GroupTheory`
GroupTheory`

GTTbMatrixElement

GTTbMatrixElement[l1,m1,l2,m2,shell]

gives the decomposition of the tight-binding three-center integral between atom1 and atom2 , when atom2 belongs to the neighborhood shell and atom2 is located in direction relative to atom1.

Details and Options

  • In tight-binding theory the following integrals
  • have to be calculated. The Hamiltonian is represented in a Löwdin basis. labels the atomic site and the angular symmetry with respect to this site. The energy integrals are expressed as a linear combination of two-center integrals in dependence on the direction cosines of the distance vector .
  • The following options can be given:
  • GOTbBasis 0Supresses superscripts with element names
    GOTbRule 1Selects substitution rules
  • See: A.V. Podolskiy, P. Vogl,Compact expression for the angular dependence of tight-binding Hamiltonian matrix elements, Phys. Rev. B 69, 233101 (2004)
  • W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 9.4

Examples

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Basic Examples  (1)

First load the package.

The tight-binding matrix element will be expressed in terms of two-center-parameters and the direction cosines of the vector between the two atoms.

At the first atom a orbital is localized, at the second atom an orbital . The distance belongs to the nearest neighbor shell.

Usually , and orbitals are used. The algorithm is general, also orbitals of higher angular momentum can be considered.

In some considerations of semiconductors an excited orbital * is included in the basis. This can be simulated using .

Options  (2)

GOTbBasis  (1)

If two different types of atoms are involved, this has to indicated by a corresponding superscript. The atom names in the superscript are given in alphabetic order. The angular momenta are ordered accordingly.

GOTbRule  (1)

Rules can be used to simplify the results. Sometimes it is useful for the comparison with results in the literature.