git-svn-id: svn://svn.icms.temple.edu/lammps-ro/trunk@14195 f3b2605a-c512-4ea7-a41b-209d697bcdaa

doc/doc2/Eqs/pair_mgpt.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+$$
+   E_{\rm tot}({\bf R}_1 \ldots {\bf R}_N) = NE_{\rm vol}(\Omega ) 
+   + \frac{1}{2} \sum _{i,j} \mbox{}^\prime \ v_2(ij;\Omega ) 
+   + \frac{1}{6} \sum _{i,j,k} \mbox{}^\prime \ v_3(ijk;\Omega ) 
+   + \frac{1}{24} \sum _{i,j,k,l} \mbox{}^\prime \ v_4(ijkl;\Omega )
+$$
+
+\end{document}

doc/doc2/Eqs/pair_smtbq1.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+E_{tot} & = & E_{ES} + E_{OO} + E_{MO} \\
+E_{ES} & = & \sum_i{\Big[ \chi_{i}^{0}Q_i + \frac{1}{2}J_{i}^{0}Q_{i}^{2} +
+ \frac{1}{2} \sum_{j\neq i}{ J_{ij}(r_{ij})f_{cut}^{R_{coul}}(r_{ij})Q_i Q_j } \Big] } \\
+E_{OO} & = & \sum_{i,j}^{i,j = O}{\Bigg[Cexp( -\frac{r_{ij}}{\rho} ) - Df_{cut}^{r_1^{OO}r_2^{OO}}(r_{ij}) exp(Br_{ij})\Bigg]}  \\
+E_{MO} & = & \sum_i{E_{cov}^{i} + \sum_{j\neq i}{ Af_{cut}^{r_{c1}r_{c2}}(r_{ij})exp\Big[-p(\frac{r_{ij}}{r_0} -1) \Big] } }  \\
+\end{eqnarray*}
+
+\end{document}

doc/doc2/Eqs/pair_smtbq2.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+E_{cov}^{i(i=M,O)} & = & - \Bigg\{\eta_i(\mu \xi^{0})^2 f_{cut}^{r_{c1}r_{c2}}(r_{ij})
+\Bigg( \sum_{j(j=O,M)}{ exp[ -2q(\frac{r_{ij}}{r_0} - 1)] } \Bigg) 
+\delta Q_i \Big( 2\frac{n_0}{\eta_i} - \delta Q_i \Big) \Bigg\}^{1/2} \\
+\delta Q_i & = & | Q_i^{F} | - | Q_i |
+\end{eqnarray*}
+
+\end{document}

doc/doc2/Eqs/pair_smtbq3.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+\xi^0 & = & \frac{\xi_O}{m} = \frac{\xi_C}{n} \\
+\frac{\beta_O}{\sqrt{m}} & = & \frac{\beta_C}{\sqrt{n}} = \xi^0 \frac{\sqrt{m}+\sqrt{n}}{2}\\
+\end{eqnarray*}
+
+\end{document}

doc/doc2/Eqs/pair_vashishta.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+  U & = & \sum_i^N \sum_{j > i}^N U_{ij}^{(2)} (r_{ij}) + 
+          \sum_i^N \sum_{j \neq i}^N \sum_{k > j, k \neq i}^N 
+          U_{ijk}^{(3)} (r_{ij}, r_{ik}, \theta_{ijk}) 
+          \\
+  U_{ij}^{(2)} (r) & = &  \frac{H_{ij}}{r^{\eta_{ij}}}
+  	                        + \frac{Z_i Z_j}{r}\exp(-r/\lambda_{1,ij})
+  	                        - \frac{D_{ij}}{r^4}\exp(-r/\lambda_{4,ij})
+	                        - \frac{W_{ij}}{r^6}, r < r_{c,{ij}}
+                   \\
+  U_{ijk}^{(3)}(r_{ij},r_{ik},\theta_{ijk}) & = & B_{ijk} 
+                    \frac{\left[ \cos \theta_{ijk} - \cos \theta_{0ijk} \right]^2}
+                    {1+C_{ijk}\left[ \cos \theta_{ijk} - \cos \theta_{0ijk} \right]^2} \times \\
+&  &                     \exp \left( \frac{\gamma_{ij}}{r_{ij} - r_{0,ij}} \right)
+                    \exp \left( \frac{\gamma_{ik}}{r_{ik} - r_{0,ik}} \right), r_{ij} < r_{0,ij}, r_{ik} < r_{0,ik}
+\end{eqnarray*}                           
+
+\end{document}

doc/doc2/Eqs/polymorphic1.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+$$
+E=\frac{1}{2}\sum_{i=1}^{i=N}\sum_{j=1}^{j=N}\left[\left(1-\delta_{ij}\right)\cdot U_{IJ}\left(r_{ij}\right)-\left(1-\eta_{ij}\right)\cdot F_{IJ}\left(r_{ij}\right)\cdot V_{IJ}\left(r_{ij}\right)\right]
+$$
+
+\end{document}

doc/doc2/Eqs/polymorphic2.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+$$
+X_{ij}=\sum_{k=i_1,k\neq i,j}^{i_N}W_{IK}\left(r_{ik}\right)\cdot G_{JIK}\left(\theta_{jik}\right)\cdot P_{IK}\left(\Delta r_{jik}\right)
+\label{X_eq2}
+$$
+
+\end{document}

doc/doc2/Eqs/polymorphic3.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+$$
+\Delta r_{jik}=r_{ij}-\xi_{IJ}\cdot r_{ik}
+\label{Dr_eq3}
+$$
+
+\end{document}

doc/doc2/Eqs/polymorphic4.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+\left\{\begin{array}{l}
+\eta_{ij}=\delta_{ij},\xi_{IJ}=0 \\
+U_{IJ}\left(r\right)=A_{IJ}\cdot\epsilon_{IJ}\cdot \left(\frac{\sigma_{IJ}}{r}\right)^q\cdot \left[B_{IJ}\cdot \left(\frac{\sigma_{IJ}}{r}\right)^{p-q}-1\right]\cdot exp\left(\frac{\sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
+V_{IJ}\left(r\right)=\sqrt{\lambda_{IJ}\cdot \epsilon_{IJ}}\cdot exp\left(\frac{\gamma_{IJ}\cdot \sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\ 
+F_{IJ}\left(X\right)=-X \\
+P_{IJ}\left(\Delta r\right)=1 \\ 
+W_{IJ}\left(r\right)=\sqrt{\lambda_{IJ}\cdot \epsilon_{IJ}}\cdot exp\left(\frac{\gamma_{IJ}\cdot \sigma_{IJ}}{r-a_{IJ}\cdot \sigma_{IJ}}\right) \\
+G_{JIK}\left(\theta\right)=\left(cos\theta+\frac{1}{3}\right)^2
+\end{array}\right.
+\end{eqnarray*} 
+
+\end{document}

doc/doc2/Eqs/polymorphic5.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+\left\{\begin{array}{l}
+\eta_{ij}=\delta_{ij},\xi_{IJ}=1 \\
+U_{IJ}\left(r\right)=\frac{D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{2S_{IJ}\left(r-r_{e,IJ}\right)}\right]\cdot f_{c,IJ}\left(r\right) \\
+V_{IJ}\left(r\right)=\frac{S_{IJ}\cdot D_{e,IJ}}{S_{IJ}-1}\cdot exp\left[-\beta_{IJ}\sqrt{\frac{2}{S_{IJ}}\left(r-r_{e,IJ}\right)}\right]\cdot f_{c,IJ}\left(r\right) \\
+F_{IJ}\left(X\right)=\left(1+X\right)^{-\frac{1}{2}} \\
+P_{IJ}\left(\Delta r\right)=exp\left(2\mu_{IK}\cdot \Delta r\right) \\ 
+W_{IJ}\left(r\right)=f_{c,IK}\left(r\right) \\
+G_{JIK}\left(\theta\right)=\gamma_{IK}\left[1+\frac{c_{IK}^2}{d_{IK}^2}-\frac{c_{IK}^2}{d_{IK}^2+\left(h_{IK}+cos\theta\right)^2}\right]
+\end{array}\right.
+\end{eqnarray*} 
+
+\end{document}

doc/doc2/Eqs/polymorphic6.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+f_{c,IJ}=\left\{\begin{array}{lr}
+1, & r\leq r_{s,IJ} \\
+\frac{1}{2}+\frac{1}{2} cos \left[\frac{\pi \left(r-r_{s,IJ}\right)}{r_{c,IJ}-r_{s,IJ}}\right], & r_{s,IJ}<r<r_{c,IJ} \\
+0, & r \geq r_{c,IJ} \\
+\end{array}\right.
+\end{eqnarray*} 
+
+\end{document}

doc/doc2/Eqs/polymorphic7.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+\left\{\begin{array}{l}
+\eta_{ij}=\delta_{ij},\xi_{IJ}=1 \\
+U_{IJ}\left(r\right)=\left\{\begin{array}{lr}
+A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right), & r\leq r_{s,1,IJ} \\
+A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right)\cdot f_{c,1,IJ}\left(r\right), & r_{s,1,IJ}<r<r_{c,1,IJ} \\
+0, & r\ge r_{c,1,IJ}
+\end{array}\right. \\
+V_{IJ}\left(r\right)=\left\{\begin{array}{lr}
+B_{IJ} \cdot exp\left(-\lambda_{2,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right), & r\le r_{s,1,IJ} \\
+B_{IJ} \cdot exp\left(-\lambda_{2,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right)+A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot & \\ ~~~~~~ f_{c,IJ}\left(r\right)\cdot \left[1-f_{c,1,IJ}\left(r\right)\right], & r_{s,1,IJ}<r<r_{c,1,IJ} \\
+B_{IJ} \cdot exp\left(-\lambda_{2,IJ}\cdot r\right)\cdot f_{c,IJ}\left(r\right)+A_{IJ}\cdot exp\left(-\lambda_{1,IJ}\cdot r\right)\cdot & \\ ~~~~~~ f_{c,IJ}\left(r\right) & r \ge r_{c,1,IJ}
+\end{array}\right. \\
+F_{IJ}\left(X\right)=\left[1+\left(\beta_{IJ}\cdot X\right)^{n_{IJ}}\right]^{-\frac{1}{2n_{IJ}}} \\
+P_{IJ}\left(\Delta r\right)=exp\left(\lambda_{3,IK}\cdot \Delta r^3\right) \\ 
+W_{IJ}\left(r\right)=f_{c,IK}\left(r\right) \\
+G_{JIK}\left(\theta\right)=1+\frac{c_{IK}^2}{d_{IK}^2}-\frac{c_{IK}^2}{d_{IK}^2+\left(h_{IK}+cos\theta\right)^2}
+\end{array}\right.
+\end{eqnarray*} 
+
+\end{document}

doc/doc2/Eqs/polymorphic8.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+f_{c,1,IJ}=\left\{\begin{array}{lr}
+1, & r\leq r_{s,1,IJ} \\
+\frac{1}{2}+\frac{1}{2} cos \left[\frac{\pi \left(r-r_{s,1,IJ}\right)}{r_{c,1,IJ}-r_{s,1,IJ}}\right], & r_{s,1,IJ}<r<r_{c,1,IJ} \\
+0, & r \geq r_{c,1,IJ} \\
+\end{array}\right.
+\end{eqnarray*} 
+
+\end{document}

doc/doc2/Eqs/polymorphic9.tex

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+\documentclass[12pt]{article}
+
+\begin{document}
+
+\begin{eqnarray*}
+\left\{\begin{array}{l}
+\eta_{ij}=1-\delta_{ij},\xi_{IJ}=0 \\
+U_{IJ}\left(r\right)=\phi_{IJ}\left(r\right) \\
+V_{IJ}\left(r\right)=1 \\
+F_{II}\left(X\right)=-2F_I\left(X\right) \\
+P_{IJ}\left(\Delta r\right)=1 \\ 
+W_{IJ}\left(r\right)=f_{K}\left(r\right) \\
+G_{JIK}\left(\theta\right)=1
+\end{array}\right.
+\end{eqnarray*} 
+
+\end{document}