# Source code for ase.thermochemistry

"""Modules for calculating thermochemical information from computational
outputs."""

import os
import sys
import numpy as np

from ase import units

class ThermoChem:
"""Base class containing common methods used in thermochemistry
calculations."""

def get_ZPE_correction(self):
"""Returns the zero-point vibrational energy correction in eV."""
zpe = 0.
for energy in self.vib_energies:
zpe += 0.5 * energy
return zpe

def _vibrational_energy_contribution(self, temperature):
"""Calculates the change in internal energy due to vibrations from
0K to the specified temperature for a set of vibrations given in
eV and a temperature given in Kelvin. Returns the energy change
in eV."""
kT = units.kB * temperature
dU = 0.
for energy in self.vib_energies:
dU += energy / (np.exp(energy / kT) - 1.)
return dU

def _vibrational_entropy_contribution(self, temperature):
"""Calculates the entropy due to vibrations for a set of vibrations
given in eV and a temperature given in Kelvin.  Returns the entropy
in eV/K."""
kT = units.kB * temperature
S_v = 0.
for energy in self.vib_energies:
x = energy / kT
S_v += x / (np.exp(x) - 1.) - np.log(1. - np.exp(-x))
S_v *= units.kB
return S_v

def _vprint(self, text):
"""Print output if verbose flag True."""
if self.verbose:
sys.stdout.write(text + os.linesep)

[docs]class HarmonicThermo(ThermoChem):
"""Class for calculating thermodynamic properties in the approximation
that all degrees of freedom are treated harmonically. Often used for

Inputs:

vib_energies : list
a list of the harmonic energies of the adsorbate (e.g., from
ase.vibrations.Vibrations.get_energies). The number of
energies should match the number of degrees of freedom of the
adsorbate; i.e., 3*n, where n is the number of atoms. Note that
this class does not check that the user has supplied the correct
number of energies. Units of energies are eV.
potentialenergy : float
the potential energy in eV (e.g., from atoms.get_potential_energy)
(if potentialenergy is unspecified, then the methods of this
class can be interpreted as the energy corrections)
"""

def __init__(self, vib_energies, potentialenergy=0.):
self.vib_energies = vib_energies
# Check for imaginary frequencies.
if sum(np.iscomplex(self.vib_energies)):
raise ValueError('Imaginary vibrational energies are present.')
else:
self.vib_energies = np.real(self.vib_energies)  # clear +0.j

self.potentialenergy = potentialenergy

[docs]    def get_internal_energy(self, temperature, verbose=True):
"""Returns the internal energy, in eV, in the harmonic approximation
at a specified temperature (K)."""

self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.3f eV'
write('Internal energy components at T = %.2f K:' % temperature)
write('=' * 31)

U = 0.

write(fmt % ('E_pot', self.potentialenergy))
U += self.potentialenergy

zpe = self.get_ZPE_correction()
write(fmt % ('E_ZPE', zpe))
U += zpe

dU_v = self._vibrational_energy_contribution(temperature)
write(fmt % ('Cv_harm (0->T)', dU_v))
U += dU_v

write('-' * 31)
write(fmt % ('U', U))
write('=' * 31)
return U

[docs]    def get_entropy(self, temperature, verbose=True):
"""Returns the entropy, in eV/K, in the harmonic approximation
at a specified temperature (K)."""

self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.7f eV/K%13.3f eV'
write('Entropy components at T = %.2f K:' % temperature)
write('=' * 49)
write('%15s%13s     %13s' % ('', 'S', 'T*S'))

S = 0.

S_v = self._vibrational_entropy_contribution(temperature)
write(fmt % ('S_harm', S_v, S_v * temperature))
S += S_v

write('-' * 49)
write(fmt % ('S', S, S * temperature))
write('=' * 49)
return S

[docs]    def get_helmholtz_energy(self, temperature, verbose=True):
"""Returns the Helmholtz free energy, in eV, in the harmonic
approximation at a specified temperature (K)."""

self.verbose = True
write = self._vprint

U = self.get_internal_energy(temperature, verbose=verbose)
write('')
S = self.get_entropy(temperature, verbose=verbose)
F = U - temperature * S

write('')
write('Free energy components at T = %.2f K:' % temperature)
write('=' * 23)
fmt = '%5s%15.3f eV'
write(fmt % ('U', U))
write(fmt % ('-T*S', -temperature * S))
write('-' * 23)
write(fmt % ('F', F))
write('=' * 23)
return F

[docs]class HinderedThermo(ThermoChem):
"""Class for calculating thermodynamic properties in the hindered
translator and hindered rotor model where all but three degrees of
freedom are treated as harmonic vibrations, two are treated as
hindered translations, and one is treated as a hindered rotation.

Inputs:

vib_energies : list
a list of all the vibrational energies of the adsorbate (e.g., from
ase.vibrations.Vibrations.get_energies). The number of energies
should match the number of degrees of freedom of the adsorbate;
i.e., 3*n, where n is the number of atoms. Note that this class does
not check that the user has supplied the correct number of energies.
Units of energies are eV.
trans_barrier_energy : float
the translational energy barrier in eV. This is the barrier for an
adsorbate to diffuse on the surface.
rot_barrier_energy : float
the rotational energy barrier in eV. This is the barrier for an
sitedensity : float
density of surface sites in cm^-2
rotationalminima : integer
the number of equivalent minima for an adsorbate's full rotation.
For example, 6 for an adsorbate on an fcc(111) top site
potentialenergy : float
the potential energy in eV (e.g., from atoms.get_potential_energy)
(if potentialenergy is unspecified, then the methods of this class
can be interpreted as the energy corrections)
mass : float
the mass of the adsorbate in amu (if mass is unspecified, then it will
be calculated from the atoms class)
inertia : float
the reduced moment of inertia of the adsorbate in amu*Ang^-2
(if inertia is unspecified, then it will be calculated from the
atoms class)
atoms : an ASE atoms object
used to calculate rotational moments of inertia and molecular mass
symmetrynumber : integer
symmetry number of the adsorbate. This is the number of symmetric arms
of the adsorbate and depends upon how it is bound to the surface.
For example, propane bound through its end carbon has a symmetry
number of 1 but propane bound through its middle carbon has a symmetry
number of 2. (if symmetrynumber is unspecified, then the default is 1)
"""

def __init__(self, vib_energies, trans_barrier_energy, rot_barrier_energy,
sitedensity, rotationalminima, potentialenergy=0.,
mass=None, inertia=None, atoms=None, symmetrynumber=1):
self.vib_energies = sorted(vib_energies, reverse=True)[:-3]
self.trans_barrier_energy = trans_barrier_energy * units._e
self.rot_barrier_energy = rot_barrier_energy * units._e
self.area = 1. / sitedensity / 100.0**2
self.rotationalminima = rotationalminima
self.potentialenergy = potentialenergy
self.atoms = atoms
self.symmetry = symmetrynumber

if (mass or atoms) and (inertia or atoms):
if mass:
self.mass = mass * units._amu
elif atoms:
self.mass = np.sum(atoms.get_masses()) * units._amu
if inertia:
self.inertia = inertia * units._amu / units.m**2
elif atoms:
self.inertia = (atoms.get_moments_of_inertia()[2] *
units._amu / units.m**2)
else:
raise RuntimeError('Either mass and inertia of the '
'adsorbate must be specified or '
'atoms must be specified.')

# Make sure no imaginary frequencies remain.
if sum(np.iscomplex(self.vib_energies)):
raise ValueError('Imaginary frequencies are present.')
else:
self.vib_energies = np.real(self.vib_energies)  # clear +0.j

# Calculate hindered translational and rotational frequencies
self.freq_t = np.sqrt(self.trans_barrier_energy / (2 * self.mass *
self.area))
self.freq_r = 1. / (2 * np.pi) * np.sqrt(self.rotationalminima**2 *
self.rot_barrier_energy /
(2 * self.inertia))

[docs]    def get_internal_energy(self, temperature, verbose=True):
"""Returns the internal energy (including the zero point energy),
in eV, in the hindered translator and hindered rotor model at a
specified temperature (K)."""

from scipy.special import iv

self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.3f eV'
write('Internal energy components at T = %.2f K:' % temperature)
write('=' * 31)

U = 0.

write(fmt % ('E_pot', self.potentialenergy))
U += self.potentialenergy

# Translational Energy
T_t = units._k * temperature / (units._hplanck * self.freq_t)
R_t = self.trans_barrier_energy / (units._hplanck * self.freq_t)
dU_t = 2 * (-1. / 2 - 1. / T_t / (2 + 16 * R_t) + R_t / 2 / T_t -
R_t / 2 / T_t *
iv(1, R_t / 2 / T_t) / iv(0, R_t / 2 / T_t) +
1. / T_t / (np.exp(1. / T_t) - 1))
dU_t *= units.kB * temperature
write(fmt % ('E_trans', dU_t))
U += dU_t

# Rotational Energy
T_r = units._k * temperature / (units._hplanck * self.freq_r)
R_r = self.rot_barrier_energy / (units._hplanck * self.freq_r)
dU_r = (-1. / 2 - 1. / T_r / (2 + 16 * R_r) + R_r / 2 / T_r -
R_r / 2 / T_r *
iv(1, R_r / 2 / T_r) / iv(0, R_r / 2 / T_r) +
1. / T_r / (np.exp(1. / T_r) - 1))
dU_r *= units.kB * temperature
write(fmt % ('E_rot', dU_r))
U += dU_r

# Vibrational Energy
dU_v = self._vibrational_energy_contribution(temperature)
write(fmt % ('E_vib', dU_v))
U += dU_v

# Zero Point Energy
dU_zpe = self.get_zero_point_energy()
write(fmt % ('E_ZPE', dU_zpe))
U += dU_zpe

write('-' * 31)
write(fmt % ('U', U))
write('=' * 31)
return U

[docs]    def get_zero_point_energy(self, verbose=True):
"""Returns the zero point energy, in eV, in the hindered
translator and hindered rotor model"""

zpe_t = 2 * (1. / 2 * self.freq_t * units._hplanck / units._e)
zpe_r = 1. / 2 * self.freq_r * units._hplanck / units._e
zpe_v = self.get_ZPE_correction()
zpe = zpe_t + zpe_r + zpe_v
return zpe

[docs]    def get_entropy(self, temperature, verbose=True):
"""Returns the entropy, in eV/K, in the hindered translator
and hindered rotor model at a specified temperature (K)."""

from scipy.special import iv

self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.7f eV/K%13.3f eV'
write('Entropy components at T = %.2f K:' % temperature)
write('=' * 49)
write('%15s%13s     %13s' % ('', 'S', 'T*S'))

S = 0.

# Translational Entropy
T_t = units._k * temperature / (units._hplanck * self.freq_t)
R_t = self.trans_barrier_energy / (units._hplanck * self.freq_t)
S_t = 2 * (-1. / 2 + 1. / 2 * np.log(np.pi * R_t / T_t) -
R_t / 2 / T_t *
iv(1, R_t / 2 / T_t) / iv(0, R_t / 2 / T_t) +
np.log(iv(0, R_t / 2 / T_t)) +
1. / T_t / (np.exp(1. / T_t) - 1) -
np.log(1 - np.exp(-1. / T_t)))
S_t *= units.kB
write(fmt % ('S_trans', S_t, S_t * temperature))
S += S_t

# Rotational Entropy
T_r = units._k * temperature / (units._hplanck * self.freq_r)
R_r = self.rot_barrier_energy / (units._hplanck * self.freq_r)
S_r = (-1. / 2 + 1. / 2 * np.log(np.pi * R_r / T_r) -
np.log(self.symmetry) -
R_r / 2 / T_r * iv(1, R_r / 2 / T_r) / iv(0, R_r / 2 / T_r) +
np.log(iv(0, R_r / 2 / T_r)) +
1. / T_r / (np.exp(1. / T_r) - 1) -
np.log(1 - np.exp(-1. / T_r)))
S_r *= units.kB
write(fmt % ('S_rot', S_r, S_r * temperature))
S += S_r

# Vibrational Entropy
S_v = self._vibrational_entropy_contribution(temperature)
write(fmt % ('S_vib', S_v, S_v * temperature))
S += S_v

# Concentration Related Entropy
N_over_A = np.exp(1. / 3) * (10.0**5 /
(units._k * temperature))**(2. / 3)
S_c = 1 - np.log(N_over_A) - np.log(self.area)
S_c *= units.kB
write(fmt % ('S_con', S_c, S_c * temperature))
S += S_c

write('-' * 49)
write(fmt % ('S', S, S * temperature))
write('=' * 49)
return S

[docs]    def get_helmholtz_energy(self, temperature, verbose=True):
"""Returns the Helmholtz free energy, in eV, in the hindered
translator and hindered rotor model at a specified temperature
(K)."""

self.verbose = True
write = self._vprint

U = self.get_internal_energy(temperature, verbose=verbose)
write('')
S = self.get_entropy(temperature, verbose=verbose)
F = U - temperature * S

write('')
write('Free energy components at T = %.2f K:' % temperature)
write('=' * 23)
fmt = '%5s%15.3f eV'
write(fmt % ('U', U))
write(fmt % ('-T*S', -temperature * S))
write('-' * 23)
write(fmt % ('F', F))
write('=' * 23)
return F

[docs]class IdealGasThermo(ThermoChem):
"""Class for calculating thermodynamic properties of a molecule
based on statistical mechanical treatments in the ideal gas
approximation.

Inputs for enthalpy calculations:

vib_energies : list
a list of the vibrational energies of the molecule (e.g., from
ase.vibrations.Vibrations.get_energies). The number of vibrations
used is automatically calculated by the geometry and the number of
atoms. If more are specified than are needed, then the lowest
numbered vibrations are neglected. If either atoms or natoms is
unspecified, then uses the entire list. Units are eV.
geometry : 'monatomic', 'linear', or 'nonlinear'
geometry of the molecule
potentialenergy : float
the potential energy in eV (e.g., from atoms.get_potential_energy)
(if potentialenergy is unspecified, then the methods of this
class can be interpreted as the energy corrections)
natoms : integer
the number of atoms, used along with 'geometry' to determine how
many vibrations to use. (Not needed if an atoms object is supplied
in 'atoms' or if the user desires the entire list of vibrations
to be used.)

Extra inputs needed for entropy / free energy calculations:

atoms : an ASE atoms object
used to calculate rotational moments of inertia and molecular mass
symmetrynumber : integer
symmetry number of the molecule. See, for example, Table 10.1 and
Appendix B of C. Cramer "Essentials of Computational Chemistry",
2nd Ed.
spin : float
the total electronic spin. (0 for molecules in which all electrons
are paired, 0.5 for a free radical with a single unpaired electron,
1.0 for a triplet with two unpaired electrons, such as O_2.)
"""

def __init__(self, vib_energies, geometry, potentialenergy=0.,
atoms=None, symmetrynumber=None, spin=None, natoms=None):
self.potentialenergy = potentialenergy
self.geometry = geometry
self.atoms = atoms
self.sigma = symmetrynumber
self.spin = spin
if natoms is None:
if atoms:
natoms = len(atoms)
# Cut the vibrations to those needed from the geometry.
if natoms:
if geometry == 'nonlinear':
self.vib_energies = vib_energies[-(3 * natoms - 6):]
elif geometry == 'linear':
self.vib_energies = vib_energies[-(3 * natoms - 5):]
elif geometry == 'monatomic':
self.vib_energies = []
else:
self.vib_energies = vib_energies
# Make sure no imaginary frequencies remain.
if sum(np.iscomplex(self.vib_energies)):
raise ValueError('Imaginary frequencies are present.')
else:
self.vib_energies = np.real(self.vib_energies)  # clear +0.j
self.referencepressure = 1.0e5  # Pa

[docs]    def get_enthalpy(self, temperature, verbose=True):
"""Returns the enthalpy, in eV, in the ideal gas approximation
at a specified temperature (K)."""

self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.3f eV'
write('Enthalpy components at T = %.2f K:' % temperature)
write('=' * 31)

H = 0.

write(fmt % ('E_pot', self.potentialenergy))
H += self.potentialenergy

zpe = self.get_ZPE_correction()
write(fmt % ('E_ZPE', zpe))
H += zpe

Cv_t = 3. / 2. * units.kB  # translational heat capacity (3-d gas)
write(fmt % ('Cv_trans (0->T)', Cv_t * temperature))
H += Cv_t * temperature

if self.geometry == 'nonlinear':  # rotational heat capacity
Cv_r = 3. / 2. * units.kB
elif self.geometry == 'linear':
Cv_r = units.kB
elif self.geometry == 'monatomic':
Cv_r = 0.
write(fmt % ('Cv_rot (0->T)', Cv_r * temperature))
H += Cv_r * temperature

dH_v = self._vibrational_energy_contribution(temperature)
write(fmt % ('Cv_vib (0->T)', dH_v))
H += dH_v

Cp_corr = units.kB * temperature
write(fmt % ('(C_v -> C_p)', Cp_corr))
H += Cp_corr

write('-' * 31)
write(fmt % ('H', H))
write('=' * 31)
return H

[docs]    def get_entropy(self, temperature, pressure, verbose=True):
"""Returns the entropy, in eV/K, in the ideal gas approximation
at a specified temperature (K) and pressure (Pa)."""

if self.atoms is None or self.sigma is None or self.spin is None:
raise RuntimeError('atoms, symmetrynumber, and spin must be '
'specified for entropy and free energy '
'calculations.')
self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.7f eV/K%13.3f eV'
write('Entropy components at T = %.2f K and P = %.1f Pa:' %
(temperature, pressure))
write('=' * 49)
write('%15s%13s     %13s' % ('', 'S', 'T*S'))

S = 0.0

# Translational entropy (term inside the log is in SI units).
mass = sum(self.atoms.get_masses()) * units._amu  # kg/molecule
S_t = (2 * np.pi * mass * units._k *
temperature / units._hplanck**2)**(3.0 / 2)
S_t *= units._k * temperature / self.referencepressure
S_t = units.kB * (np.log(S_t) + 5.0 / 2.0)
write(fmt % ('S_trans (1 bar)', S_t, S_t * temperature))
S += S_t

# Rotational entropy (term inside the log is in SI units).
if self.geometry == 'monatomic':
S_r = 0.0
elif self.geometry == 'nonlinear':
inertias = (self.atoms.get_moments_of_inertia() * units._amu /
(10.0**10)**2)  # kg m^2
S_r = np.sqrt(np.pi * np.product(inertias)) / self.sigma
S_r *= (8.0 * np.pi**2 * units._k * temperature /
units._hplanck**2)**(3.0 / 2.0)
S_r = units.kB * (np.log(S_r) + 3.0 / 2.0)
elif self.geometry == 'linear':
inertias = (self.atoms.get_moments_of_inertia() * units._amu /
(10.0**10)**2)  # kg m^2
inertia = max(inertias)  # should be two identical and one zero
S_r = (8 * np.pi**2 * inertia * units._k * temperature /
self.sigma / units._hplanck**2)
S_r = units.kB * (np.log(S_r) + 1.)
write(fmt % ('S_rot', S_r, S_r * temperature))
S += S_r

# Electronic entropy.
S_e = units.kB * np.log(2 * self.spin + 1)
write(fmt % ('S_elec', S_e, S_e * temperature))
S += S_e

# Vibrational entropy.
S_v = self._vibrational_entropy_contribution(temperature)
write(fmt % ('S_vib', S_v, S_v * temperature))
S += S_v

# Pressure correction to translational entropy.
S_p = - units.kB * np.log(pressure / self.referencepressure)
write(fmt % ('S (1 bar -> P)', S_p, S_p * temperature))
S += S_p

write('-' * 49)
write(fmt % ('S', S, S * temperature))
write('=' * 49)
return S

[docs]    def get_gibbs_energy(self, temperature, pressure, verbose=True):
"""Returns the Gibbs free energy, in eV, in the ideal gas
approximation at a specified temperature (K) and pressure (Pa)."""

self.verbose = verbose
write = self._vprint

H = self.get_enthalpy(temperature, verbose=verbose)
write('')
S = self.get_entropy(temperature, pressure, verbose=verbose)
G = H - temperature * S

write('')
write('Free energy components at T = %.2f K and P = %.1f Pa:' %
(temperature, pressure))
write('=' * 23)
fmt = '%5s%15.3f eV'
write(fmt % ('H', H))
write(fmt % ('-T*S', -temperature * S))
write('-' * 23)
write(fmt % ('G', G))
write('=' * 23)
return G

[docs]class CrystalThermo(ThermoChem):
"""Class for calculating thermodynamic properties of a crystalline
solid in the approximation that a lattice of N atoms behaves as a
system of 3N independent harmonic oscillators.

Inputs:

phonon_DOS : list
a list of the phonon density of states,
where each value represents the phonon DOS at the vibrational energy
value of the corresponding index in phonon_energies.

phonon_energies : list
a list of the range of vibrational energies (hbar*omega) over which
the phonon density of states has been evaluated. This list should be
the same length as phonon_DOS and integrating phonon_DOS over
phonon_energies should yield approximately 3N, where N is the number
of atoms per unit cell. If the first element of this list is
zero-valued it will be deleted along with the first element of
phonon_DOS. Units of vibrational energies are eV.

potentialenergy : float
the potential energy in eV (e.g., from atoms.get_potential_energy)
(if potentialenergy is unspecified, then the methods of this
class can be interpreted as the energy corrections)

formula_units : int
the number of formula units per unit cell. If unspecified, the
thermodynamic quantities calculated will be listed on a
per-unit-cell basis.
"""

def __init__(self, phonon_DOS, phonon_energies,
formula_units=None, potentialenergy=0.):
self.phonon_energies = phonon_energies
self.phonon_DOS = phonon_DOS

if formula_units:
self.formula_units = formula_units
self.potentialenergy = potentialenergy / formula_units
else:
self.formula_units = 0
self.potentialenergy = potentialenergy

[docs]    def get_internal_energy(self, temperature, verbose=True):
"""Returns the internal energy, in eV, of crystalline solid
at a specified temperature (K)."""

self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.4f eV'
if self.formula_units == 0:
write('Internal energy components at '
'T = %.2f K,\non a per-unit-cell basis:' % temperature)
else:
write('Internal energy components at '
'T = %.2f K,\non a per-formula-unit basis:' % temperature)
write('=' * 31)

U = 0.

omega_e = self.phonon_energies
dos_e = self.phonon_DOS
if omega_e[0] == 0.:
omega_e = np.delete(omega_e, 0)
dos_e = np.delete(dos_e, 0)

write(fmt % ('E_pot', self.potentialenergy))
U += self.potentialenergy

zpe_list = omega_e / 2.
if self.formula_units == 0:
zpe = np.trapz(zpe_list * dos_e, omega_e)
else:
zpe = np.trapz(zpe_list * dos_e, omega_e) / self.formula_units
write(fmt % ('E_ZPE', zpe))
U += zpe

B = 1. / (units.kB * temperature)
E_vib = omega_e / (np.exp(omega_e * B) - 1.)
if self.formula_units == 0:
E_phonon = np.trapz(E_vib * dos_e, omega_e)
else:
E_phonon = np.trapz(E_vib * dos_e, omega_e) / self.formula_units
write(fmt % ('E_phonon', E_phonon))
U += E_phonon

write('-' * 31)
write(fmt % ('U', U))
write('=' * 31)
return U

[docs]    def get_entropy(self, temperature, verbose=True):
"""Returns the entropy, in eV/K, of crystalline solid
at a specified temperature (K)."""

self.verbose = verbose
write = self._vprint
fmt = '%-15s%13.7f eV/K%13.4f eV'
if self.formula_units == 0:
write('Entropy components at '
'T = %.2f K,\non a per-unit-cell basis:' % temperature)
else:
write('Entropy components at '
'T = %.2f K,\non a per-formula-unit basis:' % temperature)
write('=' * 49)
write('%15s%13s     %13s' % ('', 'S', 'T*S'))

omega_e = self.phonon_energies
dos_e = self.phonon_DOS
if omega_e[0] == 0.:
omega_e = np.delete(omega_e, 0)
dos_e = np.delete(dos_e, 0)

B = 1. / (units.kB * temperature)
S_vib = (omega_e / (temperature * (np.exp(omega_e * B) - 1.)) -
units.kB * np.log(1. - np.exp(-omega_e * B)))
if self.formula_units == 0:
S = np.trapz(S_vib * dos_e, omega_e)
else:
S = np.trapz(S_vib * dos_e, omega_e) / self.formula_units

write('-' * 49)
write(fmt % ('S', S, S * temperature))
write('=' * 49)
return S

[docs]    def get_helmholtz_energy(self, temperature, verbose=True):
"""Returns the Helmholtz free energy, in eV, of crystalline solid
at a specified temperature (K)."""

self.verbose = True
write = self._vprint

U = self.get_internal_energy(temperature, verbose=verbose)
write('')
S = self.get_entropy(temperature, verbose=verbose)
F = U - temperature * S

write('')
if self.formula_units == 0:
write('Helmholtz free energy components at '
'T = %.2f K,\non a per-unit-cell basis:' % temperature)
else:
write('Helmholtz free energy components at '
'T = %.2f K,\non a per-formula-unit basis:' % temperature)
write('=' * 23)
fmt = '%5s%15.4f eV'
write(fmt % ('U', U))
write(fmt % ('-T*S', -temperature * S))
write('-' * 23)
write(fmt % ('F', F))
write('=' * 23)
return F