convert math in more pair styles

\documentclass[12pt]{article}

\begin{document}

$$

  \mathbf{F}_{ij}^{C} = \alpha_{ij}{\omega_{C}}(r_{ij})\mathbf{e}_{ij},

$$

$$

  \mathbf{F}_{ij}^{D} = -\gamma {\omega_{D}}(r_{ij})(\mathbf{e}_{ij} \cdot \mathbf{v}_{ij})\mathbf{e}_{ij},

$$

$$

  \mathbf{F}_{ij}^{R} = \sigma {\omega_{R}}(r_{ij}){\xi_{ij}}\Delta t^{-1/2} \mathbf{e}_{ij},

$$

$$

  \omega_{C}(r) = 1 - r/r_c,

$$

$$

  \alpha_{ij} = A\cdot k_B(T_i + T_j)/2,

$$

$$  

  \omega_{D}(r) = \omega^2_{R}(r) = (1-r/r_c)^s,

$$

$$  

  \sigma_{ij}^2 = 4\gamma k_B T_i T_j/(T_i + T_j),

$$

\end{document}

\documentclass[12pt]{article}

\begin{document}

$$

  \frac{\mathrm{d}^2 \mathbf{r}_i}{\mathrm{d} t^2}=

  \frac{\mathrm{d} \mathbf{v}_i}{\mathrm{d} t}

  =\mathbf{F}_{i}=\sum_{i\neq j}(\mathbf{F}_{ij}^{C}+\mathbf{F}_{ij}^{D}+\mathbf{F}_{ij}^{R}),

$$

$$

  C_v\frac{\mathrm{d} T_i}{\mathrm{d} t}= q_{i} = \sum_{i\neq j}(q_{ij}^{C}+q_{ij}^{V}+q_{ij}^{R}),

$$

\end{document}