- 4.4.1 Trace
- 4.4.2 Idempotency
- 4.4.3 Penalty functional
- 4.4.4 Purifying transformation
- 4.4.5 Idempotency-preserving variations

4.4 Constraints on the density-matrix

(4.26) |

4.4.2 Idempotency

The self-consistent ground-state density-matrix must display the property of
idempotency i.e. . Unless the eigenvalues of the density-matrix
(occupation numbers) remain in the interval the density-matrix
will follow unphysical ``run-away'' solutions. Unfortunately it is not
possible to work directly with the eigenvalues of the density-matrix^{4.1} to
constrain them to lie in this interval and together with the non-linearity
of the idempotency condition, this constraint turns out to be the major
problem to be tackled. We briefly outline three ways in which this constraint
can be dealt with. The first two of these are related and all three are
described in [134] in the context of Hartree-Fock calculations.

If we are considering orbitals in our scheme, then the density-matrix in
the representation of those orbitals, or of a linear combination of those
orbitals, is an Hermitian matrix of rank
. The factorisation property of idempotent
density-matrices is that an idempotent matrix may always be written

(4.27) |

in which denotes the identity matrix.

Any Hermitian matrix can be diagonalised by some unitary matrix such
that the diagonal matrix is

(4.29) |

(4.30) |

We also note that expressing the density-matrix in this way guarantees that
it is positive semi-definite

(4.31) |

(4.32) |

4.4.3 Penalty functional

Consider a matrix which is not idempotent i.e.
.
To make it so, we need to reduce the matrix to zero, which
can be achieved by minimising the (positive semi-definite) scalar quantity
, whose minimum value is zero, with respect to the
individual elements. Since

(4.33) |

The limit is a strictly idempotent matrix close to in the sense that the separation

(4.35) |

Kohn [135] has suggested the use of the square-root of this
function as a penalty functional for the density-matrix:

equals the ground-state grand potential (i.e. ) for some . In particular,

(4.37) |

4.4.4 Purifying transformation

We consider the result of one steepest descent step i.e. one iteration of
equation 4.34 which allows us to write the density-matrix
in terms of an auxiliary matrix as

(4.39) |

In the common diagonal representation of and this relationship
can be expressed in terms of the individual eigenvalues
and
:

(4.40) |

4.4.5 Idempotency-preserving variations

Finally we consider the most general change which an idempotent
matrix of rank can suffer, while maintaining that idempotency. Using
the factorisation property we write
and consider changes
in i.e.
, where, without loss of generality,

(4.41) |

where

(4.43) | |||

(4.44) |

To define a new matrix it is sufficient to define a new -dimensional
subspace. Since any vector can be decomposed according to equation
4.42, including the columns of , any new vector (of
arbitrary length) can be formed by adding a vector lying
completely outside .
arbitrary linearly-independent vectors of this kind are given by the
columns of

(4.46) |

The metric associated with the vectors of is the matrix
and so a convenient orthonormalisation is

(4.47) |

(4.48) | |||

(4.49) | |||

(4.50) | |||

(4.51) |

Thus

When represents a small change, then a convergent expansion for the inverse matrix in equation 4.52 can be used

(4.53) |

(4.54) |

By taking the expansion to first order only, we make the change
linear in (which has certain advantages e.g. in implementing conjugate gradients) and see that this
does indeed maintain idempotency to first order:

(4.55) | |||

(4.56) | |||

(4.57) |

which vanishes to first order in as required. Thus if we have the ground-state density-matrix and consider making variations consistent with idempotency to first order as described here, then the energy must increase, and again some stability against the run-away solutions is obtained.