Total energy optimisation

The total energy is a functional of the charge density
. From equation (10) we see that
the charge density is expanded in fine grid delta functions where
are the expansion coefficients. Therefore the energy will
have a variational dependence on these coefficients
provided they form an -representable charge density.
Consequently, the energy should also have a variational dependence
on the density kernel
and the
NGWF expansion coefficients
since the
are constructed from them

(18) |

should remain constant. The second is that the ground state density matrix should be idempotent, or in other words the eigenfunctions of the Kohn-Sham Hamiltonian have to be orthonormal

We choose to carry out the total energy minimisation in two nested loops, in a fashion similar to the ensemble DFT method of Marzari

with

where the minimisation with respect to the density kernel in equation (22) ensures that of equation (21) is a function of the NGWF coefficients only. In practice in equation (22) we do not just minimise the energy with respect to but we also impose the electron number and idempotency constraints (19) and (20). There are a variety of efficient methods for achieving this available in the literature, derived from the need to perform linear-scaling calculations with a localised basis [9,10,33,34,35]. Any of these methods would ensure that the density kernel in (22) adapts to the current NGWFs so that it minimises the energy within the imposed constraints.

In the present work we have used the variant of the Li, Nunes and Vanderbilt (LNV)[9] method that was developed by Millam and Scuseria[36] in calculations with Gaussian basis sets. We emphasise again, though, that any of the other available methods could have been used as well. For simplicity of presentation, our analysis from now on will assume that the energy of equation (22) is minimised without any constraints. In order to take into account the constraints, the formulae we derive will have to be modified according to the density kernel minimisation method one chooses to use. This is a straightforward but tedious exercise [37].

The minimisation of equation (22) can be performed
iteratively with the conjugate gradients method [38].
As in the simpler steepest descents method, the essential
ingredient is the gradient. It is easy to show [39] that
this quantity is equal to twice the matrix elements of the
Kohn-Sham Hamiltonian

The minimisation stage of equation (21)
is also performed iteratively with the conjugate gradients method.
In this case, one can show by using the properties
of the delta function basis set that the gradient is:

When the minimisation with respect to the density kernel of equation (22) is carried out under the electron number and idempotency constaints, equation (24) contains extra terms as a result of the constraints imposed in (22). These terms ensure that the electron number and idempotency constraints are automatically obeyed in (21) and as a result, the optimisation with respect to the support functions can be carried out in an unconstrained fashion.