[deutdeut]

In a homoatomic molecule like hydrogen, the two H atoms share electrons equally. Why is the net charge on each H atom be zero if both atoms have 2 valence electrons after sharing?

[Yggdrasil]

Didn't we have this discussion here already?

Anyway, if you look at the overall structure of H₂, you see two protons and two electrons.  Therefore the overall charge on H₂ is zero.  That means that charges on each hydrogen must add up to zero.  So if Z₁ is the charge on one hydrogen and Z₂ is the charge on the other hydrogen, Z₁ + Z₂ = 0.

Now, hydrogen is a symmetric molecule.  Because of this symmetry both hydrogens should have the same charge.  In other words, Z₁ = Z₂.

So, solving the system of equations:

Z₁ = Z₂
Z₁ + Z₂ = 0

You get that Z₁ = Z₂ = 0 and that both hydrogen atoms in H₂ are neutral.

[deutdeut]

What does it mean by symmetric molecule?

[Yggdrasil]

This is way above high school chemistry level, but:

H₂ has point group symmetry of D_{inf_h}.  A point group is basically a collection of symmetry operations (i.e. reflections, rotations, etc) which leave the molecule's structure unchanged.  This is important because the wavefunctions of the electrons must also be unchanged by these symmetry operations, which puts important restrictions on your wavefunctions and makes them easier to estimate.

[constant thinker]

I guess sometimes being able to get further into understanding something take a leap of blind faith. Then once you build enough of the knowledge that people tell you "it's just that way because I say so." Then you can get to the more advanced stuff, and actually understand why.

Take for example the quadratic formula (I'm assuming you know what that is). When your first taught it, your just told to take a quadratic equation in standard form; identify a, b, and c; then plug and chug. You generally aren't taught where it comes from, or why it works. Later on in math though once you learn about completing the square, you can finally understand how the quadratic equation is derived, and it makes sense as to why it works.

This also applies to the covalent bond topic that was previously discussed.