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The Choice of a Basis Set - Plane Waves

 

The Kohn-Sham orbitals, tex2html_wrap_inline2527 in Equation gif, may be represented in terms of any complete basis set. Many choices are possible including atomic orbitals, Gaussians, LAPWgif and plane waves, the basis set we use in practice. The use of a plane wave (PW) basis set offers a number of advantages, including the simplicity of the basis functions, which make no preconceptions regarding the form of the solution, the absence of basis set superposition error, and the ability to efficiently calculate the forces on atoms (See Section gif).

In general, the representation of an arbitrary orbital in terms of a PW basis set would require a continuous, and hence infinite, basis set. However, the imposition of periodic boundary conditions allows the use of Bloch's Theorem [20] whereby the tex2html_wrap_inline2527 may be written

equation260

where the sum is over reciprocal lattice vectors tex2html_wrap_inline2531 and tex2html_wrap_inline2533 is a symmetry label which lies within the first Brillouin zone. Thus, the basis set for a given tex2html_wrap_inline2533 will be discrete, although in principle it will still be infinite. In practice, the set of plane waves is restricted to a sphere in reciprocal space most conveniently represented in terms of a cut-off energy, tex2html_wrap_inline2537 , such that for all values of tex2html_wrap_inline2531 used in the expansion

equation266

Thus, the convergence of the calculation with respect to basis set may be ensured by variation of a single parameter, tex2html_wrap_inline2537 . This is a significant advantage over many other basis set choices, with which calculated properties often show extreme sensitivity to small changes in basis set and no systematic scheme for convergence is available.

The choice of periodic boundary conditions is natural in the case of bulk solids which exhibit perfect translational symmetry. If we wish to model an isolated molecule we must artificially introduce periodic boundary conditions by construction of a supercell (See Figure gif). In this scheme the calculations are performed on a periodic array of molecules, separated by large vacuum regions. In the limit of large separation between the periodic images, the results will be those for an isolated molecule. Therefore, care must be taken to converge the results with respect to supercell dimensions.

   figure273
Figure: A schematic illustration of a supercell geometry for a molecule. The boundaries of the supercell are depicted by dashed lines.

The electron density tex2html_wrap_inline2469 and energy are given by averaging the results for all values of tex2html_wrap_inline2533 in the first Brillouin zone,

equation280

where tex2html_wrap_inline2547 , and

equation287

In an extended system, these integrals are replaced by weighted sums over a discrete set of tex2html_wrap_inline2533 -points which must be carefully selected to ensure convergence of the results [21]. An isolated molecule will exhibit no dispersion, ie. there will be no variation of E and tex2html_wrap_inline2469 with tex2html_wrap_inline2533 . Therefore, these properties need only be calculated at a single tex2html_wrap_inline2533 -point. There has been significant discussion regarding the optimal choice of tex2html_wrap_inline2533 -point for performing calculations on isolated systems [22]. For the molecular calculations presented in this thesis, the tex2html_wrap_inline2561 point, the origin in tex2html_wrap_inline2533 -space, was chosen. This choice confers significant savings in storage and computation, as the coefficients tex2html_wrap_inline2565 may be represented as real numbers, whereas in general they are complex.

The principle disadvantage of the use of a PW basis set is the number of basis functions required to accurately represent the Kohn-Sham orbitals. This problem may be reduced by the use of pseudopotentials as described in the next section, but several hundred basis functions per atom must still be used, compared with a few tens of basis function with the use of some atom-centred basis sets.


next up previous contents
Next: Pseudopotentials Up: The Total Energy Pseudopotential Previous: The Exchange-Correlation Term

Matthew Segall
Wed Sep 24 12:24:18 BST 1997