Other topics in DFT simulations

1
Other topics in DFT simulations

• Exchange-correlation functionals
• Solving the KS equations, some tricks
• Wannier functions

Exercise (read about functionals in Parr/Yang
and/or Martin, both for general math of
functionals and for DFT details)
2
DFT functionals

• Martin p. 152
• LSDA
• The xc energies are for a uniform electron gas,
  and were obtained numerically by QMC
• We included spin up and down here which may apply
  if spins are not paired

3
GGAs

• Include info from derivatives of the density.
  Note uniform gas approximation is very extreme --
  atoms are not at all uniform
• One common form is BLYP, another is PBE, and
  there are others

4
Hybrid functionals

• The most common one is B3LYP, developed by Becke,
  which includes some degree of exact HF exchange.
• This improves accuracies quite a lot, making DFT
  about as good as MP2
• The drawback with the hybrids is that computation
  of the exchange matrix is more expensive than the
  near local GGAs

Exercise Martin references some studies which
compare these methods in tests on molecules. Look
some of those up. (p. 169 Martin)
5
Solving the KS equations

• Initial guess for wave functions --gt n(r)
• That n(r) creates a new Veff(r)
• Solve KS equations (expensive step). This --gt new
  n(r)
• Keep repeating until the process converges and
  n(r) does not change

6
Total energy functionals

• KS total energy functional
• KS eigenvalues
• Kinetic energy

Exercise understand Martins argument about the
Importance of using in and out in the right
places!
7
Explicit (Harris) functional of the density

• KS functional of the in potential
• Now express total energy in terms on in density
• Difference between two energy expressions

Exercise show this, that difference involves
only potential terms
8
Achieving self-consistency

• Update potentials or densities? The density is
  unique.
• Linear mixing
• Why not just use the new density itself? That is,
  where does the mixing come from?
• Can show
• Response function is costly calc though, and
  alternatives have been developed Broyden, DIIS

9
Wannier functions

• Localized functions used to represent electron
  states (superposition of Bloch states in periodic
  systems)
• Have found new uses as analogies for localized
  representations of electrons in large systems
  (linear scaling)
• Used to compute molecular dipole moments in
  condensed phases (e.g. water dipole changes)

10
WF definition

• Consider an extended system in periodic
  boundaries. Then Blochs theorem applies--gtbands
• Bloch states
• Translation on the lattice
• Wannier function

Exercise show that the WFs are orthogonal
11
Maximally localized WFs

• Minimize
• People have also developed ways to utilize
  non-orthogonal localized functions. It turns out
  that these functions can be shorter ranged and
  better behaved, so they have been useful in
  linear scaling algorithms (Martin)
• Linear scaling? As we will see, it is necessary
  to exploit localization to achieve this scaling.