In a paper describing a Kohn-Sham Density Functional Theory implementation, the authors describe the use of the density matrix for e.g. the calculation of the electronic density and for efficiency improvements. With a basis expansion of the Kohn-Sham states $$ |\Psi_n^\mathrm{KS}\rangle = \sum_i c_{in}\,|\phi_i\rangle $$ the density operator and matrix are \begin{align*} \hat{\rho} &= \sum_n f_n |\Psi_n\rangle\langle\Psi_n| =\sum_{ij} \rho_{ij}\, |\phi_i\rangle\langle\phi_j| \\ \rho_{ij} &= \sum_n f_n\, c^*_{in}\,c_{jn} \end{align*} with the occupation numbers $f_n$ - the electron density is then obtained via $$ \langle\mathbf{r}|\hat{\rho}|\mathbf{r}\rangle = \rho = \sum_{ij} \phi_i\,\rho_{ij}\,\phi_j. $$ When comparing this with the standard definition of the density operator $$ \hat{\rho} = \sum_n w_n |\Psi_n\rangle\langle\Psi_n|, $$ the weights $w_n$ have a different meaning, being the probabilities to find a subsystem in the state $|\Psi_n\rangle$ and fulfilling $\sum_n w_n = 1$. In my understanding, this has nothing to do with the density operator in above context where $f_n$ denote whether a single-particle orbital is occupied or not and where $\sum_n w_n \geq 1$ is possible.

Now to my question: is the density matrix in the context of DFT as explained above just a tool to ease technical/computational operations or can still any meaning in the 'classical' quantum-mechanical view be given to it?


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