NIST: Hydrocarbons - LaTeX Equations

LaTeX version of Equations


eq 1
$$ B = {\displaystyle \frac{h^2}{8\pi^2 I_b}} ~ . $$


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


eq 3
$$ B_v = B_{v_1,v_2,v_3} = B_e - \sum_{i=1,2,3} \alpha_i 
            \left( vi + \frac{d_i}{2} \right) ~ , $$ 


eq 4
$ \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 + \frac{1}{3}~ J^{\prime\,2} 
            - 2(J^\prime)^2 \, \ell^2  + \ell^4 \right] ~ . $


eq 5 
$E_{J,K} = B\,J(J+1)+(A-B)K^2 -D_J J^2(J+1)^2 -
 D_{J\,K} J(J+1) K^2 -D_K \,K^4 ~ ,$


eq 6
\begin{eqnarray*}v = 2B_{\rm o} J^\prime - 4D_J(J^\prime)^3 
-2D_{JK} J^\prime K^2 &+& H_{JJJ}(J^\prime)^3[(J^\prime+1)^3 -(J^\prime+1)^3]\\
&+& 4H_{JJk}(J^\prime)^3\, K^2 + 2H_{KKJ} \, J^\prime \, K^4 ~ .\end{eqnarray*}


eq 7
$$ \kappa = ~\frac{2B-A-C}{A-C} ~ . $$


eq 8
$ \Delta J = 0, \pm 1; ~ \Delta K_{-1} = 0, \pm 2, ...; ~ 
             \Delta K_{+1} = \pm 1, \mp 3, ... ~ , $
                

eq 9
$ \Delta J = 0, \pm 1; \quad \Delta K_{-1} = \pm 1, \pm 3, ...; \quad  
             \Delta K_{+1} = \mp 1, \mp 3, ... ~ , $


eq 10
$ \Delta J = 0, \pm 1; \quad \Delta K_{-1} = \pm 1, \pm 3; \quad  
             \Delta K_{+1} = 0, \mp 2 ~ . $
                

eq 11
$$ {\cal H} = A^\prime P_a^2 + B^\prime P_b^2 + C^\prime P_c^2 + 
           {\textstyle\frac{1}{4}} ~\sum_{\alpha,\beta} ~ \tau_{\alpha\alpha\beta\beta}^\prime 
           P_\alpha^2 P_\beta^2 ~ , $$
 

eq 12a 
$ \tau_{acac} = \tau_{bcbc} = 0  $


eq 12b
$$ \tau_{aacc} = \frac{C^2}{A^2} ~\tau_{aaaa} +  
 \frac{C^2}{B^2} ~\tau_{aabb} ~ , $$


eq 12c
$$ \tau_{bbcc} = \frac{C^2}{B^2} ~\tau_{bbbb} +  
 \frac{C^2}{A^2} ~\tau_{aabb} ~ , $$


eq 12d
$$ \tau_{cccc} = \frac{C^2}{A^2} ~\tau_{aacc} +  
 \frac{C^2}{B^2} ~\tau_{bbcc} ~ , $$


eq 13
$$\Delta_0= 3F~\frac{a_1(s)}{2}=\frac{28}{8}~F\,\omega_1(s) ~ .$$


eq 14
$$\alpha_{\rm P}=4\Sigma(-1)^s \times \frac{\langle\Pi| \,(A+2B) L_y \, 
|\Sigma\rangle \, \langle\Sigma| \,B\,L_y \,|\Pi\rangle} {E_\Sigma-E\Pi} $$


eq 15
$$\beta_{\rm P}=4\Sigma(-1)^s \times \frac{|\langle\Pi| \,B\,L_y 
\,|\Sigma\rangle|^2}{E_\Sigma-E\Pi} ~ .$$


eq 16
$a=2\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} (1/r^3)$


eq 17
$$b=-\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} 
\left\langle \frac{3\cos^2\, \chi-1}{r^3}\right\rangle +\frac{16}{3}~
\pi\,\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N}~\Psi^2(0) $$


eq 18
$$c=3\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} 
\left\langle \frac{3\cos^2\, \chi-1}{r^3}\right\rangle$$


eq 19
$$d=3\mu_{\rm B}\,g_{\rm N}\,\mu_{\rm N} 
\left\langle \frac{\sin^2\, \chi}{r^3}\right\rangle ~ .$$