The macroscopic electric polarization of a bulk system can
be calculated based on the Berry phase formalism [12].
As an example, let us illustrate a calculation of a Born
effective charge of Na in a NaCl bulk via the macroscopic
polarization.

**(1) SCF calculation**

First, perform a conventional SCF calculation using an input
file 'NaCl.dat' in the directory 'work'. Then, the following
keyword 'HS.fileout' should be switched on

HS.fileout on # on|off, default=offWhen the calculation is completed normally, then you can find an output file 'nacl.scfout' in the directory 'work'.

**(2) Calculation of macroscopic polarization**

The macroscopic polarization is calculated by a post-processing
code 'polB' of which input data is 'nacl.scfout'.
In the directory 'source', compile as follows:

% make polBWhen the compile is completed normally, then you can find an executable file 'polB' in the directory 'work'. Then, move to the directory 'work', and perform as follows:

% polB nacl.scfout or % polB nacl.scfout < in > outIn the latter case, the file 'in' contains the following ingredients:

9 9 9 1 1 1In the former case, you will be interactively asked from the program as follows:

****************************************************************** ****************************************************************** polB: code for calculating the electric polarization of bulk systems Copyright (C), 2006-2007, Fumiyuki Ishii and Taisuke Ozaki This is free software, and you are welcome to redistribute it under the constitution of the GNU-GPL. ****************************************************************** ****************************************************************** Read the scfout file (nacl.scfout) Previous eigenvalue solver = Band atomnum = 2 ChemP = -0.156250000000 (Hartree) E_Temp = 300.000000000000 (K) Total_SpinS = 0.000000000000 (K) Spin treatment = collinear spin-unpolarized r-space primitive vector (Bohr) tv1= 0.000000 5.319579 5.319579 tv2= 5.319579 0.000000 5.319579 tv3= 5.319579 5.319579 0.000000 k-space primitive vector (Bohr^-1) rtv1= -0.590572 0.590572 0.590572 rtv2= 0.590572 -0.590572 0.590572 rtv3= 0.590572 0.590572 -0.590572 Cell_Volume=301.065992 (Bohr^3) Specify the number of grids to discretize reciprocal a-, b-, and c-vectors (e.g 2 4 3) k1 0.00000 0.11111 0.22222 0.33333 0.44444 ... k2 0.00000 0.11111 0.22222 0.33333 0.44444 ... k3 0.00000 0.11111 0.22222 0.33333 0.44444 ... Specify the direction of polarization as reciprocal a-, b-, and c-vectors (e.g 0 0 1 ) 1 1 1Then, the calculation will start like this:

calculating the polarization along the a-axis .... The number of strings for Berry phase : AB mesh=81 calculating the polarization along the a-axis .... 1/ 82 calculating the polarization along the a-axis .... 2/ 82 ..... ... ******************************************************* Electric dipole (Debye) : Berry phase ******************************************************* Absolute dipole moment 163.93373639 Background Core Electron Total Dx -0.00000000 94.64718996 -0.00000338 94.64718658 Dy -0.00000000 94.64718996 -0.00000283 94.64718713 Dz -0.00000000 94.64718996 -0.00000317 94.64718679 *************************************************************** Electric polarization (muC/cm^2) : Berry phase *************************************************************** Background Core Electron Total Px -0.00000000 707.66166752 -0.00002529 707.66164223 Py -0.00000000 707.66166752 -0.00002118 707.66164633 Pz -0.00000000 707.66166752 -0.00002371 707.66164381 Elapsed time = 77.772559 (s) for myid= 0

Since the Born effective charge is defined by a tensor:

Px = 94.39497736 (Debye/unit cell) at x= -0.05 (Ang) Px = 94.64718658 (Debye/unit cell) at x= 0.0 (Ang) Px = 94.89939513 (Debye/unit cell) at x= 0.05 (Ang)

Thus,

Note that in the NaCl bulk the off-diagonal terms in the tensor of Born charge are zero, and . In Table 5 we see that the calculated value is in good agreement with the other calculation [72] and an experimental result [73]. The calculation of macroscopic polarization is supported for both the collinear and non-collinear DFT. It is also noted that the code 'polB' has been parallelized for large-scale systems where the number of processors can exceed the number of atoms in the system.

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