Principles of kinetic energy ill-conditioning

The phenomenon of kinetic energy or length-scale ill-conditioning is a familiar one within the plane wave approach to electronic structure calculations [1]. It is not, however, restricted to this approach and its effects are seen in many methods which use a large basis set [21,33,34].

The efficiency with which a function can be minimized using
iterative techniques such as steepest descents or conjugate gradients
is related to the *condition number*
, where
and
are the
extremal curvatures of the function about the
minimum [35].
Minimization is most efficient when the condition number is small
and the curvatures have a narrow range of values.
On the other hand, when the curvatures take a wide range of values,
the number of iterations required for convergence can become
unacceptably large and the minimization is said to be ill-conditioned.

The curvatures of the total energy functional are determined by the
eigenvalues of the Hamiltonian.
Hence, the condition number depends upon the ratio of
the largest and smallest eigenvalues in the basis representation that
is being used.
With a *large* systematic basis, these
eigenvalues span a broad range. As a result, the condition number is
large, rendering the problem ill-conditioned.
A significant source of this ill-conditioning is associated with the
contribution to the total energy due to the kinetic energy
, which is given by

In the plane wave approach, the effect of kinetic energy
ill-conditioning is reduced by multiplying the
steepest descents directions in reciprocal space by a diagonal
preconditioning matrix which behaves as the inverse of the kinetic
energy operator at high wave vectors and is a constant at low
wave vectors [1].
Such a preconditioner, as pointed out in
Ref. [21], is qualitatively equivalent to the
*exact* preconditioner for a model Hamiltonian
given by