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: