Thermochemistry¶
ASE contains a thermochemistry
module that lets the user derive
commonly desired thermodynamic quantities of molecules and crystalline solids
from ASE output and some user-specified parameters. Four cases are currently
handled by this module: the ideal-gas limit (in which translational and
rotational degrees of freedom are taken into account), the harmonic limit
(generally used for adsorbates, in which all degrees of freedom are treated
harmonically), the hindered translator / hindered rotor model (used for
adsorbates, in which two degrees of freedom are translational, one is
rotational, and the remaining 3N-3 are vibrational), and a crystalline solid
model (in which a lattice of N atoms is treated as a system of 3N independent
harmonic oscillators). The first three cases rely on good vibrational energies
being fed to the calculators, which can be calculated with the
vibrations
module. Likewise, the crystalline solid model depends on
an accurate phonon density of states; this is readily calculated using the
phonons
module.
Ideal-gas limit¶
The thermodynamic quantities of ideal gases are calculated by assuming that
all spatial degrees of freedom are independent and separable into
translational, rotational, and vibrational degrees of freedom. The
IdealGasThermo
class supports calculation of
enthalpy (\(H\)), entropy (\(S\)), and Gibbs free energy (\(G\)),
and has the interface listed below.
- class ase.thermochemistry.IdealGasThermo(vib_energies, geometry, potentialenergy=0.0, atoms=None, symmetrynumber=None, spin=None, natoms=None, ignore_imag_modes=False)[source]¶
Class for calculating thermodynamic properties of a molecule based on statistical mechanical treatments in the ideal gas approximation.
Inputs for enthalpy calculations:
- vib_energieslist
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
- potentialenergyfloat
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)
- natomsinteger
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:
- atomsan ASE atoms object
used to calculate rotational moments of inertia and molecular mass
- symmetrynumberinteger
symmetry number of the molecule. See, for example, Table 10.1 and Appendix B of C. Cramer “Essentials of Computational Chemistry”, 2nd Ed.
- spinfloat
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.)
- ignore_imag_modesbool
If True, any imaginary frequencies present after the 3N-5/3N-6 cut will not be included in the calculation of the thermochemical properties. If False (default), a ValueError will be raised if any imaginary frequencies remain after the 3N-5/3N-6 cut.
- get_enthalpy(temperature, verbose=True)[source]¶
Returns the enthalpy, in eV, in the ideal gas approximation at a specified temperature (K).
Example¶
The IdealGasThermo
class would generally be called after an energy
optimization and a vibrational analysis. The user needs to supply certain
parameters if the entropy or free energy are desired, such as the geometry
and symmetry number. An example on the nitrogen molecule is:
from ase.build import molecule
from ase.calculators.emt import EMT
from ase.optimize import QuasiNewton
from ase.thermochemistry import IdealGasThermo
from ase.vibrations import Vibrations
atoms = molecule('N2')
atoms.calc = EMT()
dyn = QuasiNewton(atoms)
dyn.run(fmax=0.01)
potentialenergy = atoms.get_potential_energy()
vib = Vibrations(atoms)
vib.run()
vib_energies = vib.get_energies()
thermo = IdealGasThermo(vib_energies=vib_energies,
potentialenergy=potentialenergy,
atoms=atoms,
geometry='linear',
symmetrynumber=2, spin=0)
G = thermo.get_gibbs_energy(temperature=298.15, pressure=101325.)
This will give the thermodynamic summary output:
Enthalpy components at T = 298.15 K:
===============================
E_pot 0.263 eV
E_ZPE 0.076 eV
Cv_trans (0->T) 0.039 eV
Cv_rot (0->T) 0.026 eV
Cv_vib (0->T) 0.000 eV
(C_v -> C_p) 0.026 eV
-------------------------------
H 0.429 eV
===============================
Entropy components at T = 298.15 K and P = 101325.0 Pa:
=================================================
S T*S
S_trans (1 bar) 0.0015590 eV/K 0.465 eV
S_rot 0.0004101 eV/K 0.122 eV
S_elec 0.0000000 eV/K 0.000 eV
S_vib 0.0000016 eV/K 0.000 eV
S (1 bar -> P) -0.0000011 eV/K -0.000 eV
-------------------------------------------------
S 0.0019695 eV/K 0.587 eV
=================================================
Free energy components at T = 298.15 K and P = 101325.0 Pa:
=======================
H 0.429 eV
-T*S -0.587 eV
-----------------------
G -0.158 eV
=======================
Harmonic limit¶
In the harmonic limit, all degrees of freedom are treated harmonically. The
HarmonicThermo
class supports the calculation of internal energy,
entropy, and free energy. This class returns the Helmholtz free energy; if
the user assumes the pV term (in H = U + pV) is zero this can also be
interpreted as the Gibbs free energy. This class uses all of the energies
given to it in the vib_energies list; this is a list as can be generated
with the .get_energies() method of ase.vibrations.Vibrations
, but
the user should take care that all of these energies are real
(non-imaginary). The class HarmonicThermo
has the interface
described below.
- class ase.thermochemistry.HarmonicThermo(vib_energies, potentialenergy=0.0, ignore_imag_modes=False)[source]¶
Class for calculating thermodynamic properties in the approximation that all degrees of freedom are treated harmonically. Often used for adsorbates.
Inputs:
- vib_energieslist
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.
- potentialenergyfloat
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)
- ignore_imag_modesbool
If True, any imaginary frequencies will be ignored in the calculation of the thermochemical properties. If False (default), an error will be raised if any imaginary frequencies are present.
- get_entropy(temperature, verbose=True)[source]¶
Returns the entropy, in eV/K, in the harmonic approximation at a specified temperature (K).
Hindered translator / hindered rotor model¶
The hindered translator / hindered rotor model bridges the gap between the
2D gas (i.e. free translator / free rotor) and the 2D lattice gas (i.e.
harmonic oscillator). For an adsorbate containing N atoms, two degrees of
freedom are treated as hindered translations in the two directions parallel to
the surface, one degree of freedom is treated as a hindered rotation about the
axis perpendicular to the surface, and the remaining 3N-3 degrees of freedom
are treated as vibrations. The HinderedThermo
class supports the
calculation of internal energy, entropy, free energy, and zero point energy
(included in the internal energy). All of the thermodynamic properties
calculated here are at the standard state surface concentration (defined here
such that a 2D ideal gas at that concentration has 2/3 the translational
entropy of a 3D ideal gas at 1 bar pressure, so that \(\theta^0\) = 0.012
at 298 K for a surface with \(10^{15}\) sites/cm^{2}). This class
returns the Helmholtz free energy; if the user assumes that the pV term (in G =
U + pV - TS) is zero then this free energy can also be interpreted as the Gibbs
free energy. This class depends on the user defined translation barrier
(trans_barrier_energy) and rotational barrier (rot_barrier_energy) for the
adsorbate to move on the surface in order to calculate the translational and
rotational degrees of freedom. To calculate the vibrational degrees of freedom,
all 3N vibrational energies must be supplied in the vib_energies list and the
3N-3 largest vibrational energies are used to calculate the vibrational
contribution; this is a list as can be generated with the .get_energies()
method of ase.vibrations.Vibrations
. The class HinderedThermo
has the interface described below.
- class ase.thermochemistry.HinderedThermo(vib_energies, trans_barrier_energy, rot_barrier_energy, sitedensity, rotationalminima, potentialenergy=0.0, mass=None, inertia=None, atoms=None, symmetrynumber=1, ignore_imag_modes=False)[source]¶
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_energieslist
a list of all the vibrational energies of the adsorbate (e.g., from ase.vibrations.Vibrations.get_energies). If atoms is not provided, then the number of energies must 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_energyfloat
the translational energy barrier in eV. This is the barrier for an adsorbate to diffuse on the surface.
- rot_barrier_energyfloat
the rotational energy barrier in eV. This is the barrier for an adsorbate to rotate about an axis perpendicular to the surface.
- sitedensityfloat
density of surface sites in cm^-2
- rotationalminimainteger
the number of equivalent minima for an adsorbate’s full rotation. For example, 6 for an adsorbate on an fcc(111) top site
- potentialenergyfloat
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)
- massfloat
the mass of the adsorbate in amu (if mass is unspecified, then it will be calculated from the atoms class)
- inertiafloat
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)
- atomsan ASE atoms object
used to calculate rotational moments of inertia and molecular mass
- symmetrynumberinteger
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)
- ignore_imag_modesbool
If True, any imaginary frequencies present after the 3N-3 cut will not be included in the calculation of the thermochemical properties. If False (default), an error will be raised if imaginary frequencies are present after the 3N-3 cut.
- get_entropy(temperature, verbose=True)[source]¶
Returns the entropy, in eV/K, in the hindered translator and hindered rotor model at a specified temperature (K).
- get_helmholtz_energy(temperature, verbose=True)[source]¶
Returns the Helmholtz free energy, in eV, in the hindered translator and hindered rotor model at a specified temperature (K).
Example¶
The HinderedThermo
class would generally be called after an energy
optimization and a vibrational analysis. The user needs to supply certain
parameters, such as the vibrational energies, translational energy barrier,
rotational energy barrier, surface site density, number of equivalent minima
in a full rotation, and the number of symmetric arms of the adsorbate as it
rotates on the surface. The user also needs to supply either the mass of the
adsorbate and the reduced moment of inertia of the adsorbate as it rotates on
the surface or the user can supply the atoms object from which the mass and an
approximate reduced moment of inertia may be determined. An example for ethane
on a platinum (111) surface is:
from numpy import array
from ase.thermochemistry import HinderedThermo
vibs = array([3049.060670,
3040.796863,
3001.661338,
2997.961647,
2866.153162,
2750.855460,
1436.792655,
1431.413595,
1415.952186,
1395.726300,
1358.412432,
1335.922737,
1167.009954,
1142.126116,
1013.918680,
803.400098,
783.026031,
310.448278,
136.112935,
112.939853,
103.926392,
77.262869,
60.278004,
25.825447])
vib_energies = vibs / 8065.54429 # convert to eV from cm^-1
trans_barrier_energy = 0.049313 # eV
rot_barrier_energy = 0.017675 # eV
sitedensity = 1.5e15 # cm^-2
rotationalminima = 6
symmetrynumber = 1
mass = 30.07 # amu
inertia = 73.149 # amu Ang^-2
thermo = HinderedThermo(vib_energies=vib_energies,
trans_barrier_energy=trans_barrier_energy,
rot_barrier_energy=rot_barrier_energy,
sitedensity=sitedensity,
rotationalminima=rotationalminima,
symmetrynumber=symmetrynumber,
mass=mass,
inertia=inertia)
F = thermo.get_helmholtz_energy(temperature=298.15)
This will give the thermodynamic summary output:
Internal energy components at T = 298.15 K:
===============================
E_pot 0.000 eV
E_trans 0.049 eV
E_rot 0.018 eV
E_vib 0.076 eV
E_ZPE 1.969 eV
-------------------------------
U 2.112 eV
===============================
Entropy components at T = 298.15 K:
=================================================
S T*S
S_trans 0.0005074 eV/K 0.151 eV
S_rot 0.0002287 eV/K 0.068 eV
S_vib 0.0005004 eV/K 0.149 eV
S_con 0.0005044 eV/K 0.150 eV
-------------------------------------------------
S 0.0017409 eV/K 0.519 eV
=================================================
Free energy components at T = 298.15 K:
=======================
U 2.112 eV
-T*S -0.519 eV
-----------------------
F 1.593 eV
=======================
Crystals¶
In this model a crystalline solid is treated as a periodic system of
independent harmonic oscillators. The CrystalThermo
class supports
the calculation of internal energy (\(U\)), entropy (\(S\)) and
Helmholtz free energy (\(F\)), and has the interface listed below.
- class ase.thermochemistry.CrystalThermo(phonon_DOS, phonon_energies, formula_units=None, potentialenergy=0.0)[source]¶
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_DOSlist
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_energieslist
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.
- potentialenergyfloat
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_unitsint
the number of formula units per unit cell. If unspecified, the thermodynamic quantities calculated will be listed on a per-unit-cell basis.
- get_entropy(temperature, verbose=True)[source]¶
Returns the entropy, in eV/K, of crystalline solid at a specified temperature (K).
Example¶
The CrystalThermo
class will generally be called after an energy
optimization and a phonon vibrational analysis of the crystal. An example for
bulk gold is:
from ase.calculators.emt import EMT
from ase.optimize import QuasiNewton
from ase.phonons import Phonons
from ase.spacegroup import crystal
from ase.thermochemistry import CrystalThermo
# Set up gold bulk and attach EMT calculator
a = 4.078
atoms = crystal('Au', (0., 0., 0.),
spacegroup=225,
cellpar=[a, a, a, 90, 90, 90],
pbc=(1, 1, 1))
calc = EMT()
atoms.calc = calc
qn = QuasiNewton(atoms)
qn.run(fmax=0.05)
potentialenergy = atoms.get_potential_energy()
# Phonon analysis
N = 5
ph = Phonons(atoms, calc, supercell=(N, N, N), delta=0.05)
ph.run()
ph.read(acoustic=True)
phonon_energies, phonon_DOS = ph.dos(kpts=(40, 40, 40), npts=3000,
delta=5e-4)
# Calculate the Helmholtz free energy
thermo = CrystalThermo(phonon_energies=phonon_energies,
phonon_DOS=phonon_DOS,
potentialenergy=potentialenergy,
formula_units=4)
F = thermo.get_helmholtz_energy(temperature=298.15)
This will give the thermodynamic summary output:
Internal energy components at T = 298.15 K,
on a per-formula-unit basis:
===============================
E_pot 0.0022 eV
E_ZPE 0.0135 eV
E_phonon 0.0629 eV
-------------------------------
U 0.0786 eV
===============================
Entropy components at T = 298.15 K,
on a per-formula-unit basis:
=================================================
S T*S
-------------------------------------------------
S 0.0005316 eV/K 0.1585 eV
=================================================
Helmholtz free energy components at T = 298.15 K,
on a per-formula-unit basis:
=======================
U 0.0786 eV
-T*S -0.1585 eV
-----------------------
F -0.0799 eV
=======================
Background¶
Ideal gas¶
The conversion of electronic structure calculations to thermodynamic properties
in the ideal-gas limit is well documented; see, for example, Chapter 10 of
Cramer, 2004. The key equations used in the IdealGasThermo
class are
summarized here.
C.J. Cramer. Essentials of Computational Chemistry, Second Edition. Wiley, 2004.
The ideal-gas enthalpy is calculated from extrapolation of the energy at 0 K to the relevant temperature (for an ideal gas, the enthalpy is not a function of pressure):
where the first two terms are the electronic energy and the zero-point energy, and the integral is over the constant-pressure heat capacity. The heat capacity is separable into translational, rotational, vibrational, and electronic parts (plus a term of \(k_\text{B}\) to switch from constant-volume to constant-pressure):
The translational heat capacity is 3/2 \(k_\text{B}\) for a 3-dimensional gas. The rotational heat capacity is 0 for a monatomic species, \(k_\text{B}\) for a linear molecule, and 3/2 \(k_\text{B}\) for a nonlinear molecule. In this module, the electronic component of the heat capacity is assumed to be 0. The vibrational heat capacity contains \(3N-6\) degrees of freedom for nonlinear molecules and \(3N-5\) degrees of freedom for linear molecules (where \(N\) is the number of atoms). The integrated form of the vibrational heat capacity is:
where \(\epsilon_i\) are the energies associated with the vibrational frequencies, \(\epsilon_i = h \omega_i\).
The ideal gas entropy can be calculated as a function of temperature and pressure as:
where the translational, rotational, electronic, and vibrational components are calculated as below. (Note that the translational component also includes components from the Stirling approximation, and that the vibrational degrees of freedom are enumerated the same as in the above.)
\(I_\text{A}\) through \(I_\text{C}\) are the three principle moments of inertia for a non-linear molecule. \(I\) is the degenerate moment of inertia for a linear molecule. \(\sigma\) is the symmetry number of the molecule.
The ideal-gas Gibbs free energy is then just calculated from the combination of the enthalpy and entropy:
Harmonic limit¶
The conversion of electronic structure calculation information into
thermodynamic properties is less established for adsorbates. However, the
simplest approach often taken is to treat all \(3N\) degrees of freedom of the
adsorbate harmonically since the adsorbate often has no real translational or
rotational degrees of freedom. This is the approach implemented in the
HarmonicThermo
class. Thus, the internal energy and entropy of the
adsorbate are calculated as
and the Helmholtz free energy is calculated as
In this case, the number of harmonic energies (\(\epsilon_i\)) used in the summation is generally \(3N\), where \(N\) is the number of atoms in the adsorbate. If the user assumes that the \(pV\) term in \(H = U + pV\) is negligible, then the Helmholtz free energy can be used to approximate the Gibbs free energy, as \(G = F + pV\).
Hindered translator / hindered rotor¶
The conversion of electronic structure calculations to thermodynamic properties
in the hindered translator / hindered rotor model was developed for adsorbates
on close packed surfaces and is documented by Sprowl, Campbell, and Arnadottir,
2016. The key equations used in the HinderedThermo
class are
summarized here.
L.H. Sprowl, C.T. Campbell, and L. Arnadottir. Hindered Translator and Hindered Rotor Models for Adsorbates: Partition Functions and Entropies. J. Phys. Chem. C, 2016, 120 (18), pp 9719-9731.
L.H. Sprowl, C.T. Campbell, and L. Arnadottir. Correction to “Hindered Translator and Hindered Rotor Models for Adsorbates: Partition Functions and Entropies”. J. Phys. Chem. C, 2017, 121 (17), pp 9655-9655.
C.T. Campbell, L.H. Sprowl, and L. Arnadottir. Equilibrium Constants and Rate Constants for Adsorbates: Two-Dimensional (2D) Ideal Gas, 2D Ideal Lattice Gas, and Ideal Hindered Translator Models. J. Phys. Chem. C, 2016, 120 (19), pp 10283-10297.
The \(3N-3\) largest vibrational frequencies are used to calculate the vibrational contributions to the internal energy and the entropy. The remaining three degrees of freedom are calculated from two translational contributions and one rotational contribution of the adsorbate. The energy barriers for the adsorbate to translate and rotate on a close packed surface are used to calculate the translational and rotational frequencies, respectively. From the translational and rotational frequencies, the translational and rotational contributions to the internal energy and the entropy of the adsorbate are determined. The calculation of the translational frequency is:
where \(W_{trans}\) is the translational energy barrier, \(m\) is the mass of the adsorbate, and \(A\) is the area per surface atom, or the inverse of the surface site density. The rotational frequency is calculated as:
where \(W_{rot}\) is the rotational energy barrier, \(n\) is the number of equivalent energy minima in a full rotation of the adsorbate, and \(I\) is the reduced moment of inertia of the adsorbate about its surface bond. Two variables are now introduced, a unitless temperature
and a unitless energy barrier
to ease the internal energy and entropy calculations.
The internal energy of the adsorbate is calculated as:
where \(E_{trans}\) and \(E_{rot}\) are:
where \(I_{n}\) is the nth-order modified Bessel function of the first kind. Similarly for the harmonic limit, \(E_{vib}\) is:
The entropy of the adsorbate is calculated as:
where \(S_{trans}\) and \(S_{rot}\) are:
and \(S_{vib}\) is:
\(S_{con}\) is a concentration related entropy and is calculated as:
where
The Helmholtz free energy is calculated as:
If the user assumes that the \(pV\) term in \(H = U + pV\) is negligible, then the Helmholtz free energy can be used to approximate the Gibbs free energy, as \(G = F + pV\).
Crystalline solid¶
The derivation of the partition function for a crystalline solid is fairly straight-forward and can be found, for example, in Chapter 11 of McQuarrie, 2000.
D.A. McQuarrie. Statistical Mechanics. University Science Books, 2000.
The treatment implemented in the CrystalThermo
class depends on
introducing normal coordinates to the entire crystal and treating each atom in
the lattice as an independent harmonic oscillator. This yields the partition
function
where \(\epsilon_j\) are the \(3N\) vibrational energy levels and \(E_\text{elec}\) is the electronic energy of the crystalline solid. Now, taking the logarithm of the partition function and replacing the resulting sum with an integral (assuming that the energy level spacing is essentially continuous) gives
Here \(\sigma (\epsilon)\) represents the degeneracy or phonon density of
states as a function of vibrational energy. Once this function has been
determined (i.e. using the ase.phonons
module), it is a simple matter
to calculate the canonical ensemble thermodynamic quantities; namely the
internal energy, the entropy and the Helmholtz free energy.