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V. N. OSTROVSKY
WHAT AND HOW PHYSICS CONTRIBUTES TO
UNDERSTANDING THE PERIODIC LAW
ABSTRACT. The current status of explanation worked out by Physics for the
Periodic Law is considered from philosophical and methodological points of view.
The principle gnosiological role of approximations and models in providing interpretation for complicated systems is emphasized. The achievements, deficiencies
and perspectives of the existing quantum mechanical interpretation of the Periodic Table are discussed. The mainstream ab initio theory is based on analysis of
selfconsistent one-electron effective potential. Alternative approaches employing
symmetry considerations and applying group theory usually require some empirical information. The approximate dynamic symmetry of one-electron potential
casts light on the secondary periodicity phenomenon. The periodicity patterns
found in various multiparticle systems (atoms in special situations, atomic nuclei,
clusters, particles in the traps, etc) comprise a field for comparative study of the
Periodic Laws found in nature.
1. INTRODUCTION
Since the formulation of the Periodic Law by D. I. Mendeleev1 it
attracted a large number of researchers from different fields, first of
all chemistry, but also physics, philosophy, history of science etc.
Among them physics claims to provide explanation of the origin
of periodicity on the microlevel, by using the methods of quantum
mechanics.
The present paper concerns general philosophical and methological features of the contribution by physics to understanding the
Periodic Law. Its objective is twofold. First, section 2 contains some
general remarks on the character of explanations which physics
aims to provide to the phenomena of nature. To understand the
issue properly it is essential to distinguish between the quantitative
results and explanations. These two major types of research output
in 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.
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V. N. OSTROVSKY
be in complementary relation in the spirit of the universal complementarity principle put forward by N. Bohr (1999). As discussed
below, due to its very general character, the Periodic Law evades
exact quantitative formulation, therefore it is natural that physics
seeks for its explanation.
The second goal of the present contribution is a brief outline of
the current state of research in physics concerning the phenomenon
of periodicity. Regretfully, it seems that a significant part of workers
outside physics limit the issue to the famous papers by Bohr (1977)
supplemented by the methods of quantitative atomic structure calculations developed later in modern quantum mechanics. The history
of the early developments is studied and analyzed in much detail2
albeit the latest advancements in the explanatory aspect frequently
remain beyond consideration. However, these results are relevant
when the Periodic System of Elements is concerned. In the main
body of the paper these developments are discussed from the methodological point of view. We single out more traditional approaches
relying on analysis of filling of one-electron orbitals (section 3) and
the symmetry considerations based on mathematical Group Theory
(section 4). Additionally, in section 5 we demonstrate that the Periodic System is only one (although the most widely known) member
in the family of periodicity phenomena in multi-particle systems.
Comparing various periodic laws met in nature one can better understand what physical aspects are crucial in creating different patterns
of periodicity.
Some important points are worth preliminary mentioning. First,
the Periodic Law embraces vast amounts of knowledge. It is difficult
to formulate the Law exactly and unambiguously (see, for instance,
discussion by Scerri and McIntyre (1997c)). Only part of it could be
cast in physical terms and expressed in quantitative measure (such
as atomic radii or ionization energies and also densities, specific
heat etc.). The major part of information belongs to the realm
of pure chemistry and includes a variety of facts about valences,
chemical compounds and reactivity etc. These chemical properties
are often difficult to formalize and express numerically.3 Chemical
periodicity itself has a very particular flavour. It is not exact, but
only approximate, since each chemical element possesses its individuality; 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 does
not encounter a quantitative problem but the task of explanation.
Moreover, as argued in section 4, this is a classification problem to
a large extent.
Such a conclusion is intimately related to our second point that
one deals here with complicated system, whatever exact definition
of the latter notion could be adopted.4
The third point stems from the fact that historically the Periodic Law was formulated based on empirical data before physics
was capable of approaching the problem on the microlevel. The
power of physical (and, more generally, scientific) approach to the
world is manifested most convincingly when predictions are made.5
In this respect the Periodic Law provides very limited possibilities for exploits since a major part of the work was already done
by chemists, starting from Mendeleev who had predicted several
new elements. Therefore it is understandable that sometimes the
question is asked (Scerri, 1994a): “Does physics provide a truly
deductive explanation of the periodic table, or does it simply re-state
Mendeleev’s discoveries?” Here we limit ourselves to indicating
that according to some modern historical studies (see bibliography
provided by Scerri and McIntyre (1997c)) “Mendeleev’s ability
to accommodate the already known elements may have contributed as much to the acceptance of the periodic system as did his
dramatic predictions”. This great example shows that the proper
outlook could be of very substantial importance even without much
predictions provided.6
2. EXPLANATIONS VERSUS CALCULATIONS
Sometimes it is argued (Scerri and McIntyre, 1997c) that “The
reduction of the periodic table [to physics] should mean the ability
to calculate exactly the total energies or other properties of the atoms
in the periodic table”. In our opinion, these stringent requirements7
would be more relevant in the case when reduction of not only the
Periodic Table, but entire chemistry is concerned. Anyway, in this
paper we do not deal with reduction, but with physical explanation
and interpretation. By explanation we imply (approximate) replacement of a complicated system by a system which is ‘simple’, i.e.
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V. N. OSTROVSKY
possesses well known properties and appealing to what is called
physical intuition or physical sense. A more exact definition of the
latter notions would imply serious study that is beyond the scope of
the present paper.8
Current progress in computer techniques makes it even more
acute that there is always a complementary relation between calculations and explanation. To illustrate it on relevant material let
us consider evaluation of such a basic atomic property as ionization potentials Ia of neutral atoms.9 Modern quantum mechanics is
capable of doing this numerically with rather high accuracy, taking
into account relativistic effects (important mostly for heavy atoms
in the vicinity of nucleus) and avoiding electron orbital approximation, for instance, by using the configuration mixing technique. Such
calculations inevitably produce well known quasiperiodic dependence of Ia on the element number Z (which is atomic nucleus
charge), thus reflecting the Periodic Law. The question is: can we
say that in this way we obtain the physical explanation or interpretation of the Periodic Law? Our answer would be: no. By performing
calculations of this type one carries out a kind of mathematical
experiment, results of which could be compared with the real physical experimental data. The successful comparison convinces us
once more that quantum mechanics is a valid theory being capable
of reproducing reality in its quantitative aspect. However, the qualitative explanation in most cases (i.e., for sufficiently complex
systems) cannot be achieved by precise numerical calculations. The
set of numerically obtained ionization potentials hardly contributes
more to understanding the periodicity than the set of empirical data
for the same quantity. For complicated systems the real explanation
is possible only within some hierarchy of approximations or models.
As far as we know, the principle importance of approximations in
physics was first emphasized by V. Fock (1936, 1974) who stressed
that new notion in physics appear when approximate methods are
introduced. These notions are absent in the general theory and can
be formulated only within approximations.
If one’s objective is to obtain the best numerical results, then
the approximations are something to evade and at least to limit as
much as possible in the course of scientific progress: less approximations provide better numerical output. However, if one seeks
PHYSICS AND PERIODIC LAW
149
explanations, then dropping some approximations may hopelessly
destroy the entire framework. Indeed, usually the bare exact equations for complex system provide very limited insight. This, of
course, does not necessarily mean that the same set of explanatory
approximations or models would be retained forever. On the path
of historical progress the models could be substantially modified or
even completely new models could be developed, but nevertheless
the models and approximations remain substantial and an inevitable part of the explanation for the complex system, and not some
annoying deficiency.10
Of course this situation is not specific for relation between
physics and chemistry but constitutes a substantial feature of the
entire physical approach to nature. For instance, the theory of solid
state is not normally based on ab initio calculations of electronic
structure of crystals, but on some intermediate models (although
some ab initio approaches were successfully developed in the last
decades). This in no way invalidates numerous important achievements in this branch of physics. Nuclear physics is also essentially
based not on ab initio theories, but on models. In general, physics is
reduced to a small number of theories only in principle, whereas the
practical applications are based on many cases on some models of
‘intermediate’ character, standing in between the basic theories and
concrete developments.
The approximations and models are a fully legitimate part of the
theory, but not its temporary, abominable and shameful part. Every
textbook in quantum mechanics includes some simple problem
such as bound states in one-dimensional potential wells, scattering
on potential barrier, harmonic oscillator, hydrogen atom etc. This
is mostly not because these problems are capable of accurately
describing nature, but because they allow the reader to develop
important qualitative quantum concepts (such as form of the boundstate wavefunction, tunneling phenomenon, above-barrier reflection, etc.). The basic approximations and the simple model problems
with easily grasped properties form the kind of appropriate language
in which explanation of complicated situations can be developed.
Of course, as is the case of every language, the explanations are
addressed to a knowledgeable audience.
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V. N. OSTROVSKY
The complementarity concept originated from Bohr’s works in
physics but he later applied it in fields outside physics, such as
psychology, biology and anthropology (Bohr, 1999). The epistemological significance of this concept stems from the fact that it
concerns a very general pattern of relations between subject and
object. Within the complementary pair ‘numerical calculations’ –
‘explanation’, the numerical calculations seek to reproduce a physical object with the highest possible precision whereas explanations
appeal to the subject. In particular, this means that explanation
appeals to some community of researchers and it can be different
for various communities.
The problems solvable in closed form provide an important
ingredient for creating explanatory networks. This is because
usually such problems have transparent properties, clear physical
meaning and can be employed as simple building blocks in order
to construct comprehensible approximations. However, sometimes
exactly solvable problems have unclear physical interpretation and
in such cases they prove to be useless for explanation. Note that
all exactly solvable problems in physics describe some idealized
reality. For instance, calculations of the energy spectrum for the oneelectron atom or ion by using Schrödinger equation does not account
for relativistic effects; Dirac’s equation does not include quantum
electrodynamics effects; in more exact approaches the structure of
the atomic nucleus is to be taken into account etc. In describing any
reality physics cannot avoid approximations.
3. EXPLANATION OF PERIODIC SYSTEMS BY MODERN
QUANTUM MECHANICS
3.1. Standard approximations in atomic theory
The complexity of the multi-electron atom is well illustrated by
arguments provided by Hartree (1957). The wave function of a
system with q electrons depends on q electron vector coordinates.
When one wishes to tabulate the wave function for the neutral Fe
atom with q = 26 taking only 10 values for each variable, then, even
after reducing the amount of information by using the symmetry
properties, the table still would contain 1053 entries. All the matter
contained 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 feasible
progress in computing facilities. Therefore the multi-electron atoms
cannot be treated without approximations.
The basic approximation employed exploits the idea of a selfconsistent field which presumes that the electrons move independently in some mean field created by other electrons and the atomic
nucleus. Mathematically this is a drastic simplification allowing one
to present the wave function with q vector arguments as a product
of some small number of functions with one argument, the latter
functions being known as one-electron orbitals. Due to spherical
symmetry of the self-consistent field11 the atomic orbital in fact
depends on a single scalar radial variable r, i.e. electron distance
from the nucleus. The set of orbitals employed for the construction of the wave function (so called filled, or occupied orbitals) is
named configuration of atom.12 The most exact scheme based on
one-electron orbitals and atomic configuration is the Hartree-Fock
approximation (Fock, 1930; Hartree, 1957; Slater, 1947). It properly
accounts for the Pauli principle which concerns the wave function
symmetry under particle permutation, or, in other words, for electron exchange. The self-consistent Hartree-Fock method is one of
the fundamental tools designed in quantum mechanics to tackle a
variety of multi-particle systems (not only atoms and molecules, but
also clusters, the objects studied in the condensed-matter physics,
atomic nuclei etc). When quantitatively more exact methods are
developed to account for electron correlation effects, i.e., effects
beyond the Hartree-Fock approximation (for instance, configuration
mixing, or superposition schemes, so-called Random Phase Approximation, Density Functional Theory etc.), in most cases they are
based on the Hartree-Fock method as the first approximation.
According to quantum mechanics, the orbitals in atoms are
labeled by angular quantum numbers l, m and radial quantum
number nr that is the number of nodes, i.e., zeroes in r-dependence.
The principal quantum number n is conventionally employed for
orbital labeling instead of nr , being defined as n ≡ nr + l + 1. It
is worthwhile to stress that using atomic configurations does not
mean that an orbital, or a set of quantum numbers are ascribed to
any particular electron. This is clear from the fact that the manyelectron wavefunction is constructed from a product of orbitals by
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V. N. OSTROVSKY
antisymmetrization over electron permutations. Thus the coordinate
of any electron stands in the arguments of different orbitals. The
actual meaning of the configuration is in ascribing to the whole atom
a set of approximate constants of motion, or integrals of motion,
or in other words, a set of approximate labels { nj , lj , ζj } where
the index j runs over occupied, or filled orbitals and ζj is the occupation number, i.e., number of electrons on j-the orbital. In the
non-relativistic approximation an atom has only two exact integrals
and total spin S (when
of motion: the total orbital momentum L
relativistic effects are taken into account, only the total angular
+ S is exact integral of motion). The integrals
momentum J = L
of motion play a very important role in quantum mechanics, since
they make possible the classification of states (see also section 4.1).
The number of exact integrals of motion is usually insufficient for
complete classification. Therefore, the importance of approximate
(but ‘good’, i.e., well conserved) integrals of motion is difficult to
overestimate.13
It can be noted that ‘global’ atomic quantum labels L, S and J
(that define atomic multiplet levels) cannot be uniquely deduced
from the configuration { nj , lj , ζj }: generally several sets of
‘global’ numbers are allowed. Moreover, the non-relativistic selfconsistent field depends on the choice of L and S. This means
that one-electron energies as well as the total energy of the atom
are somewhat dependent on multiplet levels. Quite often a simplified multiplet level averaged Hartree-Fock scheme is employed.
However, for heavy atoms the splitting of multiplet levels belonging
to the same configuration could be appreciable. It could exceed
the energy separation of different configurations. This circumstance
is of importance when non-regularities in the Periodic Table are
discussed, see section 3.5.
Rigorously speaking, the atomic wave function presents an
infinite sum over different configurations with the same exact
quantum numbers. However, if one term in the sum manifestly
predominates it can be safely used for classification of atomic state.
In the Hartree-Fock method each individual orbital corresponds
to electron motion in a particular potential which generally is nonlocal (due to exchange effects). Such a potential is spread over
space 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 individual orbital considered, although the motion of different electrons
is 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 complicating
peculiarities make Hartree-Fock approximation excessively sophisticated for interpretation of the Periodic Table, at least at the current
stage of our knowledge.14 In order to describe its major features
it is sufficient to consider the motion of electrons in unique (for
all the orbitals in the atom) local spherically symmetrical potential
UTF
a (r) that is obtained within the Thomas-Fermi approximation.
The latter is based on an additional simplification: the semi-classical
treatment of electrons (Slater, 1947; Gombas, 1949; Landau and
Lifshitz, 1977). Within it an atom looks like a bulb of electron gas.
For subsequent discussion it is important that the potential UTF
a (r) is
expressed in terms of a universal function χ (x):
Z
UaTF (r) = χ (kr),
r
√ 2/3
8 2
Z 1/3
k=
3π
(1)
(hereafter we use standard atomic system of units: e = = me =
1). In the potential (1) all specifics of a particular chemical element
are included in ‘potential strength’ prefactor Z and scaling factor
k which is proportional to cubic root of the nucleus charge Z.
Although the analytical expression for χ (x) is not known, this is
a well studied and tabulated function. It can also be approximated
by simple analytical expressions, as discussed in section 3.4. The
Thomas-Fermi approximation allows one to carry out some calculations relevant to the building-up of the Periodic System without
considering individual electronic orbitals. The first application of
this kind was done by Fermi (1928) who found the place in the
Periodic Table where the first electron with a given orbital quantum
number l appears for the first time. Generally this approach provides
a less detailed interpretation of the Periodic Table than that based
on analysis of orbitals. Therefore this line of development is beyond
the main scope of the present paper. Here we only refer to relatively
recent review by Bosi (1983).
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V. N. OSTROVSKY
3.2. Basic properties of effective one-particle potential as
foundations of Aufbau principle
The physical explanation of the Periodic Law is based on the
observation that the properties of elements are defined by the
filled orbitals with the lowest binding energies (outer or valence
orbitals). It should be recognized that the object of most physical studies (including the present paper) is in fact the Periodic
Table of Neutral Atoms. This is sufficient for analysis of periodicity
patterns in atomic properties such as ionization potentials, atomic
radii etc. Concerning chemistry, the situation looks substantially
more intricate, although one can refer to explanation of the valencies
of chemical elements using the idea of outer orbital hybridization for atoms bonded in chemical compounds. Complexity of
formalizing and interpreting chemical properties in physical terms
(already discussed in section 1) could be considered as a special
manifestation of non-reducibility of chemistry to physics.
To define which orbitals in an atom are filled one has to use
the building up or Aufbau rule which is a corollary of the method
used for wave function construction. Namely, since we consider the
ground state of an atom, the electron distribution over orbitals has
to correspond to the minimum energy compatible with the Pauli
principle. In accordance with other simplifications employed, the
approximate form of the Pauli principle is used: not more than two
electrons (differing by the spin projection ms = ± 12 ) can occupy
an orbital labeled by three space quantum numbers (n, l, m). The
ground state provides a minimum to the energy of whole atom that
within the Hartree-Fock method does not necessarily correspond to
the minimum sum of one-electron energies for the occupied orbitals.
This is due to the method’s peculiarities already mentioned above in
formula (1); see more discussion in the end of section 3.5. However,
within the Density Functional Theory the relation between the two
minima becomes direct due to Janak’s theorem (Janak, 1978). This
holds also for our cruder scheme based on the one-electron potential
(1).
Clearly, the result of applying the Aufbau rule directly depends
on the ordering of the levels in a one-electron effective potential
Ua (r). Depending on the form of Ua (r) a variety of very different
Periodic 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 is
frequently overlooked in the simplified exposures where implicitly
it is presumed that Ua (r) is close to a pure Coulomb potential, that
is operative in the simplest atom, that of hydrogen. Below we show
that this choice is generally unjustified and rather unfortunate since
it provides the wrong form of the Periodic Law which does not agree
with empirical observations.
Hence it is worthwhile to consider the basic properties of the
effective potential seen by the electron in an atom. Far from the
nucleus the electron is affected by an attractive Coulomb potential
of the atomic nucleus screened by all other electrons in an atom,
therefore
1
r ra ,
(2)
Ua (r) ≈ −
r
where ra is a characteristic radius of an atom. Near the nucleus the
screening effect vanishes and the electron is attracted by Coulomb
potential of the bare atomic nucleus
Z
Ua (r) ≈ −
r ra .
(3)
r
Considering the outer (valence) electrons, one could try to construct
the model of effective potential based on the −1/r approximate
behavior (2). For a pure Coulomb field (i.e., in the hydrogen atom)
2
the energy levels (EC
nl = −1/(2n )) are degenerate in the orbital
quantum number l. From formula (3) one infers that for small r
the effective potential is always somewhat deeper than −1/r potential (Z > 1). This shifts the energy levels downwards. The effect is
more pronounced for the low-l levels, since for higher l the effective
centripetal potential prevents penetration of small-r domain, i.e., in
other words, the high-l orbitals have maximum of electron density
far from the nucleus.15 Hence the degeneracy over l is lifted and one
comes to the hydrogenlike (n, l) Aufbau scheme: the one-electron
orbitals are occupied in order of increasing principal quantum
number n; for the same n the orbitals are filled in order of increasing
orbital quantum number l. The periods in this scheme correspond to
the hydrogenic n-shells. Taking into account the electron spin, one
obtains the period lengths
2n2 = 2, 8, 18, 32, 50, . . . .
(4)
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V. N. OSTROVSKY
Comparison with the detailed numerical quantum calculations for
multielectron systems and with empirical data shows that for highly
ionized atoms the (n, l) ordering rule is indeed operative. Its heuristic
derivation given above presumes that the deviation of the effective
one-electron potential Ua (r) from −1/r behavior can be treated as a
small perturbation.
However, the structure of the neutral atom is of importance
when the Periodic Law is considered. Here the (n, l) rule fails.
This implies that the effective potential exhibits strong deviations
from the Coulomb one, leading to the substantial rearrangements
of the spectrum. The overlap appears between the groups of energy
levels with different principal quantum numbers n. The n-grouping
of levels disappears, but a new type of regularity emerges in the
form of (n + l, n) to be considered in section 3.3. Its description
requires use of a non-Coulomb one-electron potential from the very
beginning.
Before concluding this subsection we emphasize again that the
notion of n-shell, i.e. states with the same principal quantum number
n, originates from the fact that for pure Coulomb potential, i.e., for
hydrogen atom, these states are degenerate in energy. If the potential differs slightly from Coulomb one, the degeneracy is lifted, but
the levels with the same n remain grouped together on the energy
scale. In this case the notion of shell remains physically meaningful. Otherwise, if deviation from the Coulomb potential is strong,
complete regrouping of levels occurs and hydrogenlike shells loose
physical 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 degenerate
in azimuthal quantum number m in any spherically symmetrical
potential.
3.3. Periodic Table as a result of ordering of one-electron levels in
atoms
The simplest and most straightforward thing to do is to take the oneelectron potential UTF
a (r), calculate the one-electron energy levels
numerically and compare the resulting Periodic Table with empirical
data. 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: the
periodicity phenomenon is reproduced by using a single universal
function describing a one-electron potential. Although based on a
set of approximations, this analysis should be classified as ab initio
since it does not employ any empirical or fitting parameters.
Additional very important and useful insight in the orbital filling
can be provided by considering an effective potential operative for
orbitals with orbital momentum l. Accounting for the centrifugal
repulsion this potential Ua (r) + l(l + 1)/(2r2 ) exhibits a double-well
structure revealed first by M. Göppert-Mayer (1941). It is important
for analysis of space localization of d and f orbitals and competition
between filling of these orbitals and s-orbitals (Connerade, 1998).
However, for one interested in explanation these results cannot
provide full satisfaction, since they do not directly interpret the
origin of (n + l, n) ordering rule. Empirically it was noticed that
the 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 to
trace the origin of this rule that looks like a kind of scientific folklore. Without giving references, Löwdin (1969)) ascribes this rule
to Bohr, but remarks that “Bohr himself was never too explicit
about his ‘Aufbau’-principle and the (n + l, n) rule is sometimes
referred to as Goudsmith-rule or Bose-rule” (no references to Goudsmith or Bose are given either).16 The rule was published only in
1936 in Madelung’s handbook (Madelung, 1936) rather implicitly
as an adopted form of the Periodic Table, although according to
Goudsmith (Goudsmith and Richards, 1964), he received private
communication from Madelung about this rule in December, 1926.
In between these dates Karapetoff (1930) used this rule to predict
configurations of transuranian elements, up to Z = 124. Later the
rule was rediscovered independently by a number of authors as an
empirical ‘lexicographic’ rule without theoretical ab initio foundation: 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,
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V. N. OSTROVSKY
1953a, 1953b, 1954, 1960, 1961, 1962) summarized in his book
(Klechkovskii, 1968) should be particularly praised since this author
studied systematically different aspects of the (n + l, n) rule in much
detail. Nevertheless the dynamical origin of the sum of principal n
and orbital l quantum numbers remained mysterious. This particular
linear combination of quantum numbers has never appeared as a
result of solution of any Schrödinger equation.17 This circumstance
induced Löwdin to write in 1969 that “it is perhaps remarkable that,
in axiomatic quantum theory, the simple energy rule (order of filling
of orbitals) has not yet been derived from first principles”. Since
that time some substantial progress has been achieved, as discussed
below.
3.4. n + l rule and orbital genesis
The origin of quantum number N = n + l was understood by Demkov
and Ostrovsky (1971b) who further simplified the Thomas-Fermi
potential (1) by using an analytical approximation for the function
χ (x):
χ (x) =
1
.
(1 + αx)2
(5)
As discussed by the authors, within the Thomas-Fermi theory this
approximation was considered before by Tietz (1954, 1955) (see
original paper (Demkov and Ostrovsky, 1971b) for a complete list
of numerous papers by this author), the parameter α ≈ 12 being
defined by applying the variational principle or normalization condition in momentum space. However, it was not noticed earlier that
the Schrödinger equation for the related potential can be solved
analytically 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 atomic
orbitals in the approximate one-electron potential obtained from
formulas (1) and (5):
Z
3π 2/3
DO
−1 −1/3
, R=α Z
.
(6)
Ua (r) = −
√
r(1 + r/R)2
8 2
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. In
the general case appearance of {n, l} bound level occurs at some
values of parameter18 Z = Znl labeled by two quantum numbers n
and l. However, a remarkable property of the potential (6) is that
the levels with the same sum N = n + l appear simultaneously at
certain critical values of the parameter Z = ZN . This at once shows
that the N-grouping is realized for this particular potential, instead
of n-grouping for a weakly distorted Coulomb potential (see section
3.2).
To reduce the second part of the (n + l, n) rule one have
to consider how the levels within the same N-group are ordered
for a deeper potential, i.e., for Z > ZN , when the levels energies
differ from zero and N-degeneracy is lifted. Such analysis was
successfully carried out by using perturbation theory (Demkov and
Ostrovsky, 1971b). This completes theoretical ab initio (i.e. not
using empirical information or fitting) derivation of the (n + l, n)
filling rule.
As shown by Demkov and Ostrovsky (1971a, 1971b), the potential (6) belongs to a broader family of potentials Uµ (r)
Uµ (r) = −
2v
+ (R/r)µ ]2
r 2 R 2 [(r/R)µ
(7)
with strength parameter v and coordinate scaling parameter R. The
potentials (7) are exactly solvable for E = 0 and exhibit degeneracy of levels with the same linear combination n + (µ−1 −
1)l of quantum numbers n and l. Thus the degeneracy pattern is
directly defined by the potential parameter µ. Some other members
of the family Uµ (r) are also physically important, for instance,
the famous Maxwell’s fish-eye (Maxwell, 1952; Demkov and
Ostrovsky, 1971a) (µ = 1) or potentials with higher µ used in the
analysis of periodicity met in clusters (Ostrovsky, 1997), see section
5. As soon as the family is known, one can derive potential (6) in an
alternative way (Demkov and Ostrovsky, 1971b), namely selecting
the (n + l) degeneracy pattern by putting µ = 12 . Remarkably, in this
way we come at once to the potential of the form (6) which exhibits
Coulomb behavior ∼ −1/r as r → 0. Such a potential singularity
is 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 which
probably urges for deeper understanding. There is no contradiction
here to the statement that the potential (6) provides an explanation
to the Aufbau Principle. First of all, the explanations are subject
to improvements, just as the numerical calculations are made more
precise; second, there are different levels in the hierarchy of explanations. From yet another points of view the potential (6) is discussed
by Wheeler (1976) and Tarbeev et al. (1997).
Before concluding this subsection we have to stress an important
circumstance. Notwithstanding different possibilities of its derivation, the potential (6) has direct physical meaning as an approximation for an effective one-electron potential. This is completely
clear already from the fact that, as mentioned above, the approximation (5) was known in Thomas-Fermi theory prior to the paper by
Demkov and Ostrovsky (1971b). There is crucial difference between
potential (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 the
meaning of the potential (6) in the literature. Kitagawara and Barut
(1983) wrongly claim that “The Demkov-Ostrovsky equation is just
a mathematical model providing the quantum number n + l and
its degeneracy. The coordinates appearing in this equation do not
have a direct physical meaning such as the spatial coordinates of
the 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 potential
came from”.19
3.5. Exceptions to the n + l rule
Although the (n + l, n-rule provides mostly correct ordering of
elements 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)
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PHYSICS AND PERIODIC LAW
The difference is shown in more detail in the following scheme
n+l=1
n+l=2
n+l=3
n+l=4
3s < 3p 4s <
2s
< 2p
1s
dim=2
dim=2
n+l=5
dim=8
(10)
dim=8
n+l=6
< 3d < 4p 5s < 4d < 5p 6s <
dim=18
n+l=7
dim=18
n+l=8
< 4f < 5d < 6p 7s < 5f < 6d < 7p 8s < . . .
dim=32
dim=32
where dimensions of (n + l)-shells are indicated with account for the
electron spin. Following Novaro (1973) and Katriel and Jorgensen
(1982) we denote by symbol the large energy gaps between the
one-electron levels. These gaps (dividing the periods in the Periodic
Table) do not coincide with the borders between the (n + l)-groups
of levels. The quantum interpretation of the difference between the
sequences (8) and (9) was suggested by Ostrovsky (1981). Briefly,
due to particular properties of weakly-bound s-states in quantum
mechanics, they are shifted to the higher-N group and join it on the
energy scale.
Comparing (8) with results of the hydrogenic (n, l)-rule (4) one
sees that the period lengths met are the same in both cases, being
equal to 2N 2 with some integer N . However, in case of (n + l,
n)-rule (8) each length (except the first one) appears twice. These
lengths doubling are an important feature to be discussed further in
section 4.
Considering individual atoms, there are 18 exceptions to the (n +
l, n)-rule as listed by Demkov and Ostrovsky (1971b), some of them
being discussed by Scerri (1991b, 1997b, 1998b, 1997a) in more
detail. In 16 cases the real configuration differs from that predicted
by the (n + l, n) rule by a single electron on occupied orbital; only in
two cases the difference is by two electrons. The exceptions could
arise due to a variety of reasons.
− First of all, as discussed in section 3.1 for given electron configuration generally there exist several multiplet levels of atom
differing by total orbital momentum L and total spin S. As
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V. N. OSTROVSKY
argued by Goudsmith and Richards (1964), “Many minor deviations from the Madelung rule can be ascribed to the large
spread of multiplet levels in the complex energy configurations. While the center of gravity of the multiplet levels may
obey the Madelung rule, one of the levels of a higher state
may be pushed down below the lower state by large exchange
interaction.”
− For heavy atoms the mixing of different configurations generally becomes more significant than for light ones. When the
number of electrons in atoms is large, different sets of occupied
orbitals can result in close values of total energy. As a general
trend the configurations are mixed stronger when they are close
in energy. The mixing might result in violation of the (n + l, n)
rule.
− The relativistic effects are not included in the model. These
effects are most important for inner electrons; and indirectly,
via the form of the effective potential created jointly by all the
electrons and nucleus, they could also affect the outer electrons.
Additionally, relativistic Dirac wave functions for ns and np1/2
states are known to have singularities at the Coulomb center,
i.e., as r → 0. This means enhancement of electron density
in the vicinity of atomic nucleus where relativistic effects are
stronger. According to Pyykkö (1988), “the relativistic change
of the atomic potential matters less than the direct dynamical
effect on the valence electron itself”.
The important question is whether the number of exceptions
is large enough to undermine or even fully discredit the physical
explanation of the Periodic System. In our opinion the situation
here corresponds to complexity of the system and relative simplicity
of the explanation based on analytical approximation (6) for oneelectron potential in atoms. The complexity of the Periodic Law
was discussed already in section 1. Carroll and Lehrman (1942)
argue that exceptions to (n + l) rule “are relatively unimportant
for chemists”. Additionally it should be recognized that for heavy
atoms sometimes it is not easy to assign elements a place in the
Periodic Table based on purely chemical information (see survey
of history for rare earth elements by Scerri (1994b)) since “The
periodic table contains many subtleties and anomalies which have
PHYSICS AND PERIODIC LAW
163
defied attempts at a complete reduction” (Scerri, 1996). Without
firm assignment based on the value of nucleus charge some discussions would probably continue till now. Therefore, regardless of
the exceptions, the situation looks satisfactory, although it would
be extremely interesting to explain20 why the exceptions occur for
the particular elements. Recently an interesting attempt (Tarbeev et
al., 1997) was made to describe an effective potential which does
not lead to exceptions; however, note that the analysis was based on
empirical data, i.e., it cannot be classified as ab initio.
As discussed above, a rather long chain of approximations is
used to come from the exact Schrödinger equation for the manyelectron atom to the approximate one-electron potential (6). From
the point of view of the Hartree-Fock method, the filling of orbitals
is defined by a rather subtle interplay of various effects. Consider
the competition between filling 4s and 3d orbitals studies in detail
by Melrose and Scerri (1996). The correct configurations of atoms
from Sc to Cu were obtained by considering the energies of oneelectron orbitals. However, the result could not be derived within
the Hartree-Fock method using additional so called frozen-orbitals
approximation. The calculations have to take into account that the
energy of each orbital depends on the occupation numbers for all
other orbitals. Therefore, the orbital energies may be changed by an
electronic 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 that
probably could be learned from these calculations is that the potential (6) effectively absorbs some features that in fact lie beyond the
simplistic 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 classification
testifies to a rather high level of knowledge. Physics often uses
the formalized method of classification provided by mathematical
Group Theory. The group-theoretical approach in physics is based
on the notion of symmetry.21 The operations which do not change
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V. N. OSTROVSKY
the system (more exactly, leave invariant its Hamiltonian operator)
comprise the symmetry group. These operations can have geometric
meaning (such as rotations in the case of spherically symmetrical
potential), but it also can evade direct geometrical interpretation
(so called hidden symmetries). By applying to the eigenstate an
operation belonging to the symmetry group we obtain another
(degenerate) eigenstate with the same energy.
Another important notion is the dynamical group which includes
the symmetry group, but contains also operators allowing one to
construct a complete set of eigenstates starting from any particular
one.
If the group is known, then mathematical techniques allow one
to produce a set of labels necessary for classification of states;
the dynamical group provides complete classification whereas the
symmetry group is able to predict degeneracy patterns met in the
system. For instance, the symmetry of central potentials is described
by the three-dimensional rotation groups designated as O(3). This
group provides l and m labels and predicts (2l + 1)-fold degeneracies
of energy levels, i.e., the independence of energies on the azimuthal
quantum number m.
Both symmetry and dynamical groups could be derived from
analysis of the Hamiltonian operator of the physical system under
consideration. In this way one can find not only rather obvious
geometrical symmetries, but also hidden symmetries, such as fourdimensional rotation group O(4) for the hydrogen atom revealed
by Fock (1935). Some work along these lines was carried out in
application to the Periodic System; it is referred to below as Atomic
Physics Approach (APA), see section 4.3.
For some physical systems it could occur that the Hamiltonian
of the system is not known (or even does not exist, at least in
the conventional sense), but the underlying symmetry or dynamical group could be somehow guessed. This was the background
for successful applications of group theory in elementary particle
theory. Usually one is interested not only in classification of states
and degeneracies, but also in the ordering of the state energies. To
achieve this, the dynamical group should be supplemented by the so
called mass formula which orders the energy levels depending on
their 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 the
ordering of levels). We refer to this type of group theory application
as Elementary Particle Approach (EPA).
4.2. Elementary particle approach to Periodic Table
With EPA the chemical elements are formally considered as various
states of some artificial object: ‘atomic matter’ (Barut, 1972) or
‘structure-less particle with inner degrees of freedom’ (Rumer and
Fet, 1972). Various states of such a system can be labeled by the
quantum numbers provided by the chosen group.22
Since EPA is necessarily a phenomenological approach, it is
particularly important (i) to specify exactly the empirical basis for
the choice of the group and mass formula and (ii) to outline how
the formal mathematical scheme can be interrelated to the observable physical objects and quantities. The authors who apply EPA
to the Periodic System, as anticipated, choose to forget about the
structure of the atom (and even about such basic ideas that an atom
consists of electron and nucleus). Nevertheless, the choice of group
is in fact based on some information outside Group Theory, and
there should be a possibility of comparing the results with empirical
data. It is highly desirable to demonstrate that the output suitable
for comparison exceeds the empirical input. From this point of view
the statement that “Within this approach we have to give up all available chemical and spectroscopic information” (Konopel’chenko and
Rumer, 1979) looks as unfounded extremism, since information of
this kind is necessary just for choice of the particular group among
the 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 group
which allows the degeneracy pattern coinciding with the empirically known period lengths (8) in the Periodic Table. The general
guideline in the search for the group is the already mentioned fact
that the degeneracies met can be expressed as 2N 2 with some
integer N . The same degeneracies are met in the Coulomb problem,
i.e., for hydrogen atom, which both symmetry and dynamical groups
have been known for a long time (see review by Ostrovsky (1981)).
Hence, the only problem is to modify the Coulomb field group so as
to incorporate the lengths doubling met in the Periodic System, see
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V. N. OSTROVSKY
discussion in section 3.5. The solution of this problem proves
to be non-unique, the specific groups suggested by different
authors23 (Barut, 1972; Rumer and Fet, 1972; Fet, 1979; Fet, 1980;
Konopel’chenko, 1972; Novaro and Berrondo, 1972; Berrondo and
Novaro, 1973; Novaro, 1973; Novaro and Wolf, 1971; Novaro,
1989) have been critically reviewed by Ostrovsky (1996).
Unfortunately, further perspectives of EPA remain unclear. Its
current achievements look like the translation of empirical information about period lengths to the mathematical language which
is probably more appealing for a certain part of the scientific
community, but nothing more. The empirical input (period lengths)
is cast in mathematical terms of the dynamical group, but hardly
anything more than period lengths return back to the interested
researcher who could aspire for some results to be compared with
experiment, or additional physical insight. For most workers in the
field such translation into specialized mathematical language24 does
not justify an accompanying sacrifice: giving up all references to
electronic structure of atoms. The situation looks very different from
that in elementary particle theory where the dynamical structure of
particles is a difficult and not completely solved problem which one
could desire to circumvent. Here the EPA approach was capable of
predicting new particles, but when applied to the Periodic Law, in
our opinion, it provides a very limited contribution.
Concluding this subsection we mention two more paper based on
mathematical technique, albeit not a group-theoretical one. Scerri et
al. (1998b) described a class of feasible ordering rules that satisfy
some natural criteria. It should be indicated that this class is quite
broad and additional restrictive criteria are needed to produce a
more limited selection. Purdela (1988) presented some empirical
arguments in favor of n + 12 l ordering.
4.3. Atomic physics approach and secondary periodicity
Contrary to EPA, the Atomic Physics Approach is not based on
empirical information but seeks to find the symmetry properties
of exact or approximate Hamiltonians that are already available
from ab initio quantum theory. Indications of deep symmetry of
the effective atomic potential (6) were revealed by Demkov and
Ostrovsky (1971b).
PHYSICS AND PERIODIC LAW
167
First of all, the classical trajectories in the potential (6) are closed
at E = 0 independent of initial conditions, i.e. of orbital momentum
l. Namely, the trajectories close after two revolutions around a force
center. As known from classical mechanics, generally a trajectory
in the central potential is not closed but covers some ring in its
plane. One of the notable exceptions is the pure Coulomb potential
where trajectories are known to close after one revolution around the
center. This property is a manifestation of hidden O(4) symmetry
of Coulomb potential (Fock, 1935) mentioned in section 4.1. The
importance 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 possess
a focusing property: all the trajectories exiting from any point r
after one revolution come through the point r/R2 . Thus in terms of
geometrical optics one can say that the rays emanating from any
source 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 onto
the four-dimensional sphere.25 In particular, this analysis allowed
him to reveal an additional integral of motion designated as T3
and defined as the following discrete transformation of the wave
function (see also Demkov and Ostrovksy (1971a)):
2
R
R
T3 ψ(r ) = ψ r 2 .
2r
r
(11)
In geometry the transformation r ⇒ r R2 /r2 is known as inversion in
the sphere of radius R. It conformally maps inner parts of the sphere
r ⇒ R on its outer part, and vice versa. The discrete operator T3
has only two eigenvalues 12 τ with τ = ±1. This situation is similar
to the electron spin projection operator ms = ± 12 , see section 3.2.
By analogy with the terminology of nuclear physics the quantum
number τ could be named atomic isospin. The eigenstates of potential (6) are also eigenstates of operator T3 . The odd and even values
of (n + l) correspond to τ = 1 and τ = −1 respectively. Thus, an
additional integral of motion is revealed, providing an additional
classifying label τ .
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V. N. OSTROVSKY
The dynamical group for the Periodic System suggested by
Ostrovsky (1981) incorporates operator T3 with a clear geometrical meaning. This is its advantage as compared with the groups
suggested within EPA. The additional atomic isospin classification
also has physical (or chemical) meaning.
For a rather long time the so called ‘secondary’ periodicity effect
was discussed in chemistry. Originally Biron (1915) noticed that for
the elements in a given group some chemical and physical properties
are reproduced most completely not in the adjacent periods, but in
every second period. As an example, he considered tendency of N,
As and Bi to be trivalent while P and Sb are pentavalent. According
to Smith (1924) “it is . . . a general observation that alternative
members of a valency group in the periodic table show the greatest
chemical resemblance”.26 In chemical literature the secondary periodicity is also refered to as an ‘alternation effect’, manifested, for
instance, for electronegativity (Sanderson, 1952, 1960), or for heats
of formation of oxides, and also for ionization potentials and sizes
of ions (Phillips and Williams, 1965). Some additional bibliography
can be found in the papers by Ostrovsky (1981) and Pyykkö (1988).
The latter reference interprets secondary periodicity in terms of
properties of Hartree-Fock orbitals. Convincing graphical illustration of the secondary periodicity can also be found in the paper by
Odabasi (1973) who depicted variation of ionization potentials and
mean value of r−2 along some groups of elements in the Periodic
Table. These properties exhibit saw-like modulation of the general
smooth trend within the Table column.
The elements within the group that belong to every second period
in the Table correspond to the same quantum number τ . The atomic
model gives here a new insight relating the secondary periodicity
to the properties of the atomic orbitals, namely, the isospin, or
T3 -symmetry. From this point of view it is particularly important
that the T3 -symmetry is stable with respect to the variation of the
atomic potential and the energy of the electron. Indeed, as shown by
Ostrovsky (1981), the realistic atomic orbitals (calculated within
a simplified version of the Hartree-Fock approximation (Herman
and Skillman, 1963)) in a good approximation possess a definite
parity under the transformation T3 (11). The possible applications
PHYSICS AND PERIODIC LAW
169
of this symmetry for the calculation of some matrix elements are
also discussed in this paper.
5. PERIODIC LAWS IN OTHER MULTIPARTICLE
PHYSICAL SYSTEMS
5.1. Atomic systems
It should be stressed once again that the ordering of the energy
levels in the effective potential is crucial for interpretation of the
Periodic System. The levels in the spherical potential are known
to be degenerate in azimuthal quantum number m, but as for {n,
l}-dependence, this can be varied in broad limits depending on the
choice of one-particle potential Ua (r) that is the key problem. In its
turn, the form of the potential Ua (r) is governed by the interactions
operative between the particles in the system. The characteristic
features of the atoms are (i) the presence of massive center of
force (atomic nucleus) and (ii) the long-range Coulomb interaction
between constituent particles. In the other systems considered in
sections 5.2–5.4 these features are absent; in particular, instead of
long range Coulomb forces in atoms one meets short range interactions in atomic nuclei and clusters. In the present subsection we start
with atomic systems that are similar to ground state neutral atoms,
but differ in some important aspects.
5.1.1. Excited states in atoms and ions
Klechkovskii (1952c) was first to notice that for some atoms and
low-charge ions27 the excited levels with the same value of the
sum (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 the
examples of overlapping and non-overlapping (n + l)-groups in the
spectra of one-electron excitations for atoms and ions. Sternheimer
(contrary to Klechkovskii) did not use a convenient representation in
terms of the quantum defects, which made his discussion redundant
(since it was sufficient to consider a small number of Rydberg series
of levels instead of a large number of individual levels). The (n + l)grouping is observed for levels with small l (l ≤ l0 ) while for larger
l it is replaced by hydrogen-like n-grouping.
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V. N. OSTROVSKY
The aforementioned authors did not give a quantum mechanical
explanation of the observed regularities (for example, Sternheimer
tentatively related them to the relativistic effects, magnetic interactions etc.). The quantum mechanical interpretation of the (n +
l)-grouping was developed by Ostrovsky (1981) via analysis of
properties of an effective one-electron potential. He found the
borderline l0 for different atoms and established the relationship
between (n + l)-grouping of excited levels and the (n + l, n) filling
rule for ground states.
5.1.2. Positively charged ions
It was already indicated in section 3.3 that for multicharged ions
the 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 intermediate
ordering scheme in between. An attempt to describe this transition using group-theoretical formalism of q-deformed algebras was
undertaken by Négadi and Kibler (1992).
5.1.3. Compressed atoms
For the atoms confined to a cavity of atomic size the ordering of
orbital energies is changed. As the calculations by Connerade et al.
(2000) show, the competition between ns and (n + 1)d orbitals disappears in favor of the former, i.e., (n + l, n) filling rule is replaced by
the hydrogenlike (n, l) rule. Thus compressing an atom changes the
situation in the same direction as its ionization, see section 5.1.2.
This similarity might be interpreted in terms of properties of an
effective one-electron potential (Connerade et al., 2000).
5.1.4. Molecules
Hefferlin with co-workers (Hefferlin et al., 1979a, b, 1984; Hefferlin
and Kuhlman, 1980a, b; Zhuvikin and Hefferlin, 1983; Karlson et
al., 1995) discussed the periodic system of diatomic molecules
formed from different atoms. These authors introduced the classification of diatomics bearing combinatorial character. Zhuvikin and
Hefferlin (1983) consider group-theoretical aspects of the problem
in 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 particular number of protons and/or neutrons are especially stable. Such
numbers, 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 of
nuclei, just as the period lengths in the Periodic Table are manifestations of electronic shells in atoms (the period lengths (8) are
also often referred to as magic number in physical literature, see,
for instance, Löwdin (1969)). The interpretation of shell structure
is based on an effective one-particle potential which in the crudest
approximation is that of harmonic oscillator. The more sophisticated
constructions which agree better with experimental data employ a
potential of trough-like shape with shallow bottom and rather abrupt
cut-off on the nucleus border (the latter notion is defined much better
for nuclei than for atoms). Here it is important to stress that the
states in this potential are labeled by quantum numbers28 n and l
just as for atoms. The difference in two sets of magic numbers (8)
and (12) is governed by the difference in the shape of one-particle
potentials.
Applications of various group-theoretical techniques in nuclear
physics are numerous and sophisticated.
5.3. Magic numbers in clusters
The clusters are relatively new object in physics, lying in between
large molecules and condensed matter. There are numerous indications of particular stability of clusters composed of some magic
number of atoms. In particular, the experiments with the sodium
clusters (up to approximately 1500 atoms) indicate some specific
type of the electronic shell structure. All valence electrons in such
clusters might be considered as moving in some effective field. For
sodium atom clusters the empirical data correspond to grouping
together the one-electron levels with the same sum 3nr + l ≡ 3n
− 2l − 3 (Martin et al., 1990, 1991a, 1991b). Generally the shape
of the effective one-particle potential in clusters is similar to that
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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 to
as ‘wine-bottle shape’. An interpretation of the 3nr + l grouping
in terms of the shape of an effective one-electron potential was
provided by Ostrovsky (1997). In particular, it was demonstrated
that the related effective potential leads to closed classical trajectories with some special pattern. The importance of the shape of
the outer wall in the potential well was emphasized by Lermé et
al. (1993a, 1993b) who considered applications to aluminium and
gallium clusters.
Recently the group-theoretical technique was applied to the
description of shell structure in clusters (Bonatsos et al., 1999,
2000). This approach essentially consists of choosing (basing on
heuristic arguments) some group-theoretical scheme (a particular qdeformed algebra) and selecting a fitting parameter that allows the
authors to reproduce some set of empirical magic numbers.
5.4. Particles in the traps
Recently, considerable interest has appeared for the experimental
and theoretical study of localization of a finite number of ions or
electrons in the traps that are created by external confining potential. The examples are radio-frequency traps for ions and electrons
in plasma, heavy-ion storage rings, electrons in quantum dots in
semiconductor structures; some key references could be found in
the paper by Bedanov and Peeters (1994). The classical calculations by the cited authors for two-dimensional parabolic and
hard-wall traps show that electrons are arranged in shells. For
large number of electrons there is a competition between ordering
into a crystal-like structure (Wigner lattice) for inner electrons
and ordering into a shell structure for outer electrons. A periodic
system of two-dimensional crystals composed of “particles in a Paul
trap” was confirmed by recent experiments (Block et al., 2000a, b).
The Aufbau principle for electrons confined by quantum dots was
studied by Franceshetti and Zunger (2000).
PHYSICS AND PERIODIC LAW
173
6. CONCLUSIONS
The main points of the present paper could be summarized as
follows.
− When one is concerned with explanation or interpretation of
numerical results or experimental data for a complex system,
the use of approximations or models is the only way to achieve
success. In particular, the approaches used to interpret the Periodic System as a whole are necessarily quite different from the
theoretical techniques employed to obtain the best numerical
results for some particular property of an individual atom.
− The current mainstream interpretation of the Periodic Law is
based on the selfconsistent effective field concept, the notion
of atomic configuration and modeling of effective field experienced by an electron in an atom. This approach achieved
considerable success by providing non-empirical, ab initio
direct explanation of the (n + l, n) filling rule. The current state
of the problem contradicts the statement that “The emergence
of quantum mechanics in 1925–1926 rather interestingly did
not provide any improved qualitative explanation” (Scerri et
al., 1998b), although it could be true that “the role played by
quantum theory and quantum mechanics in chemistry is less
dramatic than is commonly held” (Scerri, 1996).
− Remarkably, an understanding of the (n + l, n) filling rule
is even better achieved by a crude model for effective oneelectron potential in an atom than that needed for quantitative
demonstration of this rule.
− Alternative, complementary approaches to the interpretation
of the Periodic Table seek to provide classification schemes
using techniques of Group Theory. The specific group is either
restored from the empirical structure of the Periodic Table or
extracted from the analysis of atomic field description within
quantum mechanics. The effective one-electron potential in
atoms possesses hidden symmetry properties that are probably
only partially revealed by now.
− There is plenty of room for future studies, such as the explanation of exceptions from the (n + l, n) filling rule or uncovering the deep origin of symmetry properties. One can always
174
V. N. OSTROVSKY
think about the possibility of higher-level explanations, for
instance, whether the symmetry of an effective one-electron
potential can be directly deduced from the hidden symmetry
of the Coulomb potential (Fock, 1935) operative between the
electrons and nucleus.
− Periodicity phenomena seems to be a general feature of
various multiparticle systems studied in physics (ionized or
compressed atoms, atomic nuclei, clusters, particles in the
traps). All of them exhibit a trend towards what probably could
be named a tendency to self-organization, with appearance of
shells, magic numbers etc. Comparative study of these manifestations is able to cast a new light on the origin of Periodic
Laws. In all cases various patterns of periodicity are governed
by the difference in the one-particle effective potentials.
NOTES
1. The predecessors of Mendeleev were J. Döbereiner, J.-B. Dumas, E. de
Chancourtois, J. Newland and L. J. Meyer (Scerri, 1998b).
2. This allows us to essentially skip the issues of early history in the present
brief exposure. The bibliography on the history of quantum interpretation of
the 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 between
metals and non-metals as discussed by Birk (1997).
4. Regarding serious difficulties which emerge both in classical and quantum
mechanics in solution of three-body problem, it seems that definition of
complexity appropriate to our present objectives is that the system under
consideration contains three or more particles. As shown by Poincare, the
classical three-body problem has no closed-form solution since the motion
is chaotic. Of course the same refers to the systems with larger number of
particles; in quantum mechanics closed-form solutions are absent also. Threebody systems in atomic physics provide a rich variety of phenomena that
could model more complicated objects.
5. Concerning the debate regarding prediction and accommodation of data by
scientific 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 physics
of the actual system onto the classical point of observer. However, we believe
that the workers in quantum mechanics develop a special kind of ‘quantum
intuition’ that allows direct understanding of quantum objects without appeal
to classical analogues; see, for instance, monograph by Zakhar’ev (1996)
under appealing title.
9. Being introduced absolutely and rigorously, the ionization potentials, or
ionization energies are preferable to more loosely defined quantities, such
as atomic radii.
10. It is hardly necessary to stress that in education as well as in research the
exact equations and approximations employed for their solutions should be
strictly distinguished, unambiguously formulated and clearly emphasized.
11. Rigorously speaking, the spherical symmetry of self-consistent field is guaranteed only for closed-shell atoms. In other cases the spherical symmetry
appears due to the standard additional approximation that works well for
the ground state atom. In more general situations the instability of the selfconsistent field could lead to spontaneous symmetry breaking; however these
advanced issues are not important for the present discussion.
12. We do not discuss here the historical development of configuration notion
exposed by Scerri (1991a).
13. The statement that “The electronic configurations . . . cannot be derived using
quantum mechanics . . . because the fundamental equation of quantum mechanics, the Schrödinger equation, cannot be solved exactly for atoms other
than hydrogen” (Scerri, 1998a) is not exact. In modern quantum mechanics
there is no other way to derive electronic configurations than starting from
the exact Schrödinger equation and developing a scheme for its approximate solution, as briefly outlined above. Therefore it is too strong to say
that “quantum mechanics forbids any talk of electrons in orbitals and hence
electronic configuration” (Scerri, 1997a). Equally it cannot be said that the
atomic orbitals stem from approximations that logically contradict the theory
(H. Post, as cited by Scerri (1989)). The orbitals are simply the one-electron
building blocks routlinely used in quantum mechanics to construct a multielectron wave function. The Hartree-Fock method allows one to choose these
blocks in an optimal way for each particular atom.
14. It is worth indicating here that there are some phenomena in atomic physics
that cannot be explained without more sophisticated description, which
from 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 approximately
to the wave function. In partiuclar, in the theory of doubly excited atomic
states, 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
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
V. N. OSTROVSKY
Prudov and Ostrovsky (1998). Such sophisticated treatment is not necessary
for explanation of the Periodic Table.
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 exposures
of quantum analysis of the Periodic Table could give a reader impression that
a mere introduction of quantum numbers n and l gives the filling rule, which
certainly is not true.
Scerri et al. (1998b) attribute n + l rule to Bohr (1922), but we were unable
to locate formulation of the rule in Bohr’s paper.
This situation can be compared with that for pure Coulomb field where in the
course of a detailed quantum solution the combination nr + l + 1 naturally
emerges in the expression for energy, subsequently being designated as n.
Most frequently in quantum mechanics the quantized (i.e., discretized)
magnitude is the energy, although this is not the only option (recall, for
instance, quantization of angular momentum). The present case demonstrates
yet another possibility: the energy E is fixed, but the quantized parameter
is Z that defines the potential strength (pre-factor in formula (6)), and also
the coordinate scaling. Physically this situation is justified by the fact that
the valence electrons in atoms are always weakly bound, i.e., their energy is
always close to the borderline E = 0.
In this brief exposure we do not discuss some subtle aspects of using potential
(6) detailed in original publications (Demkov and Ostrovsky, 1971b; Demkov
and Berezina, 1973; Ostrovsky, 1981).
It is worthwhile to stress again that this is an explanatory problem. Quantitative description of atoms in the case of exceptions from the (n + l, n) does not
present a particular problem.
We present here only a brief qualitative discussion of the group theoretical
approach bearing in mind its applications to the Periodic Table reviewed by
Ostrovsky (1996).
In the most general terms the suggestion to apply group theory to the Periodic
Table could be found in the paper by Neubert (1970).
It is worthwhile to indicate that Novaro and Berrondo (Novaro and Berrondo,
1972; Berrondo and Novaro, 1973; Novaro, 1973; Novaro, 1989) looked
for the group which describes the chemical periods, whereas other authors
have concentrated on the (n + l) grouping (the phenomenological difference
between these patterns was discussed in the beginning of section 3.5).
Here it is worthwhile to recall that the explanation appeals to some
community of researchers, see section 2.
Kitagawara and Barut (1983, 1984) modified the scheme by Ostrovsky
(1981) to consider mapping of the (nonphysical) two-dimensional problem
on the three-dimensional sphere. In this case the treatment is much easier
due to the possibility of using well developed mathematical theories of
complex 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 radium
is more closely allied to strontium (both give intensely red flame coloration), just as thorium is most closely allied to zirconium, and uranium most
closely allied to molybdenum. It is, in fact, a general observation that alternate
members of a valency group in the periodic table show the greatest chemical 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 earth
atoms and some isoelectron ions.
28. Note that in nuclear physics the notation n is usually understood for the radial
quantum number nr that is simply related to the principal and orbital quantum
numbers, see section 3.1.
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Institute of Physics
The University of St Petersburg
198904 St Petersburg
Russia
E-mail: [email protected]