4.3 Density-matrix DFT

We consider a system with a set of orthonormalised orbitals $\{ \vert \psi_i \rangle \}$ and occupation numbers $\{f_i\}$. The single-particle density-operator ${\hat \rho}$ is defined by

$${\hat \rho} = \sum_i f_i \vert \psi_i \rangle \langle \psi_i \vert$$ (4.21)

and the density-matrix in the coordinate representation is

$$\rho({\bf r},{\bf r'}) = \langle {\bf r} \vert {\hat \rho} \... ...angle = \sum_i f_i ~ \psi_i({\bf r}) \psi_i^{\ast}({\bf r'}) .$$ (4.22)

The diagonal elements of the density-matrix are thus related to the electronic density by

$$n({\bf r}) = 2 \rho({\bf r},{\bf r})$$ (4.23)

and the generalised non-interacting kinetic energy is

$$T_{\mathrm s}^{\mathrm J}[n] = 2 \int {\mathrm d}{\bf r'} \l... ..._{\bf r}^2 \rho({\bf r},{\bf r'}) \right]_{{\bf r}={\bf r'}} .$$ (4.24)

This expression can be written as a trace of the density-matrix and the matrix elements of the kinetic energy operator ${\hat T} = -\textstyle{1 \over 2} \nabla^2$ i.e. $T_{\mathrm s}^{\mathrm J}[n] = 2 {\rm Tr} (\rho T)$. Similarly, for the energy of interaction of the electrons with the external (pseudo-) potential

$$E_{\mathrm{ps}} = 2 \int {\mathrm d}{\bf r}~{\mathrm d}{\bf ... ...r}) \rho({\bf r},{\bf r'}) = 2 {\rm Tr} (\rho V_{\mathrm{ps}})$$ (4.25)

where $V_{\mathrm{ps}}({\bf r'},{\bf r}) = \langle {\bf r'} \vert {\hat V}_{\mathrm{ps}} \vert {\bf r} \rangle$. 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.