Let 
be the electronic eigenstates
described in a plane wave basis set (These will be referred to as PW
eigenstates) resulting from the electronic structure
calculation. Also, let the atomic basis set (hereafter referred to as
the LCAO basis set) be represented 
.
The projection operator is defined
where 
are the duals of the atomic basis given by
and where in turn 
is the overlap matrix of the atomic basis
set defined:
The overlap matrix between the PW eigenstates and the atomic orbitals is given by
The spilling parameter
may be written in terms of the overlap matrixes S and T thus:
Here 
is the weight assigned to each k-point at which the bands are
calculated. The sum of for every k-point must equal 1.
As neither the PW or LCAO basis sets are complete, the overlaps of the projected eigenstates,
must be considered in order to ensure that the total charge is unchanged by the projection. This is achieved by defining the density operator as
where 
is the occupancy of band 
. The LCAO density
matrix can then be written:
the elements of which may be found from the matrixes S, R and T as follows:
