6.6 Independent particle finite size effects

The ``independent particle finite size effect'' (IPFSE) is defined by the correction term $(E_{\infty}^{\rm LDA} - E_{N}^{\rm LDA})$ . The finite size effect arising from the particular model interaction used is then the remaining ``Coulomb finite size effect'' (CFSE). In this section the IPFSE is discussed in detail. Particular attention is paid to k-point selection, or BZ sampling.

The ``finite size error'' in an LDA calculation for a perfect crystal arises from errors in the BZ integration. In an LDA calculation the computational cost is proportional to the number of k-points used. In a QMC calculation the volume of the simulation cell is proportional to the number of k-points in the primitive BZ. The computational cost increases approximately as the cube of the volume of the simulation cell, meaning that the selection of k-points is even more critical in QMC calculations.

Additional errors arise when using the supercell approximation for systems which break translational symmetry. Consider a crystal containing a point defect. In the supercell approach a finite simulation cell containing a single defect is repeated throughout space to form an infinite three-dimensional array. The supercell must be sufficiently large that the interaction between defects in different cells is negligible. This type of finite size effect is distinct from the BZ integration error because it persists even when the BZ integration is performed exactly. However, the correction to the defect formation energy from this type of finite size error should be described reasonably well by a $(E_{\infty}^{\rm LDA} - E_{N}^{\rm LDA})$ correction term, where $E_{\infty}^{\rm LDA}$ and $E_{N}^{\rm LDA}$ are, respectively, the defect formation energies for a large simulation cell and a smaller one, each containing a single defect.

A determinantal Bloch wave function suitable for use in a QMC calculation may be formed from a set of single particle orbitals at a single k-point, ${\bf k}_{\rm s}$, in the Brillouin zone of the simulation cell reciprocal lattice. The IPFSE can be greatly reduced for insulating systems by a careful choice of ${\bf k}_{\rm s}$ using the method of ``special k-points'' [93,94] borrowed from band structure theory. [95,96] Baldereschi [95] defined the ``mean-value point'', a k-point at which smooth periodic functions of wave vector accurately approximate their averages over the BZ, and clearly the Baldereschi mean-value point is a strong candidate for ${\bf k}_{\rm s}$. It is convenient to choose ${\bf k}_{\rm s}$ to be equal to half a translation vector of the simulation cell reciprocal lattice ( ${\bf k}_{\rm s}={\bf G}/2$), which allows the construction of real single-particle orbitals and hence is computationally more efficient. Some freedom is still left in the selection of ${\bf k}_{\rm s}$ and the symmetrized plane wave test of BZ integration quality introduced by Baldereschi [95] provides an accurate guide to final selection. For example, for an fcc simulation cell the half-reciprocal-lattice vectors correspond to the $\Gamma$, $X$ and $L$ points of the BZ of the simulation cell reciprocal lattice. For a crystal with the full cubic symmetry ${\bf k}_{\rm s} \equiv L$ gives the best BZ integration and ${\bf k}_{\rm s} \equiv \Gamma$ the worst.