We have shown how it is possible to reformulate density-functional theory in terms of the single-particle density-matrix, and the constraints which must be obeyed by ground-state density-matrices. However, in the coordinate representation, we note that the density-matrix is a function of two position variables and thus contains an amount of information which scales as the square of the system-size (as of course it must since it contains all of the information in the Kohn-Sham orbitals, which are functions of one position but with the number of occupied orbitals also scaling linearly with system-size). To obtain a linear-scaling method, it is necessary to impose some further restrictions on the density-matrix.

4.5.1 Separability

In practice we do not wish to deal with a function of six variables (i.e. two
three-dimensional positions).
From the factorisation property of idempotent density-matrices, or the
definition of the ground-state density-matrix in terms of the Kohn-Sham
orbitals;

(4.59) |

Although it is not necessary for the auxiliary orbitals to be orthonormal, in the
case when they are, we can consider this general form as simply a
unitary transformation of the Kohn-Sham expression (4.58):

(4.60) |

(4.61) |

When the auxiliary orbitals are not orthonormal, then they can be viewed as a more general linear combination of the Kohn-Sham orbitals (involving both a unitary and Löwdin transformation) which is described in section 4.6. Whichever case applies, there is no loss of generality here as all idempotent matrices can always be expressed in this way, and these are the density-matrices of interest to us.

Kohn [136] has proved that in one-dimensional systems with a gap, a set of exponentially decaying Wannier functions can be found in the tight-binding limit, and that this localisation is related to the square-root of the gap. His method is not easily generalised to higher numbers of dimensions, and so until recently the exact nature of the Wannier functions in general three-dimensional systems was unknown, although it was anticipated that they would decay exponentially [137,138,139]. More recent numerical and analytical studies of the localisation of the density-matrix showed the decay to be exponential and again related to the square-root of the gap [140,141], thus supporting the general validity of Kohn's result. Very recently, however, Ismail-Beigi and Arias [142] have argued that in the weak-binding limit the exponential decay varies linearly with the gap. What is now certain is that the Wannier functions and density-matrix decay exponentially in systems with a gap, and that this decay is more rapid in systems with larger gaps.

Wannier functions
are simply a unitary transformation of Bloch wave-functions with respect to
the complementary variables of Bloch wave-vector and lattice vector.
Let
be
the normalised Bloch wave-function for the th band with wave-vector
. Then the corresponding Wannier function for that band
is defined by [143]

(4.62) |

(4.63) |

The properties of Wannier functions are that they are localised in different
cells (labelled by lattice vector ) and
are orthonormal:

(4.64) |

The single-particle density-matrix in the case of full -point sampling
is given by

(4.65) |

(4.66) |

(4.67) |

We can exploit this long-range behaviour to obtain a linear-scaling method:
we introduce a spatial cut-off
and require that the
density-matrix be strictly zero when the separation of its arguments exceeds
this cut-off;

(4.68) |