4.6 Non-orthogonal orbitals

We conclude this chapter with a discussion about the representation of the
density-matrix using non-orthogonal orbitals. We consider a set of
non-orthogonal functions
which we denote
, and introduce their dual functions
defined by

(4.69) |

(4.70) |

(4.71) |

In general we will represent the density-matrix in the separable form

We can construct an orthonormalised set of orbitals
defined as linear combinations of the
by the
Löwdin transformation:

(4.74) |

(4.75) |

(4.76) |

(4.77) |

where the matrix and obeys

Using the completeness relation (4.72) we can now express the
matrix in terms of other quantities:

(4.81) |

so that

(4.82) | |||

(4.83) |

In fact, the density-kernel contains the matrix elements of the density-operator in the representation of the dual vectors of the non-orthogonal functions:

(4.84) |

If we wish to obtain the occupation numbers, we must diagonalise the matrix
which is given by

(4.85) |

Thus the eigenvalues of are the occupation numbers.

At the ground-state, the density-operator and Hamiltonian commute, and thus
both the Hamiltonian and the density-matrix can be diagonalised simultaneously.
The Hamiltonian is usually represented by its matrix elements in the
representation of the non-orthogonal orbitals. Thus

(4.86) |

(4.87) |

i.e. the eigenvalues of are those of the Kohn-Sham Hamiltonian.

The advantage of representing the density-operator and Hamiltonian in
different ways is that quantities such as the electron number and
non-interacting energy can be expressed easily:

(4.88) | |||

(4.89) |

since the factors of and cancel.

In the language of tensor analysis, the functions
are covariant vectors, and their duals the associated contravariant quantities. The overlap matrix
plays the rôle of the metric tensor to convert between covariant and contravariant quantities. This is seen by verifying the relationship

(4.90) |

In a linear-scaling scheme we will not be able to access the eigenvalues directly, since although and are sparse, and need not be, and in any case, the effort to diagonalise even a sparse matrix is . However it is important to understand the different origins and rôles of these matrices in order to analyse the equations which result when we attempt to minimise the total energy to find the ground-state.