In Density Functional Theory (DFT) we derive the Grand Potential as a functional of a so-called one particle density (OPD). I have trouble imagining what exactly that is. Could someone help me with that ?

I know the mathematical background of the performed calculus of variations but i can't figure out what this OPD represents.


The one-particle density can be viewed as the localization probability of a particle in the system, with integration over all the state vectors except that of the single particle of interest.

For example, suppose you are interested in the positions $\mathbf{x}_i$ of $N$ electrons in a many-electron system in which the $i$-th electron is in spin state $\sigma_i$. The OPD for a single electron is then given as $$ n(\mathbf{x_1})=N\int \mathrm{d}\mathbf{x}_2\mathrm{d}\mathbf{x}_3...\mathrm{d}\mathbf{x}_N\sum_{\mathbf{\sigma}} |\Psi(\mathbf{x_1}\sigma_1,\mathbf{x_2}\sigma_2,...,\mathbf{x}_N\sigma_N)|^2. $$ Similarly, you can define the two-particle density $n(\mathbf{x}_1,\mathbf{x}_2)$ with a similar integral over the vector positions $\mathbf{x}_3...\mathbf{x}_N$.

The density matrix for your full system is given by these one-particle terms on the diagonal (appropriately normalised such that the trace is equal to unity), and the pairwise exchange terms (the Pauli correlation) as the off-diagonal elements [1].

[1] http://th.physik.uni-frankfurt.de/~engel/2pd_helium.html


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