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In the linearscaling method, we have a densitymatrix represented in the
form

(8.1) 
First we represent the localised support functions by linear combinations of
planewaves. For the analytic basisset described in chapter 5
this is easily accomplished using equation 5.9 which gives
the Fourier transform of the basis functions.
Having obtained an expansion for the support functions in a complete
basisset, it is now possible to orthogonalise the support functions
by means of the Löwdin transformation to the set of orthonormal orbitals
given by

(8.2) 
Simultaneously transforming the matrix into the matrix by

(8.3) 
leaves the densitymatrix invariant in the sense that

(8.4) 
To obtain the KohnSham orbitals and occupation numbers it is necessary to
diagonalise . If this densitymatrix is a groundstate
densitymatrix, then the densityoperator and Hamiltonian commute so that
they have the same diagonal representation, and therefore diagonalising
is equivalent to diagonalising
.
Thus the unitary transformation which yields the occupation numbers
(no summation convention) also yields the
KohnSham orbitals
.
This information can then be used in a traditional planewave code, and this
is the method which was used to check the analytic results for the kinetic
energy and
nonlocal pseudopotential energy in chapter 5. The spatial
cutoff of the support regions in realspace leads to algebraicallydecaying
oscillatory behaviour for large wavevectors in reciprocalspace, so that a
single basis function needs a high planewave energy cutoff to accurately
describe this truncation.
However, for support functions which decay smoothly
to zero at the edge of the support region, the decay will be much faster, and
the planewave cutoff comparable to the cutoff for the basis
functions themselves.
Next: 8.2 Densitymatrices from KohnSham
Up: 8. Relating linearscaling and
Previous: 8. Relating linearscaling and
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Peter Haynes