# Monte Carlo¶

• Needed keys:
• type = "MonteCarlo"
• temperature (string): System temperature. The string contains the temperature with unit.
• Optional keys:
• update_frequency (positive integer): After this number of steps of a move, delta values for this move are updated. Updates use statistics of a moves’ acceptance ratio so it is recommended to choose a sufficiently high number (>100).

If you want to perform a Monte Carlo simulation, you have to set the propagator type to "MonteCarlo". Every Monte Carlo simulations needs a temperature and a set of moves (this set can consist of a single move).

You can think of a “move” as a specific instruction to generate a new trial configuration. For example: “translate a single molecule in the system”, “rotate a molecule”, or “change the cell size”. If a move is accepted (based on an acceptance criterion), the trial configuration becomes the new configuration. If the move is rejected, the system stays in its current configuration. You can add multiple moves so that the ensemble of your choice is sampled.

Here is a list of all moves that are currently implemented in Lumol:

• Translate: Change the center of mass position of a molecule.
• Rotate: Perform a rotation of a molecule about its center of mass.
• Resize: Change the size of the simulation cell.

Currently, all Monte Carlo simulations are carried out using Metropolis acceptance criteria.

You can add all necessary information after the [simulations.propagator] label.

Naturally, Monte Carlo simulations are carried out at constant temperature which is set using the temperature key.

Different from Molecular Dynamics, Monte Carlo simulations don’t carry information about the velocities of particles. As a consequence we cannot access temperature from the kinetic energy.

Example

A sample input for a Monte Carlo simulation (in the NPT ensemble) can look like so:

[simulations.propagator]
type = "MonteCarlo"
temperature = "500 K"

moves = [
{type = "Translate", delta = "1 A", frequency = 2},
{type = "Rotate", delta = "20 deg", molecule = "CO2.xyz"},
{type = "Resize", pressure = "10 bar", delta = "3 A^3", frequency = 0.001},
]


## Moves¶

All moves are specified as inline tables. You can add a move using the type key with the name of the move.

moves accept an optional frequency parameter. During a Monte Carlo simulation it is very important that a move is selected randomly from the whole set. You can increase the chance to pick a certain move (compared to all other moves) by assigning a high frequency to it. If you don’t specify a frequency, it is set to one.

Lumol normalizes frequencies after all moves are added. The easiest way to handle frequencies is to use relative values. We will explain this below in the given examples.

Some moves can be specified to act on a single molecule or particle type. These moves accept a molecule key whose value is a path to a configuration file that can be read by chemfiles.

You can add the same move multiple times. For example, you can assign different amplitudes for different species in a mixture to make sampling more efficient. You can also use the molecule type to freeze a species by assigning a move for all but the frozen species.

If you specify a molecule, it will be selected with the following algorithm:

• Read the first frame of the file;
• If the file does not contain any bonding information, try to guess the bonds;
• Use the first molecule of the frame.

moves that use a displacement (delta) can be added with the target_acceptance key. After a specific number of times a move was called (update_frequency), we compute the acceptance ratio for the current delta value, i.e. how often the move was accepted versus how often a move was attempted. If the current acceptance is far away from the target_acceptance, we compute a new value of delta based on the current acceptance. A target_acceptance can only be used in conjunction with the update\_frequency key that specifies the frequency between updates.

Sometimes a given acceptance value cannot be achieved. Either due to limits of the adjusted delta value (it makes no sense to rotate a particle by more than 180° or to translate it by multiple values of the cutoff range) or due to the nature of the system.

To summarize, using an adjustable displacement, we can increase the efficiency of our simulation, but strictly speaking we violate detailed balance and therefore the Markov chain. To make sure you get correct results from your simulations, we recommend to use adjustable displacements only for equilibration runs. You can then take the resulting values for delta and use them for a production run, where no further adjustments are made.

Example

# Equilibration of a protein in water.
[simulations.propagator]
type = "MonteCarlo"
temperature = "300 K"
# we update the maximum displacement delta after a move was called 500 times
update_frequency = 500

moves = [
# we have much more water in the system so we want to move it more often
# hence we set the frequency = 100
# after 500 calls to this translation move, we adjust delta to get to approximately 50% acceptance
{type = "Translate", delta = "2 A", molecule = "H2O.xyz", frequency = 100, target_acceptance = 0.5},
# the single protein will be displaced only by a small distance delta = "0.05 A" during the whole run
{type = "Translate", delta = "0.05 A", molecule = "protein.pdb", frequency = 1},
]


### Translate¶

The Translate move changes the position of a single, randomly selected molecule by adding a random displacement vector to its center of mass.

• Needed keys:
• type = "Translate"
• delta (string): Maximum amplitude for displacement.
• Optional keys:
• frequency (float): Move frequency.
• molecule (string): Select only the specified molecule type. The string contains the path to the configuration file of the molecule.
• target_acceptance (float): The target acceptance for this move. Value has to be greater than zero and smaller than one. Can only be used in conjunction with update_frequency.

If the molecule key is used, the move will only apply to one molecule type. If not, the move will apply to all molecule types in the system. The delta key is the maximum magnitude of the translation vector. The conjugated string contains the value with unit of distance.

Example

[simulations.propagator]
type = "MonteCarlo"
temperature = "500 K"
moves = [
# Define a translation for all molecules in the system, including He.
{type = "Translate", delta = "1 A", frequency = 2},
# For He, pick a larger displacement with half the frequency of the
# first move. Now there is a 66% chance to pick *any* molecule
# and translate it by up to 1 A. There is a 33% chance to pick He (and only He)
# and translate it by up to 10 A.
{type = "Translate", delta = "10 A", molecule = "He.xyz"},
]


### Rotate¶

The Rotate move randomly rotates a single molecule around its center of mass.

• Needed keys:
• type = "Rotate"
• delta (string): Maximum angle for rotation.
• Optional keys:
• frequency (float): Move frequency.
• molecule (string): Select only the specified molecule type. The string contains the path to the configuration file of the molecule.
• target_acceptance (float): The target acceptance for this move. Value has to be greater than zero and smaller than one. Can only be used in conjunction with update_frequency.

If the molecule key is used, the move will only apply to one molecule type. If not, the move will apply to all molecules in the system. The delta key is the maximum angle. The conjugated string contains the value and the unit of either radians or degrees (rad or deg).

Example

[simulations.propagator]
type = "MonteCarlo"
temperature = "500 K"
moves = [
{type = "Rotate", delta = "3 deg", frequency = 2},
]


### Resize¶

The Resize move can be used to isotropically change the systems’ volume.

• Needed keys:
• type = "Resize"
• pressure (string): Target pressure.
• delta (string): Amplitude.
• Optional keys:
• frequency (float): Move frequency.
• target_acceptance (float): The target acceptance for this move. Value has to be greater than zero and smaller than one. Can only be used in conjunction with update_frequency.

For a given pressure, the volume will fluctuate during the simulation. We can use this move to sample an isobaric-isothermal ensemble. The delta key sets the maximum amplitude of the volume change in units of cubic length.

By changing the volume, we effectively change all (center of mass) positions at once. This makes Resize moves computationally expensive and we recommend to use a comparatively low value for the frequency. As a rule of thumb, for a system containing $$N$$ particles, every $$N + 1$$’th move should be a Resize move, since a single volume change is approximately as expensive as $$N$$ particle translations or rotations.

Example

# Simulation of 500 molecules.
[simulations.propagator]
type = "MonteCarlo"
temperature = "500 K"
moves = [
{type = "Translate", delta = "1 A", frequency = 250},
{type = "Rotate", delta = "20 deg", frequency = 250},
{type = "Resize", pressure = "10 bar", delta = "3 A^3", frequency = 1},
]


As mentioned above, frequencies are normalized. In this example, 501 moves consist of 250 translations, 250 rotations and a single resizing of the cell on average (remember, moves are picked at random with their respective frequency). Setting up a move set like we did in this example is very convenient and in literature you’ll often find the term “cycle” (here, 1 cycle = 501 moves) to describe such a set of moves and respective frequencies.