The Periodic Table of the elements is a fascinating icon of science. It is incredibly useful and has been exploited and sexploited too in the form of a periodic table of yoga and a sexy PT. It has also been hacked apart, cut and paste into different formats, created as illuminated wall cases, woodworked into furniture, spiralled, spherized, and generally rebuilt in almost every imaginable way ever since Mendeleev first dreamed of laying out his elemental cards according to the periodicity of elemental properties.

Now, in an effort to inspire chemists to reconsider the foundations of the periodic table, chemical philosopher Eric Scerri of the University of California, Los Angeles, is building a new way to classify the chemical elements one step at a time.

Writing in the latest issue of the Journal of Chemical Education (PDF 2008, 85, 585-589), Scerri explains how the periodic table initially arose from the discovery of atomic weight triads but he now suggests that chemists should recognize the fundamental importance of atomic number triads.

This sea change in elemental attitude might enhance the periodic table by classifying the elements at a fundamental level as basic substances. As such, he and his colleagues have developed a new version of the “left-step” periodic table, which looks very different from the conventional PT. In the new layout, with its step-like pattern actinides and lanthanides are no longer relegated to a standalone box, but form the first step of the PT.

Climbing right to the transition metals (Fe, Mn, Ir, Sg et al) on the next step and then to the non- and semi-metals, such as boron carbon, oxygen, silicon etc and finally a step in which the halogens (fluorine, chlorine…), noble gases (neon, xenon…), alkali metals (potassium, sodium…) and alkaline earth metals (beryllium, calcium…) form the final highest step on the right. Hydrogen tops the halogen column and helium crowns the noble gases rather than acting as the outer beacons as with the conventional layout. (Click the graphic for a clearer, full-size view).

“The left step table has been around for some time,” Scerri told me, “but I am modifying it to accommodate two atomic number triads which would otherwise be absent. They are He, Ne, Ar which ceases to exist as a triad in the usually encountered left-step table and H, F, Cl which does not exist either in the conventional medium-long form table or the usually encountered left-step table.”

In the grander scheme of things, whatever form the Periodic Table takes in the future matters not to those of us who sing, so we end with a song, the periodic table song from Tom Lehrer (who was 80 on April 9, 2008 and gets a mention in the Official Google Blog this week), known simply as *The Elements*.

Prof. Bent, after Valery and I started interacting and I tried to rationalize what to ‘do’ with the spacer layers in the Adomah model (originally I’d wondered whether we could stick antielements in there!), I realized that the numbers of spheres NOT used was the same as the numbers that were utilized, and that the slice sizes were the same as well, just coming from the directly opposite edge as that of the s-block in the Adomah, and oriented perpendicularly.

I’d always been bothered by the fact that we had to double up elements to make a regular tetrahedron. Now with equal-count and -distribution real estate one could re-assort and have all elements get one sphere each.

A remaining problem was that this inverse Adomah set, which intercalated the first one, broke up the continuity of subperiods- that is, one edge would get, say, all the s1 electrons, and its opposite s2. In addition one still had to deal with the numerical lopsidedness of putting Z=1 on one vertex and Z=2 on another, but not 3 and 4 on the remaining two vertices (though one COULD do this and I tried it too).

As I explored this virgin territory, I found that a number of different ways to rationally distribute the quantum numbers on vertices, edges, faces, and the tetrahedral body existed that would give symmetrical results- some were just more aesthetically pleasing than others. One particular set took the center-point of two opposing edges as the place around which to put H, He, and then 119 and 120 on the other edge. This system gave one a very symmetrical result, with all Cartesian coordinates, straight lines, and right angles for the locations of subperiods. But it still broke things up in ways I found displeasing.

It was at that moment that, serendipitously, I realized suddenly that by switching the constructional assignment of two of the quantum numbers I could get around these subperiod discontinuities. But I was dubious- I’d already built an Adomah-style tetrahedron from wooden spheres (my young nephews built others for themselves from the remaining pile of balls). I physically counted, one by one, the spheres on the outer edge of the 120-sphere tetrahedron (which wasn’t THAT easy since I’d constructed the thing from slices and it would fall apart unless deliberately held together) and to my delight and astonishment I found that the outer ring contained exactly 28 spheres, enough for the full f block.

So, did this carry over to the other orbital types? The next ring held 20 spheres, but there were TWO of these. The next inward had 12 spheres- it took a bit of doing to visualize that there were three of these rings, since nothing was colored, or labeled, and the slices didn’t lend themselves easily to the counts. Well, if there were THREE, then the innermost rings would be 4 spheres apiece, and there should be four of them- as there turned out to be.

Note that the differences in sphere counts for the rings were consistently 8 different: 28, 20, 12, and 4, reflecting the fact that each ring represents two full subperiods of the same type. Having to multiply these by 1, 2, 3 and 4 respectively shows the relation to the symmetrically disposed flat Adomah system as adapted from the LSPT, and the Madelung table.

The s-block elements directly surround a rotational axis that extends from the center of one edge to the center of the opposing edge at right angles to it. In my model the counts still increase along this axis, so there is still ‘lopsidedness’ numerically (but then this seems to be a fault of all periodic depictions). A 120-sphere tetrahedron has a continuous outer ‘jacket’ of 100 spheres surrounding a core of 20 (and as discussed over on the Yahoo Group (http://tech.groups.yahoo.com/group/tetrahedronT3) everything up to Z=20 seems much more regular), so I’ve wondered whether one could actually start in the middle and move outward instead.

Just as an aside the rule for constructing these jackets is to sum successive squares of even numbers: The core has 2 sq + 4 sq, and the outer jacket has 6 sq + 8 sq, which are oriented the same way. One can also get inverted tetrahedra by summing intermediately, so 0 sq + 2 sq gives the first four elements which can be hidden inside the 20-sphere tet. What the significance of the next sum, 4 sq + 6 sq (=52), may be, I haven’t any clue. Takers?

I’ve also constructed periodic systems out of 4-sphere tetrahedra. Two such sets turned out to have been discovered first (to my chagrin, but then one has to take as well as one gives…) by Pierre Demers in Quebec some years ago.

All of these models have the advantage of assigning one element to one sphere, though they otherwise vary wildly in terms of symmetry and continuity of elements from the original periodic string.

One can also get symmetrical models utilizing the 4 vertices for symmetry, but the results are far uglier to the mind of someone used to 3D. Perhaps someone from Flatland would find such aesthetic reactions superfluous…

Because of kinships beyond primary, secondary, tertiary and some quaternary, I still think that extra dimensions might be likely, but now made up for by ‘hidden’ dimensional quality as an offset, so no NET increase on the surface- the figure folds back on itself, negating any increase in numbers of spheres (though this would possibly mean that all elements and their properties would be combinatoric in nature from contributions from higher dimensions (i.e. extra spheres), then reduced by the back-folding (so fewer spheres). Hard to wrap one’s mind around, and I can’t find any references to such a thing. But if good enough for subatomic physics, why wouldn’t it work for us?

Jess Tauber

Henry,

I have never built sliced up tetrahedron myself in real world. I built computer model of it using 2×7, 4×5, 6×3, 8×1 and 9×0 blocks. It surely works on computer. The last pair yields edge of the tetrahedron (line thickness of 0) with zero area and zero corresponding elements. I am not sure about irregular tetrahedron that you suggested. I have never tried that. I am sure that Alex will enjoy that. Jess told me once that he actually built physical model around 1979.

I find it amazing, though, that in order to make regular tetrahedron work, quantum number “ms” values have to be +/-1/2 as guessed by Pauli and later confirmed by Dirac. So, the relationship between the quantum numbers “n”, “l”, “ml” and “ms” has tetrahedral character.

I had a lot of fun while making 3D Madelung rule diagram using red and green spheres. Each red sphere has to be marked with two consecutive elements: H-He, Li-Be, B-C, N-O…. Green spheres stay unmarked. Than 3D Madelung rule diagram is built like this:

Put first red sphere (H-He) on top of a table; then put two green spheres next to the red to form triangle and put second red sphere (Li-Be) on top of the triangle. Then put three red spheres (B…Ne) on top of table so they interlock with two green spheres, put two green spheres on top of three red and next to one red and then red sphere (Na-Mg) at the very top. Start fourth layer with 4 green spheres on top of the table next to three red. Put three red (Al….Ar) on top of four green, then two green on top of three red and then one red for K and Ca. Proceed in the same manner, alternating red and green spheres to build up bi-color tetrahedral stack of spheres of any size. The tetrahedron does not have to stop, it can grow infinitely, just like 2D Madelung rule diagram. You will notice that Janet’s dyads occure due to the presence of two types of spheres, red and green (protons and neutrons?). Jess suggested once that it could be electrons and neutrinos.

I was amazed at first that such simple procedure of constructing bi-color stack of spheres mimics Madelung rule diagram in 3D so precisely and third dimension corresponds to number of “ml” values for each subshell exactly. I think that Alex will enjoy that!

Valery.

ANOTHER THOUGHT ABOUT ORBITALS

The many-body problem in classical mechanics and quantum mechanics are alike in one respect: Neither problem has been solved in the general case in closed form.

Strictly speaking there is no such thing as “the earth’s orbit” about the sun.

The earth’s motion is influenced by the locations of all the other planets.

Of course, planet-planet interactions in our solar system are much weaker, relative to sun-planet interactions, than are electron-electron interactions in atoms, relative to nucleus-electron interactions, so, of course, the orbital approximation is much better for our planetary system than it is for atoms. Still, modern electron-density functional theory yields electron-densities for frontier orbitals that many chemists take very seriously indeed. [In my electron-density models, frontier nucleophilic (electron-pair donor) sites correspond, usually, to unshared valence-shell electron-pairs and and frontier electrophilic (electron-pair acceptor) sites correspond to the models' "pockets". That simple model accounts for a huge number of organic reaction mechanisms and the stereochemistry of a huge number of inter- and intra-molecular interactions. For me, those (localized molecular) orbitals are so useful I cannot imagine doing with them!]

Valery, about the “one step that [I] overlooked”: I wasn’t concerned with creating a REGULAR tetrahedron, whereas I gather that has been one of your aims. In climbing up the front face of my tetrahedron, block heights increase in the order 2 4 6 8. In proceeding back across the bottom face, block widths decrease in the order 14 10 6 2. So “my” tetrahedron is an irregular tetrahedron. “Doubling up” locations changes the latter sequence to 7 5 3 1 and yields your “perimeter rule”: 2+7 = 4+5 = 6+3 = 8+1 = 9, and, in my case, the possibility of a regular tetrahedron. I’m not absolutely certain about all of that, since I’ve not yet made a model. If I mention it to one of my grandsons, Alex, I might receive one as a Christmas present. Several years ago, when I was in the midst of writing “New Ideas”, Alex gave me a magnificent framed large sheet of stainless steel and, backed by magnetic tape, 120 wood squares with burned into them the element’s symbols and atomic numbers, so that I can move elements around at will. It hangs in a location of honor in our upper hallway near my study. Alex is author of the remark, on seeing the left-step table without H and He, Z = 1 and 2, and asked where to put them, “It’s a no brainer”. Both of his parents are chemists as are three of his grandparents and assorted aunts and uncles, brain-washed, as you might suppose, regarding He and Be. Henry

As for myself, my undergraduate chemistry training allowed me to work in labs whenever money was tight. I even have my name on a patent for synthesizing arachidonic acid inhibitors for a pharmaceuticals firm in the ’80′s, and my last lab position was doing adhesives research.

The thinking was more fun than being a pair of hands, but mostly this was quashed by superiors. Because of this that last firm missed out on getting in on wound glues, at a time when they were focussed on clipping fractions off cents per pound costwise. Shortsighted would be putting it mildly. I remember telling them about the tetrahedral PT configuration too- the only response was derision.

Jess Tauber