The Kohn-Sham orbitals, 
in Equation 
, may be
represented in terms of any complete basis set. Many choices are
possible including atomic orbitals, Gaussians, LAPW
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
).
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
may be written
where the sum is over reciprocal lattice vectors 
and 
is a
symmetry label which lies within the first Brillouin zone. Thus, the
basis set for a given 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, 
, such that for all
values of used in the expansion
Thus, the convergence of the calculation with respect to basis set may
be ensured by variation of a single parameter, . 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
). 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.
Figure: A schematic
illustration of a supercell geometry for a molecule. The boundaries of
the supercell are depicted by dashed lines.
The electron density 
and energy are given by averaging the
results for all values of in the first Brillouin zone,
where 
, and
In an extended system, these integrals are replaced by weighted sums
over a discrete set of -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 with . Therefore, these properties need only be
calculated at a single -point. There has been significant
discussion regarding the optimal choice of -point for performing
calculations on isolated systems [22]. For the molecular
calculations presented in this thesis, the 
point, the origin
in -space, was chosen. This choice confers significant savings in
storage and computation, as the coefficients 
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.