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On a -theoretical approach to the of chemical elements Maurice Kibler

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Maurice Kibler. On a group-theoretical approach to the periodic table of chemical elements. Jun 2004, Prague. ￿hal-00002554￿

HAL Id: hal-00002554 https://hal.archives-ouvertes.fr/hal-00002554 Submitted on 16 Aug 2004

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ccsd-00002554, version 1 - 16 Aug 2004 ℓ { Ψ vectors hl oe a eue o etn paproi al htca that table periodic Made a the up how setting shown SO(4 for group is used the It be via can elements. model chemical shell of table periodic hn eg rmS()t h yaia nainegopSO(4 group invariance dynamical SO(3 the to or a SO(3) of spectrum from SO(3) go group we symmetry geometrical Then, the from starting cess re them. This of two [1-3]. review spin briefly the We labels ways. that several SU(2) group the introduce esfrtedsrt pcrmare spectrum discrete the for bers rmtetbeadtegnrllnso rgam o quant a o basis for the programme on a developed of are lines elements general chemical of the properties and table the from e words Key PACS tdigmn-lcrnaos avl,w a xett con to t expect constitutes may In number often we atomic Naively, atom, atom. atoms. the many-electron the of studying namely treatment quantum-mechanical atom, a simplest on based are ru n42dmnin o ofra ru)SO(4 group) conformal r (or special dimensions the kno 4+2 is we in atom fact, hydrogen-like of group matter a a of As group noninvariance view. of rati point be group-theoretical can a principle building-up This atom. hydrogen-like IC,ntd( noted (IRC), R fS() h ietsum direct The SO(4). of IRC Ψ 0 = nℓm nagoptertclapoc oteproi al of table periodic the to approach group-theoretical a On h rtwyi ut elkonadcrepnst symmetr a to corresponds and well-known quite is way first The oto h oenpeettoso h eidctbeo che of table periodic the of presentations modern the of Most hsppri ocre ihteapiaino h ru SO( group the of application the with concerned is paper This , 36.d 31.15.Hz 03.65.Fd, : ℓ 1 , : · · · yrgnlk tm amncoclao,ivrac and invariance oscillator, harmonic atom, hydrogen-like : nℓm n 3B u1 oebe11,F662Vleran ee,Franc Cedex, Villeurbanne F-69622 1918, Novembre 11 du Bd 43 rus i ler ne osrit,Mdln ue peri rule, Madelung constraints, under algebra Lie groups, n , and ℓ − ℓ a eognzdt pnmlilt fS()adS() h se The SO(4). and SO(3) of multiplets span to organized be can Z ,wieteset the while ), ℓ ;frfixed for 1; , ydistributing by , fixed; 2) )frtecniuu pcrm h eeatqatmnum- quantum relevant The spectrum. continuous the for 1) ⊗ NP-NSe nvri´ lueBernard Universit´e Claude et IN2P3-CNRS U2 n oeo t ugop.Qaiaierslsaeobt are results Qualitative subgroups. its of some and SU(2) ntttd hsqeNc´ar eLyon Nucl´eaire de Physique de Institut m ℓ ranging hmclelements chemical ℓ : arc .Kibler R. Maurice Introduction 1 m { Ψ Z h ℓ nℓm } n = = lcrn nteoeeeto nrylvl fa of levels energy one-electron the on electrons , eeae nirdcberpeetto class representation irreducible an generates − M n ℓ ℓ ∞ =1 ℓ, : and 1 n − M n ℓ =0 − ℓ fixed; 1 m 1 + ( ℓ ℓ ) , (with ℓ · · · and ℓ , , )o SO(4 or 2) .Tecrepnigstate corresponding The ). n h ru SO(4 group the f m nlzdadrfie from refined and onalized 1 = ℓ efrhrrationalized further be n ranging ugrl fteatomic the of rule lung a pseudo-orthogonal eal utcnb eie in derived be can sult ttv praht the to approach itative httedynamical the that w , tuta tmwith atom an struct 2 yrgnlk atom. hydrogen-like trigpitfor point starting a , 4 3 dctable odic o h discrete the for ) , , , 2) i epc,the respect, his · · · 2) ia elements mical } ⊗ ⊗ non-invariance setpro- ascent y o fixed for ; eeae an generates U2 othe to SU(2) U2 fwe if SU(2) e , 2) ⊗ SU(2). ained n t : M.R. Kibler

spanned by all the possible state vectors Ψnℓmℓ corresponds to an IRC of the de Sitter group SO(4, 1). The IRC h is also an IRC of SO(4, 2). This IRC thus remains irreducible when restricting SO(4, 2) to SO(4, 1) but splits into two IRC’s when restricting SO(4, 2) to SO(3, 2). The groups SO(4, 2), SO(4, 1) and SO(3, 2) are dynamical noninvariance groups in the sense that not all their generators commute with the Hamiltonian of the hydrogen-like atom. The second way to derive SO(4, 2) corresponds to a symmetry descent pro- cess starting from the dynamical noninvariance group Sp(8, R), the real symplectic group in 8 dimensions, for a four-dimensional isotropic harmonic oscillator. We know that there is a connection between the hydrogen-like atom in R3 and a four- dimensional oscillator in R4 [3-4]. Such a connection can be established via Lie- like methods (local or infinitesimal approach) or algebraic methods based on the so-called Kustaanheimo-Stiefel transformation (global or partial differential equa- tion approach). Both approaches give rise to a constraint and the introduction of this constraint into the Lie algebra of Sp(8, R) produces a Lie algebra under con- straints that turns out to be isomorphic with the Lie algebra of SO(4, 2). From a mathematical point of view, the latter Lie algebra is given by [5]

centsp(8,R)so(2)/so(2) = su(2, 2) ∼ so(4, 2) in terms of Lie algebras. Once we accept that the hydrogen-like atom may serve as a guide for studying the periodic table, the group SO(4, 2) and some of its sugroups play an impor- tant role in the construction of this table. This was first realized in by Barut [6] and, independently, by Konopel’chenko [7]. Later, Byakov et al. [8] further devel- oped this group-theoretical approach of the periodic chart of chemical elements by introducing the direct product SO(4, 2)⊗SU(2). The aim of this paper is to emphasize the importance of a connection between the periodic table `ala SO(4, 2)⊗SU(2) and the Madelung rule [9] of atomic spec- troscopy. The material is organized as follows. In Section 2, a periodic table is derived from the Madelung rule. In Section 3, this table is rationalized in terms of SO(4, 2)⊗SU(2). Some qualitative and quantitative aspects of the table are de- scribed in Sections 4 and 5. This paper constitutes a companion article to four previous papers by the author [10].

2 The periodic table `ala Madelung

We first describe the construction of a periodic table based on the so-called Madelung rule [9] which arises from the atomic shell model. This approach to the periodic table uses the quantum numbers occurring in the quantum-mechanical treatment of the hydrogen atom as well as of a many-electron atom. In the central-field approximation, each of the Z electrons of an atom with Z is partly characterized by the quantum numbers n, ℓ, and mℓ. The numbers ℓ and mℓ are the orbital quantum number and the magnetic quantum number, respectively. They are connected to the chain of groups SO(3)⊃SO(2): the

2 On the periodic table . . .

quantum number ℓ characterizes an IRC, of dimension 2ℓ +1, of SO(3) and mℓ a one-dimensional IRC of SO(2). The principal quantum number n is such that n − ℓ − 1 is the number of nodes of the radial wave function associated with the doublet (n,ℓ). In the case of the hydrogen atom or of a hydrogen-like atom, the number n is connected to the group SO(4): the quantum number n characterizes an IRC, of dimension n2, of SO(4). The latter IRC of SO(4) splits into the IRC’s of SO(3) corresponding to ℓ = 0, 1, · · · ,n − 1 when SO(4) is restricted to SO(3). A complete characterization of the dynamical state of each electron is provided by 1 the quartet (n,ℓ,mℓ,ms) or alternatively (n,ℓ,j,m). Here, the spin s = 2 of the electron has been introduced and ms is the z-component of the spin. Furthermore, 1 j = 2 for ℓ = 0 and j can take the values j = ℓ − s and j = ℓ + s for ℓ 6= 0. The quantum numbers j and m are connected to the chain of groups SU(2)⊃U(1): j characterizes an IRC, of dimension 2j + 1, of SU(2) and m a one-dimensional IRC of U(1). Each doublet (n,ℓ) defines an atomic shell. The ground state of the atom is obtained by distributing the Z electrons of the atom among the various atomic shells nℓ,n′ℓ′,n′′ℓ′′, · · · according to (i) an ordeing rule and (ii) the Pauli exclusion principle. A somewhat idealized situation is provided by the Madelung ordering rule: the energy of the shells increases with n+ℓ and, for a given value of n+ℓ, with n [9]. This may be depicted by Fig. 1 where the rows are labelled with n =1, 2, 3, · · · and the columns with ℓ =0, 1, 2, · · · and where the entry in the n-th row and ℓ-th column is [n + ℓ,n]. We thus have the ordering [1, 1] < [2, 2] < [3, 2] < [3, 3] < [4, 3] < [4, 4] < [5, 3] < [5, 4] < [5, 5] < [6, 4] < [6, 5] < [6, 6] < · · ·. This dictionary order corresponds to the following ordering of the nℓ shells

1s< 2s< 2p< 3s< 3p< 4s< 3d< 4p< 5s< 4d< 5p< 6s< · · · which is verified to a good extent by experimental data. (According to atomic spectroscopy, we use the letters s (for sharp), p (for principal), d (for diffuse), f (for fundamental), . . . to denote ℓ =0, 1, 2, 3, · · ·, respectively.) From these considerations of an entirely atomic character, we can construct a periodic table of chemical elements. We start from the Madelung array of Fig. 1. Here, the significance of the quantum numbers n and ℓ is abandoned. The numbers n and ℓ are now simple row and column indexes, respectively. We thus forget about the significance of the quartet n,ℓ,j,m. The various blocks [n+ℓ,n] are filled in the dictionary order, starting from [1, 1], with chemical elements of increasing atomic numbers. More precisely, the [n + ℓ,n] is filled with 2(2ℓ + 1) elements, the atomic numbers of which increase from left to right. This yields Fig. 2, where each element is denoted by its atomic number Z. For instance, the block [1, 1] is filled with 2(2 × 0 + 1) = 2 elements corresponding to Z = 1 up to Z =2. Ina similar way, the blocks [2, 2] and [3, 2] are filled with 2(2 × 0 + 1) = 2 elements and 2(2 × 1 + 1) = 6 elements corresponding to Z = 3 up to Z =4 and to Z = 5 up to Z = 10, respectively. It is to be noted, that the so obtained periodic table a priori contains an infinite number of elements: the n-th row contains 2n2 elements and each column (bounded from top) contains an infinite number of elements.

3 M.R. Kibler

0 1 2 3 4 …

1 [1,1]

2 [2,2] [3,2]

3 [3,3] [4,3] [5,3]

4 [4,4] [5,4] [6,4] [7,4]

5 [5,5] [6,5] [7,5] [8,5] […]

6 [6,6] [...]

7 […]

…

Fig. 1. The [n + ℓ, n] Madelung array. The lines are labelled by n = 1, 2, 3, · · · and the columns by ℓ = 0, 1, 2, · · ·. For fixed n, the label ℓ assumes the values ℓ = 0, 1, · · · , n − 1.

l = 0 (s-block) l = 1 (p-block) l = 2 (d-block) l = 3 (f-block) l = 4 (g-block) …

n = 1 1 up to 2

n = 2 3 up to 4 5 up to 10

n = 3 11 up to 12 13 up to 18 21 up to 30

n = 4 19 up to 20 31 up to 36 39 up to 48 57 up to 70

n = 5 37 up to 38 49 up to 54 71 up to 80 89 up to 102 121 up to 138

n = 6 55 up to 56 81 up to 86 103 up to 112 139 up to 152 …

n = 7 87 up to 88 113 up to 118 153 up to 162 …

n = 8 119 up to 120 163 up to 168 …

n = 9 169 up to 170 …

……

Fig. 2. The periodic table deduced from the Madelung array. The box [n + ℓ, n] is filled with 2(2ℓ + 1) elements. The filling of the various boxes [n + ℓ, n] is done according to the dictionary order implied by Fig. 1.

4 On the periodic table . . .

3 The periodic table `ala SO(4, 2)⊗SU(2)

We are now in a position to give a group-theoretical articulation to the periodic table of Fig. 2. For fixed n, the 2(2ℓ + 1) elements in the block [n + ℓ,n], that we shall refer to an ℓ-block, may be labelled in the following way. For ℓ = 0, the s-block in the n-th row contains two elements that we can distinguish by the number m 1 1 with m ranging from − 2 to 2 when going from left to right in the row. For ℓ 6= 0, the ℓ-block in the n-th row can be divided into two sub-blocks, one corresponding 1 1 to j = ℓ − 2 (on the left) and the other to j = ℓ + 2 (on the right). Each sub-block 1 contains 2j + 1 elements (with 2j +1=2ℓ for j = ℓ − 2 and 2j +1=2(ℓ +1) for 1 j = ℓ + 2 ) that can be distinguished by the number m with m ranging from −j to j by step of one unit when going from left to right in the row. In other words, a can be located in the table by the quartet (n,ℓ,j,m) (where 1 j = 2 for ℓ = 0). Following Byakov et al. [8], it is perhaps interesting to use an image with streets, avenues and houses in a city. Let us call Mendeleev city the city whose (west-east) streets are labelled by n and (north-south) avenues by (ℓ,j,m). In the n-th street there are n blocks of houses. The n blocks are labelled by ℓ = 0, 1, · · · ,n − 1 so that the address of a block is (n,ℓ). Each block contains one sub-block (for ℓ = 0) or two sub-blocks (for ℓ 6= 0). An address (n,ℓ,j,m) can be given to each house: n indicates the street, ℓ the block, j the sub-block and m the location inside the sub-block. The organization of the city appears in Fig. 3. At this stage, it is worthwhile to re-give to the quartet (n,ℓ,j,m) its group- theoretical significance. Then, Mendeleev city is clearly associated to the IRC class h ⊗ [2] of SO(4, 2)⊗SU(2) where [2] stands for the fundamental representation of SU(2). The whole city corresponds to the IRC

1 ∞ n−1 j=ℓ+ 2 ∞ n−1 (j)= (ℓ) ⊗ [2] n=1 ℓ=0 1 n=1 ℓ=0 ! M M j=M|ℓ− 2 | M M of SO(4, 2)⊗SU(2) in the sense that all the possible quartets (n,ℓ,j,m), or alterna- tively (n,ℓ,mℓ,ms), can be associated to state vectors spanning this IRC. (In the latter equation, (ℓ) and (j) stand for the IRC’s of SO(3) and SU(2) associated with the quantum numbers ℓ and j, respectively.) We can ask the question: how to move in Mendeleev city? Indeed, there are several bus lines to go from one house to another one? The SO(3) bus lines (also called SO(3)⊗SU(2) ladder operators) make it possible to go from one house in a given ℓ-block to another house in the same ℓ-block (see Fig. 4). The SO(4) bus lines (also called SO(4)⊗SU(2) ladder operators) and the SO(2, 1) bus lines (also called SO(2, 1) ladder operators) allow to move in a given street (see Fig. 5) and in a given avenue (see Fig. 6), respectively. Finally, it should be noted that there are taxis (also called SO(4, 2)⊗SU(2) ladder operators) to go from a given house to an arbitrary house. Another question concerns the inhabitants (also called chemical elements) of Mendeleev city. In fact, they are distinguished by a number Z (also called the

5 M.R. Kibler atomic number). The inhabitant living at the address (n,ℓ,j,m) has the number 1 1 Z(nℓjm) = (n + ℓ)[(n + ℓ)2 − 1]+ (n + ℓ + 1)2 6 2 1 − [1 + (−1)n+ℓ](n + ℓ + 1) − 4ℓ(ℓ +1)+ ℓ + j(2ℓ +1)+ m − 1 4 Each inhabitant may also have a nickname. All the inhabitants up to Z = 110 have a nickname. For example, we have Ds, or in full, for Z = 110. Not all the houses in Mendeleev city are inhabited. The inhabited houses go from Z =1to Z = 116 (the houses Z = 113 and Z = 115 are occupied since the begining of 2004 [11]). The houses corresponding to Z ≥ 117 are not presently inhabited. (When a house is not inhabited, we also say that the corresponding element has not been observed yet.) The houses from Z = 111 to Z = 116 are inhabited but have not received a nickname yet. The various inhabitants known at the present time are indicated on Fig. 7. It is not forbidden to get married in Mendeleev city. Each inhabitant may get married with one or several inhabitants (including one or several clones). For example, we know H2 (including H and its clone), HCl (including H and Cl), and H2O (including O, H and its clone). However, there is a strict rule in the city: the assemblages or married inhabitants have to leave the city. They must live in another city and go to a city sometimes referred to as a molecular city. (Only the clones can stay in Mendeleev city.)

4 Qualitative aspects of the periodic table

Going back to and , we now describe Mendeleev city as a periodic table for chemical elements. We have obtained a table with rows and columns for which the n-th row contains 2n2 elements and the (ℓ,j,m)-th column contains an infinite number of elements. A given column corresponds to a family of chemical analogs in the standard periodic table and a given row may contain several periods of the standard periodic table. The chemical elements in their ground state are considered as different states of atomic matter: each atom in the table appears as a particular partner for the (infinite-dimensional) unitary irreducible representation h ⊗ [2] of the group SO(4, 2)⊗SU(2), where SO(4, 2) is reminiscent of the hydrogen atom and SU(2) is introduced for a doubling purpose. In fact, it is possible to connect two partners of the representation h ⊗ [2] by making use of shift operators of the Lie algebra of SO(4, 2)⊗SU(2). In other words, it is possible to pass from one atom to another one by means of raising or lowering operators. The internal dynamics of each element is ignored. In other words, each neutral atom is assumed to be a noncomposite phys- ical system. By way of illustration, we give a brief description of some particular columns and rows of the table. 1 The alkali- atoms (see Fig. 8) are in the first column (with ℓ = 0, j = 2 , 1 and m = − 2 ); in the atomic shell model, they correspond to an external shell of type 1s, 2s, 3s, · · ·; we note that hydrogen (H) belongs to the alkali-metal atoms.

6 On the periodic table . . .

1 1 The second column (with ℓ = 0, j = 2 , and m = 2 ) concerns the alkaline earth (see Fig. 9) with an external atomic shell of type 1s2, 2s2, 3s2, · · ·; we note that (He) belongs to the alkaline earth metals. The sixth column corresponds to (see Fig. 10) and the seventh column to (see Fig. 11); it is to be observed that hydrogen does not belong to halogens as it is 3 often the case in usual periodic tables. The eighth column (with ℓ = 1, j = 2 , and 3 m = 2 ) gives the noble gases (see Fig. 12) with an external atomic shell of type 2p6, 3p6, 4p6, · · ·; helium (with the atomic configuration 1s22s2) does not belong to the noble gases in contrast with usual periodic tables. The d-blocks with n = 3, 4 and 5 yield the three familiar transition series (see Fig. 13): the group goes from Sc(21) to Zn(30), the group from Y(39) to Cd(48) and the group from Lu(71) to Hg(80). A fourth transition series goes from Lr(103) to Z = 112 (observed but not named yet). In the shell model, the four transition series correspond to the filling of the nd shell while the (n + 1)s shell is fully occupied, with n = 3 ( series), n = 4 (palladium group series), n = 5 ( series) and n = 6 (fourth series). The two familiar inner transition series are the f-blocks with n = 4 and n = 5 (see Fig. 14): the series goes from La(57) to Yb(70) and the series from Ac(89) to No(102). Observe that start with La(57) not Ce(58) and start with Ac(89) not Th(90). We note that lanthanides and actinides occupy a natural place in the table and are not reduced to appendages as it is generally the case in usual periodic tables in 18 columns. A superactinide series is predicted to go from Z = 139 to Z = 152 (and not from Z = 122 to Z = 153 as predicted by Seaborg [12]). In a shell model approach, the inner transition series correspond to the filling of the nf shell while the (n + 2)s shell is fully occupied, with n = 4 (lanthanides), n = 5 (actinides) and n = 6 (superactinides). The table in Fig. 7 shows that the elements from Z = 121 to Z = 138 form a new having no homologue among the known elements. In Section 3, we have noted that each ℓ-block with ℓ 6= 0 gives rise to two sub- blocks. As an example, the f-block for the lanthanides is composed of a sub-block 5 (corresponding to j = 2 ) from La(57) to Sm(62) and another one (corresponding 7 to j = 2 ) from Eu(63) to Yb(70) (see Fig. 15). This division corresponds to the classification in light or ceric rare earths and heavy or yttric rare earths. It has received justifications both from the experimental and the theoretical sides.

5 Quantitative aspects of the periodic table

To date, the use of SO(4, 2) or SO(4, 2)⊗SU(2) in connection with periodic charts has been limited to qualitative aspects only, viz., classification of neutral atoms and ions as well. We would like to give here the main lines of a programme under development (inherited from and particle physics) for dealing with quantitative aspects. The first step concerns the mathematics of the programme. The direct product group SO(4, 2)⊗SU(2) is a Lie group of order eighteen. Let us first consider the

7 M.R. Kibler … … … … … j = 7/2 l = 3 (f-block) j = 5/2 … j = 5/2 l = 2 (d-block) j = 3/2 j = 3/2 l = 1 (p-block) j = 1/2 l = 0 j = 1/2 -1 1 -1 1 -3 -1 1 3 -3 -1 1 3 -5 -3 -1 1 3 5 -5 -3 -1 1 3 5 -7 -5 -3 -1 1 3 5 7 2m n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 ……

∗ Fig. 3. Mendeleev city. The streets are labelled by n ∈ N and the avenues by (ℓ,j,m) 1 1 1 [ℓ = 0, 1, · · · , n−1; j = 2 for ℓ = 0, j = ℓ− 2 or j = ℓ+ 2 for ℓ =6 0; m = −j, −j +1, · · · , j].

8 On the periodic table . . . … … … … … j = 7/2 l = 3 (f-block) j = 5/2 … j = 5/2 l = 2 (d-block) j = 3/2 j = 3/2 l = 1 (p-block) j = 1/2 l = 0 j = 1/2 -1 1 -1 1 -3 -1 1 3 -3 -1 1 3 -5 -3 -1 1 3 5 -5 -3 -1 1 3 5 -7 -5 -3 -1 1 3 5 7 2m n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 ……

Fig. 4. How to move in Mendeleev city? To move inside a block, take an SO(3)⊗SU(2) bus line! The shaded bus line corresponds to ℓ = 3 and n = 6.

9 M.R. Kibler … … … … … j = 7/2 l = 3 (f-block) j = 5/2 … j = 5/2 l = 2 (d-block) j = 3/2 j = 3/2 l = 1 (p-block) j = 1/2 l = 0 j = 1/2 -1 1 -1 1 -3 -1 1 3 -3 -1 1 3 -5 -3 -1 1 3 5 -5 -3 -1 1 3 5 -7 -5 -3 -1 1 3 5 7 2m n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 ……

Fig. 5. How to move in Mendeleev city? To move inside a west-east street, take an SO(4)⊗SU(2) bus line! The shaded bus line corresponds to n = 4.

10 On the periodic table . . . … … … … … j = 7/2 l = 3 (f-block) j = 5/2 … j = 5/2 l = 2 (d-block) j = 3/2 j = 3/2 l = 1 (p-block) j = 1/2 l = 0 j = 1/2 -1 1 -1 1 -3 -1 1 3 -3 -1 1 3 -5 -3 -1 1 3 5 -5 -3 -1 1 3 5 -7 -5 -3 -1 1 3 5 7 2m n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 ……

Fig. 6. How to move in Mendeleev city? To move inside a north-south avenue, take an 1 1 SO(2, 1) bus line! The shaded bus line corresponds to ℓ = 0, j = 2 , and m = − 2 .

11 M.R. Kibler … … … … … … 1 2 H He 3 4Li Be 5 B 611 C 12 7 NNa 8 Mg 13 O 14 9 Al F 15 Si 16 10 Ne 19 17 P 20 18K S 31 Ca 21 Cl 32 22 Ar 33 Ga 23 Ge 3437 As 24 Sc 35 38 Se 25 Ti 36 BrRb 26 Sr V 49 Kr 27 39 50 28 Cr 40 In 51 Y 29 Mn 41 52 Fe 30 Sn55 42 Zr 53 Co Sb 56 43 Ni Nb 54 Te 44 Mo I Cu 81 Tc 45 Zn 71 82 Ru 46 Xe 72 Rh 83 47 73 Pd 84 48 Lu Ag 74 85 Cd Hf 75 86 57Fr Ta 76 La 58 W Ra 77 103 Ce 59 104 78 105 Re Pr 106 60 X? 107 Os 79 108 109 Nd X? 61 Ir 110 80 111 Pm X? 112 62 Sm Pt X? Eu 63 139 89 no 140 Gd Au 64 141 142 90 Tb no Hg 143 65 144 Dy 91 145 146 66 147 Ho 92 Ac 148 no 149 67 Er 150 93 Th no 151 152 68 Tm Pa 94 no Yb 69 U 95 no 70 96 no Np 97 Pu no 98 Am no Cm 99 Bk no Cf 100 no 101 102 Es no Fm 121 - Md 138 No no Cs Ba Tl Pb87 Bi 88 Po At 113 114 115 Rn 116 117 118 Lr 153 154 155 Rf 156 157 158 Db 159 160 Sg 161 162 Bh Hs Mt Ds X? X? no no no no no no no no no no no no no no

Fig. 7. The inhabitants of Mendeleev city. The houses up to number Z = 116 are inhabited [‘X?’ means inhabited (or observed) but not named, ‘no’ means not inhabited (or not observed)]. 12 On the periodic table . . . … … … … Ra X? X? X? X? no no no no no no no no no no no no 88 113 114 115 116 117 118 153 154 155 156 157 158 159 160 161 162 Ba Tl Pb Bi Po At Rn Lr Rf Db Sg Bh Hs Mt Ds X? X? no no no no no no no no no no no no no no 2 He 4Be 5 B 6 C 712 N 8 O 13Mg 9 14 F Al 15 10 Ne Si 16 17 P20 18 S 31Ca 21 Cl 32 22 Ga 33 Ar Ge 23 34 As 24 35 Sc Se38 25 36 Ti Br 26 Kr 49Sr V 27 39 50 28 Cr 40 51 Y In 29 41 Mn 52 30 Fe Sn Zr 42 53 Co56 Sb 43 Nb 54 Ni Te Mo 44 I Tc Cu 81 45 71 Zn Ru 82 46 72 Rh Xe 83 47 Pd 73 84 48 Ag Lu 74 85 Cd Hf 75 86 57 76 Ta La 58 77 W 103 Ce 59 104 78 105 Pr Re 106 60 107 79 Os 108 Nd 109 61 110 Ir 80 Pm 111 112 62 Sm Eu Pt 63 139 89 Gd 140 64 141 Au Tb 142 90 143 Hg 65 144 Dy 91 145 146 66 Ho 147 92 148 Ac Er 149 67 150 93 151 Th 152 Tm 68 94 Pa Yb 69 95 U 70 96 Np 97 Pu 98 Am Cm 99 Bk 100 Cf 101 102 Es 121 Fm - Md 138 No no … … Fr 87 Cs 1 H 3 Li 11 Na 19 K 37 Rb 55

Fig. 8. The inhabitants of Mendeleev city. The family of alkali metals (with an infinite 1 1 N∗ number of elements) corresponds to (ℓ = 0, j = 2 , m = − 2 ) and n ∈ .

13 M.R. Kibler … … … … 81 82 83 84 85 86 103 104 105 106 107 108 109 110 111 112 139 140 141 142 143 144 145 146 147 148 149 150 151 152 49 50 51In 52 Sn 53 Sb 54 Te I 71 Xe 72 73 Lu 74 Hf 75 Ta 76 W 77 78 Re Os 79 Ir 80 Pt 89 Au 90 Hg 91 92 Ac 93 Th 94 Pa U 95 96 Np 97 Pu 98 Am Cm 99 Bk 100 Cf 101 102 Es 121 Fm - Md 138 No no Ga Ge As Se Br Kr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb 31 32 33 34 35 36 39 40 41 42 43 44 45 46 47 48 57 58 59 60 61 62 63 64 65 66 67 68 69 70 13 14 15Al 16 Si 17 P 18 S 21 Cl 22 Ar 23 24 Sc 25 Ti 26 27 V 28 Cr 29 Mn 30 Fe Co Ni Cu Zn X? X? X? X? no no no no no no no no no no no no B C N O F Ne 113 114 115 116 117 118 153 154 155 156 157 158 159 160 161 162 Tl Pb Bi Po At Rn Lr Rf Db Sg Bh Hs Mt Ds X? X? no no no no no no no no no no no no no no 5 6 7 8 9 10 56 Sr 38 Ca 20 Mg 12 Ra Be 88 Ba 2 He 4 … … 55 Rb 37 K 19 Na 11 Fr Li 87 Cs 1 H 3

Fig. 9. The inhabitants of Mendeleev city. The family of alkaline earth metals (with an 1 1 N∗ infinite number of elements) corresponds to (ℓ = 0, j = 2 , m = 2 ) and n ∈ .

14 On the periodic table . . . … … … … Cl Ar Sc Ti V Cr Mn Fe Co Ni Cu Zn 35 36 39 40 41 42 43 44 45 46 47 48 57 58 59 60 61 62 63 64 65 66 67 68 69 70 Br Kr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb 53 54 71 72 73 74 75 76 77 78 79 80 89 90 91 92 93 94 95 96 97 98 99 100 101 102 121 - 138 I Xe Lu Hf Ta W Re Os Ir Pt Au Hg Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No no 85 86 103 104 105 106 107 108 109 110 111 112 139 140 141 142 143 144 145 146 147 148 149 150 151 152 9 10 F Ne 17 18 21 22 23 24 25 26 27 28 29 30 no no no no no no no no no no no no At Rn Lr Rf117 118 Db 153 Sg 154 155 Bh 156 157 158 Hs 159 160 Mt 161 162 Ds X? X? no no no no no no no no no no no no no no Se 52 Te 8 84 O 16 S 34 X? Po 116 … … K Ca Ga Ge As Li Be B C N 37 38 49 50 51 11 12 13 14 15 Na Mg Al Si P Rb Sr In Sn Sb 19 20 31 32 33 1 2 H He 3 4 5 6 7 55 56 81 82 83 Fr Ra X? X? X? Cs Ba Tl Pb87 Bi 88 113 114 115

Fig. 10. The inhabitants of Mendeleev city. The family of chalcogens (with an infinite 3 1 number of elements) corresponds to (ℓ = 1, j = 2 , m = − 2 ) and n = 2, 3, 4, · · ·.

15 M.R. Kibler … … … … 86 103 104 105 106 107 108 109 110 111 112 139 140 141 142 143 144 145 146 147 148 149 150 151 152 54 71Xe 72 Lu 73 Hf 74 75 Ta W 76 77 Re Os 78 Ir 79 80 Pt Au 89 Hg 90 91 Ac 92 Th 93 Pa U 94 95 Np 96 Pu 97 Am Cm 98 Bk 99 Cf 100 Es 101 102 Fm Md 121 No - 138 no 36 39Kr 40 41 Y 42 Zr 43 Nb 44 Mo 45 Tc 46 Ru 47 Rh Pd 48 Ag Cd 57 58 La 59 Ce 60 Pr 61 Nd 62 Pm Sm 63 Eu 64 Gd Tb 65 Dy 66 Ho 67 Er 68 Tm 69 Yb 70 18 21 22 23 24 25 26 27 28 29 30 Ar Sc Ti V Cr Mn Fe Co Ni Cu Zn Ne no no no no no no no no no no no 118 153 154 155 156 157 158 159 160 161 162 Rn Lr Rf Db Sg Bh Hs Mt Ds X? X? no no no no no no no no no no no no no no 10 85 53 I 35 Br Cl F 17 no 117 At 9 … … 55 56 81 82 83 84 Rb Sr In Sn Sb Te 37 38 49 50 51 52 K Ca Ga Ge As Se 19 20 31 32 33 34 Na Mg Al Si P S 11 12 13 14 15 16 Li Be B C N O Fr Ra X? X? X? X? 87 88 113 114 115 116 Cs Ba Tl Pb Bi Po 1 2 H He 3 4 5 6 7 8

Fig. 11. The inhabitants of Mendeleev city. The family of halogens (with an infinite 3 1 number of elements) corresponds to (ℓ = 1, j = 2 , m = 2 ) and n = 2, 3, 4, · · ·.

16 On the periodic table . . . … … … … Lr Rf Db Sg Bh153 154 Hs 155 156 157 Mt 158no 159 Ds 160 161 no 162 X? no X? no no no no no no no no no no no no no no no no no no no no no 21 22 23Sc Ti 24 25 V 26 Cr 2739 Mn 28 40 Fe 29 41Y Co 30 Ni 42 Zr 43 Cu Nb Zn 44 Mo 45 Tc71 46 Ru 72 47 Rh 73Lu Pd 48 74 Hf Ag 75 Ta Cd 57 76 W 58 77 La Re103 59 104 78 Ce 105 Os 106 60 107 Pr Ir 79 108 109 61 110 Nd 80 111 Pt 112 62 Pm Sm 63 Au Eu 139 89 140 Hg 64 141 Gd 142 90 143 Tb 65 144 91 145 Ac Dy 146 66 147 92 148 Ho Th 149 67 150 93 Er 151 Pa 152 68 94 U Tm 69 Yb 95 Np 70 96 Pu 97 Am Cm 98 Bk 99 Cf 100 Es 101 102 Fm Md 121 - No 138 no Rn 118 no 10 Ne 18 Ar 36 Kr 54 Xe 86 … … Fr Ra X? X? X? X? no Cs Ba Tl Pb87 Bi 88 Po At 113 114 115 116 117 1 2 H He 3 4Li Be 5 B 611 C 12 7 NNa 8 Mg 13 O 14 9 Al F 15 Si 1619 17 P 20K S 31 Ca Cl 32 33 Ga Ge 3437 As 35 38 Se Br Rb Sr 49 50 In 51 52 Sn55 53 Sb 56 Te I 81 82 83 84 85

Fig. 12. The inhabitants of Mendeleev city. The family of noble gases (with an infinite 3 3 number of elements) corresponds to (ℓ = 1, j = 2 , m = 2 ) and n = 2, 3, 4, · · ·.

17 M.R. Kibler … … … … 57 58 59La 60 Ce Pr 61 Nd 62 Pm 63 Sm89 Eu 64 90 Gd 65 91Ac Tb 66 Th 92 Dy 67 Pa Ho 93 68 Er U 94 69 Tm 95 Np Yb 70 96 Pu Am 97 Cm Bk 98 Cf 99 100 Es 101 102 Fm Md 121 No - 138 no 21 22 23Sc Ti 24 25 V 26 Cr 2739 Mn 28 Fe 40 29 Co 41Y Ni 30 42 Zr Cu 43 Zn Nb 44 Mo 45 Tc71 Ru 46 72 Rh 47 73Lu Pd 48 Ag Hf 74 Cd Ta 75 W 76 77 Re Os 78 Ir 79 80 Pt Au Hg … … Na Mg Al Si19 P 20 S 31 Cl 32 Ar 33 34 35 36 1 2 H He 3 4Li Be 5 B 611 C 12 7 N 8 13 O 14 9 F 15 16 10 Ne 17 18 K Ca Ga Ge37 As 38 Se BrRb Sr 49 Kr 50 In 51 52 Sn55 53 Sb 56 54 Te I 81 82 Xe 83 84 85 86Fr Ra 103 104 105 106 X? 107 108 109 X? 110 111 X? 112 X? 139 no 140 141 142 no 143 144 145 146 147 148 no 149 150 no 151 152 no no no no no no no no Cs Ba Tl Pb87 Bi 88 Po At 113 114 115 Rn 116 117 118 Lr 153 154 155 Rf 156 157 158 Db 159 160 Sg 161 162 Bh Hs Mt Ds X? X? no no no no no no no no no no no no no no

Fig. 13. The inhabitants of Mendeleev city. The three transition series: the iron group series [from Sc(21) to Zn(30)], the palladium group series [from Y(39) to Cd(48)], and the platinum group series [from Lu(71) to Hg(80)]. 18 On the periodic table . . . no … … 121 - 138 … … 57 58 59La Ce 60 Pr 61 Nd 62 Pm 63 Sm89 Eu 64 Gd 90 65 Tb 91Ac Dy 66 Th 92 Ho 67 Pa 93 Er U 68 94 Tm 69 Yb 95 Np139 70 140 Pu 96 141 142 Am 143 97 Cm 144no Bk 145 146 98 147 no Cf 148 149 99 no 150 Es 151 no 152 100 Fm 101 102 Md no No no no no no no no no no no … … 55 56 81 82 83 84 85 86 103 104 105 106 107 108 109 110 111 112 1 2 H He 3 4Li Be 5 B 611 C 12 7 NNa 8 Mg 13 O 14 9 Al F 15 Si 16 10 Ne 19 17 P 20 18K S 31 Ca 21 Cl 32 22 Ar 33 Ga 23 Ge 3437 As 24 Sc 35 38 Se 25 Ti 36 Br 26 V 49 Kr 27 39 50 28 Cr 40 51 Y 29 Mn 41 52 Fe 30 42 Zr 53 Co 43 Ni Nb 54 44 Mo Cu Tc 45 Zn 71 Ru 46 72 Rh 47 73 Pd 48 Ag 74 Cd 75Fr 76 Ra 77 78 X? 79 X? 80 X? X? no no no no no no no no no no no no Rb Sr In Sn Sb TeCs I Ba Xe Tl Pb87 Lu Bi 88 Hf Po Ta At 113 W 114 115 Rn 116 Re 117 118 Os Lr Ir 153 154 155 Rf 156 Pt 157 158 Db 159 160 Au Sg 161 162 Hg Bh Hs Mt Ds X? X?

Fig. 14. The inhabitants of Mendeleev city. The three inner transition series: the lanthanide series [from La(57) to Yb(70)], the actnide series [from Ac(89) to No(102)], and the superactinides series [from (139) to (152)]. 19 M.R. Kibler … … 63 64 65 66 67 68 69 70 Eu Gd Tb Dy Ho Er Tm Yb … … La Ce Pr Nd Pm Sm … … 1 2 H He 3 4Li Be 5 B 611 C 12 7 NNa 8 Mg 13 O 14 9 Al F 15 Si 16 10 Ne 19 17 P 20 18 S 31 21 Cl 32 22 Ar 33 23 34 24 Sc 35 25 Ti 36 26 V 27 39 28 Cr 40 29 Mn 41 Fe 30 42 Co 43 Ni 44 Cu 45 Zn 46 47 48 57 58 59 60 61 62 K Ca Ga Ge As Se Br Kr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd Fr Ra X? X? X? X? no no no no no no no no no no no no 37 38Rb Sr 49 50 In 51 52 Sn55 53 Sb 56 54 TeCs I Ba 81 71 82 Xe 72 Tl 83 73 84 Pb87 Lu 74 Bi 85 88 Hf 75 86 Po Ta 76 At 113 W 77 114 103 115 Rn 104 116 78 105 Re 117 106 118 107 Os 79 108 Lr 109 Ir 110 153 80 111 154 112 155 Rf 156 Pt 157 158 Db 139 89 159 140 160 Au Sg 141 161 142 90 162 Hg 143 Bh 144 91 145 Hs 146 147 92 Ac 148 Mt 149 150 93 Th Ds 151 152 Pa 94 X? U X? 95 96 Np no 97 Pu no 98 Am Cm no 99 Bk no Cf 100 101 102 no Es no Fm 121 - Md no 138 No no no no no no no no no

Fig. 15. The inhabitants of Mendeleev city. The lanthanides or rare earths can be divided into two sub-blocks: the light (or ceric) rare earths [from La(57) to Sm(62)] correspond 5 7 to j = 2 and the heavy (or yttric) rare earths [from Eu(63) to Yb(70)] to j = 2 . 20 On the periodic table . . .

SO(4, 2) part which is a semi-simple Lie group of order r = 15 and of rank ℓ = 3. It has thus fifteen generators involving three Cartan generators (i.e., generators commuting between themselves). Furthermore, it has three invariant operators or Casimir operators (i.e., independent polynomials, in the enveloping algebra of the Lie algebra of SO(4, 2), that commute with all generators of the group SO(4, 2)). Therefore, we have a set of six (3+3) operators that commute between themselves: the three Cartan generators and the three Casimir operators. Indeed, this set is not complete from the mathematical point of view. In other words, the eigenvalues of the six above-mentioned operators are not sufficient for labelling the state vectors in the representation space of SO(4, 2). According to a (not very well-known) result 1 popularised by Racah, we need to find 2 (r − 3ℓ) = 3 additional operators in order to complete the set of the six preceding operators. This yields a complete set of nine (6+3) commuting operators and this solves the state labelling problem for the group SO(4, 2). The consideration of the group SU(2) is trivial: SU(2) is a semi- 1 simple Lie group of order r = 3 and of rank ℓ = 1 so that 2 (r −3ℓ) = 0 in that case. As a result, we end up with a complete set of eleven (9 + 2) commuting operators. It is to be stressed that this result constitutes the key and original starting point of the programme. The second step establishes contact with chemical physics. Each of the eleven operators can be taken to be self-adjoint and thus, from the quantum-mechanical point of view, can describe an observable. Indeed, four of the eleven operators, namely, the three Casimir operators of SO(4, 2) and the Casimir operator of SU(2), serve for labelling the representation h ⊗ [2] of SO(4, 2)⊗SU(2) for which the var- ious chemical elements are partners. The seven remaining operators can thus be used for describing chemical and physical properties of the elements like: ; oxydation degree; electron affinity; ; melting and boiling points; specific heat; ; atomic volume; density; magnetic suscepti- bility; solubility; etc. In most cases, this can be done by expressing a chemical observable associated with a given property in terms of the seven operators which serve as an integrity basis for the various observables. Each observable can be developed as a linear combination of operators constructed from the integrity ba- sis. This is reminiscent of group-theoretical techniques used in nuclear and atomic spectroscopy (cf. the Interacting Boson Model) or in hadronic spectroscopy (cf. the Gell-Mann/Okubo mass formulas for baryons and mesons). The last step is to proceed to a diagonalisation process and then to fit the various linear combinations to experimental data. This can be achieved through fitting procedures concerning either a period of elements (taken along a same line of the periodic table) or a family of elements (taken along a same column of the periodic table). For each property this will to a formula or phenomenological law that can be used in turn for making predictions concerning the chemical elements for which no data are available. In addition, it is hoped that this will shed light on regularities and well-known (as well as recently discovered) patterns of the periodic table. This programme, referred to as the KGR programme, was briefly presented at the 2003 Harry Wiener International Conference. It is presently under progress.

21 M.R. Kibler: On the periodic table ...

6 Closing remarks

We close this paper with two remarks. Possible extensions of this work concern and molecules. The consideration of isotopes needs the introduction of the number of in the . With such an introduction we have to consider other dimensions for Mendeleev city: the city is not anymore restricted to spread in Flatland. Group-theoretical analyses of periodic systems of molecules can be achieved by considering direct products involving several copies of SO(4, 2)⊗SU(2). Several works have been already devoted to this subject [13].

Acknowledgments. The author is indebted to M.S. Antony for comments con- cerning the elements Z = 110, 113 and 115 and to Yu.Ts. Oganessian for providing him with Ref. [11]. He would like also to thank I. Berkes, M. Cox, R.A. Hefferlin and M. Laing for interesting discussions and correspondence.

References

[1] I.A. Malkin and V.I. Man’ko: Zh. Eksp. Teor. Fiz. 2 (1965) 230 [Sov. Phys. JETP Lett. 2 (1965) 146]. [2] A.O. Barut and H. Kleinert: Phys. Rev. 156 (1967) 1541; 157 (1967) 1180; 160 (1967) 1149. [3] M. Kibler and T. N´egadi: Lett. Nuovo Cimento 37 (1983) 225; J. Phys. A : Math. Gen. 16 (1983) 4265; Phys. Rev. A 29 (1984) 2891. [4] M. Boiteux: C. R. Acad. Sci. (Paris), S´er. B, 274 (1972) 867; 276 (1973) 1. [5] M. Kibler and P. Winternitz: J. Phys. A: Math. Gen. 21 (1988) 1787. See also D. Lambert and M. Kibler: J. Phys. A: Math. Gen. 21 (1988) 307. [6] A.O. Barut: in Proc. Rutherford Centennial Symp., Christchurch 1971 (Ed. B.G. Wybourne), Univ. of Canterbury Publications, Christchurch, New Zealand, 1972. [7] V.G. Konopel’chenko: Preprint 40-72 (1972), Institute of Nuclear Physics, Siberian Branch, Academy of Sciences of the USSR, Novosibirsk. A.I. Fet: Teor. i Mat. Fiz. 22 (1975) 323 [Theor. Math. Phys. 22 (1975) 227]. See also: V.G. Konopel’chenko and Yu.B. Rumer: Usp. Fiz. Nauk 129 (1979) 339 [Sov. Phys. Usp. 22 (1979) 837]. [8] V.M. Byakov, Y.I. Kulakov, Y.B. Rumer and A.I. Fet: Preprint ITEP-26, Moscow (1976); Preprint ITEP-90, Moscow (1976); Preprint ITEP-7, Moscow (1977). [9] E. Madelung: Die matematischen Hilfsmittel des Physikers, Springer, Berlin, 1936. [10] M. Kibler: J. Mol. Struct. (Theochem) 187 (1989) 83. M.R. Kibler: Mol. Phys. xxx (2004) xx; in The Periodic Table: Into the 21st Century (Eds. D.H. Rouvray and R.B. King), Research Studies Press, Baldock, United Kingdom, 2004; in The Mathematics of the Periodic Table (Eds. D. H. Rouvray and R. B. King), to be published. [11] Yu.Ts. Oganessian et al.: Phys. Rev. C 69 (2004) 021601; 69 (2004) 029902. [12] G.T. Seaborg: Phys. Scipt. A 10 (1974) 5. [13] G.V. Zhuvikin and R. Hefferlin: Joint Report SC/SPBU-1, 1994; C.M. Carlson, R.A. Hefferlin and G.V. Zhuvikin: Joint Report SC/SPBU-2, 1995.

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