Hartree–Fock RI
***************

The Hartree-Fock exchange matrix is calculated through a procedure
known as Resolution of the Identity (RI). The implementation of the RI
scheme in BAND is loosely based on work by Ren *et al.* [1]. For more
information on hybrid functionals in BAND, see the XC section.

Technical aspects of the RI scheme can be tweaked in the
"RIHartreeFock" block:

   RIHartreeFock
      Quality [Auto | VeryBasic | Basic | Normal | Good | VeryGood | Excellent]
      FitSetQuality [Auto | VeryBasic | Basic | Normal | Good | VeryGood | Excellent | FromBasisProducts]
      DependencyThreshold float
      QualityPerRegion
         Quality [VeryBasic | Basic | Normal | Good | VeryGood | Excellent]
         Region string
      End
   End

"RIHartreeFock"
   Type:
      Block

   Description:
   "Quality"
      Type:
         Multiple Choice

      Default value:
         Auto

      Options:
         [Auto, VeryBasic, Basic, Normal, Good, VeryGood, Excellent]

      Description:
         Numerical accuracy of the RI procedure. If 'Auto', the
         quality specified in the 'NumericalQuality' will be used.

   "FitSetQuality"
      Type:
         Multiple Choice

      Default value:
         Auto

      Options:
         [Auto, VeryBasic, Basic, Normal, Good, VeryGood, Excellent,
         FromBasisProducts]

      Description:
         The quality of auxiliary fit set employed in the RI scheme.
         If 'Auto', the value of the RIHartreeFock Quality option will
         be used.                      Normal quality is generally
         sufficient for basis sets up to and including TZ2P.
         For larger basis sets (or for benchmarking purposes) a
         VeryGood fit set is recommended.                      Note
         that the FitSetQuality heavily influences the computational
         cost of the calculation.

   "DependencyThreshold"
      Type:
         Float

      Default value:
         0.001

      Description:
         To improve numerical stability, almost linearly-dependent
         combination of basis functions are removed from the Hartree-
         Fock exchange matrix.                      If you obtain
         unphysically large bond energy in an Hybrid calculation, or
         an unphysically low correlation energy in an RPA, MP2, or
         double hybrid calculation, you might try setting the
         DependencyThreshold to a larger value (e.g. 3.0E-3)
         Note, that in GW calculations and GW-BSE calculations the
         default for this key is 5.0e-3.

   "QualityPerRegion"
      Type:
         Block

      Recurring:
         True

      Description:
         Sets the fit-set quality for all atoms in a region. If
         specified, this overwrites the globally set quality.

      "Quality"
         Type:
            Multiple Choice

         Options:
            [VeryBasic, Basic, Normal, Good, VeryGood, Excellent]

         Description:
            This region's quality of the auxiliary fit set employed in
            the RI scheme.

      "Region"
         Type:
            String

         Description:
            The identifier of the region for which to set the quality.

For efficiency and numerical stability reasons, it is advisable to
include:

   SoftConfinement
     Quality Basic
   End

See the Confinement of basis functions section for more info.

**Notes:** for periodic systems it is only possible to use short-range
hybrid functionals (*e.g.* HSE06) and all-electron basis sets.

Note:

  * In AMS2019.3 the fit set for "FitSetQuality Good" has been
    improved.

[1] X. Ren, P. Rinke, V. Blum, J. Wieferink, A. Tkatchenko, A.
    Sanfilippo, K. Reuter and M. Scheffler, *Resolution-of-identity
    approach to Hartree–Fock, hybrid density functionals, RPA, MP2 and
    GW with numeric atom-centered orbital basis functions*, New J.
    Phys. 14 053020.
