VacancyFormationMigration Test Driver

SUMMARY
  
  This test driver calculates the vacancy formation energy and migration energy at zero temperature and zero pressure.
  
  It works for all the elements, in either sc, fcc, bcc, diamond or hcp crystal structure.
  
METHOD
  
  The calculation consists of two steps:
  
  1. Calculate the vacancy formation energy and relaxation volume corresponding to three different sizes of supercell.
  The minimum size is 3*3*3 cubic unit cells for diamond structure (27 * 8 = 216 atoms).
  For other structures the minimum size is determined by the smallest number n, which makes n*n*n conventional unit cells has at least the same number of atoms as the above diamond supercell.
  The other two sizes are the two numbers that follows the minimum size.
  
  The Calculation of each size starts from constructing the periodic supercell, then take out one atom from the supercell.
  Then it does the relaxation of positions within the supercell to obtain vacancy formation energy.
  After that, it moves the nearest atom to the vacanted site and calculates vacancy migration energy with nudged elastic band method.
  
  2. Extrapolate from these three results to get the value corresponding to an infinitely large supercell.
  
  The extrapolation is based on the elastic theory, which gives the supercell effect on vacancy formation energy and migration energy are 1/r^3.
  Therefore, we use the two larger size to fit f(x) = a0 + a1/r^3, and obtain a0, which we use as the source value.
  
  It also estimates the uncertainty, based on: a. fitting with all the three sizes to the function above; b. minimization criteria used in positions relaxation and nudged elastic band calculation.