The quantum mechanics behind periodicity of elements
As you surmise, quantum mechanics was developed to help explain observations such as that periodicity, spectral lines and the photoelectric effect.
However, the solution of the Schrödinger equation is difficult, and not complete beyond hydrogen. For example Wikipedia states, &quot;Unlike for hydrogen, a closed-form solution to the Schrödinger equation for the helium atom has not been found.&quot; Matrix mechanics can also predict some properties, such as spectral emissions. Neither, approach, though, is likely to give a clear insight into chemical behavior for heavier elements, particularly where relativity affects inner electron behavior for the actinides.
So you've raised a good question, and as mathematics and computational speed increase, exact solutions for other elements will be found.
[To me, it seems a bit odd (no pun intended) that there is one pair of s electrons, three pairs of p, five pairs of d and seven of f, as if it's numerology. But that's my limitation. Sigh... or ψ.]