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MM Force Fields: Part 2

Abstract

Integrating over phase space

Note

Expectation value: The weighted average of the probabilistic distribution, to give a statistically expected value.

This can be considered a Boltzmann weighted distribution, however we have the input of both the momentum and the position

\[ P(r)=e^{\frac{-E(q,p)}{k_BT}} \]

Metropolis Monte Carlo

A lower energy geometry will ultimately have a higher population distribution

  1. Pick a starting geometry, propose a change and calculate that change’s energy
    • \(r_1\ce{->} r_2\)
  2. If the energy of the new geometry is less than the energy of the initial geometry, you accept the change
    • if: \(-E(r_1)<-E(r_2)\)
  3. Otherwise you calculate the probability and compare it to a randomly generated number and if it’s greater than that, you can also accept it
    • elif: \(e^{\frac{-U(r_1)-U(r_2)}{k_BT}}>random \#\:\varepsilon[0,1]\)
  4. Otherwise you reject the change and iterate again

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Simulated Annealing

We start the Monte Carlo simulation at a really high temperature and gradually decrease it. If we do this infinitely slow enough, we should easily be able to fall into the global minima

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Molecular Dynamics (MD)

We solve classical (newtonian) equations of motion to calculate the behaviour of the atoms

  • we use \(F=ma=-dU(r)/dr\)

The logic goes that if we do this for long enough, we’ll sample phase space in a weighted way

High Energy Quenching

Make a list of minima by randomly heating the system and occasionally cooling it rapidly to see where it lands

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