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Subsections
6.1 Kohn's method
As mentioned in section 4.4.3, Kohn [135] has
suggested the use of a penalty functional to impose the idempotency
condition, and has proved a variational principle based upon it.
We consider trial densitymatrices
expressed in diagonal form with real orthonormal extended orbitals
and occupation numbers
:

(6.1) 
The functional
is then formed:

(6.2) 
in which
and where is the chemical potential and is a positive real
parameter.
Kohn proves the following variational principle: that for some
, the minimum value of
is obtained for the idempotent groundstate
densitymatrix and that the minimum value is the groundstate grand potential i.e.

(6.6) 
in which the
are the exact eigenvalues of the
selfconsistent Hamiltonian, generated by the groundstate densitymatrix
.
The critical value of , denoted
, is given by

(6.7) 
in which
is the conditional minimum defined by

(6.8) 
i.e. the minimum grand potential for all trial densitymatrices which give
a penalty functional value of . Clearly

(6.9) 
although this is only a lower bound on
.
Figure 6.1:
Behaviour of Kohn's penalty functional
when a single occupation number is varied and all others
are zero or unity.

Kohn's variational principle is based on the noninteracting energy
. We now present a simple modification of this functional
based upon
selfconsistent variation of the interacting energy. Consider the
functional

(6.10) 
in which is the interacting energy, and is
a positive semidefinite trial densitymatrix. A given set of occupation
numbers
fixes the value of the penalty functional
and variation
of
with respect to the orbitals
at fixed occupation numbers and subject to the
orthonormality constraint yields KohnShamlike equations. Selfconsistent
variation of the occupation numbers (i.e. allowing the
orbitals to relax, as in section 4.2) yields

(6.11) 
In the case of idempotent densitymatrices,
for which
, we obtain the special cases

(6.12) 
For this functional the critical value of , again denoted
, is
given by

(6.13) 
where the maximum is strictly over those densitymatrices searched during the
minimisation.
For
the total functional
takes its minimum value when
for
and
respectively. In particular, for the
groundstate densitymatrix , the functional is strictly increasing
with respect to all variations in occupation numbers. The discontinuity in
the occupation number derivative of the penalty functional at idempotency is
required because of the nonvariational behaviour of the total energy
with respect to these variations (section 4.2).
The behaviour of the penalty functional for unconstrained occupation
number variation is plotted in figure 6.1, and in figure
6.2 the total functional is sketched schematically for
several representative values of the parameter . This demonstrates
how the minimising densitymatrix is idempotent only for
.
Figure 6.2:
Schematic illustration of Kohn's variational principle: behaviour of the total energy (black) and total functional (red) for representative values of .

6.1.2 Implementation problems
The conjugate gradients algorithm for minimising functions is described
in appendix B. Throughout the lengthy derivation it is clear that
the useful results obtained and the remarkably simple final result are due
to the special properties of quadratic functions.
Any function may be expanded in the form of a Taylor
series about an analytic point, and around a minimum where the first order term
from the gradient vanishes, a quadratic function is generally a good
approximation. However, we note that the Kohn penalty functional has a branch
point from the squareroot function exactly at the groundstate
minimum which we seek, and so the function cannot be Taylorexpanded there.
Local information from the gradient cannot be used to infer the global
shape of the function. This is illustrated in figure 6.3
for the case of a parabolic interpolation to find a line minimum based
upon the gradient and a trial step, but the problem is even worse in the
multidimensional space since the ``conjugate'' directions constructed from
the gradients will not point in the direction of the groundstate minimum.
Figure 6.3:
Failure of quadratic interpolation for Kohn's penalty functional.

This problem is reflected in the very poor convergence when an attempt is
made to minimise the functional using conjugate gradients: the steepest
descents method actually performs better because it does not assume global quadratic
behaviour. Also, the penalty functional does not vanish at the minimum
sufficiently quickly
as the parameter is increased. However, the rootmeansquare error in
the occupation numbers
, given by

(6.14) 
in fact decays more rapidly, so that the total energy calculated at the
minimum is
quite accurate, although it is neither variational nor an upper bound. Also, although the
total functional
decreases monotonically,
the total energy does not. Thus no advantage is gained by
using the variational property, since it can only be applied to the
total energy when
.
The variational property of the total functional is that it is minimal at the
groundstate, but this minimum is defined in terms of the functional taking
its minimum value there, not in terms of a vanishing gradient
(the gradient being undefined at the groundstate).
Because of the nonvariational behaviour of the total energy with respect to
the occupation numbers at the groundstate, it is impossible to construct
a penalty functional which has a continuous first derivative at the
groundstate and also results in a variational principle for the total
energy.
Figure 6.4:
Convergence properties of Kohn's penalty functional: behaviour of penalty functional and occupation numbers with .

In figure 6.4 we present the results of tests on an 8atom silicon cell
to demonstrate the behaviour of the functional. As the penalty functional
parameter is increased, both the contribution of the penalty
functional to the total functional
, and the root mean
square error in the occupation numbers
decrease, but
not rapidly enough with since the number of iterations required to
reach convergence increases with making the calculations too
expensive for practical applications. For example, the number of iterations
required to converge the
total functional to eV per atom increases by a factor of more than ten
when is increased from 100 eV to 1000 eV. Even with the smaller
value for , the rate of convergence is much slower than traditional
methods, and this is due to the incompatibility of the functional with the
conjugate gradients scheme.
Next: 6.2 Corrected penalty functional
Up: 6. Penalty Functionals
Previous: 6. Penalty Functionals
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Peter Haynes