Definition of electron density in DFT

Question

I read that the electron density used for density functional theory in a system of $N$ electrons with wavefunction $\psi$ is defined as

$$\rho(r)=N\int d^3r_2\dots d^3r_N \psi^*(r,r_2,\dots r_N)\psi(r,r_2,\dots r_N) $$

this definition seems weirdly asymmetrical to me. Why is the first coordinate special? How can I interpret this definition physically?

Answer 1

In a one particle system, the probability of finding the electron between $\mathbf{r}$ and $\mathbf{r}+d\mathbf{r}$ is $|\Psi(\mathbf{r})|^2d\mathbf{r}$, with the probability density $\rho(\mathbf{r})=|\Psi(\mathbf{r})|^2$.

In a many-electron system, the probability of finding electron 1 in the volume element $d\mathbf{r}_1$ at $\mathbf{r}_1$, electron 2 in the volume element $d\mathbf{r}_2$ at $\mathbf{r}_2$, and so on, is $|\Psi(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N)|^2d\mathbf{r}_1 d\mathbf{r}_2\dots d\mathbf{r}_N$. Therefore, the probability of finding electron 1 in the volume element $d\mathbf{r}_1$ at $\mathbf{r}_1$ and the other electrons elsewhere is $$P(1)=\left[\int|\Psi(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N)|^2 d\mathbf{r}_2\dots d\mathbf{r}_N\right]d\mathbf{r}_1=\rho(\mathbf{r}_1)d\mathbf{r}_1$$ This probability is exactly the same for all electrons, because they are indistinguishable, so we don't care about the label. Thus, the probability density of finding an electron in the volume element $d\mathbf{r}$ at $\mathbf{r}$ and the other $N-1$ electrons elsewhere is $N$ times the probability to find one of the electrons at that region. That is, $$\rho(\mathbf{r})=N\int|\Psi(\mathbf{r},\mathbf{r}_2,\dots,\mathbf{r}_N)|^2 d\mathbf{r}_2\dots d\mathbf{r}_N$$

Answer 2

The total density is the sum of each particle's density $\rho(r) = \sum_i^N \rho_i(r)$ with $$ \rho_i (r) = \int d^3r_1...d^3r_{i-1}d^3r_{i+1}...d^3r_n |\psi(r_1, ... r_{i-1}, r, r_{i+1}, ..., r_n)|^2. $$ In DFT, the wave function is a Slater determinant and is thus antisymmetrized, reflecting the indiscernibility of fermions as $\rho_i(r) = \rho_1(r)$. We find $$ \rho(r) = N\rho_1(r) $$ which is another form for your equation of $\rho(r)$.