electron configuration explained
When several electrons enter one atom they do not sit just anywhere. They fill rooms by a fixed set of rules. They fill the lowest-energy room first, a room holds at most two electrons of opposite spin, and the rooms of one shell are occupied one by one before any pairing. Add the way inner electrons screen the outer ones, and the reversal in which 4s fills before 3d, and the order of the periodic table becomes visible. This lesson is not about the shape of one orbital. It is about several electrons dividing the seats by rule. The rules divide the seats.
Put electrons in one by one and each falls into the lowest-energy room that is still empty. That is the aufbau principle. A room holds at most two electrons, and then the two spins must be opposite. As you add electrons the rooms fill in the order 1s, 2s, 2p, 3s, and the configuration string below, 1s² 2s² 2p⁶, grows. The low room fills first.
When one shell has several rooms, for example the three rooms of a p shell, an electron sits one to a room before any pairing. And the spins of these single electrons line up with each other. That is Hund's rule. Set that beside a wrong filling, pairing crowded into one room, and the spread-out side has the larger total spin and is the true ground state of lower energy. Because the electrons push each other away, it pays to sit in different rooms. The spread-out side is the lower one.
The shape of the space where an electron actually stays is different for each room. 1s is a round ball, 2p is a dumbbell of two lobes, and 3d is a four-leaf shape. The wave function seen here and its square |ψ|² are the probability cloud of finding the electron, and this lesson turns that cloud as a solid in 3 dimensions. A lesson elsewhere shows the same thing as a slice. Drag it around and change the shape, and the sign and the solid form show how they change. This is not a 2 dimensional slice. It is the solid. The shape shows the room.
An outer electron does not feel the whole positive charge of the nucleus. The inner electrons between them screen the nucleus. So the effective nuclear charge the outer electron actually feels is Zeff = Z − S, the number of protons with the screening taken off. Add inner electrons and the effective charge falls, and the orbital radius of the outer electron, its pull weakened, grows. Roughly r ∝ n²/Zeff. The more the screening, the farther the outer electron is pushed. Screening reduces the pull.
The energy order changes with the atomic number. Raise the number and the energy bars of 4s and 3d both fall, but 3d falls more steeply, so at some point the two cross. So in potassium, Z=19, and calcium, Z=20, 4s lies below 3d and filling starts at 4s, and from scandium, Z=21, 3d drops below 4s and 3d begins to fill. This reversal fixes where the transition metals begin. Push the number and the moment the two bars trade places appears. The crossing opens the transition metals.