Why are the sublevels within a principal level split into different energies for multi-electron atoms but not for the hydrogen atom?

The principal quantum number, denoted as 'n', is a critical concept in quantum mechanics and plays a pivotal role in dictating the energy levels within an atom. Think of it like the address of an electron within an atom, identifying the electron's shell or energy level. For instance, when we say an electron is in the 2nd energy level, we're saying its principal quantum number is 2.
The energy associated with each level increases as the principal quantum number does. So, a higher 'n' means that the electron resides further from the nucleus and has more energy. In a simple hydrogen atom, where only one electron is present, the energy levels solely rely on 'n'. However, in multi-electron atoms, things get a bit more complicated due to additional interactions, leading to sublevel energy splitting.

When we add more electrons to an atom, we introduce a force known as electron-electron repulsion. Consider a classroom where each student needs space; when it gets crowded, students need to adjust to make room. Electrons do the same. They repel each other because they carry the same negative charge.
In a multi-electron atom, this repulsion impacts the energy of electrons differently depending on their arrangement in orbitals. It causes the energy sublevels within a principal level to vary. Electrons in the same orbital don't rejoice in sharing space, so their mutual repulsion slightly raises their energy levels, resulting in a split of these sublevels.

Diving into the hydrogen atom energy levels, it's fascinating to see how simple they are compared to other elements. Since the hydrogen atom consists of just one electron and one proton, the energy levels are beautifully straightforward. They're determined by the principal quantum number 'n', and all sublevels within a principal level (known as degenerate states), hold the same energy.
In hydrogen's elegant simplicity, there's no need to worry about electron repulsion within its levels, hence the degeneracy. This lends to easy predictions of the hydrogen atom's behavior and forms the foundation for understanding more complex atoms.

The azimuthal quantum number, labeled as 'l', brings another layer of detail to an electron's address. It further divides the principal energy levels into sublevels or orbitals. Starting from 0 up to 'n-1', it defines the shape and number of sub-orbitals within a principal level. For example, for 'n' equal to 2, 'l' can be 0 or 1, corresponding to the s and p sublevels, respectively.
The azimuthal quantum number becomes particularly important in multi-electron atoms where it helps to show the differences in energy within sublevels due to the varying shapes and electron distributions. This is one of the factors behind why sublevels have different energies in atoms more complex than hydrogen.

Imagine trying to hear someone across a crowded room; the people in between you two would muffle the sound. This is akin to the shielding effect in atomic physics. Electrons in lower energy levels act as a sort of 'crowd', blocking the full attractive force of the nucleus from the outer level electrons.
This effect is significant because it plays into the energy split within sublevels. Electrons in inner shells shield the nucleus's charge, making the effective nuclear charge felt by outer electrons less. This leads to those electrons being less tightly bound to the nucleus and thus having higher potential energy, which is a part of the reason sublevels are not degenerate in multi-electron atoms unlike in a hydrogen atom.