I am keeping things incredibly simple. The problem is - if things are too simple, classical mechanics predicts electrons and protons collapse. Of course, we know by quantum mechanics that they do not - that there exist minimal distances between them. I crudely use this as a boundary condition on the distances between particles.
I am making a static model - as if the electrons and atomic nuclei simply "hang there" in space, with absolute knowledge of their positions - in complete violation of Heisenberg's Uncertainty Principle. So, no time derivative.
I am working on 2 electrons and 2 atomic nuclei right now: a tetrahedron. I am trying to prove that their configuration ought to be planar - in order to keep the 2 electrons as far apart from one another as possible and the atomic nuclei as far apart from one another.
On wikipedia, I found the determinantal formula for the volume of a tetrahedron - given the lengths of its 6 edges. If one chooses the 6 lengths arbitrarily in such a way that this formula yields a negative value, then no such tetrahedron with those edge lengths exist. The volume =0 is the degenerate / critical / boundary case.
I cannot really make use of calculus (derivative = 0) to find the minimum value of the total potential energy of this system
A point in my network is either an electron or an atomic nucleus.
E = sum over i<j of Zi*Zj*ei*ej/rij where Zi = atomic number (Zi=1 for an electron), ei=e if an electron, ei=-e if an atomic nucleus