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\documentstyle[12pt]{article}
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\begin{document}
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$$
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F_{i}^{DD}(\rho_i,\rho_j,r_{ij}) = \frac{1}{2} \omega_{DD}\left(r_{ij}\right)
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\left[A\left(\rho_i\right) + A\left(\rho_j\right)\right]e_{ij}
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$$
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\end{document}
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\documentstyle[12pt]{article}
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\begin{document}
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$$
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\omega_{DD}\left(r_{ij}\right) = \left(1+\frac{3r_{ij}}{r_{cut}}\right)\left(1+\frac{r_{ij}}{r_{cut}}\right)^3
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$$
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\end{document}
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\documentstyle[12pt]{article}
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\begin{document}
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$$
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u_{i}^{DD}\left(\rho_{i}\right) = \frac{\pi r_{cut}^4}{84} \int_{\rho_0}^{\rho_i} A\left(\rho^{'}\right) d\rho^{'}
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$$
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\end{document}
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