4.2 Partial occupation of the Kohn-Sham orbitals

In the Kohn-Sham scheme, the single-particle orbitals
were either empty or doubly occupied (two spin states). It will prove to be
useful if we now
generalise to include partial occupation [131] so that
each orbital contains electrons where
. The
electronic density is now defined as

(4.7) |

Janak's functional is defined as

(4.8) |

For a fixed set of occupation numbers, the Euler-Lagrange equations for the
variation of the functional with respect to the orbitals again yield
Schrödinger-like equations:

Multiplying by and integrating gives

We obtain the dependence of the energy functional on the occupation numbers
by varying one of the while allowing the orbitals to relax (i.e. solving equations 4.6 and 4.10 self-consistently).
We define the kinetic energy for orbital , , by

(4.13) |

(4.14) |

From the definition of (4.12) we obtain

(4.16) |

(4.17) |

(4.18) |

(4.19) |

Variation of the functional subject to the constraint of constant electron
number (i.e. unconstrained variation of
) gives

This generalised functional is not variational with respect to arbitrary variations in the occupation numbers [132]. Objections have been raised [133] to considering occupation numbers other than zero or one in zero-temperature density-functional theory, but the conclusion is still that at self-consistency, orbitals above the Fermi energy are unoccupied and orbitals below are fully occupied, and we recall that this state of affairs corresponds to an idempotent density-matrix.

If the occupation numbers are allowed to vary in the interval we see that the lowest value of the generalised functional is obtained for the correct choice of occupation numbers outlined above. However, if the occupation numbers are allowed to vary outside this interval, this result no longer holds since the energy can be lowered by over-filling ( ) orbitals below the Fermi level, or negatively filling ( ) orbitals above the Fermi level, while still keeping the sum of the occupation numbers correct. Constraining the occupation numbers to avoid these unphysical situations is discussed in section 4.4.