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