# 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

## 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