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(Assuming no relativistic effects or electron-electron interactions for this question)

First: Am I correct in thinking that Linear Combinations of Atomic Orbitals (LCAOs) are generally good approximations for molecular eigenfunctions because a sum of mostly-independent coulomb potentials is going to result in a sum of mostly-independent solutions to those coulomb potentials? And is there a better explanation for why exactly that's the case?

Second: How can we be certain that, given a limited LCAO basis set, there aren't other eigenfunctions which are similar or lower in energy than the MOs found as LCAOs which are going to affect our calculations?

  • Is there a proof that an infinite LCAO basis set will span all possible eigenfunctions, and does it come with some theorem that diminishing influence from the higher-order AOs will lead to diminishing effects on energy? I can visualize a sketch of the proof but I don't have the theory to make it rigorous.
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    $\begingroup$ (not an expert) They are a complete set so they can be used to expand any molecular orbital. Given the predominance of the Coulomb interaction you'd naively think this is a reasonable starting point. $\endgroup$ Commented Aug 24 at 18:09
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    $\begingroup$ Legendre polynomials are a complete orthogonal basis set. $\endgroup$ Commented Aug 24 at 18:32
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    $\begingroup$ To estimate how the energy gap between two orbitals affects the interaction, you may look at $\langle a | V | b \rangle = \frac{\langle a|[H,V ]| b \rangle }{E_b-E_a}$. $\endgroup$ Commented Aug 25 at 10:48

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First: Am I correct in thinking that Linear Combinations of Atomic Orbitals (LCAOs) are generally good approximations for molecular eigenfunctions because a sum of mostly-independent coulomb potentials is going to result in a sum of mostly-independent solutions to those coulomb potentials?

This is an argument that you might try to make. But it is hard to show this to be true by an honest and carefully thought out analysis.

For one thing, you are going to have a tough time dealing with electron-electron interactions and explaining why an "independent electron" approximation (or a combination of a few slater determinants) works so well.

The fact that it works well (sometimes) is really a miracle. In physical, terms the "miracle" has something to do with screening, but it is hard to describe briefly.

And is there a better explanation for why exactly that's the case?

Not really. At least not one that can be explained briefly here.

Second: How can we be certain that, given a limited LCAO basis set,

You can't be certain. Once you have truncated your basis set, you have no (a priori) reason to believe that the truncated basis can represent all the functions you might be interested in. However, you have no choice but to proceed, because there is not much else you can do. Will you miss out on some physics? Almost certainly yes.

there aren't other eigenfunctions which are similar or lower in energy than the MOs found as LCAOs which are going to affect our calculations?

There definitely could be other eigenfunctions which could affect our calculations.

Consider, for example, a solid block of aluminum. Solid state theorists might like to describe the high energy excitations as excitations of approximately atomic electrons "smeared" across the solid (e.g., to form bands a la Bloch waves). And this can work well to describe some of the physics. (E.g., optical excitations and optical properties.) But it is not going to show you how aluminum becomes a superconductor near zero temperature. So, yeah, you will miss out on certain low energy excitations. (And you will miss out on certain high energy excitations.) You will be able to describe certain "just right" Goldilocks excitations pretty well.

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  • $\begingroup$ So then, why are they so good at those goldilocks excitation? There has to be some reason. $\endgroup$ Commented Aug 24 at 20:52
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    $\begingroup$ There does not have to be a reason. (E.g., you could "reason" that $\frac{64}{16}=4$ because you can "cancel the sixes," but this would be absurd since it only works as a fluke.) But, anyways, in our case, presumably the reason is that the matrix elements of interest can be well-approximated by the linear combinations of molecular orbitals being used. $\endgroup$ Commented Aug 24 at 22:57
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    $\begingroup$ [mumbles something about the quasiparticle self-energy being small] $\endgroup$ Commented Aug 24 at 23:02
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I would say yes, “Am I correct in thinking that Linear Combinations of Atomic Orbitals (LCAOs) are generally good approximations for molecular eigenfunctions because a sum of mostly-independent coulomb potentials is going to result in a sum of mostly-independent solutions to those coulomb potentials?” And the reason this tends to work pretty well is exactly because of the last part. We purposely use the LCAO method in the tight binding regime, that is, when the coulombic potentials are in fact fairly independent, in the sense that each site’s potential is not much distorted from exact coulomb potentials. Since are not much distorted they must not be too close together and electrons can be considered to “be” in each potential well and their wave functions not overlapping much, i.e. tightly bound to sites. Versus in the other extreme of the nearly free electron model where the wave function is of significant magnitude everywhere and so you don’t think of electrons as strongly associated with individual potential wells, that is, they are not tightly bounds.

So since we use this approximation when we expect the electron wave functions at each site to be pretty close to the wave functions of isolation atoms, because the potential wells are similar to isolated atoms, we HOPE, that by using just a few atom orbitals we can get close to the small distortion in the wave functions caused by the small distortions in the potentials.

All of the above is hand wavy. And I agree with all other answers and comments that there are no proofs this will work, that there aren’t better basis functions in specific cases, and that it can miss important things. In the end it’s a method that is well motivated for the particular regime it was invented for because of properties of that regime. But in the end it’s just something that is a good thing to try first, see how well it matched experiments and that use the or basis functions or switch methods entirely.

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