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A C++ library to implement the "virtual-move" Monte Carlo algorithm for approximating the dynamical evolution of systems of strongly interacting overdamped particles.

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Copyright © 2015 Lester Hedges


A simple C++ library to implement the "virtual-move" Monte Carlo (VMMC) algorithm of Steve Whitelam and Phill Geissler, see:

We introduce a “virtual-move” Monte Carlo algorithm for systems of pairwise-interacting particles. This algorithm facilitates the simulation of particles possessing attractions of short range and arbitrary strength and geometry, an important realization being self-assembling particles endowed with strong, short-ranged, and angularly specific (“patchy”) attractions. Standard Monte Carlo techniques employ sequential updates of particles and can suffer from low acceptance rates when attractions are strong. In this event, collective motion can be strongly suppressed. Our algorithm avoids this problem by proposing simultaneous moves of collections (clusters) of particles according to gradients of interaction energies. One particle first executes a “virtual” trial move. We determine which of its neighbours move in a similar fashion by calculating individual bond energies before and after the proposed move. We iterate this procedure and update simultaneously the positions of all affected particles. Particles move according to an approximation of realistic dynamics without requiring the explicit computation of forces and without the step size restrictions required when integrating equations of motion. We employ a size- and shape-dependent damping of cluster movements, motivated by collective hydrodynamic effects neglected in simple implementations of Brownian dynamics.

Our primary goal is to make VMMC accessible to a wider audience, for whom the time required to code the algorithm poses a significant barrier to using the method. This allows the user to focus on model development.

The animation below shows a comparison of the dynamics generated by traditional single-particle Monte Carlo (SPMC) and the VMMC algorithm for a periodic two-dimensional square-well fluid. The model system consists of particles interacting via strong, short-ranged isotropic interactions. Due to the suppression of collective particle rearrangements, SPMC results in the slow Ostwald ripening of isolated clusters. In contrast, VMMC facilitates the diffusion and coalescence of particle clusters, resulting in a long-time dynamics that is dominated by the motion of a single large cluster. Both trajectories represent one billion trial moves of the respective algorithms, with the system initialised with a random, non-overlapping, particle configuration in each case.

Comparison of the single particle and virtual-move Monte Carlo algorithms.

The VMMC algorithm works by proposing the move of a single, randomly chosen, "seed" particle. If, following the move, the change in the energy of interaction between the particle and its neighbours is unfavourable, then those neighbours are recruited and moved in concert. This process is iterated recursively for each new recruit until no further particles show a tendency to move.

The animations below illustrate example VMMC translation and rotation moves taken from a real simulation. Red indicates the most recent recruit to the cluster, orange indicates the nearest neighbour to which link formation is currently being tested, and green indicates those particles that have been accepted as part of the cluster move. The animations show how a recursive depth-first search is used to iteratively link particles to the cluster. Particles are linked according to probabilities based on the pair interaction energy differences following the forward and reverse virtual move of each recruit. Computation of the reverse move is required to enforce superdetailed balance, thus ensuring that the probability of a given particle pushing or pulling on the cluster is the same.

Example VMMC translation move. Example VMMC rotation move.


A Makefile is included for building and installing LibVMMC.

To compile LibVMMC, then install the library, documentation, and demos:

$ make build
$ make install

By default, the library installs to /usr/local. Therefore, you may need admin priveleges for the final make install step above. An alternative is to change the install location:

$ make PREFIX=MY_INSTALL_DIR install

Further details on using the Makefile can be found by running make without a target, i.e.

$ make

Compiling and linking

To use LibVMMC with a C/C++ code first include the LibVMMC header file somewhere in the code.

#include <vmmc/VMMC.h>

Then to compile, we can use something like the following:

$ g++ -std=c++11 example.cpp -lvmmc

This assumes that we have used the default install location /usr/local. If we specify an install location, we would use a command more like the following:

$ g++ -std=c++11 example.cpp -I/my/path/include -L/my/path/lib -lvmmc

Note that the -std=c++11 compiler flag is needed for std::function and std::random.


LibVMMC uses the Mersenne Twister pseudorandom number generator. A C++11 implementation using std::random is included as a bundled header file, MersenneTwister.h. See the source code or generate Doxygen documentation with make doc for details on how to use it.

Callback functions

LibVMMC works via several user-defined callback functions that abstract model specific details, such as the pair potential. We make use of C++11's std::function to provide a general-purpose function wrapper, i.e. the callbacks can be free functions, member functions, etc. These callbacks allow LibVMMC to be blind to the implementation of the model, as well as the model to be blind to the details of the VMMC algorithm. The generic nature of the function wrapper provides great flexibility to the user, freeing them from a specific design choice for the model in hand. It is possible to glue together components written in different ways, or to use the callbacks themselves as C/C++ wrappers to external libraries.

An alternative version of LibVMMC that shows how to achieve the same callback functionality using a pure abstract Model base class can be found in the pure-abstract branch. While this provides a cleaner interface, the additional flexibility provided by std::function more than offsets the minimal performance cost.

Details of the callback prototypes are given below (where typedef has been used to simplify their declaration). The const keyword is used to aid readability and to prevent the user from modifying particle positions and orientations. (For those unfamiliar with C/C++, const double* p_double means that you can't modify the data pointed to by p_double although you can change what p_double points to.) For simplicity we make use of the "pointer to type" representation of array data, i.e. all pointer arguments in the callback prototypes represent arrays.

Particle energy

Calculate the total pair interaction energy felt by a particle.

typedef std::function<double (unsigned int index, const double* position,
    const double* orientation)> EnergyCallback;

index = The particle index.

position = An x, y, z (or x, y in 2D) coordinate vector for the particle.

orientation = The particle's orientation unit vector.

This callback function is currently somewhat redundant since it is possible to achieve the same outcome by combining the PairEnergyCallback and InteractionsCallback functions described below.

Pair energy

Calculate the pair interaction between two particles.

typedef std::function<double (unsigned int index1, const double* position1,
    const double* orientation1, unsigned int index2, const double* position2,
    const double* orientation2)> PairEnergyCallback;

index1 = The index of the first particle.

position1 = The coordinate vector of the first particle.

orientation1 = The orientation unit vector of the first particle.

index2 = The index of the second particle.

position2 = The coordinate vector of the second particle.

orientation2 = The orientation unit vector of the second particle.


Determine the interactions for a given particle.

typedef std::function<unsigned int (unsigned int index, const double* position,
    const double* orientation, unsigned int* interactions)> InteractionsCallback;

index = The index of the particle.

position = The coordinate vector of the particle.

orientation = The orientation unit vector of the particle.

interactions = An array to store the indices of the interactions.


Apply any post-move updates, e.g. update cell lists, or neighbour lists.

typedef std::function<void (unsigned int index, const double* position,
    const double* orientation)> PostMoveCallback;

index = The index of the particle.

position = The coordinate vector of the particle following the move.

orientation = The orientation unit vector of the particle following the move.

Non-pairwise energy (optional)

Test for non-pairwise energy contributions, such as interactions with a surface or external field.

typedef std::function<double (unsigned int, const double* position,
    const double* orientation)> NonPairwiseCallback;

index = The index of the particle.

position = The coordinate vector of the particle.

orientation = The orientation unit vector of the particle.

Boundary condition (optional)

Test for a custom boundary condition. This should return true if the particle moves outside of the boundary following the virtual move. An example showing how to implement custom boundary conditions is provided with the demonstration code.

typedef std::function<bool (unsigned int index, const double* position,
    const double* orientation)> BoundaryCallback;

index = The index of the particle.

position = The coordinate vector of the particle following the move.

orientation = The orientation unit vector of the particle following the move.

Assigning a callback

Using the callbacks above it is easy to create a function wrapper to whatever, e.g.

vmmc::EnergyCallback energyCallback = computeEnergy;

if computeEnergy were a free function, or

Foo foo;
using namespace std::placeholders;
vmmc::EnergyCallback energyCallback = std::bind(&Foo::computeEnergy, foo, _1, _2, _3);

if computeEnergy were instead a member of some object called Foo.

For simplicity we provide a container for callback functions. This simplifies assignment and makes it possible to pass a single callback argument to the constructor of the VMMC object.

struct CallbackFunctions
    EnergyCallback energyCallback;
    PairEnergyCallback pairEnergyCallback;
    InteractionsCallback InteractionsCallback;
    PostMoveCallback postMoveCallback;
    NonPairwiseCallback nonPairwiseCallback;
    BoundaryCallback boundaryCallback;

In the second example above, initialisation of the energyCallback function would become

Foo foo;
using namespace std::placeholders;
vmmc::CallbackFunctions callbacks;
callbacks.energyCallback = std::bind(&Foo::computeEnergy, foo, _1, _2, _3);

The VMMC object

To use LibVMMC you will want to create an instance of the VMMC object. This has the following constructor:

VMMC(unsigned int nParticles, unsigned int dimension, double* coordinates,
    double* orientations, double maxTrialTranslation, double maxTrialRotation,
    double probTranslate, double referenceRadius, unsigned int maxInteractions,
    double* boxSize, bool* isIsotropic, bool isRepulsive,
    const CallbackFunctions& callbacks);

nParticles = The number of particles in the simulation box, \(N\).

dimension = The dimension of the simulation box (either 2 or 3).

coordinates = An array containing coordinates for all of the particles in the system, i.e. \(x_1, y_1, z_1, x_2, y_2, z_2, \ldots , x_N, y_N, z_N.\) Coordinates should run from 0 to the box size in each dimension.

orientations = An array containing orientations (unit vectors) for all of the particles in the system, i.e. \(\hat{x}_1, \hat{y}_1, \hat{z}_1, \hat{x}_2, \hat{y}_2, \hat{z}_2, \ldots , \hat{x}_N, \hat{y}_N, \hat{z}_N.\) In the case of particles interacting via an isotropic potential, the particle orientations are entirely redundant, i.e. the orientation has no effect on the potential. This allows the use of a single set of callback functions for models with both isotropic and anisotropic potentials.

maxTrialTranslation = The maximum trial translation, in units of the particle diameter (or typical particle size).

maxTrialRotation = The maximum trial rotation in radians.

probTranslate = The probability of attempting a translation move (relative to rotations). Along with maxTrialTranslation and maxTrialRotation, probTranslate can be tuned to enforce an approximate Stokes drag. An excellent and detailed explanation of how this may be applied in practice can be found here.

referenceRadius = A reference radius for computing the approximate hydrodynamic damping factor, e.g. the radius of a typical particle in the system. When modeling a molecular system the reference radius should indicate the approximate radius of a single molecular unit.

maxInteractions = The maximum number of pair interactions that an individual particle can make. This will be used to resize LibVMMC's internal data structures and the user should assert that this limit isn't exceed in the InteractionsCallback function. The number can be chosen from the symmetry of the system, e.g. if particles can only make a certain number of patchy interactions, or by estimating the average number of neighbours within the interaction volume around a particle.

boxSize = The base length of the simulation box in each dimension.

isIsotropic = Whether the potential of each particle is isotropic. The handling of rotational moves is slightly different for moves seeded from isotropic particles, e.g. spheres, since the rotation of the seed causes no change in energy. This boolean array allows LibVMMC to handle mixed-potential systems.

isRepulsive = Whether the potential has finite energy repulsions. This should also be set to true when particle interactions contain a mixture of hard core overlaps and finite repulsions.

callbacks = The callback function container.

C-style arrays

The VMMC object constructor and callback functions use C-style arrays as arguments for simplicity and generality. This (hopefully) makes it as easy as possible for a user unfamiliar with C++ to make use of LibVMMC (although everyone should take time to learn std::vector). We can also exploit the fact that the C++ standard imposes that std::vector elements are contiguous, which allows std::vector containers to be passed as naked arrays.

For example, if we have some function called foo that accepts a C-style double array as an argument,

void foo(double* arr);        // or alternatively, void foo(double[]);

then both of the following are valid function calls

// C style
double c_arr[10];

// C++ style
std::vector<double> cpp_arr(10);

Internally, LibVMMC uses std::vector containers for its data structures, with data passed to the callback functions in the manner described above.


Energies returned by the callback functions should be in units of Boltzmann's constant multiplied by temperature, i.e. \(k_{\rm{B}}T\). Distances are measured in units of the particle (or molecular) diameter, \(\sigma\). In the demonstration code we take the statistical mechanician's prerogative of setting both \(k_{\rm{B}}T\) and \(\sigma\) equal to one.

Executing a virtual move

Once an instance of the VMMC object is created, e.g.

VMMC(...) vmmc;

then a single trial move can be executed as follows:


To perform 1000 trial moves:


The same can be achieved by using the overloaded ++ and += operators, i.e. vmmc++ for a single step, and vmmc += 1000 for 1000 steps.


The following example codes showing how to interface with LibVMMC are included in the demos directory.

When run, each of the demos output a trajectory file,, and a TcL script, vmd.tcl, that can be used to set camera and particle attributes and to draw the periodic simulation box when visualising the trajectory with VMD. To generate and view a trajectory, run, e.g.

$ ./demos/square_wellium
$ vmd -e vmd.tcl

Note that the trajectories are intended to be used for visualisation purposes only. The atomic coordinates are not saved to file with enough accuracy for them to reliably be used as restart files, i.e. overlaps may occur.

The following animation shows example trajectories generated by a selection of the demos.

Example VMMC trajectories.

The demo code also illustrates how to implement efficient, dynamically updated cell lists. See demos/src/CellList.h and demos/src/CellList.cpp for implementation details. If you are simulating a system of highly size asymmetric particles, then it might be preferable to search for interactions using a more efficient data structure, such as a bounding volume hierarchy.

Also included in the demos/python directory are examples showing how to interface with Python code via the Python C API. Note that this code is intended to be used for illustrative purposes and avoids the use of additional dependencies (e.g. NumPy). As such, it is likely that performance could be significantly improved.


A full test suite is forthcoming. This will allow a detailed comparison between VMMC and standard single-move Monte Carlo (SPMC) for various model systems at a range of state points.

Shown below are time-averaged pair distribution functions for Lennard-Jonesium and the square-well fluid taken from configurations equilibrated using the demo codes outlined above (although run for 10 times as long). In both cases the equilibrated structures are indistinguishable from those generated by SPMC. Note that the pair distribution functions don't converge to one at large particle separations since we are not considering a bulk system, rather a finite cluster in a background vapour. See the demos codes demos/lennard_jonesium.cpp and demos/square_wellium.cpp for details of the interaction parameters (Lennard-Jonesium is sampled in the liquid phase, the square-well fluid is sampled in the crystal (FCC/HCP) phase).

Comparison of pair distribution functions for configurations equilibrated with SPMC and VMMC.

Defining a model

The demo code illustrates a simple way of defining and handling different model potentials. A base class, Model, is used to declare default functionality and callbacks. Derived classes, such as LennardJonesium, are used to implement the model specific pair potential, which is declared as a virtual method in the base class.

Declaring a new user-defined model should be as easy as creating a UserModel class with public inheritance from the Model base class, then overriding the virtual computePairEnergy method. The LennardJonesium, SquareWellium, and PatchyDisc classes will serve as useful templates.

Pure isotropic systems

The default build of LibVMMC provides support for systems of isotropic and anisotropic particles, or mixtures of both. However, in the case of pure isotropic systems, e.g. spherical particles interacting via a spherically symmetric potential, such as the square-well fluid, particle orientations are entirely redundant since they have no bearing on the potential. This means that there is no need to pass orientations as arguments to callback functions, or to update particle orientations during VMMC trial moves.

We provide preprocessor directives that allow LibVMMC to be compiled as an optimised library for pure isotropic systems. This can be achieved as follows:


The isotropic version of LibVMMC provides a simplified set of callback functions that require no particle orientations. For example, the PairEnergyCallback becomes

typedef std::function<double (unsigned int index1, const double* position1,
    unsigned int index2, const double* position2)> PairEnergyCallback;

In addition, the VMMC object no longer needs the orientations or isIsotropic arrays to be passed to its constructor, which is simplified to

VMMC(unsigned int nParticles, unsigned int dimension, double* coordinates,
    double maxTrialTranslation, double maxTrialRotation, double probTranslate,
    double referenceRadius, unsigned int maxInteractions, double* boxSize,
    bool isRepulsive, const CallbackFunctions& callbacks);

The demo code shows how preprocessor directives can be used to provide support for either version of the library, e.g. for the default computeEnergy callback defined in the Model class, we have

    virtual double computeEnergy(unsigned int, const double*, const double*);
    virtual double computeEnergy(unsigned int, const double*);

The pure isotropic version of LibVMMC can provide a significant performance gain when executing rotations of large clusters in isotropic systems.

Note that the demo patchy_disc.cpp will not compile against the isotropic version of the library since the model is anisotropic and requires that particle orientations are passed to its callback functions.



In aid of generality there are several sources of redundancy that impact the efficiency of the VMMC implementation. As written, LibVMMC performs around 3-4 times worse than a fully optimised VMMC code for square-well fluids. A few efficiency considerations are listed below in case the user wishes to modify the VMMC source code in order to improve performance.



I'm more than happy to answer questions, or to help set up a custom model system, but please RTFM before getting in touch. I've tried hard to add extensive comments, documentation, and examples, so it's likely that I won't respond if you've not taken the time to read these.

To help new users, some common pitfalls are listed below.

Citing LibVMMC

If you make use of LibVMMC in any published research please cite the canonical VMMC reference:

and the symmetrised version of the algorithm described in the appendix of

Please also include a citation to the official LibVMMC page:

A properly formatted BibTex file is provided here.


Please be aware that this a working repository so the code should be used at your own risk. At present the code is being tested so expect that it will be updated fairly frequently with additional features and performance enhancements.