Results for the nonlocal real space implementation
1. Accuracy
2. Efficiency
1. 16 Si atoms - (2x2x2
k-point .5 .5. 5 offset) volume= 2090.32 25 Ry cutoff
radius error in Energy (Ry)
6.7
2.8 x 10-6
4.7
2.2 x 10-6
3.7 8.3 x 10-6
3.2 4.2 x 10-5
2.7
1.8 x 10-3
2.2
0.4
12 Ry cutoff
radius error in Energy (Ry)
6.7 2.9 x 10-5
4.7 3.8 x 10-6
4.2 4.3 x 10-5
3.7 6.8 x10-5
3.2 2.3x10-5
2.7 8.2x10-4
Significant differences do not occur until a radius of 2.7.
Interestingly the error increases
slightly from a radius of 4.7 to 6.7.
2. volume = 2330
radius error in
Energy (Ry)
6.7 9.4 x 10-6
4.7 1.4 x 10-5
3.2 3.7 x 10-5
The error in the energy does increase as the atoms move further apart.
3. Same as (1) above with several atoms moved by up to .144 bohr.
One lattice vector is also stretched by 10% making a monoclinic cell
radius error in Energy (Ry)
6.7 8.0 x 10-6
4.7 4.8 x 10-6
4.2 2.4 x 10-5
3.7 1.2 x 10-5
3.2 4.5 x 10-5
The error in energy is slightly affected again by the movement of the
atoms.
4. Same as above 4x4x4 mesh
radius Energy (Ry)
4.7 9.7 x 10-6
One can see the accuracy doesn’t change dramatically for higher k-point
meshes
5.
54
Si atoms - gamma point (3x3x3
supercell)
radius Energy (Ry)
4.7 3.1 x 10-5
3.7 1.1 x 10-4
3.2 1.8 x 10-4
2.7 6.1 x 10-3
6. 128 Si atoms gamma point (4x4x4 supercell)
radius Energy (Ry)
4.2 3.2 x
10-4
3.7 2.5 x 10-4
3.2 4.3 x 10-4
2.7 1.4
x 10-2
7. 250 atom
radius Energy (Ry)
3.2 8.5 x
104
4.2 6.2 x 10-4
1.
Same
as 1a. Ex. 3
Error in the force
(Ry/Bohr)
radius x y z
6.7 -1.23 x 10-5 1.16 x 10-5 5.80 x 10-5
4.7
1.16 x 10-4
-9.59x 10-5 -6.30 x 10-5
4.3
2.72 x 10-4
-2.19 x 10-4
-1.40 x 10-4
3.7 5.74 x 10-4 -4.43 x 10-4 -2.78 x 10-4
3.2
1.07 x 10-3 -7.68 x 10-4 -4.62 x 10-4
We can see that the necessary accuracy for the forces for a
given radius
is comparable for the forces and the energy
1. 16 Si atoms - 2x2x2 k-point mesh volume = 2090.32
Real space used in energy minimization to obtain eigenfunctions, but
stress calculated in G-space
radius stress xx=yy=zz (Gpa)
k-space 0.32512
4.7 0.32509
3.7
0.32507
3.2 0.32516
2.7 0.31534
2.2
-2.08488
Si 54 - gamma point
radius stress xx=yy=zz (Gpa)
k-space 2.42050
4.7 2.42055
3.7 2.42037
3.2 2.42062
2.7 2.41013
Si128 - gamma point
radius stress xx=yy=zz (Gpa)
Kspace 1.00722
3.7 1.00709
3.2 1.00734
2.7 0.99706
The stress is accurate up to a radius of 3.2 (the same as the nergy)
2. Same as above, but stress
calculated in real space
radius stress xx=yy=zz (Gpa)
6.7 0.34262
4.7 0.06012
Once can readily see that these elements of the stress field are much longer range
than the non-local force and energy fields.
3. Same as above but
volume=2330.32 and stress in real space
radius stress xx=yy=zz (Gpa)
k-space -8.29552
6.7 -8.28100
4.7 -8.51441
3.2 -12.18045
A radius of 3.2 which seemed acceptable for the accuracy of the energy
and forces
proves to be inadequate for these stress fields.
4. Same as 1a. Ex. 3. Stress calculated in real space
radius xx yy zz xy yz
zx
k-space 2.77314
2.57061 -2.96587 -1.97145 3.88317 -6.30879
6.7 2.76645 2.50404 -2.84039
-2.00233 3.75694 -6.37091
4.7 2.51409 2.24949 -3.18161
-2.00428 3.74466 -6.36816
4.2 2.06363 1.79952 -3.65964
-2.00231 3.73648 -6.35837
3.7 0.93295 0.67099
-4.81743 -1.99701 3.72417 -6.33734
3.2 -1.85748 -2.11362 -7.62670
-1.98453 3.70508 -6.29287
With this system, we see that it is only the xx, yy, and zz
stress fields which are
long range. This is not surprising
since if one stretches the cell in the x-direction
the grid points parallel to the x-axis will move the farthest and thus
giving a long
range stress field.
5.
Same
as above with 4x4x4 k-point mesh
radius xx yy zz
xy yz zx
k-space 2.73493
2.64246 -2.94995 -0.85818 5.00887 -5.32778
4.7 2.45031
2.34708 -3.16438 -0.82413 4.98236 -5.27374
G-sp. Str.
4.7 2.73473 2.64226 -2.95014
-0.85824 5.00890 -5.32785
We can see that increasing the k-points affects mostly the off-diagonal
terms
in the stress field. The biggest errors between the real space and
G-space stress
again are in the diagonal terms calculation are in the diagonal terms.
Also the error
is small for the energy minization in real space, but the stress in
G-space.
6. Same as (2) with 1 lattice
vector stretched by 10%
radius xx yy zz
xy yz zx
k-space -6.19177
-6.19177 -11.44270 -1.34064 3.75907 -3.75907
6.7 -6.21953 -6.21953
-11.34375 -1.39913 3.75590 -3.75590
4.7 -6.42959 -6.42959
-11.62475 -1.40273 3.74439 -3.74439
The error for the diagonal terms consistently stays at a lower value of
around -.2
For this and previous systems.
6.
In
a geometry relaxation, a radius of 4.7 did not give the lowest energy for the
smallest stress. The accuracy varied on the system between 2x10-4
and 4x10-5.
A radius of 6.7 did give
energies that decreased with lowering stress.
The final energy accuracy was 2.9 x 10-5. where 4.7 gave an
accuracy of 4.4 x 10-4
Comments: For geometry relaxations, the force and
stress calculations take a small percentage
of the total time. Given that one would need a
larger radius for accurate stresses, it is better to do the calculation in
G-space since the longer radius would cause much longer times in the energy
minimization. If one was doing molecular dynamics where the percentage cost was
appreciable
for the forces then one could do them in real space.
If one needed to have stress information
in a MD run,
a separate radius for the diagonal stress terms would be beneficial.
1. 16 Si atoms - (2x2x2
k-point .5 .5. 5 offset) volume= 2090.32
4 proc t3e
radius time spent in NL
k-space 15.097
4.7
238.671
3.7
125.840
3.2 90.199
2.7 63.092
2.2 43.319
2. 54 Si atoms -
gamma point 8 proc t3e
radius time spent in NL
126.921
4.7
442.615
3.7
273.511
3.2 197.473
2.7
219.008
3. 128 Si atoms 16 proc IBM Sp2 Seaborg
radius time spent in NL
Kspace 351.785
4.2
846.370
3.7
627.631
3.2
448.872
2.7
311.198
On seaborg the G-space is still faster for radii of 3.2 and above.
t3e mcurie
G-space - 25.75 seconds / H*Ψ
3.2 real space - 22.89 seconds / H*Ψ
On mcurie the real space is faster. These times are for all parts of the
H*Ψ procedure
4. 250 Si atoms 64 proc
radius time spent in NL
G-space 740.512
3.2 512.51
4.2 841.83
t3e mcurie
gspace - 84.67 seconds / H*
Ψ
3.2 real space - 66.7 seconds / H* Ψ