================== Molecular dynamics ================== .. module:: ase.md :synopsis: Molecular Dynamics .. contents:: 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 is discussed in the :ref:`structure_optimizations` section. A molecular dynamics object will operate on the atoms by moving them according to their forces - it integrates Newton's second law numerically. In the most basic form, this preserves the particle number, volume (actually, the shape of the unit cell), and the energy, generating an *NVE* or *microcanonical* ensemble. Other ensembles (*NVT* and *NpT*) can be generated by coupling the motion of the atoms to a simple model of the surroundings. *Simple example:* A molecular dynamics simulation in the *NVE* ensemble will use the `Velocity Verlet dynamics`_. You create the :class:`ase.md.verlet.VelocityVerlet` object, giving it the atoms and a time step, and then you perform dynamics by calling its :meth:`~verlet.VelocityVerlet.run` method:: dyn = VelocityVerlet(atoms, dt=5.0 * units.fs, trajectory='md.traj', logfile='md.log') dyn.run(1000) # take 1000 steps .. 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, maybe as small as 1 or 2 fs. 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 :mod:`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 :class:`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.) .. autoclass:: MDLogger 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 ------------------------ .. module:: ase.md.verlet .. autoclass:: VelocityVerlet ``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 :ref:`md_tutorial`. 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. Such algorithms can be stochastic or deterministic, and the coupling to the atoms typically occurs through a rescaling of the velocities. In the Langevin algorithm, a friction term and a fluctuating force is used instead. In the NVT ensemble both the kinetic energy and the total energy will fluctuate. Some algorithms do not correctly reproduce these fluctuations, this may be an issue in particular for small systems, and is noted in the overview below. **Recommended algorithms:** Langevin dynamics A friction and fluctuating force is added to the equation of motion. *Advantages*: Simple. Correctly samples the ensemble. *Disadvantage*: Stochastic. Nosé-Hoover chain In Nosé-Hoover dynamics, the velocities are rescaled at each time step. The scaling factor is itself a dynamic variable. For stable operation, a chain of thermostats is used. *Advantages*: Deterministic. Well-studied by the Stat. Mek. community. Correctly samples the ensemble. *Disadvantages*: The fluctuations tend to show a period given by the thermostat time scale. Slow to reach the correct temperature if started from a significantly wrong temperature. Bussi dynamics Rescales the velocities at each time step, using a stochastic algorithm to ensure the correct fluctuations, unlike the closely related Berendsen algorithm. *Advantages*: Simple. Correctly samples the ensemble. *Disadvantage*: Stochastic. **Not recommended algorithms:** Andersen dynamics At each time step a fraction of the atoms have their velocity replaced with one drawn from the Maxwell-Boltzmann distribution. *Disadvantage*: Dramatically alters a few atoms instead of gently perturbing them all. Velocities are artificially decorrelated over a time corresponding to the thermalization time. Berendsen NVT dynamics Rescale the velocities at each time step, so the kinetic energy exponentially approaches the correct one. *Advantage*: Very efficient for initializing a system to the correct initial temperature. *Disadvantage*: Almost completely supresses fluctuations in the total energy. Langevin dynamics ----------------- .. module:: ase.md.langevin .. autoclass:: Langevin 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 coefficient. A typical range for the friction coefficient may be 0.001–0.1 fs\ :sup:`−1` (1–100 ps\ :sup:`−1`). For example, you can give a friction coefficient of 0.01 fs\ :sup:`−1` (10 ps\ :sup:`−1`) as .. code-block:: python dyn = Langevin( atoms, timestep=5.0 * units.fs, temperature_K=300.0, # temperature in K friction=0.01 / units.fs, ) 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. Nosé-Hoover dynamics -------------------- .. module:: ase.md.nose_hoover_chain .. autoclass:: NoseHooverChainNVT 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. To give a more stable dynamics, the friction coefficient is itself thermalized in the same way, leading to a chain of thermostats (a **Nosé-Hoover chain**, in ASE per default the chain length is 3). Bussi dynamics -------------- .. module:: ase.md.bussi .. autoclass:: Bussi The Bussi class implements the Bussi dynamics, where constant temperature is imposed by a stochastic velocity rescaling algorithm. The thermostat is conceptually very close to the Berendsen thermostat, but does sample the canonical ensemble correctly. Given that the thermostat is both correct and global, it is advised to use it for production runs. Andersen dynamics ----------------- .. module:: ase.md.andersen .. autoclass:: Andersen 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) Berendsen NVT dynamics ----------------------- .. module:: ase.md.nvtberendsen .. autoclass:: NVTBerendsen 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) =========================================================== Constant pressure (or for solids, constant stress) is usually obtained by adding a barostat to one of the NVT algorithms above. ASE currently lacks a good NPT algorithm. The following two are available. **Algorithms:** Berendsen NPT dynamics This is a variation of Berendsen NVT dynamics with a barostat added. The size of the unit cell is rescaled after each time step, so the pressure / stress approaches the desired pressure. It exists in two variations, one where the shape of the unit cell is preserved and one where it is allowed to vary. *Disadvantage*: Fluctuations in both total energy and pressure are suppressed compared to the correct NPT ensemble. For large systems, this is not expected to be serious. NPT An implementation of NPT dynamics combining a Nosé-Hoover thermostat with a Parinello-Rahman barostat, according to Melchionna *et al.*, see below. **Not recommended!** The dynamics tend to be unstable, especially if started with a temperature or pressure that is different from the desired. The fluctuations seem to often be wrong. Berendsen NPT dynamics ----------------------- .. module:: ase.md.nptberendsen .. autoclass:: NPTBerendsen 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. The time scale of this barostat is given by the parameter taup. 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_au=4.57e-5 / units.bar) Nosé-Hoover-Parinello-Rahman NPT dynamics ----------------------------------------- .. module:: ase.md.npt .. autoclass:: NPT .. automethod:: run .. automethod:: set_stress .. automethod:: set_temperature .. automethod:: set_mask .. automethod:: set_fraction_traceless .. automethod:: get_strain_rate .. automethod:: set_strain_rate .. automethod:: get_time .. automethod:: initialize .. automethod:: get_gibbs_free_energy .. automethod:: zero_center_of_mass_momentum .. automethod:: attach **This module is not recommended!** There is a strong tendency for oscillations in the temperature and/or pressure, unless the starting configuration is chosen with great care. Contour Exploration =================== .. module:: ase.md.contour_exploration .. autoclass:: ContourExploration Contour Exploration evolves the system along constant potentials energy contours on the potential energy surface. The method uses curvature based extrapolation and a potentiostat to correct for potential energy errors. It is similar to molecular dynamics but with a potentiostat rather than a thermostat. Without changes in kinetic energy, it is more useful to automatically scale step sizes to the curvature of the potential energy contour via an ``angle_limit`` while enforcing a ``maxstep`` to ensure potentiostatic accuracy. [1] To escape loops on the pontential energy surface or to break symmetries, a random drift vector parallel to the contour can be applied as a fraction of the step size via ``parallel_drift``. Contour exploration cannot be used at minima since the contour is a single point. :: # Contour exploration at the current potential energy dyn = ContourExploration(atoms) References: [1] M. J. Waters and J. M. Rondinelli, *Contour Exploration with Potentiostatic Kinematics* `arXiv:2103.08054 `_. Velocity distributions ====================== A selection of functions are provided to initialize atomic velocities to the correct temperature. .. module:: ase.md.velocitydistribution .. autofunction:: MaxwellBoltzmannDistribution .. autofunction:: Stationary .. autofunction:: ZeroRotation .. autofunction:: PhononHarmonics .. autofunction:: phonon_harmonics 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. .. module:: ase.md.analysis .. autoclass:: DiffusionCoefficient