Quantitative MO theory- how is the overlap integral and the weighting coefficients of the molecular orbital actually evaluated?

Title. In my inorganic chemistry course, we learn about SALCs and qualitative MO treatment, with only a fleeting reference to the S integral and actual molecular wave functions (only hydrogen, $$\ce{H2}$$, was discussed).

I'm curious about how the S integral is evaluated for MOs in slightly more complex molecules i.e. $$\ce{BF3}$$, $$\ce{CO}$$, $$\ce{H2O}$$, and how the weighting coefficients are found, as they aren't discussed at all at my level.

Can someone explain this? Thanks!

• In essence, you're asking how we find the MOs of a chemical species, I suppose. That's quite a broad topic; there are several ways of doing it, each with different levels of rigour, you may want to check out e.g. Chpt 15 of Levine's Quantum Chemistry (7th ed.). – orthocresol Feb 12 '19 at 11:59

For a molecule like $$\ce{C2H6}$$, the computational time can vary between several days to weeks and less than a second, depending on sophistication of the computational level and its accuracy. In the end, we obtain an approximate wave-function (containing molecular orbitals), from which approximate properties such as energy or NMR shifts can be obtained. The computer is useful in this by calculating integrals (partially analytic, mostly numeric) and keeping track of many very large matrices.