def prefactor_opa() -> float:
r"""Prefactor for converting microscopic observable to decadic molar
extinction coefficient in one-photon absorption.
Returns
-------
prefactor : float
Notes
-----
This function implements the calculation of the following prefactor:
.. math::
k = \frac{4\pi^{2}N_{\mathrm{A}}}{3\times 1000\times \ln(10) (4 \pi \epsilon_{0}) n \hbar c}
The prefactor is computed in SI units and then adjusted for the fact that
we use atomic units to express microscopic observables: excitation energies
and transition dipole moments.
The refractive index :math:`n` is, in general, frequency-dependent. We
assume it to be constant and equal to 1.
"""
N_A = constants.get("Avogadro constant")
c = constants.get("speed of light in vacuum")
hbar = constants.get("Planck constant over 2 pi")
e_0 = constants.get("electric constant")
au_to_Coulomb_centimeter = constants.get(
"elementary charge") * constants.get(
"Bohr radius") * constants.conversion_factor("m", "cm")
numerator = 4.0 * np.pi**2 * N_A
denominator = 3 * 1000 * np.log(10) * (4 * np.pi * e_0) * hbar * c
return (numerator / denominator) * au_to_Coulomb_centimeter**2
def spectrum(*,
poles: Union[List[float], np.ndarray],
residues: Union[List[float], np.ndarray],
kind: str = "opa",
lineshape: str = "gaussian",
gamma: float = 0.2,
npoints: int = 5000,
out_units: str = "nm") -> Dict[str, np.ndarray]:
r"""One-photon absorption (OPA) or electronic circular dichroism (ECD)
spectra with phenomenological line broadening.
This function gives arrays of values ready to be plotted as OPA spectrum:
.. math::
\varepsilon(\omega) =
\frac{4\pi^{2}N_{\mathrm{A}}\omega}{3\times 1000\times \ln(10) (4 \pi \epsilon_{0}) n \hbar c}
\sum_{i \rightarrow j}g_{ij}(\omega)|\mathbf{\mu}_{ij}|^{2}
or ECD spectrum:
.. math::
\Delta\varepsilon(\omega) =
\frac{16\pi^{2}N_{\mathrm{A}}\omega}{3\times 1000\times \ln(10) (4 \pi \epsilon_{0}) n \hbar c^{2}}
\sum_{i \rightarrow j}g_{ij}(\omega)\Im(\mathbf{\mu}_{ij}\cdot\mathbf{m}_{ij})
in macroscopic units of :math:`\mathrm{L}\cdot\mathrm{mol}^{-1}\cdot\mathrm{cm}^{-1}`.
The lineshape function :math:`g_{ij}(\omega)` with phenomenological
broadening :math:`\gamma` is used for the convolution of the infinitely
narrow results from a linear response calculation.
Parameters
----------
poles
Poles of the response function, i.e. the excitation energies.
These are **expected** in atomic units of angular frequency.
residues
Residues of the linear response functions, i.e. transition dipole moments (OPA) and rotatory strengths (ECD).
These are **expected** in atomic units.
kind
{"opa", "ecd"}
Which kind of spectrum to generate, one-photon absorption ("opa") or electronic circular dichroism ("ecd").
Default is `opa`.
lineshape
{"gaussian", "lorentzian"}
The lineshape function to use in the fitting. Default is `gaussian`.
gamma
Full width at half maximum of the lineshape function.
Default is 0.2 au of angular frequency.
This value is **expected** in atomic units of angular frequency.
npoints
How many points to generate for the x axis. Default is 5000.
out_units
Units for the output array `x`, the x axis of the spectrum plot.
Default is wavelengths in nanometers.
Valid (and case-insensitive) values for the units are:
- `au` atomic units of angular frequency
- `Eh` atomic units of energy
- `eV`
- `nm`
- `THz`
Returns
-------
spectrum : Dict
The fitted electronic absorption spectrum, with units for the x axis specified by the `out_units` parameter.
This is a dictionary containing the convoluted (key: `convolution`) and the infinitely narrow spectra (key: `sticks`).
.. code-block:: python
{"convolution": {"x": np.ndarray, "y": np.ndarray},
"sticks": {"poles": np.ndarray, "residues": np.ndarray}}
Notes
-----
* Conversion of the broadening parameter :math:`\gamma`.
The lineshape functions are formulated as functions of the angular frequency :math:`\omega`.
When converting to other physical quantities, the broadening parameter has to be modified accordingly.
If :math:`\gamma_{\omega}` is the chosen broadening parameter then:
- Wavelength: :math:`gamma_{\lambda} = \frac{\lambda_{ij}^{2}}{2\pi c}\gamma_{\omega}`
- Frequency: :math:`gamma_{\nu} = \frac{\gamma_{\omega}}{2\pi}`
- Energy: :math:`gamma_{E} = \gamma_{\omega}\hbar`
References
----------
A. Rizzo, S. Coriani, K. Ruud, "Response Function Theory Computational Approaches to Linear and Nonlinear Optical Spectroscopy". In Computational Strategies for Spectroscopy.
"""
# Transmute inputs to np.ndarray
if isinstance(poles, list):
poles = np.array(poles)
if isinstance(residues, list):
residues = np.array(residues)
# Validate input arrays
if poles.shape != residues.shape:
raise ValueError(
f"Shapes of poles ({poles.shape}) and residues ({residues.shape}) vectors do not match!"
)
# Validate kind of spectrum
kind = kind.lower()
valid_kinds = ["opa", "ecd"]
if kind not in valid_kinds:
raise ValueError(
f"Spectrum kind {kind} not among recognized ({valid_kinds})")
# Validate output units
out_units = out_units.lower()
valid_out_units = ["au", "eh", "ev", "nm", "thz"]
if out_units not in valid_out_units:
raise ValueError(
f"Output units {out_units} not among recognized ({valid_out_units})"
)
c = constants.get("speed of light in vacuum")
c_nm = c * constants.conversion_factor("m", "nm")
hbar = constants.get("Planck constant over 2 pi")
h = constants.get("Planck constant")
Eh = constants.get("Hartree energy")
au_to_nm = 2.0 * np.pi * c_nm * hbar / Eh
au_to_THz = (Eh / h) * constants.conversion_factor("Hz", "THz")
au_to_eV = constants.get("Hartree energy in eV")
converters = {
"au": lambda x: x, # Angular frequency in atomic units
"eh": lambda x: x, # Energy in atomic units
"ev": lambda x: x * au_to_eV, # Energy in electronvolts
"nm": lambda x: au_to_nm / x, # Wavelength in nanometers
"thz": lambda x: x * au_to_THz, # Frequency in terahertz
}
# Perform conversion of poles from au of angular frequency to output units
poles = converters[out_units](poles)
# Broadening functions
gammas = {
"au": lambda x_0: gamma, # Angular frequency in atomic units
"eh": lambda x_0: gamma, # Energy in atomic units
"ev": lambda x_0: gamma * au_to_eV, # Energy in electronvolts
"nm": lambda x_0: ((x_0**2 * gamma * (Eh / hbar)) /
(2 * np.pi * c_nm)), # Wavelength in nanometers
"thz": lambda x_0: gamma * au_to_THz, # Frequency in terahertz
}
# Generate x axis
# Add a fifth of the range on each side
expand_side = (np.max(poles) - np.min(poles)) / 5
x = np.linspace(
np.min(poles) - expand_side,
np.max(poles) + expand_side, npoints)
# Validate lineshape
lineshape = lineshape.lower()
valid_lineshapes = ["gaussian", "lorentzian"]
if lineshape not in valid_lineshapes:
raise ValueError(
f"Lineshape {lineshape} not among recognized ({valid_lineshapes})")
# Obtain lineshape function
shape = Gaussian(
x, gammas[out_units]) if lineshape == "gaussian" else Lorentzian(
x, gammas[out_units])
# Generate y axis, i.e. molar decadic absorption coefficient
prefactor = prefactor_opa() if kind == "opa" else prefactor_ecd()
transform_residue = (lambda x: x**2) if kind == "opa" else (lambda x: x)
y = prefactor * x * np.sum([
transform_residue(r) * shape.lineshape(p)
for p, r in zip(poles, residues)
],
axis=0)
# Generate sticks
sticks = prefactor * np.array([
p * transform_residue(r) * shape.maximum(p)
for p, r in zip(poles, residues)
])
return {
"convolution": {
"x": x,
"y": y
},
"sticks": {
"poles": poles,
"residues": sticks
}
}