The plane-wave basis state
is a solution of the Helmholtz equation (the time-independent
free-electron Schrödinger equation)
If instead we wish to localise the basis functions, say within
spherical regions of radius , so that the function vanishes outside
these regions, then appropriate conditions would be to require the
functions to be finite within the regions and to vanish on the
boundary. The solutions to the Helmholtz equation (1)
subject to these conditions are then truncated spherical-waves
We note that these functions solve the same equation as the plane-wave basis functions, so that within the pseudopotential approximation the wave functions will be well-described by a truncated set of these basis functions. Moreover, these functions are eigenstates of the kinetic energy operator within the localisation region (i.e. in the region in which they will be used to describe the wave functions) with eigenvalue so that the same kinetic energy cut-off used to truncate the plane-wave basis can be used here to restrict the values of and .
Since the Laplacian is a self-adjoint operator under these boundary conditions, application of Sturm-Liouville theory proves that all states within the same spherical region are mutually orthogonal.
In a calculation, the electronic states are described by covering
the simulation cell with overlapping spheres, usually chosen to be
centred on the ions at positions
, and expanding the
within these spheres in this basis:
The expansion (3) is frequently written down formally, but rarely used computationally because of the inconvenience of using spherical Bessel functions in numerical work. However, the analytic results derived in the following sections offset this disadvantage.
methods are aimed at large systems, and so the Brillouin
zone sampling of the electronic states is usually restricted to the
states at the -point only. The wave functions can then be made
real without loss of generality, and so in practice we use real linear
combinations of the spherical harmonics defined below, which does not
alter any of the analysis here.
to represent a
truncated spherical-wave basis function centred at the origin and
confined to a sphere of radius :