Approximation of electron density of a small molecule Starting point for Quantum Theory of Atoms in Molecules (AIM) is the electron density $\rho(r)$. It could be determined experimentally (X-ray diffractions). I want to know how can we estimate them ab initio. In AIM
it is often emphasized that the focus is on density rather than a wave function for the later is just a mathematical function while the former is experimentally observable. Are there ways to estimate electron density without first calculating wave function (which will obviously lead to density function).

My interest is to use AIM in quantitative structure-activity relationship. I will typically deal with molecules of around 50 atoms. Usually I will already have the graph (connectivity) of the molecule (eg .SDF file, smile notation etc), how can we incorporate this to speed up electron density calculation?

It would be helpful if brief comment on the computational cost is also included in the answer.
You are looking for density functional theory. The theory is somewhat involved and you may be interested in A Chemist's Guide to Density Functional Theory or similar text book. A full description of the variety of ways to compute the (ground state) electron density is beyond the scope of an answer and better suited to a textbook.

The general gist is to optimize the energy with respect to the electron density numerically.