V. N. OSTROVSKY

WHAT AND HOW PHYSICS CONTRIBUTES TOUNDERSTANDING THE PERIODIC LAW

ABSTRACT. The current status of explanation worked out by Physics for thePeriodic Law is considered from philosophical and methodological points of view.The principle gnosiological role of approximations and models in providing inter-pretation for complicated systems is emphasized. The achievements, deficienciesand perspectives of the existing quantum mechanical interpretation of the Peri-odic Table are discussed. The mainstream ab initio theory is based on analysis ofselfconsistent one-electron effective potential. Alternative approaches employingsymmetry considerations and applying group theory usually require some empir-ical information. The approximate dynamic symmetry of one-electron potentialcasts light on the secondary periodicity phenomenon. The periodicity patternsfound in various multiparticle systems (atoms in special situations, atomic nuclei,clusters, particles in the traps, etc) comprise a field for comparative study of thePeriodic Laws found in nature.

1. INTRODUCTION

Since the formulation of the Periodic Law by D. I. Mendeleev1 itattracted a large number of researchers from different fields, first ofall chemistry, but also physics, philosophy, history of science etc.Among them physics claims to provide explanation of the originof periodicity on the microlevel, by using the methods of quantummechanics.

The present paper concerns general philosophical and metholo-gical features of the contribution by physics to understanding thePeriodic Law. Its objective is twofold. First, section 2 contains somegeneral remarks on the character of explanations which physicsaims to provide to the phenomena of nature. To understand theissue properly it is essential to distinguish between the quantitativeresults and explanations. These two major types of research outputin physics (or, broader, in any advanced natural science) seem to

Foundations of Chemistry 3: 145–182, 2001.© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

146 V. N. OSTROVSKY

be in complementary relation in the spirit of the universal comple-mentarity principle put forward by N. Bohr (1999). As discussedbelow, due to its very general character, the Periodic Law evadesexact quantitative formulation, therefore it is natural that physicsseeks for its explanation.

The second goal of the present contribution is a brief outline ofthe current state of research in physics concerning the phenomenonof periodicity. Regretfully, it seems that a significant part of workersoutside physics limit the issue to the famous papers by Bohr (1977)supplemented by the methods of quantitative atomic structure calcu-lations developed later in modern quantum mechanics. The historyof the early developments is studied and analyzed in much detail2

albeit the latest advancements in the explanatory aspect frequentlyremain beyond consideration. However, these results are relevantwhen the Periodic System of Elements is concerned. In the mainbody of the paper these developments are discussed from the meth-odological point of view. We single out more traditional approachesrelying on analysis of filling of one-electron orbitals (section 3) andthe symmetry considerations based on mathematical Group Theory(section 4). Additionally, in section 5 we demonstrate that the Peri-odic System is only one (although the most widely known) memberin the family of periodicity phenomena in multi-particle systems.Comparing various periodic laws met in nature one can better under-stand what physical aspects are crucial in creating different patternsof periodicity.

Some important points are worth preliminary mentioning. First,the Periodic Law embraces vast amounts of knowledge. It is difficultto formulate the Law exactly and unambiguously (see, for instance,discussion by Scerri and McIntyre (1997c)). Only part of it could becast in physical terms and expressed in quantitative measure (suchas atomic radii or ionization energies and also densities, specificheat etc.). The major part of information belongs to the realmof pure chemistry and includes a variety of facts about valences,chemical compounds and reactivity etc. These chemical propertiesare often difficult to formalize and express numerically.3 Chemicalperiodicity itself has a very particular flavour. It is not exact, butonly approximate, since each chemical element possesses its indi-viduality; the period is not constant but varies, albeit recurrences

PHYSICS AND PERIODIC LAW 147

occur in a regular way. All this clearly implies that physics doesnot encounter a quantitative problem but the task of explanation.Moreover, as argued in section 4, this is a classification problem toa large extent.

Such a conclusion is intimately related to our second point thatone deals here with complicated system, whatever exact definitionof the latter notion could be adopted.4

The third point stems from the fact that historically the Peri-odic Law was formulated based on empirical data before physicswas capable of approaching the problem on the microlevel. Thepower of physical (and, more generally, scientific) approach to theworld is manifested most convincingly when predictions are made.5

In this respect the Periodic Law provides very limited possibil-ities for exploits since a major part of the work was already doneby chemists, starting from Mendeleev who had predicted severalnew elements. Therefore it is understandable that sometimes thequestion is asked (Scerri, 1994a): “Does physics provide a trulydeductive explanation of the periodic table, or does it simply re-stateMendeleev’s discoveries?” Here we limit ourselves to indicatingthat according to some modern historical studies (see bibliographyprovided by Scerri and McIntyre (1997c)) “Mendeleev’s abilityto accommodate the already known elements may have contrib-uted as much to the acceptance of the periodic system as did hisdramatic predictions”. This great example shows that the properoutlook could be of very substantial importance even without muchpredictions provided.6

2. EXPLANATIONS VERSUS CALCULATIONS

Sometimes it is argued (Scerri and McIntyre, 1997c) that “Thereduction of the periodic table [to physics] should mean the abilityto calculate exactly the total energies or other properties of the atomsin the periodic table”. In our opinion, these stringent requirements7

would be more relevant in the case when reduction of not only thePeriodic Table, but entire chemistry is concerned. Anyway, in thispaper we do not deal with reduction, but with physical explanationand interpretation. By explanation we imply (approximate) replace-ment of a complicated system by a system which is ‘simple’, i.e.

148 V. N. OSTROVSKY

possesses well known properties and appealing to what is calledphysical intuition or physical sense. A more exact definition of thelatter notions would imply serious study that is beyond the scope ofthe present paper.8

Current progress in computer techniques makes it even moreacute that there is always a complementary relation between calcu-lations and explanation. To illustrate it on relevant material letus consider evaluation of such a basic atomic property as ioniza-tion potentials Ia of neutral atoms.9 Modern quantum mechanics iscapable of doing this numerically with rather high accuracy, takinginto account relativistic effects (important mostly for heavy atomsin the vicinity of nucleus) and avoiding electron orbital approxima-tion, for instance, by using the configuration mixing technique. Suchcalculations inevitably produce well known quasiperiodic depend-ence of Ia on the element number Z (which is atomic nucleuscharge), thus reflecting the Periodic Law. The question is: can wesay that in this way we obtain the physical explanation or interpret-ation of the Periodic Law? Our answer would be: no. By performingcalculations of this type one carries out a kind of mathematicalexperiment, results of which could be compared with the real phys-ical experimental data. The successful comparison convinces usonce more that quantum mechanics is a valid theory being capableof reproducing reality in its quantitative aspect. However, the qual-itative explanation in most cases (i.e., for sufficiently complexsystems) cannot be achieved by precise numerical calculations. Theset of numerically obtained ionization potentials hardly contributesmore to understanding the periodicity than the set of empirical datafor the same quantity. For complicated systems the real explanationis possible only within some hierarchy of approximations or models.As far as we know, the principle importance of approximations inphysics was first emphasized by V. Fock (1936, 1974) who stressedthat new notion in physics appear when approximate methods areintroduced. These notions are absent in the general theory and canbe formulated only within approximations.

If one’s objective is to obtain the best numerical results, thenthe approximations are something to evade and at least to limit asmuch as possible in the course of scientific progress: less approx-imations provide better numerical output. However, if one seeks

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explanations, then dropping some approximations may hopelesslydestroy the entire framework. Indeed, usually the bare exact equa-tions for complex system provide very limited insight. This, ofcourse, does not necessarily mean that the same set of explanatoryapproximations or models would be retained forever. On the pathof historical progress the models could be substantially modified oreven completely new models could be developed, but neverthelessthe models and approximations remain substantial and an inevit-able part of the explanation for the complex system, and not someannoying deficiency.10

Of course this situation is not specific for relation betweenphysics and chemistry but constitutes a substantial feature of theentire physical approach to nature. For instance, the theory of solidstate is not normally based on ab initio calculations of electronicstructure of crystals, but on some intermediate models (althoughsome ab initio approaches were successfully developed in the lastdecades). This in no way invalidates numerous important achieve-ments in this branch of physics. Nuclear physics is also essentiallybased not on ab initio theories, but on models. In general, physics isreduced to a small number of theories only in principle, whereas thepractical applications are based on many cases on some models of‘intermediate’ character, standing in between the basic theories andconcrete developments.

The approximations and models are a fully legitimate part of thetheory, but not its temporary, abominable and shameful part. Everytextbook in quantum mechanics includes some simple problemsuch as bound states in one-dimensional potential wells, scatteringon potential barrier, harmonic oscillator, hydrogen atom etc. Thisis mostly not because these problems are capable of accuratelydescribing nature, but because they allow the reader to developimportant qualitative quantum concepts (such as form of the bound-state wavefunction, tunneling phenomenon, above-barrier reflec-tion, etc.). The basic approximations and the simple model problemswith easily grasped properties form the kind of appropriate languagein which explanation of complicated situations can be developed.Of course, as is the case of every language, the explanations areaddressed to a knowledgeable audience.

150 V. N. OSTROVSKY

The complementarity concept originated from Bohr’s works inphysics but he later applied it in fields outside physics, such aspsychology, biology and anthropology (Bohr, 1999). The epistem-ological significance of this concept stems from the fact that itconcerns a very general pattern of relations between subject andobject. Within the complementary pair ‘numerical calculations’ –‘explanation’, the numerical calculations seek to reproduce a phys-ical object with the highest possible precision whereas explanationsappeal to the subject. In particular, this means that explanationappeals to some community of researchers and it can be differentfor various communities.

The problems solvable in closed form provide an importantingredient for creating explanatory networks. This is becauseusually such problems have transparent properties, clear physicalmeaning and can be employed as simple building blocks in orderto construct comprehensible approximations. However, sometimesexactly solvable problems have unclear physical interpretation andin such cases they prove to be useless for explanation. Note thatall exactly solvable problems in physics describe some idealizedreality. For instance, calculations of the energy spectrum for the one-electron atom or ion by using Schrödinger equation does not accountfor relativistic effects; Dirac’s equation does not include quantumelectrodynamics effects; in more exact approaches the structure ofthe atomic nucleus is to be taken into account etc. In describing anyreality physics cannot avoid approximations.

3. EXPLANATION OF PERIODIC SYSTEMS BY MODERNQUANTUM MECHANICS

3.1. Standard approximations in atomic theory

The complexity of the multi-electron atom is well illustrated byarguments provided by Hartree (1957). The wave function of asystem with q electrons depends on q electron vector coordinates.When one wishes to tabulate the wave function for the neutral Featom with q = 26 taking only 10 values for each variable, then, evenafter reducing the amount of information by using the symmetryproperties, the table still would contain 1053 entries. All the mattercontained in the solar system is insufficient to print such a table.

PHYSICS AND PERIODIC LAW 151

It is hopelessly huge even when one takes into account all feasibleprogress in computing facilities. Therefore the multi-electron atomscannot be treated without approximations.

The basic approximation employed exploits the idea of a self-consistent field which presumes that the electrons move independ-ently in some mean field created by other electrons and the atomicnucleus. Mathematically this is a drastic simplification allowing oneto present the wave function with q vector arguments as a productof some small number of functions with one argument, the latterfunctions being known as one-electron orbitals. Due to sphericalsymmetry of the self-consistent field11 the atomic orbital in factdepends on a single scalar radial variable r, i.e. electron distancefrom the nucleus. The set of orbitals employed for the construc-tion of the wave function (so called filled, or occupied orbitals) isnamed configuration of atom.12 The most exact scheme based onone-electron orbitals and atomic configuration is the Hartree-Fockapproximation (Fock, 1930; Hartree, 1957; Slater, 1947). It properlyaccounts for the Pauli principle which concerns the wave functionsymmetry under particle permutation, or, in other words, for elec-tron exchange. The self-consistent Hartree-Fock method is one ofthe fundamental tools designed in quantum mechanics to tackle avariety of multi-particle systems (not only atoms and molecules, butalso clusters, the objects studied in the condensed-matter physics,atomic nuclei etc). When quantitatively more exact methods aredeveloped to account for electron correlation effects, i.e., effectsbeyond the Hartree-Fock approximation (for instance, configurationmixing, or superposition schemes, so-called Random Phase Approx-imation, Density Functional Theory etc.), in most cases they arebased on the Hartree-Fock method as the first approximation.

According to quantum mechanics, the orbitals in atoms arelabeled by angular quantum numbers l, m and radial quantumnumber nr that is the number of nodes, i.e., zeroes in r-dependence.The principal quantum number n is conventionally employed fororbital labeling instead of nr , being defined as n ≡ nr + l + 1. Itis worthwhile to stress that using atomic configurations does notmean that an orbital, or a set of quantum numbers are ascribed toany particular electron. This is clear from the fact that the many-electron wavefunction is constructed from a product of orbitals by

152 V. N. OSTROVSKY

antisymmetrization over electron permutations. Thus the coordinateof any electron stands in the arguments of different orbitals. Theactual meaning of the configuration is in ascribing to the whole atoma set of approximate constants of motion, or integrals of motion,or in other words, a set of approximate labels { nj , lj , ζj } wherethe index j runs over occupied, or filled orbitals and ζj is the occu-pation number, i.e., number of electrons on j-the orbital. In thenon-relativistic approximation an atom has only two exact integralsof motion: the total orbital momentum �L and total spin �S (whenrelativistic effects are taken into account, only the total angularmomentum �J = �L + �S is exact integral of motion). The integralsof motion play a very important role in quantum mechanics, sincethey make possible the classification of states (see also section 4.1).The number of exact integrals of motion is usually insufficient forcomplete classification. Therefore, the importance of approximate(but ‘good’, i.e., well conserved) integrals of motion is difficult tooverestimate.13

It can be noted that ‘global’ atomic quantum labels L, S and J(that define atomic multiplet levels) cannot be uniquely deducedfrom the configuration { nj , lj , ζj}: generally several sets of‘global’ numbers are allowed. Moreover, the non-relativistic self-consistent field depends on the choice of L and S. This meansthat one-electron energies as well as the total energy of the atomare somewhat dependent on multiplet levels. Quite often a simpli-fied multiplet level averaged Hartree-Fock scheme is employed.However, for heavy atoms the splitting of multiplet levels belongingto the same configuration could be appreciable. It could exceedthe energy separation of different configurations. This circumstanceis of importance when non-regularities in the Periodic Table arediscussed, see section 3.5.

Rigorously speaking, the atomic wave function presents aninfinite sum over different configurations with the same exactquantum numbers. However, if one term in the sum manifestlypredominates it can be safely used for classification of atomic state.

In the Hartree-Fock method each individual orbital correspondsto electron motion in a particular potential which generally is non-local (due to exchange effects). Such a potential is spread overspace and, strictly speaking, is not a potential at all (localization

PHYSICS AND PERIODIC LAW 153

is achieved by using Density Functional approaches). Besides this,the one-electron potential is not universal but depends on an indi-vidual orbital considered, although the motion of different electronsis independent or uncorrelated. The energy of the entire system(atom) is different from the mere sum of one-electron energies(Slater, 1947; Melrose and Scerri, 1996). All these complicatingpeculiarities make Hartree-Fock approximation excessively sophis-ticated for interpretation of the Periodic Table, at least at the currentstage of our knowledge.14 In order to describe its major featuresit is sufficient to consider the motion of electrons in unique (forall the orbitals in the atom) local spherically symmetrical potentialUTFa (r) that is obtained within the Thomas-Fermi approximation.

The latter is based on an additional simplification: the semi-classicaltreatment of electrons (Slater, 1947; Gombas, 1949; Landau andLifshitz, 1977). Within it an atom looks like a bulb of electron gas.For subsequent discussion it is important that the potential UTF

a (r) isexpressed in terms of a universal function χ (x):

UTFa (r) = Z

rχ(kr), k =

(8√

2

3π

)2/3

Z1/3 (1)

(hereafter we use standard atomic system of units: e = � = me =1). In the potential (1) all specifics of a particular chemical elementare included in ‘potential strength’ prefactor Z and scaling factork which is proportional to cubic root of the nucleus charge Z.Although the analytical expression for χ (x) is not known, this isa well studied and tabulated function. It can also be approximatedby simple analytical expressions, as discussed in section 3.4. TheThomas-Fermi approximation allows one to carry out some calcu-lations relevant to the building-up of the Periodic System withoutconsidering individual electronic orbitals. The first application ofthis kind was done by Fermi (1928) who found the place in thePeriodic Table where the first electron with a given orbital quantumnumber l appears for the first time. Generally this approach providesa less detailed interpretation of the Periodic Table than that basedon analysis of orbitals. Therefore this line of development is beyondthe main scope of the present paper. Here we only refer to relativelyrecent review by Bosi (1983).

154 V. N. OSTROVSKY

3.2. Basic properties of effective one-particle potential asfoundations of Aufbau principle

The physical explanation of the Periodic Law is based on theobservation that the properties of elements are defined by thefilled orbitals with the lowest binding energies (outer or valenceorbitals). It should be recognized that the object of most phys-ical studies (including the present paper) is in fact the PeriodicTable of Neutral Atoms. This is sufficient for analysis of periodicitypatterns in atomic properties such as ionization potentials, atomicradii etc. Concerning chemistry, the situation looks substantiallymore intricate, although one can refer to explanation of the valenciesof chemical elements using the idea of outer orbital hybridiza-tion for atoms bonded in chemical compounds. Complexity offormalizing and interpreting chemical properties in physical terms(already discussed in section 1) could be considered as a specialmanifestation of non-reducibility of chemistry to physics.

To define which orbitals in an atom are filled one has to usethe building up or Aufbau rule which is a corollary of the methodused for wave function construction. Namely, since we consider theground state of an atom, the electron distribution over orbitals hasto correspond to the minimum energy compatible with the Pauliprinciple. In accordance with other simplifications employed, theapproximate form of the Pauli principle is used: not more than twoelectrons (differing by the spin projection ms = ±1

2) can occupyan orbital labeled by three space quantum numbers (n, l, m). Theground state provides a minimum to the energy of whole atom thatwithin the Hartree-Fock method does not necessarily correspond tothe minimum sum of one-electron energies for the occupied orbitals.This is due to the method’s peculiarities already mentioned above informula (1); see more discussion in the end of section 3.5. However,within the Density Functional Theory the relation between the twominima becomes direct due to Janak’s theorem (Janak, 1978). Thisholds also for our cruder scheme based on the one-electron potential(1).

Clearly, the result of applying the Aufbau rule directly dependson the ordering of the levels in a one-electron effective potentialUa(r). Depending on the form of Ua(r) a variety of very differentPeriodic Tables can be obtained which are realized in different

PHYSICS AND PERIODIC LAW 155

objects of nature as discussed also in section 5. This point isfrequently overlooked in the simplified exposures where implicitlyit is presumed that Ua(r) is close to a pure Coulomb potential, thatis operative in the simplest atom, that of hydrogen. Below we showthat this choice is generally unjustified and rather unfortunate sinceit provides the wrong form of the Periodic Law which does not agreewith empirical observations.

Hence it is worthwhile to consider the basic properties of theeffective potential seen by the electron in an atom. Far from thenucleus the electron is affected by an attractive Coulomb potentialof the atomic nucleus screened by all other electrons in an atom,therefore

Ua(r) ≈ −1

rr � ra, (2)

where ra is a characteristic radius of an atom. Near the nucleus thescreening effect vanishes and the electron is attracted by Coulombpotential of the bare atomic nucleus

Ua(r) ≈ −Z

rr ra. (3)

Considering the outer (valence) electrons, one could try to constructthe model of effective potential based on the −1/r approximatebehavior (2). For a pure Coulomb field (i.e., in the hydrogen atom)the energy levels (EC

nl = −1/(2n2)) are degenerate in the orbitalquantum number l. From formula (3) one infers that for small rthe effective potential is always somewhat deeper than −1/r poten-tial (Z > 1). This shifts the energy levels downwards. The effect ismore pronounced for the low-l levels, since for higher l the effectivecentripetal potential prevents penetration of small-r domain, i.e., inother words, the high-l orbitals have maximum of electron densityfar from the nucleus.15 Hence the degeneracy over l is lifted and onecomes to the hydrogenlike (n, l) Aufbau scheme: the one-electronorbitals are occupied in order of increasing principal quantumnumber n; for the same n the orbitals are filled in order of increasingorbital quantum number l. The periods in this scheme correspond tothe hydrogenic n-shells. Taking into account the electron spin, oneobtains the period lengths

2n2 = 2, 8, 18, 32, 50, . . . . (4)

156 V. N. OSTROVSKY

Comparison with the detailed numerical quantum calculations formultielectron systems and with empirical data shows that for highlyionized atoms the (n, l) ordering rule is indeed operative. Its heuristicderivation given above presumes that the deviation of the effectiveone-electron potential Ua(r) from −1/r behavior can be treated as asmall perturbation.

However, the structure of the neutral atom is of importancewhen the Periodic Law is considered. Here the (n, l) rule fails.This implies that the effective potential exhibits strong deviationsfrom the Coulomb one, leading to the substantial rearrangementsof the spectrum. The overlap appears between the groups of energylevels with different principal quantum numbers n. The n-groupingof levels disappears, but a new type of regularity emerges in theform of (n + l, n) to be considered in section 3.3. Its descriptionrequires use of a non-Coulomb one-electron potential from the verybeginning.

Before concluding this subsection we emphasize again that thenotion of n-shell, i.e. states with the same principal quantum numbern, originates from the fact that for pure Coulomb potential, i.e., forhydrogen atom, these states are degenerate in energy. If the poten-tial differs slightly from Coulomb one, the degeneracy is lifted, butthe levels with the same n remain grouped together on the energyscale. In this case the notion of shell remains physically mean-ingful. Otherwise, if deviation from the Coulomb potential is strong,complete regrouping of levels occurs and hydrogenlike shells loosephysical meaning becoming purely formal entities On the contrary,the notion of subshell labeled by a couple of quantum numbers {n, l}always remain valid for atoms since the energy levels are degeneratein azimuthal quantum number m in any spherically symmetricalpotential.

3.3. Periodic Table as a result of ordering of one-electron levels inatoms

The simplest and most straightforward thing to do is to take the one-electron potential UTF

a (r), calculate the one-electron energy levelsnumerically and compare the resulting Periodic Table with empiricaldata. Such a program was implemented by Latter (1955) who found

PHYSICS AND PERIODIC LAW 157

an agreement (earlier works were summarized by Gombas (1949);see also Demkov and Berezina (1973)).

This should be considered as a remarkable achievement: theperiodicity phenomenon is reproduced by using a single universalfunction describing a one-electron potential. Although based on aset of approximations, this analysis should be classified as ab initiosince it does not employ any empirical or fitting parameters.

Additional very important and useful insight in the orbital fillingcan be provided by considering an effective potential operative fororbitals with orbital momentum l. Accounting for the centrifugalrepulsion this potential Ua(r) + l(l + 1)/(2r2) exhibits a double-wellstructure revealed first by M. Göppert-Mayer (1941). It is importantfor analysis of space localization of d and f orbitals and competitionbetween filling of these orbitals and s-orbitals (Connerade, 1998).

However, for one interested in explanation these results cannotprovide full satisfaction, since they do not directly interpret theorigin of (n + l, n) ordering rule. Empirically it was noticed thatthe Periodic System is well described by the following simple rule:the orbitals are filled in the order of increasing sum N ≡ n + l,and for the fixed N in the order of increasing n. It is difficult totrace the origin of this rule that looks like a kind of scientific folk-lore. Without giving references, Löwdin (1969)) ascribes this ruleto Bohr, but remarks that “Bohr himself was never too explicitabout his ‘Aufbau’-principle and the (n + l, n) rule is sometimesreferred to as Goudsmith-rule or Bose-rule” (no references to Goud-smith or Bose are given either).16 The rule was published only in1936 in Madelung’s handbook (Madelung, 1936) rather implicitlyas an adopted form of the Periodic Table, although according toGoudsmith (Goudsmith and Richards, 1964), he received privatecommunication from Madelung about this rule in December, 1926.In between these dates Karapetoff (1930) used this rule to predictconfigurations of transuranian elements, up to Z = 124. Later therule was rediscovered independently by a number of authors as anempirical ‘lexicographic’ rule without theoretical ab initio found-ation: Carroll and Lehrman (1942), Wiswesser (1945), Yeou Ta(1946), Simmons (1947, 1948), Hakala (1952), Ausubel (1976)(see also paper by Dash (1969) and book by Condon and Odabasi(1980)). The works by Klechkovskii (1951, 1952a, 1952b, 1952c,

158 V. N. OSTROVSKY

1953a, 1953b, 1954, 1960, 1961, 1962) summarized in his book(Klechkovskii, 1968) should be particularly praised since this authorstudied systematically different aspects of the (n + l, n) rule in muchdetail. Nevertheless the dynamical origin of the sum of principal nand orbital l quantum numbers remained mysterious. This particularlinear combination of quantum numbers has never appeared as aresult of solution of any Schrödinger equation.17 This circumstanceinduced Löwdin to write in 1969 that “it is perhaps remarkable that,in axiomatic quantum theory, the simple energy rule (order of fillingof orbitals) has not yet been derived from first principles”. Sincethat time some substantial progress has been achieved, as discussedbelow.

3.4. n + l rule and orbital genesis

The origin of quantum number N = n + l was understood by Demkovand Ostrovsky (1971b) who further simplified the Thomas-Fermipotential (1) by using an analytical approximation for the functionχ (x):

χ(x) = 1

(1 + αx)2. (5)

As discussed by the authors, within the Thomas-Fermi theory thisapproximation was considered before by Tietz (1954, 1955) (seeoriginal paper (Demkov and Ostrovsky, 1971b) for a complete listof numerous papers by this author), the parameter α ≈ 1

2 beingdefined by applying the variational principle or normalization condi-tion in momentum space. However, it was not noticed earlier thatthe Schrödinger equation for the related potential can be solvedanalytically for one particular value of energy, namely for E = 0.The derivation of such a solution allowed Demkov and Ostrovsky(1971b) to consider what could be named the genesis of the atomicorbitals in the approximate one-electron potential obtained fromformulas (1) and (5):

UDOa (r) = − Z

r(1 + r/R)2, R = α−1Z−1/3

(3π

8√

2

)2/3

. (6)

The point is that as the nucleus charge Z increases, the potential well(6) becomes deeper and new energy levels appear on the border E

PHYSICS AND PERIODIC LAW 159

= 0 between the discrete energy spectrum and the continuum. Inthe general case appearance of {n, l} bound level occurs at somevalues of parameter18 Z = Znl labeled by two quantum numbers nand l. However, a remarkable property of the potential (6) is thatthe levels with the same sum N = n + l appear simultaneously atcertain critical values of the parameter Z = ZN . This at once showsthat the N-grouping is realized for this particular potential, insteadof n-grouping for a weakly distorted Coulomb potential (see section3.2).

To reduce the second part of the (n + l, n) rule one haveto consider how the levels within the same N-group are orderedfor a deeper potential, i.e., for Z > ZN , when the levels energiesdiffer from zero and N-degeneracy is lifted. Such analysis wassuccessfully carried out by using perturbation theory (Demkov andOstrovsky, 1971b). This completes theoretical ab initio (i.e. notusing empirical information or fitting) derivation of the (n + l, n)filling rule.

As shown by Demkov and Ostrovsky (1971a, 1971b), the poten-tial (6) belongs to a broader family of potentials Uµ(r)

Uµ(r) = − 2v

r2R2[(r/R)µ + (R/r)µ]2(7)

with strength parameter v and coordinate scaling parameter R. Thepotentials (7) are exactly solvable for E = 0 and exhibit degen-eracy of levels with the same linear combination n + (µ−1 −1)l of quantum numbers n and l. Thus the degeneracy pattern isdirectly defined by the potential parameter µ. Some other membersof the family Uµ(r) are also physically important, for instance,the famous Maxwell’s fish-eye (Maxwell, 1952; Demkov andOstrovsky, 1971a) (µ = 1) or potentials with higher µ used in theanalysis of periodicity met in clusters (Ostrovsky, 1997), see section5. As soon as the family is known, one can derive potential (6) in analternative way (Demkov and Ostrovsky, 1971b), namely selectingthe (n + l) degeneracy pattern by putting µ = 1

2. Remarkably, in thisway we come at once to the potential of the form (6) which exhibitsCoulomb behavior ∼ −1/r as r → 0. Such a potential singularityis absent for other potentials in the family Uµ(r). Thus the (n + l)-grouping of levels proves to be intimately related to the Coulomb

160 V. N. OSTROVSKY

attraction to the atomic nucleus – a beautiful connection whichprobably urges for deeper understanding. There is no contradictionhere to the statement that the potential (6) provides an explanationto the Aufbau Principle. First of all, the explanations are subjectto improvements, just as the numerical calculations are made moreprecise; second, there are different levels in the hierarchy of explan-ations. From yet another points of view the potential (6) is discussedby Wheeler (1976) and Tarbeev et al. (1997).

Before concluding this subsection we have to stress an importantcircumstance. Notwithstanding different possibilities of its deriva-tion, the potential (6) has direct physical meaning as an approx-imation for an effective one-electron potential. This is completelyclear already from the fact that, as mentioned above, the approxim-ation (5) was known in Thomas-Fermi theory prior to the paper byDemkov and Ostrovsky (1971b). There is crucial difference betweenpotential (5) and other analytical atomic potentials suggested ad hoc,see, for instance, Exman et al. (1975) or Kaldor (1977).

Unfortunately, there is a tradition of misinterpretation of themeaning of the potential (6) in the literature. Kitagawara and Barut(1983) wrongly claim that “The Demkov-Ostrovsky equation is justa mathematical model providing the quantum number n + l andits degeneracy. The coordinates appearing in this equation do nothave a direct physical meaning such as the spatial coordinates ofthe valence electron in an atom”. In the same spirit Scerri et al.(1998b) call this potential “heuristic” [in the sense ad hoc] that“leaves us with necessity to explain where this particular potentialcame from”.19

3.5. Exceptions to the n + l rule

Although the (n + l, n-rule provides mostly correct ordering ofelements in the Periodic Table, the dimensions of N-groups (N =n + l)

2, 2, 8, 8, 18, 18, 32, 32, . . . (8)

differ from the period lengths in the Periodic Table

2, 8, 8, 18, 18, 32, 32, . . . . (9)

PHYSICS AND PERIODIC LAW 161

The difference is shown in more detail in the following scheme

n+l=1︷︸︸︷1s︸ ︷︷ ︸

dim=2

n+l=2︷︸︸︷

2s︸ ︷︷ ︸dim=2

<

n+l=3︷ ︸︸ ︷2p 3s︸ ︷︷ ︸

dim=8

<

n+l=4︷ ︸︸ ︷3p 4s︸ ︷︷ ︸

dim=8

< (10)

<

n+l=5︷ ︸︸ ︷3d < 4p 5s︸ ︷︷ ︸

dim=18

<

n+l=6︷ ︸︸ ︷4d < 5p 6s︸ ︷︷ ︸

dim=18

<

<

n+l=7︷ ︸︸ ︷4f < 5d < 6p 7s︸ ︷︷ ︸

dim=32

<

n+l=8︷ ︸︸ ︷5f < 6d < 7p 8s︸ ︷︷ ︸

dim=32

< . . .

where dimensions of (n + l)-shells are indicated with account for theelectron spin. Following Novaro (1973) and Katriel and Jorgensen(1982) we denote by symbol the large energy gaps between theone-electron levels. These gaps (dividing the periods in the PeriodicTable) do not coincide with the borders between the (n + l)-groupsof levels. The quantum interpretation of the difference between thesequences (8) and (9) was suggested by Ostrovsky (1981). Briefly,due to particular properties of weakly-bound s-states in quantummechanics, they are shifted to the higher-N group and join it on theenergy scale.

Comparing (8) with results of the hydrogenic (n, l)-rule (4) onesees that the period lengths met are the same in both cases, beingequal to 2N 2 with some integer N . However, in case of (n + l,n)-rule (8) each length (except the first one) appears twice. Theselengths doubling are an important feature to be discussed further insection 4.

Considering individual atoms, there are 18 exceptions to the (n +l, n)-rule as listed by Demkov and Ostrovsky (1971b), some of thembeing discussed by Scerri (1991b, 1997b, 1998b, 1997a) in moredetail. In 16 cases the real configuration differs from that predictedby the (n + l, n) rule by a single electron on occupied orbital; only intwo cases the difference is by two electrons. The exceptions couldarise due to a variety of reasons.

− First of all, as discussed in section 3.1 for given electron config-uration generally there exist several multiplet levels of atomdiffering by total orbital momentum L and total spin S. As

162 V. N. OSTROVSKY

argued by Goudsmith and Richards (1964), “Many minor devi-ations from the Madelung rule can be ascribed to the largespread of multiplet levels in the complex energy configura-tions. While the center of gravity of the multiplet levels mayobey the Madelung rule, one of the levels of a higher statemay be pushed down below the lower state by large exchangeinteraction.”

− For heavy atoms the mixing of different configurations gener-ally becomes more significant than for light ones. When thenumber of electrons in atoms is large, different sets of occupiedorbitals can result in close values of total energy. As a generaltrend the configurations are mixed stronger when they are closein energy. The mixing might result in violation of the (n + l, n)rule.

− The relativistic effects are not included in the model. Theseeffects are most important for inner electrons; and indirectly,via the form of the effective potential created jointly by all theelectrons and nucleus, they could also affect the outer electrons.Additionally, relativistic Dirac wave functions for ns and np1/2states are known to have singularities at the Coulomb center,i.e., as r → 0. This means enhancement of electron densityin the vicinity of atomic nucleus where relativistic effects arestronger. According to Pyykkö (1988), “the relativistic changeof the atomic potential matters less than the direct dynamicaleffect on the valence electron itself”.

The important question is whether the number of exceptionsis large enough to undermine or even fully discredit the physicalexplanation of the Periodic System. In our opinion the situationhere corresponds to complexity of the system and relative simplicityof the explanation based on analytical approximation (6) for one-electron potential in atoms. The complexity of the Periodic Lawwas discussed already in section 1. Carroll and Lehrman (1942)argue that exceptions to (n + l) rule “are relatively unimportantfor chemists”. Additionally it should be recognized that for heavyatoms sometimes it is not easy to assign elements a place in thePeriodic Table based on purely chemical information (see surveyof history for rare earth elements by Scerri (1994b)) since “Theperiodic table contains many subtleties and anomalies which have

PHYSICS AND PERIODIC LAW 163

defied attempts at a complete reduction” (Scerri, 1996). Withoutfirm assignment based on the value of nucleus charge some discus-sions would probably continue till now. Therefore, regardless ofthe exceptions, the situation looks satisfactory, although it wouldbe extremely interesting to explain20 why the exceptions occur forthe particular elements. Recently an interesting attempt (Tarbeev etal., 1997) was made to describe an effective potential which doesnot lead to exceptions; however, note that the analysis was based onempirical data, i.e., it cannot be classified as ab initio.

As discussed above, a rather long chain of approximations isused to come from the exact Schrödinger equation for the many-electron atom to the approximate one-electron potential (6). Fromthe point of view of the Hartree-Fock method, the filling of orbitalsis defined by a rather subtle interplay of various effects. Considerthe competition between filling 4s and 3d orbitals studies in detailby Melrose and Scerri (1996). The correct configurations of atomsfrom Sc to Cu were obtained by considering the energies of one-electron orbitals. However, the result could not be derived withinthe Hartree-Fock method using additional so called frozen-orbitalsapproximation. The calculations have to take into account that theenergy of each orbital depends on the occupation numbers for allother orbitals. Therefore, the orbital energies may be changed by anelectronic transition. The calculations by Melrose and Scerri (1996)suggest some kind of explanation for deviations from the (n + l,n) rule that take place for Cr and Cu atoms. The other lesson thatprobably could be learned from these calculations is that the poten-tial (6) effectively absorbs some features that in fact lie beyond thesimplistic one-electron scheme.

4. ALTERNATIVES TO ORBITAL-FILLING APPROACH:GROUP-THEORETICAL TREATMENTS

4.1. Classification, symmetry and group theory

It is common wisdom in science that a reasonable classificationtestifies to a rather high level of knowledge. Physics often usesthe formalized method of classification provided by mathematicalGroup Theory. The group-theoretical approach in physics is basedon the notion of symmetry.21 The operations which do not change

164 V. N. OSTROVSKY

the system (more exactly, leave invariant its Hamiltonian operator)comprise the symmetry group. These operations can have geometricmeaning (such as rotations in the case of spherically symmetricalpotential), but it also can evade direct geometrical interpretation(so called hidden symmetries). By applying to the eigenstate anoperation belonging to the symmetry group we obtain another(degenerate) eigenstate with the same energy.

Another important notion is the dynamical group which includesthe symmetry group, but contains also operators allowing one toconstruct a complete set of eigenstates starting from any particularone.

If the group is known, then mathematical techniques allow oneto produce a set of labels necessary for classification of states;the dynamical group provides complete classification whereas thesymmetry group is able to predict degeneracy patterns met in thesystem. For instance, the symmetry of central potentials is describedby the three-dimensional rotation groups designated as O(3). Thisgroup provides l and m labels and predicts (2l + 1)-fold degeneraciesof energy levels, i.e., the independence of energies on the azimuthalquantum number m.

Both symmetry and dynamical groups could be derived fromanalysis of the Hamiltonian operator of the physical system underconsideration. In this way one can find not only rather obviousgeometrical symmetries, but also hidden symmetries, such as four-dimensional rotation group O(4) for the hydrogen atom revealedby Fock (1935). Some work along these lines was carried out inapplication to the Periodic System; it is referred to below as AtomicPhysics Approach (APA), see section 4.3.

For some physical systems it could occur that the Hamiltonianof the system is not known (or even does not exist, at least inthe conventional sense), but the underlying symmetry or dynam-ical group could be somehow guessed. This was the backgroundfor successful applications of group theory in elementary particletheory. Usually one is interested not only in classification of statesand degeneracies, but also in the ordering of the state energies. Toachieve this, the dynamical group should be supplemented by the socalled mass formula which orders the energy levels depending ontheir labels (in the most fortunate situation the mass formula could

PHYSICS AND PERIODIC LAW 165

directly give the energies, but it is valuable even if it only gives theordering of levels). We refer to this type of group theory applicationas Elementary Particle Approach (EPA).

4.2. Elementary particle approach to Periodic Table

With EPA the chemical elements are formally considered as variousstates of some artificial object: ‘atomic matter’ (Barut, 1972) or‘structure-less particle with inner degrees of freedom’ (Rumer andFet, 1972). Various states of such a system can be labeled by thequantum numbers provided by the chosen group.22

Since EPA is necessarily a phenomenological approach, it isparticularly important (i) to specify exactly the empirical basis forthe choice of the group and mass formula and (ii) to outline howthe formal mathematical scheme can be interrelated to the observ-able physical objects and quantities. The authors who apply EPAto the Periodic System, as anticipated, choose to forget about thestructure of the atom (and even about such basic ideas that an atomconsists of electron and nucleus). Nevertheless, the choice of groupis in fact based on some information outside Group Theory, andthere should be a possibility of comparing the results with empiricaldata. It is highly desirable to demonstrate that the output suitablefor comparison exceeds the empirical input. From this point of viewthe statement that “Within this approach we have to give up all avail-able chemical and spectroscopic information” (Konopel’chenko andRumer, 1979) looks as unfounded extremism, since information ofthis kind is necessary just for choice of the particular group amongthe infinite number provided by pure mathematics.

While some of the works in the field fail to satisfy criteria (i)–(ii),there are a number of papers that admit the goal of finding the groupwhich allows the degeneracy pattern coinciding with the empiri-cally known period lengths (8) in the Periodic Table. The generalguideline in the search for the group is the already mentioned factthat the degeneracies met can be expressed as 2N 2 with someinteger N . The same degeneracies are met in the Coulomb problem,i.e., for hydrogen atom, which both symmetry and dynamical groupshave been known for a long time (see review by Ostrovsky (1981)).Hence, the only problem is to modify the Coulomb field group so asto incorporate the lengths doubling met in the Periodic System, see

166 V. N. OSTROVSKY

discussion in section 3.5. The solution of this problem provesto be non-unique, the specific groups suggested by differentauthors23 (Barut, 1972; Rumer and Fet, 1972; Fet, 1979; Fet, 1980;Konopel’chenko, 1972; Novaro and Berrondo, 1972; Berrondo andNovaro, 1973; Novaro, 1973; Novaro and Wolf, 1971; Novaro,1989) have been critically reviewed by Ostrovsky (1996).

Unfortunately, further perspectives of EPA remain unclear. Itscurrent achievements look like the translation of empirical infor-mation about period lengths to the mathematical language whichis probably more appealing for a certain part of the scientificcommunity, but nothing more. The empirical input (period lengths)is cast in mathematical terms of the dynamical group, but hardlyanything more than period lengths return back to the interestedresearcher who could aspire for some results to be compared withexperiment, or additional physical insight. For most workers in thefield such translation into specialized mathematical language24 doesnot justify an accompanying sacrifice: giving up all references toelectronic structure of atoms. The situation looks very different fromthat in elementary particle theory where the dynamical structure ofparticles is a difficult and not completely solved problem which onecould desire to circumvent. Here the EPA approach was capable ofpredicting new particles, but when applied to the Periodic Law, inour opinion, it provides a very limited contribution.

Concluding this subsection we mention two more paper based onmathematical technique, albeit not a group-theoretical one. Scerri etal. (1998b) described a class of feasible ordering rules that satisfysome natural criteria. It should be indicated that this class is quitebroad and additional restrictive criteria are needed to produce amore limited selection. Purdela (1988) presented some empiricalarguments in favor of n + 1

2 l ordering.

4.3. Atomic physics approach and secondary periodicity

Contrary to EPA, the Atomic Physics Approach is not based onempirical information but seeks to find the symmetry propertiesof exact or approximate Hamiltonians that are already availablefrom ab initio quantum theory. Indications of deep symmetry ofthe effective atomic potential (6) were revealed by Demkov andOstrovsky (1971b).

PHYSICS AND PERIODIC LAW 167

First of all, the classical trajectories in the potential (6) are closedat E = 0 independent of initial conditions, i.e. of orbital momentuml. Namely, the trajectories close after two revolutions around a forcecenter. As known from classical mechanics, generally a trajectoryin the central potential is not closed but covers some ring in itsplane. One of the notable exceptions is the pure Coulomb potentialwhere trajectories are known to close after one revolution around thecenter. This property is a manifestation of hidden O(4) symmetryof Coulomb potential (Fock, 1935) mentioned in section 4.1. Theimportance of potentials providing “double necklace trajectories”for interpretation of the Periodic Table was stressed by Wheeler(1971) and Powers (1971).

The classical trajectories in the potential (6) at E = 0 also possessa focusing property: all the trajectories exiting from any point �rafter one revolution come through the point �r/R2. Thus in terms ofgeometrical optics one can say that the rays emanating from anysource �r are focused at the image point �r/R2.

The relevant quantum analysis was developed by Ostrovsky(1981) who mapped the three-dimensional quantum problem ontothe four-dimensional sphere.25 In particular, this analysis allowedhim to reveal an additional integral of motion designated as T3and defined as the following discrete transformation of the wavefunction (see also Demkov and Ostrovksy (1971a)):

T3ψ(�r) = R

2rψ

(�r R

2

r2

). (11)

In geometry the transformation �r ⇒ �r R2/r2 is known as inversion inthe sphere of radius R. It conformally maps inner parts of the spherer ⇒ R on its outer part, and vice versa. The discrete operator T3has only two eigenvalues 1

2τ with τ = ±1. This situation is similarto the electron spin projection operator ms = ±1

2 , see section 3.2.By analogy with the terminology of nuclear physics the quantumnumber τ could be named atomic isospin. The eigenstates of poten-tial (6) are also eigenstates of operator T3. The odd and even valuesof (n + l) correspond to τ = 1 and τ = −1 respectively. Thus, anadditional integral of motion is revealed, providing an additionalclassifying label τ .

168 V. N. OSTROVSKY

The dynamical group for the Periodic System suggested byOstrovsky (1981) incorporates operator T3 with a clear geomet-rical meaning. This is its advantage as compared with the groupssuggested within EPA. The additional atomic isospin classificationalso has physical (or chemical) meaning.

For a rather long time the so called ‘secondary’ periodicity effectwas discussed in chemistry. Originally Biron (1915) noticed that forthe elements in a given group some chemical and physical propertiesare reproduced most completely not in the adjacent periods, but inevery second period. As an example, he considered tendency of N,As and Bi to be trivalent while P and Sb are pentavalent. Accordingto Smith (1924) “it is . . . a general observation that alternativemembers of a valency group in the periodic table show the greatestchemical resemblance”.26 In chemical literature the secondary peri-odicity is also refered to as an ‘alternation effect’, manifested, forinstance, for electronegativity (Sanderson, 1952, 1960), or for heatsof formation of oxides, and also for ionization potentials and sizesof ions (Phillips and Williams, 1965). Some additional bibliographycan be found in the papers by Ostrovsky (1981) and Pyykkö (1988).The latter reference interprets secondary periodicity in terms ofproperties of Hartree-Fock orbitals. Convincing graphical illustra-tion of the secondary periodicity can also be found in the paper byOdabasi (1973) who depicted variation of ionization potentials andmean value of r−2 along some groups of elements in the PeriodicTable. These properties exhibit saw-like modulation of the generalsmooth trend within the Table column.

The elements within the group that belong to every second periodin the Table correspond to the same quantum number τ . The atomicmodel gives here a new insight relating the secondary periodicityto the properties of the atomic orbitals, namely, the isospin, orT3-symmetry. From this point of view it is particularly importantthat the T3-symmetry is stable with respect to the variation of theatomic potential and the energy of the electron. Indeed, as shown byOstrovsky (1981), the realistic atomic orbitals (calculated withina simplified version of the Hartree-Fock approximation (Hermanand Skillman, 1963)) in a good approximation possess a definiteparity under the transformation T3 (11). The possible applications

PHYSICS AND PERIODIC LAW 169

of this symmetry for the calculation of some matrix elements arealso discussed in this paper.

5. PERIODIC LAWS IN OTHER MULTIPARTICLEPHYSICAL SYSTEMS

5.1. Atomic systems

It should be stressed once again that the ordering of the energylevels in the effective potential is crucial for interpretation of thePeriodic System. The levels in the spherical potential are knownto be degenerate in azimuthal quantum number m, but as for {n,l}-dependence, this can be varied in broad limits depending on thechoice of one-particle potential Ua(r) that is the key problem. In itsturn, the form of the potential Ua(r) is governed by the interactionsoperative between the particles in the system. The characteristicfeatures of the atoms are (i) the presence of massive center offorce (atomic nucleus) and (ii) the long-range Coulomb interactionbetween constituent particles. In the other systems considered insections 5.2–5.4 these features are absent; in particular, instead oflong range Coulomb forces in atoms one meets short range interac-tions in atomic nuclei and clusters. In the present subsection we startwith atomic systems that are similar to ground state neutral atoms,but differ in some important aspects.

5.1.1. Excited states in atoms and ionsKlechkovskii (1952c) was first to notice that for some atoms andlow-charge ions27 the excited levels with the same value of thesum (n + l) are grouped together (see also Klechkovskii (1953b,1953b)). Twenty-five years later Sternheimer (1977a, 1977b, 1977c,1979) considered a vast amount of empirical material and listed theexamples of overlapping and non-overlapping (n + l)-groups in thespectra of one-electron excitations for atoms and ions. Sternheimer(contrary to Klechkovskii) did not use a convenient representation interms of the quantum defects, which made his discussion redundant(since it was sufficient to consider a small number of Rydberg seriesof levels instead of a large number of individual levels). The (n + l)-grouping is observed for levels with small l (l ≤ l0) while for largerl it is replaced by hydrogen-like n-grouping.

170 V. N. OSTROVSKY

The aforementioned authors did not give a quantum mechanicalexplanation of the observed regularities (for example, Sternheimertentatively related them to the relativistic effects, magnetic inter-actions etc.). The quantum mechanical interpretation of the (n +l)-grouping was developed by Ostrovsky (1981) via analysis ofproperties of an effective one-electron potential. He found theborderline l0 for different atoms and established the relationshipbetween (n + l)-grouping of excited levels and the (n + l, n) fillingrule for ground states.

5.1.2. Positively charged ionsIt was already indicated in section 3.3 that for multicharged ionsthe building-up scheme corresponds to the hydrogenlike (n, l) rule.As the ion charge decreases to zero, transition from (n, l) to (n +l, n) occurs (Katriel and Jorgensen, 1982) with some intermediateordering scheme in between. An attempt to describe this transi-tion using group-theoretical formalism of q-deformed algebras wasundertaken by Négadi and Kibler (1992).

5.1.3. Compressed atomsFor the atoms confined to a cavity of atomic size the ordering oforbital energies is changed. As the calculations by Connerade et al.(2000) show, the competition between ns and (n + 1)d orbitals disap-pears in favor of the former, i.e., (n + l, n) filling rule is replaced bythe hydrogenlike (n, l) rule. Thus compressing an atom changes thesituation in the same direction as its ionization, see section 5.1.2.This similarity might be interpreted in terms of properties of aneffective one-electron potential (Connerade et al., 2000).

5.1.4. MoleculesHefferlin with co-workers (Hefferlin et al., 1979a, b, 1984; Hefferlinand Kuhlman, 1980a, b; Zhuvikin and Hefferlin, 1983; Karlson etal., 1995) discussed the periodic system of diatomic moleculesformed from different atoms. These authors introduced the classi-fication of diatomics bearing combinatorial character. Zhuvikin andHefferlin (1983) consider group-theoretical aspects of the problemin the spirit of EPA.

PHYSICS AND PERIODIC LAW 171

5.2. Shell structure in atomic nuclei

It was long ago noted that the atomic nuclei with some partic-ular number of protons and/or neutrons are especially stable. Suchnumbers, known as magic numbers are (Bethe and Morrison, 1956)

2, 8, 20, 28, 50, 82, 126 . . . . (12)

The magic numbers are manifestations of the shell structure ofnuclei, just as the period lengths in the Periodic Table are mani-festations of electronic shells in atoms (the period lengths (8) arealso often referred to as magic number in physical literature, see,for instance, Löwdin (1969)). The interpretation of shell structureis based on an effective one-particle potential which in the crudestapproximation is that of harmonic oscillator. The more sophisticatedconstructions which agree better with experimental data employ apotential of trough-like shape with shallow bottom and rather abruptcut-off on the nucleus border (the latter notion is defined much betterfor nuclei than for atoms). Here it is important to stress that thestates in this potential are labeled by quantum numbers28 n and ljust as for atoms. The difference in two sets of magic numbers (8)and (12) is governed by the difference in the shape of one-particlepotentials.

Applications of various group-theoretical techniques in nuclearphysics are numerous and sophisticated.

5.3. Magic numbers in clusters

The clusters are relatively new object in physics, lying in betweenlarge molecules and condensed matter. There are numerous indica-tions of particular stability of clusters composed of some magicnumber of atoms. In particular, the experiments with the sodiumclusters (up to approximately 1500 atoms) indicate some specifictype of the electronic shell structure. All valence electrons in suchclusters might be considered as moving in some effective field. Forsodium atom clusters the empirical data correspond to groupingtogether the one-electron levels with the same sum 3nr + l ≡ 3n− 2l − 3 (Martin et al., 1990, 1991a, 1991b). Generally the shapeof the effective one-particle potential in clusters is similar to that

172 V. N. OSTROVSKY

found in atomic nuclear cases, i.e. a shallow trough. Near the origin(r → 0) the potential bottom could be raised that is referred toas ‘wine-bottle shape’. An interpretation of the 3nr + l groupingin terms of the shape of an effective one-electron potential wasprovided by Ostrovsky (1997). In particular, it was demonstratedthat the related effective potential leads to closed classical traject-ories with some special pattern. The importance of the shape ofthe outer wall in the potential well was emphasized by Lermé etal. (1993a, 1993b) who considered applications to aluminium andgallium clusters.

Recently the group-theoretical technique was applied to thedescription of shell structure in clusters (Bonatsos et al., 1999,2000). This approach essentially consists of choosing (basing onheuristic arguments) some group-theoretical scheme (a particular q-deformed algebra) and selecting a fitting parameter that allows theauthors to reproduce some set of empirical magic numbers.

5.4. Particles in the traps

Recently, considerable interest has appeared for the experimentaland theoretical study of localization of a finite number of ions orelectrons in the traps that are created by external confining poten-tial. The examples are radio-frequency traps for ions and electronsin plasma, heavy-ion storage rings, electrons in quantum dots insemiconductor structures; some key references could be found inthe paper by Bedanov and Peeters (1994). The classical calcu-lations by the cited authors for two-dimensional parabolic andhard-wall traps show that electrons are arranged in shells. Forlarge number of electrons there is a competition between orderinginto a crystal-like structure (Wigner lattice) for inner electronsand ordering into a shell structure for outer electrons. A periodicsystem of two-dimensional crystals composed of “particles in a Paultrap” was confirmed by recent experiments (Block et al., 2000a, b).The Aufbau principle for electrons confined by quantum dots wasstudied by Franceshetti and Zunger (2000).

PHYSICS AND PERIODIC LAW 173

6. CONCLUSIONS

The main points of the present paper could be summarized asfollows.

− When one is concerned with explanation or interpretation ofnumerical results or experimental data for a complex system,the use of approximations or models is the only way to achievesuccess. In particular, the approaches used to interpret the Peri-odic System as a whole are necessarily quite different from thetheoretical techniques employed to obtain the best numericalresults for some particular property of an individual atom.

− The current mainstream interpretation of the Periodic Law isbased on the selfconsistent effective field concept, the notionof atomic configuration and modeling of effective field exper-ienced by an electron in an atom. This approach achievedconsiderable success by providing non-empirical, ab initiodirect explanation of the (n + l, n) filling rule. The current stateof the problem contradicts the statement that “The emergenceof quantum mechanics in 1925–1926 rather interestingly didnot provide any improved qualitative explanation” (Scerri etal., 1998b), although it could be true that “the role played byquantum theory and quantum mechanics in chemistry is lessdramatic than is commonly held” (Scerri, 1996).

− Remarkably, an understanding of the (n + l, n) filling ruleis even better achieved by a crude model for effective one-electron potential in an atom than that needed for quantitativedemonstration of this rule.

− Alternative, complementary approaches to the interpretationof the Periodic Table seek to provide classification schemesusing techniques of Group Theory. The specific group is eitherrestored from the empirical structure of the Periodic Table orextracted from the analysis of atomic field description withinquantum mechanics. The effective one-electron potential inatoms possesses hidden symmetry properties that are probablyonly partially revealed by now.

− There is plenty of room for future studies, such as the expla-nation of exceptions from the (n + l, n) filling rule or uncov-ering the deep origin of symmetry properties. One can always

174 V. N. OSTROVSKY

think about the possibility of higher-level explanations, forinstance, whether the symmetry of an effective one-electronpotential can be directly deduced from the hidden symmetryof the Coulomb potential (Fock, 1935) operative between theelectrons and nucleus.

− Periodicity phenomena seems to be a general feature ofvarious multiparticle systems studied in physics (ionized orcompressed atoms, atomic nuclei, clusters, particles in thetraps). All of them exhibit a trend towards what probably couldbe named a tendency to self-organization, with appearance ofshells, magic numbers etc. Comparative study of these mani-festations is able to cast a new light on the origin of PeriodicLaws. In all cases various patterns of periodicity are governedby the difference in the one-particle effective potentials.

NOTES

1. The predecessors of Mendeleev were J. Döbereiner, J.-B. Dumas, E. deChancourtois, J. Newland and L. J. Meyer (Scerri, 1998b).

2. This allows us to essentially skip the issues of early history in the presentbrief exposure. The bibliography on the history of quantum interpretation ofthe Periodic Table is vast; we give only few latest references: Romanovskaya(1986), Scerri (1997b, 1998b, 1998a).

3. Consider, for instance, metallic character with metalloids lying in betweenmetals and non-metals as discussed by Birk (1997).

4. Regarding serious difficulties which emerge both in classical and quantummechanics in solution of three-body problem, it seems that definition ofcomplexity appropriate to our present objectives is that the system underconsideration contains three or more particles. As shown by Poincare, theclassical three-body problem has no closed-form solution since the motionis chaotic. Of course the same refers to the systems with larger number ofparticles; in quantum mechanics closed-form solutions are absent also. Three-body systems in atomic physics provide a rich variety of phenomena thatcould model more complicated objects.

5. Concerning the debate regarding prediction and accommodation of data byscientific theories see bibliography in the paper by Scerri and McIntyre(1997c).

6. See also an interesting discussion of these issues by Scerri (1996).7. As argued at the end of section 2, in this very demanding sense large branches

of physics prove to be non-reducible to its main framework.

PHYSICS AND PERIODIC LAW 175

8. The term ‘explanation’ has several meanings. In quantum measurements‘explanation’ is often understood as a mapping from the quantum physicsof the actual system onto the classical point of observer. However, we believethat the workers in quantum mechanics develop a special kind of ‘quantumintuition’ that allows direct understanding of quantum objects without appealto classical analogues; see, for instance, monograph by Zakhar’ev (1996)under appealing title.

9. Being introduced absolutely and rigorously, the ionization potentials, orionization energies are preferable to more loosely defined quantities, suchas atomic radii.

10. It is hardly necessary to stress that in education as well as in research theexact equations and approximations employed for their solutions should bestrictly distinguished, unambiguously formulated and clearly emphasized.

11. Rigorously speaking, the spherical symmetry of self-consistent field is guar-anteed only for closed-shell atoms. In other cases the spherical symmetryappears due to the standard additional approximation that works well forthe ground state atom. In more general situations the instability of the self-consistent field could lead to spontaneous symmetry breaking; however theseadvanced issues are not important for the present discussion.

12. We do not discuss here the historical development of configuration notionexposed by Scerri (1991a).

13. The statement that “The electronic configurations . . . cannot be derived usingquantum mechanics . . . because the fundamental equation of quantum mech-anics, the Schrödinger equation, cannot be solved exactly for atoms otherthan hydrogen” (Scerri, 1998a) is not exact. In modern quantum mechanicsthere is no other way to derive electronic configurations than starting fromthe exact Schrödinger equation and developing a scheme for its approx-imate solution, as briefly outlined above. Therefore it is too strong to saythat “quantum mechanics forbids any talk of electrons in orbitals and henceelectronic configuration” (Scerri, 1997a). Equally it cannot be said that theatomic orbitals stem from approximations that logically contradict the theory(H. Post, as cited by Scerri (1989)). The orbitals are simply the one-electronbuilding blocks routlinely used in quantum mechanics to construct a multi-electron wave function. The Hartree-Fock method allows one to choose theseblocks in an optimal way for each particular atom.

14. It is worth indicating here that there are some phenomena in atomic physicsthat cannot be explained without more sophisticated description, whichfrom the very beginning accounts for the correlated motion of electrons.This means a full breakdown of one-electron orbitals and configurations;in these cases the labels {nj , lj} cannot even be applied approximatelyto the wave function. In partiuclar, in the theory of doubly excited atomicstates, including their classification, the correlated two-electron motion (i.e.,two-electron orbitals) is to be considered as a zero order approximation,see Kellman (1995, 1996, 1997) and furthter bilbiography in the paper by

176 V. N. OSTROVSKY

Prudov and Ostrovsky (1998). Such sophisticated treatment is not necessaryfor explanation of the Periodic Table.

15. Without analysis of this type, which is often skipped in elementary exposures,it remains unclear in which order the orbitals are filled within each n-group,for instance, which of 2s and 2p orbitals is occupied earlier. Some exposuresof quantum analysis of the Periodic Table could give a reader impression thata mere introduction of quantum numbers n and l gives the filling rule, whichcertainly is not true.

16. Scerri et al. (1998b) attribute n + l rule to Bohr (1922), but we were unableto locate formulation of the rule in Bohr’s paper.

17. This situation can be compared with that for pure Coulomb field where in thecourse of a detailed quantum solution the combination nr + l + 1 naturallyemerges in the expression for energy, subsequently being designated as n.

18. Most frequently in quantum mechanics the quantized (i.e., discretized)magnitude is the energy, although this is not the only option (recall, forinstance, quantization of angular momentum). The present case demonstratesyet another possibility: the energy E is fixed, but the quantized parameteris Z that defines the potential strength (pre-factor in formula (6)), and alsothe coordinate scaling. Physically this situation is justified by the fact thatthe valence electrons in atoms are always weakly bound, i.e., their energy isalways close to the borderline E = 0.

19. In this brief exposure we do not discuss some subtle aspects of using potential(6) detailed in original publications (Demkov and Ostrovsky, 1971b; Demkovand Berezina, 1973; Ostrovsky, 1981).

20. It is worthwhile to stress again that this is an explanatory problem. Quantita-tive description of atoms in the case of exceptions from the (n + l, n) does notpresent a particular problem.

21. We present here only a brief qualitative discussion of the group theoreticalapproach bearing in mind its applications to the Periodic Table reviewed byOstrovsky (1996).

22. In the most general terms the suggestion to apply group theory to the PeriodicTable could be found in the paper by Neubert (1970).

23. It is worthwhile to indicate that Novaro and Berrondo (Novaro and Berrondo,1972; Berrondo and Novaro, 1973; Novaro, 1973; Novaro, 1989) lookedfor the group which describes the chemical periods, whereas other authorshave concentrated on the (n + l) grouping (the phenomenological differencebetween these patterns was discussed in the beginning of section 3.5).

24. Here it is worthwhile to recall that the explanation appeals to somecommunity of researchers, see section 2.

25. Kitagawara and Barut (1983, 1984) modified the scheme by Ostrovsky(1981) to consider mapping of the (nonphysical) two-dimensional problemon the three-dimensional sphere. In this case the treatment is much easierdue to the possibility of using well developed mathematical theories ofcomplex variables. Note however, that the scheme constructed by these

PHYSICS AND PERIODIC LAW 177

authors for three-dimensional problems suffers from very serious deficiencies(Ostrovsky, 1996).

26. It is worthwhile to give here an extended citation from the book by Smith(1924) that might be not easily available: “It is, however, probable that radiumis more closely allied to strontium (both give intensely red flame colora-tion), just as thorium is most closely allied to zirconium, and uranium mostclosely allied to molybdenum. It is, in fact, a general observation that alternatemembers of a valency group in the periodic table show the greatest chem-ical resemblance, for example, iodine and chlorine; bromine and fluorine;bismuth, arsenic and nitrogen; the antimony and phosphorus”.

27. More exactly, this pattern was found mostly for alkaline and alkaline earthatoms and some isoelectron ions.

28. Note that in nuclear physics the notation n is usually understood for the radialquantum number nr that is simply related to the principal and orbital quantumnumbers, see section 3.1.

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Institute of PhysicsThe University of St Petersburg198904 St PetersburgRussiaE-mail: Valentin.Ostrovsky@pobox.spbu.ru