Orbitals in a Pool of Water, The Octet Rule, and VSEPR

1 Introduction: Atoms, Molecules, and “The” Octet Rule

Introductory chemistry books spend a lot of time talking about “the octet rule”. Actually, one can find many different versions of “the” octet rule. We shall see that some versions make useful predictions, while other versions make lots of wrong predictions (sometimes more wrong predictions than right predictions) and hardly deserve to be called rules at all.

We shall see that asking “why chemical elements obey the octet rule” is not the best possible question. We need to find better questions. At the very least, we should start by asking whether chemical elements obey any such rule. More generally,

We should be clear and specific about which rule we are talking about. There are various octet-related rules, and they’re not all the same.
For each rule, we should ask to what extent the rule is obeyed. What is the rule’s range of validity? How sure are we about that? How shocked would we be to find exceptions?

Let’s explore some so-called octet rules.

The most successful “octet” rule applies to noble gasses and to hydrides that are more-or-less isoelectronic with noble gasses. That is, moving right-to-left across the second row, we have Ne, HF, H₂O, NH₃, NH₄⁺, CH₄, et cetera. These exist and are well behaved under ordinary conditions, while species that don’t follow that pattern (such as, say, CH₃) are not so well behaved.

We can call this the hydride octet rule. The most obvious limits to its validity are as follows: it applies to the second row and the third row ... and nowhere else. Obviously the first row has two columns, not eight. H₂ is more-or-less isoelectronic with He, and has a boiling point in the range you would expect for a noble gas, but certainly does not have any sort of octet. The fourth and fifth rows each have eighteen columns, and later rows have even more. People sometimes arrange the layout and/or the headings of the table so they can pretend that rows beyond the third have eight columns, but the reality remains 18, not 8.

Next we attempt another example of an octet rule. Let’s call it the atomic/periodic octet rule. We try to capture the idea that two atoms will have similar properties if and only if their atomic numbers differ by eight.

Let’s look at the data:

From He to Ne is eight steps. Ne is a noble gas, like He. Ne to Ar is another eight steps. Again Ar is a noble gas. The atomic/periodic octet rule is upheld. Similarly H, F, and Cl differ by steps of eight, and we recognize roughly homologous compounds H₂, HF, and HCl.

On the other hand, we can recognize a roughly homologous series in HF, LiF, NaF. But these are not spaced eight steps apart; there are only two steps between H and Li.

There are some very important periodicities here ... but they are not easily summarized in terms of octets. Pretty soon we are going to have to give up on octets and start thinking in terms of shells and orbitals.

For our third attempt, let’s consider the notion of octets as applied to molecules with ligands heavier than hydrogen. Let’s call this the molecular octet rule. It is intimately related to efforts to draw Lewis dot diagrams of molecules. (See reference 1 for more discussion of Lewis dot diagrams and what’s wrong with them.)

Again, let’s look at some facts:

Octet notions and Lewis-dot notions predict that there is no Ne-Ne bond, there is a single bond in F₂, a double bond in O₂, and a triple bond in N₂. That’s all in reasonable agreement with reality.

We get into trouble if we try to extend the series one more step. After Ne, F, O, and N, we come to C. Do you predict that there will be a quadruple bond in the C₂ molecule? If not, why not? In terms of octets, quadruple bonding in C₂ makes just as much sense as triple bonding in N₂.

We get into further trouble if we put O₂ in a magnetic field. We discover it is paramagnetic. This paramagnetism is due to unpaired electrons. It is not possible to construct a filled octet without pairing all the electrons. So the octet picture of O₂ is just plain wrong, irreparably wrong. It is absolutely not possible to draw a correct Lewis-dot diagram for O₂.

Things go from bad to worse when we consider the sequence Li₂, Be₂, and B₂. Li₂ exists. B₂ exists and is paramagnetic. Meanwhile Be₂ (the middle member of the trio) does not exist; trying to form Be₂ turns out to be closely analogous to trying to form He₂ ... it’s just not gonna work.

Furthermore, the existence of molecules such as SF₄ and SF₆ (as discussed in section 4) is hard to explain in terms of filled octets. Note that we are talking second- and third-row elements, where you might most strongly expect octet rules to work. Alas, they don’t.

Last but not least we mention spectroscopic data, optical spectroscopy in particular, which cannot be reconciled with any notion of molecular octets.

All these facts (single bonds, double bonds, existence versus non-existence of dimers, paramagnetism, and spectra) can be more-or-less systematically explained in terms of Molecular Orbitals. A detailed discussion of Molecular Orbitals is beyond the scope of this note.

The question often arises, given that Lewis-dot diagrams represent reality so poorly, why do they remain so popular?

Part of it has to do with the urge to make ball-and-stick models of molecules. Molecules sit right astride the boundary between the classical regime (where ball-and-stick models work) and the quantum regime (where they don’t). You can get away with making ball-and-stick models of what the nuclei are doing in almost all molecules (especially molecules heavier than ammonia), but you can never get away with ball-and-stick models of what the electrons are doing.
Part of it has to do with people’s fondness for Kiplingesque just-so stories, i.e. stories that give a fanciful or farcical explanation of “why” something is. The Lewis-dot diagram of O₂ must be considered a just-so story. It purports to explain what the electrons are doing, and it fits a few of the facts, but it conflicts with most of the facts, and therefore has no real scientific explanatory power.
A Lewis-dot diagram is an artistically appealing way of portraying certain aspects of a molecule in those cases where it happens to make sense. The only problem is, there’s no way to deal with all those other aspects and other cases where it doesn’t make sense.

It would be grossly unscientific to cite some selected data in support of the Lewis-dot model, while closing your eyes to the vast amount of data that conflicts with it.

Also: To really understand chemistry, we need to know the shape of molecules, but Lewis-dot diagrams won’t tell us that. VSEPR addresses the shape issue, as discussed in section 4.

Recommendation: Forget about octets. Anything that can be properly said in terms of octets can be said at least as well and at least as easily in terms of shells and orbitals.

In this vein: Filled shells are important. Filled shells occur at locations 2 ... 2+8 ... 2+8+8 ... 2+8+8+18 ... 2+8+8+18+18 ... et cetera. That is more complicated than a pattern containing only octets, but it isn’t terribly complicated, and it has the advantage of agreeing with the facts.

Continuing in this vein: Filled shells are important for atoms (not molecules). Atoms with filled shells are noble gasses, which are strongly homologous. Furthermore, the elements one electron in their outermost shell form another homologous series, the alkali metals. Similarly the elements one step lighter than noble gasses form another homologous series, the halogens. The alert reader will notice that I have just placed hydrogen as a member of two series, i.e. alkali metal and halogen both. This is entirely intentional. Also notice that I do not refer to the halogens as "group 7" or "column VII" or anything like that; their defining property is not “noble gas minus seven” but rather "noble gas minus one”.

Now we turn to questions about shells and orbitals. The nice thing is that these questions have answers, as we shall discuss anon.

a): Why are there shells at all?
b): How many slots are their in the outer shell? Why?
c): How do we decide what is the so-called “outer” shell anyway, and why do we care about that, as opposed to the higher-numbered and/or lower-numbered shells?

The usual glib answer is that the shells are determined by the mathematics of wave mechanics, as discussed in section 2. Alas, the story about mathematics and symmetry is not the whole story. If we just count the possible wavefunctions (the spherical harmonics, Y_lm) and group them into shells, we would predict that the first four rows of the periodic table would have 2, 8, 18, and 32 members. In the real periodic table, the first four rows have 2, 8, 8, and 18 members. That is, filling a mathematics-shell is not the same as filling a chemistry-shell. This point will be addressed in section 3.

Finally, in section 8 and section 9, we discuss why we should ask “why”, but also why that is often not exactly the optimal question. Often it is better to ask “how do we know xxx”, and to ask “how is xxx related to other things we know”.

2 Stationary States in a Pool of Water

Recommended equipment: one large round swimming pool with a radially-symmetric depth profile, shallow enough to stand in.

For classroom demonstrations, you can use water in a washtub, the bigger the better. It shouldn’t be toooo full.

*) First, stand in the middle of the pool. Hop up and down. Excite a standing wave such that the water in the middle goes down while the water at the edge goes up. There is one node, circular, a bit more than halfway out from the center.

This is the 2s orbital for the swimming pool. It is profoundly analogous to the 2s orbital for an atom.

Let that wave die away, then ....

*) Stand near one edge. Hop up and down. Excite a standing wave such that the water at one edge goes up while the water at the other edge goes down. There is one node, a straight line across the diameter of the pool.

This is the 2p_x orbital.

Let that wave die away, then ....

-> Call on a buddy who is standing also near the edge, but 90 degrees away from your position. Have him set up a standing wave just like the previously-described one. This is the 2p_y orbital. It is perpendicular to the 2p_x orbital.

-> You and your buddy working together can create a traveling wave, a big bump of water that rotates around the edge clockwise. There will be a trough directly opposite the top of the bump. There will be a node straight across the diameter of the pool. The node just rotates. This is the 2p₊ orbital.

-> You can also create the 2p_- orbital, which rotates in the other direction.

You can construct an infinitude of other 2p orbitals, but if you have more than two, they will be linearly dependent. For instance, you can readily convince yourself that the 2p₊ orbital is a superposition of 2p_x and 2p_y with a particular phase. Similarly the 2p_- orbital is a superposition of 2p_x and 2p_y with a different phase. And you can go the other way: 2p_x is a superposition of 2p₊ and 2p_-.

This gives us a complete description of the N=2 shell. In two dimensions, there are no other linearly-independent patterns that can be formed with only one node.

In three dimensions, there would be one more pattern. The obvious “rectangular” basis in D=3 is {2s, 2p_x, 2p_y, and 2p_z}. Another often-useful basis {2s, 2p₊, 2p_-, and 2p_z}, and there are infinitely many other bases.

* Wave Mechanics

If you know a little about wave mechanics, it sheds additional light on what’s going on. Otherwise you can skip to section 3.

You can think about the 2p₊ and 2p_- orbitals using the Bohr model, i.e. electrons orbiting around the nucleus like planets around the sun. This is wrong in general but OK for waves that run around the edge of the pool, such as the 2p, 3d, 4f, etc. waves. (These are the waves with the most angular momentum.) Bohr’s idea is that you need an integral number of wavelengths around the circumference of the orbit. The defining property of p orbitals is that they have one wavelength around the circumference (while d orbitals have two, f orbitals have three, et cetera). The wave equation gives us two solutions (leftward and rightward propagation, or in this case clockwise and counterclockwise) so there must be exactly two 2p orbitals in two dimensions. This is just counting, and a little bit of symmetry.

Similarly, the 2s wavefunction can be thought of as a standing wave, namely a radially-outbound wave that reflects off the edge of the pool and returns as a radially inbound wave. The net result is a standing wave with one node.

Switching back now from pools (D=2) to atoms (D=3), we have found four orbitals. That’s all there can be in the N=2 shell. It’s just geometry and symmetry and counting. Then the electron spin gives us double occupation of each orbital. That makes eight. Anything else is linearly dependent, or involves a different shell (i.e. different principal quantum number).

This is the fundamental basis for the octet rule for elements in the second row of the periodic table.

Maybe you’re not convinced. Maybe you think there ought to be another orbital just like 2s but a little bit different, having one node (so it belongs to the N=2 family) but somehow different, having the node in a different place. Well, sorry, it can’t be done in a high-Q system such as this. (See section 7 for details.) If you attempt it, you won’t be able to satisfy the boundary conditions. Recall we said the 2s orbital could be considered a running wave the reflects off the wall of the pool and returns. That only works with one very specific wavelength. If you try it with a slightly different wavelength, it will come back with the wrong phase. The phase errors will accumulate with every bounce. Over the long haul you will get a superposition of waves with all possible phases, which is just zero. This is physics: basic wave mechanics. Or you could call it mathematics: Sturm-Liouville theory and all that.

The previous paragraph pretty much addresses question (a) as posed in section 1. You can’t have something that is halfway between shell N=2 and shell N=3. If you try, you won’t be able to satisfy the boundary conditions.

Note that we have good scientific answers to questions (a) and (b). This isn’t mere phenomenology. This is physics. It’s really easy physics when N=2, mostly just geometry and counting.

3 Beyond Neon

The picture described so far works for low-numbered atoms, up to and including the second row of the periodic table.

When we consider higher atomic numbers (Z) and hence higher shells (N), we need to re-examine our answers to these questions. We immediately find that answering question (c) requires us to realize that shells get filled out of order; the outermost shell is not necessarily the highest-numbered shell. (Otherwise we wouldn’t have transition metals.)

The physics is as follows:

Because of electrostatic potential energy, an electron wants to be close to the nucleus.
Because of kinetic energy, an electron doesn’t want to be confined near the nucleus (or anywhere else), especially when its wavefunction has lots of nodes (large N).

The electron’s kinetic energy depends on the curvature of the wavefunction. A high-N wavefunction in a small region will have lots of curvature, hence lots of kinetic energy.

A high-N wavefunction far from the nucleus has an unfavorable potential energy. A high-N wavefunction near the nucleus has an unfavorable kinetic energy. Therefore we expect the small-N shells to fill up first.

For high-Z atoms, once the small-N shells are filled up, things get very complicated. Once we start filling the high-N shells, things proceed in a somewhat peculiar order. This produces transition metals among other things. Hund’s rules and all that.

The first row is easy: There is only one wavefunction, the 1s wavefunction. It just sits there. No nodes. No dynamics. Electron spin means we can have two electrons in this orbital. So the first row has two members and ends at helium, Z=2.

The second row has eight members and ends when we have filled the N=2 shell (on top of the N=1 shell), namely neon, Z=10.

So far so good.

Things get quite a bit more interesting when we get to the third row. The observed fact is that the third row has eight members and is is complete at argon, Z=18. Here is where we must explain the difference between a chemistry-shell and a mathematics-shell.

The N=3 chemistry-shell (also called valence-shell) is complete when we have filled the 3s and 3p orbitals (namely argon) ... but at this point the N=3 mathematics-shell is far from complete, because the 3d orbitals haven’t been touched.

There are some things we know about math, some things we know about physics, and some things we know about chemistry. The point here is that mathematics by itself will not correctly explain the chemistry when N=3 or beyond. Physics is needed. A closely related point is that jumping up and down in the pool accurately tells you certain things about the number of atomic orbitals, and the symmetry thereof, but it will not accurately tell you the energy thereof.

The key to understanding the third row of the periodic table is this: the 3d orbitals have a higher energy than the 3s and 3p orbitals. The 3d orbitals are members in good standing of the N=3 mathematics-shell, but they don’t¹ contribute to the N=3 chemistry-shell, because they are energetically unfavorable. So let’s try to figure out why they have a higher energy.

At this point the usual glib explanation is to say that the 3d orbitals have a node at the origin, so the 3d electrons don’t spend enough time near the nucleus and accordingly have an unfavorable potential energy. The problem is, if you believe that argument, you would predict that beryllium would be a noble gas, because the 2p orbital also has a node at the origin, so you would think the 2p electrons would be disfavoredActually the 2p electrons are somewhat disfavored relative to the 2s electrons, and the closed 2s shell in Be does have chemically-observable consequences, as you will notice if you try to make Be₂, which is not much easier than making He₂. But let’s not get carried away; you can obtain a chunk of metallic Be and/or BeO a lot more easily than metallic He and/or HeO. compared to the 2s electrons. We need a better argument.

We need to consider more than just the node at the origin. We need to consider what happens in the neighborhood of the origin. For a p-orbital, if you move away from the origin, you pick up electron amplitude to first order. For a d-orbital, you only pick up amplitude to second order. You have to go a lot farther to get significant amplitude. Also note that the nucleus is heavily screened by the electrons in the lower-N shells, so it’s not simply a question of how close you can get to the nucleus, but rather a question of whether you can get inside the inner shells, i.e. inside the screening.

You can set up some 3d orbitals in the swimming pool. The easiest one has the symmetry

                  + +     - -
                  + +     - -

                  - -     + +
                  - -     + +

and has two nodes, straight lines that cross in the middle. The water is fairly quiet in a fairly good-size region near the middle.

To summarize: the key idea is that the 3d orbitals don’t sufficiently get inside the screening. Argon would almost always prefer to be inert than to react using a 3d orbital. The same is essentially true of other third-row atoms, but the 3d orbitals can’t be dismissed entirely, as discussed in connection with SF₄ below.

Not only is the 3d orbital high compared to the 3p orbital, it even loses out to the 4s orbital in potassium and calcium. But not by much. It is competitive with 4p, which is roughly why the ten fourth-row transition metals are where they are in the periodic table, between calcium (the end of 4s) and gallium (the beginning of 4p). This placement does not, however, mean that 3d is necessarily filled before 4p is begun. Atoms in this part of the table can change their valence by shifting electrons back and forth between 3d and 4p.

4 VSEPR

VSEPR² is another ball-and-stick model of molecules. That puts it in the same general category as Lewis-dot diagrams. The primary difference is that unlike Lewis-dot diagrams, VSEPR predicts the shapes of the molecules.

VSEPR is in some ways more modern than Lewis-dot diagrams. Its construction is enlightened by some things we have learned from quantum mechanics, such as the idea of Linear Combinations of Atomic Orbitals (LCAO) e.g. sp³ hybrids. This is in contrast to fully-classical balls and sticks, which do not form linear combinations and do not hybridize.

You should not imagine that VSEPR is fully quantum-mechanical; it is only “borrowing” some QM ideas. In particular, VSEPR is hopelessly unable to explain the paramagnetism of O₂, and unable to explain much about the Li₂, Be₂, B₂ series.

As an example of VSEPR, let’s start by comparing water with acetylene. The molecules are in some ways similar: they have something in the middle plus a couple of hydrogen atoms stuck on the outside. Yet their macroscopic properties are radically different. Water boils at 373 kelvin, while acetylene boils at 189 kelvin. That’s quite a difference. And to the extent that the acetylene molecule is larger and heavier that water, you would expect it to have a higher boiling point, quite contrary to what is observed. (Compare the boiling-point trend in the alkane series: methane, ethane, propane, butane, et cetera). Evidently the molecules are not as analogous as they might seem, and we need to explain why not.

The key idea is that the acetylene molecule is linear and nonpolar, whereas water is bent and polar. You won’t learn this from any octet rule, nor from any Lewis-dot diagram. Alas you can easily draw a Lewis-dot diagram of an allegedly linear water molecule that satisfies the octet rule. Because water is polar, it exhibits lots of hydrogen bonding, which greatly affects the boiling point and other properties. The fact that water is polar is (a) well attested by observations and (b) of considerable importance to chemistry, life on earth, and other amusing things.

Some people use the term “Lewis-dot diagram” so broadly that it includes VSEPR diagrams, but I don’t recommend this. Professor Lewis died in 1946, while VSEPR dates from 1957. See reference 2 and reference 3.

We should always pay attention to the limits of validity of the things we believe. In this spirit, let’s take a look at the molecules SF₄ and SF₆. The known properties of these molecules seem to be inconsistent with the usual VSEPR notions and/or inconsistent with the idea that atoms in the third row of the periodic table do not have accessible d-type atomic orbitals.

These molecules are often touted as a triumph for VSEPR. If we believe the VSEPR picture, the sulfur atom in SF₄ has five bonds -- four ligands and one lone pair -- in blatant violation of the octet rule, so d orbitals must be involved. VSEPR predicts a remarkable see-saw (teeter-totter) structure for this molecule. Most of the fluorine-fluorine bond angles should be 90 degrees, but one pair of fluorines should make an angle of slightly less than 120 degrees. I assume this prediction is in good agreement with observations, although I’ve never seen any data one way or the other. There is no way to build such a teeter-totter without using d orbitals.

We turn now to SF₆. Again if we believe the VSEPR picture, the sulfur atom has six bonds, with the obvious octahedral symmetry. There is no way to build such bonds without involving d orbitals.

But not everybody believes the VSEPR picture. In particular, reference 5 states as follows:

The extended valence (violation of the octet rule) observed in compounds of higher main group elements has very little to do with the availability of d-AOs but is due rather to the size of these atoms and thus to the reduced steric hindrance between ligands and, to a lesser extent, also to the lower electronegativity of the heavy atoms.

I’m skeptical of that claim, so let’s investigate it.

The most charitable interpretation I can come up with applies to the SF₆ molecule. With difficulty I can imagine an S⁽⁶⁺⁾ ion surrounded by six F^- ions. That means the sulfur has been stripped down to its core, so that it is isoelectronic with neon, and the molecule has no covalent bonds at all, so VSEPR notions don’t apply. However I don’t really believe this picture. I don’t think it is energetically feasible to sixfold ionize the sulfur atom. That means the bonds in the molecule cannot be 100% ionic; they must have some covalent character. And as soon as you allow even a little bit of covalent bonding, we need to talk about the symmetry of the bonding. There is no way you can build molecular orbitals with octahedra symmetry without involving d orbitals. It’s impossible by symmetry. Consider just four bonds in the equatorial plane. How are you going to make those? The best you can do with s and p orbitals is an sp² hybrid in the plane, which has threefold symmetry. You need fourfold symmetry.

The next-most charitable interpretation is to start with the SF₄^(2-) molecular ion. I believe such a thing can be created. VSEPR predicts a simple tetrahedral structure, with four ligands and no lone pairs. So far so good. Now to make SF₆, you can attach a couple of F^- ions. That works as long as you treat the SF₄^2- tetrahedron as a hard core, with the two additional F^- ions attached to the outside ... but as soon as you try to treat all six F atoms on an equal footing. or try to create octahedral symmetry, once again you’re creating bonds with some d character.

As for SF⁴, the arguments are similar, differing only in details. Yes, I can imagine a S⁽⁴⁺⁾ core surrounded by four F^- ions in a tetrahedral arrangement ... but it seems energetically implausible. We are familiar with the fact that in NaF, the sodium atom is 100% ionized, but from there it is quite a big leap to imagine sulfur being 400% ionized. Starting from that ionic picture, if you consider the possibility of any covalent character at all, you get into trouble. If you maintain tetrahedral symmetry, the bond has sp³ character, so it involves the s-electrons from the S⁽⁴⁺⁾ core (which is isoelectronic with [Ne]3s²). That means some additional charge is being transferred onto the F^- ions, which would be quite a trick; it would involve opening up an entirely new shell, a new N value. That seems very much less likely than opening up the 3d levels on the sulfur atom and parking the extra electrons in a lone pair, as VSEPR predicts.

To summarize:

If VSEPR is even partly right, SF₄ is not tetrahedral, but has some sort of teeter-totter shape. That is sufficient to prove that d orbitals are involved.
If the experimental data is even partly right, SF⁴ is not tetrahedral. (If it were tetrahedral, presumably somebody would have noticed by now.) That is also sufficient to prove that d orbitals are involved.

You can sing similar songs about other molecules e.g. ClF₃.

Bottom line: There is plenty of evidence that d-orbitals are sometimes accessible to elements in the N=3 row of the periodic table. The counterarguments seem very weak. As a consequence there are dramatic violations of the octet rule.

5 Other Wave Models

You can set up standing waves on a metal plate. For present purposes, it’s appropriate to choose a round flat plate, supported at the center. Excite it by bowing. Make a heavy-duty bow using the frame of hacksaw or pruning saw, plus high-test kernmantel fishing line or weed-trimmer line. Put rosin on the bowstring ... it’s just like bowing a violin. If you put powder on the plate, it will move to the nodes and remain there, making the node pattern visible. This is called a Chladni pattern.

Another good demo involves waves on a string. It is best to use rotary waves, such as are seen on a jump-rope. Drive the demo with an electric egg-beater or a variable-speed electric drill; chuck up some sort of wheel and attach the string off-center. Install a swivel (available from the fishing-supplies store) to decouple the undesired "spin" from the desired "orbital" motion.

Choose a rotation rate that is slow enough that people can follow the actual motion with their eyes. A long, heavy string with minimal tension will give you lots of nodes at a low frequency.

Discussion: Rotary waves (as opposed to vibratory waves) give a better model of the time-dependence of the wavefunctions in an atom. Consider a particular point on the string: in a standing wave, the distance of that point from the axis remains constant as the point goes around and around. This is very different from the vibratory wave, where each point has a non-constant distance from the axis.

The wavefunction in quantum mechanics has exactly this sort of character, rotating rather than vibrating. The ordinate of the wavefunction is a two-component vector (commonly represented by a complex number).

The tradeoff with the string model is that by devoting two dimensions to modeling the ordinate of the wave function, we are left with only a one-dimensional abscissa, so we cannot model the rich spatial structure of atomic wavefunctions. Contrast this with the model described in section 2, which has a two-dimensional abscissa but only a one-dimensional ordinate.

The genuine quantum wavefunction for a single particle has a three-dimensional abscissa (real space) and a two-dimensional ordinate (in its own somewhat abstract space).

6 Wave Mechanics versus Quantum Mechanics

The demos we’ve been discussing are all macroscopic, involving strictly classical wave mechanics. Consider the contrast:

Classical waves were fully understood in the 19th century. Classical waves are useful as models of the atomic wavefunctions.

Quantum mechanics didn’t come along until the 20th century. There is more to quantum mechanics than wavefunctions.

Understanding waves is a prerequisite for understanding QM. It is necessary but far from sufficient.

There are two types of discreteness involved here. You can think of the two as being mutually perpendicular. In the chart below, the enumeration of the modes runs vertically, while the quantum occupation numbers run horizontally.

   wavefunction   | quantum occupation number  -->
                  |   0      1
  spatial | spin  |______________________
    mode  |       |
                  |
    2 p x   up    |         yes
                  |
    2 p x   down  |  yes
                  |
    2 p y   up    |         yes
                  |
    2 p y   down  |         yes
                  |
    etc.          |

Figure 1: Mode versus Occupation Number

Quantum mechanics takes its name from the quantization of the occupation numbers, i.e. the fact that if you design an experiment to measure the occupation number, you will always get an integer.

For fermions such as electrons, each wavefunction has an occupation number that is either 0 and 1. For bosons, such as photons in a box (or phonons on a violin string) the occupation numbers can be any integer from zero on up ... but otherwise the boson chart is the same as the fermion chart: the modes of the box (or string) run vertically, while the occupation numbers run horizontally.

The business of enumerating the spatial modes is entirely classical. You can tell it’s classical, because it doesn’t require knowing the value of hbar, and it doesn’t tell you anything about hbar.

Then, in addition to the spatial part of the wave function, there is another part -- spin -- which is part of the enumeration of states but is intrinsically nonclassical, i.e. intrinsically quantum-mechanical.

Finally, after we have enumerated the modes, the occupation of the modes is intrinsically nonclassical.

The occupation numbers for macroscopic objects such as strings are huuuge. You cannot perceive the difference between huuuge and huuuge+1, so for practical purposes the amplitude is not quantized. (And furthermore it’s not quantized even in principle, because the model breaks down due to thermal effects and other complexities we’re not going to discuss.)

7 Nonstationary States

The foregoing demos emphasize standing waves. But not all waves are standing waves.

Think about your experience with things like jump-ropes, tie-down ropes, extension cords, and so forth. You can flirt one end of a long rope and launch a perfectly fine wave with no definite number of nodes... not a standing wave.

Similarly, a duck can sit in the middle of a large pond, bobbing up and down, launching beautiful waves at any frequency whatsoever. The duck neither knows nor cares about the standing-wave modes of the pond.

If (!) you are weakly coupled to a high-Q system then you can excite the resonant waves more easily than nonresonant waves.

Atoms do in fact have some high-Q modes. This makes spectroscopy interesting. But atoms can do low-Q things as well.

There are a couple of lessons here:

If you want to demonstrate high-Q behavior, you must set up your demo accordingly. For instance, choose a heavy but flexible string, or even a fine chain, as opposed to sewing thread.
Do not assume that standing waves are the only game in town. If you drive a linear system at an off-resonance frequency, it will respond off-resonance. (Floquet’s theorem and all that.) The response may be small, but it won’t be zero, and in some systems it won’t even be small.

8 Why We Ask “Why”

Explaining the octet rule in terms of mathematics and physics is more scientific and more pedagogical than just saying “it is because it is”.

If pressed, I might cautiously and narrowly agree that knowing “what” happens is the only truly essential requirement for science. Sometimes you have no clue why something is happening or how it relates to other things. In such a case Newton said “hypotheses non fingo” and sometimes that’s where it has to end.

However, that notion is horribly open to abuse.

Even if knowing “what” happens is the only essential objective, it is not the only desirable objective. Having a clue about “how” and “why” things happen is for many people an exceedingly convenient mnemonic and a powerful tool for figuring out “what” happens.

Suppose a student asks why it is not possible to tile the floor with pentagons (even though it works with triangles, squares, or hexagons). Answering by saying “it is because it is” would be a cop-out. Much, much better answers are available ... answers that explain how this is related to other things we know about geometry and counting.

William James said that every memory is associated with others, and each association is a “hook” whereby you can fish up that memory when it has sunk below the surface.

The octet rule can usefully be associated with lots of other things we know. It is geometrically impossible to stick in extra N=2 orbitals. It’s not even a question of not having enough room; rather, it’s impossible by symmetry. No matter what you cook up, if there appear to be more than four of them, they’re linearly dependent ... or they involve something outside the N=2 family. Mathematics tells us that. In addition, physics tell us that anything with N>2 would have significantly higher energy. Understanding this allows us to figure out “what” happens not just in this situation, but in many other situations as well.

9 Why ... or, How Do We Know That?

As mentioned briefly in section 1, questions beginning with “why” seem to invite answers beginning with “because”. But we should not be in too much of a rush to give such an answer, because it might not be right.

Very commonly, rather than asking “why xxx” you would be much better off asking something else, such as:

How do we know xxx? How sure are we about that? What is the domain of validity of xxx?
How could we have predicted xxx? (Typically there is a multitude of good answers to this question.)
How is xxx related to other things we know?
What is a good mnemonic for xxx? (Things that make sense are vastly easier to remember than things that don’ make sense.)

In the case of the periodic table, mathematics and atomic physics gives us pretty strong reason to expect certain periodicities. It would be exceedingly surprising to hear reports of new element with properties halfway between hydrogen and helium.

But the converse is equally true: Human knowledge of the periodic table considerably predates human understanding of quantum mechanics. You could say that the periodic table is evidence supporting our belief in quantum mechanics just as much as vice versa.

Even better, we should say that our knowledge of the periodic table and our understanding of quantum mechanics are mutually consistent ... and also consistent with lots of other things we know.

In reference 9, Feynman wrote eloquently about how to think about “why” and “because”. He deprecated what he called the “Greek” style of reasoning, where everything is allegedly deduced from postulates and “first” principles (as is commonly done in high-school geometry classes). Instead, he advocated a “Babylonian” style of reasoning, where each known thing is related to other known things, with no implied directionality to the relationship. He compared it to a grand tapestry: if a hole develops somewhere, you can re-weave the fabric starting from the top, bottom, and/or sides of the hole.

This little book is highly recommended reading.

The goal is to be able to figure things out. The question “how do we know xxx” is tightly focussed on this goal.

A causal relationship is not usually the most appropriate way of figuring things out. (Sometimes it is, but not always or even usually.) Don’t insist on knowing a cause if you don’t need one. Don’t ask whether F causes ma, or whether ma causes F. It’s just the wrong question. For details on this, see reference 10.

10 References

1.: Fun with Lewis Dot Diagrams -- Or Not ./lewis-o2.htm
2.: Gilbert N. Lewis “The Atom and the Molecule” JACS Volume 38, pages 762-786 (1916) http://dbhs.wvusd.k12.ca.us/Chem-History/Lewis-1916/Lewis-1916.html
3.: R.J. Gillespie and R.S. Nyholm, “Inorganic Stereochemistry” Rev. Chem. Soc. volume 11, page 116 (1957).
4.: R.J. Gillespie Molecular Geometry (1972).
5.: W. Kutzelnigg, "Chemical bonding in higher main group elements" Ang. Chem. Int. Ed. Engl. 23 272-295 (1984).
6.: http://www.madsci.org/posts/archives/may99/926481344.Ch.r.html
7.: The Feynman Lectures on Physics volume III chapter 19, “The Hydrogen Atom and the Periodic Table”.
8.: Gordon Baym, Lectures on Quantum Mechanics chapter 20, “Atoms”.
9.: Richard Feynman, The Character of Physical Law
10.: "Cause and Effect" ./causation.htm
11.: "Why Pairs -- Or Not" ./why-pairs.htm

1: The N=3 chemistry-shell is conventionally assumed to have no 3d contributions, and the periodic table is structured accordingly. This is usually a good approximation.
2: The name VSEPR stands for Valence Shell Electron-Pair Repulsion ... but the name doesn’t fully describe the subject.

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