ConfinedOrbital

	ConfinedOrbital
CheckpointHandler	DoubleContourIntegralParameters

Name

ConfinedOrbital — Class for representing a confined atomic orbital.

Synopsis

Namespace: NanoLanguage

ConfinedOrbital(
principal_quantum_number,
angular_momentum,
radial_cutoff_radius,
confinement_start_radius,
additional_charge,
confinement_strength,
confinement_power,
radial_step_size
)

Description

Constructor for a ConfinedOrbital.

ConfinedOrbital Arguments

principal_quantum_number

The All-electron Principal Quantum Number ( n ) of the orbital.

Type: int

Default:


          None

angular_momentum

The Azimuthal Quantum Number ( l ) of the orbital.

Type: int

Default:


          None

radial_cutoff_radius

The distance from the core where the basis orbital is zero (compact support radius).

Type: PhysicalQuantity of type length

Default:


          None

confinement_start_radius

Radial distance to where the confinement potential starts. The confinement_start_radius must be less than the radial_cutoff_radius.

Type: PhysicalQuantity of type length

Default:


          None

additional_charge

Additional charge placed on the atom when generating the basis function.

Type: float

Default:

0.0

confinement_strength

The confinement strength of the potential confining the atomic orbital.

Type: PhysicalQuantity of type length*energy

Default:


          10 * Hartree * Bohr

confinement_power

The confinement power for the potential.

Type: PhysicalQuantity of type length*energy

Default:

radial_step_size

The radial step size determining the distance between grid points on the linear radial grid.

Type: PhysicalQuantity of type length

Default:


          0.01*Bohr

ConfinedOrbital Methods

A ConfinedOrbital object provides the following methods:

This object supports cloning. See the section called “Cloning of ATK Python objects”.
additionalCharge(): Return the additional charge used for generating this ConfinedOrbital.
angularMomentum(): Return the angular momentum.
confinementPower(): Return the power of the confinement potential.
confinementStartRadius(): Return the radius where the confinement potential starts.
confinementStrength(): Return the strength of the confinement potential.
principalQuantumNumber(): Return the principal quantum number.
radialCutoffRadius(): Return the radial cutoff radius.
radialStepSize(): Return the radial grid spacing.

Usage Examples

Define a BasisSet for Hydrogen.

hydrogen_1s = ConfinedOrbital(
    principal_quantum_number = 1,
    angular_momentum = 0,
    radial_cutoff_radius = 5.28603678847*Bohr,
    confinement_start_radius = 0.8 * 5.28603678847*Bohr,
    additional_charge = 0.0,
    confinement_strength = 20.000*Units.Hartree*Units.Bohr,
    radial_step_size = 0.001*Units.Bohr,
    )


my_hydrogen_basis = BasisSet(
   element = Hydrogen,
   orbitals = [hydrogen_1],
   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. The confinement potential is defined by the parameters, confinement_strength ( $V_0$ ), confinement_start_radius ( $r_{\text{inn}}$ ), and radial_cutoff_radius ( $r_c$ ) through the equation

$\displaystyle V_{\text{conf}}(r) = \begin{cases} 0 & \text{if} \, \, r < r_{\text{inn}} \\ V_0 \; \exp[-(r_c-r_{\text{inn}})/(r-r_{\text{inn}})]/(r_c-r) & \text{if} \, \, r_{\text{inn}} < r < r_c \\ \infty & \text{if} \, \, r_c < r \end{cases}$

Figure 6 shows the confinement potential and the corresponding basis functions used for constructing the LDA standard basis set for hydrogen.

$The lower part of the plot shows the 14 0 45e09236e8f293f5565707e3bbf241d4 l=0 effective potential for hydrogen (dashed) with the soft confinement potential (solid). The upper part shows the lowest occupied eigenstate of the confined potential (solid line), and the atomic s-wave function is indicated by the dashed curve (note that 16 5 86624f9a02139f6732279be33cdd6333 r \psi(r) are plotted) . The dotted curve shows the radial wave function with energy 0.01 Ry above the atomic eigenenergy. The position of the first node of this solution defines the position of 9 3 f5b9dfb293fbb1edbb2d00d790c0c6f2 r_c .$

Figure 6: The lower part of the plot shows the $l=0$ 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 $r \psi(r)$ 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 $r_c$ .

Further information about the basis functions can be found in the section called “LCAO basis set”.


CheckpointHandler		DoubleContourIntegralParameters