From 17fb57223f5b2c2f0b99d054f77fcde103ef17ad Mon Sep 17 00:00:00 2001 From: hackbard Date: Sat, 9 Dec 2006 02:33:55 +0000 Subject: [PATCH] aded appendix d_tersoff + some small changes --- posic/thesis/basics.tex | 59 +++++--------------- posic/thesis/d_tersoff.tex | 108 +++++++++++++++++++++++++++++++++++++ posic/thesis/thesis.tex | 1 + 3 files changed, 121 insertions(+), 47 deletions(-) create mode 100644 posic/thesis/d_tersoff.tex diff --git a/posic/thesis/basics.tex b/posic/thesis/basics.tex index c66e280..6addfab 100644 --- a/posic/thesis/basics.tex +++ b/posic/thesis/basics.tex @@ -94,16 +94,16 @@ Chosing $12$ as the exponent of the repulsive term it is just the square of the The constants $\epsilon$ and $\sigma$ are usually determined by fitting to experimental data. $\epsilon$ accounts to the depth of the potential well, where $\sigma$ is regarded as the radius of the particle, also known as the van der Waals radius. -Writing down the derivation of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$) +Writing down the derivative of the Lennard-Jones potential in respect to $x_i$ (the $i$th component of the distance vector ${\bf r}$) \begin{equation} \frac{\partial}{\partial x_i} U^{LJ}(r) = 4 \epsilon x_i \Big( -12 \frac{\sigma^{12}}{r^{14}} + 6 \frac{\sigma^6}{r^8} \Big) \label{eq:lj-d} \end{equation} one can easily identify $\sigma$ by the equilibrium distance of the atoms $r_e=\sqrt[6]{2} \sigma$. Applying the equilibrium distance into \eqref{eq:lj-p} $\epsilon$ turns out to be the negative well depth. -The $i$th component of the force $F^j$ on particle $j$ is obtained by +The $i$th component of the force is given by \begin{equation} -F_i^j = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.} +F_i = - \frac{\partial}{\partial x_i} U^{LJ}(r) \, \textrm{.} \label{eq:lj-f} \end{equation} @@ -173,61 +173,26 @@ It is of the form: b_{ij} & = & \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} \\ \zeta_{ij} & = & \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) \\ g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \\ -b_{ij} & = & \chi_{ij} \Big( 1 + \beta_i^{n_i} \Big[ \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} \big[ 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \big] \Big] \Big)^{-1/2n_i} \end{eqnarray} where $\theta_{ijk}$ is the bond angle between bonds $ij$ and $ik$. This is illustrated in Figure \ref{img:tersoff_angle}. \printimg{!h}{width=8cm}{tersoff_angle.eps}{Angle between bonds of atoms $i,j$ and $i,k$.}{img:tersoff_angle} -In order to calculate the forces the derivation of the potential with respect to $x^i_n$ (the $n$th component of the position vector of atom $i$ $\equiv$ ${\bf r}_i$) has to be known. -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^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) \textrm{ .} -\end{eqnarray} -For the implementation it is helpful to seperate the two and three body terms. -\begin{eqnarray} -F_n^i & = & \sum_{j \neq i} \Big( f_R(r_{ij}) \partial_{x_n^i} f_C(r_{ij}) + f_C(r_{ij}) \partial_{x_n^i} f_R(r_{ij}) \Big) + \nonumber\\ -& + & \sum_{j \neq i} \Big( \partial_{x_n^i} f_C(r_{ij}) b_{ij} f_A(r_{ij}) + f_C(r_{ij}) \big[ b_{ij} \partial_{x_n^i} f_A(r_{ij}) + f_A(r_{ij}) \partial_{x_n^i} b_{ij} \big] \Big) -\end{eqnarray} -The cutoff function $f_C$ derivated with repect to $x^i_n$ is -\begin{equation} -\partial_{x^i_n} f_C(r_{ij}) = - - \frac{1}{2} \sin \Big( \pi (r_{ij} - R_{ij}) / (S_{ij} - R_{ij}) \Big) \frac{\pi (x^i_n - x^j_n)}{(S_{ij} - R_{ij}) r_{ij}} -\label{eq:d_cutoff} -\end{equation} -for $R_{ij} < r_{ij} < S_{ij}$ and otherwise zero. -The derivations of the repulsive and attractive part are: -\begin{eqnarray} -\partial_{x_n^i} f_R(r_{ij}) & = & - \lambda_{ij} \frac{x_n^i - x_n^j}{r_{ij}} A_{ij} \exp (-\lambda_{ij} r_{ij})\\ -\partial_{x_n^i} f_A(r_{ij}) & = & \mu_{ij} \frac{x_n^i - x_n^j}{r_{ij}} B_{ij} \exp (-\mu_{ij} r_{ij}) \textrm{ .} -\end{eqnarray} -The cosine of the angle $\theta_{ijk}$ can be expressed by the atom distances with the law of cosines -\begin{equation} -\cos \theta_{ijk} = \Big( (r_{ij}^2 + r_{ik}^2 - r_{jk}^2)/(2 r_{ij} r_{ik}) \Big) -\end{equation} -or by the definition of the scalar product +Here comes an explanation, energy per bond monotonically decreasing with the amount of bonds and so on and so on \ldots + +The force acting on atom $i$ is given by the derivative of the potential energy. +For a three body potential ($V_{ij} \neq V{ji}$) the derivation is of the form \begin{equation} -\cos \theta_{ijk} = \frac{\vec{r}_{ij} \vec{r}_{ik}}{r_{ij} r_{ik}} \textrm{ .} +\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .} \end{equation} -The derivation of the angle $\theta_{ijk}$ with respect to $x^i_n$ is given by +The force is then given by \begin{equation} -\partial_{x^i_n} \cos \theta_{ijk} = \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ik}}{r_{ij}} \Big) (x_n^i - x_n^j) + \Big( r_{ik} r_{ij} - \vec{r}_{ij} \vec{r}_{ik} \frac{r_{ij}}{r_{ik}} \Big) (x_n^i - x_n^k) -\label{eq:d_costheta} +F^i = - \nabla_{{\bf r}_i} E \textrm{ .} \end{equation} +The details of the Tersoff potential derivative can be seen in appendix \ref{app:d_tersoff}. -Using the expressions \eqref{eq:d_cutoff} and \eqref{eq:d_costheta} 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_{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) \Bigg]^{n_i} \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} \frac{2 c_i^2 (h_i - \cos \theta_{ijk})}{(d_i^2 + (h_i - \cos \theta_{ijk})^2)^2} \partial_{x^i_n} \cos \theta_{ijk} \Bigg] -\end{eqnarray} - +And here comes why we use it. Advantages and disadvantages compared to other interaction potentials, maybe this is best at the very end of all potentials \ldots \subsubsection{The Brenner potential} diff --git a/posic/thesis/d_tersoff.tex b/posic/thesis/d_tersoff.tex new file mode 100644 index 0000000..d1f0488 --- /dev/null +++ b/posic/thesis/d_tersoff.tex @@ -0,0 +1,108 @@ +\chapter{Derivative of the three body Tersoff potential} +\label{app:d_tersoff} + + \section{Form of the Tersoff potential and its derivative} + +The Tersoff potential is of the form +\begin{eqnarray} +E & = & \sum_i E_i = \frac{1}{2} \sum_{i \ne j} V_{ij} \textrm{ ,} \\ +V_{ij} & = & f_C(r_{ij}) [ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) ] \textrm{ .} +\end{eqnarray} +The repulsive $f_R$ and attractive $f_A$ part is given by +\begin{eqnarray} +f_R(r_{ij}) & = & A_{ij} \exp (- \lambda_{ij} r_{ij} ) \textrm{ ,} \\ +f_A(r_{ij}) & = & -B_{ij} \exp (- \mu_{ij} r_{ij} ) \textrm{ .} +\end{eqnarray} +The bond order function $b_{ij}$ is +\begin{equation} +b_{ij} = \chi_{ij} (1 + \beta_i^{n_i} \zeta^{n_i}_{ij})^{-1/2n_i} +\end{equation} +with +\begin{eqnarray} +\zeta_{ij} & = & \sum_{k \ne i,j} f_C (r_{ik}) \omega_{ik} g(\theta_{ijk}) \textrm{ ,}\\ +g(\theta_{ijk}) & = & 1 + c_i^2/d_i^2 - c_i^2/[d_i^2 + (h_i - \cos \theta_{ijk})^2] \textrm{ .} +\end{eqnarray} +The cutoff function $f_C$ is taken to be +\begin{equation} +f_C(r_{ij}) = \left\{ + \begin{array}{ll} + 1, & r_{ij} < R_{ij} \\ + \frac{1}{2} + \frac{1}{2} \cos \Big[ \pi (r_{ij} - R_{ij})/(S_{ij} - R_{ij}) \Big], & + R_{ij} < r_{ij} < S_{ij} \\ + 0, & r_{ij} > S_{ij} + \end{array} \right. +\end{equation} +with $\theta_{ijk}$ being the bond angle between bonds $ij$ and $ik$ as shown in Figure \ref{img:tersoff_angle}.\\ +\\ +For a three body potential, if $V_{ij}$ is not equal to $V_{ji}$, the derivative is of the form +\begin{equation} +\nabla_{{\bf r}_i} E = \frac{1}{2} \big[ \sum_j ( \nabla_{{\bf r}_i} V_{ij} + \nabla_{{\bf r}_i} V_{ji} ) + \sum_k \sum_j \nabla_{{\bf r}_i} V_{jk} \big] \textrm{ .} +\end{equation} +In the following all the necessary derivatives to calculate $\nabla_{{\bf r}_i} E$ are done. + + \section{Derivative of $V_{ij}$ with respect to ${\bf r}_i$} + +\begin{eqnarray} +\nabla_{{\bf r}_i} V_{ij} & = & \nabla_{{\bf r}_i} f_C(r_{ij}) \big[ f_R(r_{ij}) + b_{ij} f_A(r_{ij}) \big] + \nonumber \\ + & & + f_C(r_{ij}) \big[ \nabla_{{\bf r}_i} f_R(r_{ij}) + b_{ij} \nabla_{{\bf r}_i} f_A(r_{ij}) + f_A(r_{ij}) \nabla_{{\bf r}_i} b_{ij} \big] +\end{eqnarray} +\begin{eqnarray} +\nabla_{{\bf r}_i} f_R(r_{ij}) & = & A_{ij} \lambda_{ij} \frac{{\bf r}_{ij}}{r_{ij}} \exp(-\lambda_{ij} r_{ij}) \\ +\nabla_{{\bf r}_i} f_A(r_{ij}) & = & - B_{ij} \mu_{ij} \frac{{\bf r}_{ij}}{r_{ij}} \exp(-\mu_{ij} r_{ij}) +\end{eqnarray} +\begin{equation} +\nabla_{{\bf r}_i} f_C(r_{ij}) = \left\{ + \begin{array}{ll} + \frac{1}{2} \sin \Big( \frac{\pi(r_{ij}-R_{ij})}{S_{ij}-R_{ij}} \Big) \frac{\pi}{S_{ij}-R_{ij}} \frac{{\bf r}_{ij}}{r_{ij}}, & R_{ij} < r_{ij} < S_{ij} \\ + 0, & \textrm{else.} + \end{array} \right. +\end{equation} +\begin{eqnarray} +\nabla_{{\bf r}_i} b_{ij} &= & - \frac{\chi_{ij}}{2} (1+\beta^{n_i} \zeta_{ij}^{n_i})^{-\frac{1}{2n_i}-1} \beta^{n_i} \zeta_{ij}^{n_i-1} \nabla_{{\bf r}_i} \zeta_{ij} \\ +\nabla_{{\bf r}_i} \zeta_{ij} & = & \sum_{k \neq i,j} \big( g(\theta_{ijk}) \nabla_{{\bf r}_i} f_C(r_{ik}) + f_C(r_{ik}) \nabla_{{\bf r}_i} g(\theta_{ijk}) \big) \\ +\nabla_{{\bf r}_i} g(\theta_{ijk}) & = & - \frac{2(h_i-\cos\theta_{ijk})c_i^2}{\big[d_i^2 + (h_i - \cos\theta_{ijk})^2\big]^2} \nabla_{{\bf r}_i} (\cos\theta_{ijk}) +\end{eqnarray} +\begin{eqnarray} +\nabla_{{\bf r}_i} \cos \theta_{ijk} & = & \nabla_{{\bf r}_i} \Big( \frac{{\bf r}_{ij} {\bf r}_{ik}}{r_{ij} r_{ik}} \Big) \nonumber \\ + & = & \Big[ \frac{\cos\theta_{ijk}}{r_{ij}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ij} + \Big[ \frac{\cos\theta_{ijk}}{r_{ik}^2} - \frac{1}{r_{ij} r_{ik}} \Big] {\bf r}_{ik} +\end{eqnarray} + + \section{Derivative of $V_{ji}$ with respect to ${\bf r}_i$} + +\begin{eqnarray} +\nabla_{{\bf r}_i} V_{ji} & = & \nabla_{{\bf r}_i} f_C(r_{ji}) \big[ f_R(r_{ji}) + b_{ji} f_A(r_{ji}) \big] + \nonumber \\ + & & + f_C(r_{ji}) \big[ \nabla_{{\bf r}_i} f_R(r_{ji}) + b_{ji} \nabla_{{\bf r}_i} f_A(r_{ji}) + f_A(r_{ji}) \nabla_{{\bf r}_i} b_{ji} \big] +\end{eqnarray} +\begin{eqnarray} +\nabla_{{\bf r}_i} f_R(r_{ji}) & = & - A_{ji} \lambda_{ji} \frac{{\bf r}_{ji}}{r_{ji}} \exp(-\lambda_{ji} r_{ji}) = \nabla_{{\bf r}_i} f_R(r_{ij} \\ +\nabla_{{\bf r}_i} f_A(r_{ji}) & = & + B_{ji} \mu_{ji} \frac{{\bf r}_{ji}}{r_{ji}} \exp(-\mu_{ji} r_{ji}) = \nabla_{{\bf r}_i} f_A(r_{ij}) +\end{eqnarray} +\begin{equation} +\nabla_{{\bf r}_i} f_C(r_{ij}) = f_C(r_{ij}) = \left\{ + \begin{array}{ll} + - \frac{1}{2} \sin \Big( \frac{\pi(r_{ji}-R_{ji})}{S_{ji}-R_{ji}} \Big) \frac{\pi}{S_{ji}-R_{ji}} \frac{{\bf r}_{ji}}{r_{ji}}, & R_{ji} < r_{ji} < S_{ji} \\ + 0, & \textrm{else.} + \end{array} \right. +\end{equation} +\begin{eqnarray} +\nabla_{{\bf r}_i} b_{ji} &= & - \frac{\chi_{ji}}{2} (1+\beta^{n_j} \zeta_{ji}^{n_j})^{-\frac{1}{2n_j}-1} \beta^{n_j} \zeta_{ji}^{n_j-1} \nabla_{{\bf r}_i} \zeta_{ji} \\ +\nabla_{{\bf r}_i} \zeta_{ji} & = & \sum_{k \neq j,i} \big( g(\theta_{jik}) \nabla_{{\bf r}_i} f_C(r_{jk}) + f_C(r_{jk}) \nabla_{{\bf r}_i} g(\theta_{jik}) \big) \nonumber \\ + & = & \sum_{k \neq j,i} f_C(r_{jk}) \nabla_{{\bf r}_i} g(\theta_{jik}) \quad \Big(\textrm{Reason: }\nabla_{{\bf r}_i} f_C(r_{jk}) = 0\Big) \\ +\nabla_{{\bf r}_i} g(\theta_{jik}) & = & - \frac{2(h_j-\cos\theta_{jik})c_j^2}{\big[d_j^2 + (h_j - \cos\theta_{jik})^2\big]^2} \nabla_{{\bf r}_i} (\cos\theta_{jik}) +\end{eqnarray} +\begin{eqnarray} +\nabla_{{\bf r}_i} \cos \theta_{jik} & = & \nabla_{{\bf r}_i} \Big( \frac{{\bf r}_{ji} {\bf r}_{jk}}{r_{ji} r_{jk}} \Big) \nonumber \\ + & = & \frac{1}{r_{ji} r_{jk}} {\bf r}_{jk} - \frac{\cos\theta_{jik}}{r_{ji}^2} {\bf r}_{ji} +\end{eqnarray} + + \section{Derivative of $V_{jk}$ with respect to ${\bf r}_i$} + +\begin{eqnarray} +\nabla_{{\bf r}_i} V_{jk} & = & f_C(r_{jk}) f_A(r_{jk}) \nabla_{{\bf r}_i} b_{jk} \\ +\nabla_{{\bf r}_i} b_{jk} & = & - \frac{\chi_{ji}}{2} (1+\beta^{n_j} \zeta_{jk}^{n_j})^{-\frac{1}{2n_j}-1} \beta^{n_j} \zeta_{jk}^{n_j-1} \nabla_{{\bf r}_i} \zeta_{jk} \\ +\nabla_{{\bf r}_i} \zeta_{jk} & = & \sum_{l \neq j,k} \big( g(\theta_{jkl}) \nabla_{{\bf r}_i} f_C(r_{jl}) + f_C(r_{jl}) \nabla_{{\bf r}_i} g(\theta_{jkp}) \big) \nonumber \\ + & = & f_C(r_{ji}) \nabla_{{\bf r}_i} g(\theta_{jki}) + g(\theta_jki) \nabla_{{\bf r}_i} f_C(r_{ji}) \\ +\nabla_{{\bf r}_i} g(\theta_{jki}) & = & - \frac{2(h_j-\cos\theta_{jki})c_j^2}{\big[d_j^2 + (h_j - \cos\theta_{jki})^2\big]^2} \nabla_{{\bf r}_i} (\cos\theta_{jki}) \\ +\nabla_{{\bf r}_i} \cos \theta_{jki} & = & \nabla_{{\bf r}_i} \Big( \frac{{\bf r}_{jk} {\bf r}_{ji}}{r_{jk} r_{ji}} \Big) \nonumber \\ + & = & \frac{1}{r_{jk} r_{ji}} {\bf r}_{jk} - \frac{\cos\theta_{jki}}{r_{ji}^2} {\bf r}_{ji} +\end{eqnarray} diff --git a/posic/thesis/thesis.tex b/posic/thesis/thesis.tex index 199af2b..643c9c2 100644 --- a/posic/thesis/thesis.tex +++ b/posic/thesis/thesis.tex @@ -61,6 +61,7 @@ \include{summary_outlook} \appendix{} +\include{d_tersoff} \include{publications} \backmatter{} -- 2.20.1