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soln10 - 10.34 Fall 2006 Homework#10 Solution Monte Carlo...

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10.34 – Fall 2006 Homework #10 Solution Monte Carlo Sampling within a molecular potential The purpose of this problem was to take a representative sampling of molecular configurations at a given temperature for the given molecular potential. Proper sampling from the probability of the molecule having a certain amount of energy will allow for average quantities to be estimated, such as the <1/R HH 6 > and <R HH > asked for in the problem statement. This could be done using internal coordinates of the molecules, but was essentially given to you in terms of Cartesian coordinates in the problem. The pertinent equation for the average of interest was given by the following equation. In this equation, several degrees of freedom have been removed by assuming a fixed position of one oxygen atom (O1), the other oxygen (O2) only moves along one dimension, and that one H atom (H1) is confined to a 2-D plane. This leaves a 6-D phase space to be probed, which in Cartesian coordinates can be written as the following variables: x O2 , x H1 , y H1 , x H2 , y H2 , and z H2 . 1 ∫∫∫ [ R HH ] q dx dx dy dx dz H 2 6 w ( ) O 2 H 1 H 1 H 2 dy H 2 1 = 6 ∫∫∫ w ( ) q dx O 2 dx H 1 dy H 1 dx H 2 dy H 2 dz H 2 R HH ( ) V q 2 where : w ( ) q = x O exp 2 y H 1 k T B It is easy to see that the weighting function in this case is a modified Boltzmann distribution, which was modified because the degrees of freedom were reduced. This weighting function will then be used in the Metropolis algorithm to accept or reject steps. The acceptance criterion for an uphill step in energy is given by: 2 x y H 1 rand [ ] w new ( ) q = ( O 2 0,1 < ) new exp { V new V current } Accept step q x 2 ) k T w ( ) ( O 2 B current y H 1 current Generally, a downhill step in energy is always accepted and this formula is not used (you can see that for a pure Boltzmann factor, a downhill step will always be accepted with the above formula). However, this criterion could also be used in this case for a downhill step, and for very small downhill steps in energy, there could be a small chance of rejecting the step based the geometry ratio pre-factor. Now that the acceptance criterion is defined, we can begin to take steps. In this problem, each coordinate could be changed independently by +/- 0.10 Å. Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].
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Part A: Equilibrium value at 0 K: <R_HH> = 2.5836 (Angstroms) <1/R_HH^6> = 0.0033623 (1/A^6) Number of MC steps taken: 100000 One would expect that the equilibrium 0 K distance and the <distance> at elevated are different, though the difference may be small.
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