Supercell expansion

The final geometry was obtained when the calculated forces acting on the atoms and stress on the supercell became smaller than the threshold values. To evaluate the relative expansion of the lattice induced by the interstitial hydrogen, geometry optimizations of the pure metals were also employed. 

It is conceptually simple. The size of the supercell depends on an expansion matrix that consists of integers. The matrix is 2 x 2 or 3 x 3 according to the dimensionality of the periodic system. 

Figure 54 Dependence of the formation energy of the axial BeO [Li] defect with size and shape of the adopted supercell, (i j k) are the expansion coefficients of the primitive lattice basis vectors.

The GAP results for the thermal expansion coefficients obtained from the quasiharmonic approximation show excellent agreement with the LDA values. This verifies that the potential energy surface represented by the GAP model is, in fact, close to the ab initio potential energy surface, even outside the harmonic regime. In the case of Density Functional Theory, the molecular dynamics simulation would be computationally expensive, because a large supercell has to be used to minimise finite-size effects. However, with GAP, these calculations can be easily performed and the thermal expansion coefficients obtained match the experimental values well, even at high temperatures. 

When trying to use the LCAO basis for partial optimization of the total energy for the same supercells, a markedly smaller expansion of the first coordination sphere involving the two nearest titanium ions as compared with complete optimization in... 

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