Background: In a CH4 molecule, there are 4 C-H bonds that repel each other. Essentially the mathematical problem is how to distribute 4 points on a unit sphere where the points have maximal mutual distance - or, how to distribute 4 position vectors such that the endpoint distances between them are maximized.

This question Angle between lines joining tetrahedron center to vertices shows that the angles between the points/vectors forming the vertices of a regular tetrahedron is $\arccos(-\frac13)$.

I have been told by chemistry teachers that that is the shape which maximizes mutual distance between the points. However, that link was not relevant to me because I am trying to figure out a *proof* that the tetrahedral model maximizes distance and is the *only* model which maximizes it.

Therefore, the very similar question Calculations of angles between bonds in CH₄ (Methane) molecule was also irrelevant, because the answer started off with "*Note that a regular tetrahedron can be inscribed in alternating vertices of a cube*.", and thus assuming that the tetrahedral shape was optimal without any mathematical basis.

So my question is, is $\arccos(-\frac13)$ the optimal angle, and if so, *how can I approach this question to prove it?* I have limited knowledge in vector calculus but I am willing to learn if this is some multivariable optimization problem.

Thanks!