Under construction…

The idea here is to categorize by general subject.

### DFT and DFT-related

- Piecewise linear of total energy versus particle number: detailed derivation in two ways
- Detailed derivation of DFT+U formalisms

### Electrostatics: charges, potentials, and convergence

- Electrostatics of stacked periodic 2D arrays of charges
- Ewald summation, infinite Coulomb sums and truncation issues in some detail
- Convergence of energy eigenvalues with respect to vacuum level versus cell size
- Periodized Coulomb interaction in slab (2D) geometry: long versus short range parts
- Periodized Coulomb interaction in wire (1D) geometry: long versus short range parts

### Dielectrics

- Analytical results for the potential of a point charge on or inside a finite-thickness dielectric slab
- Analytical results for the potential of a system of two charge distributions on dielectric surfaces and interfaces

### Ferroelectrics and polarization

- Some results on multi-domain versus single-domain ferroelectricity as a function of film thickness and screening parameters
- Some thoughts on how to think about/define ferroelectric surfaces polarization charges
- Total energy versus polarization: microscopic to macroscopic connection (EP or EdP?)

### Materials

- LAO/STO continuum model in periodic boundary conditions: dependences of electrostatics on thicknesses and dielectric constants
- Basic observations on how doping a Mott insulator differs from a band insulator (based on a very educational discussion with Prof. Mohit Randeria)
- Where does the dispersion of the dxy band in a perovskite (e.g. in SrTiO3) in the z direction come from?

### Math

- Should you be in need of various moments of Lorentzian integrand to some power

### Mechanics

- Simple model of how forcing a mechanical system of spring with a gap in the spectrum leads to exponential decay of response away from forcing
- Symmetries of the Hessian matrix

### Many-body physics

- Relation of quasiparticle weights (Z) to many-body states and their projection onto single-particle excitations
- Charge transfer issues in many-body versus single-particle descriptions: illustrated for a two level system
- Effective hopping model from a microscopic one: one of many ways to go from a more detailed model to an effective model, here carried out for lattice systems using Green’s function matching (not the only way, but one way; it generates a non-local and frequency dependent self-energy even for a non-interacting system showing how these properties of the self-energy are due to elimination of degrees of freedom and not necessary from interactions)

### Quantum mechanics

- 2nd order perturbation theory in full: perturbation theory carried out carefully in matrix form to 2nd order with care taken to deal with degenerate subspaces and the required unitary rotations in those subspaces.
- Low-angle or low-energy potential scattering in 1D and 3D
- Short tutorial on Berry phase polarization and Wannier functions
- Tunneling rate from a finite object to a continuum and its computation using k-point sampling
- Tunneling from a discrete state to a continuum
- Green’s function methods to deal with boundaries: how to attach a finite system to an infinite one using Green’s functions
- Impurity problems: examination of a set of simple lattice impurity models to see what their general effect is on the local density of states (at impurity and elsewhere) and how it depends on the strength of the impurity potential, coupling to neighbors, etc.

### Statistical mechanics

- Diffusion of oxygen vacancies under a voltage from a tip: practice with the diffusion equation
- Sommerfeld expansion worked out: for those who always forget the needed details when they are needed…
- Simple derivation of variable range hopping formula