Statistical Thermodynamic Models

StatMech Base Class

The StatMech class stores the classes associated with each mode. Each mode has methods that specifies how to calculate thermodynamic quantities, such as partition functions, heat capacities, internal energy, enthalpy, entropy, Helmholtz energy and Gibbs energy.

StatMech

Base class for statistical mechanic models.

The StatMech object can be initialized in two ways:

1. by each passing each mode as objects or

2. passing each mode as a class and the associated parameters.

The examples below give an equivalent result:

Example of initialization using objects

import numpy as np
from ase.build import molecule
from pmutt.statmech import StatMech, trans, vib, rot, elec

atoms = molecule('H2O')
H2O_trans = trans.FreeTrans(n_degrees=3, atoms=atoms)
H2O_vib = vib.HarmonicVib(vib_wavenumbers=[3825.434, 3710.2642, 1582.432])
H2O_rot = rot.RigidRotor(symmetrynumber=2, atoms=atoms)
H2O_elec = elec.GroundStateElec(potentialenergy=-14.2209, spin=0)
H2O_statmech = StatMech(trans_model=H2O_trans,
                        vib_model=H2O_vib,
                        rot_model=H2O_rot,
                        elec_model=H2O_elec)

Example of initialization using classes and parameters

import numpy as np
from ase.build import molecule
from pmutt.statmech import StatMech, trans, vib, rot, elec


H2O_statmech = StatMech(trans_model=trans.FreeTrans,
                        n_degrees=3,
                        vib_model=vib.HarmonicVib,
                        vib_wavenumbers=[3825.434, 3710.2642, 1582.432],
                        rot_model=rot.RigidRotor,
                        symmetrynumber=2,
                        atoms=molecule('H2O'),
                        elec_model=elec.GroundStateElec,
                        potentialenergy=-14.2209,
                        spin=0)

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Presets

If you are using a common model (e.g. the ideal gas model), then you can get the default parameters from the dictionary, pmutt.statmech.presets. The same H2O StatMech object can be specified without the need to pass all the types of modes:

from ase.build import molecule
from pmutt.statmech import StatMech, presets

idealgas_defaults = presets['idealgas']
H2O_new = StatMech(vib_wavenumbers=[3825.434, 3710.2642, 1582.432],
                   potentialenergy=-14.2209,
                   atoms=molecule('H2O'),
                   spin=0,
                   symmetrynumber=2,
                   **idealgas_defaults)

Currently supported presets are described below. The first table shows the attributes already specified, and the second table shows the attributes that are still required, and the third table shows the required and optional parameters to calculate a thermodynamic property (where the value in parentheses is the default value).

Ideal Gas (idealgas)

Useful for modeling ideal gases. Assumes the system has:

• 3 degrees of translational freedom and no interactions with other molecules

• harmonic vibrations

• rigid rotor rotations

• electronic ground state

• nuclear ground state

Set Attributes	Default Value
trans_model	FreeTrans
n_degrees	3
vib_model	HarmonicVib
elec_model	GroundStateElec
rot_model	RigidRotor

Required Attributes	Description
molecular_weight	(float) Molecular weight in g/mol
vib_wavenumbers	(list of float) Vibrational wavenumbers in 1/cm
potentialenergy	(float) Electronic potential energy in eV
spin	(float) Electron spin. 0 if all electrons are paired (e.g. N2), 0.5 if the specie is a radical (e.g. CH3.), 1 if the specie exists as a triplet (e.g. O2).
geometry	(str) Geometry of molecule
rot_temperatures	(list of float) Rotational temperatures in K
symmetrynumber	(int) Symmetry number
atoms	(ase.Atoms object) Optional. If this parameter is specified, molecular_weight, geometry, rot_temperatures, and potentialenergy do not have to be specified.

Thermodynamic Quantity	Expected Parameters	Optional Parameters
\(q\)	T	ignore_q_elec (False) P (1.01325 bar)
\(\frac {C_V} {R}\)	T
\(\frac {C_P} {R}\)	T
\(\frac {U} {RT}\)	T
\(\frac {H} {RT}\)	T
\(\frac {S} {R}\)	T	P (1.01325 bar)
\(\frac {A} {RT}\)	T	P (1.01325 bar)
\(\frac {G} {RT}\)	T	P (1.01325 bar)

Harmonic Approximation (harmonic)

Typically used to model adsorbates. Assumes the system has:

• harmonic vibrations

• electronic ground state

Parameter	Default Value
vib_model	HarmonicVib
elec_model	GroundStateElec

Required Parameters	Description
vib_wavenumbers	(list of float) Vibrational wavenumbers in 1/cm
potentialenergy	(float) Electronic potential energy in eV
spin	(float) Electron spin
atoms	(ase.Atoms object) Optional. If this parameter is specified, potentialenergy does not have to be specified.

Thermodynamic Quantity	Expected Parameters	Optional Parameters
\(q\)	T	ignore_q_elec (True)
\(\frac {C_V} {R}\)	T
\(\frac {C_P} {R}\)	T
\(\frac {U} {RT}\)	T
\(\frac {H} {RT}\)	T
\(\frac {S} {R}\)	T
\(\frac {A} {RT}\)	T
\(\frac {G} {RT}\)	T

Electronic (electronic)

Typically used to model systems where the electronic modes dominate and changes due to temperature are not important. Assumes the system has:

• electronic ground state

Parameter	Default Value
elec_model	GroundStateElec

Required Parameters	Description
potentialenergy	(float) Electronic potential energy in eV
spin	(float) Electron spin
atoms	(ase.Atoms object) Optional. If this parameter is specified, potentialenergy does not have to be specified

Thermodynamic Quantity	Expected Parameters	Optional Parameters
\(q\)	T	ignore_q_elec (True)
\(\frac {C_V} {R}\)
\(\frac {C_P} {R}\)
\(\frac {U} {RT}\)	T
\(\frac {H} {RT}\)	T
\(\frac {S} {R}\)
\(\frac {A} {RT}\)	T
\(\frac {G} {RT}\)	T

Placeholder (placeholder)

Typically used to model species that have no contribution. The partition function is set to 1 and all other thermodynamic quantities evaluate to 0.

Parameter	Default Value
trans_model	EmptyMode
vib_model	EmptyMode
rot_model	EmptyMode
elec_model	EmptyMode
nucl_model	EmptyMode

Required Parameters	Description
N/A

Thermodynamic Quantity	Expected Parameters	Optional Parameters
\(q\)
\(\frac {C_V} {R}\)
\(\frac {C_P} {R}\)
\(\frac {U} {RT}\)
\(\frac {H} {RT}\)
\(\frac {S} {R}\)
\(\frac {A} {RT}\)
\(\frac {G} {RT}\)

Constant (constant)

Arbitrarily specify a constant for each quantity. This is primarily used for testing so be careful as you can disobey some fundamental thermodynamic quantities. For example,

\(G = H - TS\)

may not be obeyed.

Parameter	Default Value
elec_model	ConstantMode

Required Parameters	Description
q	(float) Optional. Partition function. Default is 1
Cv	(float) Optional. Heat capacity at constant volume in eV/K. Default is 0
Cp	(float) Optional. Heat capacity at constant pressure in eV/K. Default is 0
U	(float) Optional. Internal energy in eV. Default is 0
H	(float) Optional. Enthalpy in eV. Default is 0
S	(float) Optional. Entropy in eV/K. Default is 0
F	(float) Optional. Helmholtz energy in eV. Default is 0
G	(float) Optional. Gibbs energy in eV. Default is 0

Thermodynamic Quantity	Expected Parameters	Optional Parameters
\(q\)
\(\frac {C_V} {R}\)
\(\frac {C_P} {R}\)
\(\frac {U} {RT}\)
\(\frac {H} {RT}\)
\(\frac {S} {R}\)
\(\frac {A} {RT}\)
\(\frac {G} {RT}\)

The pmutt.statmech.presets dictionary is flexible where you can create a new entry if you will use a model often.

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Translational Models

trans.FreeTrans([n_degrees, ...])

Translational mode using ideal gas assumption.

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Vibrational Models

vib.HarmonicVib([vib_wavenumbers, ...])	Vibrational modes using the harmonic approximation.
vib.QRRHOVib(vib_wavenumbers[, Bav, v0, ...])	Vibrational modes using the Quasi Rigid Rotor Harmonic Oscillator approximation.
vib.EinsteinVib(einstein_temperature[, ...])	Einstein model of a crystal.
vib.DebyeVib(debye_temperature, ...)	Debye model of a crystal.

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Rotational Models

rot.RigidRotor(symmetrynumber[, ...])

Rotational mode using the rigid rotor assumption.

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Electronic Models

elec.GroundStateElec([potentialenergy, ...])	Electronic modes using the ideal gas assumption.
lsr.LSR(slope, intercept, reaction[, ...])	Represents a linear scaling relationship
lsr.ExtendedLSR(slopes, intercept, reactions)	Represents an extended linear scaling relationship

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Nuclear Models

Typically these are unimportant for chemical reactions, but the module is present in case nuclear modes become important in the future.

nucl.EmptyNucl()

Nuclear modes.

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Misc.

EmptyMode()	Placeholder mode that returns 1 for partition function and 0 for all functions other thermodynamic properties.
ConstantMode([q, Cv, Cp, U, H, S, F, G, notes])	Mode where thermodynamic properties can be arbitrarily set.

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Creating New StatMech Models

New models should inherit from pmutt._ModelBase. This class already has the dimensional methods (e.g. get_Cv, get_U, get_S), and some routine methods (e.g. get_FoRT, get_GoRT, from_dict, to_dict) implemented.

For full accessibility, the following methods should be implemented:

• get_q

• get_CvoR

• get_CpoR

• get_UoRT

• get_HoRT

• get_SoR