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Subsections
8.2 Densitymatrices from KohnSham orbitals
This method is based upon work on the projection of planewave calculations
onto atomic orbitals [166], which has been used to
analyse atomic basissets [167] and obtain local atomic
properties from the extended KohnSham orbitals [168].
Here we review the method, for the special case of a point
Brillouin zone sampling.
The planewave eigenstates are denoted
and the
support functions are denoted
.
The states obtained by projecting the planewave eigenstates onto the space
spanned by the support functions are denoted
.
As in section 4.6 we also introduce the dual states
and
with the properties
outlined below.
The projection operator onto the subspace spanned by the support functions
is defined by

(8.8) 
A spilling parameter can be defined to measure how much the
subspace spanned by the planewave eigenstates falls outside the subspace
spanned by the support functions. Minimising this quantity is one method of
optimising the choice of support functions, and is described for the case of
the sphericalwave basis (chapter 5) in section 8.2.3.

(8.9) 
where is the number of bands (labelled ) considered.
The densityoperator is then defined by

(8.10) 
where the sum is taken over occupied bands only. Substitution of the results
given above then yields the following expression for the densitykernel:

(8.11) 
Defining the rectangular matrix as

(8.12) 
we give an expression for the matrix in terms of and :

(8.13) 
We can thus minimise the spilling parameter to optimise our choice
of support functions, and then calculate to obtain all of the information
required to start a linearscaling calculation.
In the case when the densitykernel is expanded in terms of an
auxiliary matrix e.g. in order to construct a positive semidefinite
densitymatrix, it is necessary to be able to calculate the auxiliary
matrix which corresponds to a given densitykernel by

(8.14) 
This can be achieved by minimising the function given by

(8.15) 
whose derivative with respect to is

(8.16) 
This derivative vanishes at the minimum, and so we find that the matrix
which minimises is the desired auxiliary matrix (the
solution corresponds to a local maximum). We therefore choose to
minimise by the conjugate gradients method to obtain the
auxiliary matrix.
8.2.3 Optimising the support functions
As mentioned above, we can optimise our choice of support
functions by minimising the spilling parameter . We describe
this process here when the support functions are themselves described in
terms of a localised basis:

(8.17) 
The spilling parameter can be written in terms of the matrices and by:
and we wish to obtain the gradients of with respect to the
expansion coefficients
.

(8.19) 
We obtain the derivative of the inverse matrix by differentiating
i.e.

(8.20) 
which can be rearranged to give

(8.21) 
Therefore (no summation over )
In the case of the set of basis functions introduced in chapter 5,
the overlap between planewave eigenstates and localised basis functions,
e.g.
, can be
calculated using the expression for the basis function Fourier transform
(5.9).
We can use these gradients to minimise the spilling parameter (by the
conjugate gradients method) to obtain the set of optimal
coefficients
which define the set of
support functions which best span the space of the occupied planewave
orbitals.
The final minimum spilling parameter value also gives an estimate of the
quality of the basisset being used.
Next: 8.3 Densitymatrix initialisation
Up: 8. Relating linearscaling and
Previous: 8.1 Wavefunctions from densitymatrices
Contents
Peter Haynes