Interpreting Electron Density Definition

[Jan Wo]

Hi
I'd like to ask you about interpretation of electron density definition. According to the "Handbook of Molecular Physics and Quantum Chemistry" it goes like:

$\rho (\textbf{r})=N \sum_{\sigma_1}^{\sigma_N} \idotsint_{\mathbb{R^3}} |\psi_V (\textbf{r}_1, \sigma_1;\dots ;\textbf{r}_N,\sigma_N)|^2 d^3\textbf{r}_2 \dots d^3\textbf{r}_N$

I know that my question may be a little imprecise so please - be patient. I'd like to know how to iterpret this definition. Why in the integral is only $d^3\textbf{r}_2 \dots d^3\textbf{r}_N$ but no $d^3\textbf{r}_1d^3\textbf{r}_2 \dots d^3\textbf{r}_N$. How to explain this equation using physical viewpoint and how to explain it methematically ?

Feel free to add any helpful information to understand this definition.
Thanks in advance.

 

[DuckAmuck]

Is your $$\rho(r)$$ supposed to be $$\rho(r_1)$$?

You have to leave at least one position variable un-integrated if you want the result to be a function of position. I think that's why.

 

[Jan Wo]

Ok, thank you for answer.
I think this $\rho(r)$ should be $\rho(r_1)$
Can you tell me how it works mathematically?
I need to have a function of position, I know that I can't to integrate every position (then it would be only a number). But how exactly it looks. Let's consider simple example of function $f(x,y)$ what would happen if I integrate it like this: $\int f(x,y)dy=?$. The result for this example would be $?=g(x)\times <number>$?
Is it true in general case?
Perhaps it is obvious but I really like to simplyfy everything and I'd like to be sure.