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DFT Exercises

produced by Gábor Csányi and Chris J. Pickard

Below is a set of example problems designed to introduce /ab initio/ density functional calculations. Rather than focusing on how to get a calculation running, which will be different for every package, each exercise introduces a new feature of DFT calculations, emphasizing the scientific value. It is intended that students will use the user guide of their package to figure out how to specify the given tasks to the program, and how to extract and visualize data from the output. Unless stated otherwise, use ultrasoft pseudopotentials (if available) and the Local Density Approximation (LDA). You should be able to complete all calculations on a recent PC in a reasonable amount time.

Thanks to Peter D. Haynes and Volker Heine for helpful suggestions.

  1. Hydrogen Atom

    Place a single H atom in the middle of a largish unit cell (start with a cube of 1 nm on each side), and compute the total energy. Plot the resultant wavefunction (or charge density) along a radial line and compare to the exact answer. How does the total energy and the wavefunction converge with increasing kinetic energy cutoff? Converge your answers with the size of the unit cell. Try a norm conserving pseudopotential. Now turn on spin polarization. How do your answers change?

  2. Hydrogen Molecule

    Try a H2 molecule, and an H2+ radical. Find the optimal bond length. Compare results of LDA with a gradient corrected functional (GGA). What is the ionization energy? Converge your answer with all parameters.

  3. Chlorine

    Compare the energy levels of the Cl atom with the experimental spectrum. Now consider Cl2 , is the convergence with energy cutoff better than for H2 ? Try Cl- , and plot its energy as a function of unit cell size. Can you observe something strange?

  4. Gold Atom

    Plot the valence orbitals (wave functions) of an isolated gold atom, and identify the orbitals with different symmetry. Are the true orbitals spherically symmetric? Compare the results with the relativistic treatment and also spin-orbit coupling.

  5. Oxygen

    Consider the oxygen molecule O2 . Try different spin configurations. Which is the ground state? Think about what singlet and triplet states really mean in a single particle context.

  6. Water

    Investigate the H2O molecule. Optimize bond lengths and the angle. What are the vibrational frequencies? How well does this match with experiment?

  7. Hydrogen bond

    Using two water molecules, figure out the binding energy of the hydrogen bond (the binding that occurs betweem an H atom of one water molecule and the O atom on the /other/ water molecule).

  8. Benzene

    Compute the ground state of the C6H6 ring molecule, by making a series of SCF calculations, each time allowing one more SCF iteration than before. Plot the individual orbitals, and observe how they converge with iteration number. Do all orbitals converge uniformly? Find the delocalized orbital that gives rise to the special properties of such aromatic compounds.

  9. Silicon

    Find the equilibrium lattice parameter and bulk modulus of crystalline Si, using an 8 atom cubic unit cell. Converge your answers with the number of k points that you sample the Brillouen zone with. Try shifting the k point mesh so that it symmetrically straddles the origin (but does not include it). Plot the band structure, and compare with experiment. How large is the band gap?

  10. Sodium

    Plot the density of states and the band structure. Compare with the free electron bands. Compare the dispersion of the semicore 2s state with the tight binding cosine band.

  11. Graphite

    Use variable occupancies to compute the band structure of graphite. Why is it called a semi-metal? Find the equilibrium lattice constant along the c axis. Compare the answers given by LDA and GGA.

  12. Bonding

    For each of the following systems, compute the ground state in the bulk phase, look at a slice of the charge density and observe the different bonding characters.

    1. Diamond - covalent (why is the charge density low near the ionic cores?)
    2. NaCl - ionic (can you observe the charge transfer? How does the plot change if you include/exclude the Na 2s state from the core?)
    3. GaAs - mixed ionic covalent
    4. Na - metallic
    5. Ice - hydrogen bonding
    6. solid cubane (C8H8) - Van der Waals

  13. Van Hove singularities

    Compute the band structure of a simple carbon nanotube (e.g. (6,0)) and plot the density of states. Compare with that of graphite and diamond. You can generate the atomic positions with TubeGen, for example.

  14. Surface reconstruction

    Take a slab of silicon (periodic boundary conditions in the x-y plane, but finite in the z direction) with an open (100) surface, and optimize the geometry of the atoms. What happens to the dangling bonds on the surface? Converge with the slab thickness and the vacuum size (distance between periodic images perpendicular to the surface)

Solutions

A set of solutions has been provided by Felipe Cervantes-Sodi

 


If you have any comments about these pages, you can email Matt Probert as mijp1 at york.ac.uk