \cdots ) + \frac{1}{2} (T_1 + T_2 + T_3 + T_4 + \cdots)^2 +  \frac{1}{6} (T_1 + T_2 + T_3 + T_4 + \cdots)^3 \cdots \right] \ket{\Phi}  \]

Solving for the coupled cluster energy is done by doing the same thing we always do:
project our reference determinant $\bra{\Phi}$ on the left of both sides
of the Schr{\"o}dinger equation.
Assuming intermediate normalization, we obtain
\[\braket{\Phi | \Hc | \Psi_{\mathrm{CC}}} = E_c \braket{\Phi|\Psi_{\mathrm{CC}}} \]
\[\braket{\Phi | \Hc \exp(T) | \Phi} = \Ec \]

Equations for the cluster amplitudes $t_{a b \cdots }^{i j \cdots}$
can be found by left-projecting the Schr{\"o}dinger equation by excited determinants
\[ \braket{ \Phi_{i j \cdots }^{a b \cdots} | \Hc \exp(T) | \Phi} =
   \Ec \braket{\Phi_{i j \cdots }^{a b \cdots} | \exp(T) | \Phi} \]
where an equation involving $t_i^a$ would be obtained by projecting by $\bra{\Phi_{i}^{a}}$,
an equation for $t_{ij}^{ab}$ by projecting by $\bra{\Phi_{ij}^{ab}}$, and so on.
To see this, one can take the right side of the above equation and expand the exponential $\exp(T) = 1 + T + \frac{T^2}{2!} + \cdots $, and only one term survives:
\[ \Ec \braket{\Phi_{i j \cdots }^{a b \cdots} | \exp(T) | \Phi} = \Ec t_{a b \cdots }^{i j \cdots}\]
Plugging in the exponential expansion of $\exp(T)$ into the energy equation
yields
\[ \braket{\Phi | \Hc | \Phi} + \braket{\Phi| \Hc T| \Phi} + \braket{\Phi | \Hc \frac{T^2}{2!} | \Phi} = \Ec \]
These terms are the only ones that survive because higher powers of
$T$ contain no terms for which one can obtain complete contractions.
An analgous equation exists for the amplitudes.

These equations is not easily solvable. The trick used in traditional coupled cluster
theory to make these equations more workable is to multiply by $\exp(-T)$ on both sides,
which effectively transforms our Hamiltonian $\exp(-T) H \exp(T)$.
We highlight the key results in blue.
\[\braket{\Phi | \Hc | \Psi_{\mathrm{CC}}} = E_c \braket{\Phi|\Psi_{\mathrm{CC}}} \]
\[ \braket{\Phi | \exp(-T) \Hc \exp(T) | \Phi } = E \braket{\Phi | \exp(-T) \exp(T) | \Phi} \]
\[ \color{blue} \braket{\Phi | \exp(-T) \Hc \exp(T) | \Phi } = \Ec \]
\quad
\\
\[ \braket{ \Phi_{i j \cdots }^{a b \cdots} | \Hc \exp(T) | \Phi} =
    \Ec \braket{\Phi_{i j \cdots }^{a b \cdots} | \exp(T) | \Phi}  \]
\[ \braket{ \Phi_{i j \cdots }^{a b \cdots} | \exp(-T) \Hc \exp(T) | \Phi} =
    \Ec \braket{\Phi_{i j \cdots }^{a b \cdots} | \exp(-T) \exp(T) | \Phi}  \]
\[ \braket{ \Phi_{i j \cdots }^{a b \cdots} | \exp(-T) \Hc \exp(T) | \Phi} =
    \Ec \braket{\Phi_{i j \cdots }^{a b \cdots} | \Phi}  \]
\[ \color{blue} \braket{ \Phi_{i j \cdots }^{a b \cdots} | \exp(-T) \Hc \exp(T) | \Phi}
   =  0  \]

This transformation decouples the amplitude equation from the energy equation
by setting the right side of the amplitude equation to zero.
In the language of diagrams, this transformation removes ``disconnected'' diagrams,
and $\Ec t_{ab \cdots}^{ij \cdots}$ is one example.
This transformation also can be simplified further via the \textit{Hausdorff expansion}.

\subsection{The Hausdorff Expansion}
The Hausdorff expansion expresses our transformed Hamiltonian in
terms of a sum over nested commutators. As we will see, upon simplification it produces
an expansion in powers of $T$ which naturally truncates, thus removing the infinite
expansion due to the exponential $\exp(T)$.
The Hausdorff expansion is,
\[ \exp(-T) \Hc \exp(T) = \Hc + [\Hc,T] + \frac{1}{2!} [[\Hc,T],T] +
  \frac{1}{3!} [[[\Hc,T],T],T] +  \frac{1}{4!} [[[[\Hc,T],T],T],T] + \cdots   \]
This is simplified by evaluating the commutators using Wick's theorem.
Generally, if two strings of operators $Q$ and $Q'$ are normal ordered,
and one of them has an even operator count, then
\[ [Q, Q'] =  N[\fctr{}{Q}{}{Q} Q Q'] - N[\fctr{}{Q}{'}{Q} Q' Q ] \]
In this specific case,
\[ [\Hc, T] = \Hc T - T\Hc = N[\Hc T] + N[\fctr{}{H}{{}_\mathrm{c}}{T}H_\mathrm{c}T]
  - N[T \Hc] - N[\fctr{}{T}{}{H} T H_\mathrm{c}] \]
\[ [\Hc, T] =  N[\fctr{}{H}{{}_\mathrm{c}}{T}H_\mathrm{c}T] - N[\fctr{}{T}{}{H} T H_\mathrm{c}] \]
The second term above is actually always 0, for any contraction.
The reason is that $T$ is composed entirely of quasi-particle creation operators,
which will be on the left for any possible contraction in $N[\fctr{}{T}{}{H} T H_\mathrm{c}]$.
The only nonzero contraction involving quasi-particle operators occurs
when the creation operator is on the \textit{right}, $ \fctr{}{b}{{}_q}{b} b_p b_q^\dagger = \delta_{pq} $.
Therefore, any possible contraction $N[\fctr{}{T}{}{H} T H_\mathrm{c}]$ must always be zero.
We are left with the following,
\[ [\Hc, T] =  N[\fctr{}{H}{{}_\mathrm{c}}{T}H_\mathrm{c}T] \]
For the nested commutators in the Hausdorff expansion,
since $T$ has no contractions with operators to its right,
we find that the only terms which survive are ones in which \textit{every single
cluster operator $T$ is contracted with the Hamiltonian at least once}.
In the language of diagrams, we call these terms \textit{connected},
since all excitation operators are connected to the Hamiltonian through a contraction.
We label such terms with a subscript capital C.
\[
  [\,\cdot\,,T]^n
=
N[
  \fctr[1.5ex]{}{H}{{}_\mathrm{c}TT\cdots}{T}
  \fctr[1.0ex]{}{H}{{}_\mathrm{c}T}{T}
  \fctr[0.5ex]{}{H}{{}_\mathrm{c}}{T}
  H_\mathrm{c} TT\cdots T
]
=
  (H_\mathrm{c} T^n)_\mathrm{C}
\]
The restriction to only connected terms naturally eliminates many terms in the Hausdorff expansion.
Furthermore, since $\Hc$ is a linear combination of one-electron and two-electron operators,
    the components of $\Hc$ can contract with at most two and four $T$'s, respectively.
Since every $T$ has to be connected to $\Hc$, we are only left with,
connected terms up to the fourth power in $T$:
\[ (\Hc \exp(T))_{\mathrm{C}} = (\Hc + \Hc T + \frac{1}{2!} \Hc T^2 +
  \frac{1}{3!} \Hc T^3 +  \frac{1}{4!} \Hc T^4)_{\mathrm{C}} \]
To convince yourself of this, you can write out the full form
of $T = T_1 + T_2 + T_3 + \cdots $, re-express each $T_k$ according to their
definition, take various powers of the expansion $T^2, T^3, T^4, T^5$,
and try contracting a few terms with the Hamiltonian.
You will find that most terms in $T^4$ cannot be such that every $T_k$ forms a contraction
with $\Hc$, and no terms in $T^5$ can have each cluster operator contract with $\Hc$.
There isn't enough ``room'' in $\Hc$ for that many contractions.

To summarize, we initially replaced $\Hc \exp(T)$ in our energy and amplitude equations
with $ \exp(-T) \Hc \exp(T)$. We then used the Hausdorff expansion.
Upon simplifying, we found that $\exp(-T) \Hc \exp(T) = (\Hc \exp(T))_{\mathrm{C}}$.
This tells us that only ``connected'' terms survive,
    and all disconnected terms necessarily cancel out.
Furthermore, our Hamiltonian only allows contractions with up to 4 $T$'s,
so the Hausdorff expansion naturally truncates after $T^4$.
This gives us our final results for the coupled cluster energy and amplitude equations,

\[\braket{\Phi |\Hc \exp(T) | \Phi }_\mathrm{C} = \Ec \]
\[\braket{ \Phi_{i j \cdots }^{a b \cdots} |\Hc \exp(T) | \Phi}_\mathrm{C}
   =  0  \]
Note that these are the same results as the equations previously derived (in blue),
except we have used the result of the Hausdorff expansion to replace
 $\exp(-T) \Hc \exp(T)$ with just connected terms $(\Hc \exp(T))_{\mathrm{C}}$.
Practically, when one uses the equations above algebraically,
one should look for complete contractions in which the Hamiltonian
has at least one contraction with every cluster operator on its right.

\subsection{Excitation Levels}
There is an alternative to considering which terms form complete contractions when
evaluating expectation values.
The \textit{excitation level} of a series of operators is equal
to the net number of particles or quasi-particles it creates, divided by 2.
The excitation levels of $T_1$, $T_2$, and $T_3$ are 1, 2, and 3.
$\tilde{a}_{abcd}^{ijkl}$, which would perhaps appear in a bra as
$\bra{\Phi_{ijkl}^{abcd}}$, would have an excitation level of -4.
\textbf{The total excitation level of a term in an expectation value must balance out
to zero}, otherwise the term is zero.
Thus, this is another way of detecting which terms cannot form complete contractions.
What is the excitation level of terms in our Hamiltonian $\Hc$?
Excitation operators in the Hamiltonian are made up of general indices $p,q,r,s$ (not necessarily occupied or virtual), so the excitation level of the Hamiltonian is variable.
The one-electron part of the Hamiltonian can have an excitation level of 0, +1, or -1.
The two-electron part of the Hamiltonian can have an excitation level of 0, +1 , -1, +2, or -2.
To see why, recall that the one-electron operator is $h_p^q \tilde{a}_q^p$, which
can take on virtual and occupied indices in four ways:
$h_a^b \tilde{a}_b^a, h_a^i \tilde{a}_i^a, h_i^a \tilde{a}_a^i, h_i^j \tilde{a}_j^i$.
In the $ h_a^i \tilde{a}_i^a$ case, it converts to two quasi-particle
creation operators $b_a^\dagger b_i^\dagger$. According to our definition of excitation
level, this has an exciation level of +1. In the case of $ h_i^a \tilde{a}_a^i $,
we have $b_i b_a $, two quasi-particle annihilation operators, so that the excitation
level is -1.
The other two cases have excitation levels of 0.
Similar arguments can be made for the two-electron cases involving $\tilde{a}_{rs}^{pq}$.

By analyzing excitation levels, we can decide if expectation values are 0 or not.
Examples,
\begin{itemize}
\item The following expectation value is nonzero. Each $T_1$ has an excitation
level of +1, $\tilde{a}_{ab}^{ij}$ has an excitation level of -2, and the Hamiltonian
excitation level can be chosen to be 0 so that 1 + 1 - 2 + 0 = 0.
\[ \braket{\Phi_{ij}^{ab} | \Hc T_1^2 | \Phi} \]
\item The following expectation value is 0. $T_4$ has an excitation level of +4,
$\tilde{a}_{a}^{i}$ has an excitation level of -1, and we can choose
the Hamiltonian to have an excitation level of -2, at best. -1 + -2 + 4 = 1.
\[ \braket{\Phi_i^a | \Hc T_4 | \Phi} \]
\end{itemize}
One could come up with similar conclusions by looking at what complete contractions
can be formed, but it is often much faster to count excitation levels.

When evaluating coupled cluster expectation values, if the excitation level
of a term is not 0, the term can be eliminated.
However, just because the excitation level is 0 does not mean the term is automatically considered.
From the Hausdorff expansion, we also must have that the Hamiltonian is connected to each
cluster operator in a given term. If the excitation levels balance to zero and the Hamiltonian
can be connected (contracted) to all cluster operators in the term, the term is nonzero.
The following example has an excitation level of 0, but there is no way to connect $\Hc$ to the cluster
operator $T_2$, so it goes to zero.
\[\braket{\Phi_{ijkl}^{abcd}| \Hc T_2 | \Phi } \]

\subsection{The Amplitude Equations}
The cleanest way to solve for the cluster amplitudes
in our amplitude equation
\[\braket{ \Phi_{i j \cdots }^{a b \cdots} |\Hc \exp(T) | \Phi}_\mathrm{C} = 0\]
requires an odd-looking partitioning of our Hamiltonian
into two parts: the diagonal part of the Fock operator, and everything else.
\[H_0 = f_p^p \tilde{a}_p^p \]
\[\Vc = \Hc - H_0 = f_p^q (1 - \delta_p^q) \tilde{a}_q^p + \frac{1}{4} \bar{g}_{pq}^{rs}
 \tilde{a}_{rs}^{pq} \]
$H_0$ can be absorbed into an excited determinant to give back
a term which depends on the orbital energies,
\begin{align}
  H_0\Phi_{i_1\cd i_k}^{a_1\cd a_k}
=
  \mathcal{E}_{i_1\cd i_k}^{a_1\cd a_k}
  \Phi_{i_1\cd i_k}^{a_1\cd a_k}
&&
  \mathcal{E}_{q_1\cd q_k}^{p_1\cd p_k}
\equiv
  \sum_{r=1}^k
  f_{p_r}^{p_r}
-
  \sum_{r=1}^k
  f_{q_r}^{q_r}
\end{align}
$\mathcal{E}_{i_1\cd i_k}^{a_1\cd a_k}$ is essentially the change
in the sum of the orbital energies from the reference state (Hartree-Fock) for the given
excited determinant. A single
$\mathcal{E}_{ij}^{ab}$ is $\mathcal{E}_{ij}^{ab} = \epsilon_a + \epsilon_b - \epsilon_i - \epsilon_j$.
Using the above equations in conjunction with our amplitude equations,
\[\braket{ \Phi_{i j \cdots }^{a b \cdots} |\Hc \exp(T) | \Phi}_\mathrm{C}
   =  0  \]
We plug in our partitioned Hamiltonian $\Hc = H_0 + \Vc$,
\[\braket{ \Phi_{i j \cdots }^{a b \cdots} |H_0 \exp(T) | \Phi}_\mathrm{C}
 + \braket{ \Phi_{i j \cdots }^{a b \cdots} |\Vc \exp(T) | \Phi}_\mathrm{C}
   =  0  \]
The left term evalutes to
\[ \braket{ \Phi_{i j \cdots }^{a b \cdots} |H_0 \exp(T) | \Phi}_\mathrm{C} =
   \braket{ \Phi_{i j \cdots }^{a b \cdots} |H_0 T | \Phi} =
\mathcal{E}_{ij \cd}^{ab \cd} \braket{\Phi_{i j \cdots }^{a b \cdots} |T|\Phi} =
\mathcal{E}_{ij \cd}^{ab \cd} t_{ab \cd}^{ij \cd} \]
where we have used the fact that only $T$ in $\exp(T) = 1 + T + \frac{1}{2}T^2 \cdots $
can create connected terms with $H_0$, since it is a one-electron operator.
Now our amplitude equation reads as
\[ \braket{ \Phi_{i j \cdots }^{a b \cdots} |\Vc \exp(T) | \Phi}_\mathrm{C}
  + \mathcal{E}_{ij \cd}^{ab \cd} t_{ab \cd}^{ij \cd}   =  0  \]
Since $\mathcal{E}_{ij \cd}^{ab \cd}$ is just a linear combination of orbital
energies, we can absorb a minus sign by swapping upper and lower indices, $-\mathcal{E}_{ij \cd}^{ab \cd} = \mathcal{E}_{ab \cd}^{ij \cd}$,  which gives us the following
clean equation for solving for the amplitudes,
\[ t_{ab \cd}^{ij \cd} = \left(\mathcal{E}_{ab \cd}^{ij \cd}\right)^{-1}
        \braket{ \Phi_{i j \cdots }^{a b \cdots} |\Vc \exp(T) | \Phi}_\mathrm{C}  \]

The amplitude equations must be solved iteratively, since $t_{ab \cd}^{ij \cd}$ is
also embedded in $\exp(T)$.

\subsection{The CCSDTQ amplitude equations}
If we truncate the cluster operator $T = T_1 + T_2 + T_3 + T_4$,
then $\exp(T) = \exp(T_1 + T_2 + T_3 + T_4)$.
Not every operator in the exponential expansion of $\exp(T_1 + T_2 + T_3 + T_4)$
is relevant;
disconnected terms are omitted, as well as terms for which the excitation level
of the expectation value does not balance out to 0.
After careful consideration, one can use the amplitude equation we derived above
to obtain amplitude equations for CCSDTQ:
\begin{align*}
  t_a^i
=& (\mc{E}_a^i)^{-1}
  \braket{\Phi_i^a| V_\mr{c} (1 + T_1 + T_2 + T_3 + \tfrac{1}{2} T_1^2 +T_1T_2 + \tfrac{1}{3!} T_1^3)|\Phi}_\mr{C}
\\[3pt]
  t_{ab}^{ij} =& (\mc{E}_{ab}^{ij})^{-1}
  \braket{\Phi_{ij}^{ab} | V_\mr{c} ( 1 + T_1 + T_2 +  T_3 + T_4 + \tfrac{1}{2}T_1^2 +  T_1T_2 +  T_1T_3 +  \tfrac{1}{2} T_2^2
    + \tfrac{1}{3!} T_1^3 +  \tfrac{1}{2} T_1^2T_2 + \tfrac{1}{4!} T_1^4 ) | \Phi}_\mr{C}
\\[3pt]
  t_{abc}^{ijk}
=& (\mc{E}_{abc}^{ijk})^{-1}
  \bra{\Phi_{ijk}^{abc}} V_\mr{c}
    (T_2 + T_3 + T_4 + T_1T_2 + T_1T_3 + \tfrac{1}{2} T_2^2 +T_1T_4 + T_2T_3 + \tfrac{1}{2}T_1^2T_2 + \tfrac{1}{2}T_1^2T_3 + \tfrac{1}{2}T_1T_2^2
    + \tfrac{1}{3!}T_1^3T_2) \ket{\Phi}_\mr{C}
\\[3pt]
  t_{abcd}^{ijkl}
=&
  (\mc{E}_{abcd}^{ijkl})^{-1} \bra{\Phi_{ijkl}^{abcd}} V_\mr{c}
    (T_3 + T_4 + T_1T_3 + \tfrac{1}{2} T_2^2 + T_1T_4 + T_2T_3 + T_2T_4 + \tfrac{1}{2} T_3^2 +
      \tfrac{1}{2} T_1^2T_3  + \tfrac{1}{2} T_1T_2^2 + \tfrac{1}{2} T_1^2T_4
\\
&
\hphantom{
  (\mc{E}_{abcd}^{ijkl})^{-1}
  \bra{\Phi_{ijkl}^{abcd}}
    V_\mr{c}
    (
      T_3\
 }
    + T_1T_2T_3 + \tfrac{1}{3!} T_2^3 +\tfrac{1}{3!}T_1^3T_3 + \tfrac{1}{2!2!}T_1^2T_2^2)
  \ket{\Phi}_\mr{C}
\end{align*}

\section{On the connection between Coupled Cluster and Configuration Interaction}
Our CI wavefunction is
\[ \ket{\Psi_{\mathrm{CI}}} = (1  + C_1 + C_2 + \cdots) \ket{\Phi}  \]
Our CC wavefunction is
\[ \ket{\Psi_{\mathrm{CC}}} = \exp(T_1 + T_2 + \cdots) \ket{\Phi}  \]
\[ \ket{\Psi_{\mathrm{CC}}} = \left[ 1 + (T_1 + T_2 + T_3 + T_4 + \cdots ) + \frac{1}{2} (T_1 + T_2 + T_3 + T_4 + \cdots)^2 +  \frac{1}{6} (T_1 + T_2 + T_3 + T_4 + \cdots)^3 \cdots \right] \ket{\Phi}  \]

Noting that each $T_k$ ($C_k$) is just made up of scalar cluster amplitudes
$t_{ab \cdots}^{ij \cdots}$ (CI coefficients $c_{ab \cdots}^{ij \cdots}$)
and a Phi-normal-ordered excitation operator,  we can compare these wavefunction
expansions term by term and find the following:

\begin{align*}
C_1 &= T_1 \\
C_2 &= T_2 + \frac{1}{2} T_1^2 \\
C_3 &= T_3 + T_1 T_2 + \frac{1}{6} T_1^3 \\
C_4 &= T_4 + \frac{1}{2}T_2^2 + T_1 T_3 +  \frac{1}{2} T_1^2 T_2 + \frac{1}{24} T_1^4 \\
\end{align*}

CI does not discriminate between different kinds of $n$-tuple excitations,
but in coupled cluster thoery, we can sub-classify various $n$-tuple excitations
in different ways.
Consider the quadruple excitation terms.
The physical interpretation of $T_4$ as an excitation event on some
quantum mechanical state is 4 electrons coming together at the same location
and, upon interacting, they all have to simultaneously excite to a new state.
$\frac{1}{2}T_2^2$, on the other hand, corresponds to an excitation event
in which a pair of 2 electrons which are close enough to locally interact
each excite to a new state, while simultaneously another distinct pair of
    electrons does the same.
Which of these physical events ($T_4$  or $\frac{1}{2}T_2^2$) are more likely
    to occur in a molecule?
Almost certainly the latter, since electrons tend to avoid each other, but they also
    tend to form electron pairs!
We conclude that $\frac{1}{2}T_2^2$ is a more important contribution to our wavefunction
than $T_4$.

Now consider the CCSD wavefunction,
\[ \ket{\Psi_{\mathrm{CCSD}}} = \exp(T_1 + T_2) \ket{\Phi} \]
\[ \ket{\Psi_{\mathrm{CCSD}}} = (1 + T_1 + T_2 + \frac{1}{2} T_1^2 + T_1 T_2 +
 \frac{1}{6} T_1^2 + \frac{1}{2} T_2^2 + \frac{1}{2} T_1^2 T_2 + \frac{1}{24} T_1^4 + ...)    \ket{\Phi} \]

We obtain all single and double excitations according to CI, but we also
obtain some triple excitations ($T_1 T_2$, $\frac{1}{6} T_1^3$) and some quadruple
excitations ($\frac{1}{2} T_2^2 + \frac{1}{2} T_1^2 T_2 + \frac{1}{24} T_1^4 $).
This is why its called coupled cluster; higher order excitations
(quadruple, $\frac{1}{2}T_2^2$) are included even in low-order truncation
of the scheme (CCSD) via \textit{coupling} two double excitations together.
To put it another way, in CCSD, we obtain some information about quadruple excitations
by considering two clusters which each doubly excite simultaneously.
Not only that, but CCSD is giving us a quadruple excitation of relatively high importance,
according to our physical argument earlier.

It is clear that CCSD is more accurate than CISD.
However, coupled cluster gets even better at higher orders.
Looking at our expressions for $C_1, C_2, C_3, C_4$ above,
we see that the majority of $C_4$ is made up of terms which
occur at CCSDT; expanding the CCSDT exponential
$\exp(T_1 + T_2 + T_3)$ will give you all of the terms in $C_4$ except
$T_4$ (!!!).
Thus, CCSDT is way more accurate than CISDT, and it's even
almost as accurate correlation-wise as CISDTQ, since all that is missing is $T_4$.
Nevermind the fact  that coupled cluster is size-consistent, which is
probably making up for more accuracy than the relative importance of $T_4$ in some cases.
In this view, it is even proper to say that CCSDT is \textit{more} accurate than CISDTQ.

\subsection{Deriving CC from CI}
Starting from full-CI,
\[ \ket{\Psi} = \lp 1 + C_1 + C_2 + C_3 + \cdots + C_n \rp \ket{\Phi} \]
\[ \ket{\Psi} = \lp 1 + C \rp \ket{\Phi} \]
We can define the cluster operator $T = T_1 + T_2 + \cdots$ as $T = \ln(1+C)$ where $C = C_1 + C_2 + \cdots$.
This way, $1 + C = e^T$, so it must be the case
\[\ket{\Psi} = e^T \ket{\Phi} \]
which is our coupled cluster wavefunction.

We can compare coupled cluster and CI operators using the Taylor expansion for $\ln (1 + z)$
\[ \ln (1 + z) = \sum_{m=1}^{\infty} \frac{ (-1)^{m+1}  z^m}{ m} \]
This implies that for $T$ we have
\[ T = \ln (1 + C) = \sum_{m=1}^{\infty} \frac{(-1)^{m+1}  C^m}{m} \]
This series always converges, since we always have a finite number of electrons in a system.
Expanding the series we get,
\[T = C - \frac{C^2}{2} + \frac{C^3}{3} + \cdots \]
\[T = C_1 + C_2 + C_3 + \cdots -  \frac{1}{2} ( C_1 + C_2 + C_3 + \cdots)^2 +
  \frac{1}{3} (C_1 + C_2 + C_3 + \cdots)^3 +  \cdots
\]

\section{On the connection between Coupled Cluster and Perturbation Theory}
TODO; diagrams pre-requisite.

\end{document}


    \filetitle{FAVAR}{Create new empty FAVAR object}{FAVAR/FAVAR}

	\paragraph{Syntax}\label{syntax}

\begin{verbatim}
F = FAVAR(YNames)
\end{verbatim}

\paragraph{Input arguments}\label{input-arguments}

\begin{itemize}
\itemsep1pt\parskip0pt\parsep0pt
\item
  \texttt{YNames} {[} cellstr \textbar{} char {]} - Names of observed
  variables in the FAVAR model.
\end{itemize}

\paragraph{Output arguments}\label{output-arguments}

\begin{itemize}
\itemsep1pt\parskip0pt\parsep0pt
\item
  \texttt{F} {[} FAVAR {]} - New FAVAR object.
\end{itemize}

\paragraph{Description}\label{description}

This function creates a new empty FAVAR object. It is usually followed
by the \href{FAVAR/estimate}{estimate} function to estimate the FAVAR
parameters on data.

\paragraph{Example}\label{example}

To estimate a FAVAR, you first need to create an empty VAR object, and
then run the \href{FAVAR/estimate}{FAVAR} function on it, e.g.

\begin{verbatim}
list = {'DLCPI','DLGDP','R'};
f = FAVAR(list);
f = estimate(f,d,range);
\end{verbatim}