Electronic Structure and the Periodic table

Mike Gunn

Why is Lithium a metal but Fluorine a diatomic insulating molecular crystal? One might expect by `particle-hole' symmetry that they would be equivalent (one extra electron over a closed shell in the case of Li and one hole in a closed shell in the case of F). This course is aimed at answering questions similar to this for people who are not prospective professionals in electronic structure calculations but will need to understand the phenomena described by and ideas behind electronic structure claculations. Understanding trends in the periodic table using electronic structure calculations will be one focus of the course.

  • The relevance of atomic structure to the solid state electronic structure. rl etc. Why the centrifugal barrier does not yield exponential decay and an excursion on WKB. Efficacy of screening intuitively of different l states, and its relation to the filling of shells.
  • Thomas Fermi theory of screening, and its relation to WKB. Linear approximation and TF theory of atoms. Passing mention of exactitude as Z tends to Infinity. The drawbacks of TF theory and the logic of the Hohenberg-Kohn theorem. Density as the fundamental variable. The effectiveness of LDA in terms of lattice parameters, bulk moduli etc and in predicting the ferromagnetic d-metals.
  • Kohn-Sham equations. The subtlety of the kinetic energy. The significance of the Kohn-Sham eigenvalues and wavefunctions, or lack of it.
  • The exchange correlation hole and its relation to the exchange-correlation functional. Angular averages as a source of robustness under approximation of the functionals.
  • Tight binding theory. Metallic bonding, variation as function of band filling and its message about vaporisation temperatures in the transition metals.
  • Rare gas solids: nonorthogonality as source of closed shell repulsion. Van der Waals interaction between a pair of atoms. The solid state treatment and pseudospins. Frenkel exciton zero-point motion as the Van der Waals binding energy of a solid.
  • Nearly free electron theory and its surprising validity for the alkali metals. Pseudopotentials as the origin of this – reason for first row in periodic table being different to the subsequent ones.
  • Why is one electron away from a full shell (the alkalis) so different from one hole (the Halogens)?
  • The former are nearly free electron metal the latter are molecular crystals. The origin of this in terms of quasi-1d behaviour from the p-bonding, there are strong Peierls effects and these lead to dimerisation. Mention of the pressure induced transition in I2.
  • Some general results about band structures. Why bands cannot stop, or go in circles. Relation of the size of band gaps at high energies to the asymptotic behaviour of the Fourier series of the periodic potential.
  • Wannier functions and their relation to atomic orbitals. Chemical pseudopotentials and their lack of use. Analogies with the 'normal' pseudopotential.
  • Hybridisation due to crystal structure and the strength of hybrid bonds. Contrast of graphite and diamond. sigma and pi bonds. sp3 hybrids and the Jones Zone view.
  • Density functional gaps and semiconductor gaps.
  • Boron, phosphorus, other exotica. The limitations of band theory – the Mott transition.

Bibliography