Latex Version of Equations
eq 1.01
$${\cal H} = {\cal H}_{\rm ev} + {\cal H}_{\rm r} ~ , $$
eq 1.02
$ | {\rm ev}; {\rm r} \rangle = | {\rm ev} \rangle | {\rm r} \rangle ~ ,$
eq 1.03
$ (L_z + S_z) \, |\Omega\rangle = \Omega \, \hbar\, |\Omega\rangle ~ .$
eq 1.04
$ L_z \, |\Omega\rangle \not= ({\rm constant}) \cdot |\Omega\rangle ~ ,$
eq 1.05
$\Omega = \Lambda + \Sigma ~ . $
eq 1.06a
$\langle \Omega | {\cal H}_{\rm ev} |\Omega^\prime\rangle = 0, ~ {\rm if}
~ |\Omega\rangle ~ {\rm and} ~ |\Omega^\prime\rangle ~
{\mbox{\rm are \,different \,functions}} ~ ,$
eq 1.06b
$\langle \Omega | {\cal H}_{\rm ev} |\Omega\rangle = ~
{\mbox{\rm the \,energy \,of \,the \,nonrotating-molecule \,state}} ~
|\Omega\rangle ~ . $
eq 1.07a
$\langle \Lambda S\Sigma| {\cal H}_{\rm ev}^0 +
A{\mbox{\bf L}}\cdot{\mbox{\bf S}}
|\Lambda^\prime S^\prime\Sigma^\prime\rangle = 0, ~ {\rm if}
~ |\Lambda S\Sigma\rangle ~ {\rm and}
~ |\Lambda^\prime S^\prime\Sigma^\prime\rangle
~ {\mbox{\rm are \,different \,functions}}$
eq 1.07b
$\langle \Lambda S\Sigma| {\cal H}_{\rm ev}^0 +
A{\mbox{\bf L}}\cdot{\mbox{\bf S}}
|\Lambda S \Sigma\rangle =
~ {\mbox{\rm the \,energy \,of \,the \,nonrotating-molecule \,state}} ~
|\Lambda S\Sigma\rangle$
eq 1.07c
$\cong {\mbox{\rm constant}} + A\Lambda\Sigma ~ . $
eq 1.08a
$L_z |\Lambda S\Sigma\rangle = \Lambda \hbar
|\Lambda S\Sigma\rangle + |\delta_1\rangle ~ , $
eq 1.08b
$S_z |\Lambda S\Sigma\rangle = \Sigma \hbar
|\Lambda S\Sigma\rangle + |\delta_2\rangle ~ , $
eq 1.09a
$$\langle L\Lambda S\Sigma| {\cal H}_{\rm ev}
|L^\prime \Lambda^\prime S^\prime\Sigma^\prime\rangle = 0, ~ {\rm if}
~ |L \Lambda S\Sigma\rangle ~ {\rm and}
~ |L^\prime \Lambda^\prime S^\prime\Sigma^\prime\rangle
~ {\mbox{\rm are \,different \,functions}} ~ ,$$
eq 1.09b
$$\langle L\Lambda S\Sigma| {\cal H}_{\rm ev}
|L \Lambda S\Sigma\rangle =
~ {\mbox{\rm the \,energy \,of \,the \,nonrotating-molecule \,state}} ~
|L \Lambda S\Sigma\rangle ~ , $$
eq 1.10
\begin{eqnarray*}{\cal H}_{\rm r} &=& B[R_x^2 + R_y^2] \\
&=& B[(J_x - L_x - S_x)^2 + (J_y - L_y - S_y)^2] ~ ,\end{eqnarray*}
eq 1.11
\begin{eqnarray*}{\cal H}_{\rm r}&=&B(J^2-J_z^2) + B(L^2-L_z^2) + B(S^2-S_z^2)\\
&~&+B(L_+S_{-}+L_{-}S_+) - B(J_+L_{-}+J_{-}L_+) - B(J_+S_{-}+J_{-}S_+) ~ ,\end{eqnarray*}
eq 1.12
\begin{eqnarray*}
|\Omega; \, \Omega J M \rangle&=&|\Omega\rangle \, |\Omega J M \rangle\\
|\Lambda S \Sigma; \, \Omega J M \rangle&=&|\Lambda S\Sigma\rangle \, |\Omega J M \rangle\\
|L\Lambda S \Sigma; \, \Omega J M \rangle&=&|L\Lambda S\Sigma\rangle \, |\Omega J M \rangle
~ .\end{eqnarray*}
eq 1.13
\begin{eqnarray*}
\langle S\Sigma| S^2 |S\Sigma\rangle&=&\hbar^2 S(S+1)\\
\langle S\Sigma| S_z |S\Sigma\rangle&=&\hbar \, \Sigma\\
\langle S\Sigma \pm 1| S_\pm |S\Sigma\rangle&=&\hbar [(S\mp\Sigma)
(S\pm\Sigma+1)]^{1/2} ~ .\end{eqnarray*}
eq 1.14
$$\langle J \, \Omega\pm 1| J_\mp |J \, \Omega\rangle =
\hbar \, [(J \mp \Omega) (J \pm \Omega +1)]^{1/2} ~ . $$
eq 1.15
$$\langle S\Sigma +1| S_+ |S\Sigma\rangle = (1 -\gamma) \hbar
[(S-\Sigma) (S+\Sigma+1)]^{1/2} ~ . $$
eq 1.16
$$\langle\Lambda^\prime S^\prime\Sigma^\prime ;
\Omega^\prime J^\prime M^\prime|{\cal H}|\Lambda S\Sigma ;\Omega J M\rangle =
\langle\Lambda^\prime S^\prime\Sigma^\prime ;
\Omega^\prime J^\prime M^\prime|{\cal H}_{\rm ev}|\Lambda S\Sigma ;
\Omega J M\rangle + \langle\Lambda^\prime S^\prime\Sigma^\prime ;
\Omega^\prime J^\prime M^\prime|{\cal H}_{\rm r}|\Lambda S\Sigma ;
\Omega J M\rangle ~ . $$
eq 1.17
$$\left[ \begin{array}{cccc}
E+\frac{1}{2} A & 0 & 0 & 0 \\
0 & E-\frac{1}{2} A & 0 & 0 \\
0 & 0 & E+\frac{1}{2} A & 0 \\
0 & 0 & 0 & E-\frac{1}{2} A \end{array} \right] \quad . $$
eq 1.18
$$\left[ \begin{array}{cccc}
B[J(J+1) - \frac{7}{4} + \langle L_\perp^2\rangle ], &
-B[(J-\frac{1}{2}) (J+\frac{3}{2})]^{1/2}, & 0 & 0 \\~\\
-B[(J-\frac{1}{2}) (J+\frac{3}{2})]^{1/2}, &
B[J(J+1) + \frac{1}{4} + \langle L_\perp^2\rangle ], & 0 & 0 \\~\\
0 & 0 & B[J(J+1) - \frac{7}{4} + \langle L_\perp^2\rangle ], &
-B[(J-\frac{1}{2}) (J+\frac{3}{2})]^{1/2} \\ ~\\
0 & 0 & -B[(J-\frac{1}{2}) (J+\frac{3}{2})]^{1/2}, &
B[J(J+1) + \frac{1}{4} + \langle L_\perp^2\rangle ]~ .
\end{array} \right] $$
eq 1.19
\begin{eqnarray*}
E + B\langle L_\perp^2\rangle + B[(J + \frac{1}{2})^2 - 1] &+& \frac{1}{2} [A(A-4B) + 4B^2(J+\frac{1}{2})^2]^{1/2}\\
E + B\langle L_\perp^2\rangle + B[(J + \frac{1}{2})^2 - 1] &-& \frac{1}{2} [A(A-4B) + 4B^2(J+\frac{1}{2})^2]^{1/2}\\
\end{eqnarray*}
eq 1.20
\begin{eqnarray*}
\left[A(A-4B)+4B^2(J+{\textstyle\frac{1}{2}})^2\right]^{1/2}
&=&\left\{A^2\left[1-4B/A+(4B^2/A^2) (J+{\textstyle\frac{1}{2}})^2\right]\right\}^{1/2} \\
&=&|A|\left\{1+{\textstyle\frac{1}{2}} (-4B/A)+{\textstyle\frac{1}{2}}
(4B^2/A^2) (J+{\textstyle\frac{1}{2}})^2 - {\textstyle\frac{1}{8}}
(-4B/A)^2 + O(B/A)^3 \right\}\\
&\cong&|A|\left\{1-2B/A+{\textstyle\frac{1}{2}}(4B^2/A^2)
\left[(J+{\textstyle\frac{1}{2}})^2 -1\right]\right\} ~ .
\end{eqnarray*}
eq 1.21a
$$ +{\textstyle\frac{1}{2}} A+B\left[J(J+1)- {\textstyle\frac{7}{4}}\right]$$
eq 1.21b
$$ -{\textstyle\frac{1}{2}} A+B\left[J(J+1)+ {\textstyle\frac{1}{4}}\right] ~ ,$$
eq 1.22a
$$ B\left[(J+{\textstyle\frac{1}{2}}) (J+{\textstyle\frac{3}{2}})-1\right] $$
eq 1.22b
$B\left[(J-{\textstyle\frac{1}{2}}) (J+{\textstyle\frac{1}{2}})-1\right] ~ .$
eq 1.23
\begin{eqnarray*}
|+1 \, {\textstyle\frac{1}{2}} \, N=J+{\textstyle\frac{1}{2}} \, JM\rangle =
&+&\left[(J-{\textstyle\frac{1}{2}})/(2J+1)\right]^{1/2} \,
|+1 \, {\textstyle\frac{1}{2}}+{\textstyle\frac{1}{2}}; +{\textstyle\frac{3}{2}} \, J M \rangle)\\
&-&\left[(J+{\textstyle\frac{3}{2}})/(2J+1)\right]^{1/2} \,
|+1 \, {\textstyle\frac{1}{2}}-{\textstyle\frac{1}{2}}; +{\textstyle\frac{1}{2}} \, J M \rangle)\\
|+1 \, {\textstyle\frac{1}{2}} \, N=J+{\textstyle\frac{1}{2}} \, JM\rangle =
&+&\left[(J+{\textstyle\frac{3}{2}})/(2J+1)\right]^{1/2} \,
|+1 \, {\textstyle\frac{1}{2}}+{\textstyle\frac{1}{2}}; +{\textstyle\frac{3}{2}} \, J M \rangle)\\
&+&\left[(J-{\textstyle\frac{1}{2}})/(2J+1)\right]^{1/2} \,
|+1 \, {\textstyle\frac{1}{2}}-{\textstyle\frac{1}{2}}; +{\textstyle\frac{1}{2}} \, J M \rangle)
\end{eqnarray*}
eq 1.24
$$\left[ \begin{array}{ccc}
E & 0 & 0\\ 0 & E - 2\lambda & 0\\
0 & 0 & E \end{array} \right] $$
eq 1.25
$$\left[ \begin{array}{ccc}
BJ(J+1)+B\langle L_\perp^2\rangle & -B[2J(J+1)]^{1/2} & 0\\
-B[2J(J+1)]^{1/2} & BJ(J+1)+2B+B\langle L_\perp^2\rangle & -B[2J(J+1)]^{1/2} \\
0 & -B[2J(J+1)]^{1/2} & BJ(J+1)+B\langle L_\perp^2\rangle \end{array} \right]
~ , $$
eq 1.26
$$\langle 2^{-1/2}(a+b)| {\cal H} |2^{-1/2}(c+d)\rangle =
{\textstyle\frac{1}{2}} [\langle a| {\cal H} |c\rangle +
\langle a| {\cal H} |d\rangle + \langle b| {\cal H} |c\rangle +
\langle b| {\cal H} |d\rangle ] ~ . $$
eq 1.27
$$\left[ \begin{array}{ccc}
BJ(J+1)+B\langle L_\perp^2\rangle & -2B[J(J+1)]^{1/2} & 0 \\
-2B[J(J+1)]^{1/2} & BJ(J+1)+2B+B\langle L_\perp^2\rangle & 0 \\
0 & 0 & BJ(J+1)+B\langle L_\perp^2\rangle \end{array} \right]
~ . $$
eq 1.28
\begin{eqnarray*} E+B\langle L_\perp^2\rangle &+& BJ(J+1)\\
E+B\langle L_\perp^2\rangle &+& BJ(J+1) + (B-\lambda) \pm [(B-\lambda)^2 +
(B-{\textstyle\frac{1}{2}} \gamma)^2 4J(J+1) ]^{1/2} ~ . \end{eqnarray*}
eq 1.29
\begin{eqnarray*}
[(B-\lambda)^2&+&(B-{\textstyle\frac{1}{2}} \gamma)^2 4J(J+1) ]^{1/2} \\
&\cong&[\lambda^2 - 2B\lambda + B^2(2J+1)^2 - 4\gamma BJ(J+1) ]^{1/2}\\
&\cong& [\lambda^2 - 2B\lambda + B^2(2J+1)^2]^{1/2} \,
\{1-2\gamma BJ(J+1)/[\lambda^2 - 2B\lambda + B^2(2J+1)^2]\} \\
&\cong& [\lambda^2 - 2B\lambda + B^2(2J+1)^2]^{1/2} -
\gamma(J+{\textstyle\frac{1}{2}}) ~ . \end{eqnarray*}
eq 1.30
\begin{eqnarray*}
BN(N&+&1) \\
BN(N&+&1)-(2N-1) B-\lambda+[\lambda^2-2B\lambda +B^2(2N-1)^2]^{1/2}-\gamma
N+{\textstyle\frac{1}{2}} \gamma\\
BN(N&+&1) + (2N+3)B-\lambda- [\lambda^2-2B\lambda+B^2(2N+3)^2]^{1/2}+\gamma
(N+ 1) + {\textstyle\frac{1}{2}} \gamma
~ , \end{eqnarray*}
Chapter 2
eq2.01
\begin{eqnarray*}
x_{\rm e} &=& \rho_{\rm e} \sin \theta_{\rm e} \cos\varphi_{\rm e} \\
y_{\rm e} &=& \rho_{\rm e} \sin \theta_{\rm e} \sin\varphi_{\rm e} \\
z_{\rm e} &=& \rho_{\rm e} \cos \theta_{\rm e} \end{eqnarray*}
eq2.02
\begin{eqnarray*}
C_\infty^\epsilon(z)\cdot i\cdot C_2(y)\cdot f(\ldots,\varphi_{\rm e},\ldots)
&=& C_\infty^\epsilon(z)\cdot i\cdot f(\ldots,\pi-\varphi_{\rm e},\ldots) \\
&=& C_\infty^\epsilon(z)\cdot f(\ldots,\pi-(\pi+\varphi_{\rm e}),\ldots) \\
&=& f(\ldots,-\varphi_{\rm e}-\epsilon,\ldots) ~ .\end{eqnarray*}
eq2.03
\begin{eqnarray*}
\left[ \begin{array}{c} X_{\rm e}\\ Y_{\rm e}\\ Z_{\rm e} \end{array} \right]
&=& \left[ \begin{array}{ccc} -\sin\varphi & -\cos\theta\cos\varphi & \sin\theta\cos\varphi \\
\cos\varphi & -\cos\theta\sin\varphi & \sin\theta\cos\varphi \\
0 & \sin\theta\ & \cos\theta \end{array} \right]
\left[ \begin{array}{c} x_{\rm e}\\ y_{\rm e}\\ z_{\rm e} \end{array} \right]\\
\left[ \begin{array}{c} X_1\\ Y_1\\ Z_1 \end{array} \right]
&=& \left[ \begin{array}{ccc} -\sin\varphi & -\cos\theta\cos\varphi & \sin\theta\cos\varphi \\
\cos\varphi & -\cos\theta\sin\varphi & \sin\theta\cos\varphi \\
0 & \sin\theta\ & \cos\theta \end{array} \right]
\left[ \begin{array}{r} d_{1x}\\ d_{1y}\\ -(\mu/m_1)r_{\rm e} + d_{1z} \end{array} \right]\\
\left[ \begin{array}{c} X_2\\ Y_2\\ Z_2 \end{array} \right]
&=& \left[ \begin{array}{ccc} -\sin\varphi & -\cos\theta\cos\varphi & \sin\theta\cos\varphi \\
\cos\varphi & -\cos\theta\sin\varphi & \sin\theta\cos\varphi \\
0 & \sin\theta\ & \cos\theta \end{array} \right]
\left[ \begin{array}{r} d_{2x}\\ d_{2y}\\ +(\mu/m_2)r_{\rm e} + d_{2z} \end{array} \right]\\
\end{eqnarray*}
eq2.04
\begin{eqnarray*}
|p\Sigma\rangle &=& +f(\rho_{\rm e})\cos\theta_{\rm e}\\
|p\Pi_\pm\rangle &=& \mp f(\rho_{\rm e})~2^{-1/2}\sin \theta_{\rm e})
~{\rm e}^{\pm i\varphi_{\rm e}} ~ ,\end{eqnarray*}
eq2.05
\begin{eqnarray*}
\sigma_v|p\Sigma\rangle &=& +|p\Sigma\rangle\\
\sigma_v|p\Pi_\pm\rangle &=& -|p\Pi_\mp\rangle ~ ,\end{eqnarray*}
eq2.06
$\sigma_v|L\Lambda\rangle = -(-1)^{L-\Lambda} ~ |L-\Lambda\rangle ~ . $
eq2.07
$\sigma_v|\Sigma^\pm\rangle = \pm |\Sigma^\pm\rangle ~ .$
eq2.08
\begin{eqnarray*}|np ~ n^\prime p P \Sigma\rangle &=&
+2^{-3/2} f_1(\rho_{\rm e1}) \, \sin \theta_{\rm e1} ~
{\rm e}^{-i\varphi_{\rm e1}} f_2(\rho_{\rm e2}) \,\sin \theta_{\rm e2} ~
{\rm e}^{+i\varphi_{\rm e2}}\\
&~& -2^{-3/2} f_1(\rho_{\rm e1}) \, \sin \theta_{\rm e1} ~
{\rm e}^{+i\varphi_{\rm e1}} f_2(\rho_{\rm e2}) \,\sin \theta_{\rm e2} ~
{\rm e}^{-i\varphi_{\rm e2}}\\
|np ~ n^\prime p P \Pi_+\rangle &=&
+\frac{1}{2} f_1(\rho_{\rm e1}) \, \cos \theta_{\rm e1} ~
f_2(\rho_{\rm e2}) \, \sin \theta_{\rm e2} ~ {\rm e}^{+i\varphi_{\rm e2}}\\
&~& -\frac{1}{2} f_1(\rho_{\rm e1}) \, \sin \theta_{\rm e1} ~
{\rm e}^{+i\varphi_{\rm e1}}
f_2(\rho_{\rm e2}) \, \cos \theta_{\rm e2} \\
|np ~ n^\prime p P \Pi_-\rangle &=&
-\frac{1}{2} f_1(\rho_{\rm e1}) \, \sin \theta_{\rm e1} ~
{\rm e}^{-i\varphi_{\rm e1}}
f_2(\rho_{\rm e2}) \, \cos \theta_{\rm e2} \\
&~& +\frac{1}{2} f_1(\rho_{\rm e1}) \, \cos \theta_{\rm e1} ~
f_2(\rho_{\rm e2}) \, \sin \theta_{\rm e2} ~ {\rm e}^{-i\varphi_{\rm e2}} ~ ,
\end{eqnarray*}
eq2.09
\begin{eqnarray*}
\sigma_v|np ~ n^\prime p P \Sigma\rangle &=& -|np ~ n^\prime p P \Sigma\rangle\\
\sigma_v|np ~ n^\prime p P \Pi_\pm \rangle&=&+|np ~ n^\prime p P \Pi_\mp \rangle ,
\end{eqnarray*}
eq2.10
$\sigma_v|L\Lambda\rangle = +(-1)^{L-\Lambda} ~ |L-\Lambda\rangle\quad .$
eq2.11a
$\sigma_v|L\Lambda\rangle = \pm(-1)^{L-\Lambda} ~ |L-\Lambda\rangle $
eq2.11b
$\sigma_v|S\Sigma\rangle = (-1)^{S-\Sigma} ~ |S-\Sigma\rangle$
eq2.11c
$\sigma_v|J\Omega\rangle = ~ (-1)^{J-\Omega} ~ |J-\Omega\rangle\quad ,$
eq2.12
$\sigma_v|L\Lambda S\Sigma; \Omega J M\rangle = \pm
(-1)^{L-\Lambda+S-\Sigma+J-\Omega} ~ |L-\Lambda S-\Sigma; -\Omega J M\rangle\quad .$
eq2.13
$\sigma_v|0^- \, 0 \, 0; \, 0 J M\rangle = - (-1)^J ~
|0^- \, 0 \, 0; \, 0 J M\rangle\quad .$
eq2.14
\begin{eqnarray*}
\sigma_v \{2^{-1/2} [ | 0^+ \, 1 \, 1; 1 JM\rangle &+& | 0^+ \, 1-1; -1 JM \rangle]\}\\
&=& (-1)^{J-1} \{2^{-1/2} [ | 0^+ \, 1-1; -1 JM\rangle + | 0^+ \, 1 \, 1; 1 JM\rangle]\}\\
&=& (-1)^{J-1} \{2^{-1/2} [ | 0^+ \, 1 \, 1; 1 JM\rangle + | 0^+ \, 1-1; -1 JM\rangle]\}\\
\sigma_v [ | 0^+ \, 1 \, 0; 0 JM\rangle &=& (-1)^{J+1}| 0^+ \, 1 \, 0; 0 JM\rangle \\
\sigma_v \{2^{-1/2} [ | 0^+ \, 1 \, 1; 1 JM\rangle &-& | 0^+ \, 1-1; -1 JM \rangle]\}\\
&=& (-1)^{J-1} \{2^{-1/2} [ | 0^+ \, 1-1; -1 JM\rangle - | 0^+ \, 1 \, 1; 1 JM\rangle]\}\\
&=& (-1)^J \{2^{-1/2} [ | 0^+ \, 1 \, 1; 1 JM\rangle - | 0^+ \, 1-1; -1 JM\rangle]\}\\
\end{eqnarray*}
eq2.15
\begin{eqnarray*}
i[|\Sigma_g\rangle, |\Pi_g\rangle, |\Delta_g\rangle, \, \ldots ] &=&
+ [|\Sigma_g\rangle, |\Pi_g\rangle, |\Delta_g\rangle, \, \ldots ]\\
i[|\Sigma_u\rangle, |\Pi_u\rangle, |\Delta_u\rangle, \, \ldots ] &=&
- [|\Sigma_u\rangle, |\Pi_u\rangle, |\Delta_u\rangle, \, \ldots ] \quad .
\end{eqnarray*}
eq2.16a
$C_2 |L\Lambda\rangle = (-1)^{L-\Lambda} |L-\Lambda\rangle$
eq2.16b
$C_2 |S\Sigma\rangle = (-1)^{S-\Sigma} |S-\Sigma\rangle$
eq2.16c
$C_2 |J\Omega\rangle = (-1)^{J-\Omega} |J-\Omega\rangle$
eq2.17
\begin{eqnarray*}
C_2 |\Sigma_g^+\rangle &=& +|\Sigma_g^+\rangle \qquad
C_2 |\Sigma_g^-\rangle = -|\Sigma_g^-\rangle \\
C_2 |\Sigma_u^-\rangle &=& +|\Sigma_u^-\rangle \qquad
C_2 |\Sigma_u^+\rangle = -|\Sigma_u^+\rangle \quad .
\end{eqnarray*}
eq2.18
$2^{-1/2} [|1_u \, 0 \, 0; 1 JM\rangle \pm |- 1_u \, 0 \, 0; -1 JM\rangle] \quad . $
eq2.19
\begin{eqnarray*} \sigma_v \{2^{-1/2} [| 1_u\, 0 \, 0; 1 JM\rangle
&\pm& |-1_u \, 0 \, 0; -1 JM\rangle]\} \\
&=& \mp (-1)^J \, \{2^{-1/2} [| 1_u\, 0 \, 0; 1 JM\rangle
\pm |-1_u \, 0 \, 0; -1 JM\rangle]\} \quad . \end{eqnarray*}
eq2.20
\begin{eqnarray*} C_2 \{2^{-1/2} [| 1_u\, 0 \, 0; 1 JM\rangle
&\pm& |-1_u \, 0 \, 0; -1 JM\rangle]\} \\
&=& \pm (-1)^J \, \{2^{-1/2} [| 1_u\, 0 \, 0; 1 JM\rangle
\pm |-1_u \, 0 \, 0; -1 JM\rangle]\} \quad . \end{eqnarray*}
eq2.21
\begin{eqnarray*}
\sigma\Psi_1 &=& (-1)^{n1}\,\Psi_1\\
\sigma\Psi_2 &=& (-1)^{n2}\,\Psi_2\\
\sigma L &=& (-1)^{n3}\,L \quad . \end{eqnarray*}
eq2.22
$\int \Psi_1^* L \Psi_2 d\tau =
\int [\sigma(\Psi_1^* L \Psi_2)]d\tau = (-1)^{n_1+n_2+n_3}
\int \Psi_1^* L \Psi_2 d\tau \quad .$
eq2.23
$$
\langle L^\prime\Lambda^\prime S^\prime\Sigma^\prime| {\cal H}_{{\rm e}v}
|L\Lambda S\Sigma\rangle =
[\pm (-1)^{L^\prime-\Lambda^\prime+S^\prime-\Sigma^\prime}]
[\pm (-1)^{L-\Lambda+S-\Sigma}]
\langle L^\prime-\Lambda^\prime -S^\prime-\Sigma^\prime| {\cal H}_{{\rm e}v}
|L-\Lambda \, S-\Sigma\rangle \quad . $$
eq2.24
\begin{eqnarray*}
\sigma_v S_\pm &=& - S_\mp\\
\sigma_v S_z &=& - S_z\\
\sigma_v S^2 &=& + S^2 \quad . \end{eqnarray*}
eq2.25
\begin{eqnarray*}\langle 1 \, \frac{1}{2} \, \frac{1}{2}; \frac{3}{2} J M |
L^2 - L_z^2 | 1 \, \frac{1}{2} \, \frac{1}{2}; \frac{3}{2} J M \rangle &=&
+ \langle - 1 \, \frac{1}{2} - \frac{1}{2}; -\frac{3}{2} J M |
L^2 - L_z^2 | - 1 \, \frac{1}{2} \, - \frac{1}{2}; - \frac{3}{2} J M \rangle\\
\langle 1 \, \frac{1}{2} \, - \frac{1}{2}; \frac{1}{2} J M |
L^2 - L_z^2 | 1 \, \frac{1}{2} \, - \frac{1}{2}; \frac{1}{2} J M \rangle &=&
+ \langle - 1 \, \frac{1}{2} \frac{1}{2}; -\frac{1}{2} J M |
L^2 - L_z^2 | - 1 \, \frac{1}{2} \, \frac{1}{2}; - \frac{1}{2} J M \rangle
\quad . \end{eqnarray*}
eq2.26
$\langle 1 \, \frac{1}{2} \, \frac{1}{2}; \frac{3}{2} J M |
L^2 - L_z^2 | 1 \, \frac{1}{2} \, \frac{1}{2}; \frac{3}{2} J M \rangle =
+ \langle 1 \, \frac{1}{2} \, - \frac{1}{2}; \frac{1}{2} J M \rangle
L^2 - L_z^2 | 1 \, \frac{1}{2} \, - \frac{1}{2}; \frac{1}{2} J M \rangle
\quad . $
eq2.27
$\theta k=k^* \quad , $
eq2.28
\begin{eqnarray*}
\theta\bf L &=& - \bf L\theta \\
\theta\bf S &=& - \bf S\theta \\
\theta\bf J &=& - \bf J\theta \quad .\end{eqnarray*}
eq2.29
$\theta L_z|L\Lambda\rangle = \theta\Lambda\hbar|L\Lambda\rangle =
\Lambda\hbar[\theta|L\Lambda\rangle] = -L_z[\theta|L\Lambda\rangle]$
eq2.30
\begin{eqnarray*}
\theta| L\Lambda\rangle &\propto& |L-\Lambda\rangle \\
\theta| S\Sigma\rangle &\propto& |S-\Sigma\rangle \\
\theta| \Omega J M\rangle &\propto& |-\Omega J-M\rangle \quad .\end{eqnarray*}
eq2.31
\begin{eqnarray*}
\theta| L 0\rangle &=& +| L 0\rangle \\
\theta| S 0\rangle &=& +| S 0\rangle \\
\theta| 0 J 0\rangle &=& +| 0 J 0\rangle \quad .\end{eqnarray*}
eq2.32
\begin{eqnarray*} \theta| S\frac{1}{2}\rangle
&=& + | S-\frac{1}{2} \rangle\\
\theta|\frac{1}{2} J \frac{1}{2}\rangle
&=& + |-\frac{1}{2} J-\frac{1}{2}\rangle \quad .\end{eqnarray*}
eq2.33
\begin{eqnarray*}
\mid S ~ \frac{1}{2}+m\rangle &=& k_1 (S_+)^m | S\frac{1}{2} \rangle\\
\mid S ~ \frac{1}{2}-n\rangle &=& k_2 (S_-)^n | S\frac{1}{2} \rangle
\quad .\end{eqnarray*}
eq2.34
\begin{eqnarray*}
\theta|S ~ 1/2+m\rangle &=& \theta k_1(S_+)^m |S ~ 1/2\rangle = (-1)^m k_1(S_-)^m |S~ -1/2\rangle \\
&=& (-1)^m |S ~ -1/2-m\rangle \\
\theta|S ~ 1/2-n\rangle &=& \theta k_2(S_-)^n |S ~ 1/2\rangle = (-1)^n k_2(S_+)^n |S~ -1/2\rangle \\
&=& (-1)^n |S ~ -1/2+n\rangle\quad .\end{eqnarray*}
eq2.35
$\theta|S\Sigma\rangle = (-1)^{\Sigma-1/2} |S-\Sigma\rangle $
eq2.36
\begin{eqnarray*}\theta|1/2+m~J~1/2+n\rangle
&=& \theta k_3(J_x-iJ_y)^m (J_X+iJ_Y)^n |1/2~J~1/2\rangle\\
&=& (-1)^{m+n} k_3(J_x+iJ_y)^m (J_X-iJ_Y)^n |-1/2~J~-1/2\rangle\\
&=& (-1)^{m+n} |-1/2-m ~ J~-1/2-n\rangle \quad ,\end{eqnarray*}
eq2.37
$\theta|\Omega J M\rangle = (-1)^{\Omega+M-1} |-\Omega J-M\rangle $
eq2.38
\begin{eqnarray*}
\theta| L\Lambda\rangle &=& (-1)^\Lambda |L-\Lambda\rangle\\
\theta|S\Sigma\rangle &=& (-1)^\Sigma |S-\Sigma\rangle\\
\theta|\Omega J M\rangle &=&(-1)^{\Omega+M} |-\Omega J-M\rangle
\quad .\end{eqnarray*}
Chapter 3
eq3.01
$\mu_R = \sum_s \alpha_{Rs} \mu_s \quad , $
eq3.02
\begin{eqnarray*}
\sigma_v \mu_R &=&-\mu_R \\
i \mu_R &=&-\mu_R \\
C_2 \mu_R &=&+\mu_R \qquad R = X, ~Y,~Z \quad .
\end{eqnarray*}
eq3.03
\begin{eqnarray*}
\langle\Omega + 1&|& \mu_x + i\mu_y |\Omega\rangle\\
\langle\Omega - 1&|& \mu_x - i\mu_y |\Omega\rangle\\
\langle\Omega &|& \mu_z |\Omega\rangle \quad ,
\end{eqnarray*}
eq3.04
\begin{eqnarray*}
\sigma_v(\mu_x\pm i\mu_y) &=& + (\mu_x\mp i\mu_y)\\
\sigma_v(\mu_z) &=& + (\mu_z)\\
i(\mu_x\pm i\mu_y) &=& - (\mu_x\pm i\mu_y)\\
i(\mu_z) &=& - (\mu_z)\\
C_2(\mu_x\pm i\mu_y) &=& - (\mu_x\mp i\mu_y)\\
C_2(\mu_z) &=& - (\mu_z) \quad .
\end{eqnarray*}
eq3.05
$$ |\Lambda S \Sigma\rangle = |\Lambda\rangle ~ | S \Sigma\rangle \quad ,$$
eq3.06
$$\langle\Lambda^\prime S^\prime \Sigma^\prime |~ \mu_i~
|\Lambda S \Sigma\rangle = \langle\Lambda^\prime |~ \mu_i~
|\Lambda\rangle ~ \delta_{S^\prime S} \delta_{\Sigma^\prime\Sigma} \quad ,$$
eq3.07
\begin{eqnarray*}
\langle\Lambda + 1 S \Sigma &|& ~ \mu_x + i\mu_y ~ |\Lambda S\Sigma\rangle\\
\langle\Lambda - 1 S \Sigma &|& ~ \mu_x - i\mu_y ~ |\Lambda S\Sigma\rangle\\
\langle\Lambda S \Sigma &|& ~ \mu_z ~ |\Lambda S\Sigma\rangle \quad ,
\end{eqnarray*}
eq3.08
\begin{eqnarray*}
\langle L^\prime\Lambda + 1 S \Sigma &|& ~ \mu_x + i\mu_y ~ |L \Lambda S\Sigma\rangle\\
\langle L^\prime\Lambda - 1 S \Sigma &|& ~ \mu_x - i\mu_y ~ |L \Lambda S\Sigma\rangle\\
\langle L^\prime\Lambda S \Sigma &|& ~ \mu_z ~ |L \Lambda S\Sigma\rangle \quad ,
\end{eqnarray*}
eq3.09
\begin{eqnarray*}
(\sigma_v \mbox{ or } C_2) (\alpha_{Rx}\pm i\alpha_{Ry}) &=& -(\alpha_{Rx}\mp i\alpha_{Ry})\\
(\sigma_v \mbox{ or } C_2) (\alpha_{Rz}) &=& -(\alpha_{Rz})
\end{eqnarray*}
eq3.10a
$|0^+ 0~ 0; 0~ J M \rangle $
eq3.10b
$2^{-1/2} [| 1~ 0~ 0; 1~ J M\rangle \pm | -1~ 0~ 0; -1~ J M\rangle \quad .$
eq3.11a
$+ (-1)^J $
eq3.11b
$\pm (-1)^J \quad ,$
eq3.12
$\mu_Z = 1/2 (\alpha_{Zx} - i\alpha_{Zy}) (\mu_x + i\mu_y) +
1/2 (\alpha_{Zx} + i\alpha_{Zy}) (\mu_x - i\mu_y) + \alpha_{Zz} \mu_z \quad . $
eq3.13
\begin{eqnarray*}
\langle 2^{-1/2} [ \langle 1~ 0~ 0; 1~ J^\prime M| &\pm&
\langle -1~ 0~ 0; 1 J^\prime M|] | ~ \mu_z ~
|0^+~ 0~ 0; 0~ J M \rangle \\
&=& 2^{-1/2} \langle 1~ 0~ 0; 1 J^\prime M|
1/2 (\alpha_{Zx} - i\alpha_{Zy}) (\mu_x + i\mu_y) ~
|0^+~ 0~ 0; 0~ J M \rangle \\
&\pm& 2^{-1/2} \langle -1~ 0~ 0; -1 J^\prime M|
1/2 (\alpha_{Zx} + i\alpha_{Zy}) (\mu_x - i\mu_y) ~
|0^+~ 0~ 0; 0~ J M \rangle \quad .
\end{eqnarray*}
eq3.14
$2^{-1/2} \langle 1~ 0~ 0; 1 J^\prime M| ~
(\alpha_{Zx} - i\alpha_{Zy}) (\mu_x + i\mu_y) ~
|0^+~ 0~ 0; 0~ J M \rangle \quad . $
eq3.15
$2^{-1/2} \langle 1~ 0~ 0| ~ \mu_x + i\mu_y ~ |0^+~ 0~ 0\rangle
\langle 1~ J^\prime M| ~ \alpha_{Zx} - i\alpha_{Zy} ~ |0~ J M \rangle \quad . $
eq3.16
$$\mu_\perp = 2^{-1/2}
\langle 1~ 0~ 0| ~ \mu_x + i\mu_y ~ |0^+~ 0~ 0\rangle \quad , $$
eq3.17
$$ I \propto \mu_\perp^2 \sum_M | \langle 1~ J^\prime ~M| ~
\alpha_{Zx} - i\alpha_{Zy} ~ |0~ J~ M\rangle |^2 \quad .$$
eq3.18
\begin{eqnarray*}
I(R \mbox{ branch } \propto \mu_\perp^2 \sum_M
&|& \left\{ 4(J+1) [(2J+1) (2J+3)]^{1/2} \right\}^{-1}\\
&\times& \left\{ -2[(J+0+1) (J+0+2)]^{1/2} \right\}\\
&\times& \left\{ 2[(J+M+1) (J-M+1)]^{1/2} \right\} |^2 \quad ,
\end{eqnarray*}
eq3.19
\begin{eqnarray*}
\sum_{M=-J}^{+J} 1 &=& (2J+1)\\
\sum_{M=-J}^{J} M^2 &=& (2J+1) J(J+1)/3
\end{eqnarray*}
eq3.20
$$ I(R \mbox{ branch}) \propto {1\over3} \mu_\perp^2 (J+2) \quad .$$
eq3.21
\begin{eqnarray*}
I(Q \mbox{ branch}) &\propto& {1\over3} \mu_\perp^2 (2J+1) \\
I(P \mbox{ branch}) &\propto& {1\over3} \mu_\perp^2 (J-1) \quad .
\end{eqnarray*}
eq3.22a
\begin{eqnarray*}
-[J/(2J+1)]^{1/2}~ 2^{-1/2}[|0^-~ 1~ 1; 1~J~M\rangle
&+&|0^-~ 1~ 1; -1~J~M\rangle]\\
&+& [(J+1)/(2J+1)]^{1/2}~ |0^-~ 1~ 0; 0~J~M\rangle
\end{eqnarray*}
eq3.22b
$$2^{-1/2}[|0^-~ 1~ 1; 1~J~M\rangle - |0^-~ 1~ -1; -1~J~M\rangle] $$
eq3.22c
\begin{eqnarray*}
[(J+1)/(2J+1)]^{1/2}~ 2^{-1/2}[|0^-~ 1~ 1; 1~J~M\rangle&+&|0^-~ 1~ -1; -1~J~M\rangle]\\
&+& [J/(2J+1)]^{1/2}~ |0^-~ 1~ 0; 0~J~M\rangle \quad .
\end{eqnarray*}
eq3.23
\begin{eqnarray*}
-[(J+1)/(2J+3)]^{1/2}~ 2^{-1/2} \langle 0^-~ 1~ 1; 1~J+1~M| &\mu_Z& |0^+~ 0~ 0; 0~J~M\rangle \\
-[(J+1)/(2J+3)]^{1/2}~ 2^{-1/2} \langle 0^-~ -1~ -1; -1~J+1~M| &\mu_Z& |0^+~ 0~ 0; 0~J~M\rangle \\
+[(J+2)/(2J+3)]^{1/2}~ \langle 0^-~ 1~ 0; 0~J+1~M| &\mu_Z& |0^+~ 0~ 0; 0~J~M\rangle \quad .
\end{eqnarray*}
eq3.24
\begin{eqnarray*}
\langle 0^-~ 1~ 1| &\mu_x + i\mu_y& |0^+~ 0~ 0\rangle \\
\langle 0^-~ 1~ 0| &\mu_z& |0^+~ 0~ 0\rangle\\
\langle 0^-~ 1~-~ 1| &\mu_x - i\mu_y& |0^+~ 0~ 0\rangle \quad .
\end{eqnarray*}
eq3.25
\begin{eqnarray*}
\sigma_v |0^+~ 0~ 0\rangle &=& + |0^+~ 0~ 0\rangle\\
\sigma_v |0^-~ 1~ 0\rangle &=& +|0^-~ 1~ 0\rangle \quad .
\end{eqnarray*}
eq3.26
\begin{eqnarray*}
-[(J+1)/(2J+3)]^{1/2}~ 2^{-1/2} \langle 0^-~ 1~ 1| \mu_x&+& i\mu_y
|0^+~ 0~ 0\rangle \langle 1~J+1~M|\alpha_{Zx}-i\alpha_{Zy} |0~J~M\rangle \\
+[(J+2)/(2J+3)]^{1/2}~ \langle 0^-~ 1~ 0| &\mu_Z& |0^+~ 0~ 0\rangle
\langle 0~J+1~M|\alpha_{Zz} |0~J~M\rangle \quad .
\end{eqnarray*}
eq3.27
\begin{eqnarray*}
\mu_\parallel &=& \langle 0^-~ 1~ 0| ~\mu_z ~|0^+~ 0~ 0\rangle\\
\mu_\perp &=& 2^{-1/2} \langle 0^-~ 1~ 1| ~\mu_x + i\mu_y ~|0^+~ 0~ 0\rangle \quad ,
\end{eqnarray*}
eq3.28
\begin{eqnarray*}
\theta |0^- ~ 1 ~0\rangle = + |0^- ~ 1 ~0\rangle\\
\theta |0^+ ~ 0 ~0\rangle = + |0^+ ~ 0 ~0\rangle \quad .
\end{eqnarray*}
eq3.29
\begin{eqnarray*}
\theta\mu_\parallel = \mu_\parallel^* &=& +\langle 0^-~ 1~ 0| ~\mu_z^* ~|0^+~ 0~ 0\rangle\\
&=& +\langle 0^-~ 1~ 0| ~\mu_z ~|0^+~ 0~ 0\rangle = + \mu_\parallel \quad .
\end{eqnarray*}$ $
eq3.30
$\theta\mu_\perp = \mu_\perp^* = -2^{-1/2} \langle 0^-~ 1~ -1|
~\mu_x - i\mu_y ~|0^+~ 0~ 0\rangle \quad . $
eq3.31
\begin{eqnarray*}
-2^{-1/2} \langle 0^-~ 1~-~ 0| &\mu_x - i\mu_y& ~|0^+~ 0~ 0\rangle = \\
+2^{-1/2} \langle 0^-~ 1~ 1| &\mu_x + i\mu_y& ~|0^+~ 0~ 0\rangle = + \mu_\perp \quad .
\end{eqnarray*}
eq3.32
\begin{eqnarray*}
-[(J+1)/(2J+3)]^{1/2}~ \mu_\perp \{4(J+1)[(2J+1)(2J+3)]^{1/2}\}^{-1}
&\times& (-2) [(J+0+1)(J+0+2)]^{1/2}\\
&\times& (2)[(J+M+1)(J-M+1)]^{1/2}\\
+[(J+2)/(2J+3)]^{1/2}~ \mu_\parallel \{4(J+1)[(2J+1)(2J+3)]^{1/2}\}^{-1}
&\times& (2) [(J+0+1)(J-0+1)]^{1/2}\\
&\times& (2) [(J+M+1)(J-M+1)]^{1/2} \quad ,
\end{eqnarray*}
eq3.33
$I(^S{R}~ {\rm branch}) \propto [+\mu_\parallel +\mu_\perp]^2 (J+1) (J+2)/3 (2J+3) \quad. $
eq3.34
\begin{eqnarray*}
I(^Q{R}~ {\rm branch}) &\propto& [+\mu_\parallel(J+1) - \mu_\perp(J+2)]^2 /3 (2J+3)\\
I(^Q{Q}~ {\rm branch}) &\propto& (+\mu_\perp)^2 (2J+1)/3 \\
I(^Q{P}~ {\rm branch}) &\propto& [+\mu_\parallel(J) -\mu_\perp(J-1)]^2 /3 (2J-1)\\
I(^O{P}~ {\rm branch}) &\propto& [+\mu_\parallel + \mu_\perp]^2 J(J-1)/3 (2J-1) \quad .
\end{eqnarray*}
eq3.35
\begin{eqnarray*}
U^{-1}(J^\prime) &H_u(J^\prime)& U(J^\prime)\\
L^{-1}(J^{\prime\prime}) &H_l(J^{\prime\prime})& L(J^{\prime\prime}) \quad ,
\end{eqnarray*}
eq3.36
\begin{eqnarray*}
\langle\Lambda^\prime S^\prime \Sigma^\prime;
\Omega^\prime J^\prime M\mid &\mu_Z &
\mid\Lambda^{\prime\prime} S^{\prime\prime} \Sigma^{\prime\prime};
\Omega^{\prime\prime} J^{\prime\prime} M \rangle =\\
&+& 1/2 \langle\Lambda^\prime S^\prime \Sigma^\prime\mid \mu_x + i\mu_y
\mid\Lambda^{\prime\prime} S^{\prime\prime} \Sigma^{\prime\prime}\rangle
\langle\Omega^\prime J^\prime M\mid \alpha_{Zx}-i\alpha_{Zy}
\mid\Omega^{\prime\prime} J^{\prime\prime} M \rangle \\
&+& 1/2 \langle\Lambda^\prime S^\prime \Sigma^\prime\mid \mu_x - i\mu_y
\mid\Lambda^{\prime\prime} S^{\prime\prime} \Sigma^{\prime\prime}\rangle
\langle\Omega^\prime J^\prime M\mid \alpha_{Zx}+i\alpha_{Zy}
\mid\Omega^{\prime\prime} J^{\prime\prime} M \rangle \\
&+& \langle\Lambda^\prime S^\prime \Sigma^\prime\mid \mu_z
\mid\Lambda^{\prime\prime} S^{\prime\prime} \Sigma^{\prime\prime}\rangle
\langle\Omega^\prime J^\prime M\mid \alpha_{Zz}
\mid\Omega^{\prime\prime} J^{\prime\prime} M \rangle \quad .\\
\end{eqnarray*}
eq3.37
\begin{eqnarray*}
\mu_1 &=& 2^{-1/2}\langle 2 ~\frac{3}{2} ~\frac{3}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~\frac{5}{2} \rangle =
-2^{-1/2}\langle -2 ~\frac{3}{2} ~-\frac{3}{2} \mid \mu_x - i\mu_y
\mid 0^+ ~\frac{5}{2} - \frac{5}{2} \rangle\\
\mu_2 &=& 2^{-1/2}\langle 2 ~\frac{3}{2} ~\frac{1}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~\frac{3}{2} \rangle =
-2^{-1/2}\langle -2 ~\frac{3}{2} ~-\frac{1}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~-\frac{3}{2} \rangle\\
\mu_3 &=& 2^{-1/2}\langle 2 ~\frac{3}{2} ~- \frac{1}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~\frac{1}{2} \rangle =
-2^{-1/2}\langle -2 ~\frac{3}{2} ~\frac{1}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} - \frac{1}{2} \rangle\\
\mu_4 &=& 2^{-1/2}\langle 2 ~ \frac{3}{2} ~- \frac{3}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~-\frac{1}{2} \rangle =
-2^{-1/2}\langle -2 ~ \frac{3}{2} ~\frac{3}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} - \frac{1}{2} \rangle\\
\mu_5 &=& 2^{-1/2}\langle -2 ~ \frac{3}{2} ~\frac{3}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~-\frac{3}{2} \rangle =
-2^{-1/2}\langle 2 ~ \frac{3}{2} ~-\frac{3}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~- \frac{3}{2} \rangle\\
\mu_6 &=& 2^{-1/2}\langle -2 ~ \frac{3}{2} ~\frac{1}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~-\frac{5}{2} \rangle =
-2^{-1/2}\langle 2 ~\frac{3}{2} ~-\frac{1}{2} \mid \mu_x + i\mu_y
\mid 0^+ ~\frac{5}{2} ~\frac{5}{2} \rangle\\
\mu_7 &=& \langle 2 ~ \frac{3}{2} ~\frac{1}{2} \mid \mu_z
\mid 0^+ ~\frac{5}{2} ~\frac{5}{2} \rangle =
+\langle -2 ~ \frac{3}{2} ~-\frac{1}{2} \mid \mu_z
\mid 0^+ ~\frac{5}{2} ~-\frac{5}{2} \rangle\\
\mu_8 &=& \langle 2 ~ \frac{3}{2} ~-\frac{1}{2} \mid \mu_z
\mid 0^+ ~ \frac{5}{2} ~ \frac{3}{2} \rangle =
+\langle -2 ~ \frac{3}{2} ~ \frac{1}{2} \mid \mu_z
\mid 0^+ ~\frac{5}{2}~ -\frac{3}{2} \rangle\\
\mu_9 &=& \langle 2 ~\frac{3}{2} ~-\frac{3}{2} \mid \mu_z
\mid 0^+ ~\frac{5}{2} ~\frac{1}{2} \rangle =
+\langle -2 ~ \frac{3}{2} ~\frac{3}{2} \mid \mu_z
\mid 0^+ ~\frac{5}{2} ~ -\frac{1}{2} \rangle \quad .
\end{eqnarray*}
eq3.38
$$h_Z (J^\prime ; J^{\prime\prime}) = \left[ \sum_M \,
h_Z^2(J^\prime , M; J^{\prime\prime} , M) \right]^{1/2} \quad .
$$
eq3.39
\begin{eqnarray*}
\left[ \begin{array}{cccccc}
\mu_1 c_1 & 0 & 0 & 0 & 0 & 0 \\
\mu_7 c_7 & \mu_2 c_2 & 0 & 0 & 0 & 0 \\
-\mu_6 c_6 & \mu_8 c_8 & \mu_3 c_3 & 0 & 0 & 0 \\
0 &-\mu_5 c_5 & \mu_9 c_9 & \mu_4 c_4 & 0 & 0 \\
0 & 0 &-\mu_4 c_4 &-\mu_9 c_9 & \mu_5 c_5 & 0 \\
0 & 0 & 0 &-\mu_3 c_3 &-\mu_8 c_8 & \mu_6 c_6 \\
0 & 0 & 0 & 0 &-\mu_2 c_2 &-\mu_7 c_7 \\
0 & 0 & 0 & 0 & 0 &-\mu_1 c_1
\end{array} \right] \quad ,
\end{eqnarray*}
eq3.40
\begin{eqnarray*}
c_1 &=& [(J+{1\over2}) (J-{5\over2}) (J+{7\over2}) / 3J(J+1)]^{1/2}\\
c_2 &=& [(J+{1\over2}) (J-{3\over2}) (J+{5\over2}) / 3J(J+1)]^{1/2}\\
c_3 &=& [(J+{1\over2}) (J-{1\over2}) (J+{3\over2}) / 3J(J+1)]^{1/2}\\
c_4 &=& [(J+{1\over2}) (J+{1\over2}) (J+{1\over2}) / 3J(J+1)]^{1/2}\\
c_5 &=& [(J+{1\over2}) (J+{3\over2}) (J-{1\over2}) / 3J(J+1)]^{1/2}\\
c_6 &=& [(J+{1\over2}) (J+{5\over2}) (J-{3\over2}) / 3J(J+1)]^{1/2}\\
c_7 &=& +5[(J+{1\over2}) / 6J(J+1)]^{1/2}\\
c_8 &=& +3[(J+{1\over2}) / 6J(J+1)]^{1/2}\\
c_9 &=& +1[(J+{1\over2}) / 6J(J+1)]^{1/2} \quad .
\end{eqnarray*}
eq3.41
$$\mu (J^\prime, J^{\prime\prime}; M) = U^{-1} (J^\prime) \cdot \mu_b
(J^\prime, J^{\prime\prime}; M) \cdot L(J^{\prime\prime}) \quad . $$
eq3.42
$$ I(\alpha J^\prime , \beta J^{\prime\prime}) \propto \sum_M
\vert\mu(J^\prime , J^{\prime\prime}; M)_{\alpha\beta} \vert^2 \quad .$$
eq3.43
$$ I(\alpha J^\prime , \beta J^{\prime\prime}) \propto \sum_M
\vert \mu(J^\prime , J^{\prime\prime}; M)_{\alpha\beta} \vert^2
\equiv \vert \mu(J^\prime , J^{\prime\prime})_{\alpha\beta} \vert^2 \quad ,$$
Chapter 4
eq4.01
\begin{eqnarray*}
\left[ \begin{array}{ccc}
B_\Pi [J(J+1)-1] + B_\Pi \langle L_\perp^2\rangle_\Pi & 0 &
-B\langle\Pi\mid L_+ \mid\Sigma\rangle ~ [J(J+1)]^{1/2} \\
0 & B_\Pi [J(J+1)-1] + B_\Pi \langle L_\perp^2\rangle_\Pi &
-B\langle\Pi \mid L_+ \mid\Sigma\rangle ~ [J(J+1)]^{1/2} \\
-B\langle\Pi \mid L_+ \mid \Sigma\rangle ~ [J(J+1)]^{1/2} &
-B\langle\Pi\mid L_+ \mid\Sigma\rangle ~ [J(J+1)]^{1/2} &
E_\Sigma + B_\Sigma J(J+1) + B_\Sigma \langle L_\perp^2\rangle_\Sigma
\end{array} \right]
\end{eqnarray*}
eq4.02
$\langle 1~0~0;~1~J~M\mid L_+J_- \mid 0^+~0~0;~0~J~M\rangle =
\pm \langle -1~0~0;~-1~J~M\mid L_-J_+ \mid 0^+~0~0;~0~J~M\rangle\quad,$
eq4.03
$B\langle\Pi\mid L_+ \mid\Sigma\rangle \equiv
B\langle 1~0~0\mid L_+ \mid 0^+~0~0\rangle\quad, $
eq4.04
$\langle\Pi\mid L_+ \mid\Sigma\rangle =
\langle L, \Lambda = +1\mid L_+ \mid L, \Lambda = 0\rangle =
\hbar [L(L+1)]^{1/2} \quad {\rm or} \quad 0 \quad .$
eq4.05
\begin{eqnarray*}
\left[ \begin{array}{cccc}
B_\Delta [J(J+1)-8] + B_\Delta\langle L_\perp^2\rangle_\Delta + 2A &
-B_\Delta [2(J-2) (J+3)]^{1/2} & 0 & 0 \\
-B_\Delta [2(J-2) (J+3)]^{1/2} &
B_\Delta [J(J+1)-2] + B_\Delta\langle L_\perp^2\rangle_\Delta &
-B_\Delta [2 (J-1) (J+2)]^{1/2} & 0 \\
0 & -B_\Delta [2 (J-1) (J+2)]^{1/2} &
B_\Delta \, J(J+1) + B_\Delta\langle L_\perp^2\rangle_\Delta - 2A & \eta \\
0 & 0 & \eta &
E_\Pi + B_\Pi [J(J+1)-1] + B_\Pi \langle L_\perp^2\rangle_\Pi
\end{array} \right]
\end{eqnarray*}
eq4.06
\begin{eqnarray*}
\eta &=& \langle 2~ 1~ -1;~ 1~ J~ M\mid \sum_i \xi(r_i) \mbox{\bf l}_i \cdot
\mbox{\bf s}_i \mid 1~ 0 ~0;~ 1~ J~ M\rangle\\
&=& \langle 2~ 1~ -1\mid \sum_i \xi(r_i) \mbox{\bf l}_i \cdot
\mbox{\bf s}_i \mid 1~ 0 ~0\rangle\\
&=& \pm \langle -2~ 1~ 1\mid \sum_i \xi(r_i) \mbox{\bf l}_i \cdot
\mbox{\bf s}_i \mid -1~ 0 ~0\rangle \quad ,
\end{eqnarray*}
eq4.07
$H = H_0+H_1 +H_2 $
eq4.08
$\langle i\mid H_0 \mid j \rangle +
\langle i\mid H_1 \mid j \rangle +
\langle i\mid H_2 \mid j \rangle +
\sum_{k \not=i,j}
\langle i\mid H_1 \mid k \rangle
\langle k\mid H_1 \mid j \rangle
\textstyle{\frac {[{1\over2} (E_i^{\rm o} + E_j^{\rm o}) - E_k^{\rm o}]}
{(E_i^{\rm o} - E_k^{\rm o}) (E_j^{\rm o} - E_k^{\rm o})}} \quad .$
eq4.09
$-[B^2\mid \langle\Pi\mid L_+\mid\Sigma\rangle \mid^2/E_\Sigma] J(J+1) $
eq4.10
\begin{eqnarray*}
E_{\rm rot} &=& B_\Pi \, J(J+1) - B_\Pi +B_\Pi \langle L_\perp^2\rangle_\Pi \\
E_{\rm rot} &=& [B_\Pi -2B^2\mid \langle\Pi\mid L_+\mid\Sigma\rangle
\mid^2/E_\Sigma] J(J+1) - B_\Pi +B_\Pi \langle L_\perp^2\rangle_\Pi
\end{eqnarray*}
eq4.11
\begin{eqnarray*}{\cal H}_{\rm r} &=& B(Q) \hbar^{-2} [R_x^2 + R_y^2] \\
&=& B(Q) \hbar^{-2} [(J_x - L_x - S_x)^2 + (J_y - L_y - S_y)^2] ~ ,\end{eqnarray*}
eq4.12
\begin{eqnarray*}
B(Q) &=& \hbar^2/2\mu(r_e+Q)^2 = [\hbar^2/2\mu r_e^2] \, [1+Q/r_e]^{-2} \\
&=& B_e - 2(B_e/r_e) \,Q+3(B_e/r_e^2) \,Q^2 - \ldots ~ .\end{eqnarray*}
eq4.13
$$
\sum_{v^{\prime\prime}}~ \sum_{\Lambda^{\prime\prime} \Sigma^{\prime\prime}}
\frac{\langle L\Lambda S\Sigma; v; \Omega JM\mid H_r \mid L\Lambda^{\prime\prime}
S\Sigma^{\prime\prime}; v^{\prime\prime}; \Omega^{\prime\prime} JM\rangle \,
\langle L\Lambda^{\prime\prime} S\Sigma^{\prime\prime}; v^{\prime\prime};
\Omega^{\prime\prime} JM\mid H_r \mid L\Lambda^\prime S\Sigma^\prime; v;
\Omega^\prime JM\rangle} {(E_v - E_{v^{\prime\prime}})} \quad ,
$$
eq4.14
\begin{eqnarray*}
\sum_{v^{\prime\prime}}&~&\langle v\mid -2(B_e/r_e)Q\mid v^{\prime\prime}\rangle
\langle v^{\prime\prime}\mid -2(B_e/r_e)Q\mid v\rangle
(E_v-E_{v^{\prime\prime}})^{-1} \times\\
\sum_{\Lambda^{\prime\prime} \Sigma^{\prime\prime}}&~&
\langle L\Lambda S\Sigma;\Omega JM\mid \hbar^{-2}[R_x^2 + R_y^2]
\mid L\Lambda^{\prime\prime} S\Sigma^{\prime\prime}; \Omega^{\prime\prime}JM\rangle
\langle L\Lambda^{\prime\prime} S\Sigma^{\prime\prime};\Omega^{\prime\prime}
JM\mid \hbar^{-2}[R_x^2 + R_y^2]
\mid L\Lambda^\prime S\Sigma^\prime; \Omega^\prime JM\rangle \quad .
\end{eqnarray*}
eq4.15
\begin{eqnarray*}
\langle v+1\mid Q\mid v\rangle &=&[(v+1)\hbar/4\pi\mu\nu ]^{1/2}\\
\langle v-1\mid Q\mid v\rangle &=&[(v\hbar/4\pi\mu\nu ]^{1/2}
\end{eqnarray*}
eq4.16
$ -D_e = -4B_e^3/(h\nu)^2 $
eq4.17
$D = 4B^3/\omega^2 \quad . $
eq4.18
$\langle L\Lambda S\Sigma; \Omega JM\mid
\hbar^{-4} [R_x^2 +R_y^2]^2 \mid L\Lambda^\prime S\Sigma^\prime;
\Omega^\prime JM\rangle \quad .$
eq4.19
$\langle L\Lambda S\Sigma; \, v; \, \Omega JM\mid -D \hbar^{-4}
[R_x^2 +R_y^2]^2 \mid L\Lambda^\prime S\Sigma^\prime; \, v; \,
\Omega^\prime JM\rangle $
eq4.20
$-D\hbar^{-4}[(J_x-L_x-S_x)^2 + (J_y-L_y-S_y)^2]^2 $
eq4.21
$$
\sum_{v^{\prime\prime}}~ \sum_{\Lambda^{\prime\prime} \Sigma^{\prime\prime}}
\frac{\langle\Lambda S\Sigma; v; \Omega JM\mid H_r \mid \Lambda^{\prime\prime}
S\Sigma^{\prime\prime}; v^{\prime\prime}; \Omega^{\prime\prime} JM\rangle \,
\langle\Lambda^{\prime\prime} S\Sigma^{\prime\prime}; v^{\prime\prime};
\Omega^{\prime\prime} JM\mid H_r \mid \Lambda^\prime S\Sigma^\prime; v;
\Omega^\prime JM\rangle} {(E_v - E_{v^{\prime\prime}})} \quad ,
$$
eq4.22
$-D\hbar^{-4}[(J_x-S_x)^2 + (J_y-S_y)^2 + \langle L_\perp^2\rangle]^2 $
eq4.23
$-D\hbar^{-4}[\mbox{\bf J}^4 -4(J_x S_x + J_y S_y)
\mbox{\bf J}^2] \quad . $
