SlaterOrbital¶

class SlaterOrbital(principal_quantum_number, angular_momentum, slater_coefficients, weights)

Constructor for the Slater orbitals.

Parameters: principal_quantum_number (positive int) – The Principal quantum number (n) of the orbital. angular_momentum (non-negative int) – The Azimuthal quantum number (l) of the orbital. slater_coefficients (PhysicalQuantity of type inverse length) – The Slater coefficients as inverse length. Maximum two coefficients can be given. Every entry must be positive. weights (numpy.array) – The weight for each of the Slater coefficients. Each entry should be positive.
angularMomentum()
Returns: The angular momentum for the orbital. int
principalQuantumNumber()
Returns: The principal quantum number n for the orbital. int
slaterCoefficients()
Returns: The Slater coefficients as inverse length. list of PhysicalQuantity of type inverse length

Usage Examples¶

Define a 1s SlaterOrbital from a single exponential function

carbon_2s = SlaterOrbital(
principal_quantum_number=2,
angular_momentum=0,
slater_coefficients=[2.0249*1/Bohr],
weights=[0.76422]
)


Define a 2p SlaterOrbital as superposition of two exponential functions

carbon_2p = SlaterOrbital(
principal_quantum_number=2,
angular_momentum=1,
slater_coefficients=[1.62412*1/Bohr , 2.17687*1/Bohr],
weights=[0.27152, 0.73886]
)


Notes¶

Within the extended Hückel model [eSPS+10], the electronic structure is expanded in a basis formed by a linear combination of atomic orbitals (LCAO)

$\phi_{nlm}({\bf r}) = R_{nl}(r) Y_{lm}(\hat{r}),$

where $$Y_{lm}$$ is a spherical harmonic and $$R_{nl}$$ is a Slater orbital

$R_{nl}(r) = \frac{r^{n-1}}{\sqrt{(2n)!}} \left[C_1 (2 \eta_1)^{n+\frac{1}{2}} e^{-\eta_1 \, r}+C_2 (2 \eta_2)^{n+\frac{1}{2}} e^{-\eta_2 \, r} \right].$

The SlaterOrbital is described by the adjustable parameters $$\eta_1$$, $$\eta_2$$, $$C_1$$, and $$C_2$$. These parameters must be defined for each angular shell of valence orbitals for each element.

Table 37 Slater orbital parameters.
Symbol SlaterOrbital parameters
$$n$$ principal_quantum_number
$$l$$ angular_momentum
$$\eta$$ slater_coefficients
$$C$$ weights

In the current version QuantumATK comes with built-in Hoffmann and Müller parameter sets which are appropriate for organic molecules, and Cerda parameters [eCS00] which are appropriate for crystals. The parameter set are available with the keyword HoffmannHuckelParameters.ElementName, MullerHuckelParameters.ElementName and CerdaHuckelParameters.ElementName, where ElementName is the name of the element.

Reference¶

 [eCS00] J. Cerdá and F. Soria. Accurate and transferable extended hückel-type tight-binding parameters. Phys. Rev. B, 61:7965–7971, Mar 2000. doi:10.1103/PhysRevB.61.7965.
 [eSPS+10] K. Stokbro, D. E. Petersen, S. Smidstrup, A. Blom, M. Ipsen, and K. Kaasbjerg. Semiempirical model for nanoscale device simulations. Phys. Rev. B, 82:075420, Aug 2010. doi:10.1103/PhysRevB.82.075420.