The question is similar to this one, which I created a few days ago, but the answer wasn't enough: https://www.quora.com/Why-do-atoms-combine-Whats-the-math-behind-it-What-is-an-example-of-math-solving-that-shows-that-these-atoms-will-combine .
Isn't Schroedinger's equation just the description of wave-like properties of an electron? I don't quite see in the Schroedinger equation anything indicating that atoms will combine.
The math is conceptually straightforward, but may come across as technical. It was launched in 1927 with the dihydrogen cation, $H^+_2$, the simplest molecular ion, that is a Hydrogen atom combining with a proton (hydrogen ion) into a molecular ion.
One year after its invention, Ø. Burrau applied the time-independent Schroedinger eigenvalue equation for the motion of an electron around two protons distant from each other by a fixed distance R, in atomic units, $$ \left( -\tfrac12 \nabla^2 + V \right) \psi = E \psi \qquad \mbox{with} \qquad V = -\frac{1}{r_a} - \frac{1}{r_b} \; . $$ The solution of this second order PDE might appear technical, but is routine for atomic physics, and the lowest eigenvalue E found is the so-called ground state. The eigenfunction $\psi$ yields a probability density for the presence of the electron around the two protons, which looks like this,
providing the effective "glue" between the two protons, which would normally repel each other to infinity, without the benefit of the electron.
Now the properties of the solution dictates that the potential energy, when varying R, has a minimum at some specific R, the optimum distance for the two protons from each other; so these stay put at this optimal R: it would take energy (force) to dislodge them closer or apart!
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The green curve minimum, a possibly recondite feature of the explicit eigenfunction solutions to this dispersive wave PDE, then, is the smoking gun for the two protons and the electron combining (binding) together.