# Advanced MC algorithms and free energy integration

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MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 1

Material for these lecture notes was compiled from the references below

9MIT’s 3.320 course notes

9A.P. Leach, Molecular Modeling: Principles and Applications

Advanced MC algorithms and Free energy integration

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 2

Simple sampling

¾We introduced earlier MC simulation with importance samplingand simple sampling. We will now discuss other practical methods in betweenthese.

¾Recall that in simple sampling you sample randomly and then you weight with the correct probability.

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 3

Importance sampling

¾Importance sampling is sampling with the correct Boltzmann probability and then just averaging the quantity you are sampling.

¾You can actually sample with any (not the true) Hamiltonian H 0 of the system and then correct for it in the probabilities.

¾If you sample with H0 , the states you sample have to be corrected with the relative probability of that state.

¾The relative probability is the one that you would get with the correct

Hamiltonian Hversus what you get with the Hamiltonian H 0 with which you decided to sample:

Sample with any Hamiltonian H 0

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 4

Non-Boltzmann sampling ¾This is called non-Boltzmann sampling.

¾Non-Boltzmann sampling is useful if the quantity of the system we’re interested is not determined by the the states where the system spends most of its time.

¾If we are looking at average energy (or volume or magnetization), then what we really need to know is what is the energy of the states that the system spends most of its time in.

What if the system spends only a small amount of time in certain states but those have the relevant property that we are interested to sample?

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 5

Non-Boltzmann sampling ¾For example, let’s say that this is the phase space.

 minimum. But maybe the states that live in the region marked are what you

¾With importance sampling you are being drawn towards the states near the are interested (e.g. optically active states). If you want to collect information of the optical activity of the material, you may want to build a Hamiltonian that drives you towards these states (bias towards the region of phase space where you want to get).

¾However, you will need to correct for the proper probability to get a proper ensemble. That’s non-Boltzmann sampling.

e.g. optically active states

With Boltzmann sampling, you are sampling in this red region

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 6

¾You can also use non-Boltzmann sampling to sample phase space more efficiently.If you have a phase space with a lot of local minima (blue line), you may want to define a new (flat) Hamiltonian (red line).

¾This is especially relevant if the MC simulation has some form of dynamics in it. In the blue (true) Hamiltonian, it may be hard to escape from one minimum to the next one. If you raise the potential well (red Hamiltonian), it’s going to be much easier to get out of these local minima.

¾You are essentially flattening your phase space with the new Hamiltonian and becomes easier to get out of local minima. Finally, we need to correct for that in the probability.

¾There are all kinds of MC and MD schemes that are built on this idea of lifting up the potential wells and, then, correcting the relative probability or the relative vibration frequency or the relative time you spend in each potential well.

Non-Boltzmann sampling

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 7

Non-Metropolis Monte Carlo

¾Recall that in the Metropolis step in the Markov chain the rate at which you pick the i state from the j one as a potential next step is the same as the rate at which you pick a j state from the i state. This means that the a-priori probabilities were equal.

¾You can make the rates at which you try one state from the other nonsymmetric and dependent on the Hamiltonian.

ij ji W =

In non-Metropolis MC, we allow non-equal a-priori probabilities to get less possible moves that are not accepted

MAE 715 –Atomistic Modeling of Materials

N. Zabaras (4/27/2009) 8

Non-Metropolis Monte Carlo

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