Figure 2: Spherical Harmonics
There are various ways of modelling and/or depicting atomic wavfunctions. You can use waves in a pool of water, as discussed in section 2. You can also use Chladni patterns and waves on a string, as discussed in section 5. You can also use pictures, including animated pictures and animated scatter plots, as discussed in section 6 and section 11.
Simple mathematics represents some (but not all) of what happens in real atoms, as discussed in section 3 and section 4.
The term “orbital” is often used in this context, but it is somewhat ambiguous. Sometimes it refers to the wavefunction in general, i.e. whatever the actual electron is actually doing ... but sometimes it is refers only to basis wavefunctions.
| 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. Sometimes you can get a small, bottom-of-the line “wading pool” that is the right size for an indoor hands-on demo. In a lecture situation, you can put a round flat-bottomed glass dish on the overhead projector. |
| In any case, the container shouldn’t be toooo full. |
| To launch the waves, stand in the pool and hop up and down. | For a tub or dish of water, excite the motion by pushing some smallish object up and down in the water. For a small dish, the frequency can be inconveniently high, in which case a motorized excitor comes in handy. |
This is the 2s basis wavefunction for the pool of water. It is profoundly analogous to the 2s orbital for an atom.
Let that wave die away, then ....
This is the 2px basis wavefunction.
Let that wave die away, then ....
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 2px and 2py with a particular phase. Similarly the 2p– orbital is a superposition of 2px and 2py with a different phase.
| (1) |
You can easily go the other way, writing 2px as a superposition of 2p+ and 2p–. That is to say, the set {2px, 2py} is not the only possible basis; the set {2p+, 2p–} is also a perfectly good basis.
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, 2px, 2py, and 2pz}. Another often-useful basis is {2s, 2p+, 2p–, and 2pz}, and there are infinitely many other bases.
If you know a little about wave mechanics, we can use it to shed some additional light on what’s going on. Otherwise you can skip to section 4.
Very roughly speaking, you can think about the 2p+ and 2p– wavefunctions using the Bohr model, i.e. electrons orbiting around the nucleus like planets around the sun. This is wrong in general, but serves as a rough first approximation 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, constructed from a radially-outbound running wave that reflects off the edge of the pool and returns as a radially-inbound running wave. Combining the two running waves gives us a standing wave with one node.
Switching back now from pools (two dimensions) to atoms (three dimensions), we have identified four possible wavefunctions the form a basis for the N=2 shell: 2s, 2px, 2py, and 2pz. No matter what basis we choose, there cannot be more than four basis functions when N=2. 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 as it pertains to individual atoms in the second row of the periodic table, in particular to their ionization potentials and electron affinities. There is something special about having eight electrons around an individual atom. (This must not be taken as an endorsement of anything resembling an octet rule for molecules; see reference 1 for details.)
Maybe you’re not convinced. Maybe you think there ought to be another wavefunction 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 resonant system such as this. (See section 9 for details.) If you attempt it, you won’t be able to satisfy the boundary conditions. Recall we said the 2s wavefunction could be constructed from a running wave that 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 adds up to zero. This is physics: basic wave mechanics. Or you could call it mathematics: Sturm-Liouville theory and all that.
The previous paragraph pretty much answers the question of why atoms have discrete shells. 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.
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 (anything beyond neon) and hence higher shells N (the third row of the periodic table and beyond), we need to think much more carefully about the relationship between mathematics and atomic physics. In particular, observation tells us that in terms of physics, i.e. in terms of energy, that shells get filled out of order relative to the naive mathematical numbering. We are talking about really basic observations here, starting with the existence of transition metals.
The physics is as follows:
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 wavefunctions (namely argon) ... but at this point the N=3 mathematics-shell is far from complete, because the 3d wavefunctions 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 basis wavefunctions, 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 electrons have a higher energy than the 3s and 3p electrons. The 3d electrons 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 wavefunctions 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 wavefunctions also has a node at the origin, so you would think the 2p electrons would be disfavored2 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-wave, if you move away from the origin, you pick up electron amplitude to first order. For a d-wave, 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 wavefunctions in the pool of water. 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 wavefunctions don’t sufficiently get inside the screening, so they have an unfavorable potential energy. Argon would almost always prefer to be inert than to react using a 3d wavefunction. The same is essentially true of other third-row atoms ... although the 3d wavefunctions can’t be dismissed entirely, as discussed in connection with SF4 below.
Not only do the 3d wavefunctions have high energy compared to the 3p wavefunctions, they even lose out to the 4s wavefunction 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.
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" aka "twist" 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 11, 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).
We are faced with two incompatible ideas:
For simplicity, we confine attention to hydrogenic atoms, i.e. atoms (or ions) with only a single electron.
These two descriptions are incompatible in the Heisenberg sense. For any given atom, any attempt to ascertain the position will randomize the future wavefunction, and any attempt to ascertain the wavefunction will randomize the future position.
This isn’t as much of a problem as it could be, if we have a large supply of identically-prepared atoms. We can measure one atom, throw it away, measure another atom and throw it away, and so forth. By collecting enough such measurements, we can gradually work out what positions are consistent with which wavefunctions.
I cobbled up a java applet that collects data from 10,000 simulated atoms to demonstrate how positional probability data can be extracted from wavefunctions. At present, the applet only implements the 1s, 2s, and 2px basis wavefunctions.3 Push the appropriate Go button.
The scale bar in the lower left corner has length a0, where a0 is the Bohr radius, namely
| (2) |
The scale bar gradually turns from red to black, serving as a progress meter.
Credit: The idea of using an animated scatter plot to show the probability density for an atomic wavefunction is an oldie but a goodie. I got it from a film somebody (possibly PSSC?) made in the 1960s, back when using computers to make educational animations was a lot more exotic than it is now.
Chemistry straddles the quantum/classical boundary:
| With rare exceptions, it is possible and useful to make classical ball-and-stick models of what atoms and/or ion cores are doing. | It is never possible to make a good classical model of what the electrons in an atom are doing. Electrons in this situation are highly quantum mechanical. |
Electrons weigh 1836 times less than protons, and it matters. It means that electrons will be higly quantum mechanical under conditions where anything heavier than a proton can be considered classical. In particular, the conditions I have in mind involve atomic length-scales, chemical energy-scales, and ordinary non-cryogenic temperatures.
Also note that any basis wavefunction other than the 1s orbital will have one or more nodes. A node is a place where the probability density goes to zero. The node in the 2s basis wavefunction is a sphere of radius 2a0. The node in the 2px basis wavefunction is the plane located at x=0.
It would be particularly hard to make a ball-and-stick model that explains the existence of such nodes. Consider the 2px for a moment: The electron spends half its time on the left and half its time on the right ... but never crosses the middle. Trying to explain this in terms of particles would violate the Bolzano theorem, because the classical laws of motion tell us the particle’s world line is supposed to be continuous. Position is supposed to be a continuous function of time.
Do not confuse quantum mechanical orbitals with classical orbits, such as the orbit of the earth around the sun. The earth is classical; electrons in atoms are not classical.
We can understand nodes in terms of waves. Imagine some water sloshing in a circular disk, as discussed in section 2. The water on the left has energy, and the water on the right has energy, but along the midline the energy density is zero, because the water is stationary there.
The astute reader will have noticed that the centers of the s wavefunctions are overexposed. That’s partly a reflection of the fact that the probability there is very much higher than the probability farther out, and partly a reflection of the limited dynamic range of human perception. By the time the outlying areas are dense enough to be readily perceptible, the center is necessarily overexposed. If you don’t want the center to be overexposed, you can push the Pause button to stop the simulation early ... at the cost of leaving the outlying areas underexposed and barely perceptible to the human eye.
If you want to compare one wavefunction with another, the easiest procedure is to put multiple copies of this document on your screen.
These images are not “artist’s impressions”. The probabilities are calculated accurately, directly from the Schrödinger equation. The trick for calculating the dot-positions, given the probabilities, is explained in reference 2.
Technical note: The probability plotted by this applet is not the total probability. It is a conditional probability, namely the probability in a thin slice centered on the z=0 plane. Dots falling above or below this slice are not plotted, not accounted for, and not projected onto the plane.Since the three wavefunctions implemented here are all rotationally invariant about the x-axis, i.e. about the contour of constant y=0, z=0, you can imagine rotating the figures about that axis to get an idea of the full three-dimensional distribution.
As mentioned in reference 3, when people talk about the size and shape of an atom, they usually mean the size and shape of the atom’s distribution of electrons.
This distribution is probabilistic, not deterministic. It is best visualized as a somewhat fluffly cloud.
The distribution can be formalized in terms of wavefunctions. Even though they are sometimes called orbitals, you should not imagine that the wavefunction is analogous to the orbit of a planet going around the sun (except perhaps for certain atypical orbitals such as are found in Rydberg atoms). We will have little to say about the Bohr model of the atom, except to say that it is not consistent with a modern understanding of quantum mechanics in general or atoms in particular.
Consider for example the pz wavefunction. Suppose we prepare an electron in the pz state and then measure its position. We repeat this many times. The result is that the electron is above the z=0 plane in half the observations, and below the z=0 plane in the other half of the observations. There is zero probability of finding the electron right at z=0. (Not just zero probability, but zero probability density.)
This result is incompatible with a classical “particle” model of the electron, for several reasons.
In contrast, these observations are consistent with a wave model. Water sloshing in a px pattern has energy density on the +x side of the pool and energy density on the −x side of the pool, but zero energy density along the node at x=0.
You should not imagine that this means what waves are “right” or that particles are “wrong”. Quantum mechanics tells us that in reality, there is no such thing as classical waves, and no such thing as classical particles. There is only stuff. All stuff is capable of acting like a wave and acting like a particle. The behavior you see will be wave-like and/or particle-like, depending on how you set up the experiment.
In particular, the statement that the electron is in a pz wavefunction is incompatible with the statement that the electron is above (or below) the z=0 plane. By this I mean incompatible in the Heisenberg sense. That is, you can design an experiment to determine that the electron is in the pz wavefunction, and you can design an experiment to determine whether the electron is above the z=0 plane, but you cannot determine both things at the same time.
Therefore asking whether/how the pz electron crosses from above to below the z=0 plane is a profoundly wrong question. The question is predicated on incorrect assumptions about the equations of motion.
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. |
There are two types of discreteness involved here. You can think of the two as being mutually perpendicular. Figure 1 shows the modes and occupation numbers that an atom might have. 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. |
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.)
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:
Questions beginning with “why” seem to invite answers beginning with “because”. But such an answer might be suboptimal or even impossible, for reasons discussed in reference 4.
Very commonly, when people ask “why xxx” they are really asking – or should be asking – something else, such as:
In some sense, knowing what happens is all you need to know to get through life. However, for a scientific explanation of what’s happening, it is strongly recommended that you don’t just explain what is happening, but also explain how you know.
If someone insists on narrowly asking “why” and demanding an answer in terms of “because” (as opposed to “how do you know”), that narrow question may have no scientific answer. In such cases we may need to follow Newton’s example and say “hypotheses non fingo” and leave it at that.
By way of analogy, 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. A better approach is morph the question into a how-do-we-know question, and answer that. We can explain how tiling is related to other facts 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 behavior of atoms is analogous: We know that atoms in the N=2 and N=3 rows exhibit “octets”, and we have explained how this is related to other facts we know about geometry, counting, and energy. This explanation in terms of mathematics and physics is more scientific and more pedagogical than just saying “it is because it is”.
It is physically impossible to stick in extra N=2 wavefunctions. 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.
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 5, 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. When contemplating Newton’s law F=ma, it doesn’t pay to ask whether F causes ma, or whether ma causes F. It’s just the wrong question. For details on this, see reference 4.
Here is a side view of the first few spherical harmonics. The l quantum number increases from top to bottom, and the ml quantum number increases from left to right.
Note that this gives a side view of each sphere. In contrast, the patterns in the pool of water – in particular the 2p+ and 2p– wavefunctions discussed in section 2 and section 3 – correspond more closely to a top view of the spheres in the figure.
Note that all the spheres in figure 2 are rotating at the same angular rate. If some of them appear to be moving slower than others, or not moving at all, it is only because they have less-detailed markings. Your eye is sensitive to change per unit time, and that depends not only on the rotation rate (angle per unit time) but also on the markings (change per unit angle).
For each l value, there are 2l+1 different values that ml can take on, of which only l+1 are shown in the figure. The other values of ml can be visualized as backwards-rotating versions of the spheres shown in the figure. I leave it to your imagination to mirror the diagram in a vertical plane, to form a left/right symmetric arrangement with one in the first row, three in the second row, five in the third row, et cetera.
The spherical harmonics are useful as a basis set. Anything that is function of angle (i.e. a function of position on the surface of the sphere) can be well approximated as a weighted sum of spherical harmonics. This is analogous to the way that any function of x can (with mild restrictions) be approximated by a polynomial, which is just a weighted sum of powers of x.
In particular, functions that are smooth, slowly varying functions of angle can be well represented by a sum containing only low-order spherical harmonics, and therefore relatively few terms in the sum.
The electron wavefunction in the vicinity of an atom is usually a slowly-varying function of angle. (Rapidly-varying functions are disfavored because they would have higher energy.) Therefore the electron wavefunction can be written as a sum of spherical harmonics, and usually as a sum with relatively few terms.
It should go without saying that atoms are not “really” little spheres with colored markings on them.
This figure does not attempt to portray the radial dependence of the atomic wavefunctions. For a full description of the basis wavefunctions, you would need to multiply the angular dependence (as described by the spherical harmonics) by the appropriate function of radius. Some information about this is depicted in reference 6.
We now discuss an even stronger connection that can sometimes be made between the spherical harmonics and real atoms. There are some situations – certainly not all situations – where the stationary states of the atom, i.e. the states of definite energy, have the same symmetry as a single spherical harmonic. Here we are no longer talking about a sum of spherical harmonics; we are now talking about just one particular spherical harmonic. An isolated atom in a magnetic field is an example of such a situation. The detailed shape of the stationary state may not be exactly the same as the spherical harmonic, but the symmetry is the same.
Therefore looking at figure 2 and appreciating the symmetry of the various drawings is worth some effort.