quantum-espresso/examples/example14

This example shows the use of th D3 code to calculate
the third-order expansion coefficients with respect to 
atomic displacement for Silicon.

First a normal self-consistent calculation is done.
Then a phonon calculation for the Gamma point is performed.
With these preliminary steps the coefficients C(0,0,0) are calculated.

For the X-point a non self-consistent calculation of groundstate
and the phonon calculation for this point is done.
Afterwards C(0,X,-X) is calculated.

By displacing one atom, one can get also these coefficients by a 
finite-difference mathod. We give first the values obtained by 
the 2n+1 method, then the values by the finite-differences.
All units are in Ryd/(a_b)^3.

  tensor                 |    2n+1       |  fin. dif.
------------------------------------------------------------
C_{x,y,z} (0,0,0|1,1,1)  |   0.38314     |    0.38446
------------------------------------------------------------
C_{x,y,z} (0,X,-X|1,1,1) |   0.34043     |    0.34109
C_{x,x,z} (0,X,-X|1,1,2) |  -0.25316     |   -0.25296
C_{z,x,y} (0,X,-X|1,1,1) |   0.35781     |    0.35767
C_{z,x,x} (0,X,-X|1,1,2) |  -0.25706     |   -0.25491
C_{z,z,z} (0,X,-X|1,1,2) |  -0.13133     |   -0.12813


The results of these calculations are also compared with the ones of
Debernardi given in paranthesis, see Debernardi, PhD thesis (1995), 
page 81, available at 
http://www.sissa.it/cm/phd.php
The units are in eV/(Angstrom)^3.

B_xyz = -281.43 (-284.25)
I_zaa =  225.82  (227.37)
I_zbb =  -37.00  (-37.64)
I_zcc =   48.23   (49.91)
I_xac =  436.01  (441.32)
I_ybc =  -64.10  (-63.91)