sisl.SphericalOrbital

class sisl.SphericalOrbital

Bases: Orbital

An arbitrary orbital class which only contains the harmonical part of the wavefunction where $\varphi (\mathbf{r})=f(|\mathbf{r}|)Y_l^m(\theta ,\phi )$

Note that in this case the used spherical harmonics is:

$$Y_l^m(\theta ,\phi )=(-1{)}^m\sqrt{\frac{2l+1}{4\pi }\frac{(l-m)!}{(l+m)!}}{e}^{im\theta }{P}_l^m(\cos (\phi ))$$

The resulting orbital is

$${\varphi }_{lmn}(\mathbf{r})=f(|\mathbf{r}|)Y_l^m(\theta ,\phi )$$

where typically $f(|\mathbf{r}|)\equiv {\varphi }_{ln}(|\mathbf{r}|)$. 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 ${\varphi }_{ln}(|\mathbf{r}|)$ function.

Parameters:

• l – azimuthal quantum number

• rf_or_func – radial components as a tuple/list, or the function which can interpolate to any R See set_radial for details.

• R – See Orbital for details.

• q0 – initial charge

• tag – user defined tag

• **kwargs – arguments passed directly to set_radial(rf_or_func, **kwargs)

f

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

Type:

func

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

Methods

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 $\varphi (\mathbf{r})$ at a given point (or more points)
psi_spher(r, theta, phi[, m, cos_phi])	Calculate $\varphi (|\mathbf{r}|,\theta ,\varphi )$ at a given point (in spherical coordinates)
radial(r, *args, **kwargs)	Calculate the radial part of spherical orbital $R(\mathbf{r})$
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, zeta, P, q0])	Create a list of AtomicOrbital objects
toGrid([precision, c, R, dtype, atom])	Create a Grid with only this orbital wavefunction on it
toSphere([center])	Return a sphere with radius equal to the orbital size

Attributes

R	Maxmimum radius of orbital
l	$l$ quantum number
q0	Initial charge
tag	Named tag of orbital

copy()

Create an exact copy of this object

Parameters:

orbital (SphericalOrbital)

Return type:

SphericalOrbital

equal(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

name(tex=False)

Return a named specification of the orbital (tag)

psi(r, m=0)

Calculate $\varphi (\mathbf{r})$ at a given point (or more points)

The position r is a vector from the origin of this orbital.

Parameters:

• r (ndarray of (:, 3)) – vector from the orbital origin

• m (int) – 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)

Calculate $\varphi (|\mathbf{r}|,\theta ,\varphi )$ at a given point (in spherical coordinates)

This is equivalent to psi however, the input is given in spherical coordinates.

Parameters:

• r (ndarray) – the radius from the orbital origin

• theta (ndarray) – azimuthal angle in the $xy$ plane (from $x$)

• phi (ndarray) – polar angle from $z$ axis

• m (int) – magnetic quantum number, must be in range -self.l <= m <= self.l

• cos_phi (bool) – whether phi is actually $cos(\varphi )$ which will be faster because cos is not necessary to call.

Returns:

basis function value at point r

Return type:

numpy.ndarray

radial(r, *args, **kwargs)

Calculate the radial part of spherical orbital $R(\mathbf{r})$

The position r is a vector from the origin of this orbital.

Parameters:

• r (ndarray) – radius from the orbital origin

• *args – arguments passed to the radial function

• **args – keyword arguments passed to the radial function

Returns:

radial orbital value at point r

Return type:

numpy.ndarray

scale(scale)

Scale the orbital by extending R by scale

Parameters:

• orbital (Orbital)

• scale (float)

Return type:

Orbital

set_radial(*args, **kwargs)

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 (numpy.ndarray) – the radial positions and the radial function values at 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.

Return type:

None

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.radial(R)
>>> o.set_radial(r, f, interp=i_interp1d)
>>> f_interp1d = o.radial(R)
>>> o.set_radial(r, f, interp=i_spline)
>>> f_spline = o.radial(R)
>>> np.allclose(f_univariate, f_interp1d)
True
>>> np.allclose(f_univariate, f_spline)
True

spher(theta, phi, m=0, cos_phi=False)

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

Parameters:

• theta (ndarray) – azimuthal angle in the $xy$ plane (from $x$)

• phi (ndarray) – polar angle from $z$ axis

• m (int) – magnetic quantum number, must be in range -self.l <= m <= self.l

• cos_phi (bool) – whether phi is actually $cos(\varphi )$ which will be faster because cos is not necessary to call.

Returns:

spherical harmonics at angles $\theta$ and $\varphi$ and given quantum number m

Return type:

numpy.ndarray

toAtomicOrbital(m=None, n=None, zeta=1, P=False, q0=None)

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

• zeta (int) – the specified $\zeta$-shell

• n (int | None) – specify the $n$ quantum number

• P (bool) – whether the orbitals are polarized.

• q0 (float | None) – 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 :math:`min[-l;l] an atomic orbital will be returned in the list`)

toGrid(precision=0.05, c=1.0, R=None, dtype=np.float64, atom=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

• atom (optional) – atom associated with the grid; either an atom instance or something that Atom(atom) would convert to a proper atom.

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

property R

Maxmimum radius of orbital

property l

$l$ quantum number

property q0

Initial charge

property tag

Named tag of orbital