# To check for integers
from functools import partial
from numbers import Integral
from math import pi
from math import sqrt as msqrt
from math import factorial as fact
import numpy as np
from numpy import cos, sin
from numpy import take, sqrt, square
from scipy.special import lpmv
from scipy.interpolate import UnivariateSpline
from ._internal import set_module
from . import _plot as plt
from . import _array as _a
from .shape import Sphere
from .utils.mathematics import cart2spher
__all__ = ['Orbital', 'SphericalOrbital', 'AtomicOrbital']
# Create the factor table for the real spherical harmonics
def _rfact(l, m):
pi4 = 4 * pi
if m == 0:
return msqrt((2*l + 1)/pi4)
elif m < 0:
return -msqrt(2*(2*l + 1)/pi4 * fact(l-m)/fact(l+m)) * (-1) ** m
return msqrt(2*(2*l + 1)/pi4 * fact(l-m)/fact(l+m))
# This is a list of dict
# [0]{0} is l==0, m==0
# [1]{-1} is l==1, m==-1
# [1]{1} is l==1, m==1
# and so on.
# Calculate it up to l == 5 which is the h shell
_rspher_harm_fact = [{m: _rfact(l, m) for m in range(-l, l+1)} for l in range(6)]
# Clean-up
del _rfact
def _rspherical_harm(m, l, theta, cos_phi):
r""" Calculates the real spherical harmonics using :math:`Y_l^m(\theta, \varphi)` with :math:`\mathbf R\to \{r, \theta, \varphi\}`.
These real spherical harmonics are via these equations:
.. math::
Y^m_l(\theta,\varphi) &= -(-1)^m\sqrt{2\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}
P^{m}_l (\cos(\varphi)) \sin(m \theta) & m < 0\\
Y^m_l(\theta,\varphi) &= \sqrt{\frac{2l+1}{4\pi}} P^{m}_l (\cos(\varphi)) & m = 0\\
Y^m_l(\theta,\varphi) &= \sqrt{2\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}
P^{m}_l (\cos(\varphi)) \cos(m \theta) & m > 0
Parameters
----------
m : int
order of the spherical harmonics
l : int
degree of the spherical harmonics
theta : array_like
angle in :math:`x-y` plane (azimuthal)
cos_phi : array_like
cos(phi) to angle from :math:`z` axis (polar)
"""
# Calculate the associated Legendre polynomial
# Since the real spherical harmonics has slight differences
# for positive and negative m, we have to implement them individually.
# Currently this is a re-write of what Inelastica does and a combination of
# learned lessons from Denchar.
# As such the choice of these real spherical harmonics is that of Siesta.
if m == 0:
return _rspher_harm_fact[l][m] * lpmv(m, l, cos_phi)
elif m < 0:
return _rspher_harm_fact[l][m] * (lpmv(m, l, cos_phi) * sin(m*theta))
return _rspher_harm_fact[l][m] * (lpmv(m, l, cos_phi) * cos(m*theta))
@set_module("sisl")
class Orbital:
""" Base class for orbital information.
The orbital class is still in an experimental stage and will probably evolve over some time.
Parameters
----------
R : float
maximum radius
q0 : float, optional
initial charge
tag : str, optional
user defined tag
Attributes
----------
R : float
maximum radius (in Ang)
q0 : float
initial electronic charge
tag : str
user defined tag
Examples
--------
>>> orb = Orbital(1)
>>> orb_tag = Orbital(2, tag='range=2')
>>> orb.R == orb_tag.R / 2
True
>>> orbq = Orbital(2, 1)
>>> orbq.q0
1.
"""
__slots__ = ['R', 'tag', 'q0']
def __init__(self, R, q0=0., tag=''):
""" Initialize orbital object """
if R < 0:
self.R = int(R)
else:
self.R = float(R)
self.q0 = float(q0)
self.tag = tag
def __str__(self):
""" A string representation of the object """
if len(self.tag) > 0:
return self.__class__.__name__ + f'{{R: {self.R:.5f}, q0: {self.q0}, tag: {self.tag}}}'
return self.__class__.__name__ + f'{{R: {self.R:.5f}, q0: {self.q0}}}'
def __repr__(self):
if self.tag:
return f"<{self.__module__}.{self.__class__.__name__} R={self.R:.3f}, q0={self.q0}, tag={self.tag}>"
return f"<{self.__module__}.{self.__class__.__name__} R={self.R:.3f}, q0={self.q0}>"
def name(self, tex=False):
""" Return a named specification of the orbital (`tag`) """
return self.tag
def toSphere(self, center=None):
""" Return a sphere with radius equal to the orbital size
Returns
-------
~sisl.shape.Sphere
sphere with a radius equal to the radius of this orbital
"""
return Sphere(self.R, center)
def equal(self, other, psi=False, radial=False):
""" Compare two orbitals by comparing their radius, and possibly the radial and psi functions
When comparing two orbital radius they are considered *equal* with a precision of 1e-4 Ang.
Parameters
----------
other : Orbital
comparison orbital
psi : bool, optional
also compare that the full psi are the same
radial : bool, optional
also compare that the radial parts are the same
"""
if not isinstance(other, Orbital):
return False
same = abs(self.R - other.R) <= 1e-4 and abs(self.q0 - other.q0) < 1e-4
if not same:
# Quick return
return False
if same and radial:
# Ensure they also have the same fill-values
r = np.linspace(0, self.R * 2, 500)
same &= np.allclose(self.radial(r), other.radial(r))
if same and psi:
xyz = np.linspace(0, self.R * 2, 999).reshape(-1, 3)
same &= np.allclose(self.psi(xyz), other.psi(xyz))
return same and self.tag == other.tag
def copy(self):
""" Create an exact copy of this object """
return self.__class__(self.R, self.q0, self.tag)
def scale(self, scale):
""" Scale the orbital by extending R by `scale` """
R = self.R * scale
if R < 0:
R = -1.
return self.__class__(R, self.q0, self.tag)
def __eq__(self, other):
return self.equal(other)
def radial(self, r, *args, **kwargs):
r""" Calculate the radial part of the wavefunction :math:`f(\mathbf R)` """
raise NotImplementedError
def spher(self, theta, phi, *args, **kwargs):
r""" Calculate the spherical harmonics of this orbital at a given point (in spherical coordinates) """
raise NotImplementedError
def psi(self, r, *args, **kwargs):
r""" Calculate :math:`\phi(\mathbf R)` for Cartesian coordinates """
raise NotImplementedError
def psi_spher(self, r, theta, phi, *args, **kwargs):
r""" Calculate :math:`\phi(|\mathbf R|, \theta, \phi)` for spherical coordinates """
raise NotImplementedError
def __plot__(self, harmonics=False, axes=False, *args, **kwargs):
""" Plot the orbital radial/spherical harmonics
Parameters
----------
harmonics : bool, optional
if `True` the spherical harmonics will be plotted in a 3D only plot a subset of the axis, defaults to all axis
axes : bool or matplotlib.Axes, optional
the figure axes to plot in (if ``matplotlib.Axes`` object).
If ``True`` it will create a new figure to plot in.
If ``False`` it will try and grap the current figure and the current axes.
"""
d = dict()
if harmonics:
# We are plotting the harmonic part
d['projection'] = 'polar'
axes = plt.get_axes(axes, **d)
# Add plots
if harmonics:
# Calculate the spherical harmonics
theta, phi = np.meshgrid(np.arange(360), np.arange(180) - 90)
s = self.spher(np.radians(theta), np.radians(phi))
# Plot data
cax = axes.contourf(theta, phi, s, *args, **kwargs)
cab = axes.get_figure().colorbar(cax)
cab.set_clim(s.min(), s.max())
axes.set_title(r'${}$'.format(self.name(True)))
# I don't know how exactly to handle this...
#axes.set_xlabel(r'Azimuthal angle $\theta$')
#axes.set_ylabel(r'Polar angle $\phi$')
else:
# Plot the radial function and 5% above 0 value
r = np.linspace(0, self.R * 1.05, 1000)
f = self.radial(r)
axes.plot(r, f, *args, **kwargs)
axes.set_xlim(left=0)
axes.set_xlabel('Radius [Ang]')
axes.set_ylabel(r'$f(r)$ [1/Ang$^{3/2}$]')
return axes
def toGrid(self, precision=0.05, c=1., R=None, dtype=np.float64, Z=1):
""" Create a Grid with *only* this orbital wavefunction on it
Parameters
----------
precision : float, optional
used separation in the `Grid` between voxels (in Ang)
c : float or complex, optional
coefficient for the orbital
R : float, optional
box size of the grid (default to the orbital range)
dtype : numpy.dtype, optional
the used separation in the `Grid` between voxels
Z : int, optional
atomic number associated with the grid
"""
if R is None:
R = self.R
if R < 0:
raise ValueError(self.__class__.__name__ + ".toGrid was unable to create "
"the orbital grid for plotting, the box size is negative.")
# Since all these things depend on other elements
# we will simply import them here.
from .supercell import SuperCell
from .geometry import Geometry
from .grid import Grid
from .atom import Atom
from .physics.electron import wavefunction
sc = SuperCell(R*2, origo=[-R] * 3)
g = Geometry([0] * 3, Atom(Z, self), sc=sc)
G = Grid(precision, dtype=dtype, geometry=g)
wavefunction(np.full(1, c), G, geometry=g)
return G
def __getstate__(self):
""" Return the state of this object """
return {'R': self.R, 'q0': self.q0, 'tag': self.tag}
def __setstate__(self, d):
""" Re-create the state of this object """
self.__init__(d['R'], q0=d['q0'], tag=d['tag'])
@set_module("sisl")
class SphericalOrbital(Orbital):
r""" An *arbitrary* orbital class where :math:`\phi(\mathbf r)=f(|\mathbf r|)Y_l^m(\theta,\varphi)`
Note that in this case the used spherical harmonics is:
.. math::
Y^m_l(\theta,\varphi) = (-1)^m\sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}}
e^{i m \theta} P^m_l(\cos(\varphi))
The resulting orbital is
.. math::
\phi_{lmn}(\mathbf r) = f(|\mathbf r|) Y^m_l(\theta, \varphi)
where typically :math:`f(|\mathbf r|)\equiv\phi_{ln}(|\mathbf r|)`. The above equation
clarifies that this class is only intended for each :math:`l`, and that subsequent
:math:`m` orders may be extracted by altering the spherical harmonic. Also, the quantum
number :math:`n` is not necessary as that value is implicit in the
:math:`\phi_{ln}(|\mathbf r|)` function.
Parameters
----------
l : int
azimuthal quantum number
rf_or_func : tuple of (r, f) or func
radial components as a tuple/list, or the function which can interpolate to any R
See `set_radial` for details.
q0 : float, optional
initial charge
tag : str, optional
user defined tag
Attributes
----------
R : float
maximum radius (in Ang)
q0 : float
initial electronic charge
l : int
azimuthal quantum number
f : func
interpolation function that returns `f(r)` for a given radius
tag : str
user defined tag
Examples
--------
>>> from scipy.interpolate import interp1d
>>> orb = SphericalOrbital(1, (np.arange(10.), np.arange(10.)))
>>> orb.equal(SphericalOrbital(1, interp1d(np.arange(10.), np.arange(10.),
... fill_value=(0., 0.), kind='cubic', bounds_error=False)))
True
"""
# Additional slots (inherited classes retain the same slots)
__slots__ = ['l', 'f']
def __init__(self, l, rf_or_func, q0=0., tag='', **kwargs):
""" Initialize spherical orbital object """
self.l = l
# Set the internal function
if callable(rf_or_func):
self.set_radial(rf_or_func, **kwargs)
else:
# it must be two arguments
self.set_radial(rf_or_func[0], rf_or_func[1], **kwargs)
# Initialize R and tag through the parent
# Note that the maximum range of the orbital will be the
# maximum value in r.
super().__init__(self.R, q0, tag)
def copy(self):
""" Create an exact copy of this object """
return self.__class__(self.l, self.f, self.q0, self.tag)
def equal(self, other, psi=False, radial=False):
""" Compare two orbitals by comparing their radius, and possibly the radial and psi functions
Parameters
----------
other : Orbital
comparison orbital
psi : bool, optional
also compare that the full psi are the same
radial : bool, optional
also compare that the radial parts are the same
"""
same = super().equal(other, psi, radial)
if not same:
return False
if isinstance(other, SphericalOrbital):
same &= self.l == other.l
return same
def set_radial(self, *args, **kwargs):
r""" Update the internal radial function used as a :math:`f(|\mathbf r|)`
This can be called in several ways:
set_radial(r, f)
which uses ``scipy.interpolate.UnivariateSpline(r, f, k=3, s=0, ext=1, check_finite=False)``
to define the interpolation function (see `interp` keyword).
Here the maximum radius of the orbital is the maximum `r` value,
regardless of ``f(r)`` is zero for smaller `r`.
set_radial(func)
which sets the interpolation function directly.
The maximum orbital range is determined automatically to a precision
of 0.0001 AA.
Parameters
----------
r, f : numpy.ndarray
the radial positions and the radial function values at `r`.
func : callable
a function which enables evaluation of the radial function. The function should
accept a single array and return a single array.
interp : callable, optional
When two non-keyword arguments are passed this keyword will be used.
It is the interpolation function which should return the equivalent of
`func`. By using this one can define a custom interpolation routine.
It should accept two arguments, ``interp(r, f)`` and return a callable
that returns interpolation values.
See examples for different interpolation routines.
Examples
--------
>>> from scipy import interpolate as interp
>>> o = SphericalOrbital(1, lambda x:x)
>>> r = np.linspace(0, 4, 300)
>>> f = np.exp(-r)
>>> def i_univariate(r, f):
... return interp.UnivariateSpline(r, f, k=3, s=0, ext=1, check_finite=False)
>>> def i_interp1d(r, f):
... return interp.interp1d(r, f, kind='cubic', fill_value=(f[0], 0.), bounds_error=False)
>>> def i_spline(r, f):
... from functools import partial
... tck = interp.splrep(r, f, k=3, s=0)
... return partial(interp.splev, tck=tck, der=0, ext=1)
>>> R = np.linspace(0, 4, 400)
>>> o.set_radial(r, f, interp=i_univariate)
>>> f_univariate = o.f(R)
>>> o.set_radial(r, f, interp=i_interp1d)
>>> f_interp1d = o.f(R)
>>> o.set_radial(r, f, interp=i_spline)
>>> f_spline = o.f(R)
>>> np.allclose(f_univariate, f_interp1d)
True
>>> np.allclose(f_univariate, f_spline)
True
"""
if len(args) == 0:
# Return immediately
def f0(R):
return R * 0.
self.set_radial(f0)
if 'R' in kwargs:
self.R = kwargs['R']
elif len(args) == 1 and callable(args[0]):
self.f = args[0]
# Determine the maximum R
# We should never expect a radial components above
# 50 Ang (is this not fine? ;))
# Precision of 0.05 A
r = np.linspace(0.05, 50, 1000)
f = square(self.f(r))
# Find maximum R and focus around this point
idx = (f > 0).nonzero()[0]
if len(idx) > 0:
idx = idx.max()
# Assert that we actually hit where there are zeros
if idx < len(r) - 1:
idx += 1
# Preset R
self.R = r[idx]
# This should give us a precision of 0.0001 A
r = np.linspace(r[idx]-0.055+0.0001, r[idx]+0.055, 1100)
f = square(self.f(r))
# Find minimum R and focus around this point
idx = (f > 0).nonzero()[0]
if len(idx) > 0:
idx = idx.max()
if idx < len(r) - 1:
idx += 1
self.R = r[idx]
else:
# The orbital radius
# Is undefined, no values are above 0 in a range
# of 50 A
self.R = -1
elif len(args) > 1:
# A radial and function component has been passed
r = _a.asarrayd(args[0])
f = _a.asarrayd(args[1])
# Sort r and f
idx = np.argsort(r)
r = r[idx]
f = f[idx]
# k = 3 == cubic spline
# ext = 1 == return zero outside of bounds.
# s, smoothing factor. If 0, smooth through all points
# I can see that this function is *much* faster than
# interp1d, AND it yields same results with these arguments.
interp = partial(UnivariateSpline, k=3, s=0, ext=1, check_finite=False)
interp = kwargs.get('interp', interp)
self.set_radial(interp(r, f))
elif 'R' in kwargs:
self.R = kwargs.get('R')
else:
raise ValueError('Arguments for set_radial are in-correct, please see the documentation of SphericalOrbital.set_radial')
def __str__(self):
""" A string representation of the object """
if len(self.tag) > 0:
return self.__class__.__name__ + f'{{l: {self.l}, R: {self.R}, q0: {self.q0}, tag: {self.tag}}}'
return self.__class__.__name__ + f'{{l: {self.l}, R: {self.R}, q0: {self.q0}}}'
def __repr__(self):
if self.tag:
return f"<{self.__module__}.{self.__class__.__name__} l={self.l}, R={self.R:.3f}, q0={self.q0}, tag={self.tag}>"
return f"<{self.__module__}.{self.__class__.__name__} l={self.l}, R={self.R:.3f}, q0={self.q0}>"
def radial(self, r, is_radius=True):
r""" Calculate the radial part of the wavefunction :math:`f(\mathbf R)`
The position `r` is a vector from the origin of this orbital.
Parameters
-----------
r : array_like
radius from the orbital origin, for ``is_radius=False`` `r` must be vectors
is_radius : bool, optional
whether `r` is a vector or the radius
Returns
-------
numpy.ndarray
radial orbital value at point `r`
"""
r = _a.asarrayd(r).ravel()
if is_radius:
s = r.shape
else:
r = sqrt(square(r.reshape(-1, 3)).sum(-1))
s = r.shape
r.shape = (-1,)
n = len(r)
# Only calculate where it makes sense, all other points are removed and set to zero
idx = (r <= self.R).nonzero()[0]
# Reduce memory immediately
r = take(r, idx)
p = _a.zerosd(n)
if len(idx) > 0:
p[idx] = self.f(r)
p.shape = s
return p
def psi(self, r, m=0):
r""" Calculate :math:`\phi(\mathbf R)` at a given point (or more points)
The position `r` is a vector from the origin of this orbital.
Parameters
-----------
r : array_like of (:, 3)
the vector from the orbital origin
m : int, optional
magnetic quantum number, must be in range ``-self.l <= m <= self.l``
Returns
-------
numpy.ndarray
basis function value at point `r`
"""
r = _a.asarrayd(r)
s = r.shape[:-1]
# Convert to spherical coordinates
n, idx, r, theta, phi = cart2spher(r, theta=m != 0, cos_phi=True, maxR=self.R)
p = _a.zerosd(n)
if len(idx) > 0:
p[idx] = self.psi_spher(r, theta, phi, m, cos_phi=True)
# Reduce memory immediately
del idx, r, theta, phi
p.shape = s
return p
def spher(self, theta, phi, m=0, cos_phi=False):
r""" Calculate the spherical harmonics of this orbital at a given point (in spherical coordinates)
Parameters
-----------
theta : array_like
azimuthal angle in the :math:`x-y` plane (from :math:`x`)
phi : array_like
polar angle from :math:`z` axis
m : int, optional
magnetic quantum number, must be in range ``-self.l <= m <= self.l``
cos_phi : bool, optional
whether `phi` is actually :math:`cos(\phi)` which will be faster because
`cos` is not necessary to call.
Returns
-------
numpy.ndarray
spherical harmonics at angles :math:`\theta` and :math:`\phi` and given quantum number `m`
"""
if cos_phi:
return _rspherical_harm(m, self.l, theta, phi)
return _rspherical_harm(m, self.l, theta, cos(phi))
def psi_spher(self, r, theta, phi, m=0, cos_phi=False):
r""" Calculate :math:`\phi(|\mathbf R|, \theta, \phi)` at a given point (in spherical coordinates)
This is equivalent to `psi` however, the input is given in spherical coordinates.
Parameters
-----------
r : array_like
the radius from the orbital origin
theta : array_like
azimuthal angle in the :math:`x-y` plane (from :math:`x`)
phi : array_like
polar angle from :math:`z` axis
m : int, optional
magnetic quantum number, must be in range ``-self.l <= m <= self.l``
cos_phi : bool, optional
whether `phi` is actually :math:`cos(\phi)` which will be faster because
`cos` is not necessary to call.
Returns
-------
numpy.ndarray
basis function value at point `r`
"""
return self.f(r) * self.spher(theta, phi, m, cos_phi)
def toAtomicOrbital(self, m=None, n=None, Z=1, P=False, q0=None):
r""" Create a list of `AtomicOrbital` objects
This defaults to create a list of `AtomicOrbital` objects for every `m` (for m in -l:l).
One may optionally specify the sub-set of `m` to retrieve.
Parameters
----------
m : int or list or None
if ``None`` it defaults to ``-l:l``, else only for the requested `m`
Z : int, optional
the specified zeta-shell
P : bool, optional
whether the orbitals are polarized.
q0 : float, optional
the initial charge per orbital, initially :math:`q_0 / (2l+1)` with :math:`q_0` from this object
Returns
-------
AtomicOrbital : for passed `m` an atomic orbital will be returned
list of AtomicOrbital : for each :math:`m\in[-l;l]` an atomic orbital will be returned in the list
"""
# Initial charge
if q0 is None:
q0 = self.q0 / (2 * self.l + 1)
if m is None:
m = range(-self.l, self.l + 1)
elif isinstance(m, Integral):
return AtomicOrbital(n=n, l=self.l, m=m, Z=Z, P=P, spherical=self, q0=q0)
return [AtomicOrbital(n=n, l=self.l, m=mm, Z=Z, P=P, spherical=self, q0=q0) for mm in m]
def __getstate__(self):
""" Return the state of this object """
# A function is not necessarily pickable, so we store interpolated
# data which *should* ensure the correct pickable state (to close agreement)
r = np.linspace(0, self.R, 1000)
f = self.f(r)
return {'l': self.l, 'r': r, 'f': f, 'q0': self.q0, 'tag': self.tag}
def __setstate__(self, d):
""" Re-create the state of this object """
self.__init__(d['l'], (d['r'], d['f']), q0=d['q0'], tag=d['tag'])
@set_module("sisl")
class AtomicOrbital(Orbital):
r""" A projected atomic orbital consisting of real harmonics
The `AtomicOrbital` is a specification of the `SphericalOrbital` by
assigning the magnetic quantum number :math:`m` to the object.
`AtomicOrbital` should always be preferred over the
`SphericalOrbital` because it explicitly contains *all* quantum numbers.
Parameters
----------
*args : list of arguments
list of arguments can be in different input options
q0 : float, optional
initial charge
tag : str, optional
user defined tag
Attributes
----------
R : float
maximum radius (in Ang)
n : int
principal quantum number
l : int
azimuthal quantum number
m : int
magnetic quantum number
Z : int
zeta shell
P : bool
whether this corresponds to a polarized shell (``True``)
f : func
interpolation function that returns `f(r)` for a given radius
q0 : float
initial electronic charge
tag : str
user defined tag
Examples
--------
>>> r = np.linspace(0, 5, 50)
>>> f = np.exp(-r)
>>> # n, l, m, [Z, [P]]
>>> orb1 = AtomicOrbital(2, 1, 0, 1, (r, f))
>>> orb2 = AtomicOrbital(n=2, l=1, m=0, Z=1, (r, f))
>>> orb3 = AtomicOrbital('2pzZ', (r, f))
>>> orb4 = AtomicOrbital('2pzZ1', (r, f))
>>> orb5 = AtomicOrbital('pz', (r, f))
>>> orb2 == orb3
True
>>> orb2 == orb4
True
>>> orb2 == orb5
True
"""
# All of these follow standard notation:
# n = principal quantum number
# l = azimuthal quantum number
# m = magnetic quantum number
# Z = zeta shell
# P = polarization shell or not
# orb is the SphericalOrbital class that retains the radial
# grid and enables to calculate psi(r)
__slots__ = ['n', 'l', 'm', 'Z', 'P', 'orb']
def __init__(self, *args, **kwargs):
""" Initialize atomic orbital object """
super().__init__(kwargs.get('R', 0.), q0=kwargs.get('q0', 0.), tag=kwargs.get('tag', ''))
# Ensure args is a list (to be able to pop)
args = list(args)
self.orb = None
# Extract shell information
n = kwargs.get('n', None)
l = kwargs.get('l', None)
m = kwargs.get('m', None)
Z = kwargs.get('Z', 1)
P = kwargs.get('P', False)
if len(args) > 0:
if isinstance(args[0], str):
# String specification of the atomic orbital
s = args.pop(0)
_n = {'s': 1, 'p': 2, 'd': 3, 'f': 4, 'g': 5}
_l = {'s': 0, 'p': 1, 'd': 2, 'f': 3, 'g': 4}
_m = {'s': 0,
'pz': 0, 'px': 1, 'py': -1,
'dxy': -2, 'dyz': -1, 'dz2': 0, 'dxz': 1, 'dx2-y2': 2,
'fy(3x2-y2)': -3, 'fxyz': -2, 'fz2y': -1, 'fz3': 0,
'fz2x': 1, 'fz(x2-y2)': 2, 'fx(x2-3y2)': 3,
'gxy(x2-y2)': -4, 'gzy(3x2-y2)': -3, 'gz2xy': -2, 'gz3y': -1, 'gz4': 0,
'gz3x': 1, 'gz2(x2-y2)': 2, 'gzx(x2-3y2)': 3, 'gx4+y4': 4,
}
# First remove a P for polarization
P = 'P' in s
s = s.replace('P', '')
# Try and figure out the input
# 2s => n=2, l=0, m=0, z=1, P=False
# 2sZ2P => n=2, l=0, m=0, z=2, P=True
# 2pxZ2P => n=2, l=0, m=0, z=2, P=True
# By default a non-"n" specification takes the lowest value allowed
# s => n=1
# p => n=2
# ...
try:
n = int(s[0])
# Remove n specification
s = s[1:]
except:
n = _n.get(s[0])
# Get l
l = _l.get(s[0])
# Get number of zeta shell
iZ = s.find('Z')
if iZ >= 0:
# Currently we know that we are limited to 9 zeta shells.
# However, for now we assume this is enough (could easily
# be extended by a reg-exp)
try:
Z = int(s[iZ+1])
# Remove Z + int
s = s[:iZ] + s[iZ+2:]
except:
Z = 1
s = s[:iZ] + s[iZ+1:]
# We should be left with m specification
m = _m.get(s)
# Now we should figure out how the spherical orbital
# has been passed.
# There are two options:
# 1. The radial function is passed as two arrays: r, f
# 2. The SphericalOrbital-class is passed which already contains
# the relevant information.
# Figure out if it is a sphericalorbital
if len(args) > 0:
if isinstance(args[0], SphericalOrbital):
self.orb = args.pop(0)
else:
self.orb = SphericalOrbital(l, args.pop(0))
else:
# Arguments *have* to be
# n, l, [m (only for l>0)] [, Z [, P]]
if n is None and len(args) > 0:
n = args.pop(0)
if l is None and len(args) > 0:
l = args.pop(0)
if l > 0:
if m is None and len(args) > 0:
m = args.pop(0)
else:
m = 0
# Now we need to figure out if they are shell
# information or radial functions
if len(args) > 0:
if isinstance(args[0], Integral):
Z = args.pop(0)
if len(args) > 0:
if isinstance(args[0], bool):
P = args.pop(0)
# Figure out if it is a sphericalorbital
if len(args) > 0:
if isinstance(args[0], SphericalOrbital):
self.orb = args.pop(0)
else:
self.orb = SphericalOrbital(l, args.pop(0))
# Still if n is None, we assign the default (lowest) quantum number
if n is None:
n = l + 1
# Copy over information
self.n = n
self.l = l
self.m = m
self.Z = Z
self.P = P
if self.l > 4:
raise ValueError(self.__class__.__name__ + ' does not implement shell h and above!')
if abs(self.m) > self.l:
raise ValueError(self.__class__.__name__ + ' requires |m| <= l.')
# Retrieve user-passed spherical orbital
s = kwargs.get('spherical', None)
if s is None:
# Expect the orbital to already be set
pass
elif isinstance(s, Orbital):
self.orb = s
else:
self.orb = SphericalOrbital(l, s)
if self.orb is None:
# Default orbital to none, this will not create any radial functions
# But any use of the orbital will still work
self.orb = Orbital(self.R)
self.R = self.orb.R
@property
def f(self):
return self.orb.f
def copy(self):
""" Create an exact copy of this object """
return self.__class__(n=self.n, l=self.l, m=self.m, Z=self.Z, P=self.P, spherical=self.orb.copy(), q0=self.q0, tag=self.tag)
def equal(self, other, psi=False, radial=False):
""" Compare two orbitals by comparing their radius, and possibly the radial and psi functions
Parameters
----------
other : Orbital
comparison orbital
psi : bool, optional
also compare that the full psi are the same
radial : bool, optional
also compare that the radial parts are the same
"""
if isinstance(other, AtomicOrbital):
same = self.orb.equal(other.orb, psi, radial)
same &= self.n == other.n
same &= self.l == other.l
same &= self.m == other.m
same &= self.Z == other.Z
same &= self.P == other.P
elif isinstance(other, Orbital):
same = self.orb.equal(other)
else:
return False
return same
def name(self, tex=False):
""" Return named specification of the atomic orbital """
if tex:
name = '{}{}'.format(self.n, {0: 's', 1: 'p', 2: 'd', 3: 'f', 4: 'g'}.get(self.l))
if self.l == 1:
name += {0: '_z', 1: '_x', -1: '_y'}.get(self.m)
elif self.l == 2:
name += {-2: '_{xy}', -1: '_{yz}', 0: '_{z^2}', 1: '_{xz}', 2: '_{x^2-y^2}'}.get(self.m)
elif self.l == 3:
name += {-3: '_{y(3x^2-y^2)}', -2: '_{xyz}', -1: '_{z^2y}', 0: '_{z^3}',
1: '_{z^2x}', 2: '_{z(x^2-y^2)}', 3: '_{x(x^2-3y^2)}'}.get(self.m)
elif self.l == 4:
name += {-4: '_{_{xy(x^2-y^2)}}', -3: '_{zy(3x^2-y^2)}', -2: '_{z^2xy}', -1: '_{z^3y}', 0: '_{z^4}',
1: '_{z^3x}', 2: '_{z^2(x^2-y^2)}', 3: '_{zx(x^2-3y^2)}', 4: '_{x^4+y^4}'}.get(self.m)
if self.P:
return name + fr'\zeta^{self.Z}\mathrm{{P}}'
return name + fr'\zeta^{self.Z}'
name = '{}{}'.format(self.n, {0: 's', 1: 'p', 2: 'd', 3: 'f', 4: 'g'}.get(self.l))
if self.l == 1:
name += {0: 'z', 1: 'x', -1: 'y'}.get(self.m)
elif self.l == 2:
name += {-2: 'xy', -1: 'yz', 0: 'z2', 1: 'xz', 2: 'x2-y2'}.get(self.m)
elif self.l == 3:
name += {-3: 'y(3x2-y2)', -2: 'xyz', -1: 'z2y', 0: 'z3',
1: 'z2x', 2: 'z(x2-y2)', 3: 'x(x2-3y2)'}.get(self.m)
elif self.l == 4:
name += {-4: 'xy(x2-y2)', -3: 'zy(3x2-y2)', -2: 'z2xy', -1: 'z3y', 0: 'z4',
1: 'z3x', 2: 'z2(x2-y2)', 3: 'zx(x2-3y2)', 4: 'x4+y4'}.get(self.m)
if self.P:
return name + f'Z{self.Z}P'
return name + f'Z{self.Z}'
def __str__(self):
""" A string representation of the object """
if len(self.tag) > 0:
return self.__class__.__name__ + '{{{0}, q0: {1}, tag: {2}, {3}}}'.format(self.name(), self.q0, self.tag, str(self.orb))
return self.__class__.__name__ + '{{{0}, q0: {1}, {2}}}'.format(self.name(), self.q0, str(self.orb))
def __repr__(self):
if self.tag:
return f"<{self.__module__}.{self.__class__.__name__} {self.name()} q0={self.q0}, tag={self.tag}>"
return f"<{self.__module__}.{self.__class__.__name__} {self.name()} q0={self.q0}>"
def set_radial(self, *args):
r""" Update the internal radial function used as a :math:`f(|\mathbf r|)`
See `SphericalOrbital.set_radial` where these arguments are passed to.
"""
self.orb.set_radial(*args)
self.R = self.orb.R
def radial(self, r, is_radius=True):
r""" Calculate the radial part of the wavefunction :math:`f(\mathbf R)`
The position `r` is a vector from the origin of this orbital.
Parameters
-----------
r : array_like
radius from the orbital origin, for ``is_radius=False`` `r` must be vectors
is_radius : bool, optional
whether `r` is a vector or the radius
Returns
-------
numpy.ndarray
radial orbital value at point `r`
"""
return self.orb.radial(r, is_radius=is_radius)
def psi(self, r):
r""" Calculate :math:`\phi(\mathbf r)` at a given point (or more points)
The position `r` is a vector from the origin of this orbital.
Parameters
-----------
r : array_like
the vector from the orbital origin
Returns
-------
numpy.ndarray
basis function value at point `r`
"""
return self.orb.psi(r, self.m)
def spher(self, theta, phi, cos_phi=False):
r""" Calculate the spherical harmonics of this orbital at a given point (in spherical coordinates)
Parameters
-----------
theta : array_like
azimuthal angle in the :math:`x-y` plane (from :math:`x`)
phi : array_like
polar angle from :math:`z` axis
cos_phi : bool, optional
whether `phi` is actually :math:`cos(\phi)` which will be faster because
`cos` is not necessary to call.
Returns
-------
numpy.ndarray
spherical harmonics at angles :math:`\theta` and :math:`\phi`
"""
return self.orb.spher(theta, phi, self.m, cos_phi)
def psi_spher(self, r, theta, phi, cos_phi=False):
r""" Calculate :math:`\phi(|\mathbf R|, \theta, \phi)` at a given point (in spherical coordinates)
This is equivalent to `psi` however, the input is given in spherical coordinates.
Parameters
-----------
r : array_like
the radius from the orbital origin
theta : array_like
azimuthal angle in the :math:`x-y` plane (from :math:`x`)
phi : array_like
polar angle from :math:`z` axis
cos_phi : bool, optional
whether `phi` is actually :math:`cos(\phi)` which will be faster because
`cos` is not necessary to call.
Returns
-------
numpy.ndarray
basis function value at point `r`
"""
return self.orb.psi_spher(r, theta, phi, self.m, cos_phi)
def __getstate__(self):
""" Return the state of this object """
# A function is not necessarily pickable, so we store interpolated
# data which *should* ensure the correct pickable state (to close agreement)
try:
# this will tricker the AttributeError
# before we create the data-array
f = self.orb.f
r = np.linspace(0, self.R, 1000)
f = f(r)
except AttributeError:
r, f = None, None
return {'name': self.name(), 'r': r, 'f': f, 'q0': self.q0, 'tag': self.tag}
def __setstate__(self, d):
""" Re-create the state of this object """
if d["r"] is None:
self.__init__(d['name'], q0=d['q0'], tag=d['tag'])
else:
self.__init__(d['name'], (d['r'], d['f']), q0=d['q0'], tag=d['tag'])
