Value of Fine Structure Constant
An atom’s energy levels cannot be accounted for properly without including the fine structure effects, a fact which resurfaced a decade after Bohr when the Schrodinger equation came into the scene.
Just as the Bohr model failed to reproduce Hydrogen atom’s energy levels properly, so did Schrodinger equation.
It was quickly discovered that there were 3 reasons for this.
The Schrodinger equation is fundamentally non-relativistic, but electrons and other quantum particles can move close to the speed of light and that effect must be included.
Electrons don’t simply orbit atoms, but they also have an intrinsic angular momentum inherent to them: spin, with value of h/2, that can neither be aligned nor anti-aligned with the rest of atom’s angular momentum.
Electron’s also exhibit an inherent set of quantum fluctuations to their motion known as zitterbewegung; this also contributes to fine structure of atoms.
When you include all of these effects, you can successfully reproduce both gross and fine structure of matter.
The reason these corrections are so small is because the value of Fine Structure Constant, α, is also very small.
According to the best measurements, value of α = .007297352569, where only last digit is uncertain.