#atomic orbitals

Principles to remember when filling Atomic orbitals

Aufbau: lowest energy level is filled first. 1s before 2s.

Pauli Exclusion: 2 electrons max for each orbital. Furthermore, the electrons must have opposite spin.

I guess I’m kind of doing a series on electron configurations, energy levels, and orbitals.   I don’t want to get bogged down in recaps, so I’m going to assume anyone following along has read my explanation of energy levels here, and my explanation of the Madelung Rule here.

In brief, electrons surround atomic nuclei like tiny planets orbiting tiny stars, but they follow very different rules. Let’s say an argon atom is formed inside a star (a real one, not a tiny one).  The intense heat and pressure of a stellar core allows smaller nuclei to crash into each other and form larger nuclei, which we call atomic fusion.  Under these conditions, electrons don’t really stick to any particular nucleus, as explained by They Might Be Giants.   But, if the argon nucleus makes it out into space, it can cool down, and eventually 18 electrons will be attracted to its 18 protons.  When they happens, they have to file into specific regions that are defined by quantum mechanics.  These are called orbitals, and some are higher in energy than others.  The electrons would prefer to occupy the orbital at the lowest energy level, but they can’t all go in there.  So most of them have to settle for what they can get, because even a higher energy level orbital is better than floating free of the nucleus’ sweet, sweet positive charge. 

It’s sort of like a movie theater.  When the movie is popular, the theater fills quickly, and the first people who arrive will grab the best seats.   If you show up during the trailers, you’ll have to settle for the far left side of the front row, but you’d rather do that than miss the feature altogether.  A less popular film draws a smaller audience, and they naturally tend to take the best seats because they can.  Personally, I kind of like sitting in the front row, because I get more legroom and no one can block my view, but people are different.  Electrons are all the same. 

So the order in which the electrons fill up the orbitals of an argon atom are described by its electron configuration.  In this case: 1s22s22p63s23p6.   The numbers represent electron shells, and the letters represent subshells containing different types of orbitals.  By reading left to right, you can see what order the electrons would file into  the atom.  First 1s, then 2s, then 2p, then 3s, then 3p.  This lowest-to-highest system was described by the Aufbau principle, which comes from a German word meaning “construction”.  Just as you have to build a house from the ground up, you have to fill the lowest energy level (1s) before you can move on to filling the higher ones. 

However, the elements beyond argon don’t always follow the same pattern.  There’s a 3d subshell that argon doesn’t use in its ground state, because there aren’t enough electrons in argon to fill it.  But a potassium atom has 19 electrons, so we might expect that extra electron to file into 3d.  But it doesn’t; it actually skips ahead to the next shell and goes into 4s.  The 20th electron in calcium does the same thing.  Once the 4s subshell is full, then the electrons start to file into 3d, as is the case for the 21st electron in scandium the first of the transition metals.  This continues until the last transition metal, zinc, after which any additional electrons would go into 4p. 

Incidentally, this is what the transition metals are transitioning to.  From scandium to zinc, the Aufbau principle is suspended while a d subshell is filled.  Once the d-subshell is filled with ten electrons, the Aufbau principle takes hold again.  This is also why the transition metal section of the periodic table is called the “d-block”. 

This suspension of the Aufbau principle is described by the Madelung Rule, which uses a different system to order the various subshells by priority.  The Aufbau principle assumes that electrons seek out the lowest principle quantum number, n, and then they seek out the lowest azimuthal quantum number, ℓ.  Under that ideal, 3d is better than 4s, because 4>3, and d vs. s isn’t taken into consideration.  The Madelung Rule holds that electrons actually take both quantum numbers into consideration, and the sum of n+ℓ  is the deciding factor.  For 3d, n+ℓ =5, while for 4s, n+ℓ =4, so 4s is actually a better deal than 3d. 

I should explain here that neither the Aufbau or Madelung systems are any sort of scientific law or anything.  These were simply attempts to explain and predict electron configuration.  The Aufbau principle made sense until experimental data showed that the transition metals refused to obey it.  So the Madelung rule was devised to try to rationalize the concept (quantum numbers) to the observations (the experimental data). This means that the Madelung Rule can, and does, have its own exceptions, like copper.

If we went by the Aufbau principle, copper ought to have an electron configuration like this: 1s22s22p63s2 3p63d104s1.  The 3d subshell would fill first, and the last electron would go into 4s.  But we know the Madelung rule overrides this order, so we have to fill 4s first, and then 3d. 

So using the Madelung rule, copper’s electron configuration should look like this: 1s22s22p63s2 3p63d94s2.  Now we’re talkin’.  The 4s shell is filled, then we use the final nine electrons in 3d. 

But, experimental data shows copper’s electron configuration actually looks like this: 1s22s22p63s2 3p63d104s1.  What the hell?  That’s just the Aufbau principle again, right?  Not really.  To demonstrate this, we need to look at another example: chromium.

Electron configuration of chromium (Aufbau version): 1s22s22p63s2 3p63d6. 

Electron configuration of chromium (Madelung version):  1s22s22p63s2 3p63d44s2. 

Electron configuration of chromium (Real world):  1s22s22p63s2 3p63d54s1.

The key point to both the Aufbau and Madelung systems is that each subshell has to be full before moving on to the next one, whichever order that may be.  But chromium doesn’t give a crap.  It doesn’t fill the 4s subshell, and it can’t fill 3d, so instead it just leaves both subshells half-full.  

Now there’s really only two possible explanations for this.  One is that the experimental data is incorrect, and chromium uses some other electron configuration we haven’t learned to accurately observe yet.  Like the man said, failure is always an option, but we shouldn’t rush to assume that unexpected outcomes are invalid.  The failure could lie in our assumptions, rather than in the experiment’s refusal to confirm those assumptions. 

The second explanation is that chromium exists in this ground state because it’s more stable that way.  There’s no other motive here.  The Aufbau and Madelung rules try to predict the most efficient way to pack electrons around a nucleus, but chromium actually has to do the thing.  So if it has a better way then we have to incorporate that into our rules. 

Unfortunately, there doesn’t seem to be a cool name for the rule, but it comes down to this: half-full d-subshells are favorable than full s-subshells, and full d-subshells are even more favorable.  Given the choice, a chromium atom would rather have 3d5 than 3d4 or 3d6.  Achieving this means borrowing an electron from the 4s subshell, but apparently it’s preferable that way, so chromium does it.  Copper does the same thing, only in its case it’s borrowing a electron that could have gone to 4s, in order to upgrade from 3d9 to 3d10. 

Having that d-subshell occupied so close to the outermost electron shell appears to have a stabilizing effect.  Potassium and calcium have none, and they react violently on exposure to water.  Scandium only has one, and it’s fairly reactive, but it’s not “explode-in-water” bad.  Then you’ve got elements like chromium, iron, and nickel, which are actually used as building materials.  They do react to things, but they don’t react violently to stuff like water because their d electrons make them stabler in their metallic form.  Then you have copper and zinc with a full set of d electrons, and they’re pretty hardy.  They’re not inert by any means, but when’s the last time you saw penny crumble apart?

So it appears that having that any 3d electrons just beneath 4s is a bonus.  Having a full 3d subshell is even better, but half-full is a good compromise.  The advantage of a half-full subshell is that each electron can have its own orbital all to itself.  According to the Pauli exclusion principle, two electrons can’t have the same quantum numbers in the same atom.  The only reason orbitals can hold two electrons apiece is because those two electrons can have different spins.  One of them has a spin quantum number of +1/2, and the other can have -1/2, and that way they both fit.  But an electron would rather just have an orbital all to itself.  It’s like having a seat all to yourself on a school bus.  There’s room for two people, but you have room to stretch out if you’re all by yourself. 

This concept is described by Hund’s rule of maximum multiplicity.   When electrons fill a subshell containing multiple orbitals, such as p, d, or f subshells, they don’t share orbitals until they have to.  That way, the electrons can be as far apart from each other as they can, which is good for cutting down energy costs, since electrons naturally repel one another.   They also don’t have to change their spins to accommodate each other, but it saves energy if they do it anyway. 

Having mutliple electrons in a subshell with the same spin is also good because it promotes exchange energy.  I’m just now learning about this concept as I’m writing this, but the basic idea is that having two or more electrons in a subshell with the same spin allows those electrons to swap places, and this releases energy.  I guess it’s sort of like pushing two bus seats together, and now you and your friend in the other seat have even more room between you.  The more same-spin electrons, the more exchange energy is released, and the more stable the subshell becomes. 

So exchange energy is at it’s maximum when the subhsell is full.  In a d-subshell, this would mean ten electrons, five with one spin, and five with the other.  Nine would be okay too, because you have five of one spin and four of the other.  But ten would always be better than nine, and that’s why copper does what it does.

As for chromium, a half-full d-subshell is still favorable because you’d have five same-spin electrons, without the hassle of them sharing orbitals and pushing against each other.  So it borrows an electron from 4s to get five d-electrons, because five gives you more exchange energy than four.  However, it doesn’t grab the other 4s electron to get six d-electrons, because that won’t add any exchange energy benefit.  That sixth d-electron would have to adopt the opposite spin from the other five, and it would be alone in this status, so no exchange energy.  Further, it would have to share an orbital with somebody, and that would cost a little energy to overcome the repulsion between like-charges. 

So there are a lot of competing conditions here.  This is perhaps best exemplified with nickel.  I looked it up to see what it’s configuration would be, and Wikipedia wasn’t sure. 

Electron configuration of nickel (Aufbau version):  1s22s22p63s2 3p63d10.

Electron configuration of chromium (Madelung version):  1s22s22p63s2 3p63d84s2.

Electron configuration of chromium (research literature):  1s22s22p63s2 3p63d94s1.

There’s an argument to be made for all three of these, really.  The Madelung rule would put eight electrons in the 3d subshell because that’s the last one to be filled.  Cool.  Wikipedia says that some literature uses this as the official configuration, and it’s very low in energy. Fair enough.

But why couldn’t nickel borrow its two 4s electrons to pull off a full set of ten 3d electrons?  This isn’t a flight of fantasy, because nickel’s big brother palladium actually does this.  It means leaving its 5s subshell empty, but the 4d subshell is full, and consequently palladium is pretty darn unreactive.  I’m guessing nickel doesn’t do this because there’s some sort of line in the sand.  Copper can borrow one 4s electron, and that’s all it needs, but nickel would have to take both, and maybe that arrangement costs more energy than it saves.   Again, palladium can do it, but in palladium’s case, we’re talking about 4d and 5s instead of 3d and 4s.  There may be different energy tradeoffs involved there. 

The compromise seems to be borrowing just one 4s electron and settling for a nine-electron set in 3d.  It’s not full, but you still get plenty of exchange energy, and still more than you’d get from just eight.  And it doesn’t leave the 4s subshell completely empty, for whatever that’s worth.  The dispute here seems to be that this configuration is very close in energy cost to the one predicted by the Madelung Rule, and maybe the difference is so slim that we can’t accurately measure which is better.  It’s worth noting that nickel’s bigger brother platinum uses this configuration.  So it’s not a cut-and-dried thing.  Each transition metal has to find the right balance between exchange energy, adherence to the Madelung rule, and the energy lost by pairing electrons in the same d-orbitals.  And there might be additional factors I’m not even taking into account.  So when you look at the electron configurations on a periodic table, there won’t necessarily be a lot of easily predictable patterns in the d-block. 

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