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7.4 Physical interpretation
At this stage we examine the energy gradients derived in section
7.2. At the minimum of the total functional
,

(7.44) 
Making the Löwdin transformation of this gradient into the representation
of a set of orthonormal
orbitals (using the results of section 4.6) yields

(7.45) 
which (pre and postmultiplying by
) simplifies to

(7.46) 
This result shows that at the minimum, and can be
diagonalised simultaneously, and will therefore commute. The result of
the variation of the densitykernel is to make the densitymatrix commute
with the Hamiltonian in the representation of the current support functions.
Transforming to the diagonal frame by making a
unitary transformation (the eigenvalues of being and
those of being ) we obtain the following
relationship:

(7.47) 
For the derivative with respect to the support functions we have

(7.48) 
which can again be transformed first into an orthonormal representation
defined by the Löwdin transformation:

(7.49) 
to obtain
Assuming that we have performed the minimisation with respect to the
densitykernel for the current support functions, transforming to the
representation which simultaneously
diagonalises the Hamiltonian and densitymatrix, by the unitary
transformation
,
yields
which is a KohnShamlike equation, but where the energy eigenvalue
does not explicitly appear since no orthonormalisation
constraint is explicitly applied. Using equation 7.47, however,
yields

(7.52) 
and so, at least for , we see that the support function
variations are equivalent to making the related wavefunctions obey the
KohnSham equations.
The factor of will slow this convergence for unoccupied bands, since
for the gradient is small. In the next section (7.5) we therefore turn our attention to a potential method for eliminating this
problem.
Next: 7.5 Occupation number preconditioning
Up: 7. Computational implementation
Previous: 7.3 Penalty functional and
Contents
Peter Haynes