After a compact history of the PT, from Döbereiner’s triads to the theoretical predictions up to element 172, a number of particular issues is discussed: Why may Z = 172 be a limit for stable electron shells? What are the expected stability limits of the nuclear isotopes? When are formally empty atomic orbitals used in molecular electronic structures? What is ‘Secondary Periodicity’? When do the elements (Ir, Pt, Au), at the end of a bond, simulate (N, O, I), respectively? Some new suggestions for alternative PTs are commented upon. As a local connection, Johan Gadolin’s 1794 analysis of the Ytterby mineral is mentioned.
A periodic table of elements (PT) arranges chemical elements as a function of their properties – how so? Any student might answer: by their nuclear charge, Z. Currently the elements with Z=1–118 have been found in nature or artificially produced. What did people use in the 19th century, before they knew about nuclei or nuclear charges? Atomic weights, m. Except for a few anomalies, arrangement by m gave the same running order towards heavier elements, as by Z.
History in a nutshell. The periodic behaviour of chemical elements slowly became apparent in fragments, perhaps beginning with Döbereiner’s triads, such as (Ca, Sr, Ba) in 1817 or (Li, Na, K), (S, Se, Te) and (Cl, Br. I) in 1829. In each of these triads, the m of the second element is approximately equal to the average of the first and third.
In 1843, Gmelin had a table of 55 elements, with oxygen in the correct place. Following the spiralized ‘telluric screw’ (de Chancourtois 1862) and the Law of ‘octaves’ (Newlands 1863, 1865), Meyer constructed a square table of 28 elements (with gaps) in 1864. In 1869, Mendeleev wrote two articles (one in Russian and the other – a short summary – in German) explicitly predicting the existence of three missing elements with the atomic weights 45, 68 and 70. These were discovered in 1879, 1875 and 1886, and are now known as scandium (Sc; atomic weight 44.956), gallium (Ga; atomic weight 69.723) and germanium (Ge; atomic weight 72.640), respectively.
Mendeleev’s articles (1869) also overlapped with the writing of his textbook Fundamentals of Chemistry (see Kaji ).
In 1900 Ramsay suggested that the new (nearly) noble gases should form a separate group, which is currently referred to as Group 18. Similarly, in 1945, Seaborg proposed that the newly discovered actinides should form their own row below the lanthanides. He purposely chose the elemental names europium, americium (Eu, Am), gadolinium, curium (Gd, Cm), and terbium, berkelium (Tb, Bk) to emphasize the (4f, 5f) analogy by selecting for these actinides names of a continent, a celebrated scientist, and a town.
Accelerator experiments have now completed the 6d series Rf–Cn and the 7p series Nh–Og, resulting in the currently accepted PT (highlighted in yellow in Fig. 1).
The theoretical predictions (shown in white in Fig. 1) by the present author ,  support the idea of two 8 s elements (E119 and 120), an overlapping ‘grey’ zone of additional shells (8p, 7d, and 6f) at E121–124, and then a systematic sequence of increasing 5g occupation numbers for the elements E125 onwards. Nominally, all of the elements E121–138 are assigned to a ‘5g’ series. As a parallel case, recall that we regard Th as an actinide, although the free thorium atom has no 5f orbital occupation. Accordingly, a name pre-f was recently introduced .
As a further data point supporting Fig. 1, Indelicato et al. [6a] find an 8s28p25g18 configuration for the cations E1433+ to E1488+. The atomic calculations on the superheavy elements (SHE) range from the Dirac-Slater (DS) ones by Fricke et al. [6b], and the Dirac-Fock (DF) ones by Desclaux [6c] to massive multiconfiguration DF (MCDF) approaches, such as [6a], and sophisticated many-body approaches [6d], [6e]. As an ultimate example, see  below. As to molecules, theoretical calculations on hypothetical octahedral hexafluorides, MF6, support the onset of the expected 5g occupation from E125 to at least E129. Note that the metal atom, M, delivers six electrons to the six fluorides. The other valence electrons go to the 5g shell. These molecular relativistic density-functional calculations were reported by Dognon and Pyykkö 
For more comprehensive treatises on the history of the PT, see Gordin , Kaji  or Scerri .
Why must Z be ≤172?
The current border of the PT at Z=118 is set by the nuclear instability of the existing isotopes and by their small nucleosynthetic cross sections. For discussing chemical properties, even if the nuclei existed, there may also be limits, arising from the chemical reactivity of the vacuum in strong Coulomb fields, due to quantum electrodynamics, QED.
For any elements beyond E172, or so, the lowest or 1s shell would dive to the lower, positron-like continuum of the Dirac equation. It is not yet fully understood what would physically happen. Another way to study the question is to consider heavy-atom collisions . For one, point-like nucleus this diving would already take place at Z=137. For the earlier literature on this question, see , p. 162.
Relativity vs. QED
It is well-known that the (Dirac) relativistic effects contract and stabilize the ns and np* (=np1/2) shells while the ensuing indirect relativistic effects expand and destabilize the d and f shells. As previously discussed , the QED effects, dominated by the vacuum fluctuations (the zero-point oscillations of the electromagnetic field), cancel about −1per cent of the previous effect, for the heavier elements. The other lowest-order contribution, of opposite sign, is vacuum polarization. One could say that the Dirac-Fock-Breit Hamiltonian is ‘101 percent correct’ (cp. [6a]). The QED effects can be seen in accurate quantitative comparisons but have so far not led to qualitative chemical changes.
Which orbitals to use in chemistry?
The chemical behaviour of the elements in Fig. 1 is mostly driven by the orbitals, occupied in the atomic ground state, and given in the right-hand marginal. Sometimes also other orbitals, which are unoccupied in the atomic ground state but energetically accessible for bond formation, can participate. Thus we can have the predicted  pre-s Og− anion, the pre-p Be, Mg; Zn, making bonds with their ns+np orbitals, the pre-d Ca, Sr, Ba; Cs and the pre-f Th.
A recent example was the synthesis of [Ba(CO)8], which fulfils an 18-electron rule by using the originally empty 5d shell of the central barium atom . The atomic ground state does not always explain the molecular outcome. Moreover, remember that electron correlation can make the concept of electron configurations diffuse.
Biron  pointed out in 1915 that every second period has specific properties. Taking the Group 15 as an example, along the series (N, P, As, Sb, Bi) (Biron, p. 971), the dominant oxidation states are III, V, III, V, III, respectively. The anomaly at As can be attributed to partial screening by the filled 3d shell. The anomaly at Bi is due to both an analogous partial screening by the 4f shell and to relativistic effects .
Another vertical anomaly is the small radius of every atomic shell (1s, 2p, 3d, 4f, 5g) with a new orbital angular momentum quantum number, l. The author  used the name ‘primogenic repulsion’ for its effect on the higher shells. In Russian literature the term ‘kainosymmetric’ is often used for these ‘first-born’ shells .
The inert-pair effect
Sidgwick [17a] called attention in 1933 to a decrease of the main oxidation state by two units for 6th-period elements, take Pb(II) as an example. A first explanation would be the relativistic 6s stabilization. Closer studies involve the hybridization of the metal (6s, 6p) orbitals with the ligand np orbitals [17b].
The gold atom is almost as electronegative as iodine; we can see its outermost shell as either a 6s1 electron, or a 6s−1 hole. A wide chemistry of the auride ion, Au−, is known . For a comparison of aurides with other ‘halides’, see also .
Going one step left from gold, solid Cs2Pt and other Pt(−II) compounds were studied in the group of Jansen . In molecules, in addition to σ bonding, also analogous 2pπ and 5dπ bonding was identified between OCO and PtCO, respectively, leading to multiple bonding .
In the uranyl-like isoelectronic series, OUN+ and OUIr+ were found to have similar triple bonds  and the latter species was later produced in mass-spectroscopy . These later chemical analogies were initially unexpected.
The chemical predictions quoted here are based on theoretical, relativistic quantum chemical calculations using established electronic Hamiltonians. The nuclei are simply assumed to exist, with a realistic, finite nuclear size. The synthesis of heavier nuclei, up to E118 (Oganesson), is demanding. The most recent nuclear syntheses were completed in a friendly collaboration between laboratories in Oak Ridge and Dubna. The lifetimes of these nuclides are short; for example, the present Og isotopes have lifetimes below a millisecond. The most challenging production bottleneck is, however, not the short lifetime but the small nucleosynthetic cross-section for these elements. If the experiment runs for a year and yields less than a handful of desired product nuclei, this creates an obvious problem, even with nearly 4π detection (all scattering directions seen) and an almost noiseless apparatus. The current situation on superheavy elements is discussed by Giuliani et al. .
A quantum chemist can always assume a finite nucleus of realistic size for any Z<173 and do ab initio calculations for that theoretical model, whether or not such nuclei or their compounds are ever made. One even could claim that some general conclusions could possibly be drawn, such as the possible existence of a 5g series or the vast spin-orbit effects.
In defence of the current PT: what are the choices?
Figure 1 illustrates a recent choice of IUPAC PT layout (highlighted in yellow). Note the placement of all lanthanides and actinides in Group 3. One could, however argue certain points:
H is now in Group 1, because it is often manifested as H+1, or neutral. If an emphasis was placed on hydrides (H−1), one could argue for having hydrogen also in Group 17.
The noble gas He is now in Group 18 with the other noble gases. Apart from spectroscopic species, like the astrophysical, diatomic HeH+, or high-pressure compounds, like Na2He , helium is still a very noble gas. As stressed by the authors, this compound is a sodium electride containing He(0). If one would like to emphasize heliums 1s2 electronic structure, one could also have it in Group 2. Its chemical behaviour is, however, not that of an alkaline earth. Note the easy ns-np-(n-1)d hybridization among heavier Group-2 elements, lacking for helium.
How long f-element rows? One now has a 15-element lanthanide (Ln) row from La to Lu. All of these elements are (mostly) trivalent. Their ionic radii or ionization potentials exhibit a systematic relationship along the series. It is entirely plausible to count from 4f0 to 4f14, and to leave a hole in Group 3 of Period 6. Moreover this completely avoids the heated argument on which end should one cut off – La or Lu. A clear advantage is then having all these, mostly trivalent, rare earths in the single Group 3, corresponding to three valence electrons.
Impressive experiments and computations  have recently verified that a free Lr atom has a 7s27p1 configuration, different from the 6s25d1 for a Lu atom. Computations for a handful of molecules, however, find a complete analogy between Lu and Lr , see Fig. 2. That said, if our PT is to be driven by chemistry, there are no reasons to change Fig. 1.
The 32-column option? Some desirable properties of a PT could be:
The column (or ‘Group’) number g (or g-10) is the maximum number of valence electrons. The oxidation state is then the same number or, counting holes in the spirit of Abegg’s contravalence , the negative number, g-18.
The valence atomic orbitals along a row are constant in a block and equal to those in the right-hand margin.
The nuclear charge, Z, increases systematically towards right.
The shape should be typographically convenient.
Of these properties, Fig. 1 fulfils 1, 2 and 4 but violates 3 for certain superheavy elements when relativistic effects so require. Conversely, the 32-column ‘long-form’ PT favoured by Scerri  (see his Fig. 1) violates (1) by having very many potentially Group-3 columns and may also violate (4). It does satisfy (3).
The ‘Madelung rule’
The order of filling shells in a neutral atom is approximately the one given in Fig. 3. It was extended up to Z=172 in . This mnemonic device is usually called the Madelung or (n+l, n) rule , although its shape appears to be first presented by Janet [32a] or actually Sommerfeld [32c].
Models for reproducing the PT
How to explain this approximate order of level filling in neutral, or nearly neutral atoms? For an electron in a Coulomb field, each new n introduces a new lmax and this degeneracy of all levels with the same n was discussed by Fock  using momentum-space wave functions in a four-dimensional space. Ostrovsky  used coordinate space.
Concerning the physics of many-electron atoms, the Dirac-Fock-Breit (DFB) Hamiltonian, supplemented with some estimate of leading quantum electrodynamic (QED) effects, gives an excellent description. As an example, Pašteka et al.  calculated the ionization potential and electron affinity of a gold atom with milli-electronvolt (meV) accuracy. The bottleneck rather was in the handling of electron correlation. Coupled-cluster methods with up to pentuple excitations were used. Thus no surprises are expected here. Atoms follow Physics.
A much simpler task is the question of the filling order of one-electron levels in some effective-potential model for a many-electron atom, such as a Thomas-Fermi one. This has been tried since Fermi  or Goeppert Mayer , and various versions have been included in textbooks, such as Sommerfeld , Gombás , or Landau and Lifshitz . Some examples on these studies are , . The predictions for filling new shells are quite similar, see Table 1. In a screened-Coulomb potential, the attraction must be sufficient to balance the centrifugal potential for l. This type of reasoning was used by Goeppert Mayer  to discuss the Z where 4f and 5f states are first occupied. For reviews see Ostrovsky below. The T–F treatment yields the Z values for a new l at
|l||Fermi ||Goeppert ||Iwanenko ||Landau ||Essén ||DF |
|3||≥55, 58||≥57; ≥91||58||58||57||58|
with the pre-coefficient chosen by Landau and Lifshitz . The results in Table 1 are in a surprisingly close agreement with more exact results. Note that the T–F potential is just one of the various screened-Coulomb potentials, which do the job. The literature on which T–F potential at which Z starts a given l is broad, see Table 1. Essén  gives for filling the first state of l the nuclear charge
The doubling of periods with two periods of lengths 8, 18 and 32 each has been noticed by many authors, Löwdin , Demkov and Ostrovsky , Odabasi , Ostrovsky , Essén , Katriel , Kitagawara and Barut , Novaro  and Kibler  have discussed possible underlying dynamical symmetries. An alternative is that there is no deep symmetry reason and that one only needs the quoted ‘nuts and bolts’ of the DFB Hamiltonian.
For later reviews on the doubling question, see Ostrovsky .
Concluding, of the existing literature on the PT, we would still like to remind the reader of the articles by Schwarz , ,  and of a factor-analytical search for chemical similarities by Leal and Restrepo .
Did our department contribute to the story?
The new Chemical Laboratory building of the old Kungliga Åbo Akademi (our direct legal predecessor before the removal from Turku to Helsinki in 1828) was inaugurated on 13 April 1764 under Professor Pehr Gadd. The 250th anniversary was celebrated in 2014 by having business as usual.
Gadd’s successor, Johan Gadolin (1760–1852)  published the first chemical analysis of a black mineral from Ytterby (near Stockholm) in 1794 (German version published in 1796). The paper was translated to modern chemical terminology by Pyykkö and Orama .
Starting from this mineral, which was essentially FeBe2Y2Si2O10, he obtained a new ‘earth’, i.e. oxide, essentially Y2O3, possibly with traces of other rare earths. The mineral was later named gadolinite. Although one cannot necessarily claim that Gadolin found a specific element, he did contribute to finding an entire family of 16 elements. Organizing these kept chemists busy for another century. In 1886 one of them was given the name gadolinium (Gd).
Circumstantial evidence from the parallel case of samarium (Sm) by the same authors suggests that Lecoq de Boisbaudran and Marignac considered both the mineral and the eponym, or the person behind its name. For more details on Gd, see  and supplements to it. In 1994 a bicentennial conference, 2-ICFE, was held in Helsinki, with a conference excursion to Ytterby, see Fig. 4.
A collection of invited papers based on presentations at “Mendeleev 150”: 4th International Conference on the Periodic Table (Mendeleev 150), held at ITMO University in Saint Petersburg, Russian Federation, 26–28 July 2019.
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