Wannier Functions

In order to get a feel for chemical bonds in molecules and solids, we can transform the Kohn-Sham orbitals into a set of maximally localized Wannier functions. We have cheated a little bit and prepared a file for bulk silicon and a Benzene molecule so that you only have to concentrate on the wannier analysis of the molecules.

Start by running si.py and make sure you agree with the way the diamond structure is set up. The resulting .gpw file is used as input to wannier-si.py which transforms the Kohn-Sham orbitals to maximally localized wannier functions and plot the atoms along with the centers of the wannier functions. Note that the wannier centers are treated as “X” atoms which are plotted as small red spheres. How many covalent bonds do you expect in a unit cell with 8 tetravalent Silicon atoms?

The script benzene.py produces a .gpw that can be used as input to create wannier functions. Convince yourself that the chosen number of bands matches the number of occupied orbitals in the molecule. How many covalent bonds do you expect in Benzene? Look at wannier-benzene.py and figure out what it does. Run it and look at the graphical representation. Note in particular the alternating single/double bonds between the carbon atoms. What happens if also you include one or two unoccupied bands? The script also produces two .cube files. One contains the wavefunction of the Highest Occupied Molecular Orbital (HOMO) and the other contains a wannier function centered between a Carbon and a Hydrogen atom. Study these with VMD and determine which type of orbitals they represent (\(\sigma\) or \(\pi\)).

Now repeat the wannier function analysis on the following molecules

  • H2O : use your own files from the vibrational exercise, but make sure the number of bands is equal to the number of occupied orbitals.
  • CO : use your own .gpw file from the wavefunction exercise. Is it a single, double or triple bond?

or study your own favorite molecule.

Hint

To be able to see the Wannier centers, it might be necessary to decrease the atomic radii, so the spheres don’t overlap. With ase gui this can be done by choosing View ‣ Settings, and then decrease the scaling factor of the covalent radii.