Predictions of physical properties

The converged value of the total energy agrees with the CASTEP value (again with equivalent energy cut-off, Brillouin zone sampling and pseudopotential) only within 3%. However, if energy differences are caused by density-matrix variations which are short-ranged, then they will be much better converged than absolute energies. In figure 9.7 we plot the total energy against volume $V$. For a lattice parameter $a = 5.430$ Å, which was used for the calculations in section 9.1.1, we used values of $r_{\mathrm{reg}} = 3.10$ Å, $r_K = 6.00$ Å and an energy cut-off of 200 eV. As the volume $V$ was changed, the support region radius $r_{\mathrm{reg}}$, density-kernel cut-off $r_K$ and energy cut-off $E_{\mathrm{cut}}$ were changed proportionally. The parameters used are listed in table 9.1.

Table 9.1: Parameters used to calculate energy-volume curve.

$a$ / Å	$V$ / Å$^3$	$r_{\mathrm{reg}}$ / Å	$r_K$ / Å	$E_{\mathrm{cut}}$ / eV
5.31	149.72	3.03	5.87	209
5.35	153.13	3.05	5.91	206
5.39	156.59	3.08	5.96	203
5.43	160.10	3.10	6.00	200
5.47	163.67	3.12	6.04	197
5.51	167.28	3.15	6.09	194

Figure 9.7: Energy-volume curve for silicon. The line is the best fit to the Birch-Murnaghan equation of state.

Fitting the data to the Birch-Murnaghan equation of state allows values for the equilibrium volume $V$ and bulk modulus $B$ to be calculated, which are compared with the results obtained from a CASTEP calculation using the same pseudopotential and to experiment [174] in table 9.2. Generally we expect to obtain lattice parameters to within 2% and bulk moduli to within 10%. The bulk modulus is very sensitive to the data, so that whereas the prediction of the lattice parameter is in excellent agreement with both the CASTEP and experimental values, the value of the bulk modulus predicted by the linear-scaling method is about 8% too large. These results indicate that the linear-scaling calculation is not quite fully converged with the set of parameters used, but are very encouraging overall.

Table 9.2: Comparison of calculated and experimental data for silicon.

Calculation	Linear-scaling	CASTEP	Experiment
$a$ / Å	5.423	5.390	5.430
$V$ / Å$^3$	159.47	156.56	160.10
$B$ / GPa	108.8	101.7	100.0