Janet LSPT Program

Programmatically mine atomic number sequences from Janet Left-Step Periodic Table.

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Instructions

What is computed?

Periodic tables are mental constructs of the Periodic System of Elements (PSE). These are aimed at emphasizing chemical or physical property preferences, to inject the illusion of order.
So far, IUPAC has no official form of the PSE, and an IUPAC-approved form of a periodic table does not exist. The only specific recommendation IUPAC has made concerning the periodic table covers the Group numbering of 1-18 (Leigh, 2009). Some prominent IUPAC members accept that said convention is not ideal for every purpose, that teachers might find that it does not fulfill all their requirements, and that it is not strictly consistent with some notions of electronic structure. But then they stick to tradition, making their case more an appeal to authority issue, or a tribute to old glory days, than anything else.

Over the years, all kind of periodic tables have been published: 18-group labeled, 32-group labeled, unlabeled, rectangulars, triangulars, tetrahedrals, cubic, rounded, stacked, spiral-like, galaxy-like, and even fractal-like (Bradley, 2011; Stewart, 2007). Some of these so-called "tables" are not even tabular arrangements. At the time of writing, there is no consensus on which periodic table is the "correct", "best", or "optimal" one, and IUPAC requisites do not really help.
About this tool
This tool was developed, not to validate any particular version of the periodic table, or to produce one that accounts for electron configuration anomalies, but to investigate whether those already proposed, "optimal" or not, can be modeled with algorithms; i.e., if an abstraction of the PSE can be transmitted as a few lines of code and reconstructed through a computer algorithm.
By doing so, by no means our goal is to reduce chemical periodicity to math operations, but to identify when this could be done. If a model is found, it could be used to generate, store, retrieve, and mine atomic sequences, build new tools based on those sequences, and make predictions about the existence of yet-undiscovered elements and their relative positions in an abstraction of the PSE.
Modeling Periodicity
Although modeling chemical periodicity is not easy, it is not an impossible or illusory task. Actually this has been done in the past, but as exercises for finding triads (Scerri, 2008) and atomic number series, rather than for modeling entire periodic tables or the PSE.
In the quest for finding mathematical frameworks for modeling atomic number sequences, we have identified a few interesting ones. One is described by Tsimmerman (2012); the others by Garai (Judge, 2009a, 2009b; Garai, 2003, 2008a, 2008b, 2010, 2011a, 2011b, 2015), and Perez (1997, 2015a, 2015b, 2015c). These authors kindly sent us copy of their work. Our current tool is based onTsimmerman's. Research work along other frameworks is currently under way.
Tsimmerman's Framework
Tsimmerman (2012) has described a mathematical expression of Mendeleev's Periodic Law that generates atomic number (Z) sequences. This approach consists in first computing atomic numbers (A) of alkaline earth metals from period numbers (p), and then adding group numbers (g), like this:
Z = A + g

A = (1/6)[(p+1)³ + (1/2)(p+1) + (3/2)(p+1)(-1)^p-1]

where g is a value with boundaries

(1 - L) ≤ g ≤ 0

L = (1/8)[2p + 1 + (-1)^p-1]²

p = 1,2,3...

and where L is the length of a period.

The convention adopted here is to define g as a negative value, in order for the periodic table to be left step.
Framework Evaluation
We have written a computer program that evaluates the above expressions by looping over "p" (p = 1, 2, ...) and "g" (g = 0, -1, g = -2, ...) values. Figure 1 shows its raw output.

Figure 1. Program raw output.

When we organized the output shown in Figure 1 in a tabular form and shifted all cells to the right, the result was a Janet Left-Step Periodic Table (Wikipedia, 2015a; 2015b; Siegfried, 2015; Stewart, 2009). See Figure 2.

Figure 2. Program output resembling Janet table.

Consequently, we can transmit a left-step representation of the Periodic System of Elements, not by sending a file, but by executing a few lines of codes. The program can then be used as a base framework for building chemistry tools on top of it.
Output Labeling Conventions
We color-coded table cells and did not show element symbols to grab the readers attention to the spdf-blocks. Why? Please keep reading.
After running the program, you may realize that the program labels columns from right to left, starting not at g = 0, g = -1, g = -2..., but at g = 1, g = 2, g = 3, and so forth. Essentially the program rescales the indices by applying the affine transformation g = a*g + b, where a = -1 and b = 1; i.e., by multiplying g label values by -1 and adding 1. No reordering was involved.

Said rescaling avoids any reference to a "group" 0 of elements and "negative groups" of elements. This is a cosmetic change.

It is true that, to conform to IUPAC numbering convention of groups, we could have relabelled the first two columns at the right to read g = 2 and then g = 1. We did not do that for two reasons:
1. The output of our program is what it is. It orders chemical elements according to Tsimmerman's framework; no more, no less.
2. We believe that simply relabeling the first two right columns is a subjective exercise based on a chemistry pre-knowledge, and one that defeats the gist of our research: The testing of mathematical models that attempt to make predictions.
On the other hand, we want to acknowledge IUPAC accepted labels, so we finally settled for a happy medium and included them after a forward slash. These are the red labels that you see at the top-right corner of the table.

However, we stick to the thesis that whether IUPAC or chemists reorder or label periods and groups of the PSE according to their preferences, or chemistry preknowledge, is a subjective exercise after the facts.

Some proponents of Janet's LSPT adopt an IUPAC-like numbering system, but then hold back and do no label columns after the 18 mark, failing to be consistent with their own mental construct of equating columns to groups.

To address these notation issues, and other usability issues, we have implemented several modifications to the program. These are detailed below.
Recent Modifications
As of January 1 of 2016, we added the following alternative system which consists in labeling groups based on the filling of outermost subshells according to Madelung Rule; i.e., by assuming that atoms are free neutral particles in vacuum (Wikipedia, 2015b).

f¹f²f³f⁴f⁵f⁶f⁷f⁸f⁹f¹⁰f¹¹f¹²f¹³f¹⁴d¹d²d³d⁴d⁵d⁶d⁷d⁸d⁹d¹⁰p¹p²p³p⁴p⁵p⁶s¹s²

We can expand this system in a straightforward manner to label groups of superheavy and not yet discovered elements, up to Z = 220 (Pershina, 2009; Saito, 2000; Umemoto and Saito, 1996).

The notation used is more descriptive and convey more information. For instance, any reference to the "s¹ group" means exactly this:

"The group of chemical elements with one electron in their outermost s subshell."

Many teachers and chemists will find that labeling groups in this way is more consistent with their notions of electronic structure.
Previous Modifications

A form-based user interface was initially added so users can retrieve specific Z sequences of known and not-yet discovered elements.

As it is now, the program output can be used to easily generate Janet periodic tables and other types of periodic tables. For instance, we can map atomic numbers to chemical symbols and label columns using IUPAC 18-group numbering convention, if we wish to do so. In this way we can reproduce one of the Janet periodic table versions referenced by Leigh (Leigh, 2009). We can also use the program to develop interactive tools on top of it and that accept atomic numbers as their input.
Remarks
Janet Left-Step Periodic table organizes elements according to orbital (subshell) filling. The order in which these are filled is given by the n + l rule, also known as the Madelung, Janet, or Klechkowski Rule. Adomah PT is also essentially the Madelung Rule, and even more so than Janet's LSPT.
However, as indicated in a recent tutorial, the Madelung Rule assumes a non-realistic scenario wherein atoms are considered free neutral particles in vacuum with no interactions with other atoms or themselves. Obviously, the electron configuration of a chemically bonded atom can be different from the one predicted by the Madelung Rule. To Madelung's credit, his rule is only good for groups 1 and 2 (i.e., s- and p-block elements).

Diagrams describing Madelung Rule have been used as mnemonics for writing electron configurations, can be found elsewhere, and look like this

Figure 3. Electron filling diagram.

So it is not surprising that rotating by 180 degrees Figure 2 produces a shape that, statistically speaking, resembles those diagrams:

Figure 4. Inverted Janet-like table, with content removed to highlight its similarity with Figure 3.

Of the several two-dimensional mental constructs of the PSE currently available, this one resembles, at a large scale level, the subshell filling pattern observed at atomic scale levels, with element anomalies ignored (Scerri, 2012a; 2012b; 2013). The notion of statistically similar patterns emerging at different length scales of observations is the signature of fractal geometry.

Figure 4 can be used to help students understand the connection between electron orbital filling and Janet LSPT. For those that still want to follow IUPAC, they can use their 18-group numbering system. They can also use a shorter version by placing lanthanoides and actinoides as separate sections resembling footers or headers, at the expense of diluting the above self-similarity argument, like this

Figure 5. Inverted Janet-like tables with sections resembling footers and headers.

If these flipped versions are not visually appealing to teachers or students, or do not satisfy their needs, another alternative is to keep the original orientation and flip instead Figure 3, the electron filling diagram.

Note. We proposed these ideas back in November of 2015, when this tool was first developed. In April of 2019, a study suggested the very same thing, but for the traditional version of the periodic table (Poliakoff, Makin, Tang, & Poliakoff, 2019).

Last, but not least, we want to emphasize that Madelung Rule is an idealized limiting rule; hence, it cannot be applied arbitrarily to all atoms of the periodic table, and certainly, in most cases, not to chemically bonded atoms—for that you need to use the Rydberg Rule which also maximizes (n + l) values.

To write empirically correct electron configurations, we now recommend Madelung Rule for groups 1 and 2, and Rydberg Rule for groups 3 to 18 (according to the IUPAC numbering system). The tutorial A novel mnemonic for the Rydberg Rule explains the conceptual differences between these two rules and provides useful mnemonics for properly writing electron configurations.

Acknowledgements

Updates

References

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