# ConfinedOrbital¶

class ConfinedOrbital(principal_quantum_number=None, angular_momentum=None, radial_cutoff_radius=None, confinement_start_radius=None, additional_charge=None, confinement_strength=None, confinement_power=None, radial_step_size=None)

Class for representing a confined atomic orbital.

Parameters: principal_quantum_number (int) – The all electron principal quantum number (n) of the orbital. angular_momentum (int) – The angular momentum quantum Number (l) of the orbital. radial_cutoff_radius (PhysicalQuantity of type length) – The distance from the core where the basis orbital is zero (compact support radius). confinement_start_radius (PhysicalQuantity of type length) – Radial distance to where the confinement potential starts. The confinement_start_radius must be less than or equal to the radial_cutoff_radius. additional_charge (float) – Charge of the atom when generating the basis function. Default: 0.0 confinement_strength (PhysicalQuantity of type energy) – The confinement strength of the potential confining the atomic orbital. Default: 20 * Hartree confinement_power (int) – The confinement power for the potential. Default: 1 radial_step_size (PhysicalQuantity of type length) – The radial step size determining the distance between grid points on the linear radial grid. Default: 0.01 * Bohr
additionalCharge()
Returns: The additional charge used for generating this ConfinedOrbital instance. float
angularMomentum()
Returns: The angular momentum. int
confinementPower()
Returns: The power of the confinement potential. int
confinementStartRadius()
Returns: The radius where the confinement potential starts. PhysicalQuantity of type length
confinementStrength()
Returns: The strength of the confinement potential. PhysicalQuantity of type energy
principalQuantumNumber()
Returns: The principal quantum number. int
radialCutoffRadius()
radialStepSize()
Returns: The radial grid spacing. PhysicalQuantity of type length

## Usage Examples¶

Define a BasisSet for Hydrogen:

hydrogen_1s = ConfinedOrbital(
principal_quantum_number = 1,
angular_momentum = 0,
confinement_strength = 20.000*Hartree,
confinement_power = 1,
)

my_hydrogen_basis = BasisSet(
element = Hydrogen,
orbitals = [hydrogen_1s],
occupations = [1.0],
pseudopotential = NormConservingPseudoPotential('normconserving/H.LDAPZ.zip'),
)


## Notes¶

The basis functions are found by solving the radial Schrödinger equation for the atom with a confinement potential $$V_{\text{c}}(r)$$. The confinement potential is defined by the parameters confinement_strength ($$V_0$$), confinement_power ($$n$$), confinement_start_radius ($$r_i$$), and radial_cutoff_radius ($$r_c$$ ) through the equation

$\begin{split}V_{\text{c}}(r) = \begin{cases} 0 &\text{if} \, \, r < r_i \\ V_0 \; \left(\frac{r_c-r_i}{r_c-r}\right)^n \ \; \exp[-\frac{r_c-r_i}{r-r_i}] &\text{if} \, \, r_i < r < r_c \\ \infty &\text{if} \, \, r_c < r \end{cases}\end{split}$

Fig. 123 shows the confinement potential and the corresponding basis functions used for constructing the LDA standard basis set for hydrogen.

Fig. 123 The lower part of the plot shows the $$\ell=0$$ effective potential for hydrogen (dashed) with the soft confinement potential (solid). The upper part shows the lowest occupied eigenstate of the confined potential (solid line), and the atomic s-wave function is indicated by the dashed curve (note that $$r\cdot \psi(r)$$ are plotted).The dotted curve shows the radial wave function with energy 0.01 Ry above the atomic eigenenergy. The position of the first node of this solution defines the position of $$r_c$$ .

For backwards compatibility, if $$V_0$$ is given in units of energy$$\times$$length (as was the convention in QuantumATK 12.2 and earlier, where a slightly different expression was used for the confinement potential), the rescaled value $$V_0/(r_c-r_i)$$ is used for the confinement strength, to conform to the new formula given above. Further information about the basis functions can be found in LCAO basis set.