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Unformatted text preview: 10.34 – Fall 2006 Homework #10 Due Date: Friday, December 1 st , 2006 – 9 AM ** Note: Please read the entire problem set before starting, there is important information throughout, even at the very end. For this problem, you do NOT need to have the Matlab code generate all of the results for part A-E by running it once. However, it should be able to take a temperature and number of points and generate all of the desired plots for that set of inputs. The most popular way to experimentally test a proposed geometrical structure for a large molecule (such as a protein) is by X-ray crystallography. However, some proteins are hard to crystallize; for these proteins, proposed geometrical structures can be tested using nuclear magnetic resonance (NMR). NMR measures the through-space magnetic coupling between two atoms which are not directly bonded to each other; this magnetic coupling is proportional to <1/R 6 >, where R is the distance between the two atoms. The symbol < > means the Boltzmann average over all the possible molecular geometries; in the classical limit and neglecting some minor complications due to the integral over the kinetic energy we can write: 1 ⎡ − ( 1 , x 2 ,..., x N ) 3 3 3 1 V x ⎤ 1 = ∫∫∫ 6 ⋅ exp ⎢ ⎥ d x d x 1 2 ... d x N R 6 ⎤ k T Q ⎡ ⎣ R x ( 1 , x 2 ,..., x N ) ⎦ ⎣ B ⎦ where Q is the classical partition function: ∫∫∫ ⎢ ⎡ − V x ( 1 , k T x 2 ,..., x N ) ⎥ ⎤ 3 1 3 2 3 N Q = exp d x d x ... d x ⎣ B ⎦ This high-dimensional integral can be computed for a proposed structure using Monte Carlo techniques. Of course for a molecule with a large number of atoms this can be quite challenging. Here we instead ask you to compute this integral for a small molecule. Note that it is very easy to figure out the equilibrium geometry from this analytical expression for V (note V = 0 at the equilibrium geometry). We suggest you use Metropolis’s method, and start your Monte Carlo steps from the equilibrium geometry. Write a set of Matlab functions which use Monte Carlo integration to compute <1/R HH 6 > at a given Temperature, where R HH is the distance between the two H atoms in HOOH....
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 10.302 taught by Professor Clarkcolton during the Fall '04 term at MIT.
- Fall '04