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Consider a molecule such as a porphyrin ring. For many purposes, a simple tight-binding model in which only pi-electrons are included provide a very good description of the electronic levels, shape of the molecular orbitals, etc. Now imagine there's a metal atom in the center of the ring. DFT analysis of the projected DOS show that, around the Fermi level, only the pz orbitals of the non-metal atoms and some of the d orbitals of the metal play a role (I am assumming the molecule is on the XY plane). This suggests also that it should be possible to construct a very simple model (similar to the one in which only pi-electrons are treated in hydrocarbons) with hopping between the pz and d orbitals (of course, many-body terms will be important in the d-shell, but focus now on the one-electron part and the coupling with the rest of atoms). Are there standard tight-binding hamiltonians for such systems, with parametrizations for the pz-d hoppings, something akin to Huckel hamiltonians but for systems with metal atoms? And how to count then the number of "pi'' electrons that enter the calculation? (I was guessing by looking at the number occupations of lowdin orbitals in a previous DFT calculation)

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As far as I am aware there are no universal parameters for p-d only calculations, but tight-binding models were widely used in solid-state physics to described oxides and other compounds.

A more chemically relevant method is DFTB, a tight-binding method that has extensive parametrization for a wide range of elements. DFTB has several implementation, including an efficient open-source version, DFTB+.

Another closely related tight-binding method is Grimme's xTB, which is also implemented in the DFTB+ package.

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