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  Let tex2html_wrap_inline2684 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 tex2html_wrap_inline2688 .

The projection operator is defined


where tex2html_wrap_inline3474 are the duals of the atomic basis given by


and where in turn tex2html_wrap_inline2690 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 tex2html_wrap_inline2694 is the weight assigned to each k-point at which the bands are calculated. The sum of tex2html_wrap_inline2694 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 tex2html_wrap_inline2708 is the occupancy of band tex2html_wrap_inline2653 . The LCAO density matrix can then be written:


the elements of which may be found from the matrixes S, R and T as follows:


Matthew Segall
Wed Sep 24 12:24:18 BST 1997