Physical Origin of Chemical Periodicities in the System of Elements

Pure Appl. Chem. 2019; 91(12): 1969–1999

Conference paper

Chang-Su Cao, Han-Shi Hu, Jun Li* and W. H. Eugen Schwarz* Physical origin of chemical periodicities in the system of elements https://doi.org/10.1515/pac-2019-0901

Abstract: The Periodic Law, one of the great discoveries in human history, is magnificent in the art of chemistry. Different arrangements of chemical elements in differently shaped Periodic Tables serve for different purposes. “Can this Periodic Table be derived from quantum chemistry or physics?” can only be answered positively, if the internal structure of the Periodic Table is explicitly connected to facts and data from chemistry. Quantum chemical rationalization of such a Periodic Tables is achieved by explaining the details of energies and radii of atomic core and valence orbitals in the leading electron configurations of chemically bonded atoms. The coarse horizontal pseudo-periodicity in seven rows of 2, 8, 8, 18, 18, 32, 32 members is triggered by the low energy of and large gap above the 1s and nsp valence shells (2 ≤ n ≤ 6 !). The pseudo-periodicity, in particular the wavy variation of the elemental properties in the four longer rows, is due to the different behaviors of the s and p vs. d and f pairs of atomic valence shells along the ordered array of elements. The so-called secondary or vertical periodicity is related to pseudo-periodic changes of the atomic core shells. The Periodic Law of the naturally given System of Elements describes the trends of the many chemical properties displayed inside the Chemical Periodic Tables. While the general physical laws of quantum mechanics form a simple network, their application to the unlimited field of chemical materials under ambient ‘human’ conditions results in a complex and somewhat accidental structure inside the Table that fits to some more or less symmetric outer shape. Periodic Tables designed after some creative concept for the overall appearance are of interest in non-chemical fields of wisdom and art.

Article note: 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.

*Corresponding authors: Jun Li, Department of Chemistry, Theoretical Chemistry Center, Tsinghua University, Beijing 100084 China; and Department of Chemistry, Southern University of Science and Technology, Shenzhen 518055 China, e-mail: [email protected]; and W. H. Eugen Schwarz, Department of Chemistry, Theoretical Chemistry Center, Tsinghua University, Beijing 100084 China; and Physical Chemistry Lab, S&T Faculty, Siegen University, Siegen 57068 Germany, e-mail: [email protected] Chang-Su Cao and Han-Shi Hu: Department of Chemistry, Theoretical Chemistry Center, Tsinghua University, Beijing 100084 China

Open Access. © 2019 Chang-Su Cao et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 Public License. 1970 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements

Keywords: atomic orbitals; chemical concepts; chemical elements; chemistry under ambient conditions; Mendeleev 150; periodic table for chemical science; periodic table, inner structure; quantum chemistry.

Introduction

To begin with, we analyze important chemical observations and draw conceptual conclusions on the ‘Natural System’ of ‘Chemical Elements’. Then, we work out what may be meant by ‘Periodicity of Elements’ and by the question “Can the ‘Periodic System’ of Chemical Elements be derived from basic physics”. The actual variations of various general properties of elements in chemical compounds under ambient conditions of temperature and pressure along an appropriately ordered array of elements are displayed in ‘Chemical Peri- odic Tables’. These property variations determine the inner structure of a Chemical Periodic Table, and that has to be derived quantum chemically. We present several examples of how variations of empirically derived properties can be derived quantum theoretically. On this basis we summarize on the Physical Origin of the Periodicities in the Natural System of Chemical Elements and comment in this context on several conceptual issues, brought up in the communities of chemists, educators, philosophers and artists. The latter ones create new outer shapes of Periodic Tables with apparent symmetries that are independent of its inner structure that is coupled to the chemical facts.

Historical and conceptual background of the natural system of chemical elements

Chemical elements and their similarities

With the 118 now known elements, the concept of element has evolved through a long history. During the early period of enlightenment in ancient Greece, various rational, though speculative philosophic concepts of elements were proposed by Demokritos and Lucretius, different ones by Plato, Aristoteles and others [1, 2]. After the beginning of the second period of rational enlightenment, i.e. in early modern Europe, and after the combination with the experimental scientific method, English Robert Boyle [3] formulated an empirically based concept of chemical elements around 1660, where he defined the elements as the conserved entities in chemical transformations, in the laboratory. As scientists, we assume the same nature working inside and outside the laboratory. More than a century later in the 1780s, just before the chaotic political revolution, and by a peaceful evo- lutionary scientific jump, French Antoine Lavoisier with his wife Marie-Anne Paulze and a group of Parisian

Fig. 1: System of Elements of Gmelin [6], redrawn by Leach [16]. In addition, we have marked the groups on the left and right sides, which are now called halogens, chalcogens, pnicogens and alkali and alkaline earth metals, forming the empirically given ‘backbone’ of the later Periodic System. The empirically unique H and 2nd row elements (Be–O, baptized типичecкий by Mendeleev) [17–19], are here highlighted by bold boxes. Neither Leopold Gmelin nor Lothar Meyer [20] had found a systematic arrangement for the plenty of chemically more similar transition metals, here at the middle bottom; only Mendeleev achieved it in 1869 [17, 18, 21]. The invention of Periodic Tables focused the view of subsequent generations more on kinships within the groups, and also between adjacent groups, but less so on other relations among similar elements. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1971 scientists implemented Boyle’s concept into practical reality [4]. The group identified more than 30 elements [5]. In subsequent decades, further elements were discovered. In the early 1840s, German Leopold Gmelin [6] arranged more than 50 elements in a two-dimensional table with groups of chemically similar elements by exploiting common qualitative viewpoints of the practicing chemists (Fig. 1). Basic parts of the structure of the Natural System of Chemical Elements were already found thereby, with the groups of the electropo- sitive metals on one side and of the electronegative non-metallic halogens, chalcogens and pnicogens (in present day parlance) on the other side. While these similarity clusters of elements were based on ‘chemical feeling’, it was only recently (in the 2000s) that several groups of mathematical chemists [7–15] verified and refined those chemical similarity concepts. By applying chemo-metric approaches to big empirical data sets of molecules, the felt similarities between elements could be partially or fully quantified.

The meaning of ‘chemical element’

The word element or its synonyms were used in various cultures since antiquity for two different concepts. The first meaning was some real, simply composed matter, such as liquid water or yellow sulfur. The second meaning was a principle or an abstract substance that is the dominating component in the real matter, and is conserved in some sense in chemical transformations of real compounds. Mendeleev was one of the first scholars in modern times to stress this difference and systematically distinguished the two meanings [21, 28]. Accordingly, the IUPAC [29] defines an element (i) as the concept of an abstract species in a compound with “all atoms with the same number of protons in the atomic nucleus”, and (ii) also as a real, elementary or simple substance, i.e. as a “pure chemical substance composed of atoms with the same number of protons”. Referring to the definition of the abstract elements, it is sometimes argued that an element has no properties except the given number of nuclear protons. We here hold that the physical nuclear charge number Z specifies (defines) an element, which carries many further ‘secondary’ chemical properties. Some properties have well-defined integer values. Examples are Z itself, but interpreted by the chemists as the number of electrons per neutral atom; the number of core electrons Ncore, determining the number of valence active electrons Nval = Z – Ncore, which are energetically well above the closed, chemically inert core shells; Nval determines the group number g in a Chemical Periodic Table. Some properties of the elements lie within narrow real intervals, such as the averaged atomic masses of the isotopic mixtures common on earth, which are given to us through astrophysical history [30]. There are also properties defined in a more fuzzy manner, namely the averaged properties of atoms in different compounds, such as the electron-attracting power or electronegativity; the average atomic increments for interatomic distances, meaning the various effective covalent, ionic and van der Waals radii; the magnetic susceptibility increments; etc. There are even properties of the elements which are only defined by a loose wording such as being biologically toxic, or being of metallic, metalloid or nonmetallic character, or being solid, liquid or gaseous as a simple substance at ambient conditions. They all may be displayed in Chemical Periodic Tables [31–37].

Chemical elements and their ordinal numbers

Sicilian Stanislao Cannizzaro [38] had refined some earlier ideas from the 1810s of Italian Amedeo Avoga- dro and French André-Marie Ampère about the relations of atoms and molecules in gases, [24] and popu- larized them at the first international scientific congress (on concepts in chemistry) in Karlsruhe in 1860 [39, 40]. From then on it was possible to distinguish equivalents, atoms and molecules, derive the correct empirical and molecular formulas, and arrange the chemical elements on a well-defined linear array of real-numbered atomic masses A. In the early 1910s, Dutch lawyer and amateur scientist Antonius van den Broek [41] suggested the nuclear charge as the integer ordinal number Z for the elements. When 1972 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements

English Henry Moseley [42] experimentally determined and analyzed the atomic inner shell X-ray spectra, and Danish Niels Bohr [43] invented his atomic model, the empirical and theoretical basis for the element number was found as the number of protons in the nucleus for the physicist, and as the number of electrons of the neutral atom for the chemist [44]. This reflects a conceptual difference: a ‘physical element’ is a stable or slowly or even quickly decaying nucleus; a ‘chemical element’ contains atoms consisting of nuclei and electrons, and the ‘chemical’ nuclei should even live long enough to form stable molecules in their electronic ground states. A minimum lifetime of 10−14 s suggested as a compromise of nuclear physicists and chemists [45] appears too short to obtain chemical properties of the element in respective molecules. Z has proven useful as an ordinal number to order the nuclei on an isotopic chart, and also to obtain a chemically meaningful order of the chemical elements known so far (i.e. up to Z = 118; concerning higher Z values, see below).

Combination of the two concepts of order and similarity gives the periodic law

Since the early 19th century, various chemists had observed qualitative similarities of static and reactive properties among “groups” of elements, and also some quantitative relations between the atomic equivalent masses labelled the triad and the octave rules [26, 27, 46, 47]. Further, in the 1860s, several scholars observed that basic properties exhibited pronounced variations along the array of atomic masses A in a pseudo-periodic manner. This holds for the typical and the extremal valence and oxidation numbers, which were inde- pendently chosen as main criteria for grouping the elements by German Lothar Meyer and by English John Newlands in 1864 and by Russian D. Mendeleev in 1869 [17–21]. The qualitative electro-negativities or elec- trochemical potentials (discussed already in the early 19th century, among others by Swede Berzelius) [48] played a role, and in particular the quantitative atomic volumes (Lothar Meyer [49] see Fig. 2). Already before these publications, the generally known chemical facts had by themselves suggested to several scholars (including French geologist and administrator Alexandre-Émile Béguyer de Chancourtois, 1862, [50–52] and subsequent researchers) that one should wind the mass-ordered array of elements up on a cone or cylinder in such a manner that chemically similar elements appeared in vertical columns (=groups) one below the other. One can also deform the spiral on the cone to obtain a flat spiral that is more easily printed (Fig. 3). One may cut the continuous array or spiral of elements so that one can design a rectangular table, which is easily printable, handy in the lab, and in addition offers the possibility of displaying relevant and well- readable elemental properties in the boxes [31–37]. There is empirical-experience based consensus over the

Fig. 2: Redrawn version [53] of Lothar Meyer’s [49] plot of atomic volumes (in cm3 mol−1) vs. atomic number Z (instead of the original fractional atomic mass number A). This graphic had acted as a convincing demonstration of the systematic pseudo- periodic variation of the elemental properties. (The noble gases and the lanthanoids were still missing at Meyer’s time.) C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1973

Fig. 3: The periodic spiral of chemical elements, after Longman 1951 and Stewart [54]. The bent arrows indicate the relations between the sp and the inserted d (and f) blocks. The group of noble gases forms the ‘backbone’ of periodicity on the array of elements [55–58]. centuries among practicing and theoretical chemists that the three most important groups of chemical properties of the elements, already mentioned above, are (i) the maximum valences and oxidation states (OS), (ii) the effective atomic bonded and van-der-Waals radii (R), and (iii) the electronegativities (EN) [24, 26, 40, 59, 60]. A physical elucidation and thorough statistical analysis of these relations and correlations is still wanted. The elemental properties vary smoothly along the array of elements on each circle from ‘backbone’ to ‘backbone’, where they jump between minimal and maximal values (see also Figs. 11, 13 and 14 below). It is then a matter of usefulness and suitability (though not of necessity) to perform the cut of the spiral at the ‘backbone’, where these properties undergo big jumps from the halogens through the noble gases (discovered in the 1890s) to the alkali metals. The trends of chemical properties are then easy to visualize (see the Abstract Fig.). One must pay for this benefit by losing the continuity of the spiral-array of elements at the row-ends. Figure 3 shows that each block of elements naturally has two neighbor blocks. In particular the s-block has on the side of lower Z values, along the noble gas ‘backbone’, the p-block with largely different properties, while on the side of higher Z values, there is a p-, d- or f-block on the inner, middle or outer part of the spiral, respectively, with smoothly varying property values, as sketched on the Abstract Fig. The valence s and p orbitals are energetically comparatively near (see below) so that the elements of conventional groups 1 and 2 and 12–18 form a joint sp block of elements, from the chemical point of view, with smoothly varying properties [58, 61–64]. The d and f blocks appear inserted in the heavier sp rows after group 3. We note that earth element Al is chemically more similar to the other earth-elements Sc, Y, La (group 3) than to Ga, In, Tl (group 13) [7, 9, 10]. The design of a periodic table obviously depends on one’s chosen priority of chemical properties and of one’s formal, semiotic, esthetic or symmetry views. In education, it is very common to simplify the trends as in Fig. 4, while in reality the mentioned properties vary in a more or less wavy manner (Abstract Fig.), though depending somewhat on the row in the Peri- odic Table and on the property. Most designers of new types of Periodic Tables do not explicitly consider the chemical property trends of the elements inside the table, which is the essence of the chemical Periodic Law, while for chemical lay(wo)men the esthetic appearance of the Table’s outer shape has more attractive appeal than conveying detailed chemical information. 1974 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements

Fig. 4: A common display, simplifying the trends of basic chemical properties of the elements (electron affinity, effective atomic radii, metallic character), implemented on IUPAC’s Red-Book standard template of the Periodic Table [65].

The birth of periodic tables

Dmitri Mendeleev was particularly successful in designing and popularizing comprehensive tables of short and medium long form for fruitful use in chemistry [17–19]. To convince the chemists’ community of the idea of a systematic natural order of the elements, i.e. of some Law or natural principle behind the plethora of chemical observations, three points had been instrumental: (i) Meyer’s graphical plot of atomic volumes vs.

Fig. 5: One of Mendeleev’s early ‘short’ tables [19]. The elements were horizontally ordered according to increasing atomic mass A (except for (Co,Ni) and (Te,I); for (Ru,Rh,Pd), (Os,Ir,Pt,Au) with their yet incorrectly known A; Mendeleev preferred the ‘chemically’ correct, current order corresponding to then unknown Z). The array was cut so that elements of same maximal valence in oxides and hydrides showed up in the same vertical column. Cu,Ag,Au were chemical correctly placed at the end of the iron-platinum group 8, and tentatively in parentheses also in group 1. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1975 atomic masses A (Fig. 2) clearly exhibited the phenomenon of pseudo-periodicity (Fig. 4); (ii) Mendeleev’s systematic grouping of all known elements (short version in Fig. 5) into empirical similarity classes dem- onstrated the consistency of his Periodic Law with chemical reality; (iii) Mendeleev’s correct prediction of various properties of three elements unknown at that time, which were successfully discovered several years later (31Gallium 1875, 21Scandium 1879, 32Germanium 1886), exhibited the fertility of the concept of periodicity

[24, 26]. Mendeleev’s additional correct predictions (verified with delay: 84Po 1898, 91Pa 1913, 75Re 1925, 43Tc

1937, 87Fr 1939), and his various incorrect predictions did not play much role in the further scientific development of Periodic Tables [66].

The linear Z-ordering is basic for chemistry

Originally, Periodic Tables were designed for the benefit of the teaching and practicing chemists, getting the diversity of chemical variability of matter under control. Since the 1860s, the ordering of elements according to the average atomic mass numbers A, and later, better fitting to chemistry, the nuclear charge numbers Z, has proven its practical value. From the theoretical point of view, Z is the number of electrons of the neutral atoms of an element, and the consequential periodically varying number g of outer valence electrons is the very determinant of the elements’ chemistries. Indeed, the columns of the first Periodic Tables were labeled by “g-valent” by Meyer [20], and by the respective “EHg” and “E2Og” formulas of elements E in group g by Mendeleev [17–19] and some of his coevals. The common valence numbers of the elements are fundamental for standard chemistry in our human world (also including a part of geo- and cosmo-chemistry in our large- scale surrounding), while the difference of core and valence shells fades away under very high pressure such as in the inner of the planets [67–70]. In the 19th century, simple ordering of the known elements by the A-numbers gave smooth trends of their chemical properties, except in a few cases (Co & Ni, Te & I, and Ar & K – after argon had been discovered in 1895). After Moseley and Bohr [42, 43], the Z-numbers gave a better ordering-criterion for the elements from the point of view of their chemical properties. The atomic model of a nucleus with Z protons, surrounded by Z electrons in core and valence shells was applied to rationalize a large part of chemistry. This holds until today with 118 observed elements. It is yet an open question whether the next row 8 of ca. 50 super-heavy elements with Z > 118 may exist in the chemical sense. The nuclear lifetimes are expected much shorter than a ms, and that may be too short to form fully developed atoms in their ground states, to form vibrationally relaxed molecules in collisions, not to speak of real bulk chemical substances in a bottle. Yet, the chemical behavior in the 8th row has been investigated [71–79]. Because of the huge relativistic stabilization of s½ and p½ orbitals, the 8p½ falls energetically below the 7d, and the 9s½, 9p½ both fall below the 8p3/2. When ordering the elements far above Z = 100 according to Z, it will not yield a periodic recurrence of blocks with chemically similar valence shells. The problem is somewhat similar to ordering the elements below Z = 100 (or A = 250) according to A. If elements above Z = 118 should exist in the chemical sense, one must rethink how to order them ‘horizontally’ if one wants to obtain ‘vertical’ similarity groups.

The primary purpose of periodic tables in chemistry

The trends of atomic maximum valence number, electronegativity and space requirements [80] are the main determinants of the chemical behavior of elements in chemical compounds [60]. Their trends (Abstract Fig. and Fig. 2) have to be disclosed in any chemically useful Periodic Table. This can optimally be achieved with arrangements, where the minimal and maximal values of the three meta-properties appear at the borders of a rectangular table. An appropriate template (with symbols, names, and A-ranges of the elements) was suggested in IUPAC’s “Red Book” [81], we here call it the “standard” template for Chemical Periodic Tables. It had 1976 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements been distributed by the IUPAC on the internet until the end of 2015, now to be found in their archives [82]. It is essential for application and discussion in chemistry that the main chemical meta-properties are displayed in Chemical Periodic Tables, which is indeed the case for many Periodic Table charts available commercially or on the internet [32–37].

Conceptual clarifications

Nature has created the Chemical Elements during its cosmic history in the form of various isotopes [30, 83– 85, 189]. Now we have that Natural System of Chemical Elements in the accessible outer shell of our earth. The varying isotopic abundances still yield reasonably reproducible average atomic masses A within narrow limits. There are about 90 different Z numbers of the natural elements, and in addition up to 30 ones for artificial elements with lifetimes that may offer the formation of chemical compounds and their chemical investigation. We humans are using and investigating these nearly 120 elements under ‘human’ conditions, i.e. inparticular at temperatures T around 300 K × 10±1 under pressures p in the range of 1 at × 10±3 during time- spans t of 1 h × 10±6. The huge body of chemical experiences under these restricted conditions can be largely rationalized on the basis of the three mentioned meta-properties. If Z is chosen as the order parameter for the elements, their properties vary smoothly and regularly in a saw-toothed manner from Z = 1 to ca. 100. This can be well displayed by the standard setting of the Table as promulgated in the IUPAC’s Red Book and archives (Fig. 6) [81, 82]. Quantum Physics and Statistical Physics (i.e. the subset of physics needed here and labelled theoretical chemistry) govern the behavior of the materials made up of the ca. 100 elements. The finding of a somewhat fuzzy empirical systematics behind the behavior of the chemical materials is called the Periodic Law. The nature-given System of Elements spans the high-dimensional Chemical Space of properties [86]. For human chemists it is useful to project it onto a two-dimensional rectangular Periodic Table. We have distinguished between the different concepts of Natural System, Periodic Law and Periodic Table. The Standard Setting is appropriate for general chemical purposes while the bulk of Other Designs

Fig. 6: Standard Chemical Periodic Table from IUPAC’s Red Book and archives (version until end of 2015) [81, 82]. Note the chemically motivated placement of zero-valent closed-shell He above zero-valent closed-shell Ne, and of the df block beginning with LaIII, CeIV, PrV and AcIII, ThIV, PaV, UVI, NpVII under d elements ScIII, TiIV, VV, CrVI, MnVII. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1977

Fig. 7: A multi-connected two-dimensional Periodic Table in three-dimensional Eukleidean space, commonly called a three- dimensional Table. Designed by Alexander [88].

may be appropriate for other purposes, ranging from specific chemical interests over enjoyment to esthetics, pop-art and commercial advertisement. The majority of tables is two-dimensional, the elements being speci- fied by two integers for the horizontal row and the vertical group. However topologically different carriers for the table are possible, if the surface is cut and glued together (Fig. 7). Conventionally, chemists and artists label the dimensionality of the table according to the space, into which the multi-connected two-dimensional array can be represented in an Eukleidean manner, usually our 3D space.

Guiding principles for a physical explanation of chemical periodic tables

The sense of the question

Chemists sometimes ask “Has the Periodic Table been derived from physical principles”, and some chemists feel satisfied that the apparently important so-called Madelung rule communicated in most textbooks has not been derived from physics, as physical quantum chemist Löwdin had pointed out at the Periodic Table’s centenary [89]. Thus, chemistry does not come under physical regime. There are however better arguments against the supposed inferiority of chemistry such as the complexity of chemistry or the fuzziness of many fruitful chemical rules of thumb [90–92]. Concerning the possible physical explanation of Chemical Periodic Tables, two points must be stressed. (1) The scientific issue is not the outer shape of some table, or the corresponding symmetry behind that design, but the observed empirical facts showing pseudo-periodicity of properties of elements in compounds. Of course, that largely determines and restricts the shape of Chemical Periodic Tables. From the scientific point of view, first, the observed trends of chemical properties must be deduced from the basic physical laws, and second the general frame generated by chemical reality for the table design must be worked out conform- ing to common logic (see Abstract Fig.) [79]. (2) In any community, some convincing stories have become accepted by tradition and are then passed on as established truth, in the chemical community too [93]. It is therefore necessary to figure out at first, which stories in the context of Chemical Periodicity have an empirical reference. 1978 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements

Textbook narratives

We mention five intimately connected points, communicated in most introductory chemical textbooks, which need adjustment before any physical reduction of the Chemical Periodic Table can be undertaken. We cite from the Journal of Chemical education of 1972: [93] “The electron configurations [we teach] are the wrong ones … there is a regrettable result … students [and text book authors too] come to believe in them.” (1) A fixed order of atomic valence orbital energy levels is usually alleged, and visualized by the so- called (n + ℓ, n) mnemonic with a long history [94–97] also labeled Bohr (1923), Bose, Sommerfeld (1925),

Fig. 8: The (n + ℓ, n) mnemonic (from Commons Wikimedia [106], modified), suggesting an occupying order of (nℓ) atomic orbital levels, at first following the blue arrow for increasing (n + ℓ), then following the red arrows for increasing (n). This orbital filling scheme corresponds to the left step table in Fig. 15 below. – The energetically adjacent (nℓ) levels successively filled in a period of a common Chemical Periodic Table (Fig. 6), here framed by dotted green lines, are obviously not well displayed by the mnemonic. It is also not obvious from the mnemonic that ns is filled in groups 1 and 2, while the filling order in any group 3–18 is primarily [(n – 2)f],(n – 1)d,ns,np.

Fig. 9: Atomic orbital energies, i.e. empirical one-electron ionization energies (in terms of √IE, in units of Rydberg½, 1 Rydberg = ½ atomic unit ≈ 13.6 eV), versus nuclear charge Z = 1 through 92. Modern orbital symbols are added to the presented lower cut of Bohr’s graphic of 1922/3 [94, 98]. The added orbital quantum numbers at the top, right side and bottom highlight the adjoining pair of ns&np levels, in the core energy regions (upper part of the graphic) including the approaching nd and nf levels; the ellipses indicate the inversion of the (n – 1)d/ns and (n – 2)f/ns,(n – 1)s,p,d levels along the array of elements with increasing nuclear charge. The bold double-arrows above the abscissa indicate the orbital energy gaps determining the periodic jumps of chemical properties. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1979

Madelung (1926, 1936), Janet (1928), Goudsmit (1932), Moore/Catalán (1949), Klechkovsky (1951) or Moeller (1952) scheme (Fig. 8). In reality the order of s and p versus d and f orbitals strongly varies along a period. This was known in principle since orbital energies in atoms were known (Fig. 9) [94, 98]. (2) While in reality the different sizes of the orbital energy gaps determine the periodicity, [55, 56, 99] little is mentioned about this most crucial point in the textbooks. (3) The electrons of a given atom are filled in by applying a simplified Aufbau rule, where a basic point is neglected (well known in transition metal chemistry and ligand field theory), [100–104] namely that energetically near-degenerate orbitals are both populated partially, if the energy gap is of the same order or smaller than the exchange-interaction energy. (4) While in chemistry, bonded atoms in chemical compounds are of interest, most textbooks choose chemically non-bonded atoms in physical vacuum as the paradigm for the electronic structure of atoms in compounds. A “provocative” question of 1972 was “How many students (or even chemists) ever work with isolated atoms?” [93] Atomic configurations, more stable in vacuum than the chemically relevant configurations, sometimes by less than a kcal/mole, [105] are sold as ‘THE electron configurations of the chemical elements’ in most textbooks. (5) Besides the orbital energies, the orbital radii too are chemically relevant. The radii of the valence and outer-core orbitals determine the covalent overlap interactions and non-bonded repulsions. The variation of orbital energies and orbital radii is often not in parallel at all.

Chemical facts

The atomic orbital energies and their patterns over the system of elements became first generally known through Niels Bohr’s Nobel Lecture of 1922 [94]. While for increasing Z, the ns and np levels become occupied and energetically vary smoothly in adjacent parallelism, as to be expected for near-hydrogen-like systems, the occupation of the d and f valence levels of atoms with occupied core shells sets in later than naively expected, but reaches a more hydrogen-like pattern again for higher Z values (Fig. 9, based on empirical

1.0 10 Z 100 0.1 ε

1.0

s, p (& d) levels are crossing with 3d at 4d at 4f at

Fig. 10: Quantum-chemical Kohn–Sham spin-averaged orbital energies ε of neutral configuration-averaged atoms (logarithmic scale for ε in −13.6 eV) vs. nuclear charge Z number (logarithmic scale), after Latter [107], and Levine [108], reproducing the orbital energy inversions, already spectroscopically derived by Bohr in 1922 (Fig. 9) [94]. E.g. ε(3d) hardly varies for increasing

Z up to 18Ar, 19K, where the 4s, 4p and 5s levels have fallen below 3d (crossings at blue dots ; for higher Z (from 21Sc onward) the screening situation changes and 3d falls below 5s, 4p and 4s again; for an enlarged cutout see below Fig. 18. Similarly, 4d ℓ crosses 5s, 5p and 6s forth and back, falling down in two steps (lilac dots ). The (n + , n) rule holds for group-2 elements (20Ca,

38Sr, 56Ba, 88Ra), and for a few super-heavy 6d-elements [111]. The 4f level remains hydrogen-like up to 54Xe, 55Cs, where 5s, 5p, 5d, 6s, 6p, 6d, 7s, 7p, 8s have fallen below the 4f (green dots ). 1980 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements spectroscopy). Due to nuclear screening by shells of various ℓ-values, there occur several inversions of the order of orbital energy-levels, forth and back (Fig. 10, based on quantum chemical Kohn–Sham density functional calculations) [107, 108]. In free space around a single atomic core, the valence electrons can circulate freely, while their angular momenta coupling is strongly perturbed in chemically bonded atoms. Further, valence electrons of free atoms can move in extended Rydberg orbitals, which are also strongly perturbed for atoms in molecules and solids What matters in chemistry that are the electron configurations of bonded atoms in compounds, i.e. the most frequent populations of the overlapping valence orbitals. This depends more on the radii and less so on the energies of the orbitals [109, 110]. For example, the dominant configuration of a bonded carbon atom is usually some C-1s22s12p3 configuration average, be it in methane, polyethylene, acetylene, benzene, 3 e 2 2 2 graphene or diamond, while the ground state of an unperturbed C atom is Po (1s 2s 2p ). The deviation of chemically relevant atomic configurations from the ones of single unbound atoms is even more serious for the ‘outer’ d and ‘inner’ df transition elements. For example, the usually listed ground state of a free Ni atom 3 e 8 2 3 e 9 1 is F 4 (3d 4s ), while the first excited state D 3 (3d 4s ) is only 2.4 kJ/mol higher and contributes more to the 1 e 10 0 electronic structure of bonded Ni atoms, besides the chemically most relevant S 0 (3d 4s ) configuration.

The structure of chemical periodicity

We are now in the position to formulate what Chemical Periodicity means. Including all so far discovered elements up to Z = 118, the standard template of Periodic Tables in Fig. 6 (or the Abstract Fig., showing the group of noble gases with numbers 0 and 18 on both left and right edges, respectively) can well display the two basic aspects of the Natural System of Elements. One is the variation of properties along the appropriately ordered array of elements, yielding the ‘horizontal’ periodicity. The other is the similarity of elements in the ‘vertical’ groups, when the array of elements is wound up appropriately (Fig. 3), or periodically cut and arranged in a table (Fig. 4). Chemistry is determined by the valence electrons, the number of which is the difference of the atomic number Z and the number of electrons in so-called core shells, here defined as the chemically inactive ones under common conditions (up to polarization and dispersion effects, causing the ‘secondary’ chemical interactions). Admittedly, the definition of cores is somewhat fuzzy and debatable. The first main task is to establish the numbers of core electrons, which then gives the numbers of valence electrons.

Fig. 11: Number of electrons in valence-active shells of neutral atoms (defined as Z minus number of electrons in core shells, chemically inactive under ambient conditions). The numbers increase from zero for the lighter noble gases up to some maximum value (minimum and maximum values in red). Yellow highlight and a bold vertical bar indicate, from where onward an s2, p6, d10 or f14 valence shell first becomes an inert core shell. The question marks indicate insufficient chemical knowledge of some heavy elements. Note the complex though systematic pattern of periodicity at this most basic theoretical-chemical level. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1981

The common number of valence electrons is displayed in Fig. 11. The dominant aspect of periodicity is the respective division of elements into a single row of 2 members and then of two rows each of 8, 18 and 32 members. The division is suggested by the 1s2, 2sp8 and 3sp8 shells becoming chemically inert from He, Ne and Ar onward, respectively, and of the 4sp8, 5sp8, and 6sp8 shells becoming chemically inert from one element after the heavy noble gases (Rb, Cs, Fr) onward. There are, however, theoretical indications that the 6p6 shell of Fr may still be chemically active [111, 112]. While experimentally no chemistry of the last member of the noble gas group, oganesson, is known, owing to its chemical negligibly short lifetime, quantum theory predicts that it is a chemically reactive semi-conducting metalloid solid. This is caused by the relativistic 4 0 destabilization of the outer 7p3/2 shell and the relativistic stabilization and contraction of the 8s1/2 shell [113– 8 115, 189]. It is yet unknown from which element onward the 7(s1/27p1/27p3/2) shell will become a chemically inactive core shell under ambient conditions. Chemical facts clearly demonstrate that there is only one case of a chemically inert closed s2 shell (He 1s2), while the energetic proximity of ns and np for n ≥ 2 leads to joint nsp shells, from chemical points of view [63, 99, 111]. The formal ns2 configurations of the elements in groups 2 (without a d core shell) and 12 (with a d10 core shell) are strongly mixed in the free atoms with the np2 configurations due to two-electron interactions, and for bonded atoms in addition by polarization or hybridization due to chemical interactions. Neither chemical experience nor quantum theory suggest the appearance of an s2 core shell under the valence shell, except the 1s2 core shell under the 2s valence shell in row 2 (and the 2s2 shell in F under its 2p5 valence shell). These facts do not support the so-called left-step table (Fig. 15 below) as an optimal choice to represent chemical periodicity. The d and the f shells, however, do individually form closed inert shells, though becoming inactive only two and three elements, respectively, after their first complete filling (Fig. 11). Concerning the d shells, chemical experience with the elements of groups g = 1–11 of the transition rows shows that the principal valence configurations are [(n-1)dns]g [103, 104]. The s part of the valence shell dominates only in groups 1 and 2, while the d contribution dominates from group 3 onward. The nsp block is thereby split up after group 2 (from row 4 onward after Ca,Sr,Ba,Ra) by the inserted d block, beginning with Sc, Y, La, Ac, as emphasized in the Alexander design (Fig. 7). And from group 12 onward, 10 of the valence electrons first become inactive in an inert d10 shell. In the 7th row, this may even happen only from group 13 (Nh) onward, [116–118, 189] while Hg in row 6 is a border case [119, 120]. Concerning the f block, it is inserted at the beginning of the d block (Fig. 7). The gth element in one of the lowest two rows has the principal valence configuration [(n – 2)f(n – 1)dns]g. As before, the s part dominates in groups 1 and 2, the d contribution dominates from group 3 onwards, and from Ce (in row n = 6) and Pa (in row n = 7) onward the f orbitals also play a role [121–123]. For most lanthanoids, and for the later half of the actinoids, the f-shell basically acts as an inner storage for electrons, while in Ce (and Pr) and in Pa to Pu (or Am), the f shell participates also in structure-determining bond-interactions [124–126]. With a total of 17 valence

Fig. 12: Electron configurations of the atomic core and valence shells in the 7 rows of the ℓ blocks of the Periodic Table: The s and p blocks in groups 1–2 and 12–18 in blue; the ‘s contaminated’ d block in groups 3–11 in red; the ‘s and f contaminated’ d block, called the f block, with elements La–Yb and Ac–No in red-brown. 1982 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements electrons, Lu and Lr reach a filled f14 shell that has become chemically inert, with 3 additional electrons in the valence d(sp) shell. La and Ac are d elements with a chemically inactive and yet empty f0 shell, while Lu and Lr are d elements with a chemically inactive, filled f14 shell and a lanthanide/actinide contracted core.

The basic features of a chemical periodic table, from the empirical point of view

We have summarized the deductions from the empirical chemical findings on the core and valence shells in Fig. 12. Each block of elements in each row has a different valence shell well separated by a chemically relevant energy gap from a different core shell. Every element has a different nuclear charge and a different number of valence electrons. The largest chemical property jumps occur after the filling of the 1s and 2sp to 6sp shells. This is the basis of the richness of chemistry. The Chemical Periodic Table is characterized by a complex internal pattern, as long as we need not simplify it for pedagogical or esthetic purposes. To physically explain the complex inner structure of the Periodic Table, the features of Fig. 12 must be derived quantum-chemically. First, it must be explained, why the Periodic Table is governed by the ‘backbone feature’ [55–57] of the exceptional halogen and alkali-metal groups around the noble-gas group (Figs. 2, 13 and 14), the nsp8 shell becoming an inert core with particularly large jumps of the chemical properties? Why is He-1s2 in the same group as the elements with outer ns2np6 shell (2 ≤ n ≤ 6; row number n = principal quantum number n)? Second, why there are secondary divides due to outer ns2np6nd10 shells (3 ≤ n ≤ 6) becoming inert from group g = 12 onward (di-cationic cores), and due to outer nf14(n + 1)s2(n + 1)p6 shells (4 ≤ n ≤ 5) becoming inert from group g = 13 onward (tri-cationic cores)? The 1s2 and (nsp)8 shells become inert in rows 1–3 first for the three light noble-gas neutral atoms of group 0 (He, Ne, Ar), but in the following rows only from group 1 and 2 onward, i.e. not for compound-forming Kr and Xe, but for the mono-cations of alkali-metals Rb and Cs, and not for Rn and Og, but apparently only for di-cationic group-2 elements Ra and E120 [111–114]. Third, each block in each row has its own specific property variations, which have to be rationalized. A side- question is, why there is no general subshell closure phenomenon for ns2 (n ≥ 2) but only for the n = 1 case (He-1s2)? And why there are smooth property variations through the nsp series without any typical jumps from formal ns2np0 (group 2) to ns2n(p,d)1 (group 3)?

5 2 88 10 + 8 10 + 8 14 + 10 + 8 He Ne 4 Ar Kr 3 Xe Rn

EN Ag Au 2 H Ni Hg 1 Zn Cd Ln An Li Na KRb Fr 0 Cs 0 20 40 60 80 100 120 Z

2.5 Cs Fr Rb K 2.0 Lv Na Bi 1.5 Li Ga In Ln An Og /Å Rn σ Xe R 1.0 Kr Pd Ir Rg Ar Ni 0.5 O Ne He H 0.0 0204060 80 100 120 Z

Fig. 13: Top: Allen-Electronegativity [127] EN vs. element number Z. Bottom: Pyykkö’s [128] single bond radii Rσ (in Å). Ln and An mean Lanthanoids and Actinoids, i.e. elements with variable number of f electrons. Note the singular points of the noble gases He to Rn or Og, and the empirically given ‘periodic’ row-lengths of 1 × 2 and 2 × 8, 18, 32. There is no empirical feature (large jump of value) at the purely formal ns2 closure except for He-1s2. There are wiggles near the d10 closure (in groups 10–12) and kinks at the beginning, middle and end of the 4f series (in group 3), determining the chemical periods. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1983

Fig. 14: Numbers (in blue) of highest valence or oxidation-state of the elements under ambient conditions [129, 130]. For each block (sp, ds, fds) in each row, the numbers increase and then decrease for increasing Z, between minimal and maximal values (in red, highlighted in yellow and blue, respectively). Note the complex structure inside the Periodic Table.

Admittedly, all facts and interpretations collected above are not new, each one has several times been mentioned in the literature. In 1986 for instance, Jensen [130] pointed to the relevance of the maximum valences and their variation for the design of Chemical Periodic Tables (Fig. 14), with citations back to the 19th century. Jensen also confirmed that the lanthanoids and actinoids are clearly members within the d-transition series. Very recently, Zadeh [131] confirmed purely empirically that the nsp together form individual valence shells as do 1s, nd and nf, and stressed that the orbital energy order changes from the neutral atoms to the cations. We accordingly discuss the structure of Periodic Tables based on empirical chemical reality being known since decades, and not so much the principles and rules communicated in the textbooks. We basically address four topics: (1) Why is the horizontal row structure 1 × 1s, 2 × nsp, 2 ×(n – 1)dnsp, 2 × (n – 2) f(n – 1)dnsp? (2) Where are the minima and maxima of the valences in the blocks of each row, and what are their values? (3) Why are the first elements of each vertical group somewhat special? (4) Why is the Chemical Periodic Table complete with 7 rows? By the way we will also annotate some of the alternative opinions.

Examples for physical explanations of systematics of the chemical elements

What may be possible or not

Applied to the unlimited number of collections from the hundred different nuclei and the corresponding number of electrons in 3-dimensional space, the simple set of Dirac + Maxwell + Newton and respective statistical equations (if applied correctly, and if application has become computationally feasible) can reproduce and predict and elucidate the unlimited diversity of chemical phenomena. When we cut out the sector of chemistry under ambient conditions and arrange it under a human sense of order, we can design a Chemi- cal Periodic Table that is restricted by the mentioned empirical (observed or calculated) property variations. From the theoretical point of view, it seems unrealistic that such a selection of data and its ‘accidental’ structure can be derived by paper and pencil from physics. We can only reduce chemistry to some intermediate theoretical concepts such as atomic orbital energies and radii of the valence and outer core shells, and derive the latter values by numerical quantum computations or spectroscopic observations, and rationalize the obtained trends on the basis of physical arguments. 1984 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements

Fig. 15: Janet’s left step periodic table, corresponding to the chemical Madelung rule [96, 132–134]. The elements with typical chemical noble-gas, sp, d, df or f behavior are here differently colored. Note that the border lines between the chemically defined blocks are somewhat fuzzy in reality. The heavy vertical lines indicate, where the big jumps of the properties and/or of the electronic valence shells occur, fixing the periodicity of the elements.

Concerning the popular (n + ℓ, n) rule (Fig. 8), there are four important points. As has been pointed out by Scerri, [132, 133] the rule fits to the so-called Janet left-step Periodic Table (Fig. 15), which exhibits two short periods at the top, while the present authors only know of chemical properties showing one single short period from H to He (Figs. 11–14). Second, the ns2 subshells were taken as the end groups of the periods, while 2 g in reality n(spdf) just forms one member of the open-shell n(spdf) series with smooth variation of properties from g = 1 to 7 or 8 (Figs. 2 and 13). Third, the (n + ℓ, n) rule does not give any hint, at which various places in the Janet table the singular points of periodicity might show up (i.e. the positions of the noble gases, in Fig. 15 highlighted by black bold letters). Fourth, basic physical laws determine the rules of general chemical behavior, but hardly the very special selection of rules for chemistry under human ambient conditions around 300 K and 1 at, and even less so the sense of symmetry that creative minds impose to obtain certain favored design. While quantum physics has a tight relation to the whole of reality, the (n + ℓ, n) rule and the left-step design of Periodic Tables appears to be non-derivable from physics. There is some amount of contingency, i.e. the physical derivation of a Chemical Periodic Table will be partially based on physically understandable or derivable trends, and partially on empirical fitting to calculated or observed numerical values in human chemistry.

The hinge of periodicity and the uniqueness of the filled n(sp)8 (2 ≤ n ≤ 6) shells

The majority of elements in the middle of the standard Periodic Table (i.e. the outer and inner transition elements) show many vertical and horizontal and diagonal relations, substantiated by recent chemometric methods [7–15]. Both Gmelin in 1843 (Fig. 1) and Meyer in 1864 could only form well-defined groups of what we now call the main group elements (groups 14–17 and 1–2 around the yet undiscovered noble gases of group 0 ≡ 18, where there was just a vertical line in Meyer’s table. Of course, at Meyer’s times, the concept of (dominant) zero chemical valence in chemistry did not make much sense. Yet, sitting between the halogens and the alkali-metals highlights the central role of the noble gases in general chemistry, although they form only few chemical compounds. The reason for the latter is the large orbital energy gap above any nsp shell (2 ≤ n ≤ 6, see Figs. 10 and 16). Before the next shell becomes populated along the element array, the nsp shell is already comparatively low in energy and small in radius and yields, if just filled, the mono-atomic noble gas elements. For the heavier members (Kr to Xe to Rn), the closed nsp8 shell is less strongly closed and less compact and can already be broken up to form bonds (Fig. 14). Anyway, the antecedent elements (halogens) with a hole in the (sp)7 shell are strongly electronegative with small atomic volume, while the following elements (alkali metals) have a more or less inert (sp)8 shell and a loosely bound electron with large radius. That yields the large jumps of many properties of the elements (electronegativities and effective single bond radii in Fig. 13, atomic volumes in Fig. 2, maximum valences in Fig. 14), which determines and fixes the periodicity. For oganesson however 4 the relativistic effects are so large and the gap between the spin-orbit coupled 7p3/2 and (n + 1)s1/2 subshells is so small that Og is no longer a noble gas but a solid semiconductor, as mentioned above [114, 115, 189]. Fran- cium forms a border case. The common chemical periodicity seems to end with the 7th row. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1985

4p 4s –5 3s 3p

–10 1s 2p

–15

0 4 812 16 20 24 28 32 36 Ne+ Ar+ Zn+ Be+ O+ Mg+ S+ Ca+ Cr+ Ni+ Ge+ Kr+

+ + Fig. 16: One-electron energies ε of the 1s and n(sp) (n = 2–4) shells of the elemental cations, from 2He to 36Kr , displayed on a square root energy scale, √ | ε | in (e · V)½. ε is here defined by the experimental one-electron ionization for the two states of M+ to M2+ with leading electron configurations differing by only one orbital (derived from spectroscopic NIST data) [135]. Concerning orbital radii and the order of orbital energies, singly ionized atoms fit better as paradigms for chemically bonded atoms than the free neutral atoms. Note the energetic adjacency of ns and np, as compared to the next (n + 1)sp shell. The energy pairs vary comparatively smoothly with increasing nuclear charge. Slight kinks appear where a new (nuclear shielding) shell becomes populated.

The uniqueness of the 1s shell

Hydrogen is different from all other elements [6, 7, 146, 189]. It has no electronic core, only a naked nucleus. Its valence shell consists of a single nodeless orbital only. Hydrogen has one valence electron, not loosely bound as in the alkali metals. Hydrogen has one hole in its valence shell, not strongly electron attracting as in the halogens. There is little chemical sense in advocating for a place of H in any specific column. Concerning helium, it has a stable, closed 1s2 shell with a large orbital energy gap above, comparable to the other noble gases with a stable, closed n(sp)8 shell (Fig. 16). Therefore, He behaves similar to the other noble gases and forms the endpoint of a period. The semiotic aspect of the apparently similar symbols of He-1s2, Be-2s2, Mg-3s2, Ca-4s2, Zn-4s2 etc. is not relevant in chemistry. More relevant is that for n ≥ 2, the ns2 elements have open n(sp)2 shell two-configuration character with leading ns2 and secondary np2 configuration and with significant nsp hybridization upon interaction with adjacent atoms, while He has only a single dominant and weakly polarizable 1s2 configuration.

The uniqueness of the second period

Gmelin (Fig. 1) did not find analogous elements of H (rationalized above), of N and O, and he co-housed beryllium with group-3 elements Y and La, and B with group-4 elements C and Si (the well-known diagonal relation). Mendeleev [19] grouped elements Li to F as we do it now, but he also recognized them as unusually special, as we do it too. First, there are the diagonal similarities of Be–Al instead of Be–Mg, B–Si instead of B–Al, C–P instead of C–Si; second the small 2nd row atoms have rarely higher coordination numbers than 4, nd compare stable molecules CF4 and SF6; third the later 2 row atoms have only low maximum oxidation states, compare N3+, O2+ and F1+ with P5+, S6+ and Cl7+. A century later, Jørgensen [136, 137] finally correlated the empirical uniqueness of the first elements in any block of the Periodic Table with the 1s, 2p, 3d, 4f orbitals having no radial nodes and being comparatively 1986 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements small in radial extensions.1 The ‘Radial Orbital Node Effect in Chemistry’ was then comprehensively discussed by Shchukarev, [138–141] explaining an impressive bulk of empirical findings in connection with the singular- ity of those elements, where an orbital angular momentum appears for the first time. He baptized the 1s, 2p, 3d, 4f orbitals as kaino-symmetric (Greek: καινός, kainos = new) and Pyykkö [142, 143] as primo-genic (Latin: primus = ‘first’, genitus = ‘born’). Kutzelnigg [109, 110] pointed out that the distinct tendency of hybridization of bonded atoms B to O, predominantly C and N, is mainly due to the similarity of the s and p valence orbitals’ radii and less so of their energies. The macroscopic chemical experiences had thereby been connected to the microscopic quantum chemical level of orbital radii and energies. The chemical insights were repeatedly reviewed and extended by Kaupp, in the textbook of Huheey, and elsewhere [62–64, 129, 144–157]. Concerning the microscopic level, a common observation on the radii of atomic valence shells of main group elements was that rnp is significantly more extended than rns, except for n = 2. The general feeling emerged that rns < rnp is quite natural because of the centrifugal force of p electrons with angular momentum quantum number ℓ = 1 in comparison to s electrons with ℓ = 0. Sometimes, also a parallelism of orbital radii and energies was assumed implicitly. However the detailed facts are not that simple. It was then argued that some exception occurs for the lowest possible n quantum numbers. For instance, because there is no inner 1p core shell of same angular symmetry as 2p, in contrast to 1s and 2s, there is no respective orthogonality requirement and no corresponding Pauli pseudopotential repulsion for the 2p. Therefore, the centrifugal force acts on 2p, the Pauli core repulsion acts on 2s, so r2s ≈ r2p results. However, the long-known orbital radii of hydrogen-like systems do not fit into this narrative (Table 1). Shchukarev [141] had suggested two different competing physical mechanisms causing the uniqueness of the first element of a group, one mechanism concerning the potential energy and the other the kinetic energy. The classical screening of the nuclear Coulomb-potential attraction by the occupied core shells is more effective for valence orbitals with larger principal and angular quantum numbers. Already decades ago Clem- enti et al. [158] had noted a chemically relevant deviation from Slater’s approximate screening rules [159, 160]. p Orbitals are somewhat better shielded from nuclear attraction than s-orbitals, which penetrate deeper into the atomic core shells. Shchukarev tentatively suggested a second mechanism, which he labeled as the quantum mechanical Pauli ‘pseudopotential’ repulsion. We elucidate it, beginning with a discussion of the well-known explicit formulas of hydrogen-like orbitals [161, 162].

Radial and angular electronic kinematics in hydrogen-like atoms

In a centrosymmetric (e.g. atomic) potential, the motion can be split up in a radial oscillatory and an angular rotational part. The one-electronic energies ε = –E (E positive) of hydrogen-like orbitals are given in quantum mechanics by:

E =+EE +=EQ()ρ++ 1 ⋅ (1) radial rotational Heisenberg n

Table 1: Qualitative trends of s and p valence orbital radii of hydrogen-like systems (from analytical formulas) and of p-block elements (from numerical computations, Table 2).

Hydrogen-like Main group systems elements n = 2 r2s > r2p r2s ≈ r2p n ≥ 3 rns ≈ rnp rns < rnp

1 In his thesis of 1957 [136], the 3d and 4f cases were analyzed. In the book of 1969 [137], all 1s, 2p, 3d and 4f cases were treated, and even 5g mentioned. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1987

ℓ ℓ Table 2: Ratios rns/rnp of valence orbital radii rnℓ = ⟨n | r | n ⟩ of some light and heavy main-group elements, at the Dirac-Fock level [111].

Ratio of radii H Group 13 Group 15 Group 17 Mean of groups ΔMG/H r2s/r2p 1.20 B: 0.90 N: 0.92 F: 0.92 0.91 −0.29 r3s/r3p 1.08 Al: 0.76 P: 0.81 Cl: 0.84 0.81 −0.27 r4s/r4p 1.04 Ga: 0.72 As: 0.78 Br: 0.81 0.77 −0.27 r5s/r5p 1.03 In: 0.72 Sb: 0.78 I: 0.81 0.78 −0.25

Δ is the change from the hydrogenic value to the Mean of the elements of groups 13–17. where ρ = 0, 1, 2, … is the number of excitations of radial motion (equal to the number of radial orbital nodes), ℓ = 0, 1, 2, … is the number of excitations of angular motion (equal to the number of angular nodes), and the third contribution “ + 1” is due to quantum-mechanical Heisenberg Uncertainty. n = (ρ + ℓ + 1) is the prin- 2 3 cipal quantum number, and Qn = Z /2n is the respective quantum energy in Hartree units (1 Ha ≈ 27.212 eV ~ 2625.5 kJ/mol). The energy in eq. (1) varies linearly and in the same manner with the radial and angular quantum numbers.

The radius ⟨r⟩ = ⟨ϕnℓ | r | ϕnℓ⟩ of orbital ϕnℓ depends quadratically and with different prefactors on the radial and angular components of motion:

rZ=⋅/2En[1.5– + 1/2 2 ] ( ) ( ) (2) =+[(31ρρ)62 +⋅ ++2/ 2.5 ] 2Z ( )

ℓ (Z/2E) = Ro is the classical radius of pure rotational motion (corresponding to ρ = 0, = n − 1). Equation (2) shows that at a fixed orbital energy ε = –E, the orbital radius becomes smaller for more angular (rotational) and less radial motion, see the negative ℓ(ℓ + 1) term in first line of eq. (2), or the prefactors of 3 and 2 before the radial ρ2 and angular ℓ2 terms in the second line. The classical situation is depicted in Fig. 17. The relevance of classical mechanics even for low quantum numbers has been shown by Herschbach et al. [163]. Because

Circular orbit of

radius 1Ro = Z/2E with energy –E 1.5 R & maximum o angular Average 1.0 R momentum o nuclear distance

+Z Core is 1.5Ro shell = 3Z/4E

Largest distance is 2.0 Ro Z/E = 2R o Linear s-type orbit with energy –E and zero angular momentum

Fig. 17: A classical non-relativistic electron moving at energy ε = −E in the field of a Coulomb charge +Ze. Dashed red circle: Rotational motion with maximum angular momentum, corresponding to quantum-mechanical primogenic 2p, 3d, 4f orbitals ℓ ( = n − 1). The rotational radius is Ro = Z/2E (in atomic units of Bohr = 52.9177 pm). Full blue lines: Motion at same energy without any rotation, i.e. radial oscillation to and from the nucleus, corresponding to s orbitals. Because of the flatness of the Coulomb potential ~−1/r at large r, the classical outer turning point is at the large nuclear distance of 2 · Ro, the time-averaged mean distance being R↔ = 1.5 · Ro. An s-type orbit(al) is spatially extended (classically up to 2 · Ro) but also core-penetrating, becoming more attracted by the nucleus of a heavy atom with occupied core shells and therefore becoming energetically stabilized and radially contracted. (Bohr’s model of a circular orbit for the non-rotational, radially oscillating 1s state is obviously inappropriate. Yet, Bohr obtained correct energies because of fortuitous compensation of the two errors of incorrect angular momentum, and of a flat instead of a 3-dimensional atom.) 1988 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements the Coulomb potential ~ −1/r is deep for small but flat for large nuclear distances r (in contrast e.g. to a har- monic oscillator potential, or the potential for the nucleons), the average electron-nucleus distance becomes large for a radially oscillating motion. The integration of hydrogen-like states (n, ℓ = 0) (s orbitals) and of (n, ℓ max = n − 1) for large quantum numbers n, or of a classical radial-oscillatory and a circular-rotating trajectory gives the same ratio of radii (Bohr’ correspondence principle). For the same energy, the radial motion in a Coulomb potential is on the average 50 % more extended than the radius of a pure rotational motion.

R↔ = 1.5Ro (3) Concerning the pairs ns and np orbitals, one may argue that 2p is particularly smaller than 2s, because 2p has no radial motion (up to the statistical contributions due to Heisenberg uncertainty), i.e. no radial node (the ‘Radial Node Effect’ mentioned above, e.g. [144]).

Core electronic screening in many-electron atoms

Slater [159, 160] had suggested approximate screening rules for the groups of (s,p) and of d and f atomic orbitals. Indeed, there is a large qualitative difference between the groups of s,p and d,f orbitals concerning the variation of their energies and radii with nuclear or ionic charge. However, sometimes also the differences between s and p play a role. The s orbitals better penetrate into the atomic core shells than the p orbitals. Therefore the p orbitals are somewhat better shielded from nuclear attraction by the core shells than the s orbitals. Dirac-Fock values are displayed in Table 2 [111] (similar trends are obtained for the radial orbital 2 density maxima rmax or for the √⟨r ⟩ expectation values) [164]. For all main-group elements of groups 13–17, the core electrons’ shielding reduces the hydrogenic ratio of radii of valence s over p orbitals by ca. 25 %, remarkably independent of 1s2 or 1s22s22p6 or 1s2-3s 23p63d10 or 1s2-4s2 4p64d10 cores. Due to the kinematics in a Coulomb potential, hydrogenic 2s is remarkably bigger than 2p, while the higher ns are only slightly bigger than the np. But due to the differential core shielding for s and p valence orbitals in many-electron atoms,

–5 4p

3s 3p 3d

–10+ Z 0 4 81216 20 24 28 32 36 Be+ Mg+ Ca+ Zn+

Fig. 18: One-electron energies |ε | (displayed on a square root scale, √ε in (e · V)½) of n = 3 and 4 orbitals of elemental cations, + + from 2He to 36Kr . |ε | is defined as the experimental one-electron ionization energy with same leading configuration in initial and final state, derived from the data of the NIST [135]. Note the energetic adjacency of s and p with common fixed order 3s < 3p << 4s < 4p vs. the varying order of the d and f orbitals. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1989

the s orbitals of p-block elements become more contracted than the p orbitals, obtaining r2s≈ r2p for B to F, but pronouncedly rns < rnp for the heavier homologs with n ≥ 3 (Table 1). The different valence orbital radii then cause the different covalent actions of the light and heavier main group elements as explained by Kutzelnigg [109, 110].

Insertion of d and df blocks into the sp block

A most remarkable feature of the Chemical Periodic Tables is the appearance of d transition elements after group 2, from row 4 onward, and of the df lanthanoid and actinoid elements after groups 3 and 4, respectively, from row 6 onward. The empirical explanation goes with the help of the s,p and d and f orbital energy variations along the array of elements (Fig. 18). While the s,p pairs of orbital levels monotonously decrease along the element array (Fig. 16), the d and f levels at first hardly change because of efficient nuclear screening by the 1s, 2s, 2p, 3s and 3p shells. Up to Ar and K the 4s, 4p and even 5s levels have fallen below the 3d. When the 3d and 4s levels become populated, the 3d ‘collapses’ below 5s, 4p and 4s, where it stays until the

4p becomes populated [165, 166]. Then the 3d collapses in a second step, until it eventually reaches its 3s1/2,

3p1/2 and 3p3/2 relatives. This behavior has been deduced empirically by analyzing experimentally derived (spectroscopic: Figs. 9, 16, 18) or computed (Fig. 10) data. Applying the Thomas–Fermi density functional approximation and some additional fitting models, a crude picture of the development of the general orbital energy pattern along the array of elements can also be obtained [167–172]. Similar explanations can be given for the slightly more complex orbital variations in the heavier transition rows. Sc, Y, La, Ac are typical ‘early d transition’ elements, as are Lu and Lr. The latter have an additional f14 shell in their lanthanoid- or actinoid-contracted atomic cores and are more similar to the following d elements than La and Ac. Concerning the df inner transition elements, the f orbital collapse occurs for 4f around group 4 (Ce) and for 5f around group 5 (Pa). This is the common right-shift of periodicity, when one goes down in the Periodic Table (Figs. 11 and 14). Elements Ce and Pa to Am, in particular U, Np and Pu, are somewhat special (Figs. 14 and 15) in so far as they have both d and f orbitals available for overlapping bond formation [71, 173].

Which atomic electron configurations are relevant for chemistry?

We had noticed the discrepancy concerning the general atomic orbital energy order, between the accepted stories in chemistry textbooks on the one hand (ns <(n − 1)d < np, Fig. 8; used in the classroom) and the dominant electron configurations in transition metal and heavy main group chemistry on the other hand ((n − 1) d < ns < np, Fig. 18; useful in the labs). Because of the efficient core screening for d orbitals, ns, np and often even (n + 1)s have fallen below the (n − 1)d, when the (n − 1)sp8 shell has just been filled in groups 18/0 and 1:

−<−<<< << + − (,nn1s)(1p)(ns np n 1s)((dn 1) in groups 0 and 1.) (4a)

Here < and << mean ‘small’ and ‘large’ energy differences, respectively. When the (n − 1)d and ns orbitals become populated, the (n − 1)d orbital begins to collapse spatially and energetically (Fig. 18) from a diffuse and loosely bound Rydberg orbital to an ordinary valence orbital. The chemical textbook order is reached in group 2: −<−<<<− <<<+ ()nn1s ()1p ns ()n 1 d np ()n 1 s ()in group 2 . (4b)

The chemistry of the heavy alkali metal atoms is known to be influenced by some d-orbital participation [174–176]. Except for transition metal atoms carrying a negative partial charge, the general rule in transition metal complex chemistry [103, 177] is: 1990 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements

Mn ∆ ε /eV

5 (Mn+ + 12He)

(Mn+ + 6He)

4 (Mn+ + 2He)

3 2 3 4 5 R/Å

Fig. 19: Average 3d < 4s orbital energy gap Δε (in eV) of Mn+3d54s1, surrounded by 2 or more He atoms (Kohn–Sham DFT approach). The Pauli repulsion of the closed shell He “ligands” destabilizes 4s relative to 3d. After [178].

−<−<<<− <<<− ()nn1s ()1p ()n 1 d ns np ()in groups 311. (4c)

Finally, for the heavy main group elements with closed d10 shell after group 11: −<−<− << <<<− (1nn)s ((1p)8()n 1 d ns np in groups 12 1.) (4d)

A rather similar story holds for the (n − 2) f orbital levels. One aspect of the ns valence orbital of transition metal atoms M still needs to be mentioned. If M is surrounded by the typically bigger closed shells of ligand atoms, the ns orbital becomes destabilized by Pauli non-bonded repulsion, which reduces the relevance of s orbitals in most transition element complex chemistry. The effect is displayed in Fig. 19. Even closed shells of small ligands have a remarkable effect. If however M is surrounded by many other M atoms with ns orbitals such as in metallic phases, a broad s-band is formed with center above the d-band but with bottom penetrating and mixing with the d-band, yielding partial ns population that is relevant already at the lowest level of modeling. In summary, a rule for the valence configurations of (neutral) atoms in the nth row that is simple enough to be taught by textbooks in classrooms and realistic enough to be acceptable by practicing laboratory and computational chemists is:

Groups gn=−0 to 2 : sng[(pn,1)]d 0 (5a)

Groups gn=−3 to 11: ()1 d[gg()−1) ns,]np 0(1 (5b)

Groups gn=−12 to 18: ()1d10[]ns,np g−10 (5c)

To keep the connection with chemical reality, it seems inevitable to note the large orbital energy gap above a filled (n − 1)(sp)8 noble-gas shell, and the flip of the orbital energy order for the next higher orbital levels from ns <(n − 1)d to (n − 1)d < ns after group 2. The facts are empirical details from experiment or calcu- lation, the general trends can be rationalized qualitatively by analyzing the shape of the effective potential for atomic electrons and the core shell screening and valence shell centrifugal potentials. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1991

Fig. 20: Contour plot of DIVA diagnostic values (eq. (6), in VÅ/e) for oxidation states of I to X (vertical axis) of transition elements (horizontal axis). Left: 3d row. Right: 5d row. The colors are empirically chosen so that deep magenta/lilac/blue indicates feasible oxidation states under ambient chemical conditions, while non-feasible oxidation states are indicated by green/yellow/orange/red. In the 3d row, the maximum feasible oxidation state increases from (I for K and) II for Ca to VII for Mn, then decreases slowly (finally reaching II for Zn). In the 5d row, it increases from (I for Cs and) II for Ba to VIII for Os and Ir, then decreases quickly.

From which group onward, valence does no longer increase?

A most important aspect of chemical periodicity is the number g of valence-active electrons in group g, which increases along the array of elements after each noble gas in any s-p, s-d or s-df block. The chemical valence was the main criterion for Meyer and for Mendeleev when grouping the elements in their first Chemical Peri- odic Tables. The valence electrons can be either delocalized over adjacent atoms forming covalent bonds, or transferred to other atoms forming ionic bonds, the reality being a mixture. The highest effective valence however reaches a maximum in each row and then decreases again despite further increase of g. In the 2nd row, g increases from 1 for Li-2s1 to 8 for Ne-2(sp)8, but the highest valence or oxidation state is reached at 5 for N-2(sp)5, while the maximum valences of O-2(sp)6, F-2(sp)7 and Ne-2(sp)8 are 2, 1 and 0, respectively. Each row and each block has a slightly different pattern (Fig. 14). The highest oxidation state of an atom depends on the one hand on the atomic ionization cost and the overlap and ionic bond energy gains, on the other hand on the environmental conditions. Pyykkö asked “Can we relate the ionization energies of the free ions to the maximum oxidation states in compounds? [May we] assume that the sum of the ionization energies is paid back by combined interionic Coulomb attractions and covalent bonding?” [73]. However, at low temperatures or extreme pressures, ‘exotic’ oxidation states may become feasible, which are no longer stable under ambient conditions. Therefore this question of practical chemistry, the answer of which forms an important aspect of the internal structure of the Chemical Periodic Table, is very complex and will not be easily answered by theory. We see no simple definite solution. Following Pyykkö’s second question, we define a certain DIagnostic parameter for the highest VAlence, DIVA, where we compare two quantities: 0 qq+ (a) the sum of successive ionization energies of atom Z for Z → ZI, [ Σi=1 PZ(),i ] (in eV), with (b) a term proportional to the ionic interaction energy, q2/[R(Z,q) + 1.5Å] (in e2/Å), where IP(Z,i) is the ith ionization energy of atom Z, R(Z,q) is the effective radius of Zq+, and 1.5 Å is taken as the representative bond radius of typical ligands. The unit of DIVA is [V Å e−1].

q 2 DIVAZ(),:qI=×[(Σ PZ,iR)] [,(/Zq)5+ 1. Å] q (6) i=1

DIVA plots for the 3d and 5d transition elements are displayed in Fig. 20 [179]. If the relative ionization cost, measured by DIVA, is too high to be compensated by covalent energy contributions and secondary interactions, the oxidation state q will not be feasible. We do not know the critical DIVA value for a given class of 1992 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements molecules under ambient conditions, but we can check in how far DIVA correlates with the empirical findings. According to Fig. 20, a critical DIVA = 16¼ VÅ/e for 3d-element compounds corresponds to maximum valence of 7 for Mn and 6 for Fe, while for 5d-elements DIVA = 13½ VÅ/e corresponds to maximum valence of 8 for Os and Ir. The green regions indicate possible stability at cryogenic conditions, such as valence 8 for Fe, or valence 9 for Ir and 8 for Pt [180–185].

The grand picture

Chemical elements or principles mean, since antiquity, the abstract entities that build up the materials of the real world and that are more or less well conserved in chemical transformations. Nowadays they are speci- fied by their nuclear or electronic ‘atomic number’ Z. We stick to this classical and IUPAC’s first definition of Chemical Element. It comprises also IUPAC’s second definition, meaning all allotropes of real substances consisting of one element Z only. In common chemical literature, element may highlight other aspects of the general concept of abstract elements in matter, for instance, the chemical element as a single chemi-

Fig. 21: Change of atomic-volumes (in cm3 mol−1) vs. atomic number Z upon increasing the pressure from ambient values (bottom) to 1 Mb (middle) to 10 Mb (top). After Al’tshuler et al. [67]. The periodic structure largely disappears at pressures typical for the centers of planets.

Fig. 22: Variation of Pearson-Electronegativity EN (in V) vs. atomic number between ambient pressures (upper curve) and 5 Mb (lower curve), typical for planets’ centers. The basic periodicity remains, while the detailed values vary. After Oganov et al. [68]. C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1993 cally unbound atom in physical vacuum, which may considerably differ from bound atoms in chemical compounds. Properties of the elements can only be derived from the observations on their compounds, in the form of more or less fuzzy values. This works well, at least when we confine chemistry to ambient conditions. To stress this latter point, we display the graphs of atomic volume and of electronegativity up to pressures occurring inside the planets (Figs. 21 and 22). They indicate that the basic chemical rules may change quite a bit with the variation of the environmental conditions [67–70]. One must be careful when comparing elemental properties such as valence number or effective atomic radius with those under non-ambient conditions. Periodic Tables of Chemical Elements of use in pure and applied chemistry are obtained in a cycle of three self-consistent steps [13]. (a) Order all elements on a linear array according to a property appropriate for step (c). (b) Wind that array up on a two dimensional, possibly multi-connected surface (Figs. 3 and 7). (c) Select a set of properties most relevant in the intended field of application, and arrange the array of elements so that respective similarity groups are obtained ‘vertical’ to the elemental array. A well-fitting triple of (i) element-order (a) corresponding to the chosen property, e.g. A or Z, (ii) chemical similarity groups w.r.t. some other property, or better a set of properties, (c), such as valence numbers, electronegativities, atomic radii, etc. and (iii) a systematic repeated variation of those properties along order (a) – that yields chemical periodicities.

The historical development had begun with the composition of similarity groups, culminating in Gmelin’s table of elements in 1843 (Fig. 1). Then an ordering parameter was searched for. First equivalence weights, then atomic weights, and currently the atomic number Z has been chosen. There seems to have always been some informal agreement in the community of chemists that among the chemically most relevant elemental properties are the electronegativities introduced by Berzelius in 1811 (see also Fig. 13 top), the valence numbers well-defined since Cannizzaro in 1860 (see Fig. 14), and the effective atomic volumes of Meyer since 1870 (Fig. 2, nowadays radii, Fig. 13 bottom). The trends of basic properties of chemical elements inside the table are the essential aspects of a Chemi- cal Periodic Table. Only those Periodic Tables, which reflect those details of reality, can honestly be derived from physics. A clear definition of the actual chemical periodicity that is displayed in the given table, and the analysis of its chemical meaning is the prerequisite for the elucidation of its physical basis. The physical derivation of chemical periodicity then consists in the explanation of the property variations, foremost the big jumps of properties at the 1s2 and n(sp)8 (2 ≤ n ≤ 6) shell closures, and why it’s just these shell closures. With closure we here mean the transition of a chemically active valence shell to a chemically inert outer core shell. One should not say that a Periodic Table has been derived from physics, when nothing more than the correct lengths of the periods in that table have been derived on the basis of some additional assumptions. The chemical periodicity determines the block structure. There is the sp block without any obvious sign of a remarkable break between s and p parts (Fig. 13 and Refs. [62–64, 186]). The reason is the energetic near- degeneracy of the ns and np orbitals (Fig. 16) and their more or less pronounced similarity in radial extension. A remarkable feature is the appearance of the df block after group 2, and the abrupt change of properties of the compounds of the elements around group 18. According to the empirical findings of practicing chemists, the sp block comprises the nine groups 1, 2 and 12–18, while the d(f) block comprises the nine groups 3–11. Another rather involved feature is the appearance of the f block at the beginning of the d block in group 3 with 15 elements per row. Two elements (La and Lu) in row 6, and three ones (Ac, Th and Lr) in row 7 are rather typical d elements. Ce in row 6 and four ones (Pa, U, Np, Pu) in row 7 are elements sui generis with d and f orbitals overlapping with adjacent atoms. The other f-block members of group 3 are, so to say, d-elements with a storage capacity of (g′ −3 ± 1) valence electrons in the inner f-shell. The chemical periodicity is naturally well represented by the 1994 C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements spiral of elements (Figs. 3 and 7), which considers both ‘vertical similarities’ and ‘horizontal periodicities’ of an uninterrupted array of elements. Cuts to obtain a handier rectangular table are not prescribed by nature. Meyer in 1862 chose the cut where we now have group 3, and Janet in 1928 chose the cut after group 2, and both placed the ‘backbone of periodicity’ of the halogen to the alkali-metal group somewhere in the lateral middle (Fig. 15). There are many important aspects of chemical periodicity. Besides the sp8 shell closure, the other shell closures must be analyzed and explained, i.e. why the (n − 1)d10 closure does not occur before group 12, why the (n − 2)f14 closure does not occur before group 3’, and why there is no remarkable ns2 closure at all except for 1s2. Further, why do the shell closures shift to the right in the lowest rows. The n(sp)8 noble-gas shell becomes closed from row 5 downward only in group 1, but the 6(sp)8 closure may happen only for Ra in group 2. Where the 7(sp)8 shell becomes chemically inert is presently unknown. The (n − 1)d10 closure, known to happen in group 12, shifts in row 7 to group 13. All these details can be explained as the result of competition of orbital stabilization with increasing Z, destabilization with increasing n, and various relativistic modifications in the lowest rows. Further points are the secondary extrema such as the highest valence in row 2 for N in group15; or the highest valence in rows 3–5 for Mn, Ru, Ir in groups 7–9. Further points are the so-called vertical secondary periodicity first discussed by Biron [187] already in 1915, which is related to the kainosymmetric/ primogenic effect when 1s, 2p, 3d or 4f orbitals first appear in the valence shell. At present, some of these phenomena are deeply understood, while some still need more theoretical elucidation. The Chemical Periodic Table presents a plethora of contents that must be addressed before one can say, one has derived the table from physics. We have reviewed a set of steps in this direction. And we have indicated some open problems to be solved in the future. Most important points in this context are the variation of orbital energies with increasing Z, smoothly for the sp shells (Figs. 9, 10, 18) and in steps for the d and f shells (Figs. 9, 10, 18), and a different variation of orbital radii and orbital energies. Inacceptable in pure and applied chemistry is assuming and teaching a single fixed order of orbital energies and radii for all rows. This may be acceptable however for pedagogical reasons. When such a drastic deviation from reality is assumed as by the (n + ℓ, n) mnemonic, the derivability from physics is lost of course. One may also assume some mathematical model construct and derive from it the lengths of rows, corresponding to or differing from reality (as is the case for the left step table following from the (n + ℓ, n) rule). Trivially, an axiomatic model may have its value, but is decoupled from general physical theory.

Resume

Chemical Periodic Tables as handy tools for application in pure and applied chemistry shall display a comprehensive overview of chemistry in the real human world under ambient conditions. IUPAC Fig. 6 (i.e. the ‘old’ layout with the first elements of the f-block (La,Ac and Ce,Th) placed correctly below groups 3, 4) is well suited as a template of apt format for the display of a large body of data on the elements, exhibiting the wavy trends (Abstract Fig.). Various publishing companies use this template for the benefit of the practicing chemists. The chemical facts determine the internal structure of these tables and thereby also their outer shape. Understanding the coarse structure that underlies empirical chemistry, i.e. its observed and computed data, directs to what should be derived in detail from quantum chemistry and chemical thermodynamics. We have presented several examples of this enterprise. Because of the complexity of the chemical world, qualitative classification such as that of the elements in rows, blocks and groups, and trends of various properties in these sets are eminently useful. But since the borders between different classes are fuzzy and somewhat different trends show up in each row or group, fixing strict borders in the table and fighting for them is sense- less inside the field of scientific chemistry. Any modification of the present standard form of the Chemical Periodic Table should account for the grand picture outlined above. Pedagogic Periodic Tables exhibit less information, just what is needed to educate the common population of later voters, journalists, politicians, managers and administrators, to let them more easily grasp what is extremely important for the survival of mankind. Educators may discuss simplifying the trends inside the tables, such as in Fig. 4, and use tables with modified outer shapes to achieve their so important C.-S. Cao et al.: Physical origin of chemical periodicities in the system of elements 1995 educational aims. Other arguments than the handiness for daily use in the chemical laboratory are relevant. The physical derivation of pedagogically designed tables that aim at eliminating common misconcep- tions in present societies does neither appear straightforward nor necessary as in the case of the Chemical Periodic Tables. Designer Periodic Tables are connected to the liberation of creative minds from the ‘tyranny of reality’. Little information from the real chemical world is accounted for inside such tables, when seeking for a beauti- ful design of its outer shape, or for some symmetry from an artificial mathematical model that generates an endless table. Any realistic Chemical Periodic Table is for the ca. hundred chemical elements with nuclear lifetimes above a second or millisecond and has 7 rows only. Accidentally, nonrelativistic periodicity breaks down after row 7 because of the s-p-d-f-g level crowding, where also relativistic electronic effects and nuclear lifetimes set an end. The unlimited Janet left step table in Fig. 15, corresponding to the ‘Madelung rule’, is a typical example of reduced connection to the laws of real chemistry and physics. Therefore these creative tables appear non-derivable from quantum mechanics, because human creativity isn’t too. Obviously, hard efforts are required for the human mind to achieve small progress in the rational analysis of even the simplest systems of reality, that is, the material systems of physics and chemistry. Concerning the development of basic scientific concepts, scientific genius can sometimes contribute, and sometimes even laymen, provided they are not too poor in knowing some relevant background details. In contrast, missing knowledge and alternatives to logic and facts sometimes seem helpful. Everything goes (at least during the searching process in scientific research), as Feyerabend told us [188]. Scientists need open minds to grasp useful ideas that might emerge from time to time in other circles of the society. At present however, there seem to exist no rational arguments why to improve on the IUPAC’s Periodic Table of 2015. Its empirically based internal structure appears to be derivable from theory. The first steps presented here need to be continued with a positive outlook.

Acknowledgements: We gratefully acknowledge discussions with Arnout Ceulemans, Naum Imyanitov, Guill- ermo Restrepo, Pekka Pyykkö, and Eric Scerri. This work was supported by the National Natural Science Foundation of China (NSFC Nos. 21433005 and 21590792). The calculations were performed at the Tsinghua National Laboratory for Information Science and Technology, China. WHES thanks for hospitality at Tsing- hua and Siegen Universities.

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