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We consider a system with a set of orthonormalised orbitals
and occupation numbers . The single-particle density-operator is defined by
|
(4.21) |
and the density-matrix in the coordinate representation is
|
(4.22) |
The diagonal elements of the density-matrix are thus related to the electronic
density by
|
(4.23) |
and the generalised non-interacting kinetic energy is
|
(4.24) |
This expression can be written as a trace of the density-matrix and the
matrix elements of the kinetic energy operator
i.e.
. Similarly, for the
energy of interaction of the electrons with the external (pseudo-) potential
|
(4.25) |
where
. The definitions of the Hartree
and exchange-correlation energies in terms of the electronic density
(now defined in terms of the density-matrix by equation 4.23)
remain unchanged. Thus we can express the total energy of both
interacting and non-interacting systems in terms of the density-matrix.
By minimising the energy with respect to the density-matrix (subject to
appropriate constraints to be discussed) we can thus find the ground-state
properties of the system.
Next: 4.4 Constraints on the
Up: 4. Density-Matrix Formulation
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Peter Haynes