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) |

(7.54) |

(7.55) | |||

(7.56) |

where we have adopted the Fletcher-Reeves method (B.22) for calculating . The line minimum is given by

(7.57) |

(7.58) |

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) |

(7.60) |

(7.61) |

(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) |

We now transform the preconditioned gradient back to the support function
representation using

(7.64) |

(7.65) |

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

(7.67) |

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.