Paradigms and paradoxes: the ionization potential of atomic astatine (Z =85), polonium (Z = 84), and some other elements—what does this value tell us about the energetics of atomic and diatomic halogens

The newly measured ionization potential of atomic astatine is discussed and compared with that of the recently determined value for polonium and for the other atomic halogens. Regularities in these atomic values are discussed and applied to the understanding of the energetics of diatomic halogens. Some surprises remain.


Introduction
Let us talk of species with chemical simplicity. Atoms are simpler than molecules. The simplest of elements are those for which s and p orbitals are "adequate" for their description of their atoms and for the formation of diatomic and larger molecules therefrom. We recognize these simplest elements as "main group," "non-transition," or "s-" and "p-block," as opposed to the more complicated "transition metals" or "dblock" elements, never mind the "lanthanides (rare earth) and actinides" or "f-block." In the current study, we limited our discussion mostly to the halogens, F, Cl, Br, I, and At.
Among the uniquely defined and conceptually simplest properties of any element is its ionization potential (IP), the least amount of energy to remove one electron from the gaseous atomic species. This quantity is formally a simple chemical property and thus is definitionally not affected by compound formation, solution, salt, and any other intra-or intermolecular environment. Homonuclear diatomics (F 2 , Cl 2 , Br 2 , I 2 , At 2 ) and the related ions formed by the loss of an electron to form the related radical cation are among the simplest molecules to understand. Recently, HOMO-LUMO energy difference was related to between the IP and electron affinity (EA) difference to assist in deducing the colors and chromophores in elemental nonmetals focusing on compounds with lone pair bonds and σ electrons (X = F 2 , Cl 2 , Br 2 , I 2 , S 8 , P 4 ) [1]. The EAs of the halogens were discussed as part of the understanding of the oxidizing powers of these elements [2].
In this study, we seek regularities in the molecular ionization potentials, in particular for the understanding of the chemistry of high Z elements. All energy quantities are given in electron volts, where we remind the reader that 1 eV ≈ 23.06 kcal mol -1 ≈ 96.5 kJ mol -1 .

Atomic ionization potentials: measurements and regularities
The measurements of the ionization potential of the majority of the main group elements have been made and then have been collected and presented in a single, quite brief, now 50-year-old database [3]. In this now classic compendium, there are but two non-transition elements that are lacking their measured ionization potentials. These are the heavy halogen, astatine (At) with atomic number (Z = 85), and the even heavier alkali metal (Fr) with Z = 87. Quite recently, this long-term missing value for astatine has been experimentally determined to high precision as 9.31751 ± 0.00008 eV [4]. The literature value of the ionization potential of astatine's neighboring elements, polonium (Po) with Z = 84, is even more recent and even more precise presented as 8.418072 ± 0.000015eV [5], while that of francium (Fr) with Z = 87 the likewise highly precise as 4.0712 ± 0.00004 eV [6].
Soon after the publication of the aforementioned database ago [3], one of the current paper's authors (JFL) published a brief note presenting simple numerical regularities for the values of the ionization potentials of the main group elements [7]. One such pattern asserts that for a given row in the periodic table, the sum of the ionization potential for the relevant alkali metal (group 1) and of the corresponding halogen (group 17) is nearly equal to those for the alkaline earth (group 2) and that of the chalcogen (group 16) elements. The rule can be written as For example, IP(Li) + IP(F) ≈ IP(Be) + IP(O). Numerically, the two sums equal 22.81 and 22.91 eV, respectively. From the use of this approximation, we therefore suggest that Using the suggested ionization potentials of all four elements from the above sources [3][4][5], we derive the desired sums IP(Cs) + IP(At) and IP(Ba) + IP(Po) as 13.21 and 13.63 eV, respectively. The difference of the two sums is 0.42 eV and exceeds the combined uncertainties from experiment. Nonetheless, it is but a 3% difference, encouragingly small give how little effort was needed to derive the sums of interest.
This encourages us now to consider the sum corresponding to that of the neighboring (inter-row) and even higher atomic number elements of the periodic table. In the particular, let us consider IP(Fr, Z = 87), IP(Ts, Z = 117), IP(Ra, Z = 88), and IP(Lv, Z =116). More explicitly, we would expect IP(Fr) + IP(Ts) ≈ IP(Ra) + IP(Lv). Neither surprisingly nor disappointingly, the input values for the ionization potentials of the high Z elements (Z =116 and Z =117) remain unmeasured, and thus we accept the high-level calculated values of 6.855 and 7.654 eV, respectively [8]. The experimental values of 5.279 eV [3] and 4.0712 ± 0.00004 eV [6] were taken for IP(Ra) and IP(Fr). The two sums are 11.73 and 12.13 eV, respectively. The difference of the two sums is 0.40 eV which exceeds the uncertainties from experiment but less than 4%, again encouragingly small.
The aforementioned rule (Eq. 1) can be extended [7] to isoelectronic and thus even more cationic species. For example, using this rule and recognizing Be + is isoelectronic with Li and Ne + with F, B + with Be, and F + with O, we can correctly suggest IP(Be + ) + IP(Ne + ) ≈ IP(B + ) + IP(F + ). The two sums are 59.17 and 60.13 eV, respectively. The sum regularity so continues through at least the 17th ionization potentials of elements with appropriate atomic numbers.
What about anions where the relevant energy quantity is the electron affinity (EA, binding energy of an added electron, IP(X -) = EA(X))? What about the corresponding sums (IP(He -) + IP(O -)) and (IP(Li -) + IP(N -)), quantities more often written as (EA(He) + EA(O)) and (EA(Li) + EA(N)), respectively? Both Heand Nare unbound relative to the loss of an electron to form the neutrals He and N: the electron affinities (EA) of these atoms are negative [9], and so we are thwarted in the application of our regularity.
Using Hess' law, we derive the difference of the dissociation energy of a species [X 2 ] + and X 2 as the difference of the ionization potentials of X and X 2 . Accordingly, from the numbers given earlier in this study, we find that the dissociation energy of [F 2 ] + is 1.72 eV higher than that of F 2 . For the corresponding species with Cl, Br, and I, the differences of the cation and neutral are 1.42,1.29, and 1.15 eV, respectively. The diatomic cation has a dissociation energy higher than that of the corresponding neutral is sensible. The ionization process corresponds to the "removal" of an antibonding π* electron from the neutral. It may be said that the diatomic [X 2 ] + species have one fewer atom with 8 electrons than the diatomic neutral X 2 species and so, [F 2 ] + gains stabilization relative to F 2 consistent with the logic of Politzer [14].
In relation to ionization potentials of the dihalogens, we now consider the electron affinities and the corresponding anions [X 2 ] -. The electron affinity equals 3.01 eV for F 2 [22], 2.50 eV for Cl 2 [23], 2.62 eV for Br 2 [24,25], and 2.52 eV for I 2 [26]. In other words, the electron affinity of the dihalogens decreases in the order F 2 > Cl 2 ≈ Br 2 ≈ I 2 . This seems reasonable until it is remembered that the electron affinities of the atomic halogens follow the order F (3.40 eV) < Cl (3.61 eV) > Br (3.36 eV) > I (3.06 eV. In all cases, the electron affinity of the diatomic halogen is smaller than that of the atomic constituent. Thus, the bond energy for [X 2 ]is smaller than that of neutral X 2 . This is a logical consequence of a half-occupied antibonding ο* molecular orbital in the anion that was hitherto unoccupied in the neutral diatomic. Said differently, the formal reaction is exothermic by 0.39, 0.99, 0.84, and 0.54 eV for F, Cl, Br, and I, respectively. This reaction has the least exothermicity for X = F. From these values, we deduce that completing an octet for atomic fluorine, cf. F -, provides less stabilization than the other atomic halogens, cf. Cl -, Br -, I -. This finding is consistent with Politzer [14] and also our earlier analysis made during discussion of the energetics of atomic and diatomic halogen cations.

Conclusion
We conclude our paper on the energetics of high Z atoms (Z = 84 and Z = 85 and their even higher Z congeners) and of small and simple halogen-containing ions (in the particular, monoatomic, and diatomic species). Simple additivity regularities for ionization potentials have been shown in this study. Fluorine paradoxes were discussed, e.g., lower EA of F (3.401190 ± 0.000004 eV) in comparison to that of Cl (3.612642 ± 0.000027 eV) and less stabilization on completing an octet for atomic fluorine, cf. F -, than the other atomic halogens, cf. Cl -, Br -, I -. Thus, the study of small and simple species is rightfully recognized as wonderfully rich and complicated.
Funding MPS gratefully acknowledges the Slovenian Research Agency (ARRS Grant P1-0045, Inorganic Chemistry and Technology) for financial support.

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Declarations We did not perform any experiments when preparing this article, so neither ethics review nor informed consent was necessary.
Consent to participate All authors agreed with participation in research and publication of the results.