Next: 4.4 Constraints on the
Up: 4. DensityMatrix Formulation
Previous: 4.2 Partial occupation of
Contents
We consider a system with a set of orthonormalised orbitals
and occupation numbers . The singleparticle densityoperator is defined by

(4.21) 
and the densitymatrix in the coordinate representation is

(4.22) 
The diagonal elements of the densitymatrix are thus related to the electronic
density by

(4.23) 
and the generalised noninteracting kinetic energy is

(4.24) 
This expression can be written as a trace of the densitymatrix 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 exchangecorrelation energies in terms of the electronic density
(now defined in terms of the densitymatrix by equation 4.23)
remain unchanged. Thus we can express the total energy of both
interacting and noninteracting systems in terms of the densitymatrix.
By minimising the energy with respect to the densitymatrix (subject to
appropriate constraints to be discussed) we can thus find the groundstate
properties of the system.
Next: 4.4 Constraints on the
Up: 4. DensityMatrix Formulation
Previous: 4.2 Partial occupation of
Contents
Peter Haynes