Dipole moment and Berry Phase
*****************************

By default, Band computes (and prints to the output) only the
components of the dipole moment orthogonal to the periodic
direction(s).

Since the position operator is ill-defined in periodic systems,
computing the longitudinal components of the dipole moment (i.e. the
components along periodic directions) is not completely trivial. To
obtain the longitudinal components of the dipole moment, you can
perform a Berry Phase calculation:

   BerryPhase Yes/No

"BerryPhase"
   Type:
      Bool

   Default value:
      No

   Description:
      Boolean that determines whether the dipole as determined through
      the Berry phase approach should be calculated.

In the graphical user interface (AMSinput) you can find the Berry
Phase checkbox in the **Details → Expert BAND** panel.

In a Berry phase calculation, the dipole moment of a unit cell is
calculated with the help of the geometric phase within the unit cell.
The theoretical framework of the calculations originates from the
*'Modern theory of polarization'*  that was come up with in the early
1990s by King-Smith, Vanderbilt and Resta. [1] [2] The implementation
in BAND is a generalization of the one-dimensional Berry phase
approach for quantum chemistry codes with local basis sets devised by
Kudin and Car. [3]

Warning:

  The Berry phase implementation in BAND has been thoroughly for 1D
  systems. Nonetheless, the implementation seems to break down for 2D
  and 3D systems, thus requiring careful testing and validation of the
  calculations for such systems.

Warning:

  * Berry phase calculations require orthorhombic unit cells,
    differently shaped unit cells are not currently supported.

  * For Berry Phase calculations you should use an all electron basis
    set (i.e. Set the "core" to "none")

  * Using a good k-space sampling is recommended for Berry Phase
    calculations

An option that can be useful when validating the Berry Phase
calculation is "ShiftCoordinates" in the AMS System block:

   System header
   End

"System"

[1] R. King-Smith, D. Vanderbilt, *Theory of polarization of
    crystalline solids.* Physical Review B 47, 1651 (1993).

[2] R. Resta, *Macroscopic polarization in crystalline dielectrics:
    the geometric phase approach.* Reviews of Modern Physics 66, 899
    (1994).

[3] K. Kuding, R. Car, *Berry phase approach to longitudinal dipole
    moments of infinite chains in electronic-structure methods with
    local basis sets* Journal of Chemical Physics 126, 234101 (2007).
