tersoff fixes

diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex
index d3c91aa..1de8827 100644 (file)
--- a/posic/thesis/basics.tex
+++ b/posic/thesis/basics.tex
@@ -183,7 +183,7 @@ In order to calculate the forces the derivation of the potential with respect to
 This is gradually done in the following.
 The $n$th component of the force acting on atom $i$ is
 \begin{eqnarray}
-F_n^i & = & - \frac{\partial}{\partial x_n} \sum_{j \neq i} V_{ij} \nonumber\\
+F_n^i & = & - \frac{\partial}{\partial x_n^i} \sum_{j \neq i} V_{ij} \nonumber\\
  & = & \sum_{j \neq i} \Big( \partial_{x_n^i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber\\
 & & + f_C(r_{ij}) \big[ \partial_{x_n^i} f_R(r_{ij}) + b_{ij} \partial_{x_n^i} f_A(r_{ij}) + f_A(r_{ij}) \partial_{x_n^i} b_{ij} \big] \Big)
 \end{eqnarray}
@@ -209,11 +209,11 @@ The angle $\theta_{ijk}$ can be expressed by the atom distances with the law of
 Using the expressions \eqref{eq:d_cutoff} and \eqref{eq:d_theta} the derivation of $b_{ij}$ with respect to $x^i_n$ can be written as:
 \begin{eqnarray}
 \partial_{x^i_n} b_{ij} & = &
-- \frac{1}{2n_i} \chi_{ij} \Bigg( 1 + \beta_i^{n_i} \Bigg[ \sum_{k \ne i,j} \bigg( f_C(r_{ij}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \bigg)^{n_i} \Bigg] \Bigg)^{-\frac{1}{2n_i} - 1} \times \nonumber\\
-&& \times n_i \beta_i^{n_i} \Bigg[ \sum_{k \ne i,j} f_C(r_{ik}) \omega_{ik} \Big( 1 \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2}  \Big) \Bigg]^{n_i -1} \times \nonumber\\
-&& \times \sum_{k \ne i,j} \Bigg( \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \partial_{x^i_n} f_C(r_{ik}) + \nonumber\\
-&& + f_C(r_{ik}) \omega_{ik} (-1) \frac{c_i^2}{(d_i^2 + (h_i - \cos \theta_{ijk})^2)^2} \times \nonumber\\
-&& \times 2 \Big( h_i - \cos \theta_{ijk} \Big) \sin \theta_{ijk} \partial_{x^i_n} \theta_{ijk} \Bigg)
+- \frac{1}{2n_i} \chi_{ij} \Bigg( 1 + \beta_i^{n_i} \Bigg[ \sum_{k \ne i,j} \bigg( f_C(r_{ik}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \bigg)^{n_i} \Bigg] \Bigg)^{-\frac{1}{2n_i} - 1} \times \nonumber\\
+&& \times n_i \beta_i^{n_i} \sum_{k \ne i,j} \Bigg( \Bigg[ f_C(r_{ik}) \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2}  \Big) \Bigg]^{n_i -1} \times \nonumber\\
+&& \times \Bigg[ \omega_{ik} \Big( 1 + \frac{c_i^2}{d_i^2} - \frac{c_i^2}{d_i^2 + (h_i - \cos \theta_{ijk})^2} \Big) \partial_{x^i_n} f_C(r_{ik}) + \nonumber\\
+&& + f_C(r_{ik}) \omega_{ik} \frac{c_i^2}{(d_i^2 + (h_i - \cos \theta_{ijk})^2)^2} \times \nonumber\\
+&& \times 2 \Big( h_i - \cos \theta_{ijk} \Big) \sin \theta_{ijk} \partial_{x^i_n} \theta_{ijk} \Bigg] \Bigg)
 \end{eqnarray}