SphericalOrbital¶

class sisl.SphericalOrbital(l, rf_or_func, q0=0.0, tag='')[source]¶

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

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 : 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

Attributes

`R`
`f`
`l`
`q0`
`tag`

Methods

`__init__`(l, rf_or_func[, q0, tag])	Initialize spherical orbital object
`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:	psi : the orbital value at point r

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:	psi : the orbital value at point r

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:	f : the orbital value at point r

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:

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

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:	spher : the spherical harmonics at angles \(\theta\) and \(\phi\).

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 : the sphere with a radius equal to the radius of this orbital