Electric Polarization by Berry Phase: Ver. 1.1
Taisuke Ozaki, RCIS, JAIST
The polarization coming from the electric contribution is given by



(1) 
can be evaluated by the following Berry phase
formula [1,2]:
where means that the integral over the first Brillouin
zone of which volume is .
The quantum phase
is given by



(3) 
where and run over the occupied states.
The integration and derivative in Eq. (2) are approximated by
a discretization:



(4) 
Noting that
the overlap matrix
in Eq. (3)
is evaluated as
Defining that



(7) 
we have
Since each term depends on only the relative position
, Eq. (8) becomes
The exponential function in Eq. (9) can be approximated by



(10) 
Thus, Eq. (9) becomes
where the overlap integral is evaluated in momentum space, and the
expectation value for the position operator is evaluated using
the same real space mesh as for the solution of Poisson's equation
in OpenMX.

 1

R. D. KingSmith, and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993).
 2

R. Resta , Rev. Mod. Phys. 66, 899 (1994).
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