sisl.HydrogenicOrbital

class sisl.HydrogenicOrbital

Bases: AtomicOrbital

A hydrogen-like atomic orbital defined by an effective atomic number Z in addition to the usual quantum numbers (n, l, m).

A hydrogenic atom (Hydrogen-like) is an atom with a single valence electron.

The returned orbital is properly normalized, see [HydrogenicO] for details.

The orbital has the familiar spherical shape

$$\begin{array}{rl}{Y}_l^m(\theta ,\phi )& =(-1{)}^m\sqrt{\frac{2l+1}{4\pi }\frac{(l-m)!}{(l+m)!}}{e}^{im\theta }{P}_l^m(\cos (\phi ))\\ {\varphi }_{lmn}(\mathbf{r})& =R_{nl}(|\mathbf{r}|)Y_l^m(\theta ,\phi )\\ R_{nl}(|\mathbf{r}|)& =-\sqrt{(\frac{2Z}{n{a}_0}{)}^3\frac{(n-l-1)!}{2n(n+l)!}}{e}^{-Zr/(n{a}_0)}(\frac{2Zr}{n{a}_0}{)}^l{L}_{n-l-1}^{(2l+1)}(\frac{2Zr}{n{a}_0})\end{array}$$

With $L_{n-l-1}^{(2l+1)}$ is the generalized Laguerre polynomials.

References

[HydrogenicO]

https://en.wikipedia.org/wiki/Hydrogen-like_atom

Parameters:

• n – principal quantum number

• l – angular momentum quantum number

• m – magnetic quantum number

• Z – effective atomic number

• **kwargs – See Orbital for details.

Examples

>>> carbon_pz = HydrogenicOrbital(2, 1, 0, 3.2)

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 named specification of the atomic orbital
psi(r)	Calculate $\varphi (\mathbf{r})$ at a given point (or more points)
psi_spher(r, theta, phi[, cos_phi])	Calculate $\varphi (|\mathbf{r}|,\theta ,\varphi )$ at a given point (in spherical coordinates)
radial(r, *args, **kwargs)	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[, cos_phi])	Calculate the spherical harmonics of this orbital at a given point (in spherical coordinates)
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

P	Whether this is polarized shell or not
R	Maxmimum radius of orbital
l	$l$ quantum number
m	$m$ quantum number
n	$n$ shell
orb	Orbital with radial part
q0	Initial charge
tag	Named tag of orbital
zeta	$\zeta$ shell

copy()

Create an exact copy of this object

Parameters:

orbital (HydrogenicOrbital)

Return type:

HydrogenicOrbital

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) – also compare that the full psi are the same

• radial (bool) – also compare that the radial parts are the same

name(tex=False)

Return named specification of the atomic orbital

psi(r)

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) – the vector from the orbital origin

Returns:

basis function value at point r

Return type:

numpy.ndarray

psi_spher(r, theta, phi, 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

• 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 the wavefunction $f(\mathbf{r})$

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

Parameters:

r (ndarray) – radius from the orbital origin

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 $f(|\mathbf{r}|)$

See SphericalOrbital.set_radial where these arguments are passed to.

spher(theta, phi, 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

• 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 $\varphi$

Return type:

numpy.ndarray

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 P

Whether this is polarized shell or not

property R

Maxmimum radius of orbital

property l

$l$ quantum number

property m

$m$ quantum number

property n

$n$ shell

property orb

Orbital with radial part

property q0

Initial charge

property tag

Named tag of orbital

property zeta

$\zeta$ shell