The quantum mechanics behind periodicity of elements Especially in high school/first-year undergraduate chemistry courses, we learn with great dedication the periodicities along groups and periods. There are various useful and interesting trends.
I wanted, however, to find the theoretical framework behind it which explains the trends through mathematical analyses. I skimmed through the book Physical Chemistry by Peter Atkins, but couldn't find much information.
So I was also considering the possibility that the trends were mainly discovered and investigated through experiments, and quantum mechanics was there only to provide some rough theoretical yet non-quantitative interpretations. (For example, explaining the fact that the $4s$ orbital has a lower energy level than $3p$ orbital using what's called the penetration effect, which says that there is a comparably significant value of $|\psi|^2$ for $4s$ than $3p$ for smaller radius despite the probability densities at higher radius or expectation values of radius)
Is there any qualitative analysis of such periodic trends using quantum theory? If so, where may I find it?
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.]