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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)

in which the summation convention is assumed and the matrix is the inverse of the overlap matrix defined by
 (4.70)

We have assumed from now on that we are only calculating wave-functions at the -point and so can assume that everything is real. By construction, the dual states obey
 (4.71)

and the completeness relation is expressed as
 (4.72)

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

 (4.73)

and note that the density-kernel because of the non-orthogonality.

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

 (4.74)

such that
 (4.75)

At the ground-state, these orthonormal orbitals will be a unitary transformation of the Kohn-Sham orbitals , so that the density-kernel defined as the density-matrix in the representation of the orthonormalised orbitals ,
 (4.76)

can be diagonalised by a unitary transformation as described in section 4.5.1 i.e. (no summation convention). The following relationship also holds;
 (4.77)

and the relationship between the non-orthogonal orbitals and the Kohn-Sham orbitals is of the general form
 (4.78)

where the matrix and obeys
 (4.79) (4.80)

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)

hence the superscript notation.

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)

in contrast to the definition of . In this case, to obtain the eigenvalues of the Hamiltonian it is necessary to diagonalise the matrix
 (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)

For the orthogonal functions , the metric tensor is the identity and so there is no distinction between covariant and contravariant quantities.

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.

Next: 5. Localised basis-set Up: 4. Density-Matrix Formulation Previous: 4.5 Requirements for linear-scaling   Contents
Peter Haynes