Síntesis del caso practico

(4.1)

The basis sets used in plane wave calculations are very different to the atom-centred Gaussian contractions just examined: instead the electron density is modelled using tens of thousands of periodic functions, spanning the entire unit cell. These individual plane waves are not localised onto any particular ion in the material, and the description of highly localised bonding features requires the combination of a great many plane waves. For this reason, such calculations are often used to model metallic systems in which the electrons in the valence band are not associated with any particular ion.

The core states o f an ion form a set o f very highly localised energy states - they are centred around its nucleus. To correctly describe core electrons using only plane waves would require a huge number of functions at great computational cost: instead the plane wave basis is augmented using a collection of atom-centred functions to describe the core states with electrons in the inter-ion regions o f space described by the plane wave basis. By ensuring that the two types o f function remain continuous as we move away from the nucleus the description o f the system need not be reduced: usually the primary concern lies with the valence electrons, since these dictate the chemistry of the material.

Chapter 4: Fluorite

This approach is similar to the pseudopotential method explained in section 2.3.3: the cost o f the calculations is reduced through the use o f multiple functional forms to describe the space used in the construction o f the Hamiltonian matrix.

The CASTEP calculations performed here made use of an ultrasoft pseudopotential on both the metal and oxygen ions. The electronic configuration o f the species present in these calculations were Ce {5s^5p^^f'6s^) and O (2s^2p^). The cut-off radius (rcut) used in these calculations was dependent on the /-value o f the orbital on the cerium ions: values of 1.60, 1.80 and 2 . 0 0 a.u. for the 5, and / were employed here, on the oxygen ions a cut-off of 1.30 a.u. was used for all functions.

Optimisation of a plane wave basis set is very different to that detailed in the previous section: here we need to determine the minimum number of functions which must be included into the plane wave basis in order to correctly model the system. Often a series o f calculations are performed with various plane wave cut-off points: each function has a unique periodicity in a similar way as the Bloch functions constructed in the CRYSTAL calculations.

This periodicity is associated with a given energy: functions with low periodicity represent low energy functions, while a high periodicity is associated with a large energy. By setting an upper limit on the periodicity o f the wave, and defining the step in periodicity between adjacent functions, we can construct a set of N distinct plane waves to use as the basis for the following calculations. The higher this energy cut-off point lies, the greater the number of functions which need including into the set - improving the overall accuracy of the calculations but dramatically increasing the cost. To balance the two, a series of calculations are performed on the bulk phase using a range o f energy cut-offs, the plot of calculated internal energy against this cut-off energy (an example of which is given in figure 4.4.

Chapter 4: Fluorite B Q> lU (0

I

5

Maximum cut off point needed for

required Internal Energy convergence

Cut Off Energy

Figure 4.4: Typical internal energy versus plane wave energy cut-off plot. The internal energy is calculated for a series of cut-off energies, and the optimal

configuration used in future calculations.

The point at which the calculated internal energy has converged (ensuring the other properties of the system are correctly described) is then used in future calculations as the optimal balance between the number o f plane waves which must be included and the accuracy of the calculations. In addition to this, we have also to optimise the density of the grid sampling reciprocal space: periodic DFT calculations are solved in exactly the same way as the periodic HF, with the Kohn-Sham equations being solved at a discrete grid o f points spanning the whole of the irreducible Brillouin zone.

Table 4.4: Energy minimised lattice parameter and internal energy calculated

Plane-wave cut-off

energy (eV) ^cub (Â) Erriin (cV)

370 5.4560 -7805.9059

400 5.4541 -7805.9523

415 5.4197 -7805.9882

430 5.4195 -7806.0108

445 5.4195 -7806.0443

Chapter 4: Fluorite

Hamiltonian. We consider as converged the results obtained with a cut-off energy of 430 eV, and shall employ this value in all future bulk phase DFT calculations on ceria performed with the CASTEP code. We should note also that the energy minimised lattice parameter appears in excellent agreement to the experimental (room temperature) figure o f 5.411 Â.

Table 4.5 reports the changes calculated in the equilibrium lattice parameter and internal energy for increasing A:-space shrinking factors: we see from these results that the current grid of 2x2x2 is already converged in the bulk systems, and there are no benefits to increasing this any further.

Table 4.5: E n e i^ minimised lattice parameter and internal energy calculated

A:-space shrinking factor

(nxnxn points) ^cub (^) Emir, (eV)

2 5.4195 -7806.0107612

4 5.4195 -7806.0107618

8 5.4195 -7806.0107676