# Source code for ase.md.velocitydistribution

```# VelocityDistributions.py -- set up a velocity distribution

"""Module for setting up velocity distributions such as Maxwell–Boltzmann.

Currently, only a few functions are defined, such as
MaxwellBoltzmannDistribution, which sets the momenta of a list of
atoms according to a Maxwell-Boltzmann distribution at a given
temperature.

"""

import numpy as np
from ase.parallel import world
from ase import units
from ase.md.md import process_temperature

# define a ``zero'' temperature to avoid divisions by zero
eps_temp = 1e-12

class UnitError(Exception):
"""Exception raised when wrong units are specified"""

def force_temperature(atoms, temperature, unit="K"):
""" force (nucl.) temperature to have a precise value

Parameters:
atoms: ase.Atoms
the structure
temperature: float
nuclear temperature to set
unit: str
'K' or 'eV' as unit for the temperature
"""

if unit == "K":
E_temp = temperature * units.kB
elif unit == "eV":
E_temp = temperature
else:
raise UnitError("'{}' is not supported, use 'K' or 'eV'.".format(unit))

if temperature > eps_temp:
E_kin0 = atoms.get_kinetic_energy() / len(atoms) / 1.5
gamma = E_temp / E_kin0
else:
gamma = 0.0
atoms.set_momenta(atoms.get_momenta() * np.sqrt(gamma))

def _maxwellboltzmanndistribution(masses, temp, communicator=None, rng=None):
"""Return a Maxwell-Boltzmann distribution with a given temperature.

Paremeters:

masses: float
The atomic masses.

temp: float
The temperature in electron volt.

communicator: MPI communicator (optional)
Communicator used to distribute an identical distribution to
all tasks.  Set to 'serial' to disable communication (setting to None
gives the default).  Default: ase.parallel.world

rng: numpy RNG (optional)
The random number generator.  Default: np.random

Returns:

A numpy array with Maxwell-Boltzmann distributed momenta.
"""
if rng is None:
rng = np.random
if communicator is None:
communicator = world
xi = rng.standard_normal((len(masses), 3))
if communicator != 'serial':
momenta = xi * np.sqrt(masses * temp)[:, np.newaxis]
return momenta

[docs]def MaxwellBoltzmannDistribution(
atoms, temp=None, *, temperature_K=None,
communicator=None, force_temp=False,
rng=None
):
"""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
"""
temp = units.kB * process_temperature(temp, temperature_K, 'eV')
masses = atoms.get_masses()
momenta = _maxwellboltzmanndistribution(masses, temp, communicator, rng)
atoms.set_momenta(momenta)
if force_temp:
force_temperature(atoms, temperature=temp, unit="eV")

[docs]def Stationary(atoms, preserve_temperature=True):
"Sets the center-of-mass momentum to zero."

# Save initial temperature
temp0 = atoms.get_temperature()

p = atoms.get_momenta()
p0 = np.sum(p, 0)
# We should add a constant velocity, not momentum, to the atoms
m = atoms.get_masses()
mtot = np.sum(m)
v0 = p0 / mtot
p -= v0 * m[:, np.newaxis]
atoms.set_momenta(p)

if preserve_temperature:
force_temperature(atoms, temp0)

[docs]def ZeroRotation(atoms, preserve_temperature=True):
"Sets the total angular momentum to zero by counteracting rigid rotations."

# Save initial temperature
temp0 = atoms.get_temperature()

# Find the principal moments of inertia and principal axes basis vectors
Ip, basis = atoms.get_moments_of_inertia(vectors=True)
# Calculate the total angular momentum and transform to principal basis
Lp = np.dot(basis, atoms.get_angular_momentum())
# Calculate the rotation velocity vector in the principal basis, avoiding
# zero division, and transform it back to the cartesian coordinate system
omega = np.dot(np.linalg.inv(basis), np.select([Ip > 0], [Lp / Ip]))
# We subtract a rigid rotation corresponding to this rotation vector
com = atoms.get_center_of_mass()
positions = atoms.get_positions()
positions -= com  # translate center of mass to origin
velocities = atoms.get_velocities()
atoms.set_velocities(velocities - np.cross(omega, positions))

if preserve_temperature:
force_temperature(atoms, temp0)

def n_BE(temp, omega):
"""Bose-Einstein distribution function.

Args:
temp: temperature converted to eV (*units.kB)
omega: sequence of frequencies converted to eV

Returns:
Value of Bose-Einstein distribution function for each energy

"""

omega = np.asarray(omega)

# 0K limit
if temp < eps_temp:
n = np.zeros_like(omega)
else:
n = 1 / (np.exp(omega / (temp)) - 1)
return n

[docs]def phonon_harmonics(
force_constants,
masses,
temp=None,
*,
temperature_K=None,
rng=np.random.rand,
quantum=False,
plus_minus=False,
return_eigensolution=False,
failfast=True,
):
r"""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/Å²

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

.. code-block:: none

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)]
"""

# Handle the temperature units
temp = units.kB * process_temperature(temp, temperature_K, 'eV')

# Build dynamical matrix
rminv = (masses ** -0.5).repeat(3)
dynamical_matrix = force_constants * rminv[:, None] * rminv[None, :]

# Solve eigenvalue problem to compute phonon spectrum and eigenvectors
w2_s, X_is = np.linalg.eigh(dynamical_matrix)

# Check for soft modes
if failfast:
zeros = w2_s[:3]
worst_zero = np.abs(zeros).max()
if worst_zero > 1e-3:
msg = "Translational deviate from 0 significantly: "
raise ValueError(msg + "{}".format(w2_s[:3]))

w2min = w2_s[3:].min()
if w2min < 0:
msg = "Dynamical matrix has negative eigenvalues such as "
raise ValueError(msg + "{}".format(w2min))

# First three modes are translational so ignore:
nw = len(w2_s) - 3
n_atoms = len(masses)
w_s = np.sqrt(w2_s[3:])
X_acs = X_is[:, 3:].reshape(n_atoms, 3, nw)

# Assign the amplitudes according to Bose-Einstein distribution
# or high temperature (== classical) limit
if quantum:
hbar = units._hbar * units.J * units.s
A_s = np.sqrt(hbar * (2 * n_BE(temp, hbar * w_s) + 1) / (2 * w_s))
else:
A_s = np.sqrt(temp) / w_s

if plus_minus:
# create samples by multiplying the amplitude with +/-
# according to Eq. 5 in PRB 94, 075125

# gauge eigenvectors: largest value always positive
for ii in range(X_acs.shape[-1]):
vec = X_acs[:, :, ii]
max_arg = np.argmax(abs(vec))
X_acs[:, :, ii] *= np.sign(vec.flat[max_arg])

# Create velocities und displacements from the amplitudes and
# eigenvectors
phi_s = 2.0 * np.pi * rng(nw)

# Assign velocities, sqrt(2) compensates for missing sin(phi) in
# amplitude for displacement
v_ac = (w_s * A_s * np.sqrt(2) * np.cos(phi_s) * X_acs).sum(axis=2)
v_ac /= np.sqrt(masses)[:, None]

# Assign displacements
d_ac = (A_s * X_acs).sum(axis=2)
d_ac /= np.sqrt(masses)[:, None]

else:
# compute the gaussian distribution for the amplitudes
# We need 0 < P <= 1.0 and not 0 0 <= P < 1.0 for the logarithm
# to avoid (highly improbable) NaN.

# Box Muller [en.wikipedia.org/wiki/Box–Muller_transform]:
spread = np.sqrt(-2.0 * np.log(1.0 - rng(nw)))

# assign amplitudes and phases
phi_s = 2.0 * np.pi * rng(nw)

# Assign velocities and displacements
v_ac = (w_s * A_s * np.cos(phi_s) * X_acs).sum(axis=2)
v_ac /= np.sqrt(masses)[:, None]

d_ac = (A_s * np.sin(phi_s) * X_acs).sum(axis=2)
d_ac /= np.sqrt(masses)[:, None]

if return_eigensolution:
return d_ac, v_ac, w2_s, X_is
# else
return d_ac, v_ac

[docs]def PhononHarmonics(
atoms,
force_constants,
temp=None,
*,
temperature_K=None,
rng=np.random,
quantum=False,
plus_minus=False,
return_eigensolution=False,
failfast=True,
):
r"""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/Å²

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.

"""

# Receive displacements and velocities from phonon_harmonics()
d_ac, v_ac = phonon_harmonics(
force_constants=force_constants,
masses=atoms.get_masses(),
temp=temp,
temperature_K=temperature_K,
rng=rng.rand,
plus_minus=plus_minus,
quantum=quantum,
failfast=failfast,
return_eigensolution=False,
)

# Assign new positions (with displacements) and velocities
atoms.positions += d_ac
atoms.set_velocities(v_ac)
```