Source code for psi4.driver.p4util.spectrum

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__all__ = [
    "Gaussian",
    "Lineshape",
    "Lorentzian",
    "prefactor_ecd",
    "prefactor_opa",
    "spectrum",
]

from abc import abstractmethod
from dataclasses import dataclass
from typing import Callable, Dict, List, Union

import numpy as np
from psi4.driver import constants


[docs]@dataclass class Lineshape: """Lineshape ABC Attributes ---------- domain Domain of the spectral band. gamma A function returning the broadening factor. Notes ----- Why do we use a callable broadening factor? For plots in the *wavelength domain*, the broadening factor depends on the location of the band's maximum. """ domain: Union[np.ndarray, List[float]] gamma: Callable[[float], float]
[docs] @abstractmethod def lineshape(self, x_0: float) -> np.ndarray: pass
[docs] @abstractmethod def maximum(self, x_0: float) -> float: pass
[docs]class Gaussian(Lineshape): r"""Gaussian function on `domain`, centered at `x_0` with broadening `gamma`. Parameters ---------- domain The domain of the Gaussian profile. gamma Broadening parameter. This is related to the full width at half maximum as :math:`\mathrm{FWHM} = \gamma \sqrt{2\ln 2}` Notes ----- Use this profile to model inhomegenous broadening. """
[docs] def lineshape(self, x_0: float) -> np.ndarray: """Gaussian function on :py:attr:`Lineshape.domain`, centered at `x_0` with broadening :py:attr:`Lineshape.gamma`. Parameters ---------- x_0 Center of the Gaussian, i.e. its maximum. Returns ------- numpy.ndarray The Gaussian profile. """ prefactor = 2.0 / (self.gamma(x_0) * np.sqrt(2.0 * np.pi)) exponent = -2.0 * ((self.domain - x_0) / self.gamma(x_0))**2 return prefactor * np.exp(exponent)
[docs] def maximum(self, x_0: float) -> float: """Maximum value of Gaussian profile centered at `x_0`. Parameters ---------- x_0 Center of the Lorentzian, i.e. its maximum. """ return 2.0 / (self.gamma(x_0) * np.sqrt(2.0 * np.pi))
[docs]class Lorentzian(Lineshape): """Lorentzian function on `domain`, centered at `x_0` with broadening `gamma`. Parameters ---------- domain The domain of the Lorentzian profile. gamma Broadening parameter. This is the full width at half maximum (FWHM). Notes ----- Use this profile to model homogeneous broadening. """
[docs] def lineshape(self, x_0: float) -> np.ndarray: """Lorentzian function on :py:attr:`Lineshape.domain`, centered at `x_0` with broadening :py:attr:`Lineshape.gamma`. Parameters ---------- x_0 Center of the Lorentzian, i.e. its maximum. Returns ------- numpy.ndarray The Lorentzian profile. """ prefactor = 1.0 / np.pi numerator = self.gamma(x_0) / 2.0 denominator = (self.domain - x_0)**2 + numerator**2 return prefactor * (numerator / denominator)
[docs] def maximum(self, x_0: float) -> float: """Maximum value of Lorentzian profile centered at `x_0`. Parameters ---------- x_0 Center of the Lorentzian, i.e. its maximum. """ return 2.0 / (np.pi * self.gamma(x_0))
[docs]def prefactor_opa() -> float: r"""Prefactor for converting microscopic observable to decadic molar extinction coefficient in one-photon absorption. 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
[docs]def prefactor_ecd() -> float: r"""Prefactor for converting microscopic observable to decadic molar extinction coefficient in electronic circular dichroism. Notes ----- This function implements the calculation of the following prefactor: .. math:: k = \frac{16\pi^{2}N_{\mathrm{A}}}{3\times 1000\times \ln(10) (4 \pi \epsilon_{0}) n \hbar c^{2}} 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") au_to_Joule_inverse_Tesla = 2.0 * constants.get("Bohr magneton") * constants.conversion_factor("m", "cm") conversion = au_to_Coulomb_centimeter * au_to_Joule_inverse_Tesla numerator = 16.0 * np.pi**2 * N_A denominator = 3 * 1000 * np.log(10) * (4 * np.pi * e_0) * hbar * c**2 return (numerator / denominator) * conversion
[docs]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[str, numpy.ndarray] 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. https://doi.org/10.1002/9781118008720.ch2 """ # 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}}