Next: 7.6 Tensor properties of
Up: 7. Computational implementation
Previous: 7.4 Physical interpretation
  Contents
7.5 Occupation number preconditioning
The eigenvalues of the Hessian at a stationary point determine the nature
and shape of that stationary point. Thus the shape of the ground-state
minimum of an energy functional is determined by the eigenvalues of that
functional. For the Kohn-Sham scheme these eigenvalues are the
and the narrower the eigenvalue spectrum, the more
``spherical'' the minimum, and the easier the functional is to minimise.
From equation 4.9 we note that when partial occupation
numbers are
introduced, the relevant eigenvalue spectrum becomes
.
When conduction bands are included in a calculation, their occupation numbers
will be vanishingly small near the ground-state minimum, which will therefore be
very aspherical, and convergence of these bands will become very slow.
This problem has been addressed in the study of metallic systems
[164,165] by the method of preconditioning which
changes the metric of the parameter space to compress the eigenvalue
spectrum and make the minimum more spherical.
With reference to the results in appendix B,
we introduce a metric, represented by the matrix ,
such that a new set of
variables (denoted by a tilde) is introduced:
|
(7.53) |
and so that the new gradients are related to the old gradients by
|
(7.54) |
In the new metric, the conjugate directions are defined (see appendix B) by
where we have adopted the Fletcher-Reeves method (B.22) for
calculating .
The line minimum is given by
|
(7.57) |
which can be rewritten in terms of the original variables as
This identifies
as the set of preconditioned
conjugate gradients for the original variables in the original space.
These directions
are thus given by
|
(7.59) |
so that the gradients to be used for the preconditioned search are
|
(7.60) |
with mixing factor
|
(7.61) |
It is observed that defining
in terms of the
preconditioned gradients alone does not interfere with the minimisation
procedure, so that in practice
|
(7.62) |
In order to apply this scheme here, we choose to make the metric
diagonal
in the representation of the Kohn-Sham orbitals. In the original variables
(the subscript labels a component of a vector)
the minimum can be expanded as
so that the scaled variables
defined by
(where
)
produce the desired compression since in terms of the new variables, the
minimum has the form
.
In the representation of the Kohn-Sham orbitals, the gradient of the functional
(7.52) becomes
|
(7.63) |
in which we see that the factor of in front of the
operator has been cancelled so that the effect of the gradient is now the
same on both occupied and unoccupied bands, and these bands should now
converge at the same rate.
We now transform the preconditioned gradient back to the support function
representation using
|
(7.64) |
where
from 4.78 and is a unitary
matrix.
Thus the preconditioned gradient we require is
|
(7.65) |
from the properties of the matrix (4.79, 4.80).
We see that the gradient has thus been pre-multiplied by the matrix
i.e. in the support function representation, the metric is
.
Although the overlap matrix is a sparse matrix for localised support
functions, its inverse is not sparse in general, so that this scheme is
not straightforward to implement.
For a sufficiently diagonally dominant overlap matrix, it is possible to
approximate the inverse in the following manner. We write
where contains only the diagonal elements of and contains
the off-diagonal elements. is thus trivial to diagonalise.
Writing
we have
and if is diagonally dominant, the elements of the matrix
are small so that we can approximate the inverse of the term in
brackets. If the elements of a matrix are small then
When the first few terms of equation 7.66 are applied to the
inverse overlap matrix we obtain
This expression could be used to obtain a good approximation to the inverse
overlap matrix, which may be sufficient for preconditioning, but there is
the danger that, particularly for large systems, the overlap matrix may
become singular and the performance of the algorithm would deteriorate.
In the following section, however, we show that the correct preconditioned
gradient does not involve the inverse of the overlap matrix.
Next: 7.6 Tensor properties of
Up: 7. Computational implementation
Previous: 7.4 Physical interpretation
  Contents
Peter Haynes