8.3 Density-matrix initialisation

Finally we discuss the subject of constructing an initial density-matrix
for our calculations, which is related to the other work in this chapter.
Although any linear-scaling method will obviously be more efficient than a
traditional method for a sufficiently large system, the
* cross-over* (i.e. the system-size at which the linear-scaling
method beats
the traditional method) may be very large. If it is larger than the largest
system which can currently be simulated by traditional methods, then there
is obviously little practical use for such a method. The methods
described in this dissertation are not that inefficient, but neither is the
cross-over sufficiently small that some advantage cannot be obtained by
using some physical insight to assist the calculation e.g. by imposing
appropriate symmetries (although we must take care not to impose
symmetries which subsequently prevent the method from reaching the
ground-state).

For example, consider a vacancy in an otherwise perfect crystal. Running a traditional calculation on the bulk crystal would allow us, using the methods in section 8.2, to obtain the density-matrix elements for the bulk crystal which could be used to initialise the density-matrix for a simulation of the large system with the vacancy. This would avoid wasting computational effort converging the density-matrix from a random position when the general form can be guessed from the bulk case.

A second example is that of a molecule interacting with a solid surface. In this case, the density-matrix for molecule and surface could be converged separately to obtain an estimate for the complete system. Obviously this method will be more successful the more weakly bound the molecule is to the surface.

The transferability of localised orbitals between closely-related systems has been studied in hydrocarbon chains [169] and methods have also been developed to calculate generalised Wannier functions [170,171] which may also be transferred between systems [172].

One important consideration is that of local charge neutrality, often used as an approximation to self-consistency in non-self-consistent tight-binding calculations [173]. Because the density-matrix is truncated in real-space, long-wavelength fluctuations in the electronic density are suppressed. This has the advantage that it prevents ``charge-sloshing'' instabilities, so problematic in early traditional plane-wave methods on large systems. However, if the system is not initially charge neutral locally, it takes a very large number of iterations to transfer charge across the system, and this results in very poor convergence. We thus need to at least consider initialising our density-matrix to correspond to isolated atoms brought together, which are then allowed to interact and form bonds etc. This is simply achieved using the projection method described here and leads to a great improvement in performance.