Next: 7.5 Occupation number preconditioning Up: 7. Computational implementation Previous: 7.3 Penalty functional and   Contents

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 post-multiplying 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 density-kernel is to make the density-matrix 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
 (7.50)

Assuming that we have performed the minimisation with respect to the density-kernel for the current support functions, transforming to the representation which simultaneously diagonalises the Hamiltonian and density-matrix, by the unitary transformation , yields
 (7.51)

which is a Kohn-Sham-like 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 wave-functions obey the Kohn-Sham 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