Molecular dynamics

Typical computer simulations involve moving the atoms around, either to optimize a structure (energy minimization) or to do molecular dynamics. This chapter discusses molecular dynamics, energy minimization algorithms will be discussed in the ase.optimize section.

A molecular dynamics object will operate on the atoms by moving them according to their forces - it integrates Newton’s second law numerically. A typical molecular dynamics simulation will use the Velocity Verlet dynamics. You create the ase.md.verlet.VelocityVerlet object, giving it the atoms and a time step, and then you perform dynamics by calling its run() method:

dyn = VelocityVerlet(atoms, dt=5.0 * units.fs,
                     trajectory='md.traj', logfile='md.log')
dyn.run(1000)  # take 1000 steps

A number of algorithms can be used to perform molecular dynamics, with slightly different results.

Note

Prior to ASE version 3.21.0, inconsistent units were used to specify temperature. Some modules expected kT (in eV), others T (in Kelvin). From ASE 3.21.0, all molecular dynamics modules expecting a temperature take a parameter temperature_K which is the temperature in Kelvin. For compatibility, they still accept the temperature parameter in the same unit as previous versions (eV or K), but using the old parameter will issue a warning.

Choosing the time step

All the dynamics objects need a time step. Choosing it too small will waste computer time, choosing it too large will make the dynamics unstable, typically the energy increases dramatically (the system “blows up”). If the time step is only a little to large, the lack of energy conservation is most obvious in Velocity Verlet dynamics, where energy should otherwise be conserved.

Experience has shown that 5 femtoseconds is a good choice for most metallic systems. Systems with light atoms (e.g. hydrogen) and/or with strong bonds (carbon) will need a smaller time step.

All the dynamics objects documented here are sufficiently related to have the same optimal time step.

File output

The time evolution of the system can be saved in a trajectory file, by creating a trajectory object, and attaching it to the dynamics object. This is documented in the module ase.io.trajectory. You can attach the trajectory explicitly to the dynamics object, and you may want to use the optional interval argument, so every time step is not written to the file.

Alternatively, you can just use the trajectory keyword when instantiating the dynamics object as in the example above. In this case, a loginterval keyword may also be supplied to specify the frequency of writing to the trajectory. The loginterval keyword will apply to both the trajectory and the logfile.

Logging

A logging mechanism is provided, printing time; total, potential and kinetic energy; and temperature (calculated from the kinetic energy). It is enabled by giving the logfile argument when the dynamics object is created, logfile may be an open file, a filename or the string ‘-’ meaning standard output. Per default, a line is printed for each timestep, specifying the loginterval argument will chance this to a more reasonable frequency.

The logging can be customized by explicitly attaching a MDLogger object to the dynamics:

from ase.md import MDLogger
dyn = VelocityVerlet(atoms, dt=2*ase.units.fs)
dyn.attach(MDLogger(dyn, atoms, 'md.log', header=False, stress=False,
           peratom=True, mode="a"), interval=1000)

This example will skip the header line and write energies per atom instead of total energies. The parameters are

header: Print a header line defining the columns.

stress: Print the six components of the stress tensor.

peratom: Print energy per atom instead of total energy.

mode: If ‘a’, append to existing file, if ‘w’ overwrite existing file.

Despite appearances, attaching a logger like this does not create a cyclic reference to the dynamics.

Note

If building your own logging class, be sure not to attach the dynamics object directly to the logging object. Instead, create a weak reference using the proxy method of the weakref package. See the ase.md.MDLogger source code for an example. (If this is not done, a cyclic reference may be created which can cause certain calculators to not terminate correctly.)

class ase.md.MDLogger(dyn, atoms, logfile, header=True, stress=False, peratom=False, mode='a')[source]

Class for logging molecular dynamics simulations.

Parameters: dyn: The dynamics. Only a weak reference is kept.

atoms: The atoms.

logfile: File name or open file, “-” meaning standard output.

stress=False: Include stress in log.

peratom=False: Write energies per atom.

mode=”a”: How the file is opened if logfile is a filename.

Constant NVE simulations (the microcanonical ensemble)

Newton’s second law preserves the total energy of the system, and a straightforward integration of Newton’s second law therefore leads to simulations preserving the total energy of the system (E), the number of atoms (N) and the volume of the system (V). The most appropriate algorithm for doing this is velocity Verlet dynamics, since it gives very good long-term stability of the total energy even with quite large time steps. Fancier algorithms such as Runge-Kutta may give very good short-term energy preservation, but at the price of a slow drift in energy over longer timescales, causing trouble for long simulations.

In a typical NVE simulation, the temperature will remain approximately constant, but if significant structural changes occurs they may result in temperature changes. If external work is done on the system, the temperature is likely to rise significantly.

Velocity Verlet dynamics

class ase.md.verlet.VelocityVerlet(atoms, timestep=None, trajectory=None, logfile=None, loginterval=1, dt=None, append_trajectory=False)[source]

Molecular Dynamics object.

Parameters:

atoms: Atoms object

The Atoms object to operate on.

timestep: float

The time step in ASE time units.

trajectory: Trajectory object or str (optional)

Attach trajectory object. If trajectory is a string a Trajectory will be constructed. Default: None.

logfile: file object or str (optional)

If logfile is a string, a file with that name will be opened. Use ‘-’ for stdout. Default: None.

loginterval: int (optional)

Only write a log line for every loginterval time steps. Default: 1

append_trajectory: boolean

Defaults to False, which causes the trajectory file to be overwriten each time the dynamics is restarted from scratch. If True, the new structures are appended to the trajectory file instead.

dt: float (deprecated)

Alias for timestep.

VelocityVerlet is the only dynamics implementing the NVE ensemble. It requires two arguments, the atoms and the time step. Choosing a too large time step will immediately be obvious, as the energy will increase with time, often very rapidly.

Example: See the tutorial Molecular dynamics.

Constant NVT simulations (the canonical ensemble)

Since Newton’s second law conserves energy and not temperature, simulations at constant temperature will somehow involve coupling the system to a heat bath. This cannot help being somewhat artificial. Two different approaches are possible within ASE. In Langevin dynamics, each atom is coupled to a heat bath through a fluctuating force and a friction term. In Nosé-Hoover dynamics, a term representing the heat bath through a single degree of freedom is introduced into the Hamiltonian.

Langevin dynamics

class ase.md.langevin.Langevin(atoms, timestep, temperature=None, friction=None, fixcm=True, *, temperature_K=None, trajectory=None, logfile=None, loginterval=1, communicator=<ase.parallel.MPI object>, rng=None, append_trajectory=False)[source]

Langevin (constant N, V, T) molecular dynamics.

Parameters:

atoms: Atoms object

The list of atoms.

timestep: float

The time step in ASE time units.

temperature: float (deprecated)

The desired temperature, in electron volt.

temperature_K: float

The desired temperature, in Kelvin.

friction: float

A friction coefficient, typically 1e-4 to 1e-2.

fixcm: bool (optional)

If True, the position and momentum of the center of mass is kept unperturbed. Default: True.

rng: RNG object (optional)

Random number generator, by default numpy.random. Must have a standard_normal method matching the signature of numpy.random.standard_normal.

logfile: file object or str (optional)

If logfile is a string, a file with that name will be opened. Use ‘-’ for stdout.

trajectory: Trajectory object or str (optional)

Attach trajectory object. If trajectory is a string a Trajectory will be constructed. Use None (the default) for no trajectory.

communicator: MPI communicator (optional)

Communicator used to distribute random numbers to all tasks. Default: ase.parallel.world. Set to None to disable communication.

append_trajectory: bool (optional)

Defaults to False, which causes the trajectory file to be overwritten each time the dynamics is restarted from scratch. If True, the new structures are appended to the trajectory file instead.

The temperature and friction are normally scalars, but in principle one quantity per atom could be specified by giving an array.

RATTLE constraints can be used with these propagators, see: E. V.-Eijnden, and G. Ciccotti, Chem. Phys. Lett. 429, 310 (2006)

The propagator is Equation 23 (Eq. 39 if RATTLE constraints are used) of the above reference. That reference also contains another propagator in Eq. 21/34; but that propagator is not quasi-symplectic and gives a systematic offset in the temperature at large time steps.

The Langevin class implements Langevin dynamics, where a (small) friction term and a fluctuating force are added to Newton’s second law which is then integrated numerically. The temperature of the heat bath and magnitude of the friction is specified by the user, the amplitude of the fluctuating force is then calculated to give that temperature. This procedure has some physical justification: in a real metal the atoms are (weakly) coupled to the electron gas, and the electron gas therefore acts like a heat bath for the atoms. If heat is produced locally, the atoms locally get a temperature that is higher than the temperature of the electrons, heat is transferred to the electrons and then rapidly transported away by them. A Langevin equation is probably a reasonable model for this process.

A disadvantage of using Langevin dynamics is that if significant heat is produced in the simulation, then the temperature will stabilize at a value higher than the specified temperature of the heat bath, since a temperature difference between the system and the heat bath is necessary to get a finite heat flow. Another disadvantage is that the fluctuating force is stochastic in nature, so repeating the simulation will not give exactly the same trajectory.

When the Langevin object is created, you must specify a time step, a temperature (in Kelvin) and a friction. Typical values for the friction are 0.01-0.02 atomic units.

# Room temperature simulation
dyn = Langevin(atoms, 5 * units.fs, 300, 0.002)

Both the friction and the temperature can be replaced with arrays giving per-atom values. This is mostly useful for the friction, where one can choose a rather high friction near the boundaries, and set it to zero in the part of the system where the phenomenon being studied is located.

Andersen dynamics

class ase.md.andersen.Andersen(atoms, timestep, temperature_K, andersen_prob, fixcm=True, trajectory=None, logfile=None, loginterval=1, communicator=<ase.parallel.MPI object>, rng=<module 'numpy.random' from '/tmp/ase-docs-web-page/venv/lib/python3.6/site-packages/numpy/random/__init__.py'>, append_trajectory=False)[source]

Andersen (constant N, V, T) molecular dynamics.

” Parameters:

atoms: Atoms object

The list of atoms.

timestep: float

The time step in ASE time units.

temperature_K: float

The desired temperature, in Kelvin.

andersen_prob: float

A random collision probability, typically 1e-4 to 1e-1. With this probability atoms get assigned random velocity components.

fixcm: bool (optional)

If True, the position and momentum of the center of mass is kept unperturbed. Default: True.

rng: RNG object (optional)

Random number generator, by default numpy.random. Must have a random_sample method matching the signature of numpy.random.random_sample.

logfile: file object or str (optional)

If logfile is a string, a file with that name will be opened. Use ‘-’ for stdout.

trajectory: Trajectory object or str (optional)

Attach trajectory object. If trajectory is a string a Trajectory will be constructed. Use None (the default) for no trajectory.

communicator: MPI communicator (optional)

Communicator used to distribute random numbers to all tasks. Default: ase.parallel.world. Set to None to disable communication.

append_trajectory: bool (optional)

Defaults to False, which causes the trajectory file to be overwritten each time the dynamics is restarted from scratch. If True, the new structures are appended to the trajectory file instead.

The temperature is imposed by stochastic collisions with a heat bath that acts on velocity components of randomly chosen particles. The algorithm randomly decorrelates velocities, so dynamical properties like diffusion or viscosity cannot be properly measured.

    1. Andersen, J. Chem. Phys. 72 (4), 2384–2393 (1980)

The Andersen class implements Andersen dynamics, where constant temperature is imposed by stochastic collisions with a heat bath. With a (small) probability (\(andersen_prob\)) the collisions act occasionally on velocity components of randomly selected particles Upon a collision the new velocity is drawn from the Maxwell-Boltzmann distribution at the corresponding temperature. The system is then integrated numerically at constant energy according to the Newtonian laws of motion. The collision probability is defined as the average number of collisions per atom and timestep. The algorithm generates a canonical distribution. [1] However, due to the random decorrelation of velocities, the dynamics are unphysical and cannot represent dynamical properties like e.g. diffusion or viscosity. Another disadvantage is that the collisions are stochastic in nature, so repeating the simulation will not give exactly the same trajectory.

When the Andersen object is created, you must specify a time step, a temperature (in Kelvin) and a collision probability. Typical values for this probability are in the order of 1e-4 to 1e-1.

# Room temperature simulation (300 Kelvin, Andersen probability: 0.002)
dyn = Andersen(atoms, 5 * units.fs, 300, 0.002)

References:

[1] D. Frenkel and B. Smit, Understanding Molecular Simulation (Academic Press, London, 1996)

Nosé-Hoover dynamics

In Nosé-Hoover dynamics, an extra term is added to the Hamiltonian representing the coupling to the heat bath. From a pragmatic point of view one can regard Nosé-Hoover dynamics as adding a friction term to Newton’s second law, but dynamically changing the friction coefficient to move the system towards the desired temperature. Typically the “friction coefficient” will fluctuate around zero.

Nosé-Hoover dynamics is not implemented as a separate class, but is a special case of NPT dynamics.

Berendsen NVT dynamics

class ase.md.nvtberendsen.NVTBerendsen(atoms, timestep, temperature=None, taut=None, fixcm=True, *, temperature_K=None, trajectory=None, logfile=None, loginterval=1, communicator=<ase.parallel.MPI object>, append_trajectory=False)[source]

Berendsen (constant N, V, T) molecular dynamics.

Parameters:

atoms: Atoms object

The list of atoms.

timestep: float

The time step in ASE time units.

temperature: float

The desired temperature, in Kelvin.

temperature_K: float

Alias for temperature

taut: float

Time constant for Berendsen temperature coupling in ASE time units.

fixcm: bool (optional)

If True, the position and momentum of the center of mass is kept unperturbed. Default: True.

trajectory: Trajectory object or str (optional)

Attach trajectory object. If trajectory is a string a Trajectory will be constructed. Use None for no trajectory.

logfile: file object or str (optional)

If logfile is a string, a file with that name will be opened. Use ‘-’ for stdout.

loginterval: int (optional)

Only write a log line for every loginterval time steps. Default: 1

append_trajectory: boolean (optional)

Defaults to False, which causes the trajectory file to be overwriten each time the dynamics is restarted from scratch. If True, the new structures are appended to the trajectory file instead.

In Berendsen NVT simulations the velocities are scaled to achieve the desired temperature. The speed of the scaling is determined by the parameter taut.

This method does not result proper NVT sampling but it usually is sufficiently good in practice (with large taut). For discussion see the gromacs manual at www.gromacs.org.

# Room temperature simulation (300K, 0.1 fs time step)
dyn = NVTBerendsen(atoms, 0.1 * units.fs, 300, taut=0.5*1000*units.fs)

Constant NPT simulations (the isothermal-isobaric ensemble)

class ase.md.npt.NPT(atoms, timestep, temperature=None, externalstress=None, ttime=None, pfactor=None, *, temperature_K=None, mask=None, trajectory=None, logfile=None, loginterval=1, append_trajectory=False)[source]

Constant pressure/stress and temperature dynamics.

Combined Nose-Hoover and Parrinello-Rahman dynamics, creating an NPT (or N,stress,T) ensemble.

The method is the one proposed by Melchionna et al. [1] and later modified by Melchionna [2]. The differential equations are integrated using a centered difference method [3]. See also NPTdynamics.tex

The dynamics object is called with the following parameters:

atoms: Atoms object

The list of atoms.

timestep: float

The timestep in units matching eV, Å, u.

temperature: float (deprecated)

The desired temperature in eV.

temperature_K: float

The desired temperature in K.

externalstress: float or nparray

The external stress in eV/A^3. Either a symmetric 3x3 tensor, a 6-vector representing the same, or a scalar representing the pressure. Note that the stress is positive in tension whereas the pressure is positive in compression: giving a scalar p is equivalent to giving the tensor (-p, -p, -p, 0, 0, 0).

ttime: float

Characteristic timescale of the thermostat, in ASE internal units Set to None to disable the thermostat.

pfactor: float

A constant in the barostat differential equation. If a characteristic barostat timescale of ptime is desired, set pfactor to ptime^2 * B (where ptime is in units matching eV, Å, u; and B is the Bulk Modulus, given in eV/Å^3). Set to None to disable the barostat. Typical metallic bulk moduli are of the order of 100 GPa or 0.6 eV/A^3.

mask: None or 3-tuple or 3x3 nparray (optional)

Optional argument. A tuple of three integers (0 or 1), indicating if the system can change size along the three Cartesian axes. Set to (1,1,1) or None to allow a fully flexible computational box. Set to (1,1,0) to disallow elongations along the z-axis etc. mask may also be specified as a symmetric 3x3 array indicating which strain values may change.

Useful parameter values:

  • The same timestep can be used as in Verlet dynamics, i.e. 5 fs is fine for bulk copper.

  • The ttime and pfactor are quite critical[4], too small values may cause instabilites and/or wrong fluctuations in T / p. Too large values cause an oscillation which is slow to die. Good values for the characteristic times seem to be 25 fs for ttime, and 75 fs for ptime (used to calculate pfactor), at least for bulk copper with 15000-200000 atoms. But this is not well tested, it is IMPORTANT to monitor the temperature and stress/pressure fluctuations.

References:

  1. S. Melchionna, G. Ciccotti and B. L. Holian, Molecular Physics 78, p. 533 (1993).

  2. S. Melchionna, Physical Review E 61, p. 6165 (2000).

  3. B. L. Holian, A. J. De Groot, W. G. Hoover, and C. G. Hoover, Physical Review A 41, p. 4552 (1990).

  4. F. D. Di Tolla and M. Ronchetti, Physical Review E 48, p. 1726 (1993).

run(steps)[source]

Perform a number of time steps.

set_stress(stress)[source]

Set the applied stress.

Must be a symmetric 3x3 tensor, a 6-vector representing a symmetric 3x3 tensor, or a number representing the pressure.

Use with care, it is better to set the correct stress when creating the object.

set_temperature(temperature=None, *, temperature_K=None)[source]

Set the temperature.

Parameters:

temperature: float (deprecated)

The new temperature in eV. Deprecated, use temperature_K.

temperature_K: float (keyword-only argument)

The new temperature, in K.

set_mask(mask)[source]

Set the mask indicating dynamic elements of the computational box.

If set to None, all elements may change. If set to a 3-vector of ones and zeros, elements which are zero specify directions along which the size of the computational box cannot change. For example, if mask = (1,1,0) the length of the system along the z-axis cannot change, although xz and yz shear is still possible. May also be specified as a symmetric 3x3 array indicating which strain values may change.

Use with care, as you may “freeze in” a fluctuation in the strain rate.

set_fraction_traceless(fracTraceless)[source]

set what fraction of the traceless part of the force on eta is kept.

By setting this to zero, the volume may change but the shape may not.

get_strain_rate()[source]

Get the strain rate as an upper-triangular 3x3 matrix.

This includes the fluctuations in the shape of the computational box.

set_strain_rate(rate)[source]

Set the strain rate. Must be an upper triangular 3x3 matrix.

If you set a strain rate along a direction that is “masked out” (see set_mask), the strain rate along that direction will be maintained constantly.

get_time()[source]

Get the elapsed time.

initialize()[source]

Initialize the dynamics.

The dynamics requires positions etc for the two last times to do a timestep, so the algorithm is not self-starting. This method performs a ‘backwards’ timestep to generate a configuration before the current.

This is called automatically the first time run() is called.

get_gibbs_free_energy()[source]

Return the Gibb’s free energy, which is supposed to be conserved.

Requires that the energies of the atoms are up to date.

This is mainly intended as a diagnostic tool. If called before the first timestep, Initialize will be called.

zero_center_of_mass_momentum(verbose=0)[source]

Set the center of mass momentum to zero.

attach(function, interval=1, *args, **kwargs)[source]

Attach callback function or trajectory.

At every interval steps, call function with arguments args and keyword arguments kwargs.

If function is a trajectory object, its write() method is attached, but if function is a BundleTrajectory (or another trajectory supporting set_extra_data(), said method is first used to instruct the trajectory to also save internal data from the NPT dynamics object.

Berendsen NPT dynamics

class ase.md.nptberendsen.NPTBerendsen(atoms, timestep, temperature=None, *, temperature_K=None, pressure=None, pressure_au=None, taut=49.11347394232032, taup=98.22694788464064, compressibility=None, compressibility_au=None, fixcm=True, trajectory=None, logfile=None, loginterval=1, append_trajectory=False)[source]

Berendsen (constant N, P, T) molecular dynamics.

This dynamics scale the velocities and volumes to maintain a constant pressure and temperature. The shape of the simulation cell is not altered, if that is desired use Inhomogenous_NPTBerendsen.

Parameters:

atoms: Atoms object

The list of atoms.

timestep: float

The time step in ASE time units.

temperature: float

The desired temperature, in Kelvin.

temperature_K: float

Alias for temperature.

pressure: float (deprecated)

The desired pressure, in bar (1 bar = 1e5 Pa). Deprecated, use pressure_au instead.

pressure: float

The desired pressure, in atomic units (eV/Å^3).

taut: float

Time constant for Berendsen temperature coupling in ASE time units. Default: 0.5 ps.

taup: float

Time constant for Berendsen pressure coupling. Default: 1 ps.

compressibility: float (deprecated)

The compressibility of the material, in bar-1. Deprecated, use compressibility_au instead.

compressibility_au: float

The compressibility of the material, in atomic units (Å^3/eV).

fixcm: bool (optional)

If True, the position and momentum of the center of mass is kept unperturbed. Default: True.

trajectory: Trajectory object or str (optional)

Attach trajectory object. If trajectory is a string a Trajectory will be constructed. Use None for no trajectory.

logfile: file object or str (optional)

If logfile is a string, a file with that name will be opened. Use ‘-’ for stdout.

loginterval: int (optional)

Only write a log line for every loginterval time steps. Default: 1

append_trajectory: boolean (optional)

Defaults to False, which causes the trajectory file to be overwriten each time the dynamics is restarted from scratch. If True, the new structures are appended to the trajectory file instead.

In Berendsen NPT simulations the velocities are scaled to achieve the desired temperature. The speed of the scaling is determined by the parameter taut.

The atom positions and the simulation cell are scaled in order to achieve the desired pressure.

This method does not result proper NPT sampling but it usually is sufficiently good in practice (with large taut and taup). For discussion see the gromacs manual at www.gromacs.org. or amber at ambermd.org

# Room temperature simulation (300K, 0.1 fs time step, atmospheric pressure)
dyn = NPTBerendsen(atoms, timestep=0.1 * units.fs, temperature_K=300,
                   taut=100 * units.fs, pressure_au=1.01325 * units.bar,
                   taup=1000 * units.fs, compressibility=4.57e-5 / units.bar)

Velocity distributions

A selection of functions are provided to initialize atomic velocities to the correct temperature.

ase.md.velocitydistribution.MaxwellBoltzmannDistribution(atoms, temp=None, *, temperature_K=None, communicator=None, force_temp=False, rng=None)[source]

Sets the momenta to a Maxwell-Boltzmann distribution.

Parameters:

atoms: Atoms object

The atoms. Their momenta will be modified.

temp: float (deprecated)

The temperature in eV. Deprecated, used temperature_K instead.

temperature_K: float

The temperature in Kelvin.

communicator: MPI communicator (optional)

Communicator used to distribute an identical distribution to all tasks. Set to ‘serial’ to disable communication. Leave as None to get the default: ase.parallel.world

force_temp: bool (optinal, default: False)

If True, random the momenta are rescaled so the kinetic energy is exactly 3/2 N k T. This is a slight deviation from the correct Maxwell-Boltzmann distribution.

rng: Numpy RNG (optional)

Random number generator. Default: numpy.random

ase.md.velocitydistribution.Stationary(atoms, preserve_temperature=True)[source]

Sets the center-of-mass momentum to zero.

ase.md.velocitydistribution.ZeroRotation(atoms, preserve_temperature=True)[source]

Sets the total angular momentum to zero by counteracting rigid rotations.

ase.md.velocitydistribution.PhononHarmonics(atoms, force_constants, temp=None, *, temperature_K=None, rng=<module 'numpy.random' from '/tmp/ase-docs-web-page/venv/lib/python3.6/site-packages/numpy/random/__init__.py'>, quantum=False, plus_minus=False, return_eigensolution=False, failfast=True)[source]

Excite phonon modes to specified temperature.

This will displace atomic positions and set the velocities so as to produce a random, phononically correct state with the requested temperature.

Parameters:

atoms: ase.atoms.Atoms() object

Positions and momenta of this object are perturbed.

force_constants: ndarray of size 3N x 3N

Force constants for the the structure represented by atoms in eV/Ų

temp: float (deprecated).

Temperature in eV. Deprecated, use temperature_K instead.

temperature_K: float

Temperature in Kelvin.

rng: Random number generator

RandomState or other random number generator, e.g., np.random.rand

quantum: bool

True for Bose-Einstein distribution, False for Maxwell-Boltzmann (classical limit)

failfast: bool

True for sanity checking the phonon spectrum for negative frequencies at Gamma.

ase.md.velocitydistribution.phonon_harmonics(force_constants, masses, temp=None, *, temperature_K=None, rng=<built-in method rand of mtrand.RandomState object>, quantum=False, plus_minus=False, return_eigensolution=False, failfast=True)[source]

Return displacements and velocities that produce a given temperature.

Parameters:

force_constants: array of size 3N x 3N

force constants (Hessian) of the system in eV/Ų

masses: array of length N

masses of the structure in amu

temp: float (deprecated)

Temperature converted to eV (T * units.kB). Deprecated, use temperature_K.

temperature_K: float

Temperature in Kelvin.

rng: function

Random number generator function, e.g., np.random.rand

quantum: bool

True for Bose-Einstein distribution, False for Maxwell-Boltzmann (classical limit)

plus_minus: bool

Displace atoms with +/- the amplitude accoding to PRB 94, 075125

return_eigensolution: bool

return eigenvalues and eigenvectors of the dynamical matrix

failfast: bool

True for sanity checking the phonon spectrum for negative frequencies at Gamma

Returns:

Displacements, velocities generated from the eigenmodes, (optional: eigenvalues, eigenvectors of dynamical matrix)

Purpose:

Excite phonon modes to specified temperature.

This excites all phonon modes randomly so that each contributes, on average, equally to the given temperature. Both potential energy and kinetic energy will be consistent with the phononic vibrations characteristic of the specified temperature.

In other words the system will be equilibrated for an MD run at that temperature.

force_constants should be the matrix as force constants, e.g., as computed by the ase.phonons module.

Let X_ai be the phonon modes indexed by atom and mode, w_i the phonon frequencies, and let 0 < Q_i <= 1 and 0 <= R_i < 1 be uniformly random numbers. Then

             1/2
_     / k T \     ---  1  _             1/2
R  += | --- |      >  --- X   (-2 ln Q )    cos (2 pi R )
 a    \  m  /     ---  w   ai         i                i
          a        i    i


             1/2
_     / k T \     --- _            1/2
v   = | --- |      >  X  (-2 ln Q )    sin (2 pi R )
 a    \  m  /     ---  ai        i                i
          a        i

Reference: [West, Estreicher; PRL 96, 22 (2006)]

Post-simulation Analysis

Functionality is provided to perform analysis of atomic/molecular behaviour as calculation in a molecular dynamics simulation. Currently, this is presented as a class to address the Einstein equation for diffusivity.

class ase.md.analysis.DiffusionCoefficient(traj, timestep, atom_indices=None, molecule=False)[source]

This class calculates the Diffusion Coefficient for the given Trajectory using the Einstein Equation:

..math:: left langle left | r(t) - r(0) right | ^{2} right rangle = 2nDt

where r(t) is the position of atom at time t, n is the degrees of freedom and D is the Diffusion Coefficient

Solved herein by fitting with \(y = mx + c\), i.e. \(\frac{1}{2n} \left \langle \left | r(t) - r(0) \right | ^{2} \right \rangle = Dt\), with m = D and c = 0

wiki : https://en.wikibooks.org/wiki/Molecular_Simulation/Diffusion_Coefficients

Parameters
  • traj (Trajectory) – Trajectory of atoms objects (images)

  • timestep (Float) – Timestep between each image in the trajectory, in ASE timestep units (For an MD simulation with timestep of N, and images written every M iterations, our timestep here is N * M)

  • atom_indices (List of Int) – The indices of atoms whose Diffusion Coefficient is to be calculated explicitly

  • molecule (Boolean) – Indicate if we are studying a molecule instead of atoms, therefore use centre of mass in calculations