We have implemented the projection technique of Sanchez-Portal et. al.
[8]. The eigenstates 
resulting from the PW calculations are projected onto a localised
basis set 
. In our case, the natural choice of
basis set is that of atomic pseudo-orbitals generated from the
pseudopotentials used in the calculation. However, care must be taken
when performing the projection because this basis set is neither
orthonormal nor complete.
The overlap matrix of the localised basis set, 
, is
defined :
We compute these overlaps in reciprocal space as the imposition of periodic boundary conditions implies that the orbitals need only be evaluated on a discrete reciprocal space grid. The overlaps between orbitals on different atomic sites may be calculated on the same grid with the application of a phase factor. The representations of the atomic pseudo-orbitals on the discrete reciprocal space grid may be calculated by Fourier transformation of the real space wavefunctions generated during construction of the pseudopotentials, or can be generated using the CASTEP code.
The overlap between the plane wave states and the basis functions,
were also calculated in reciprocal space as this is the natural representation of the plane wave states.
The quality of the projection may be assessed by the calculation of a spilling parameter,
where 
is the number of PW eigenstates, 
the
weights associated with the calculated 
points in the Brillouin
zone and 
is the projection operator of Bloch functions with
wave vector 
generated by the atomic basis.
where 
are the duals of the atomic basis
states such that
The spilling parameter varies between one in the case that the atomic
basis set is orthogonal to the PW eigenstates and zero when the projected
wavefunctions perfectly represent the PW states. Unlike the procedure
of Sanchez-Portal et al. [8] we have not scaled
the basis functions in order to optimise 
. This gives
spilling parameters in the region of 
which were found to be
adequate for our purposes.
The density operator may be defined,
where 
are the occupancies of the PW states (
=1,2),
are the projected eigenstates 
and 
are the
duals of these states. From this, the density matrix for the atomic
basis set may be calculated as follows:
We find that 
may be calculated in a time proportional to
the cube of the system size, the same order of scaling as the original
PW calculations.
The density matrix 
and the overlap matrix 
are sufficient to perform population analysis of the
electronic distribution. In Mulliken analysis [6] the
charge associated with a given atom A is given by
and the overlap population between two atoms A and B is
Similarly in Löwdin analysis [7] these quantities are given by
and
In equations (10) and (11) the matrix
is calculated as a Taylor expansion of
where 
.
