### LaTeX version of Equations


eq01
$B = {\displaystyle \frac{h^2}{8\pi^2 I_b}} ~ .$

eq02
$F(J) = B_v[J(J+1) - \ell^2] - D_v[J(J+1) - \ell^2]^2 + H_v[J(J+1) - \ell^2]^3 ~ ,$

eq03
$$B_v = B_{v_1,v_2,v_3} = B_{\rm e} - \sum_{i=1,2,3} \alpha_i \left( v_i + \frac{d_i}{2} \right) + ... ~ ,$$

eq04
$\nu_{J^\prime\leftarrow J^{\prime\prime}} = 2B_v J^\prime - 4 D_v \left[ (J^\prime)^3 - J^\prime \, \ell^2 \right] + 6 J^\prime H_v \left[ (J^\prime)^4 - 2(J^\prime)^2 \, \ell^2 + \ell^4 \right] ~ .$

eq05
$$W_{\rm hfs} = -eQq_v \left[ \frac{3\ell^2}{J(J+1)} ~ -1 \right] f(I, J, F) + (c/2)~ [F(F+1) - I(I+1) - J(J+1)] ~ ,$$

eq06a
$$F_1 = J+I_1, ~ J+I_1-1 , ~ {\rm etc.}$$

eq06b
$$F = F_1 +I_2, ~ F_1+I_2-1 , ~ {\rm etc.}$$

eq07
$$\kappa = ~ \frac{2B\,-\,A-C}{A-C} ~ .$$

eq08a
$\Delta J = 0, \pm 1; ~ \Delta K_{-1} = 0, \pm 2, ...; ~ \Delta K_{+1} = \pm 1, \pm 3, ... ~ ,$

eq08b
$\Delta J = 0, \pm 1; \quad \Delta K_{-1} = \pm 1, \pm 3; \quad \Delta K_{+1} = \mp 1, \mp 3, ... ~ .$

eq09
$${\cal H} = A^\prime P_a^2 + B^\prime P_b^2 + C^\prime P_c^2 + {\textstyle\frac{1}{4}} ~\sum_{\alpha,\beta} ~ \tau_{\alpha\alpha\beta\beta}^\prime P_\alpha^2 P_\beta^2$$

eq10
$\tau_{acac} = \tau_{bcbc} = 0$

eq11
$$\tau_{aacc} = \frac{C^2}{A^2} ~\tau_{aaaa} + \frac{C^2}{B^2} ~\tau_{aabb}$$

eq12
$$\tau_{bbcc} = \frac{C^2}{B^2} ~\tau_{bbbb} + \frac{C^2}{A^2} ~\tau_{aabb}$$

eq13
$$\tau_{cccc} = \frac{C^2}{A^2} ~\tau_{aacc} + \frac{C^2}{B^2} ~\tau_{bbcc}$$

eq14
$$\alpha_p = 4\Sigma(-1)^S \times~ \frac{\langle \Pi|\, (A+2B) L_y\, |\Sigma\rangle \, \langle\Sigma|\, BL_y\, |\Pi\rangle} {E_\Sigma - E_\Pi}$$

eq15
$$\beta_p = 4\Sigma(-1)^S \times ~ \frac {| \langle\Pi|\, BL_y\, |\Sigma\rangle |^2} {E_\Sigma - E_\Pi}$$

eq16
$$a = 2\mu_{\rm B} g_{\rm N}\mu_{\rm N} \langle 1/r^3\rangle$$

eq17
$$b = - \mu_{\rm B}g_{\rm N}\mu_{\rm N} ~\left\langle {\displaystyle \frac{3\cos^2\chi-1}{r^3}}\right\rangle + {\displaystyle \frac{16}{3}} ~ \pi \mu_{\rm B} g_{\rm N}\mu_{\rm N} ~ \Psi^2(0)$$

eq18
$$c = 3\mu_{\rm B}g_{\rm N}\mu_{\rm N} ~\left\langle \frac{3\cos^2\chi-1}{r^3}\right\rangle$$

eq19
$$d = 3\mu_{\rm B}g_{\rm N}\mu_{\rm N} ~\left\langle \frac{\sin^2\chi}{r^3}\right\rangle$$