"""
Date: 11/26/2024
Author: Martin E. Liza
File: quantum_mechanics.py
Def: Contains Quantum Mechanics functions.
"""
import molmass
import numpy as np
import scipy.constants as s_consts
from haot import constants_tables
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def wavenumber_to_electronvolt(wavenumber_cm):
"""Convert wavenumber [cm^-1] to energy in Joules [J]."""
return wavenumber_to_joules(wavenumber_cm) / s_consts.eV
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def wavenumber_to_joules(wavenumber_cm):
"""Convert wavenumber [cm^-1] to energy in electron volts [eV]."""
return wavenumber_cm * s_consts.c * 100 * s_consts.h
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def molar_mass_to_kilogram(molar_mass_gmol):
"""Convert molar mass [g/mol] to mass [kg]."""
return molar_mass_gmol * 1e-3 / s_consts.N_A
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def zero_point_energy(molecule):
"""
Calculates zero-point energy (ZPE) using spectroscopy constants for
diatomic molecules
Parameters:
molecule (string): NO+, N2+, O2+, NO, N2, O2
Reference:
Experimental Vibrational Zero-Point Energies: Diatomic Molecules
doi.org/10.1063/1.2436891
Returns:
zpe (float): zero point energy [cm^-1]
"""
spectroscopy_const = constants_tables.spectroscopy_constants(molecule)
scope_var = spectroscopy_const["alpha_e"]
scope_var *= spectroscopy_const["omega_e"]
scope_var /= spectroscopy_const["B_e"]
zpe = spectroscopy_const["omega_e"] / 2
zpe -= spectroscopy_const["omega_xe"] / 2
zpe += spectroscopy_const["omega_ye"] / 8
zpe += spectroscopy_const["B_e"] / 4
zpe += scope_var / 12
zpe += scope_var**2 / (144 * spectroscopy_const["B_e"])
return zpe # [1/cm]
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def vibrational_partition_function(vibrational_number, temperature_K, molecule):
"""
Calculates the vibrational partition function base in the harmonic
terms only for diatomic molecules.
Parameters:
vibrational_number (int): vibrational quantum number
temperature_K (float):
molecule (string): NO+, N2+, O2+, NO, N2, O2
Returns:
z_vib (float): vibrational partition function
"""
z_vib = 0.0
for v in range(vibrational_number + 1):
z_vib += boltzman_factor(temperature_K, molecule, vibrational_number=v)
return z_vib
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def rotational_partition_function(rotational_number, temperature_K, molecule):
"""
Calculates the rotational partition function base in the harmonic
terms only for diatomic molecules.
Parameters:
rotational_number (int): rotational quantum number
temperature_K (float):
molecule (string): NO+, N2+, O2+, NO, N2, O2
Returns:
z_rot (float): rotational partition function
"""
z_rot = 0.0
for j in range(rotational_number + 1):
z_rot += boltzman_factor(temperature_K, molecule, rotational_number=j)
return z_rot
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def born_oppenheimer_partition_function(
vibrational_number, rotational_number, temperature_K, molecule
):
"""Calculates the partition function using the Born-Oppenheimer
approximation"""
z_bo = 0.0
for j in range(rotational_number + 1):
for v in range(vibrational_number + 1):
z_bo += boltzman_factor(
temperature_K,
molecule,
vibrational_number=v,
rotational_number=j,
born_opp_flag=True,
)
return z_bo
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def potential_dunham_coef_012(molecule):
"""Calculates the 0th, 1st, and 2nd Dunham potential coefficients.
Using: Ogilvie (https://doi.org/10.1016/0022-2852(76)90323-4)
and Herschbach (https://doi.org/10.1063/1.1731952)."""
spectroscopy_const = constants_tables.spectroscopy_constants(molecule)
a_0 = spectroscopy_const["omega_e"] ** 2 / (4 * spectroscopy_const["B_e"])
a_1 = -(
spectroscopy_const["alpha_e"]
* spectroscopy_const["omega_e"]
/ (6 * spectroscopy_const["B_e"] ** 2)
+ 1
)
a_2 = (5 / 4) * a_1**2 - (2 / 3) * (
spectroscopy_const["omega_xe"] / spectroscopy_const["B_e"]
)
return (a_0, a_1, a_2)
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def potential_dunham_coeff_m(a_1, a_2, m):
"""Calculates the higher order Dunham potential coefficients, using
Morizadeh work (https://doi.org/10.1016/j.theochem.2003.12.003)."""
tmp = (12 / a_1) ** (m - 2)
tmp *= 2 ** (m + 1) - 1
tmp *= (a_2 / 7) ** (m - 1)
for i in range(m - 2):
tmp *= 1 / (m + 2 - i)
return tmp
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def boltzman_factor(
temperature_K,
molecule,
vibrational_number=None,
rotational_number=None,
born_opp_flag=False,
):
"""Calculates the Boltzman factor at a given vibrational_number and/or
rotational_number. If the born_opp_flag is provided, it will calculate the
total energy using the Born-Oppenheimer approximation"""
# Initialize energy terms, degeneracy and thermal beta
energy_vib_k = 0
energy_rot_k = 0
degeneracy_rotation = 1
thermal_beta = 1 / (s_consts.k * temperature_K)
# Calculates Energy levels
if not born_opp_flag:
if vibrational_number is not None:
energy_vib_k = vibrational_energy_k(vibrational_number, molecule)
if rotational_number is not None:
energy_rot_k = rotational_energy_k(rotational_number, molecule)
degeneracy_rotation = 2 * rotational_number + 1
tot_energy = wavenumber_to_joules(energy_vib_k + energy_rot_k)
else:
degeneracy_rotation = 2 * rotational_number + 1
tot_energy = wavenumber_to_joules(
born_oppenheimer_approximation(
vibrational_number, rotational_number, molecule
)
)
return degeneracy_rotation * np.exp(-tot_energy * thermal_beta)
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def distribution_function(
temperature_K,
molecule,
vibrational_number=None,
rotational_number=None,
born_opp_flag=False,
):
"""Compute the population distribution function."""
# Calculates partition functions if vibrational or rotational numbers are provided
if not born_opp_flag:
z_rot = 1
z_vib = 1
if vibrational_number is not None:
z_vib = vibrational_partition_function(
vibrational_number, temperature_K, molecule
)
if rotational_number is not None:
z_rot = rotational_partition_function(
rotational_number, temperature_K, molecule
)
z_tot = z_rot * z_vib
else:
z_tot = born_oppenheimer_partition_function(
vibrational_number, rotational_number, temperature_K, molecule
)
# Create the distribution array based on the inputs provided
if vibrational_number and rotational_number:
tmp = np.zeros([vibrational_number + 1, rotational_number + 1])
for j in range(rotational_number + 1):
for v in range(vibrational_number + 1):
tmp[v][j] = boltzman_factor(
temperature_K=temperature_K,
molecule=molecule,
vibrational_number=v,
rotational_number=j,
born_opp_flag=born_opp_flag,
)
elif vibrational_number:
tmp = np.zeros(vibrational_number + 1)
for v in range(vibrational_number + 1):
tmp[v] = boltzman_factor(
temperature_K=temperature_K, molecule=molecule, vibrational_number=v
)
elif rotational_number:
tmp = np.zeros(rotational_number + 1)
for j in range(rotational_number + 1):
tmp[j] = boltzman_factor(
temperature_K=temperature_K, molecule=molecule, rotational_number=j
)
return tmp / z_tot
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def born_oppenheimer_approximation(vibrational_number, rotational_number, molecule):
"""Calculates the energy at a rotational and vibrational quantum number,
using the Born-Oppenheimer approximation."""
spectroscopy_constants = constants_tables.spectroscopy_constants(molecule)
vib_levels = vibrational_number + 1 / 2
rot_levels = rotational_number * (rotational_number + 1)
# Harmonic vibration and rotation terms
harmonic = spectroscopy_constants["omega_e"] * vib_levels
harmonic += spectroscopy_constants["B_e"] * rot_levels
# Anharmonic vibration and rotation terms
anharmonic = spectroscopy_constants["omega_xe"] * vib_levels**2
anharmonic += spectroscopy_constants["D_e"] * rot_levels**2
# Interaction between vibration and rotation modes
interaction = spectroscopy_constants["alpha_e"] * vib_levels * rot_levels
return harmonic - anharmonic - interaction # [cm^1]
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def vibrational_energy_k(vibrational_number, molecule):
"""Calculates the vibrational energy at a given vibrational quantum number,
using for the harmonic terms"""
spectroscopy_constants = constants_tables.spectroscopy_constants(molecule)
# Calculates the vibrational energy in units of wave number
vib_levels = vibrational_number + 1 / 2
return spectroscopy_constants["omega_e"] * vib_levels # [cm^-1]
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def rotational_energy_k(rotational_number, molecule):
"""Calculates the rotational energy at a given rotational quantum number,
using for the harmonic terms"""
spectroscopy_constants = constants_tables.spectroscopy_constants(molecule)
# Calculates the rotational energy in units of wave number
rot_levels = rotational_number * (rotational_number + 1)
return spectroscopy_constants["B_e"] * rot_levels # [cm^-1]
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def reduced_mass_kg(molecule_1, molecule_2):
"""Calculates the molar reduced mass and returns it in kg of two
elements"""
m_1 = molmass.Formula(molecule_1).mass
m_2 = molmass.Formula(molecule_2).mass
mu = m_1 * m_2 / (m_1 + m_2)
return molar_mass_to_kilogram(mu)
# TODO: Missing Translational Energy
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def tranlational_energy(principal_number_x, principal_number_y, principal_number_z):
print("TODO: Missing implementation of this function")