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\textsc{\Large CS 4220 / MATH 4260: Project 2}\\
Instructor: Anil Damle\\
Due: April 27, 2018\\
\subsection*{Policies}
For this project, each individual will be turning in their own write up and code. However, you may work in groups of up to three on the project, this includes helping each other write and debug code, and reading and editing each others reports. This is a bit beyond the level of collaboration allowed on the HW. Within a group you can look at ``solutions'' and discuss them in detail. However, you cannot simply copy each others. Furthermore, please list your collaborators (if any) in your project report. Part of the purpose of this project is to provide you with an opportunity to practice writing more free form reports and carefully choosing what plots, results, etc. are needed to convince us your solution is correct. Therefore, some of the goals are leading, but do not give a concrete list of exactly what has to be included. Please submit your code along with the report via CMS.\\
\hrule
\subsection*{Preamble}
For this project we are going to consider solving a simplified configuration problem. More specifically, we are going to look for minimal energy configurations of non-bonded atoms in three dimensional space using a simple model for atomic interactions. Practically, a molecular configuration problem contains many more components then we will omit for simplicity. If you are curious, I would encourage you to look further into this problem and I would be happy to point you to resources.
\subsection*{A simple model}
For the purpose of this problem, we are going to consider $N$ atoms in three dimensional Euclidean space and try to find energy minimizing configurations. A common model for the interaction of neutral atoms is the so-called Lennard-Jones potential. Specifically, given two atomic locations $x_i$ and $x_j$ in $\mathbb{R}^3$ we define $r_{ij} = \|x_i-x_j\|_2$ and the potential between atoms as
\[
V_{ij} = \frac{1}{r^{12}} - \frac{2}{r^6}.
\]
Note that more generally there are several parameters in this model that define the optimal distance between two atoms, and the optimal energy. Here, since we are using this as a model problem we have simply set those coefficients for you.
Now, given $N$ atoms defined by their locations $\{x_i\}_{i=1}^N$the problem we wish to solve is
\[
\min_{x_1,\ldots,x_N} \sum_{i