SphericalOrbital¶

class sisl.SphericalOrbital(l, rf_or_func, q0=0.0, tag='', **kwargs)¶

Bases: sisl.Orbital

An arbitrary orbital class where \(\phi(\mathbf r)=f(|\mathbf r|)Y_l^m(\theta,\varphi)\)

Note that in this case the used spherical harmonics is:

\[Y^m_l(\theta,\varphi) = (-1)^m\sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!}} e^{i m \theta} P^m_l(\cos(\varphi))\]

The resulting orbital is

\[\phi_{lmn}(\mathbf r) = f(|\mathbf r|) Y^m_l(\theta, \varphi)\]

where typically \(f(|\mathbf r|)\equiv\phi_{ln}(|\mathbf r|)\). The above equation clarifies that this class is only intended for each \(l\), and that subsequent \(m\) orders may be extracted by altering the spherical harmonic. Also, the quantum number \(n\) is not necessary as that value is implicit in the \(\phi_{ln}(|\mathbf r|)\) 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

R¶

maximum radius (in Ang)

Type: float

q0¶

initial electronic charge

Type: float

l¶

azimuthal quantum number

Type: int

f¶

interpolation function that returns f(r) for a given radius

Type: func

tag¶

user defined tag

Type: str

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

Attributes

`R`
`__doc__`
`__hash__`
`__module__`
`__slots__`
`f`
`l`
`q0`
`tag`

Methods

`__delattr__`	Implement delattr(self, name).
`__dir__`	Default dir() implementation.
`__eq__`(other)	Return self==value.
`__format__`	Default object formatter.
`__ge__`	Return self>=value.
`__getattribute__`	Return getattr(self, name).
`__getstate__`()	Return the state of this object
`__gt__`	Return self>value.
`__init__`(l, rf_or_func[, q0, tag])	Initialize spherical orbital object
`__init_subclass__`	This method is called when a class is subclassed.
`__le__`	Return self<=value.
`__lt__`	Return self<value.
`__ne__`	Return self!=value.
`__new__`	Create and return a new object.
`__plot__`([harmonics, axes])	Plot the orbital radial/spherical harmonics
`__reduce__`	Helper for pickle.
`__reduce_ex__`	Helper for pickle.
`__repr__`()	Return repr(self).
`__setattr__`	Implement setattr(self, name, value).
`__setstate__`(d)	Re-create the state of this object
`__sizeof__`	Size of object in memory, in bytes.
`__str__`()	A string representation of the object
`__subclasshook__`	Abstract classes can override this to customize issubclass().
`copy`()	Create an exact copy of this object
`equal`(other[, psi, radial])	Compare two orbitals by comparing their radius, and possibly the radial and psi functions
`name`([tex])	Return a named specification of the orbital (`tag`)
`psi`(r[, m])	Calculate \(\phi(\mathbf R)\) at a given point (or more points)
`psi_spher`(r, theta, phi[, m, cos_phi])	Calculate \(\phi(\|\mathbf R\|, \theta, \phi)\) at a given point (in spherical coordinates)
`radial`(r[, is_radius])	Calculate the radial part of the wavefunction \(f(\mathbf R)\)
`scale`(scale)	Scale the orbital by extending R by `scale`
`set_radial`(args, *kwargs)	Update the internal radial function used as a \(f(\|\mathbf r\|)\)
`spher`(theta, phi[, m, cos_phi])	Calculate the spherical harmonics of this orbital at a given point (in spherical coordinates)
`toAtomicOrbital`([m, n, Z, P, q0])	Create a list of `AtomicOrbital` objects
`toGrid`([precision, c, R, dtype, Z])	Create a Grid with only this orbital wavefunction on it
`toSphere`([center])	Return a sphere with radius equal to the orbital size

R¶

copy()[source]¶: Create an exact copy of this object

equal(other, psi=False, radial=False)[source]¶

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

f¶

l¶

name(tex=False)¶: Return a named specification of the orbital (tag)

psi(r, m=0)[source]¶

Calculate \(\phi(\mathbf R)\) 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

basis function value at point r

Return type

numpy.ndarray

psi_spher(r, theta, phi, m=0, cos_phi=False)[source]¶

Calculate \(\phi(|\mathbf R|, \theta, \phi)\) 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 \(x-y\) plane (from \(x\))
phi (array_like) – polar angle from \(z\) axis
m (int, optional) – magnetic quantum number, must be in range -self.l <= m <= self.l
cos_phi (bool, optional) – whether phi is actually \(cos(\phi)\) which will be faster because cos is not necessary to call.

Returns

basis function value at point r

Return type

numpy.ndarray

q0¶

radial(r, is_radius=True)[source]¶

Calculate the radial part of the wavefunction \(f(\mathbf R)\)

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

radial orbital value at point r

Return type

numpy.ndarray

scale(scale)¶: Scale the orbital by extending R by scale

set_radial(*args, **kwargs)[source]¶

Update the internal radial function used as a \(f(|\mathbf r|)\)

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

f (r,) – 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

spher(theta, phi, m=0, cos_phi=False)[source]¶

Calculate the spherical harmonics of this orbital at a given point (in spherical coordinates)

Parameters

theta (array_like) – azimuthal angle in the \(x-y\) plane (from \(x\))
phi (array_like) – polar angle from \(z\) axis
m (int, optional) – magnetic quantum number, must be in range -self.l <= m <= self.l
cos_phi (bool, optional) – whether phi is actually \(cos(\phi)\) which will be faster because cos is not necessary to call.

Returns

spherical harmonics at angles \(\theta\) and \(\phi\) and given quantum number m

Return type

numpy.ndarray

tag¶

toAtomicOrbital(m=None, n=None, Z=1, P=False, q0=None)[source]¶

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 \(q_0 / (2l+1)\) with \(q_0\) from this object

Returns

AtomicOrbital (for passed m an atomic orbital will be returned)
list of AtomicOrbital (for each \(m\in[-l;l]\) an atomic orbital will be returned in the list)

toGrid(precision=0.05, c=1.0, R=None, dtype=<class 'numpy.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

toSphere(center=None)¶

Return a sphere with radius equal to the orbital size

Returns: sphere with a radius equal to the radius of this orbital
Return type: Sphere