List the quantum-mechanical orbitals through \(5 s\), in the correct energy order for multi-electron atoms.

Quantum numbers are the GPS of electrons in an atom. They tell us the exact 'address' of an electron including its energy state, position, and spin. There are four types of quantum numbers. The principal quantum number ()), indicates the energy level and distance from the nucleus. Larger represents electrons that are further away and typically have higher energy. Next is the azimuthal quantum number ()), defining the shape of the orbital, with each shape linked to a specific subshell (s, p, d, f). The magnetic quantum number ()) pinpoints the orientation of the orbital in space, and finally the spin quantum number ()), describes the direction of the electron’s spin, which can be either up or down. Understanding these numbers is vital for visualizing electron configuration and predicting chemical behaviors of elements.

For instance, the quantum numbers for the first electron in hydrogen are ) for the principal quantum number, ) for the azimuthal quantum number, ) for the magnetic quantum number, and +1/2 or -1/2 for the spin quantum number. This concise notation fully characterizes the electron's state in the atom.

The Aufbau Principle is like a rulebook for electron accommodation in an atom. It says: start from scratch and build up. In simple terms, electrons fill up orbitals starting from the lowest energy level and move up as they go. It's like filling up a hotel from the lower to the higher floors. This principle helps us predict how electrons populate orbitals, and it is crucial for writing electron configurations.

However, nature likes to throw a curveball, energy levels aren’t always straightforward. Orbitals with the same principal quantum number ()) don’t always increase in energy sequentially. This is where the knowledge of orbital energy levels and exceptions in the Aufbau order become important. For example, the 4s orbital is filled before the 3d orbital, even though '4' is higher than '3'. This is because the 4s orbital is actually lower in energy than the 3d orbital due to the complexities of electron interactions in multi-electron atoms.

Electron configuration is the address book of electrons in an atom. It details who lives where, in terms of orbitals. Like a seating chart, it follows rules to ensure electrons are organized according to their energy levels. The patterns in this chart come from the quantum numbers and the Aufbau Principle. Configuration notation lists orbitals with their corresponding number of electrons, like an apartment listing with the number of occupants. For example, the electron configuration for carbon is written as 1s² 2s² 2p².

Furthermore, there's something called Hund's Rule which means 'spread out before pairing up.' When electrons occupy orbitals of the same energy, they fill them singly first, with parallel spins, before they double up. This is because electrons, like people, prefer their own space and only pair up when they must.

Orbital energy levels are likened to the steps on a ladder – each step represents a different energy level. In a single-electron atom like hydrogen, the energy levels increase predictably as you go higher in principal quantum number ()). But throw in more electrons, and things get trickier due to electron-electron interactions. Instead of a straight climb, some steps might be closer than others. This alters the energy arrangement, especially visible in the 's', 'p', 'd', and 'f' subshells.

In multi-electron atoms, the energy of an orbital is affected not just by the principal quantum number ()) but also by the azimuthal quantum number ()). For orbitals with similar +) values, the one with the lower ) is generally of lower energy. That’s how we get the energy order of 1s, 2s, 2p, 3s, 3p, 4s, and then things get more interesting with 3d, 4p, 5s where the 'expected order' doesn’t always apply due to the specific energies of the orbitals involved.