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MM Force Fields: Part 2


Integrating over phase space


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


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


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