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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!

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    $\begingroup$ 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.). $\endgroup$ – orthocresol Feb 12 '19 at 11:59
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In essence, one picks a computational level, chooses a system to run the calculation on (typically a molecule) and lets a computer and an appropriate program do the heavy numeric and algebraic lifting. No really, that's it.

In more detail: One would want to solve the multi-electron, multi-nucleus Schrödinger equation (that's already an approximation, because we ignore relativistic effects here). That's really difficult. So we make more approximations, such as fixed nuclei, a product ansatz for the wave-function (made up of one-electron wave-functions) and often a finite-size basis set. A basis set are functions that can be linearly combined into molecular orbitals; a large one usually gives better results, but make the computation longer. This choice is part of the computational level. Other, more mathematical and technical choices make up the other part (and this is a science in itself).

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.

At times, the problems can be worked out using pen and paper, making use of high symmetry and tiny basis sets. This leads to easily interpreted, but not very accurate wave-functions.

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