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.

Return type:

int

principalQuantumNumber()
Returns:

The principal quantum number n for the orbital.

Return type:

int

slaterCoefficients()
Returns:

The Slater coefficients as inverse length.

Return type:

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 [1], 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 34 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 [2] 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