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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
![\begin{displaymath}
{\hat \rho} = \sum_i f_i \vert \psi_i \rangle \langle \psi_i \vert
\end{displaymath}](img424.gif) |
(4.21) |
and the density-matrix in the coordinate representation is
![\begin{displaymath}
\rho({\bf r},{\bf r'}) = \langle {\bf r} \vert {\hat \rho} \...
...angle =
\sum_i f_i ~ \psi_i({\bf r}) \psi_i^{\ast}({\bf r'}) .
\end{displaymath}](img425.gif) |
(4.22) |
The diagonal elements of the density-matrix are thus related to the electronic
density by
![\begin{displaymath}
n({\bf r}) = 2 \rho({\bf r},{\bf r})
\end{displaymath}](img426.gif) |
(4.23) |
and the generalised non-interacting kinetic energy is
![\begin{displaymath}
T_{\mathrm s}^{\mathrm J}[n] = 2 \int {\mathrm d}{\bf r'} \l...
..._{\bf r}^2 \rho({\bf r},{\bf r'})
\right]_{{\bf r}={\bf r'}} .
\end{displaymath}](img427.gif) |
(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
![\begin{displaymath}
E_{\mathrm{ps}} = 2 \int {\mathrm d}{\bf r}~{\mathrm d}{\bf ...
...r}) \rho({\bf r},{\bf r'}) = 2 {\rm Tr} (\rho
V_{\mathrm{ps}})
\end{displaymath}](img430.gif) |
(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.
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Up: 4. Density-Matrix Formulation
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Peter Haynes