The transition metals are more complicated than lighter elements.
Why?
Because they’re the first whose electron wavefunctions are described by quadratic functions of and — not just linear or constant. These are called ‘d orbitals’, and they look sort of like this:
More precisely: the wavefunctions of electrons in atoms depend on the distance from the nucleus and also the angles The angular dependence is described by ‘spherical harmonics’, certain functions on the sphere. These are gotten by taking certain polynomials in and restricting them to the unit sphere. Chemists have their own jargon for this:
• constant polynomial: s orbital
• linear polynomial: p orbital
• quadratic polynomial: d orbital
• cubic polynomial: f orbital
and so on.
To be even more precise, a spherical harmonic is an
eigenfunction of the Laplacian on the sphere. Any such function is the restriction to the sphere of some homogeneous polynomial in whose Laplacian in 3d space is zero. This polynomial can be constant, linear, etc.
The dimension of the space of spherical harmonics goes like 1, 3, 5, 7,… as we increase the degree of the polynomial starting from 0:
• constant:
• linear:
• quadratic:
etcetera. So, we get one s orbital, three p orbitals, five d orbitals and so on. Here I’ve arbitrarily chosen a basis of the space of quadratic polynomials with vanishing Laplacian, and I’m not claiming this matches the d orbitals in the pictures!
The transition metals are the first to use the d orbitals. This is why they’re so different than lighter elements.
Although there are 5 d orbitals, an electron occupying such an orbital can have spin up or down. This is why there are 10 transition metals per row!
This chart doesn’t show the last row of highly radioactive transition metals, just the ones you’re likely to see:
Look: 10 per row, all because there’s a 5d space of quadratic polynomials in with vanishing Laplacian. Math becomes matter.
The Madelung rules
Can we understand why the first transition element, scandium, has 21 electrons? Yes, if we’re willing to use the ‘Madelung rules’ explained last time. Let me review them rapidly here.
You’ll notice this chart has axes called and
As I just explained, the angular dependence of an orbital is determined by a homogeneous polynomial with vanishing Laplacian. In the above chart, the degree of this polynomial is called The space of such polynomials has dimension
But an orbital has an additional radial dependence, described using a number called The math, which I won’t go into, requires that That gives the above chart its roughly triangular appearance.
The letters s, p, d, f are just chemistry jargon for
Thanks to spin and the Pauli exclusion principle, we can pack at most electrons into the orbitals with a given choice of and This bunch of orbitals is called a ‘subshell’.
The Madelung rules say the order in which subshells get filled:
 Electrons are assigned to subshells in order of increasing values of .

For subshells with the same value of , electrons are assigned first to the subshell with lower
So let’s see what happens. Only when we hit will we get transition metals!
This is called the 1s subshell, and we can put 2 electrons in here. First we get hydrogen with 1 electron, then helium with 2. At this point all the subshells are full, so the ‘1st shell’ is complete, and helium is called a ‘noble gas’.
This is called the 2s subshell, and we can put 2 more electrons in here. We get lithium with 3 electrons, and then beryllium with 4.
This is called the 2p subshell, and we can put 6 more electrons in here. We get:
◦ boron with 5 electrons,
◦ carbon with 6,
◦ nitrogen with 7,
◦ oxygen with 8,
◦ fluorine with 9,
◦ neon with 10.
At this point all the subshells are full, so the 2nd shell is complete and neon is another noble gas.
This is is called the 3s subshell, and we can put 2 more electrons in here. We get sodium with 11 electrons, and magnesium with 12.
This is called the 4p subshell, and we can put 6 more electrons in here. We get:
◦ aluminum with 13 electrons,
◦ silicon with 14,
◦ phosphorus with 15,
◦ sulfur with 16,
◦ chlorine with 17,
◦ argon with 18.
At this point all the subshells are full, so the 3rd shell is complete and argon is another noble gas.
This is called the 4s subshell, and we can put 2 more electrons in here. We get potassium with 19 electrons and calcium with 20.
This is called the 3d subshell, and we can put 10 electrons in here. Since now we’ve finally hit and thus a d subshell, these are transition metals! We get:
◦ scandium with 21 electrons,
◦ titanium with 22,
◦ vanadium with 23,
◦ chromium with 24,
◦ manganese with 25,
◦ iron with 26,
◦ cobalt with 27,
◦ nickel with 28,
◦ copper with 29,
◦ zinc with 30.
And the story continues—but at least we’ve seen why the first batch of transition elements starts where it does!
The scandal of scandium
For a strong attack on the Madelung rules, see:
• Eric Scerri, The problem with the Aufbau principle for finding electronic configurations, 24 June 2012.
But it’s important to realize that he’s attacking a version of the Madelung rules that is different, and stronger than the version stated above. My version only concerned atoms, not ions. The stronger version claims that you can use the Madelung rules not only to determine the ground state of an atom, but also those of the positive ions obtained by taking that atom and removing some electrons!
This stronger version breaks down if you consider scandium with one electron removed. As we’ve just seen, scandium has the electrons as in argon together with three more: two in the 4s orbital and one in the 3d orbital. This conforms to the Madelung rules.
But when you ionize scandium and remove one electron, it’s not the 3d electron that leaves—it’s one of the 4s electrons! This breaks the stronger version of the Madelung rules.
The weaker version of the Madelung rules also breaks down, but later in the transition metals. The first problem is with chromium, the second is with copper:
By the Madelung rules, chromium should have 2 electrons in the 4s shell and 4 in the 3d shell. But in fact it has just 1 in the 4s and 5 in the 3d.
The second is with copper. By the Madelung rules, this should have 2 electrons in the 4s shell and 9 in the 3d. But in fact it has just 1 in the 4s and 10 in the 3d.
There are also other breakdowns in heavier transition metals, listed here:
• Wikipedia, Aufbau principle: exceptions in the d block.
These subtleties can only be understood by digging a lot deeper into how the electrons in an atom interact with each other. That’s above my pay grade right now. If you know a good place to learn more about this, let me know! I’m only interested in atoms here, not molecules.
Oxidation states of transition metals
Transition metals get some of their special properties because the electrons in the d subshell are easily removed. For example, this is why the transition metals conduct electricity.
Also, when reacting chemically with other elements, they lose different numbers of electrons. The different possibilities are called ‘oxidation states’.
For example, scandium has all the electrons of argon (Ar) plus two in an s orbital and one in a d orbital. It can easily lose 3 electrons, giving an oxidation state called Sc^{3+}. Titanium has one more electron, so it can lose 4 and form Ti^{4+}. And so on:
This accounts for the most obvious pattern in the chart below: the diagonal lines sloping up.
The red dots are common oxidation states, while the white dots are rarer oxidation states. For example iron (Fe) can lose 2 electrons, 3 electrons, 4 electrons (more rarely), 5 electrons, or 6 electrons (more rarely).
The diagonal lines sloping up come from the simple fact that as we move through a group of transition metals, there are more and more electrons in the d subshell, so more can be easily be removed. But everything is complicated by the fact that electrons interact! So the trend doesn’t go on forever: manganese gives up 8 electrons but iron doesn’t easily give up 8, only at most 6. And there’s much more going on, too.
Note also that the two charts above don’t actually agree: the chart in color includes more rare oxidation states.
References
For a bit more, read:
• Wikipedia, Transition metals.
• Oxidation states of transition metals, Chemistry LibreTexts.
The colored chart of oxidation states in this post is from Wikicommons,
made by Felix Wan, corrected to include the two most common oxidation
states of ruthenium. The blackandwhite chart is from the Chemistry
Libretexts webpage.