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Results and assessment

Figure 9: Energy (inset RMS gradient) convergence of the NGWF optimisation for a variety of systems - figures in parentheses in legend indicate number of atoms.

The ONETEP method has been parallelised, and results are available elsewhere [12,57]. By construction, the cost of a single self-consistent iteration in the determination of the ground state scales linearly with system-size. However, as pointed out in Sec. 1, this is not sufficient to guarantee that the calculation of physical properties scales linearly with system-size, since the number of iterations required has been observed to increase in some methods.

Figure 9 shows the convergence of the ONETEP method with respect to iterations of the NGWF optimisation procedure. A wide variety of systems with different properties have been chosen: a supercell of crystalline silicon, a hydrogen-bonded formaldehyde-water complex, a (20,0) carbon nanotube, a protein complex and a supercell of the zeolite ZSM5. Both the gain in total energy and the length of the residual vector are plotted. In all cases, the convergence criteria are satisfied within 10-20 iterations, independently of the number of atoms in the system. Thus in ONETEP, the total calculation time, not just the time per iteration, scales linearly with system-size. This is believed to result from the choice of an orthogonal basis set and the careful and consistent treatment of all terms in the energy.

Figure 10: Interaction potentials (negated binding energies) for a hydrogen-bonded water dimer.

The accuracy of ONETEP calculations has been compared with traditional plane-wave pseudopotential and all-electron methods elsewhere [13]. Figure 10 shows the interaction potential for a hydrogen-bonded water dimer as calculated by the CASTEP [58] code and by ONETEP using the same pseudopotential, gradient-corrected exchange-correlation functional [59] and equivalent energy cut-offs for the basis sets [60]. In two of the ONETEP calculations, only the density-kernel has been optimised, the NGWFs have been fixed as a minimal set of atomic-type orbitals. This calculation exhibits basis set superposition error (BSSE) [61] for which in this system the counterpoise correction [62] is only partially successful. While ONETEP is capable of performing calculations at this level, for weakly bound systems such as this one, high accuracy is required. When the NGWFs are optimised in addition to the density-kernel, no BSSE is observed and the agreement is vastly improved. With fully converged basis sets, the equilibrium bond lengths and binding energies obtained by ONETEP and CASTEP agree to 0.3% and 2% respectively.

Together with the results presented in Fig. 4, it can be seen that the accuracy of calculations in ONETEP can be controlled, and that plane-wave accuracy can be achieved routinely. The twin aims of overall linear scaling and controlled accuracy have thus been achieved.

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