Category talk:Electronic minimization

• Add a more precise definition of "electronic minimization" (from the lecture).

• The page Algorithms used in VASP to calculate the electronic groundstate describes the class of minimizers based on a combination of the SCF cycle and density mixing. This, however, is not the only class of methods available (there's the direct minimizers as well).

• Several "Theory" pages mentioned here are off-topic:
  • Move "k-point integration" elsewhere
  • Move "Wrap-around errors" elsewhere

• All "How to" pages mentioned here are off-topic (should be moved elsewhere)

• Subcategory "density of states" is off-topic

• Several (probably all) pages on electronic minimizers need to be reworked. In some of them references like "in the previous section" are used, which are meaningless in a wiki environment. Examples: Efficient single band eigenvalue-minimization and Conjugate gradient optimization.

• Do not use the TAG template to link to non-tag related articles.

We seek to minimize the Kohn-Sham free energy:

[math]\displaystyle{ F = \sum_n f_n \epsilon_n -E_{\rm H}\left[ \rho \right] + E_{\rm xc} \left[ \rho \right] -\int V_{\rm xc}({\bf r})\rho({\bf r})d{\bf r} - \sum_n \sigma S \left( \frac{\epsilon_n - \mu}{\sigma} \right) }[/math]

where the electronic density is given by:

[math]\displaystyle{ \rho({\bf r})= \sum_n f_{n} |\psi_{n}({\bf r})|^2 }[/math]

and the Kohn-Sham orbitals and eigenenergies, [math]\displaystyle{ \{\psi_n, \epsilon_n \} }[/math] are solutions to the Kohn-Sham equations:

[math]\displaystyle{ H \left[ \rho \right] \psi_n = \epsilon_n S \psi_n }[/math]

under the constraint that the orbitals are S-orthonormal:

[math]\displaystyle{ \langle \psi_m | S | \psi_n \rangle = \delta_{mn} }[/math]