Comparisons of efficiency with VASP

128 atom unit cell of GaAs

Fig. 1 comparison of progressive error in total energy for different cutoff energies for PARATEC and VASP

Since VASP uses ultra-soft pseudopotentials, we cannot compare calculations using the same cutoff energy. We need to compare calculations done at the same level of accuracy. In Fig 1, we present a graph of the cutoff energies vs. the error in total energy from fully converged calculations. By inspection we see that a cutoff energy of 340 eV (25 Ryd) for PARATEC has the same accuracy as a 150 EV (11 Ryd) cutoff for VASP. We now proceed to compare the efficiency of the two codes with these energy cutoffs.

Fig. 2 comparison of efficiency for VASP and PARATEC for runs of comparable accuracy

Even though an energy cutoff of twice that of VASP is used for PARATEC, Fig. 2 shows comparable efficiency. This was done on a 16 node shared memory machine so there are no communication issues. The main reason for the surprising performance of PARATEC is in the algorithm for electronic minimization (i.e. obtaining self-consistency). VASP uses a band-by-band conjugate gradient technique. A band-by-band method is inherently slower than a subspace (block) method since the former can onlu use BLAS2 while the latter can implement BLAS3, which performs much faster. Subspace methods also usually outperform band-by-band methods in the number of iterations since the interaction of bands is accounted for when the system is looked at as a whole. In addition PARATEC uses a subspace diagonalization technique for the initial guess. This is superior to the random initial guess that VASP uses.

20 layer surface of Al

Fig. 3 comparison of progressive error in total energy for different cutoff energies for PARATEC and VASP

In Fig 3, we present a graph of the cutoff energies vs. the error in total energy from fully converged calculations. By inspection we see that a cutoff energy of 275 eV (20 Ryd) for PARATEC has the same accuracy as a 130 EV (9.5 Ryd) cutoff for VASP. We now proceed to compare the efficiency of the two codes with these energy cutoffs. A k-point mesh of 12x12x12 was used.

Fig. 4 comparison of efficiency for VASP and PARATEC for runs of comparable accuracy for a 20 layer Al surface

For this system, PARATEC actually outperforms VASP even though it uses more than twice the cutoff of VASP. This is due to in part to the Grassmann metal algorithm for the electronic minization, but mostly due to the improved Pulay-Thomas-Fermi mixing method. The same problems in convergence exhibited by VASP in Fig. 3 also occur in PARATEC when the Pulay-Kerker (equivalent to the mixing method in VASP) is used. A k-point mesh of 4x12x12 was used.

10 layers surface of Pd

Fig. 5 comparison of progressive error in total energy for different cutoff energies for PARATEC and VASP

In Fig 5, we present a graph of the cutoff energies vs. the error in total energy from fully converged calculations. We see that the convergence curve for VASP is very flat. Reasonable energy convergence is found around 175 eV (12.9Ry). The same energy convergence is not achieved for PARATEC (and the norm-conserving pseudopotenitals) until 680 eV (50Ry). The stress for VASP at 175 eV is not correct and does not reach acceptable values until 225 eV (16.54 Ry). We now proceed to compare the efficiency of the two codes with energy cutoffs of VASP 16.54 Ry and PARATEC 50 Ry.

Fig. 6 comparison of efficiency for VASP and PARATEC for runs of comparable accuracy for a 10 layer Pd surface

The difference in energy cutoffs of about 3 times results in about 5 times as many plane waves. Since the energy cutoff is relatively greater here than for Al we do not expect the performance of PARATEC to be as good. Fig. 6 shows this to be the case. PARATEC converges slightly less than 2 times as slow due to the much larger energy cutoff. Given that 5 times as many plane waves are used, a difference of slightly less than 2 still seems pretty good. The added comfort of the accuracy of norm-conserving is also something to consider. A k-point mesh of 2x12x12 was used.