Benchmark calculations

A couple of examples as benchmark calculations are shown below:


Si
The real part of dielectric function of Si bulk is shown for a series of k-grids in Fig. 81. We see that as increasing k-grid from $10\times 10\times 10$ to $100\times 100\times 100$, the real part of dielectric function is getting converged. It is found that we need to have a fine grid for the k-points to obtain a well converged result. In Tables 14 and 15, we show the computational time and parallel efficiency in the calculation of the conductivity and dielectric function for supercells of Si bulk. The results suggest that it might be possible to treat systems including 1000 atoms if 1000 CPU cores are available.



Table 14: Computational time of conductivity and dielectric function of Si crystal.
  # of Si atoms Supercell Diagonalization k-Grid
  Total time (s)  
  (CPUs=128)  
  Total time (s)  
  (CPUs=256)  
  Total time (s)  
  (CPUs=512)  
  Total time (s)  
  (CPUs=1024)  
  Total time (s)  
  (CPUs=2048)  
 
  512 atoms 4x4x4 Cluster 1x1x1 3367.16826 1755.60797 919.21912 464.27761 253.32210  
  4x4x4 ScaLAPACK 2x1x1 6819.30193 3499.51872 1838.64406 948.87978 513.79250  
  4x4x4 Band 2x2x2 15300.58350 10217.17765 5953.19907 3518.84650 1747.20171  
  1000 atoms 5x5x5 Cluster 1x1x1 6900.35370 3511.85143 1778.33693  
  5x5x5 ScaLAPACK 2x1x1 12994.17818 6817.43990 3460.76787  
  5x5x5 Band 2x2x2 43676.20392 26055.12739 13318.14587  




Table 15: Calculation time of conductivity and dielectric function of Si crystal (512 atoms, $4\times 4\times 4$ supercell, k-grid= $1\times 1\times 1$). The speed-up ratio with repect to the case with 128 CPU cores is shown in the last column.
  # of CPUs
 Total time of calculating  
  conductivity and dielectric function (s)  
 Ratio of  
  total time by # of CPUs  
  to total time by 128 CPUs  
 
  128 3367.16826 1.000  
  256 1755.60797 1.918  
  512 919.21912 3.663  
  1024 464.27761 7.252  
  2048 253.32210 13.292  



$\beta $-PVDF
The real part of dielectric function of $\beta $-PVDF (polyvinylidene fluoride) is shown for a series of k-grids in Fig. 82. We see that the k-grid of $6\times 9\times 21$ is required to get the convergent result. In Tables 16 and 17, we show the computational time and parallel efficiency in the calculation of the conductivity and dielectric function for supercells of $\beta $-PVDF. It is confirmed that the parallel efficiency is reasonably good, and the elapsed time is less than one hour when the CPU cores of 256 are used. In Fig. 82, we show the $xx$, $yy$, and $zz$ components of real part of dielectric function of $\beta $-PVDF for your reference.



Table 16: Total time of calculating conductivity and dielectric function of $\beta $-PVDF (polyvinylidene fluoride) consisting of 540 atoms which corresponds to the $3\times 3\times 5$ supercell.
  Diagonalization k-Grid
  Total time (s)  
  (CPUs=128)  
  Total time (s)  
  (CPUs=256)  
  Total time (s)  
  (CPUs=512)  
  Total time (s)  
  (CPUs=1024)  
  Total time (s)  
  (CPUs=2048)  
 
  Cluster 1x1x1 4215.67238 2086.76841 1061.06835 565.92209 329.42010  
  ScaLAPACK 1x1x2 3330.66855 1711.98413 877.61705 599.73266 330.11080  
  Band 2x2x2 12854.63816 7244.26768 3591.33618 1866.03405 998.07756  



Table 17: Calculation time of conductivity and dielectric function of $\beta $-PVDF (polyvinylidene fluoride) consisting of 540 atoms which corresponds to the $3\times 3\times 5$ supercell, where k-grid of $1\times 1\times 1$ was used. The speed-up ratio with repect to the case with 128 CPU cores is shown in the last column.
 # of CPUs
 Total time of calculating  
 conductivity and dielectric function (s)  
 Ratio of  
 total time by # of CPUs  
 to total time by 128 CPUs  
 
 128 4215.67238 1.000  
 256 2086.76841 2.020  
 512 1061.06835 3.973  
 1024 565.92209 7.449  
 2048 329.42010 12.797  

Figure 82: The real part of dielectric function of $\beta $-PVDF (polyvinylidene fluoride) consisting of 6 atoms in the $1\times 1\times 1$ cell for a series of k-grids. CDDF.FWHM=0.2 eV was used.
Figure 83: The $xx$, $yy$, and $zz$ components of real parts of dielectric function of $\beta $-PVDF (6 atoms, $1\times 1\times 1$ cell).
\includegraphics[width=14.0cm]{CDDF-Fig3.eps} \includegraphics[width=14.0cm]{CDDF-Fig4.eps}



VO$_2$ in the R phase
The real and imaginary parts of dielectric function of VO$_2$ in the R phase are shown for a series of k-grids in Fig. 84. We see that the k-grid of $16\times 16\times 16$ is required to get the convergent result. Tables 18 and 19 show the computational time and parallel efficiency in the calculation of the conductivity and dielectric function for supercells of VO$_2$ in the R phase. It is confirmed that the parallel efficiency is reasonably good, allowing us to treat large-scale systems in an elapsed time of 1 hour.

Figure 84: The real and imaginary parts of dielectric function of VO$_2$ (R phase, 6 atoms, $1\times 1\times 1$ cell) with a series of k-grids.
\includegraphics[width=17.0cm]{CDDF-Fig5.eps}



Table 18: Calculation time of conductivity and dielectric function of VO$_2$ (R phase, 384 atoms, $4\times 4\times 4$ supercell)
  Diagonalization k-Grid
  Total time (s)  
  (CPUs=128)  
  Total time (s)  
  (CPUs=256)  
  Total time (s)  
  (CPUs=512)  
  Total time (s)  
  (CPUs=1024)  
  Total time (s)  
  (CPUs=2048)  
 
  Cluster 1x1x1 5328.56778 2719.77511 1382.56556 704.32679 360.64294  
  ScaLAPACK 1x1x2 5276.74287 2755.58509 1395.48266 718.42770 368.24621  
  Band 2x2x2 21031.96431 10686.37446 5586.12815 2781.05698 1431.91243  



Table 19: Total time of calculating conductivity and dielectric function of VO$_2$ (R phase, 384 atoms, $4\times 4\times 4$ supercell, k-grid= $1\times 1\times 1$). The speed-up ratio with repect to the case with 128 CPU cores is shown in the last column.
  # of CPUs
 Total time of calculating  
  conductivity and dielectric function (s)  
 Ratio of  
  total time by # of CPUs  
  to total time by 128 CPUs  
 
  128 5328.56778 1.000  
  256 2719.77511 1.959  
  512 1382.56556 3.854  
  1024 704.32679 7.565  
  2048 360.64294 14.775