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\begin{document}

\title{Birefringence}
\maketitle

\section{Why Birefringence}

From WMAP Komatsu paper:
Since the temperature and E-mode polarization are
parity-even and the B-mode polarization is parity-odd, the TB and EB correlations should vanish in a universe that conserves parity. For this reason the TB and EB correlations are usually used to check for systematics, and not widely used as a cosmological probe.
Polarization of photons offers a powerful way of probing the cosmological parity violation, or the “cosmological birefringence”. Let us consider a parity-violating interaction term in the Lagrangian such as the Chern-Simons term ${\cal L}_{CS}= -\frac{1}{2}p_{\alpha}A_{\beta}{\tilde F}^{\alpha \beta}$ where $F^{\alpha \beta}$ and $A_{\beta}$ are the usual electromagnetic tensor and vector potential, ${\tilde F}^{\alpha \beta}$ is the dual tensor and $p_{\alpha}$ is an arbitrary timelike vector. This Chern-Simons term makes two polarization states of photons propagate with different group velocities, causing the polarization plane to rotate by an angle $\alpha$ (Carroll 1990). $p_{\alpha}$ can be sources by a scalar field. Such a field might have something to do with dark energy, for example. We are therefore looking at a potential parity-violating interaction between the visible section (i.e., photons) and dark sector (i.e., dark energy)

\section{Effects of Birefringence on the Power Spectra}

(from 1904.12440)
When polarisation angles are rotated uniformly over the sky by an angle $\alpha$, spherical harmonics coefficients of the observed E- and B-mode polarisation, denoted by "o", are related
to the intrinsic ones by
\ba
a_{\ell m}^{E, o} &=& a_{\ell m}^{E} \cos 2\alpha - a_{\ell m}^{B} \sin 2\alpha  \nonumber \\
a_{\ell m}^{B, o} &=& a_{\ell m}^{E} \sin 2\alpha + a_{\ell m}^{B} \cos 2\alpha
\ea
For the power spectra, we obtained
\ba
C^{EE,o}_{\ell} &=&  C^{EE}_{\ell} \cos^{2} 2\alpha + C^{BB}_{\ell} \sin^{2} 2\alpha -  C^{EB}_{\ell} \sin 4\alpha \nonumber \\
C^{BB,o}_{\ell} &=&  C^{EE}_{\ell} \sin^{2} 2\alpha + C^{BB}_{\ell} \cos^{2} 2\alpha + C^{EB}_{\ell} \sin 4\alpha \nonumber \\
C^{EB,o}_{\ell} &=&  C^{EE}_{\ell} \sin 2\alpha  \cos 2\alpha. - C^{BB}_{\ell} \sin 2\alpha  \cos 2\alpha + C^{EB}_{\ell}(\cos^{2} 2\alpha - \sin^{2} 2\alpha) \nonumber \\
C^{EB,o}_{\ell} &=&  \frac{\sin 4 \alpha}{2} (C^{EE}_{\ell} . - C^{BB}_{\ell} ) + \cos 4\alpha \ C^{EB}_{\ell}
\ea
We can relate the observed EB power spectrum to the observed EE and BB power spectrum
\ba
C^{EB,o}_{\ell} &=&  \frac{1}{2} ( C^{EE,o}_{\ell} - C^{BB,o}_{\ell} ) \tan 4\alpha  + \frac{ C^{EB}_{\ell}}{\cos 4 \alpha}
\ea
indeed
\ba
C^{EB,o}_{\ell} &=&  \frac{1}{2} ( C^{EE}_{\ell} \cos^{2} 2\alpha + C^{BB}_{\ell} \sin^{2} 2\alpha -  C^{EB}_{\ell} \sin 4\alpha - C^{EE}_{\ell} \sin^{2} 2\alpha - C^{BB}_{\ell} \cos^{2} 2\alpha - C^{EB}_{\ell} \sin 4\alpha \nonumber) \tan 4\alpha  \nonumber \\
 &+& \frac{ C^{EB}_{\ell}}{\cos 4 \alpha} \nonumber \\
&=&   \frac{1}{2} ( C^{EE}_{\ell}  - C^{BB}_{\ell} ) \sin 4\alpha + C^{EB}_{\ell}  (\frac{1 - \sin^{2}  4 \alpha}{\cos 4 \alpha} ) = \frac{1}{2} ( C^{EE}_{\ell}  - C^{BB}_{\ell} ) \sin 4\alpha + C^{EB}_{\ell}  \cos 4 \alpha
\ea

\section{Likelihood}

Assuming the intrinsic EB is equal to zero, we can therefore form two different $\chi^{2}$
\ba
\chi^{2} (\alpha)_{\rm data} &=& \sum_{bb'} \left[C^{EB,o}_{b} -  \frac{1}{2} ( C^{EE,o}_{b} - C^{BB,o}_{b} ) \tan 4\alpha \right] \Xi^{-1}_{bb'} \left[ C^{EB,o}_{b'} -  \frac{1}{2} ( C^{EE,o}_{b'} - C^{BB,o}_{b'} ) \tan 4\alpha \right] \\
\chi^{2} (\alpha)_{\rm th} &=& \sum_{bb'} \left[C^{EB, o}_{b} -  \frac{1}{2} ( C^{EE}_{b} - C^{BB}_{b} ) \sin 4\alpha \right] \Xi^{-1}_{bb'} \left[C^{EB, o}_{b} -  \frac{1}{2} ( C^{EE}_{b} - C^{BB}_{b} ) \sin 4\alpha \right]
\ea

\section{Fisher}
\ba
F_{\alpha \alpha} &=& \left \langle \frac{ \partial^{2} {\cal L} }{ \partial^{2}{\alpha} }\right \rangle \Bigr\rvert_{\alpha=0} = 4 \sum_{bb'}( C^{EE,o}_{b} - C^{BB,o}_{b} ) \Xi^{-1}_{bb'} ( C^{EE,o}_{b'} - C^{BB,o}_{b'} ) \\
\sigma_{\alpha} &=& \frac{1}{2} ( \sum_{bb'}( C^{EE,o}_{b} - C^{BB,o}_{b} ) \Xi^{-1}_{bb'} ( C^{EE,o}_{b'} - C^{BB,o}_{b'} ))^{-1/2}
\ea

\section{Separating CMB and foregrounds}

The foregrounds are sensitive to the instrumental miscalibrated angle but is not sensitive to the cosmic birefringence (because fg is at redshift zero) we can therefore separate CMB and fg.
Now let's call $\alpha$ the instrumental miscalibrated angle and $\beta$ the birefringence angle
\ba
a_{\ell m}^{E, o} &=& a^{\rm fg, E}_{\ell m} \cos (2\alpha) - a^{\rm fg, B}_{\ell m} \sin (2\alpha)  + a^{\rm CMB, E}_{\ell m} \cos (2\alpha + 2\beta) - a^{\rm CMB, B}_{\ell m} \sin (2\alpha + 2\beta) + a_{\ell m}^{E, N} \nonumber \\
a_{\ell m}^{B, o} &=& a^{\rm fg, E}_{\ell m} \sin (2\alpha) + a^{\rm fg, B}_{\ell m} \cos (2\alpha) + a^{\rm CMB, E}_{\ell m} \sin (2\alpha + 2\beta ) + a^{\rm CMB, B}_{\ell m} \cos (2\alpha + 2\beta) + a_{\ell m}^{B, N}
\ea
Assuming we form cross spectra
\ba
C^{EE,o}_{\ell} &=& \langle( a^{\rm fg, E}_{\ell m} \cos (2\alpha) - a^{\rm fg, B}_{\ell m} \sin (2\alpha)  + a^{\rm CMB, E}_{\ell m} \cos (2\alpha + 2\beta) - a^{\rm CMB, B}_{\ell m} \sin (2\alpha + 2\beta)) \nonumber \\
&\times& (a^{\rm fg, E}_{\ell m} \cos (2\alpha) - a^{\rm fg, B}_{\ell m} \sin (2\alpha)  + a^{\rm CMB, E}_{\ell m} \cos (2\alpha + 2\beta) - a^{\rm CMB, B}_{\ell m} \sin (2\alpha + 2\beta)) \rangle \nonumber \\
&=& C^{\rm EE, fg}_{\ell} \cos^{2} (2\alpha) + C^{\rm BB, fg}_{\ell} \sin^{2} (2\alpha) + C^{\rm EE, CMB}_{\ell} \cos^{2} (2\alpha + 2\beta)   + C^{\rm BB, CMB}_{\ell} \sin^{2} (2\alpha + 2\beta)  \nonumber \\
&-& 2 C^{\rm EB, fg}_{\ell} \cos (2\alpha)\sin (2\alpha) -  2 C^{\rm EB, CMB}_{\ell} \cos (2\alpha + 2\beta)\sin (2\alpha + 2\beta)  \nonumber \\
C^{BB,o}_{\ell}  &=& C^{\rm EE, fg}_{\ell} \sin^{2} (2\alpha) + C^{\rm BB, fg}_{\ell} \cos^{2} (2\alpha) + C^{\rm EE, CMB}_{\ell} \sin^{2} (2\alpha + 2\beta)   + C^{\rm BB, CMB}_{\ell} \cos^{2} (2\alpha + 2\beta)  \nonumber \\
&+& 2 C^{\rm EB, fg}_{\ell} \cos (2\alpha)\sin (2\alpha) +  2 C^{\rm EB, CMB}_{\ell} \cos (2\alpha + 2\beta)\sin (2\alpha + 2\beta)  \nonumber \\
\ea
And
\ba
C^{EB,o}_{\ell} &=&  \langle( a^{\rm fg, E}_{\ell m} \cos (2\alpha) - a^{\rm fg, B}_{\ell m} \sin (2\alpha)  + a^{\rm CMB, E}_{\ell m} \cos (2\alpha + 2\beta) - a^{\rm CMB, B}_{\ell m} \sin (2\alpha + 2\beta)) \nonumber \\
&\times& (a^{\rm fg, E}_{\ell m} \sin (2\alpha) + a^{\rm fg, B}_{\ell m} \cos (2\alpha)  + a^{\rm CMB, E}_{\ell m} \sin (2\alpha + 2\beta) + a^{\rm CMB, B}_{\ell m} \cos (2\alpha + 2\beta)) \rangle \nonumber \\
&=&  C^{\rm EE, fg}_{\ell} \cos (2\alpha) \sin (2\alpha)  - C^{\rm BB, fg}_{\ell} \sin (2\alpha) \cos (2\alpha) + C^{\rm EE, CMB}_{\ell} \cos (2\alpha +2 \beta) \sin (2\alpha +2 \beta)  \nonumber \\
&-& C^{\rm BB, CMB}_{\ell} \sin (2\alpha + 2\beta) \cos (2\alpha + 2\beta)  +  C^{\rm EB, fg}_{\ell} ( \cos^{2} (2\alpha) - \sin^{2} (2\alpha) )  \nonumber \\
&+&  C^{\rm EB, CMB}_{\ell} ( \cos^{2} (2\alpha + 2\beta) - \sin^{2} (2\alpha + 2 \beta)) \nonumber \\
C^{EB,o}_{\ell} &=& \frac{\sin(4\alpha)}{2}(C^{\rm EE, fg}_{\ell}  - C^{\rm BB, fg}_{\ell} ) + \frac{\sin(4\alpha + 4\beta)}{2}(C^{\rm EE, CMB}_{\ell}  - C^{\rm BB, CMB}_{\ell} )  \nonumber \\
&+& \cos(4\alpha) C^{\rm EB, fg}_{\ell} + \cos(4\alpha + 4\beta) C^{\rm EB, CMB}_{\ell}
\ea
Let's demonstrate that we can refactorize $C^{EB,o}_{\ell}$ as
\ba
C^{EB,o}_{\ell} &=& \frac{\tan(4\alpha)}{2} \left( C^{EB,o}_{\ell} - C^{BB,o}_{\ell} \right) + \frac{ \sin(4\beta)}{2 \cos(4\alpha)}( C^{\rm EE, CMB}_{\ell} - C^{\rm BB, CMB}_{\ell}) \nonumber \\
&+& \frac{1}{\cos(4\alpha)} C^{\rm EB, fg}_{\ell}  + \frac{ \cos(4\beta)}{ \cos(4\alpha)} C^{\rm EB, CMB}_{\ell}
\ea
We can expand
\ba
 \left( C^{EE,o}_{\ell} - C^{BB,o}_{\ell} \right) &=&   C^{\rm EE, fg}_{\ell} \cos^{2} (2\alpha) + C^{\rm BB, fg}_{\ell} \sin^{2} (2\alpha) + C^{\rm EE, CMB}_{\ell} \cos^{2} (2\alpha + 2\beta)   \nonumber \\
 &+& C^{\rm BB, CMB}_{\ell} \sin^{2} (2\alpha + 2\beta)  - 2 C^{\rm EB, fg}_{\ell} \cos (2\alpha)\sin (2\alpha) -  2 C^{\rm EB, CMB}_{\ell} \cos (2\alpha + 2\beta)\sin (2\alpha + 2\beta)  \nonumber \\
&-& C^{\rm EE, fg}_{\ell} \sin^{2} (2\alpha) - C^{\rm BB, fg}_{\ell} \cos^{2} (2\alpha) - C^{\rm EE, CMB}_{\ell} \sin^{2} (2\alpha + 2\beta)   - C^{\rm BB, CMB}_{\ell} \cos^{2} (2\alpha + 2\beta)  \nonumber \\
&-& 2 C^{\rm EB, fg}_{\ell} \cos (2\alpha)\sin (2\alpha) -  2 C^{\rm EB, CMB}_{\ell} \cos (2\alpha + 2\beta)\sin (2\alpha + 2\beta) \nonumber \\
&=&   \cos (4\alpha) (C^{\rm EE, fg}_{\ell}  - C^{\rm BB, fg}_{\ell} ) + \cos (4\alpha + 4\beta) (C^{\rm EE, CMB}_{\ell}  - C^{\rm BB, CMB}_{\ell} ) \nonumber \\
&-& 2 C^{\rm EB, fg}_{\ell} \sin (4\alpha) - 2 C^{\rm EB, CMB}_{\ell} \sin (4\alpha + 4\beta)
\ea
so that
\ba
C^{EB,o}_{\ell} &=& \frac{\sin(4\alpha)}{2}  (C^{\rm EE, fg}_{\ell}  - C^{\rm BB, fg}_{\ell} ) +  \frac{\tan(4\alpha)}{2} \cos (4\alpha + 4\beta) (C^{\rm EE, CMB}_{\ell}  - C^{\rm BB, CMB}_{\ell} ) \nonumber \\
&-& C^{\rm EB, fg}_{\ell} \sin (4\alpha )\tan(4\alpha)  - C^{\rm EB, CMB}_{\ell} \sin (4\alpha + 4\beta)\tan(4\alpha)  + \frac{ \sin(4\beta)}{2 \cos(4\alpha)}( C^{\rm EE, CMB}_{\ell} - C^{\rm BB, CMB}_{\ell}) \nonumber \\
&+& \frac{1}{\cos(4\alpha)} C^{\rm EB, fg}_{\ell}  + \frac{ \cos(4\beta)}{ \cos(4\alpha)} C^{\rm EB, CMB}_{\ell}  \nonumber \\
&=& \frac{\sin(4\alpha)}{2}  (C^{\rm EE, fg}_{\ell}  - C^{\rm BB, fg}_{\ell} )  +  (C^{\rm EE, CMB}_{\ell}  - C^{\rm BB, CMB}_{\ell} )\left( \frac{\tan(4\alpha)}{2} \cos (4\alpha + 4\beta) + \frac{ \sin(4\beta)}{2 \cos(4\alpha)}\right) \nonumber \\
&+& \left( \frac{1}{\cos(4\alpha)} - \sin (4\alpha )\tan(4\alpha) \right) C^{\rm EB, fg}_{\ell} + \left(\frac{ \cos(4\beta)}{ \cos(4\alpha)} - \sin (4\alpha + 4\beta)\tan(4\alpha) \right) C^{\rm EB, CMB}_{\ell}
\ea
Let's do some trigometry
\ba
 \frac{1}{\cos(4\alpha)} - \sin (4\alpha )\tan(4\alpha)  &=& \cos(4\alpha) \nonumber \\
 \frac{ \cos(4\beta)}{ \cos(4\alpha)} - \sin (4\alpha + 4\beta)\tan(4\alpha)   &=&  \frac{ \cos(4\beta)}{ \cos(4\alpha)}  - (\sin (4\alpha) \cos(4\beta)+ \sin (4\beta) \cos(4\alpha) )\tan(4\alpha) \nonumber \\
 &=&  \frac{ \cos(4\beta)}{ \cos(4\alpha)}   - \left(\sin^{2} (4\alpha)\frac{ \cos(4\beta)}{ \cos(4\alpha)}+ \sin (4\beta) \sin(4\alpha) \right) \nonumber \\
 &=&  \frac{ \cos(4\beta)}{ \cos(4\alpha)}(1 - \sin^{2} (4\alpha)) - \sin (4\beta) \sin(4\alpha) )  \nonumber \\
 &=&  \cos(4\beta) \cos(4\alpha) -  \sin (4\beta) \sin(4\alpha)  = \cos(4\beta + 4\alpha) \nonumber \\
  \frac{\tan(4\alpha)}{2} \cos (4\alpha + 4\beta) + \frac{ \sin(4\beta)}{2 \cos(4\alpha)} &=&    \frac{\tan(4\alpha)}{2} [\cos (4\alpha) \cos( 4\beta)- \sin(4\alpha) \sin( 4\beta) ]+ \frac{ \sin(4\beta)}{2 \cos(4\alpha)} \nonumber \\
 &=& \frac{\sin (4\alpha) \cos( 4\beta)}{2} - \frac{ \sin^{2}(4\alpha) \sin( 4\beta) }{2 \cos(4 \alpha)} + \frac{ \sin(4\beta)}{2 \cos(4\alpha)} \nonumber \\
 &=& \frac{\sin (4\alpha) \cos( 4\beta)}{2}  -  \frac{ \sin(4\beta)}{2 \cos(4\alpha)}  ( \sin^{2}(4\alpha)-1)  \nonumber \\
 &=& \frac{\sin (4\alpha) \cos( 4\beta)}{2}  + \frac{\cos (4\alpha) \sin( 4\beta)}{2} = \frac{ \sin(4\alpha + 4\beta)}{2}
\ea
With this, we demonstrate that
\ba
C^{EB,o}_{\ell} &=& \frac{\tan(4\alpha)}{2} \left( C^{EB,o}_{\ell} - C^{BB,o}_{\ell} \right) + \frac{ \sin(4\beta)}{2 \cos(4\alpha)}( C^{\rm EE, CMB}_{\ell} - C^{\rm BB, CMB}_{\ell}) \nonumber \\
&+& \frac{1}{\cos(4\alpha)} C^{\rm EB, fg}_{\ell}  + \frac{ \cos(4\beta)}{ \cos(4\alpha)} C^{\rm EB, CMB}_{\ell}
\ea
is indeed equal to
\ba
C^{EB,o}_{\ell} &=& \frac{\sin(4\alpha)}{2}(C^{\rm EE, fg}_{\ell}  - C^{\rm BB, fg}_{\ell} ) + \frac{\sin(4\alpha + 4\beta)}{2}(C^{\rm EE, CMB}_{\ell}  - C^{\rm BB, CMB}_{\ell} )  \nonumber \\
&+& \cos(4\alpha) C^{\rm EB, fg}_{\ell} + \cos(4\alpha + 4\beta) C^{\rm EB, CMB}_{\ell}
\ea
which complete the proof.

\section{A Multifrequency likelihood}
We can rewrite
\ba
C^{EE,o}_{\ell} &=& C^{\rm EE, fg}_{\ell} \cos^{2} (2\alpha) + C^{\rm BB, fg}_{\ell} \sin^{2} (2\alpha) + C^{\rm EE, CMB}_{\ell} \cos^{2} (2\alpha + 2\beta)   + C^{\rm BB, CMB}_{\ell} \sin^{2} (2\alpha + 2\beta)  \nonumber \\
&-& 2 C^{\rm EB, fg}_{\ell} \cos (2\alpha)\sin (2\alpha) -  2 C^{\rm EB, CMB}_{\ell} \cos (2\alpha + 2\beta)\sin (2\alpha + 2\beta)  \nonumber \\
C^{BB,o}_{\ell}  &=& C^{\rm EE, fg}_{\ell} \sin^{2} (2\alpha) + C^{\rm BB, fg}_{\ell} \cos^{2} (2\alpha) + C^{\rm EE, CMB}_{\ell} \sin^{2} (2\alpha + 2\beta)   + C^{\rm BB, CMB}_{\ell} \cos^{2} (2\alpha + 2\beta)  \nonumber \\
&+& 2 C^{\rm EB, fg}_{\ell} \cos (2\alpha)\sin (2\alpha) +  2 C^{\rm EB, CMB}_{\ell} \cos (2\alpha + 2\beta)\sin (2\alpha + 2\beta)  \nonumber \\
C^{EB,o}_{\ell} &=& \frac{\sin(4\alpha)}{2}(C^{\rm EE, fg}_{\ell}  - C^{\rm BB, fg}_{\ell} ) + \frac{\sin(4\alpha + 4\beta)}{2}(C^{\rm EE, CMB}_{\ell}  - C^{\rm BB, CMB}_{\ell} )  \nonumber \\
&+& \cos(4\alpha) C^{\rm EB, fg}_{\ell} + \cos(4\alpha + 4\beta) C^{\rm EB, CMB}_{\ell}
\ea
In a matricial form, let's neglect intrinsic EB correlation
\ba
\begin{pmatrix}
C^{\rm E_{i}E_{j},o}_{\ell} \cr
C^{\rm B_{i}B_{j},o}_{\ell}
\end{pmatrix}
&=&
{\bm R}(\alpha_{i}, \alpha_{j})
\begin{pmatrix}
C^{\rm E_{i}E_{j},fg}_{\ell} \cr
C^{\rm B_{i}B_{j},fg}_{\ell}
\end{pmatrix}
+
{\bm R}(\alpha_{i} + \beta, \alpha_{j}+ \beta)
\begin{pmatrix}
C^{\rm E_{i}E_{j},CMB}_{\ell} \cr
C^{\rm B_{i}B_{j},CMB}_{\ell}
\end{pmatrix} \nonumber \\
C^{\rm E_{i}B_{j},o}_{\ell} &=& \vec{R}^{T} (\alpha_{i}, \alpha_{j})
\begin{pmatrix}
C^{\rm E_{i}E_{j},fg}_{\ell} \cr
C^{\rm B_{i}B_{j},fg}_{\ell}
\end{pmatrix}
+
\vec{R}^{T} (\alpha_{i} + \beta, \alpha_{j}+ \beta )
\begin{pmatrix}
C^{\rm E_{i}E_{j},CMB}_{\ell} \cr
C^{\rm B_{i}B_{j},CMB}_{\ell}
\end{pmatrix}
\ea
With  ${\bm R}$ a rotation matrix and $\vec{R}$ a rotation vector
\ba
{\bm R}(\alpha_{i}, \alpha_{j})  =
\begin{pmatrix}
\cos(2\alpha_{i}) \cos(2\alpha_{j}) &
\sin(2\alpha_{i}) \sin(2\alpha_{j}) \cr
\sin(2\alpha_{i}) \sin(2\alpha_{j}) &
\cos(2\alpha_{i}) \cos(2\alpha_{j})
\end{pmatrix}
\ea
\ba
\vec{R} (\alpha_{i}, \alpha_{j}) =
\begin{pmatrix}
\cos(2\alpha_{i}) \sin(2\alpha_{j}) \cr
-\sin(2\alpha_{i}) \cos(2\alpha_{j})
\end{pmatrix}
\ea
We can simply re-express $C^{EB,o}_{\ell} $ as a function of $C^{EE,o}_{\ell} $ and $C^{BB,o}_{\ell} $ using the equation above
\ba
C^{\rm E_{i}B_{j},o}_{\ell} &=& \vec{R}^{T} (\alpha_{i}, \alpha_{j}) \left[ {\bm R}^{-1}(\alpha_{i}, \alpha_{j})
\begin{pmatrix}
C^{\rm E_{i}E_{j},o}_{\ell} \cr
C^{\rm B_{i}B_{j},o}_{\ell}
\end{pmatrix}
- {\bm R}^{-1}(\alpha_{i}, \alpha_{j})
{\bm R}(\alpha_{i} + \beta, \alpha_{j}+ \beta)
\begin{pmatrix}
C^{\rm E_{i}E_{j},CMB}_{\ell} \cr
C^{\rm B_{i}B_{j},CMB}_{\ell}
\end{pmatrix}  \right] \nonumber \\
&+&
\vec{R}^{T} (\alpha_{i} + \beta, \alpha_{j}+ \beta )
\begin{pmatrix}
C^{\rm E_{i}E_{j},CMB}_{\ell} \cr
C^{\rm B_{i}B_{j},CMB}_{\ell}
\end{pmatrix} \nonumber \\
&=& \vec{R}^{T} (\alpha_{i}, \alpha_{j})  {\bm R}^{-1}(\alpha_{i}, \alpha_{j})
\begin{pmatrix}
C^{\rm E_{i}E_{j},o}_{\ell} \cr
C^{\rm B_{i}B_{j},o}_{\ell}
\end{pmatrix}  \nonumber \\
&+& \left[  \vec{R}^{T} (\alpha_{i} + \beta, \alpha_{j}+ \beta ) -  \vec{R}^{T} (\alpha_{i}, \alpha_{j})  {\bm R}^{-1}(\alpha_{i}, \alpha_{j})  {\bm R}(\alpha_{i} + \beta, \alpha_{j}+ \beta) \right]
\begin{pmatrix}
C^{\rm E_{i}E_{j},CMB}_{\ell} \cr
C^{\rm B_{i}B_{j},CMB}_{\ell}
\end{pmatrix}
\ea
This can finally be rewritten
\ba
C^{\rm E_{i}B_{j},o}_{\ell} - {\bm A}
\begin{pmatrix}
C^{\rm E_{i}E_{j},o}_{\ell} \cr
C^{\rm B_{i}B_{j},o}_{\ell}
\end{pmatrix}
-   {\bm B}
\begin{pmatrix}
C^{\rm E_{i}E_{j},CMB}_{\ell} \cr
C^{\rm B_{i}B_{j},CMB}_{\ell}
\end{pmatrix}
= 0
\ea
with
\ba
{\bm A} =
\begin{pmatrix}
 \vec{R}^{T} (\alpha_{i}, \alpha_{j})  {\bm R}^{-1}(\alpha_{i}, \alpha_{j})
\end{pmatrix}
\ea
and
\ba
{\bm B} = \vec{R}^{T} (\alpha_{i} + \beta, \alpha_{j}+ \beta ) -  \vec{R}^{T} (\alpha_{i}, \alpha_{j})  {\bm R}^{-1}(\alpha_{i}, \alpha_{j})  {\bm R}(\alpha_{i} + \beta, \alpha_{j}+ \beta)
\ea
%A multifrequency likelihood can be written
%\ba
%-2\ln {\cal L} &=& \sum_{\ell}  ({\bm A}  \vec{C_{\ell}}^{\rm o}  - {\bm B}\vec{C_{\ell}}^{\rm CMB} )^{T} {\bm C}^{-1}({\bm A}  \vec{C_{\ell}}^{\rm o}  - {\bm B}\vec{C_{\ell}}^{\rm CMB} ) \nonumber \\
%{\bm C} &=& {\bm A} {\rm Cov}( \vec{C_{\ell}}^{\rm o}, \vec{C_{\ell}}^{\rm o}) {\bm A}^{T}
%\ea

\end{document}


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\title{\textbf{Modern Quantum Chemistry, Szabo \& Ostlund}\\HW}
\author{wsr
\vspace{5pt}\\
}
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\begin{document}
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\tableofcontents

\newpage

\setcounter{section}{5}

\section{Many-body Perturbation Theory}
\subsection{RS Perturbation Theory}

\subsection{Diagrammatic Representation of RS Perturbation Theory}
\subsubsection{Diagrammatic Perturbation Theory for Two States}
\ex{6.1}
~\\
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\end{minipage}
\begin{minipage}{0.7\linewidth}
	$ = (-1)^5 \dfrac{V_{12}V_{21}V_{11}^3}{(E_1^{(0)} - E_2^{(0)})^4} = -\dfrac{V_{12}V_{21}V_{11}^3}{(E_1^{(0)} - E_2^{(0)})^4} $\\
\end{minipage}
~\vspace{15pt}\\
\begin{minipage}{0.3\linewidth}
	\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(e) -- [fermion, quarter left, edge label=2] (d)
		-- [fermion, quarter left, edge label=2] (c)
		-- [fermion, quarter left, edge label=2] (b)
		-- [fermion, quarter left, edge label=2] (a)
		-- [fermion, quarter left, edge label=1] (e),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.7\linewidth}
	$ = (-1)^2 \dfrac{V_{12}V_{21}V_{22}^3}{(E_1^{(0)} - E_2^{(0)})^4} = \dfrac{V_{12}V_{21}V_{22}^3}{(E_1^{(0)} - E_2^{(0)})^4} $\\
\end{minipage}

~\vspace{15pt}\\
\begin{minipage}{0.3\linewidth}
	\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, quarter right, edge label'=1] (b)
		-- [fermion, quarter right, edge label'=1] (d)
		-- [fermion, quarter right, edge label'=1] (e)
		-- [fermion, quarter right, edge label'=2] (c)
		-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.7\linewidth}
	$ = (-1)^4 \dfrac{V_{12}V_{21}V_{11}^2 V_{22}}{(E_1^{(0)} - E_2^{(0)})^4} = \dfrac{V_{12}V_{21}V_{11}^2 V_{22}}{(E_1^{(0)} - E_2^{(0)})^4} $\\
\end{minipage}

~\vspace{15pt}\\
\begin{minipage}{0.3\linewidth}
	\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, quarter left, edge label=1] (c)
		-- [fermion, quarter left, edge label=1] (e)
		-- [fermion, quarter left, edge label=2] (d)
		-- [fermion, quarter left, edge label=2] (b)
		-- [fermion, quarter left, edge label=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.7\linewidth}
	$ = (-1)^3 \dfrac{V_{12}V_{21}V_{11} V_{22}^2}{(E_1^{(0)} - E_2^{(0)})^4} = -\dfrac{V_{12}V_{21}V_{11} V_{22}^2}{(E_1^{(0)} - E_2^{(0)})^4} $\\
\end{minipage}

\newpage
Similarly,\\
%~\vspace{15pt}\\
\begin{minipage}{0.2\linewidth}
	\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, quarter right, edge label'=1] (c)
		-- [fermion, quarter right, edge label'=1] (d)
		-- [fermion, quarter right, edge label'=1] (e)
		-- [fermion, quarter right, edge label'=2] (b)
		-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
,
\begin{minipage}{0.15\linewidth}
	%\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, quarter right, edge label'=1] (b)
		-- [fermion, quarter right, edge label'=1] (c)
		-- [fermion, quarter right, edge label'=1] (e)
		-- [fermion, quarter right, edge label'=2] (d)
		-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.6\linewidth}
	$ = \dfrac{V_{12}V_{21}V_{11}^2 V_{22}}{(E_1^{(0)} - E_2^{(0)})^4} $\\
\end{minipage}

~\vspace{15pt}\\
\begin{minipage}{0.2\linewidth}
	\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, quarter left, edge label=1] (b)
		-- [fermion, quarter left, edge label=1] (e)
		-- [fermion, quarter left, edge label=2] (d)
		-- [fermion, quarter left, edge label=2] (c)
		-- [fermion, quarter left, edge label=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
,
\begin{minipage}{0.15\linewidth}
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, quarter left, edge label=1] (d)
		-- [fermion, quarter left, edge label=1] (e)
		-- [fermion, quarter left, edge label=2] (c)
		-- [fermion, quarter left, edge label=2] (b)
		-- [fermion, quarter left, edge label=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.6\linewidth}
	$ = -\dfrac{V_{12}V_{21}V_{11} V_{22}^2}{(E_1^{(0)} - E_2^{(0)})^4} $\\
\end{minipage}
~\vspace{10pt}\\
thus, the sum of above terms is
\begin{align}
\dfrac{V_{12}V_{21}(V_{22}^3 - V_{11}^3)}{(E_1^{(0)} - E_2^{(0)})^4} + 3\times\dfrac{V_{12}V_{21}(V_{11}^2 V_{22} - V_{11}V_{22}^2)}{(E_1^{(0)} - E_2^{(0)})^4}
= \dfrac{V_{12}V_{21}(V_{22} - V_{11})^3}{(E_1^{(0)} - E_2^{(0)})^4}
\end{align}

\subsubsection{Diagrammatic Perturbation Theory for $ N $ States}
\ex{6.2}
The 4th-order perturbation energy of state $ i $ can be expressed as
\begin{align}
&\sum_{k,n,m\neq i} \dfrac{V_{ki}V_{nk}V_{mn}V_{im}}{(E_i^{(0)}-E_k^{(0)})(E_i^{(0)}-E_n^{(0)})(E_i^{(0)}-E_m^{(0)})} + \sum_{n\neq i} \dfrac{V_{ii}^2 V_{ni}V_{in}}{(E_i^{(0)}-E_n^{(0)})^3} - \sum_{m,n\neq i} \dfrac{V_{ii}V_{mi}V_{in}V_{nm}}{(E_i^{(0)}-E_m^{(0)})^2(E_i^{(0)}-E_n^{(0)})} \notag\\
&- \sum_{m,n\neq i} \dfrac{V_{ii}V_{ni}V_{im}V_{mn}}{(E_i^{(0)}-E_m^{(0)})^2(E_i^{(0)}-E_n^{(0)})} - \sum_{m,n\neq i} \dfrac{V_{mi}V_{im}V_{in}V_{ni}}{(E_i^{(0)}-E_m^{(0)})(E_i^{(0)}-E_n^{(0)})(2E_i^{(0)}-E_n^{(0)}-E_m^{(0)})} \notag\\
&- \sum_{m,n\neq i} \dfrac{V_{mi}V_{im}V_{in}V_{ni}}{(E_i^{(0)}-E_n^{(0)})^2(2E_i^{(0)}-E_n^{(0)}-E_m^{(0)})} \notag\\
= &\sum_{k,n,m\neq i} \dfrac{V_{ki}V_{nk}V_{mn}V_{im}}{(E_i^{(0)}-E_k^{(0)})(E_i^{(0)}-E_n^{(0)})(E_i^{(0)}-E_m^{(0)})} + \sum_{n\neq i} \dfrac{V_{ii}^2 V_{ni}V_{in}}{(E_i^{(0)}-E_n^{(0)})^3} - 2\sum_{m,n\neq i} \dfrac{V_{ii}V_{mi}V_{in}V_{nm}}{(E_i^{(0)}-E_m^{(0)})^2(E_i^{(0)}-E_n^{(0)})} \notag\\
&%- \sum_{m,n\neq i} \dfrac{V_{ii}V_{ni}V_{im}V_{mn}}{(E_i^{(0)}-E_m^{(0)})^2(E_i^{(0)}-E_n^{(0)})}
- \sum_{m,n\neq i} \dfrac{V_{mi}V_{im}V_{in}V_{ni}}{(E_i^{(0)}-E_m^{(0)})(E_i^{(0)}-E_n^{(0)})^2}
\end{align}
while
\begin{equation}\label{key}
\Braket{n | \sH | \Psi_i^{(3)}} + \Braket{n | \mathscr{V} | \Psi_i^{(2)}} = E_i^{(0)} \Braket{n|\Psi_i^{(3)}} + E_i^{(1)} \Braket{n|\Psi_i^{(2)}} + E_i^{(2)} \Braket{n|\Psi_i^{(1)}}
\end{equation}
\begin{align}\label{key}
(E_i^{(0)} - E_n^{(0)})\Braket{n | \Psi_i^{(3)}}
&= \Braket{n | \mathscr{V} | \Psi_i^{(2)}} - E_i^{(1)} \Braket{n|\Psi_i^{(2)}} - E_i^{(2)} \Braket{n|\Psi_i^{(1)}} \notag\\
&= \Braket{n | \mathscr{V} | \Psi_i^{(2)}}
- E_i^{(1)} \dfrac{\Braket{n | \mathscr{V} | \Psi_i^{(1)}} - E_i^{(1)}\Braket{n|\Psi_i^{(1)}}}{E_i^{(0)} - E_n^{(0)}}
- E_i^{(2)} \Braket{n|\Psi_i^{(1)}} \notag\\
&= \Braket{n | \mathscr{V} | \Psi_i^{(2)}}
- E_i^{(1)} \dfrac{\Braket{n | \mathscr{V} | \Psi_i^{(1)}}}{E_i^{(0)} - E_n^{(0)}}
+ \qty[E_i^{(1)}]^2 \dfrac{\Braket{n|\mathscr{V}|i}}{\qty[E_i^{(0)} - E_n^{(0)}]^2}
- E_i^{(2)} \dfrac{\Braket{n|\mathscr{V}|i}}{E_i^{(0)} - E_n^{(0)}}
\end{align}
\begin{align}
E_i^{(4)} &= \Braket{i | \mathscr{V} | \Psi_i^{(3)}} \notag\\
&= \sum_{n\neq i} \dfrac{\Braket{i | \mathscr{V} | n}}{E_i^{(0)} - E_n^{(0)}} \qty{\Braket{n | \mathscr{V} | \Psi_i^{(2)}}
- E_i^{(1)} \dfrac{\Braket{n | \mathscr{V} | \Psi_i^{(1)}}}{E_i^{(0)} - E_n^{(0)}}
+ \qty[E_i^{(1)}]^2 \dfrac{\Braket{n|\mathscr{V}|i}}{\qty[E_i^{(0)} - E_n^{(0)}]^2}
- E_i^{(2)} \dfrac{\Braket{n|\mathscr{V}|i}}{E_i^{(0)} - E_n^{(0)}} }\notag\\
&= \sum_{n\neq i} \dfrac{\Braket{i | \mathscr{V} | n}}{E_i^{(0)} - E_n^{(0)}} \Braket{n | \mathscr{V} | \Psi_i^{(2)}}
- E_i^{(1)} \sum_{n\neq i} \dfrac{\Braket{i | \mathscr{V} | n}}{\qty[E_i^{(0)} - E_n^{(0)}]^2} \Braket{n | \mathscr{V} | \Psi_i^{(1)}} \notag\\
&{}\quad +  \qty[E_i^{(1)}]^2 \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^3}
-  E_i^{(2)} \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^2} \notag\\
&= \sum_{n,m\neq i} \dfrac{\Braket{i | \mathscr{V} | n}}{E_i^{(0)} - E_n^{(0)}} \Braket{n | \mathscr{V} | m} \Braket{m |\Psi_i^{(2)}}
- E_i^{(1)} \sum_{n,m\neq i} \dfrac{\Braket{i | \mathscr{V} | n}\Braket{n | \mathscr{V} | m}\Braket{m | \mathscr{V} | i} }{\qty[E_i^{(0)} - E_n^{(0)}]^2\qty[E_i^{(0)} - E_m^{(0)}]}  \notag\\
&{}\quad +  \qty[E_i^{(1)}]^2 \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^3}
-  E_i^{(2)} \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^2} \notag\\
&= \sum_{n,m\neq i} \dfrac{V_{in} V_{nm}}{E_i^{(0)} - E_n^{(0)}} \dfrac{\Braket{m | \mathscr{V} | \Psi_i^{(1)}} - E_i^{(1)}\Braket{m|\Psi_i^{(1)}}}{E_i^{(0)} - E_m^{(0)}}
- E_i^{(1)} \sum_{n,m\neq i} \dfrac{V_{in}V_{nm}V_{mi} }{\qty[E_i^{(0)} - E_n^{(0)}]^2\qty[E_i^{(0)} - E_m^{(0)}]}  \notag\\
&{}\quad +  \qty[E_i^{(1)}]^2 \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^3}
-  E_i^{(2)} \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^2} \notag\\
&= \sum_{n,m,k\neq i} \dfrac{V_{in} V_{nm}}{E_i^{(0)} - E_n^{(0)}} \dfrac{\Braket{m | \mathscr{V} | k} \Braket{k| \mathscr{V} | i} }{\qty[E_i^{(0)} - E_m^{(0)}]\qty[E_i^{(0)} - E_k^{(0)}]}
- E_i^{(1)}\sum_{n,m\neq i} \dfrac{V_{in} V_{nm}}{E_i^{(0)} - E_n^{(0)}} \dfrac{ \Braket{m| \mathscr{V} |i}}{\qty[E_i^{(0)} - E_m^{(0)}]^2} \notag\\
&{}\quad- E_i^{(1)} \sum_{n,m\neq i} \dfrac{V_{in}V_{nm}V_{mi} }{\qty[E_i^{(0)} - E_n^{(0)}]^2\qty[E_i^{(0)} - E_m^{(0)}]}
 +  \qty[E_i^{(1)}]^2 \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^3}
-  E_i^{(2)} \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^2} \notag\\
&= \sum_{n,m,k\neq i} \dfrac{V_{in} V_{nm}V_{mk}V_{ki}}{\qty[E_i^{(0)} - E_n^{(0)}]\qty[E_i^{(0)} - E_m^{(0)}]\qty[E_i^{(0)} - E_k^{(0)}]}
- 2V_{ii}\sum_{n,m\neq i} \dfrac{V_{in} V_{nm} V_{mi}}{\qty[E_i^{(0)} - E_n^{(0)}]\qty[E_i^{(0)} - E_m^{(0)}]^2}  \notag\\
&{}\quad %- V_{ii} \sum_{n,m\neq i} \dfrac{V_{in}V_{nm}V_{mi} }{\qty[E_i^{(0)} - E_n^{(0)}]^2\qty[E_i^{(0)} - E_m^{(0)}]}
+  V_{ii}^2 \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^3}
-  \sum_{m\neq i} \dfrac{V_{mi}V_{im}}{\qty[E_i^{(0)} - E_m^{(0)}]} \sum_{n\neq i} \dfrac{V_{in}V_{ni}}{\qty[E_i^{(0)} - E_n^{(0)}]^2}
\end{align}
which agrees with diagrammatic results above.

\subsubsection{Summation of Diagrams}

\newpage
\subsection{Orbital Perturbation Theory: One-Particle Perturbations}
\ex{6.3}
Since $ n\neq 0 $ and $ v(i) $ is one-particle operator, $ n $ must be single-excited, i.e. $ \ket{\Psi_a^r} $. Thus,
\begin{align}
E_0^{(2)} &= \sum_{a,r} \dfrac{\abs{\Braket{\Psi_0 | \sum_i v(i) | \Psi_a^r}}^2}{\Braket{\Psi_0 | \sH | \Psi_0} - \Braket{\Psi_a^r| \sH | \Psi_a^r}} \notag\\
&= \sum_{a,r} \dfrac{v_{ar} v_{ra}}{\sum_b \varepsilon_b^{(0)} - \qty(\sum_{b\neq a}\varepsilon_b^{(0)} + \varepsilon_r^{(0)})} \notag\\
&= \sum_{a,r} \dfrac{v_{ar} v_{ra}}{\varepsilon_a^{(0)} - \varepsilon_r^{(0)}}
\end{align}

\ex{6.4}
Eq 6.15 in textbook gives
\begin{align}
E_i^{(3)} &= \sum_{n,m\neq i} \dfrac{\Braket{i|\mathscr{V}|n} \Braket{n|\mathscr{V}|m} \Braket{m|\mathscr{V}|i}}{(E_i^{(0)}-E_n^{(0)}) (E_i^{(0)}-E_m^{(0)})}
- E_i^{(1)}\sum_{n\neq i} \dfrac{\abs{\Braket{i|\mathscr{V}|n}}^2 }{(E_i^{(0)}-E_n^{(0)})^2} \notag\\
&= A_i^{(3)} + B_i^{(3)}
\end{align}
\subex{a.}
\begin{align}
B_0^{(3)} &= - E_0^{(1)}\sum_{n\neq 0} \dfrac{\abs{\Braket{\Psi_0|\mathscr{V}|n}}^2 }{(E_0^{(0)}-E_n^{(0)})^2} \notag\\
&= -\sum_b v_{bb} \sum_{a,r}\dfrac{v_{ar}v_{ra}}{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)})^2} \notag\\
&= - \sum_{a,b,r}\dfrac{v_{aa} v_{br}v_{rb}}{(\varepsilon_b^{(0)} - \varepsilon_r^{(0)})^2}
\end{align}
\subex{b.}
\begin{align}
A_0^{(3)} &= \sum_{n,m\neq 0} \dfrac{\Braket{\Psi_0|\mathscr{V}|n} \Braket{n|\mathscr{V}|m} \Braket{m|\mathscr{V}|\Psi_0}}{(E_0^{(0)}-E_n^{(0)}) (E_0^{(0)}-E_m^{(0)})} \notag\\
&= \sum_{a,r,b,s} \dfrac{\Braket{\Psi_0|\mathscr{V}|\Psi_a^r} \Braket{\Psi_a^r|\mathscr{V}|\Psi_b^s} \Braket{\Psi_b^s|\mathscr{V}|\Psi_0}}{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_s^{(0)})} \notag\\
&= \sum_{a,r,b,s} \dfrac{v_{ar}v_{sb} \Braket{\Psi_a^r|\mathscr{V}|\Psi_b^s} }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_s^{(0)})}
\end{align}
\subex{c.}
Clearly, if $ a\neq b, r\neq s $
\begin{align}
\Braket{\Psi_a^r | \mathscr{V} | \Psi_b^s} = 0
\end{align}
If $ a=b, r\neq s $,
\begin{align}
\Braket{\Psi_a^r | \mathscr{V} | \Psi_b^s}
&= \Braket{r| v |s} \notag\\
&= v_{rs}
\end{align}
If $ a\neq b, r= s $,
\begin{align}
\Braket{\Psi_a^r | \mathscr{V} | \Psi_b^s}
&= \Braket{\Psi_a^r | \mathscr{V} | \Psi_b^r} \notag\\
&= \Braket{\Psi_a^r | \mathscr{V} | -\Psi_{ab}^{ra}} \notag\\
&= -\Braket{b| v |a} \notag\\
&= -v_{ba}
\end{align}
If $ a= b, r= s $,
\begin{align}
\Braket{\Psi_a^r | \mathscr{V} | \Psi_b^s}
&= \Braket{\Psi_a^r | \mathscr{V} | \Psi_a^r} \notag\\
&= \sum_c v_{cc} - v_{aa} + v_{rr}
\end{align}

\subex{d.}
\begin{align}
E_0^{(3)} &= A_0^{(3)} + B_0^{(3)} \notag\\
&= \sum_{a,r,b,s}
\dfrac{v_{ar}v_{sb} \Braket{\Psi_a^r|\mathscr{V}|\Psi_b^s} }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_s^{(0)})}
- \sum_{a,b,r}\dfrac{v_{aa} v_{br}v_{rb}}{(\varepsilon_b - \varepsilon_r)^2}
\notag\\
&=  \sum_{a,r\neq s}
\dfrac{v_{ar}v_{sa} v_{rs} }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_a^{(0)} - \varepsilon_s^{(0)})}
+ \sum_{a\neq b,r}
\dfrac{v_{ar}v_{rb}(-v_{ba}) }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_r^{(0)})}
\notag\\
&\quad {} + \sum_{a,r}
\dfrac{v_{ar}v_{ra} (\sum_c v_{cc} - v_{aa} + v_{rr}) }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)})^2}
- \sum_{a,b,r}\dfrac{v_{aa} v_{br}v_{rb}}{(\varepsilon_b^{(0)} - \varepsilon_r^{(0)})^2}
\notag\\
&=  \sum_{a,r\neq s}
\dfrac{v_{ar}v_{sa} v_{rs} }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_a^{(0)} - \varepsilon_s^{(0)})}
+ \sum_{a\neq b,r}
\dfrac{v_{ar}v_{rb}(-v_{ba}) }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_r^{(0)})}
\notag\\
&\quad {} + \sum_{a,r}
\dfrac{v_{ar}v_{ra} (\sum_c v_{cc} - v_{aa} + v_{rr}) }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)})^2}
- \sum_{a,r}\dfrac{\sum_c v_{cc} v_{ar}v_{ra}}{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)})^2}
\notag\\
&=  \sum_{a,r\neq s}
\dfrac{v_{ar}v_{sa} v_{rs} }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_a^{(0)} - \varepsilon_s^{(0)})}
+ \sum_{a\neq b,r}
\dfrac{v_{ar}v_{rb}(-v_{ba}) }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_r^{(0)})}
+ \sum_{a,r}
\dfrac{v_{ar}v_{ra} ( - v_{aa} + v_{rr}) }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)})^2}
\notag\\
&=  \sum_{a,r,s}
\dfrac{v_{ar}v_{sa} v_{rs} }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_a^{(0)} - \varepsilon_s^{(0)})}
- \sum_{a,b,r}
\dfrac{v_{ar}v_{rb}v_{ba} }{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_r^{(0)})}
\end{align}

\subex{e.}
That's obvious.

\ex{6.5}
Since $ a,b $ run over all $ n $ occupied orbitals $ i,j $ and $ r $ runs over all $ n $ unoccupied orbitals $ k^* $, we have
\begin{align}
-2\sum_{a,b,r}^{N/2} \dfrac{v_{ra}v_{ab}v_{br}}{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)}) (\varepsilon_b^{(0)} - \varepsilon_r^{(0)})}
&= -\dfrac{2}{(2\beta)^2} \sum_i^n \sum_j^n \sum_k^n \Braket{i|v|j} \Braket{j|v|k^*} \Braket{k^*|v|i} \notag\\
&= -\dfrac{2}{(2\beta)^2} \sum_i^3
\mqty[\Braket{i|v|i+1} \Braket{i+1|v|(i+2)^*} \Braket{(i+2)^*|v|i} \\
+ \Braket{i|v|i+2} \Braket{i+2|v|(i+1)^*} \Braket{(i+1)^*|v|i} ] \notag\\
&= -\dfrac{2}{(2\beta)^2} \sum_i^3 \qty[(\beta/2)(\beta/2)(-\beta/2) + (\beta/2)(-\beta/2)(\beta/2)] \notag\\
&= -\dfrac{2}{(2\beta)^2} \times 3\times (-\beta^3/4) \notag\\
&= 3\beta/8
\end{align}

\ex{6.6}
\subex{a.}
Using the general expression, we get
\begin{align}
\mathscr{E}_0 &= 6\alpha - 2\sum_{j=-1}^1 (\beta_1^2 + \beta_2^2 + 2\beta_1\beta_2\cos\dfrac{2j\pi}{3})^{1/2} \notag\\
&= 6\alpha - 2(\beta_1^2 + \beta_2^2 + 2\beta_1\beta_2\cos\dfrac{-2\pi}{3})^{1/2}
- 2(\beta_1^2 + \beta_2^2 + 2\beta_1\beta_2\cos 0)^{1/2}
- 2(\beta_1^2 + \beta_2^2 + 2\beta_1\beta_2\cos\dfrac{2\pi}{3})^{1/2}
\notag\\
&= 6\alpha - 2(\beta_1^2 + \beta_2^2 - \beta_1\beta_2)^{1/2}
- 2(\beta_1^2 + \beta_2^2 + 2\beta_1\beta_2)^{1/2}
- 2(\beta_1^2 + \beta_2^2 - \beta_1\beta_2)^{1/2} \notag\\
&= 6\alpha
- 2\abs{\beta_1 + \beta_2}
- 4(\beta_1^2 + \beta_2^2 - \beta_1\beta_2)^{1/2} \notag\\
&= 6\alpha + 2(\beta_1 + \beta_2)
- 4(\beta_1^2 + \beta_2^2 - \beta_1\beta_2)^{1/2}
\end{align}

Using H\"uckel matrix:
\begin{align}
\vb{H} = \mqty(
\alpha  & \beta_1 & 0       & 0       & 0       & \beta_2\\
\beta_1 & \alpha  & \beta_2 & 0       & 0       & 0      \\
0       & \beta_2 & \alpha  & \beta_1 & 0       & 0      \\
0       & 0       & \beta_1 & \alpha  & \beta_2 & 0      \\
0       & 0       & 0       & \beta_2 & \alpha  & \beta_1\\
\beta_2 & 0       & 0       & 0       & \beta_1 & \alpha
)
\end{align}
%Diagonalize $ \vb{H} - \bm\varepsilon $, we get
Eigenvalues of $ \vb{H} $ are
\begin{align}\label{key}
&\alpha + (\beta_1 + \beta_2), \notag\\
&\alpha - \sqrt{\beta_1^2 + \beta_2^2 - \beta_1\beta_2} \;\text{ (2-fold)}, \notag\\
&\alpha + \sqrt{\beta_1^2 + \beta_2^2 - \beta_1\beta_2} \;\text{ (2-fold)}, \notag\\
&\alpha - (\beta_1 + \beta_2),
\end{align}
thus
\begin{align}
\mathscr{E}_0 &= 2\qty[\alpha + (\beta_1 + \beta_2)] + 4\qty[\alpha - \sqrt{\beta_1^2 + \beta_2^2 - \beta_1\beta_2}] \notag\\
&= 6\alpha + 2(\beta_1 + \beta_2) - 4\sqrt{\beta_1^2 + \beta_2^2 - \beta_1\beta_2}
\end{align}

\subex{b.}
\begin{align}
E_R &= \mathscr{E}_0 - (N\alpha + N\beta) \notag\\
&= 6\alpha + 2(\beta_1 + \beta_2) - 4\sqrt{\beta_1^2 + \beta_2^2 - \beta_1\beta_2} - (6\alpha + 6\beta) \notag\\
&= -4\beta_1 + 2\beta_2 - 4\sqrt{\beta_1^2 + \beta_2^2 - \beta_1\beta_2} \notag\\
&= 4\beta\qty(-1 + \dfrac{1}{2}x + \sqrt{1 + x^2 - x})
\end{align}

\subex{c.}
\begin{align}
E_R &= 4\beta\qty(-1 + \dfrac{1}{2}x + \sqrt{1 + x^2 - x}) \notag\\
&= 4\beta\qty[ -1 + \dfrac{1}{2}x
+ 1 + \dfrac{1}{2}(x^2-x) - \dfrac{1}{8}(x^2-x)^2 + \dfrac{1}{16}(x^2-x)^3 - \dfrac{5}{128}(x^2-x)^4 ] \notag\\
&= 4\beta\qty[
\dfrac{1}{2}x^2 - \dfrac{1}{8}(x^4 + x^2 - 2x^3)
+ \dfrac{1}{16}(-x^3 + 3x^4) - \dfrac{5}{128}x^4 + \cdots ] \notag\\
&= 4\beta\qty[
\dfrac{3}{8}x^2 + \dfrac{3}{16}x^3 + \dfrac{3}{128}x^4 + \cdots ] \notag\\
&= \beta\qty[
\dfrac{3}{2}x^2 + \dfrac{3}{4}x^3 + \dfrac{3}{32}x^4 + \cdots ]
\end{align}

\newpage
\subsection{Diagrammatic Representation of Orbital Perturbation Theory}
\ex{6.7}
\subex{a.}
~\\
\begin{minipage}{0.2\linewidth}
		\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	%\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, quarter right, edge label'=a] (b)
		-- [fermion, quarter right, edge label'=b] (d)
		-- [fermion, quarter right, edge label'=s] (c)
		-- [fermion, quarter right, edge label'=r] (a)
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
	\\
	\vspace{25pt}\\
\end{minipage}
\begin{minipage}{0.8\linewidth}
	\vspace{25pt}~\\
	\begin{align}
	&= - \sum_{a,b,r,s} \dfrac{v_{ab}v_{bs}v_{sr}v_{ra}}{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)})(\varepsilon_b^{(0)} - \varepsilon_r^{(0)})(\varepsilon_b^{(0)} - \varepsilon_b^{(0)})}  &\notag\\
	&= -\dfrac{1}{(2\beta)^3} \sum_{i,j,k,l} \Braket{i|v|j}\Braket{j|v|k^*}\Braket{k^*|v|l^*}\Braket{l^*|v|i} & \notag\\
	&= -\dfrac{2}{(2\beta)^3} \sum_i^{N/2} [-1 + 1 -1 -1 +1 -1]\times (\beta/2)^4  & \notag\\
	&= \dfrac{N\beta}{64}  & \notag\\
	\end{align}
\end{minipage}
%\tikzfeynmanset{
%}
The pictorial representation of the summation are as follows\\
\begin{center}
	\begin{tikzpicture}
	\begin{feynman}[nodes=circle, large]
	\vertex (a) {$ (i) $};
	\vertex [below right=of a] (b1) {$ (i-1) $};
	\vertex [above right=of a] (b2) {$ (i+1) $};
	\vertex [below right=of b1] (c1) {$ (i-2)^* $};
	\vertex [above right=of b1] (c2) {$ (i)^* $};
	\vertex [above right=of b2] (c3) {$ (i+2)^* $};
	\vertex [above right=of c1] (d1) {$ (i-1)^* $};
	\vertex [above right=of c2] (d2) {$ (i+1)^* $};
	\vertex [above right=of d1] (e) {$ (i) $};
	\diagram[inline=(c)]{
		(a) -- [plain, edge label'=+] (b1);
		(a) -- [plain, edge label=+] (b2);
		(b1) -- [plain, edge label'=--] (c1);
		(b1) -- [plain, edge label'=+] (c2);
		(b2) -- [plain, edge label=--] (c2);
		(b2) -- [plain, edge label=+] (c3);
		(c1) -- [plain, edge label'=--] (d1);
		(c2) -- [plain, edge label'=--] (d1);
		(c2) -- [plain, edge label=--] (d2);
		(c3) -- [plain, edge label=--] (d2);
		(d1) -- [plain, edge label'=--] (e);
		(d2) -- [plain, edge label=+] (e);
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{center}

\begin{minipage}{0.2\linewidth}
	\hfill
	\begin{tikzpicture}
	\begin{feynman}[small]
	\vertex (a);
	\vertex [below=of a] (b);
	\vertex [below=of b] (c);
	\vertex [below=of c] (d);
	%\vertex [below=of d] (e);
	\diagram[inline=(c)]{
		(a) -- [fermion, out=200, in=150, edge label'=a] (d)
		-- [fermion, out=45, in=330, edge label'=r] (b)
		-- [fermion, quarter left, edge label'=b] (c)
		-- [fermion, quarter left, edge label=s] (a)
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
	\\
	\vspace{25pt}\\
\end{minipage}
\begin{minipage}{0.8\linewidth}
	\vspace{25pt}~\\
	\begin{align}
	&= - \sum_{a,r,b,s} \dfrac{v_{ar}v_{rb}v_{bs}v_{sa}}{(\varepsilon_a^{(0)} - \varepsilon_r^{(0)})(\varepsilon_a^{(0)} - \varepsilon_s^{(0)})(\varepsilon_a^{(0)} + \varepsilon_b^{(0)} - \varepsilon_r^{(0)} - \varepsilon_s^{(0)})}  &\notag\\
	&= -\dfrac{1}{(2\beta)^2\times 4\beta} \sum_{i,j,k,l} \Braket{i|v|j^*}\Braket{j^*|v|k}\Braket{k|v|l^*}\Braket{l^*|v|i} & \notag\\
	&= -\dfrac{2}{(2\beta)^2\times 4\beta} \sum_i^{N/2} 6\times (\beta/2)^4  & \notag\\
	&= -\dfrac{3N\beta}{128}  & \notag\\
	\end{align}
\end{minipage}
%\tikzfeynmanset{
%}
The pictorial representation of the summation are as follows\\
\begin{center}
	\begin{tikzpicture}
	\begin{feynman}[nodes=circle, large]
	\vertex (a) {$ (i) $};
	\vertex [below right=of a] (b1) {$ (i-1)^* $};
	\vertex [above right=of a] (b2) {$ (i+1)^* $};
	\vertex [below right=of b1] (c1) {$ (i-2) $};
	\vertex [above right=of b1] (c2) {$ (i) $};
	\vertex [above right=of b2] (c3) {$ (i+2) $};
	\vertex [above right=of c1] (d1) {$ (i-1)^* $};
	\vertex [above right=of c2] (d2) {$ (i+1)^* $};
	\vertex [above right=of d1] (e) {$ (i) $};
	\diagram[inline=(c)]{
		(a) -- [plain, edge label'=--] (b1);
		(a) -- [plain, edge label=+] (b2);
		(b1) -- [plain, edge label'=+] (c1);
		(b1) -- [plain, edge label'=--] (c2);
		(b2) -- [plain, edge label=+] (c2);
		(b2) -- [plain, edge label=--] (c3);
		(c1) -- [plain, edge label'=+] (d1);
		(c2) -- [plain, edge label'=--] (d1);
		(c2) -- [plain, edge label=+] (d2);
		(c3) -- [plain, edge label=--] (d2);
		(d1) -- [plain, edge label'=--] (e);
		(d2) -- [plain, edge label=+] (e);
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{center}
thus
\begin{equation}\label{key}
E_0^{(4)} = 4\times \dfrac{N\beta}{64} + 3\times\qty(-\dfrac{3N\beta}{128}) =  \dfrac{N\beta}{64}
\end{equation}

\subex{b.}
Let $ N=6 $, we get
\begin{equation}\label{key}
E_0^{(4)} = \dfrac{3\beta}{32}
\end{equation}
which agrees with the result in Ex 6.6.

\subsection{Perturbation Expansion of the Correlation Energy}
\ex{6.8}
\begin{align}
E_0^{(2)} &=
\dfrac{1}{4}\sum_{a,b,r,s} \dfrac{\abs{\Braket{ab||rs}}^2}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s} \notag\\
&= \dfrac{1}{4}\sum_{a,b,r,s}
\dfrac{\qty(\Braket{ab|rs} - \Braket{ab|sr}) \qty(\Braket{rs|ab} - \Braket{sr|ab})}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s} \notag\\
&= \dfrac{1}{4}\sum_{a,b,r,s}
\dfrac{\Braket{ab|rs}\Braket{rs|ab} - \Braket{ab|sr}\Braket{rs|ab}  - \Braket{ab|rs}\Braket{sr|ab} + \Braket{ab|sr}\Braket{sr|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s} \notag\\
&= \dfrac{1}{4}\qty[\sum_{a,b,r,s}
\dfrac{\Braket{ab|rs}\Braket{rs|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
- \sum_{a,b,r,s}
\dfrac{\Braket{ab|sr}\Braket{rs|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
- \sum_{a,b,r,s}
\dfrac{ \Braket{ab|rs}\Braket{sr|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
+ \sum_{a,b,r,s}
\dfrac{\Braket{ab|sr}\Braket{sr|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
] \notag\\
&= \dfrac{1}{4}\qty[2\sum_{a,b,r,s}
\dfrac{\Braket{ab|rs}\Braket{rs|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
- 2\sum_{a,b,r,s}
\dfrac{\Braket{ab|rs}\Braket{sr|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
] \notag\\
&= \dfrac{1}{2}\sum_{a,b,r,s}
\dfrac{\Braket{ab|rs}\Braket{rs|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
- \dfrac{1}{2}\sum_{a,b,r,s}
\dfrac{\Braket{ab|rs}\Braket{rs|ba}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
\end{align}
For a closed-shell system, the possible spin part of $ a,b,r,s $ of the non-zero terms are\\
first term: $ \alpha,\alpha,\alpha,\alpha $;\quad $ \alpha,\beta,\alpha,\beta $;\quad $ \beta,\alpha,\beta,\alpha $;\quad $ \beta,\beta,\beta,\beta $\\
second term: $ \alpha,\alpha,\alpha,\alpha $;\quad $ \beta,\beta,\beta,\beta $\\
thus
\begin{align}
E_0^{(2)} &= 2\sum_{a,b,r,s}^{N/2}
\dfrac{\Braket{ab|rs}\Braket{rs|ab}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
- \sum_{a,b,r,s}^{N/2}
\dfrac{\Braket{ab|rs}\Braket{rs|ba}}{\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_s}
\end{align}

\ex{6.9}
\begin{align}
E_{\corr} &= \Delta - (\Delta^2 + K_{12}^2)^{1/2} \notag\\
&= \Delta - \qty[\Delta + \dfrac{K_{12}^2}{2\Delta}] \notag\\
&= - \dfrac{K_{12}^2}{2\Delta} \notag\\
&= - \dfrac{K_{12}^2}{2(\varepsilon_2-\varepsilon_1) + J_{11} + J_{22} - 4J_{12} + 2K_{12}} \notag\\
&= - K_{12}^2 \qty(\dfrac{1}{2(\varepsilon_2-\varepsilon_1)} - \dfrac{ J_{11} + J_{22} - 4J_{12} + 2K_{12}}{4(\varepsilon_2-\varepsilon_1)^2}) \notag\\
&= \dfrac{K_{12}^2}{2(\varepsilon_1-\varepsilon_2)} + \dfrac{K_{12}^2 (J_{11} + J_{22} - 4J_{12} + 2K_{12})}{4(\varepsilon_1-\varepsilon_2)^2})
\end{align}

\subsection{The $ N $-dependence of the RS Perturbation Expansion}
\ex{6.10}
From Eq 6.68, we get
\begin{align}
E_0^{(1)} &= \Braket{\Psi_0 | \mathscr{V} | \Psi_0} = -\dfrac{1}{2}\sum_{ab}\Braket{ab||ab} \notag\\
&= -\dfrac{1}{2}\sum_{i=1}^N \qty[\Braket{1_i \bar{1}_i|| 1_i \bar{1}_i} + \Braket{\bar{1}_i 1_i || \bar{1}_i 1_i}] \notag\\
&= -\dfrac{1}{2}\sum_{i=1}^N
\qty[\Braket{1_i \bar{1}_i| 1_i \bar{1}_i}
- \Braket{1_i \bar{1}_i| \bar{1}_i 1_i}
+ \Braket{\bar{1}_i 1_i | \bar{1}_i 1_i}
- \Braket{\bar{1}_i 1_i | 1_i \bar{1}_i} ] \notag\\
&= -\dfrac{1}{2}\times 2N [1_i 1_i| 1_i 1_i] \notag\\
&= -N J_{11}
\end{align}
\begin{align}
\Braket{\Psi_{1_i\bar{1}_i}^{2_i\bar{2}_i} | \mathscr{V} | \Psi_{1_i\bar{1}_i}^{2_i\bar{2}_i}}
&= \Braket{\Psi_{1_i\bar{1}_i}^{2_i\bar{2}_i} | \sH | \Psi_{1_i\bar{1}_i}^{2_i\bar{2}_i}}
- \Braket{\Psi_{1_i\bar{1}_i}^{2_i\bar{2}_i} | \sH_0 | \Psi_{1_i\bar{1}_i}^{2_i\bar{2}_i}} \notag\\
&= (2N-2) h_{11} + 2h_{22} + (N-1)J_{11} + J_{22} - (2N-2)\varepsilon_1 - 2\varepsilon_2 \notag\\
&= (2N-2) h_{11} + 2h_{22} + (N-1)J_{11} + J_{22} - (2N-2)(h_{11} + J_{11}) - 2(h_{22} + 2J_{12} - K_{12}) \notag\\
&= -(N-1)J_{11} + J_{22} - 4J_{12} + 2K_{12}
\end{align}

\subsection{Diagrammatic Representation of the Perturbation Expansion of the Correlation Energy}

\subsubsection{Hugenholtz Diagrams}
\ex{6.11}
The numerator and denominator are obvious.\\
$ h=5 $, and $ l=2 $ since closed loops are $ r\ra a\ra d\ra t\ra e\ra r; \; s\ra c\ra b\ra s $. The number of quivalent line pairs is one ($ r,s $). Thus the pre-factor is $ -\dfrac{1}{2} $.

\subsubsection{Goldstone Diagrams}
\ex{6.12}
~\\
1
\begin{minipage}{0.4\linewidth}
	\centering
\begin{tikzpicture}
\begin{feynman}[large]
\vertex (a); \vertex [below=of a] (b);
\vertex [right=of a] (c); \vertex [right=of b] (d);

\diagram[%inline=(e.base)
]{
	(a) -- [fermion,  quarter left, edge label=r] (b)
	-- [fermion, quarter left, edge label=a] (a),
	(c)-- [fermion, quarter left, edge label=s] (d)
	-- [fermion, quarter left, edge label=b] (c),
	(a) -- [scalar] (c),
	(b) -- [scalar] (d),
	%-- [fermion, quarter right, edge label'=2] (a),
};
\end{feynman}
\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=of a] (b);
	%\vertex [right=of a] (c); %\vertex [right=of b] (d);
	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion,  half left, edge label=r] (b)
		-- [fermion, half left, edge label=a] (a),
		(a)-- [fermion, quarter left, edge label=s] (b)
		-- [fermion, quarter left, edge label=b] (a),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
2
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=of a] (b);
	\vertex [right=of a] (c); \vertex [right=of b] (d);

	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion, out=-60, in=150, edge label'=b] (d)
		-- [fermion, quarter right, edge label=s] (c),
		(c)-- [fermion, out=240, in=30, edge label=a] (b)
		-- [fermion, quarter left, edge label=r] (a),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=of a] (b);
	%\vertex [right=of a] (c); %\vertex [right=of b] (d);
	\diagram[%inline=(e.base)
	]{
		(b) -- [fermion,  half right, edge label'=r] (a),
		(b) -- [fermion, half left, edge label'=s] (a),
		(a)-- [fermion, quarter left, edge label=a] (b),
		(a) -- [fermion, quarter right, edge label=b] (b),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
1
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [anti fermion,  quarter left, edge label=t] (e)
		-- [anti fermion,  quarter left, edge label=r] (b)
		-- [anti fermion, quarter left, edge label=a] (a),
		(c) -- [fermion, quarter left, edge label=b] (d)
		-- [fermion, quarter left, edge label=u] (f)
		-- [fermion, quarter left, edge label=s] (c),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=of a] (b);
	%\vertex [right=of a] (c); %\vertex [right=of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e);
	\diagram[%inline=(e.base)
	]{
		(a) -- [anti fermion,  quarter left, edge label=t] (e)
		-- [anti fermion,  quarter left, edge label=r] (b)
		-- [anti fermion, half left, edge label=a] (a),
		(a) -- [fermion, half left, edge label=b] (b)
		-- [fermion, quarter left, edge label=u] (e)
		-- [fermion, quarter left, edge label=s] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
2
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion,  quarter left, edge label=c] (e)
		-- [ fermion,  quarter left, edge label=a] (b)
		-- [ fermion, quarter left, edge label=r] (a),
		(c) -- [anti fermion, quarter left, edge label=s] (d)
		-- [anti fermion, quarter left, edge label=d] (f)
		-- [anti fermion, quarter left, edge label=b] (c),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e);
	%\vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion,  quarter left, edge label=c] (e)
		-- [ fermion,  quarter left, edge label=a] (b)
		-- [ fermion, half left, edge label=r] (a),
		(a) -- [anti fermion, half left, edge label=s] (b)
		-- [anti fermion, quarter left, edge label=d] (e)
		-- [anti fermion, quarter left, edge label=b] (a),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
3
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [anti fermion,  quarter left, edge label=s] (e)
		-- [anti fermion,  quarter left, edge label=r] (b)
		-- [anti fermion, quarter left, edge label=a] (a),
		(c) -- [anti fermion, quarter left, edge label=t] (d)
		-- [anti fermion, quarter left, edge label=c] (f)
		-- [anti fermion, quarter left, edge label=b] (c),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e);
	%\vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [anti fermion,  quarter left, edge label=s] (e)
		-- [anti fermion,  quarter left, edge label=r] (b)
		-- [anti fermion, half left, edge label=a] (a),
		(a) -- [anti fermion, half left, edge label=t] (b)
		-- [anti fermion, quarter left, edge label=c] (e)
		-- [anti fermion, quarter left, edge label=b] (a),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
4
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (-0.7cm, 0)$) (e); \vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion,  quarter left, edge label=a] (f)
		-- [ fermion,  quarter left, edge label=b] (b)
		-- [ fermion, quarter left, edge label=r] (e)
		-- [ fermion, quarter left, edge label=s] (a),
		(c) -- [fermion, quarter left, edge label=c] (d)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, quarter left, edge label=t] (c),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e);
	%\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion,  quarter left, edge label=a] (e)
		-- [ fermion,  quarter left, edge label=b] (b)
		-- [ fermion, quarter left, edge label=r] (e)
		-- [ fermion, quarter left, edge label=s] (a),
		(a) -- [fermion, half left, edge label=c] (b)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, half left, edge label=t] (a),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
5
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right= of a] (c); \vertex [right=4cm of b] (d);
	\vertex [below=1.2cm of c] (e); \vertex [right= of e] (f);

	\diagram[%inline=(e.base)
	]{
		(a) %-- [fermion,  quarter left, edge label=a] (f)
		-- [ fermion,  quarter left, edge label=a] (b)
		%-- [ fermion, quarter left, edge label=r] (e)
		-- [ fermion, quarter left, edge label=r] (a),
		(c) -- [fermion, quarter left, edge label=b] (e)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, quarter left, edge label=s] (c),
		(f) -- [fermion, quarter left, edge label=c] (d)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, quarter left, edge label=t] (f),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right= of a] (c); \vertex [right=4cm of b] (d);
	\vertex [below=1.2cm of a] (e); %\vertex [right= of e] (f);

	\diagram[%inline=(e.base)
	]{
		(a) %-- [fermion,  quarter left, edge label=a] (f)
		-- [ fermion,  half left, edge label=a] (b)
		%-- [ fermion, quarter left, edge label=r] (e)
		-- [ fermion, half left, edge label=r] (a),
		(a) -- [fermion, quarter left, edge label=b] (e)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, quarter left, edge label=s] (a),
		(e) -- [fermion, quarter left, edge label=c] (b)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, quarter left, edge label=t] (e),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}

~\\
~\vspace{10pt}\\
6
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (0.6cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (-0.6cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion,  quarter left, edge label=a] (e)
		-- [ fermion,  edge label=s] (c)
		-- [ fermion, quarter left, edge label=b] (d)
		-- [ fermion, quarter left, edge label=t] (f)
		-- [fermion, edge label=c] (b)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, quarter left, edge label=r] (a),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b)$) (e); %\vertex at ($(c)!0.5!(d) + (-0.6cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [fermion,  quarter left, edge label=a] (e)
		-- [ fermion,  quarter left, edge label=s] (a)
		-- [ fermion, half left, edge label=b] (b)
		-- [ fermion, quarter left, edge label=t] (e)
		-- [fermion, quarter left, edge label=c] (b)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, half left, edge label=r] (a),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
7
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (-0.6cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (0.6cm, 0)$) (f);

	\diagram[]{
		(a) -- [fermion,  quarter right, edge label'=d] (e)
		-- [ fermion,  quarter right,  edge label'=a] (b)
		-- [ fermion,out=20, in=-120,  edge label'=r] (c)
		-- [ fermion, quarter left, edge label=b] (f)
		-- [fermion, quarter left, edge label=c] (d)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion,out=160, in=-60, edge label=s] (a),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e); %\vertex at ($(c)!0.5!(d) + (0.6cm, 0)$) (f);

	\diagram[]{
		(a) -- [fermion,  \qr, \el'=d] (e) -- [ fermion,  \qr,  \el'=a] (b)
		-- [ fermion, \hfr,  \el'=r] (a) -- [ fermion, \ql, \el=b] (e)
		-- [fermion, \ql, \el=c] (b)
		%-- [anti fermion, quarter left, edge label=d] (f)
		-- [fermion, \hfl, \el=s] (a),
		%(a) -- [scalar] (c),
		%(b) -- [scalar] (d),
		%(e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
8
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);
	\diagram[]{
		(a) -- [fermion,  \qr, \el'=a] (b) -- [ fermion, \el=r] (f)
		-- [ fermion,  \el'=s] (c) -- [ fermion, \ql, \el=b] (d)
		-- [fermion,  \el'=t] (e) -- [fermion, \el=u] (a),
		(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e); %\vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);
	\diagram[]{
		(a) -- [fermion,  \hfr, \el'=a] (b) -- [ fermion,\qr, \el'=r] (e)
		-- [ fermion, \qr, \el'=s] (a) -- [ fermion, \hfl, \el=b] (b)
		-- [fermion, \ql, \el=t] (e) -- [fermion, \ql,\el=u] (a),
		%(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
9
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (-0.7cm, 0)$) (e); \vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (f);
	\diagram[]{
		(a) -- [fermion,  \qr, \el'=a] (e) -- [ fermion, \qr, \el'=b] (b)
		-- [ fermion,  \el'=r] (f) -- [ fermion, \el=s] (c)
		-- [fermion, \ql, \el'=c] (d) -- [fermion, \el'=t] (a),
		(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e); %\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (f);
	\diagram[]{
		(a) -- [fermion,  \qr, \el'=a] (e) -- [ fermion, \qr, \el'=b] (b)
		-- [ fermion, \qr, \el'=r] (e) -- [ fermion, \qr, \el'=s] (a)
		-- [fermion, \hfl, \el=c] (b) -- [fermion, \hfl, \el=t] (a),
		%(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
10
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (-0.7cm, 0)$) (e); \vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (f);
	\diagram[]{
		(a) -- [fermion,  \el'=s] (f) -- [ fermion, \el'=r] (d)
		-- [ fermion, \qr,  \el'=c] (c) -- [ fermion, \el'=t] (b)
		-- [fermion, \ql, \el=b] (e) -- [fermion,\ql, \el=a] (a),
		(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e); %\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (f);
	\diagram[]{
		(a) -- [fermion, \ql, \el=s] (e) -- [ fermion,\ql, \el=r] (b)
		-- [ fermion, \hfr,  \el'=c] (a) -- [ fermion, \hfr, \el'=t] (b)
		-- [fermion, \ql, \el=b] (e) -- [fermion,\ql, \el=a] (a),
		%(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\vspace{10pt}\\
11
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=3.5cm of a] (c); \vertex [right= of b] (d);
	\vertex [above=1.2cm of d] (e); \vertex [below=1.2cm of c] (f);
	\diagram[]{
		(a) -- [fermion,  \el'=b] (d) -- [ fermion, \qr, \el'=s] (e)
		-- [ fermion,  \el=a] (b) -- [ fermion, \ql, \el=r] (a),
		(c) -- [fermion, \ql, \el=c] (f) -- [fermion,\ql, \el=t] (c),
		(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=3.5cm of a] (c); \vertex [right= of b] (d);
	\vertex at ($(a)!0.5!(b)$) (e); %\vertex [below=1.2cm of c] (f);
	\diagram[]{
		(a) -- [fermion, \hfl, \el=b] (b) -- [ fermion, \qr, \el'=s] (e)
		-- [ fermion, \qr, \el'=a] (b) -- [ fermion, \hfl, \el=r] (a),
		(a) -- [fermion, \ql, \el=c] (e) -- [fermion,\ql, \el=t] (a),
		%(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\~\vspace{10pt}\\
12
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=of a] (c); \vertex [right=3.5cm of b] (d);
	\vertex [above=1.2cm of d] (f); \vertex [below=1.2cm of c] (e);
	\diagram[]{
		(a) -- [fermion,  \el'=a] (e) -- [ fermion, \qr, \el'=s] (c)
		-- [ fermion,  \el=b] (b) -- [ fermion, \ql, \el=r] (a),
		(f) -- [fermion, \ql, \el=c] (d) -- [fermion,\ql, \el=t] (f),
		(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
$ \lra  $
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	%\vertex [right=of a] (c); \vertex [right=3.5cm of b] (d);
	%\vertex [above=1.2cm of d] (f);
	\vertex [below=1.2cm of a] (e);
	\diagram[]{
		(a) -- [fermion, \qr, \el'=a] (e) -- [ fermion, \qr, \el'=s] (a)
		-- [ fermion, \hfl, \el=b] (b) -- [ fermion, \hfl, \el=r] (a),
		(e) -- [fermion, \ql, \el=c] (b) -- [fermion,\ql, \el=t] (e),
		%(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
~\\
~\\~\\
For the Hugenholtz diagram provided, its value is\\
\begin{minipage}{0.3\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=of a] (b);
	%\vertex [right=of a] (c); %\vertex [right=of b] (d);
	\vertex at ($(a)!0.5!(b) $) (e);
	\diagram[%inline=(e.base)
	]{
		(a) -- [anti fermion,  quarter left, edge label=t] (e)
		-- [anti fermion,  quarter left, edge label=r] (b)
		-- [anti fermion, half left, edge label=a] (a),
		(a) -- [fermion, half left, edge label=b] (b)
		-- [fermion, quarter left, edge label=u] (e)
		-- [fermion, quarter left, edge label=s] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
\begin{minipage}{0.6\linewidth}
	\vskip0.6cm
\begin{align}
&= \qty(\dfrac{1}{2})^3 (-1)^{2+2} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab||ru} \Braket{ru||ts}\Braket{ts||ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} &&& \notag\\
& =\dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab||ru} \Braket{ru||ts}\Braket{ts||ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}  &&&
\notag
\end{align}
\end{minipage}
\begin{align}
&=\dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{(\Braket{ab|ru}-\Braket{ab|ur}) (\Braket{ru|ts}-\Braket{ru|st}) (\Braket{ts|ab}-\Braket{ts|ba})}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}  \notag\\
&= \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
- \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&\quad{} - \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|st} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
+ \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ru|st} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
\notag\\
&\quad{} - \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
+ \dfrac{1}{8} \sum_{a,b,r,s,u,t} + \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&\quad{} + \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|st} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
- \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ru|st} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}  \notag\\
&= \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
- \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&\quad{} - \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ur|st} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_u)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
+ \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ur|st} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_u)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
\notag\\
&\quad{} - \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
+ \dfrac{1}{8} \sum_{a,b,r,s,u,t} + \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&\quad{} + \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ur|st} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_u)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
- \dfrac{1}{8} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ur|st} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_u)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}  \notag\\
&= \dfrac{1}{4} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
- \dfrac{1}{4} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&\quad{} - \dfrac{1}{4} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
+ \dfrac{1}{4} \sum_{a,b,r,s,u,t} + \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ba}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&= \dfrac{1}{4} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
- \dfrac{1}{4} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&\quad{} - \dfrac{1}{4} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ba|ru} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
+ \dfrac{1}{4} \sum_{a,b,r,s,u,t} + \dfrac{\Braket{ba|ur} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)} \notag\\
&= \dfrac{1}{2} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ru} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
- \dfrac{1}{2} \sum_{a,b,r,s,u,t} \dfrac{\Braket{ab|ur} \Braket{ru|ts} \Braket{ts|ab}}{(\varepsilon_a + \varepsilon_b - \varepsilon_u - \varepsilon_r)(\varepsilon_a + \varepsilon_b - \varepsilon_s - \varepsilon_t)}
\end{align}
=
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);

	\diagram[%inline=(e.base)
	]{
		(a) -- [anti fermion,  quarter left, edge label=t] (e)
		-- [anti fermion,  quarter left, edge label=r] (b)
		-- [anti fermion, quarter left, edge label=a] (a),
		(c) -- [fermion, quarter left, edge label=b] (d)
		-- [fermion, quarter left, edge label=u] (f)
		-- [fermion, quarter left, edge label=s] (c),
		(a) -- [scalar] (c),
		(b) -- [scalar] (d),
		(e) -- [scalar] (f),
		%-- [fermion, quarter right, edge label'=2] (a),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}
+
\begin{minipage}{0.4\linewidth}
	\centering
	\begin{tikzpicture}
	\begin{feynman}[large]
	\vertex (a); \vertex [below=2.4cm of a] (b);
	\vertex [right=2.5cm of a] (c); \vertex [right=2.5cm of b] (d);
	\vertex at ($(a)!0.5!(b) + (0.7cm, 0)$) (e); \vertex at ($(c)!0.5!(d) + (-0.7cm, 0)$) (f);
	\diagram[]{
		(a) -- [fermion,  \qr, \el'=a] (b) -- [ fermion, \el=u] (f)
		-- [ fermion,  \el'=s] (c) -- [ fermion, \ql, \el=b] (d)
		-- [fermion,  \el'=r] (e) -- [fermion, \el=t] (a),
		(a) -- [scalar] (c), 	(b) -- [scalar] (d), (e) -- [scalar] (f),
	};
	\end{feynman}
	\end{tikzpicture}
\end{minipage}

\subsubsection{Summation of Diagrams}

\subsubsection{What Is the Linked-Cluster Theorem?}
\ex{6.13}
For the 3rd-order Goldstone diagrams in Table 6.2,
\begin{equation}\label{key}
\text{diagram1} = (-1)^4 \qty(\dfrac{1}{2}) \sum_{ab}\sum_{rsut} \dfrac{\Braket{ab|ru}\Braket{ru|ts}\Braket{ts|ab} }{(\varepsilon_a + \varepsilon_b - \varepsilon_r - \varepsilon_u)(\varepsilon_a + \varepsilon_b - \varepsilon_t - \varepsilon_t)}
\end{equation}
$ a,b,r,s,u,t $ must come from 1 or 2 molecules. If they come from 2 molecules, $ \Braket{ru|ts} $ must be zero. Thus they only come from 1 molecule, i.e. the value of each Goldstone diagram is $ N $ times the result for a single molecule.

\subsection{Some Illustrative Calculations}

\end{document}


\section*{Reduction}

The
\emph{program reducer} $\Rrightarrow$ is a partial function from
programs to
programs
and $\Rrightarrow^*$ is its reflexive transitive closure.
Within the definition of $\Rrightarrow$, we make use of
the following two groups of analogous partial functions:
The statement
\emph{reducer} $\Rightarrow$ is a partial function from
statements to statements
and $\Rightarrow^*$ is its reflexive transitive closure.
The expression
\emph{reducer} $\rightarrow$ is a partial function from
expressions to expressions
and $\rightarrow^*$ is its reflexive transitive closure.

A \emph{program reduction} is a sequence of programs
$p_1 \Rrightarrow \cdots \Rrightarrow p_n$,
where $p_n$ is not reducible, i.e. there is no
program $q$ such that $p_n \Rightarrow q$.
Here, the program $p_n$ is called the \emph{result
of reducing} $p_1$.

A \emph{value} is a primitive number expression,
primitive boolean expression,
a primitive string expression, a function definition
expression or a function declaration statement.

The \emph{substitution} function
$p [ n \leftarrow v ]$ on programs/statements/expressions
replaces every free occurrence of the name $n$
in statement $p$ by value $v$. Care must be taken to introduce
and preserve
co-references in this process; substitution can introduce
cyclic references in the result of the substitution. For example,
$n$ may occur free in $v$, in which case
every occurrence of $n$ in $p$
will be replaced by $v$ such that $n$ in $v$ refers cyclically
to the node at which the replacement happens.

In summary:

\begin{enumerate}
\item $\Rrightarrow$: \emph{program reducer}
\item $\Rrightarrow^*$: transitive closure of \emph{program reducer}
\item $\Rightarrow$: \emph{statement reducer}
\item $\Rightarrow^*$: transitive closure of \emph{statement reducer}
\item $\rightarrow$: \emph{expression reducer}
\item $\rightarrow^*$: transitive closure of \emph{expression reducer}
\item $p [ n \leftarrow v ]$: substitution function
\end{enumerate}

\subsection*{How to read inference rules}

To read the rules below, think of each "fraction" as an
"if we know X then we can infer Y" implication statement.
For example,

\[
\frac{
  X = 5
}{
  X + 1 = 5 + 1
}
\]

corresponds to the statement
"If we know $X$ is equal to $5$,
then we can infer that $X + 1$ is equal to $5 + 1$".

\subsection*{Programs}

\textbf{First-statement}: In a sequence of statements, we can always
reduce the first one.
\[
\frac{
  \textit{statement}\ \Rightarrow\ \textit{statement}'
}{
  \textit{statement} \ldots
  \Rrightarrow
  \textit{statement}' \ldots
}
\]

\vspace{10mm}

\textbf{Eliminate-function-declaration}: Function declarations as first
statements are substituted in the remaining statements.
\[
\frac{
             f = \textbf{\texttt{function}}\  \textit{name} \
                 \textbf{\texttt{(}}\  \textit{parameters}
                 \ \textbf{\texttt{)}}\ \textit{block}
}{
f\ \textit{statement} \ldots\
  \Rrightarrow\
  \textit{statement} \ldots[\textit{name} \leftarrow f]
}
\]

\vspace{10mm}
\textbf{Eliminate-Values}: Values as first statemments are discarded, if
they are preceding one or more statements in a statement sequence.
\[
\frac{
v \ \mbox{is a value}
}{
v \textbf{\texttt{;}} \textit{statement}+\
   \Rrightarrow  \
  \textit{statement}+
}
\]

\subsection*{Statements: Expression statements}

\textbf{Expression-statement-reduce}: An expression statement
is reducible if its expression is reducible.
\[
\frac{
  e\ \rightarrow\ e'
}{
  e \textbf{\texttt{;}}
  \ \Rightarrow \
  e' \textbf{\texttt{;}}
}
\]

\subsection*{Expressions: Binary operators}

\textbf{Left-binary-reduce}: An expression with binary operator
can be reduced if its left sub-expression can be reduced.
\[
\frac{
  e_1 \ \rightarrow \ e_1'
}{
  e_1\  \textit{binary-operator} \ e_2
  \ \rightarrow \
  e_1'\  \textit{binary-operator} \ e_2
}
\]

\vspace{10mm}
\textbf{And-shortcut-false}: An expression with binary operator
$\textbf{\texttt{\&\&}}$ whose left sub-expression is
$\textbf{\texttt{false}}$ can be reduced to
$\textbf{\texttt{false}}$.
\[
\frac{
}{
  \textbf{\texttt{false}}\  \textbf{\texttt{\&\&}}\ e
  \ \rightarrow \
  \textbf{\texttt{false}}
}
\]

\vspace{10mm}
\textbf{And-shortcut-true}: An expression with binary operator
$\textbf{\texttt{\&\&}}$ whose left sub-expression is
$\textbf{\texttt{true}}$ can be reduced to
the right sub-expression.
\[
\frac{
}{
  \textbf{\texttt{true}}\  \textbf{\texttt{\&\&}}\ e
  \ \rightarrow \
  e
}
\]

\vspace{10mm}
\textbf{Or-shortcut-true}: An expression with binary operator
$\textbf{\texttt{||}}$ whose left sub-expression is
$\textbf{\texttt{true}}$ can be reduced to
$\textbf{\texttt{true}}$.
\[
\frac{
}{
  \textbf{\texttt{true}}\  \textbf{\texttt{||}}\ e
  \ \rightarrow \
  \textbf{\texttt{true}}
}
\]

\vspace{10mm}
\textbf{Or-shortcut-false}: An expression with binary operator
$\textbf{\texttt{||}}$ whose left sub-expression is
$\textbf{\texttt{false}}$ can be reduced to
the right sub-expression.
\[
\frac{
}{
  \textbf{\texttt{false}}\  \textbf{\texttt{||}}\ e
  \ \rightarrow \
  e
}
\]

\vspace{10mm}
\textbf{Right-binary-reduce}: An expression with binary operator
can be reduced if its left sub-expression is a value and its right
sub-expression can be reduced.
\[
\frac{
  e_2\ \rightarrow\ e_2', \textrm{and}\ \textit{binary-operator}
  \ \mbox{is not}\ \textbf{\texttt{\&\&}}\ \textrm{or}\ \texttt{\textbf{||}}
}{
  v\  \textit{binary-operator} \ e_2
  \ \rightarrow \
  v\  \textit{binary-operator} \ e_2'
}
\]

\vspace{10mm}
\textbf{Prim-binary-reduce}: An expression with binary operator
can be reduced if its left and right sub-expressions are values and
the corresponding function is defined for those values.
\[
\frac{
  v\ \mbox{is result of}\ v_1\  \textit{binary-operator} \ v_2
}{
  v_1\  \textit{binary-operator} \ v_2
  \ \rightarrow \
  v
}
\]

\subsection*{Expressions: Unary operators}

\textbf{Unary-reduce}: An expression with unary operator
can be reduced if its sub-expression can be reduced.
\[
\frac{
  e \ \rightarrow \ e'
}{
  \textit{unary-operator} \ e
  \ \rightarrow \
  \textit{unary-operator} \ e'
}
\]

\vspace{10mm}
\textbf{Prim-unary-reduce}: An expression with unary operator
can be reduced if its sub-expression is a value and
the corresponding function is defined for that value.
\[
\frac{
  v'\ \mbox{is result of}\ \textit{unary-operator} \ v
}{
  \textit{unary-operator} \ v
  \ \rightarrow \
  v'
}
\]

\subsection*{Expressions: conditionals}

\textbf{Conditional-predicate-reduce}: A conditional
expression can be reduced, if its predicate can be reduced.
\[
\frac{
  e_1 \ \rightarrow \ e_1'
}{
  e_1\  \textbf{\texttt{?}}\ e_2\ \textbf{\texttt{:}}\ e_3
  \ \rightarrow \
  e_1'\ \textbf{\texttt{?}}\ e_2\ \textbf{\texttt{:}}\ e_3
}
\]

\vspace{10mm}
\textbf{Conditional-true-reduce}: A conditional
expression whose predicate is the value
$\textbf{\texttt{true}}$
can be reduced to its consequent expression.
\[
\frac{
}{
  \textbf{\texttt{true}}\  \textbf{\texttt{?}}\ e_1\ \textbf{\texttt{:}}\ e_2
  \ \rightarrow \
  e_1
}
\]

\vspace{10mm}
\textbf{Conditional-false-reduce}: A conditional
expression whose predicate is the value
$\textbf{\texttt{false}}$
can be reduced to its alternative expression.
\[
\frac{
}{
  \textbf{\texttt{false}}\  \textbf{\texttt{?}}\ e_1\ \textbf{\texttt{:}}\ e_2
  \ \rightarrow \
  e_2
}
\]

\subsection*{Expressions: function application}

\textbf{Application-functor-reduce}: A function application
can be reduced if its functor expression can be reduced.
\[
\frac{
  e \ \rightarrow \ e'
}{
  e\  \textbf{\texttt{(}}\ \textit{expressions} \ \textbf{\texttt{)}}
  \ \rightarrow \
  e'\  \textbf{\texttt{(}}\ \textit{expressions} \ \textbf{\texttt{)}}
}
\]

\vspace{10mm}
\textbf{Application-argument-reduce}: A function application
can be reduced if one of its argument expressions can be reduced and all
preceding arguments are values.
\[
\frac{
  e \ \rightarrow \ e'
}{
  v\  \textbf{\texttt{(}}\ v_1 \ldots v_i \ e \ldots\ \textbf{\texttt{)}}
  \ \rightarrow \
  v\  \textbf{\texttt{(}}\ v_1 \ldots v_i \ e' \ldots\ \textbf{\texttt{)}}
}
\]

\vspace{10mm}
\textbf{Function-declaration-application-reduce}:
The application of a function declaration
can be reduced, if all
arguments are values.
\[
\frac{
  f = \textbf{\texttt{function}}\  \textit{n} \
                 \textbf{\texttt{(}}\  x_1 \ldots x_n
                 \ \textbf{\texttt{)\ \{\ return}}\ \textit{expression}
                 \ \textbf{\texttt{;\ \}}}
}{
  f\ \textbf{\texttt{(}}\ v_1 \ldots v_n\ \textbf{\texttt{)}}
  \ \rightarrow \
  \textit{expression} [x_1 \leftarrow v_1]\ldots[x_n \leftarrow v_n]
  [n \leftarrow f]
}
\]

    \end{document}