def free_energy(temperature: Scalar,
q_weights: Vector,
static_energies: Vector,
frequencies: Array3D,
static_only: bool = False) -> Vector:
"""
The total free energy at a certain temperature.
:param temperature: A ``float`` that represents the temperature at which the total free energy is calculated. This
value is in unit 'Kelvin'.
:param q_weights: An :math:`m \\times 1` vector that represents the weight of each q-point in Brillouin zone
sampling. This vector can be non-normalized since a normalization will be done internally.
:param static_energies: An :math:`n \\times 1` vector that represents the static energy of the system with
:math:`n` different volumes. This should be the same unit as user-defined in the "settings" file.
:param frequencies: An :math:`n \\times m \\times l` 3D array that represents the frequency read from file.
:param static_only: If ``True``, directly return the static energies, i.e., the *static_energies* parameter itself.
This is useful when the user just wants to see static result while keeping all other functions unchanged.
:return: An :math:`n \\times 1` vector that represents the total free energy of the system with :math:`n` different
volumes. The default unit is the same as in function ``ho_free_energy``.
"""
if not np.all(np.greater_equal(q_weights, 0)):
raise ValueError('Weights should all be greater equal than 0!')
if static_only:
return static_energies
scaled_q_weights: Vector = q_weights / np.sum(q_weights)
vibrational_energies: Vector = np.dot(
ho_free_energy(temperature, frequencies).sum(axis=2), scaled_q_weights)
return static_energies + vibrational_energies
def on_band(self, i: int) -> Matrix:
"""
Sample free energy on the :math:`i` th band.
:param i: An integer labeling :math:`i` th band.
:return: The accumulated free energy on the :math:`i`th q-point.
"""
return ho_free_energy(self.temperature, self.omegas[:, :, i])
def harmonic_part(self) -> Vector:
"""
Calculate the harmonic contribution to the free energy.
:return: The harmonic contribution on the temperature-volume grid.
"""
sum_modes = np.sum(ho_free_energy(self.temperature, self.frequencies),
axis=2)
return np.dot(sum_modes, self._scaled_q_weights)
def on_volume(self, i: int) -> Scalar:
"""
Sample free energy on the :math:`i` th volume.
:param i: An integer labeling :math:`i` th volume.
:return: The accumulated free energy on the :math:`i` th volume.
"""
return np.vdot(
ho_free_energy(self.temperature, self.omegas[i]).sum(axis=1),
self._scaled_q_weights)
def on_all_volumes(self) -> Vector:
"""
Sample free energies on every volume.
:return: A vector with length equals the number of volumes. Each element is the free energy of
one volume.
"""
# First calculate free energies on a 3D array, then sum along the third axis (bands),
# at last contract weights and free energies on all q-points.
return np.dot(
ho_free_energy(self.temperature, self.omegas).sum(axis=2),
self._scaled_q_weights)
def test_ho_free_energy(self):
self.assertEqual(ho_free_energy(0, 0), 0)
self.assertEqual(ho_free_energy(1, -2), 0)
self.assertAlmostEqual(ho_free_energy(0, 1000), 0.004556299262079407)
self.assertAlmostEqual(ho_free_energy(100, 1000),
0.0045562989049199466)