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In my high school chemistry class, we talked about the angles between bonds in molecules. One that caught my attention was the CH₄ molecule. I asked my teacher how to calculate this result, he said that I would learn it in my math classes, so I put my curiosity on hold. I am going into my second year of university and I still have not been able to prove it. I tackled the problem in 2 ways:

First I tried to view the problem as an optimization problem. In this case, placing four points on a sphere as to minimize their distance. This is not working for me since I am having trouble coming up with the actual function.

Secondly, I tried studying a specific case of n points on a circle and generalizing from there. I found an interesting link between representation of roots in the Cauchy-Argand plane and the minimum spacing of n points on a circle, but I could no rigorously prove it. Even if I could, I have no idea how to extend the Cauchy-Argand plane to 3 dimensions.

I have a ''hunch'' that manifolds are a natural fit here but I am not sure. Are there any tools that would help me find the angles?

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CH4 is tetrahedral. The question of how to compute the angle between two vertices has already been asked and answered. Also, if you're unaware of how to work in three dimensions, you'll want to read about vectors. –  anon Aug 23 '11 at 8:05
    
In any event: see this. –  J. M. Aug 24 '11 at 4:11
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Astoundingly, this question was asked by another user - that is, the question of how to find the angles. I refer you to my preferred answer, given by Mark Bennet, and briefly summarize it here.

Note that a regular tetrahedron can be inscribed in alternating vertices of a cube. Then use vectors and the dot product to calculate.

I wanted to talk a bit more about it though - methane is highly symmetric, and so the actual angle is almost exactly 109.5, as predicted by geometry. But other molecules aren't as symmetric. Ammonia should also have a tetrahedral-type structure (it's pyramidal because of the lone pair of electrons, but the predicted angle is still 109.5), but it's actual angle is less. The repulsion from the lone pair squishes the hydrogens. Or a molecule like the highly unstable $\text{CH}_2\text{Li}_2$ will have no angles at 109.5, but will have different sets of angles as the lithium and the hydrogen have unbalanced effects on eachother.

ADDED
All models and predictions that I mention are per the VSEPR model of predicting the structure of chemical compounds.

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Sorry to disagree with you for the second time today; I assure you it's merely a coincidence ;-) There's no reason for ammonia (or CH$_2$Li$_2$) to have tetrahedral symmetry. When you say "the predicted angle is still $109.5$", what prediction method are you referring to? Also, when you say about methane "the actual angle is almost exactly 109.5, as predicted by geometry", are you merely referring to the fact that the tetrahedral angle isn't exactly 109.5, or are you implying that methane isn't exactly tetrahedral? –  joriki Aug 23 '11 at 10:17
    
@joriki: I always love it when people check over my work. I was basing my work on the same method I assumed that the OP learned in his high school chemistry class (or that I used up to my organic chem classes before I realized that math was the place for me) - the VSEPR model. This would predict both ammonia and the carbon-hydro-lithium molecules to have tetrahedral symmetry. With respect to methane, I refer to the fact that experimental results in chemistry suggest it's not exactly tetrahedral. –  mixedmath Aug 23 '11 at 10:23
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In the same sense, you could say that the predicted angle in H$_2$O is 109.5 :-). It's just that in H$_2$O the effects favouring more space for the lone pairs are greater. Considering the accuracy with which the actual angles can be predicted by quantum chemistry, it seems a bit misleading to say that "the" predicted angle is tetrahedral without mentioning that this is a prediction on the basis of a simple non-quantitative model that makes no distinction between bonding electrons and lone pairs. With respect to methane, that's news to me; do you know a reference for that? –  joriki Aug 23 '11 at 10:40
    
@joriki: Yes, you could predict that for water. I will update my answer to include explicit reference to the VSEPR model - quantum predictions are a different beast. And looking into methane I realized that I was assuming many methane molecules - confusing different sorts of stresses on the bond angles. So I can't really give you a reference ;p –  mixedmath Aug 23 '11 at 10:50
    
I really appreciate your answers, thank you very much! However, I might not have phrase my question correctly. What I am looking for a prediction as to ''why'' it is a tetrahedral shape. I saw a proof that explained why the shape of a drop is a circle, because the circle is the shape that maximizes the area with a given perimeter (I think it is called the isoperimetric inequality). I kind of wanted that same reasoning as to why is the methane tetrahedral and why no other shape is possible. Again, this is entirely my fault as my question was posed in very vague way. –  Stéphane Aug 23 '11 at 15:07
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