An essay on periodic tables

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 [10]. For one, point-like nucleus this diving would already take place at Z=137. For the earlier literature on this question, see [3], p. 162.

Relativity vs. QED

It is well-known that the (Dirac) relativistic effects contract and stabilize the ns and np* (=np_1/2) shells while the ensuing indirect relativistic effects expand and destabilize the d and f shells. As previously discussed [5], 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 [11] 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 [12]. The atomic ground state does not always explain the molecular outcome. Moreover, remember that electron correlation can make the concept of electron configurations diffuse.

‘Secondary periodicity’

Biron [13] 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 [14].

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 [15] 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 [16].

The inert-pair effect

Sidgwick [17a] called attention in 1933 to a decrease of the main oxidation state by two units for 6^th-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].

‘False friends’

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 [18]. For a comparison of aurides with other ‘halides’, see also [19].

Going one step left from gold, solid Cs₂Pt and other Pt^(−II) compounds were studied in the group of Jansen [20]. In molecules, in addition to σ bonding, also analogous 2pπ and 5dπ bonding was identified between OCO and PtCO, respectively, leading to multiple bonding [21].

In the uranyl-like isoelectronic series, OUN⁺ and OUIr⁺ were found to have similar triple bonds [22] and the latter species was later produced in mass-spectroscopy [23]. These later chemical analogies were initially unexpected.

Nuclear stability

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. [24].

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.

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 [4]. This mnemonic device is usually called the Madelung or (n+l, n) rule [31], although its shape appears to be first presented by Janet [32a] or actually Sommerfeld [32c].

Fig. 3:

The ‘Madelung rule’ for filling atomic orbitals up to Z=172, corresponding to Fig. 1 (reproduced from Pyykkö [4]).

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 l_max and this degeneracy of all levels with the same n was discussed by Fock [33] using momentum-space wave functions in a four-dimensional space. Ostrovsky [34] 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. [35] 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 [36] or Goeppert Mayer [37], and various versions have been included in textbooks, such as Sommerfeld [38], Gombás [39], or Landau and Lifshitz [40]. Some examples on these studies are [41], [42]. 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 [37] 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

with the pre-coefficient chosen by Landau and Lifshitz [40]. 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 [43] 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 [44], Demkov and Ostrovsky [45], Odabasi [46], Ostrovsky [47], Essén [43], Katriel [48], Kitagawara and Barut [49], Novaro [50] and Kibler [51] 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 [52].

Concluding, of the existing literature on the PT, we would still like to remind the reader of the articles by Schwarz [53], [54], [55] and of a factor-analytical search for chemical similarities by Leal and Restrepo [56].