Checking the absolute convergence of our Bulk Silicon Calculation

The example calculation with bulk silicon does not produce a very accurate answer. Can it be improved? There are two main sources of error in our calculation;

• Fundamental errors associated with the approximations used in the calculation such as the use of the LDA.
• Systematic errors associated with inaccuracies in the wavefunctions introduced by using a finite basis set and wavefunction sampling in reciprocal space.

The latter of these must be reduced to a minimum. To show how this might be done I have repeated the calculation several times with increasing precision in the basis set and with various MP grid densities. Figure 3 below shows the total energy as a function of the basis set size for calculations with 1, 10, 28 and 60 K-Points. (See the section on variable cell calculations for a neat trick to efficiently perform many singlepoint energy calculations at a variety of cutoff energies at a fixed k-point density).

Figure 3: Convergence of the bulk Si calculation with respect to the basis set

We can see that gamma point sampling with a single k-point produces a structure with a much higher energy than sampling with other k-point densities. Figure 4 below shows a plot of figure 3 with the 1 k-point data removed for greater clarity.

Figure 4: Convergence of the basis set showing only the higher k-point densities

The cutoff energy is a variational parameter and as it is increased the energy will converge asymptotically on the ground state from above. However, it is important to remember that the sampling set size is not a variational parameter. It entirely possible to increase the total final energy of the system by increasing the density of the sampling grid. If we know more information about the wavefunction it does not mean that this information will provide us with a lower energy structure. The key point about convergence with respect to the sampling grid density is that the difference in energy from one grid density to the next should be minimised. We can see from figure 4 that increasing the MP grid density from 28 to 60 k-points has a negligible effect on the total final energy. Indeed, the two curves lie on top of one another and cannot be separately resolved. Also, the total energy is converged with respect to the cutoff energy when this is above about 280eV.

In summary, our fully converged basis set parameters are; 28 k-points in the reciprocal space sampling grid and a cutoff energy of 280eV or higher. Has the reduction of systematic error helped? The bond length for the calculation with the optimal set of parameters for both accurate wavefunction modelling and conservation of computational resources is 2.28753 Å. This is only 1.9% away from the acepted value of 2.332 Å. Convergence has resulted in an improvement of 6% on the answer obtained using an arbitrary set of basis set parameters.

One final point on sampling grids. An important factor in dictating the required sampling grid density is the size of the supercell. A large real space supercell might require only a very sparse reciprocal space sampling grid. Our small bulk silicon unit cell requires quite a dense MP grid. In the later silicon (100) surface example the supercells contain 20 atoms but require only a 9 k-point sampling grid. Of course, the electronic complexity of the constituent atoms also plays a role in determining the required sampling grid density.